Notation

Physical AI draws on three disciplines at once. Machine learning supplies the foundation models, vision-language-action architectures, diffusion policies, and action chunking that propose what a robot should do. Robotics and control supply the mechanics that decide what it can do: articulated dynamics, the Lie group $\SpecialEuclideanGroup$, contact and friction, Lyapunov stability, control barrier functions, and actuator electromechanics. Embedded and real-time systems supply the machinery that decides when it happens, from interconnect and memory contention through zero-order hold, real-time scheduling, leases, hardware watchdogs, and functional safety.

Because each discipline developed its mathematical conventions independently over decades, combining them creates symbol collisions that cyber-physical systems engineering has long lived with. For example, writing $\nabla_{\Glyph{\theta}} \Policy(\Glyph{\mathbf{a}} \mid \Glyph{\mathbf{q}})$ when $\Glyph{\theta}$ also denotes joint angles, or writing $\Glyph{\mathbf{a}}_t$ for an action when $\Glyph{\mathbf{a}}$ denotes Cartesian acceleration or barrier constraints, confuses spatial mechanics with parameter optimization and control execution authority.

The sections that follow state the cross-discipline disambiguation rules, the governing relationships of embodied systems, and the symbol tables used throughout this book. Every symbol of the series core keeps its foundational meaning: $\Time$ is time, $\Bandwidth$ bandwidth, $\Latency$ latency, $\PeakRate$ peak rate, $\Operations$ operations, $\ArithmeticIntensity$ arithmetic intensity, $\DataVolume$ bytes moved, $\LossFunction$ loss, $\LearningRate$ learning rate, and $\mathcal{O}(\cdot)$ asymptotic cost. Serving and memory arithmetic on the robot’s onboard computer uses those symbols, with a subscript naming the instance ($\SustainedMemoryBandwidth$, $\PrefillOperations$).

Cross-Discipline Disambiguation

When physical plant dynamics meet deep neural networks and real-time embedded hardware, common mathematical shorthand breaks down. The conventions below resolve these cross-domain collisions systematically:

Symbol	Canonical Convention	Colliding Meaning	Disambiguation Rule
$\Glyph{\theta}$	Learned model (policy) parameters, as in $\Policy$	Joint angle, trajectory polynomial $\Glyph{\theta}(\Glyph{t})$, heading, azimuth, attitude or mounting error, clock offset, or thermal resistance $\Glyph{\theta_{JA}}$	Bare $\Glyph{\theta}$ is always the parameter collection of a learned model, set italic (never bold), as the series core defines it. A geometric angle always carries a subscript that names it: $\JointAngle$ for one joint ($\Glyph{\theta_{\text{motor}}}$ and $\Glyph{\theta_{\text{link}}}$ on either side of a reducer, $\EncoderAngle$ for an encoder reading), $\HeadingAngle$ for planar heading, $\AzimuthAngle$ for azimuth, $\AttitudeError$ for attitude error, $\ExtrinsicMisalignmentAngle$ for extrinsic misalignment, $\TiltAngle$ for tilt from vertical, $\OrientationTolerance$ for an orientation tolerance, and $\ElectricalRotorAngle$ for the electrical rotor angle; increments keep the subscript ($\FiniteChange\JointAngle$, $\SmallChange\EncoderAngle$). Joint configurations are the generalized coordinates $\GeneralizedCoordinates \in \mathbb{R}^\GeneralizedCoordinateCount$ with entries $\JointCoordinate_i$, and a joint trajectory polynomial is $\JointCoordinate(\TimeInstant) = \sum \PolynomialTrajectoryCoefficient_k \TimeInstant^{\Glyph{k}}$ with boundary values $\JointCoordinate_0, \JointCoordinate_1$ (never $\Glyph{\theta}(\Glyph{t})$). A clock offset is $\ClockOffset$ (in seconds). Thermal resistance is $\ThermalResistance$ or $\JunctionAmbientThermalResistance$ (in $\text{K/W}$), never $\Glyph{\theta_{JA}}$.
$\Glyph{\mathbf{a}}$ vs. $\NominalControlCommand$ vs. $\CertifiedControlAction$ vs. $\JointTorque$	Four-stage authority chain ($\CognitivePolicyAction_{t} \to \NominalControlCommand \to \CertifiedControlAction \to \JointTorque$)	Generic “action” symbol colliding with deceleration $\CredibleBrakingDeceleration$ or a barrier constraint row	Disambiguate execution authority: $\CognitivePolicyAction_{t}$ is the unprivileged policy action (dimensionless/normalized); $\NominalControlCommand$ is the proposed plant control setpoint; $\CertifiedControlAction \in \AdmissibleControlSet$ is the certified, CBF-filtered control command; $\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount$ is realized physical actuator torque. Control commands are bold $\ControlCommand$ (with time index $\ControlCommand_t$). Credible linear deceleration is $\CredibleBrakingDeceleration$ ($\text{m/s}^2$); stacked barrier constraints are $\Glyph{\mathbf{A}_{\text{cbf}}}(\MechanicalPlantState)\ControlCommand \le \Glyph{\mathbf{b}_{\text{cbf}}}(\MechanicalPlantState)$, and a single constraint row is $\BarrierConstraintNormal(\MechanicalPlantState)^\top\ControlCommand \le \BarrierConstraintBound(\MechanicalPlantState)$.
$\JointTorque$ vs. $\Glyph{\tau}$	Actuator torque vector $\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount$ ($\text{N}\cdot\text{m}$ or $\text{N}$)	Time constants ($\ElectricalTimeConstant$, $\ThermalTimeConstant$), delays ($\PrebrakeDelay$, $\ZohTransportDelay$), observation age, lease durations, or real-time task names	Bare or bold $\Glyph{\tau}$ is always torque ($\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount$, scalar joint torque $\CommandedJointTorque_i$, component torques such as $\MotorTorque$, $\HoldingTorque$, $\Glyph{\tau_{\text{bias}}}$), with $\MotorTorque = \MotorTorqueConstant \TorqueProducingCurrent$ at the motor. A subscripted Greek $\Glyph{\tau}$ marks a time only for these named constants and delays: $\PrebrakeDelay$ (pre-brake delay, excluding $\DispatchAdmissionInterval$) and its ceiling $\PrebrakeDelayCeiling$, $\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance$ (electrical), $\ThermalTimeConstant = \ThermalResistance \StatorThermalCapacitance$ (thermal, with stage constants such as $\CopperThermalTimeConstant$ and $\CaseThermalTimeConstant$), $\Glyph{\tau_{\text{fall}}}$ (balance divergence), $\ZohTransportDelay$ (sample-and-hold delay), $\EvidenceHorizon$ (evidence horizon), and $\BlendingTransitionWindow$ (authority blend window). Every other interval uses Latin $\Glyph{T}$ or $\Glyph{t}$: observation age is $\ObservationAge$, never $\Glyph{\tau}$; leases are $\ChunkLease$, $\ManualLease$, and $\IntentAdministrativeLease$; compute and fieldbus times are $\EnforcerComputeTime$ and $\CommandTransportTime$; a pure transport delay is $\PureTransportDelay$. A real-time task is named “task $\Glyph{i}$”, never $\Glyph{\tau}_i$.
$\MechanicalPlantState$ vs. $\Glyph{s}$ vs. $\CartesianPosition$ vs. $\VitImagePatch$	Mechanical plant state $\MechanicalPlantState = [\GeneralizedCoordinates^\top, \GeneralizedVelocities^\top]^\top \in \mathbb{R}^{2\GeneralizedCoordinateCount}$	World state $\Glyph{s} \in \Glyph{\mathcal{S}}$, 3D position $\CartesianPosition = [\CartesianXCoordinate, \CartesianYCoordinate, \CartesianZCoordinate]^\top$, image point, or ViT image patch $\VitImagePatch$	Continuous mechanical plant state is bold vector $\MechanicalPlantState \in \mathbb{R}^{2\GeneralizedCoordinateCount}$. Ground-truth latent world state is italic $\PhysicalWorldState_{t}$ (a mixed-type state, never bold). 3D Cartesian position is vector $\CartesianPosition \in \mathbb{R}^3$, including a camera-frame point $\CameraFramePoint$ and a voxel center $\VoxelCenterPosition$; the translation of a rigid transform is $\RigidTransformTranslation$. An image point in homogeneous pixel coordinates is $\HomogeneousPixelCoordinates = [\PixelColumnCoordinate, \PixelRowCoordinate, 1]^\top$, never $\Glyph{\mathbf{x}}$ or $\Glyph{\mathbf{u}}$. Vision transformer patch embeddings use explicitly subscripted $\VitImagePatch$ or latent feature $\VitPatchTokenEmbedding$.
$\Parameters$	Model parameter count $\Parameters$ (series core)	Power ($\WindingLossPower, \Glyph{P_{\text{peak}}}$), estimation covariance $\EstimationErrorCovariance$, failure probability or rate, tail-latency percentile, or 3D point $\CartesianPosition$	Use $\Parameters$ for parameter count. Electrical/mechanical power always carries a subscript ($\WindingLossPower, \Glyph{P_{\text{peak}}}, \HeatDissipationPower$, in Watts); the estimator’s error covariance is bold matrix $\EstimationErrorCovariance$, and a generic Gaussian covariance is $\GaussianCovarianceMatrix$; a per-trial probability is lowercase italic $\Glyph{p}$ ($\TrialFailureProbability, \TargetTrialSuccessProbability$), a rate per hour is $\Glyph{\lambda}$ ($\SystemHazardRate, \TargetHazardRate$), and an event probability is $\Pr(\cdot)$; percentiles are upright p50, p95, and p99 (never $\Glyph{P}_{99}$); 3D Cartesian position is bold lowercase $\CartesianPosition = [\CartesianXCoordinate, \CartesianYCoordinate, \CartesianZCoordinate]^\top$.
$\BatchSize$	Batch size $\BatchSize$ (series core)	Bandwidth ($\Bandwidth$), Body frame ($\BodyReferenceFrame$), bus transfer size, SPRT boundary, or input matrix $\StateSpaceInputMatrix$	Batch size is $\BatchSize$; sustained bandwidth is $\Bandwidth$ (per the single-device decomposition), with a subscript naming the path ($\SustainedMemoryBandwidth$, $\Glyph{\Bandwidth_{\text{chirp}}}$); coordinate reference frames use calligraphic script ($\WorldReferenceFrame, \BodyReferenceFrame, \Glyph{\mathcal{F}_E}$); the bytes of bus transaction $\Glyph{k}$ are ${\DataVolume}_{\Glyph{k}}$; SPRT boundaries are $\SprtUpperBoundary$ and $\SprtLowerBoundary$; state-space control input matrix is bold capital $\StateSpaceInputMatrix \in \mathbb{R}^{\StateDimension \times \ControlInputDimension}$.
$\BumplessTransferBlend$	Bumpless transfer blending function $\BumplessTransferBlend(\BumplessTransferProgress) \in [0, 1]$	CBF decay ($\LinearBarrierDecayRate$), significance ($\HypothesisTestSignificanceLevel$), angular acceleration ($\AngularAcceleration$), copper tempco ($\CopperTemperatureCoefficient$), or noise schedule ($\DiffusionCumulativeSchedule_{k}$)	Authority transfer uses scalar function $\BumplessTransferBlend(\BumplessTransferProgress) \in [0, 1]$ of normalized time $\BumplessTransferProgress \in [0, 1]$; barrier linear decay uses $\LinearBarrierDecayRate > 0$; statistical confidence significance uses $\HypothesisTestSignificanceLevel = 1 - \text{CL}$; angular acceleration is bold vector $\AngularAcceleration \in \mathbb{R}^3$ ($\text{rad/s}^2$); copper resistance tempco is $\CopperTemperatureCoefficient$; diffusion noise schedules use indexed $\DiffusionCumulativeSchedule_{k}$.
$\CoriolisCentrifugalMatrix$	Coriolis matrix $\CoriolisCentrifugalMatrix(\GeneralizedCoordinates, \GeneralizedVelocities) \in \mathbb{R}^{\GeneralizedCoordinateCount \times \GeneralizedCoordinateCount}$	Safe set $\ForwardInvariantSafeSet$, conformal prediction set, assurance claim, state-space output matrix, controllability matrix, capacitance ($\StatorThermalCapacitance, \DcBusBulkCapacitance$), CoM $\MassCenterPosition$, or shield coverage $\ShieldInterceptionCoverage$	Coriolis/centrifugal forces use matrix function $\CoriolisCentrifugalMatrix(\GeneralizedCoordinates, \GeneralizedVelocities)$, always written with its arguments; forward-invariant safe sets use calligraphic $\ForwardInvariantSafeSet \subset \mathbb{R}^{2\GeneralizedCoordinateCount}$, defined on the plant state, and the conformal prediction set $\ConformalPredictionSet$ of target grounding shares that glyph but never appears beside a safe set; the top-level claim of an assurance argument is italic $\AssuranceClaim$; the state-space output map is $\MeasurementMatrix$ (the same map as the Kalman measurement map), never $\Glyph{\mathbf{C}}$; the controllability matrix is $\ControllabilityMatrix$; electrical/thermal capacitance uses italic $\Glyph{C}$ with a subscript; center of mass is bold vector $\MassCenterPosition \in \mathbb{R}^3$; safety shield coverage of hazard class $\HazardClassIndex$ uses $\ShieldInterceptionCoverage \in [0, 1]$.
$\ActionChunk$	Action chunk $\ActionChunk_t$, $\Glyph{\mathbf{A}_{t:t+H}}$, and its denoising iterate $\ActionChunk_t^\DiffusionStepIndex$	Continuous LTI state matrix, barrier constraint matrix	Bold $\ActionChunk$ with a time or denoising index is the action chunk. The continuous LTI state matrix is $\StateSpaceSystemMatrix$ (its discretizations $\ForwardEulerTransitionMatrix$ and $\BackwardEulerTransitionMatrix$, the discrete Kalman transition $\StateTransitionMatrix$); stacked barrier constraints use $\Glyph{\mathbf{A}_{\text{cbf}}}$.
$\Velocity$	Velocity ($\Velocity$, end-effector velocity $\Glyph{\mathbf{v}_e}$)	Measurement noise, voxel	Bold $\Velocity$ is a velocity. Measurement noise is $\MeasurementNoiseVector$ with covariance $\MeasurementNoiseCovariance$ (bold $\Glyph{\mathbf{R}}$ is reserved for rotation); a voxel is addressed by its center $\VoxelCenterPosition$.
$\CameraIntrinsicCalibrationMatrix$	Camera intrinsic calibration matrix $\CameraIntrinsicCalibrationMatrix \in \mathbb{R}^{3 \times 3}$	Contact stiffness ($\ContactStiffness$), control gains ($\PidProportionalGain, \PidDerivativeGain$, LQR gain), motor constants ($\MotorTorqueConstant, \MotorBackEmfConstant$), ensembling window $\EnsemblingWindowLength$, or diffusion step $\DiffusionStepIndex$	Camera intrinsics use bold matrix $\CameraIntrinsicCalibrationMatrix$; contact stiffness is $\ContactStiffness$, with body-region stiffness $\HertzianContactStiffness$ ($\text{N/m}$); feedback control gains use capitalized scalars $\PidProportionalGain, \PidDerivativeGain$, and a state-feedback gain matrix is $\LqrFeedbackGain$; motor torque/back-EMF constants use $\MotorTorqueConstant, \MotorBackEmfConstant$ (per joint ${\MotorTorqueConstant}_{i}$); temporal ensembling window length uses integer $\EnsemblingWindowLength$; the number of denoising or solver evaluations per chunk is $\DecoderEvaluationCount$, never $\Glyph{K}$; diffusion noise steps use lowercase index $\DiffusionStepIndex$; the Kalman gain is $\KalmanGain_{k}$.
$\GeneralizedInertiaMatrix$	Articulated inertia matrix $\GeneralizedInertiaMatrix(\GeneralizedCoordinates) \in \mathbb{R}^{\GeneralizedCoordinateCount \times \GeneralizedCoordinateCount}$	Mass ($\Mass, \EffectiveImpactMass$), memory footprint ($\WeightMemory, \KvCacheMemory$), or fleet size ($\FleetRobotCount$)	Articulated generalized mass/inertia uses bold matrix $\GeneralizedInertiaMatrix(\GeneralizedCoordinates)$; physical body mass is lowercase italic $\Mass$ ($\text{kg}$); effective impact mass is $\EffectiveImpactMass$; hardware memory storage uses capital $\Glyph{M}$ with descriptive subscript (in bytes); robot fleet size uses $\FleetRobotCount$.
$\Time$	Time (series core)	Temperature	Bare $\Glyph{T}$ is always time. A temperature always carries a subscript that names where it is measured: $\WindingTemperature$ (winding), $\StatorCoreTemperature$, $\JunctionTemperature$ (junction), $\CaseTemperature$, $\AmbientTemperature$ (ambient), $\WindingTemperatureLimit$ (rated limit), $\ThrottleTripTemperature$, $\EquilibriumWindingTemperature$ (equilibrium), and $\CopperReferenceTemperature$ (the reference temperature of $\CopperTemperatureCoefficient$). A temperature rise is $\JouleanTemperatureRise$ (in $\text{K}$); a time interval is $\FiniteChange \TimeInstant$, never $\Glyph{\Delta} \Glyph{T}$.
SIL (abbreviation)	Safety integrity level (IEC 61508 family), a grade of a safety function	Software-in-the-loop, the first rung of the verification ladder	Spell out at first use in each chapter. “SIL 2” and similar graded forms always mean safety integrity level; the verification ladder writes software-in-the-loop (SIL) only where the ladder’s abbreviations MIL, SIL, PIL, and HIL appear together, and never beside a safety integrity level.

Governing Relationships

Physical intelligence operates subject to thermodynamic, mechanical, kinematic, and communications constraints. The relationships below state those constraints in the symbols this book uses. They are working relationships, not laws: the book’s four bedrock laws are the ones The Four Bedrock Laws states.

The sense-to-actuation delay budget

Delay converts directly into travel before any retarding force begins. The pre-brake delay $\PrebrakeDelay$ runs from the capture of the frame that first shows a hazard to retarding force, excluding the dispatch-to-admission interval $\DispatchAdmissionInterval$, which the stopping budget charges against its timing allowance. The stop it leaves, whose travel is $\ResidentStopDistance(\SpeedAtDetection)$, must fit inside the clear distance, and the spare that remains must pay for $\SpeedAtDetection\,\DispatchAdmissionInterval$:

\[\PrebrakeDelay = \ObservationAge + \ChunkLease + \PermissionLoopPeriod + \CommandTransportTime + \ActuatorResponseTime\]

\[\SpeedAtDetection\,\PrebrakeDelay + \ResidentStopDistance(\SpeedAtDetection) + \LocalizationBound + \ProtectiveClearanceMargin + \TrackingErrorBound \le \ClearDistance\]

Inference time is spent on the proposal side of the proposal boundary. It ages the evidence behind a proposal, which the permission path checks against the proposal’s expiry, but it is neither a term of $\PrebrakeDelay$ nor part of the permission loop’s period $\PermissionLoopPeriod$, because the fallback does not wait for the proposer.

The appendices regroup the same interval for two specific stop paths, and each form defines its own terms. For a stop triggered by lease expiry, $\PrebrakeDelay = \ChunkLease + \DetectionInterval + \FallbackHold + \ObservationAge + \ActuatorResponseTime$; for a stop triggered by a hazard detector, $\PrebrakeDelay = \ObservationAge + \DetectionInterval + \StopDecisionComputeBound + \CommandTransportTime + \ActuatorResponseTime$.

Symbol	Definition	Unit	Notes
$\PrebrakeDelay$	Pre-Brake Delay	$\text{ms}$	Time from capture of the frame that first shows a hazard to the onset of retarding force, excluding $\DispatchAdmissionInterval$; the machine travels $\SpeedAtDetection\,(\PrebrakeDelay + \DispatchAdmissionInterval)$ before braking begins. Its allowed ceiling is $\PrebrakeDelayCeiling$.
$\ObservationAge$	Observation Age	$\text{ms}$	Age of the state the permission path acts on, from the exposure midpoint at the sensor to the decision, which enters $\PrebrakeDelay$. Its perception-side terms are the exposure midpoint $\tfrac{1}{2}\ExposureDuration$, sensor readout, transfer, and perception processing; a bound on it is $\Glyph{t_{\text{age},\max}}$.
$\ChunkLease$	Chunk Lease	$\text{ms}$	Longest interval an admitted chunk may keep driving the actuators without renewal, which enters $\PrebrakeDelay$; chosen, then checked against the ceiling the stopping budget derives.
$\ChunkRenewalPeriod$	Chunk Renewal Period	$\text{ms}$	Interval between successful admissions of the chunk policy; with $\RenewalJitterAllowance$ it sets the floor of the lease rule, $\ChunkRenewalPeriod + \RenewalJitterAllowance < \ChunkLease$ (equation 4.4).
$\RenewalJitterAllowance$	Renewal-Jitter Allowance	$\text{ms}$	Allowance for variation in the interval between successful admissions; chosen, then checked against measured renewal intervals.
$\PermissionLoopPeriod$	Permission-Loop Period	$\text{ms}$	Period of the permission loop, which enters $\PrebrakeDelay$ because a hazard may wait up to one period before it is checked.
$\CommandTransportTime$	Command Transport Time	$\text{ms}$	Fieldbus delivery time from the permission path to the drive, which enters $\PrebrakeDelay$.
$\ActuatorResponseTime$	Actuator Response Time	$\text{ms}$	Time from a delivered stop command to retarding force, including gate-drive and current rise, which enters $\PrebrakeDelay$.
$\DetectionInterval$	Detection Interval	$\text{ms}$	Recognition or expiry-detection time after the decision-start epoch, not already charged to $\ObservationAge$; a term of the lease-expiry and hazard-detection forms of $\PrebrakeDelay$.
$\FallbackHold$	Fallback Hold	$\text{ms}$	Optional, separately authorized hold or fault-confirmation interval before the stop command reaches the drive; zero when the stop path has none.
$\ProposerInferenceTime$	Proposer Inference Time	$\text{ms}$	Time for one complete evaluation of the learned proposer (prefill and decode); spent above the proposal boundary and outside the permission loop’s period.
$\DispatchAdmissionInterval$	Dispatch-to-Admission Interval	$\text{ms}$	Elapsed time from perception dispatch to the admission of the candidate it conditions; evidence keeps aging through it. Not a term of $\PrebrakeDelay$; its travel $\SpeedAtDetection\,\DispatchAdmissionInterval$ is charged against the stopping budget’s timing allowance.
$\TakeoverResponseTime, \DisengagedTakeoverResponseTime$	Takeover Response Time	$\text{ms}, \text{s}$	Human response plus console transport for an in-loop ($\TakeoverResponseTime$) and an out-of-loop ($\DisengagedTakeoverResponseTime$) takeover; it adds to $\PrebrakeDelay$ while the machine waits for the person.
$\CompletionDeadline$	Deadline	$\text{ms}, \text{s}$	A duration budget a computation or handover must meet (the series symbol); qualifiers name the instance, as in $\FallbackDeadline$, the latest permissible wait before the fallback must begin. A deadline stated as a clock instant is lowercase, $\DeadlineInstant$ or $\Glyph{t_{\text{deadline,stop}}}$.
$\HandoverTimeout$	Handover Timeout	$\text{s}$	Timer an arbiter sets for an unacknowledged handover, $\HandoverTimeout = \max(0, \min(\HandoverTimeoutCeiling, \FallbackDeadline))$, where $\HandoverTimeoutCeiling$ is the protocol ceiling.
$\CommitmentInstant$	Commitment Instant	$\text{s}$	Instant after which a receding-horizon plan’s prefix is committed and can no longer be replaced by a new proposal.
$\HazardOnset$	Hazard Onset	$\text{s}$	Instant at which a hazard first becomes physically present, the reference from which detection and stopping intervals are measured.

Leases

A lease is authority that lapses unless it is renewed. The book uses six kinds:

Lease	Symbol	Distinction
Execution lease	remaining $\RemainingExecutionLease$	Any authority an admitted command source holds to keep driving the actuators without renewal; it begins at admission, and its expiry invokes the state-matched fallback.
Chunk lease	$\ChunkLease$	The execution lease of the autonomous chunk policy, renewed by each admitted chunk; it enters $\PrebrakeDelay$.
Manual lease	$\ManualLease$	The execution lease of a human operator in manual mode, renewed by authenticated heartbeats; it replaces the chunk lease in $\PrebrakeDelay$ while the operator drives.
Administrative lease	$\IntentAdministrativeLease$	The host-silence bound inside an intent lease: how long the application processor may hold a goal without renewing it.
Intent lease	expiry $\IntentLeaseExpiry$	The record that grants a grounded target and effort ceilings, never a path; it expires at the earliest of its evidence horizon, its administrative lease, and its stopping deadline.
Hardware lease	(mechanism)	The countdown timer on the safety microcontroller that enforces any of these expiries even when the process that issued the authority has hung.

Single-device execution time

In a serial model without overlap, one evaluation of a learned proposer on the robot’s onboard accelerator takes a time that splits into three terms: moving bytes through memory, executing operations, and a fixed overhead. The chunk-cadence and proposer-inference relationships below reuse these symbols, with a subscript naming the instance ($\SustainedMemoryBandwidth$, $\PrefillOperations$):

\[\Time = \underbrace{\frac{\DataVolume}{\Bandwidth}}_{\text{Memory Time}} + \underbrace{\frac{\Operations}{\PeakRate \cdot \EfficiencyHw}}_{\text{Compute Time}} + \underbrace{\Latency}_{\text{Latency Overhead}}\]

Symbol	Definition	Unit	Notes
$\Time$	Execution Time	seconds	Total execution time of one evaluation; bare $\Glyph{T}$ is always time.
$\Bandwidth$	Bandwidth	bytes/s	Sustained bytes per second across a named path; never bare $\Glyph{B}$, which is batch size $\BatchSize$.
$\DataVolume$	Data Volume	bytes	Bytes moved through memory in one evaluation; never bare $\Glyph{D}$, which is dataset size.
$\EfficiencyHw$	Efficiency	—	Fraction of the peak rate an evaluation achieves, $0 \le \EfficiencyHw \le 1$; never bare $\Glyph{\eta}$, which is the learning rate.
$\Latency$	Latency	seconds	Fixed overhead per evaluation (kernel launch, transfer setup); the subscript keeps it distinct from the loss $\LossFunction$.
$\Operations$	Operations	FLOPs	Floating-point operations one evaluation executes; a subscript names the phase ($\PrefillOperations$). Asymptotic cost is $\mathcal{O}(\cdot)$.
$\PeakRate$	Peak Rate	FLOP/s	Peak arithmetic rate of the processor at the stated precision; never bare $\Glyph{P}$, which is parameter count.

Action-chunk cadence and proposer inference

Large neural policies that stream their weights from memory may exceed a kilohertz control period. Action chunking predicts an action trajectory of horizon $\ActionChunkHorizon$ at a lower cognitive frequency $\Glyph{f_{\text{policy}}}$, while the chunk’s waypoints are consumed at $\ActionExecutionRate$. In the serial model with one weight-streaming pass per chunk and no additional fixed overhead, memory streaming must fit inside the chunk execution window after reserving the compute term $\Glyph{T_{\text{compute}}}$ of the single-device decomposition:

\[\ParameterStreamingTime = \frac{\WeightMemory}{\SustainedMemoryBandwidth} \le \frac{\ActionChunkHorizon}{\ActionExecutionRate} - \Glyph{T_{\text{compute}}}\]

A decoder that streams its nonresident weights once per emitted token spends, for $\EmittedTokenCount$ emitted tokens, the inference time

\[\ProposerInferenceTime \approx \max\!\left(\frac{\PrefillOperations}{\EfficiencyHw\,\PeakRate},\ \frac{\WeightMemory}{\SustainedMemoryBandwidth}\right) + (\EmittedTokenCount - 1)\,\frac{\WeightMemory}{\SustainedMemoryBandwidth}\]

which caps the fresh-evaluation rate at $\ProposerEvaluationRate \le \SustainedMemoryBandwidth/\WeightMemory$ when every evaluation streams the weights. The symbols are those of the single-device decomposition: $\PrefillOperations$ counts the prefill’s operations, $\PeakRate$ is the processor’s peak rate at the activation precision, $\EfficiencyHw$ is the fraction of it the prefill achieves, and $\ArithmeticIntensity$ is the arithmetic intensity that decides which term binds.

Symbol	Definition	Unit	Notes
$\ActionChunkHorizon$	Horizon	Time steps	Number of contiguous future action steps generated in a single policy forward pass ($\ActionChunkHorizon=16$ on the mobile manipulator; 16 to 64 steps in the literature).
$\ActionDimension$	Action Dimension	Integer	Number of components of one action $\CognitivePolicyAction_{t}$; a chunk $\Glyph{\mathbf{A}_{t:t+H}}$ has $\ActionChunkHorizon \times \ActionDimension$ entries.
$\SustainedMemoryBandwidth$	Sustained Memory Bandwidth	bytes/s	Achievable memory throughput of the SoC or accelerator memory subsystem (e.g., LPDDR5, HBM); the core’s $\Bandwidth$ at the memory interface.
$\ParameterStreamingTime$	Parameter Streaming Time	$\text{ms}$	Time required to stream non-resident model weights into execution registers.
$\ActionExecutionRate$	Action Execution Rate	$\text{Hz}$	Temporal consumption rate of chunked waypoints along the planned trajectory ($100\text{ Hz}$ to $1\text{ kHz}$).
$\EmittedTokenCount$	Emitted Token Count	tokens	Output tokens an autoregressive decoder emits per evaluation; the core’s sequence length $\SequenceLength$ for the decoded part. Never $\Glyph{D}$, which is dataset size.

Stopping, braking, and safe-set invariants

For the stopping-based safety envelope, the machine must retain enough clearance to arrest its approach before crossing the protected boundary. The relationship below specializes the resident stop to one-dimensional motion starting and ending at zero acceleration, with symmetric jerk ramps and a constant-deceleration segment. The stopping distance combines travel during the pre-brake delay with braking travel; the delay-budget inequality adds the spatial margins:

\[\StoppingDistance(\SpeedAtDetection) = \SpeedAtDetection\,\PrebrakeDelay + \ResidentStopDistance(\SpeedAtDetection), \qquad \ResidentStopDistance(\SpeedAtDetection) = \frac{\SpeedAtDetection^2}{2\CredibleBrakingDeceleration} + \frac{\SpeedAtDetection\,\CredibleBrakingDeceleration}{2\JerkLimit} \quad (\SpeedAtDetection \ge \CredibleBrakingDeceleration^2/\JerkLimit)\]

Symbol	Definition	Unit	Notes
$\StoppingDistance$	Stopping Distance	$\text{m}$	Distance required to bring the plant to rest from speed $\SpeedAtDetection$.
$\StoppingDuration$	Stopping Duration	$\text{s}$	Elapsed time from braking onset to standstill in the protective-separation calculation; distinct from controller sampling period.
$\ClearDistance$	Clear Distance	$\text{m}$	Clear distance from the machine to the boundary it must not cross; the subscript keeps it distinct from dataset size $\Glyph{D}$.
$\SpeedAtDetection$	Speed at Detection	$\text{m/s}$	Speed of the machine at the moment the hazard is detected; in stopping, pre-brake, and impact algebra, the scalar travel or approach speed $\SpeedAtDetection$.
$\SpeedBound$	Speed Bound	$\text{m/s}$	Largest speed the machine, a tracked object, or an approaching body is credited with in a budget; the bounding speed in $\TelemetryFreshnessBound \le \PositionErrorCeiling/\SpeedBound$.
$\CredibleBrakingDeceleration$	Credible Deceleration	$\text{m/s}^2$	Validated low tail of the deceleration the machine achieves under its stated conditions, not the datasheet peak.
$\ProfileAccelerationBound$	Profile Acceleration Bound	$\text{m/s}^2$	Largest acceleration or deceleration a planned or interpolated profile may command; for the base’s resident stop it is set to $\CredibleBrakingDeceleration$.
$\NetBrakingDeceleration$	Net Braking Deceleration	$\text{m/s}^2, \text{rad/s}^2$	Linear or angular deceleration left after a gravity or load bias torque $\Glyph{\tau_{\text{bias}}}$ is subtracted from the actuator’s braking capacity, in the direction of travel.
$\ResidentStopDistance(\SpeedAtDetection)$	Resident Stop Distance	$\text{m}$	Travel of the resident stop from speed $\SpeedAtDetection$: $\SpeedAtDetection^2/(2\CredibleBrakingDeceleration)$ for constant-deceleration braking; for the jerk-limited stop of equation 11.17, that term plus the jerk ramps’ $\SpeedAtDetection\,\CredibleBrakingDeceleration/(2\JerkLimit)$ once $\SpeedAtDetection \ge \CredibleBrakingDeceleration^2/\JerkLimit$.
$\LocalizationBound$	Localization Bound	$\text{m}$	Bound on localization and state-estimation error.
$\ProtectiveClearanceMargin$	Protective Clearance	$\text{m}$	Fixed, nonzero protective clearance between the machine envelope and the boundary.
$\TrackingErrorBound$	Tracking-Error Bound	$\text{m}$	Bound on the deviation between the commanded and the executed trajectory, stated by the enforcement record.
$\HumanWalkingSpeedBound$	Human Walking-Speed Bound	$\text{m/s}$	Walking-speed bound for a person approaching the machine, used in the form its safety standard states.
$\JerkLimit$	Jerk Limit	$\text{m/s}^3$	Maximum rate of change of deceleration; sets the jerk ramps of the resident stop in $\ResidentStopDistance(\SpeedAtDetection)$ (equation 11.17) and bounds the bumpless-transfer window $\BlendingTransitionWindow$.

Zero-order hold and phase margin

At crossover frequencies well below the sampling frequency, holding discrete control commands constant over sampling period $\SamplingPeriod = 1/\SamplingRate$ contributes approximately half a sample period of delay. With fixed modeled enforcer and fieldbus delays, the relationship below estimates phase-margin erosion; its phase-lag term is in radians, so all phase margins must use the same angular unit before subtraction:

\[\ZohTransportDelay = \frac{\SamplingPeriod}{2}, \qquad \DelayPhaseLag = \CrossoverFrequency \left( \frac{\SamplingPeriod}{2} + \EnforcerComputeTime + \CommandTransportTime \right), \qquad \Glyph{\text{PM}_{\text{net}}} = \text{PM}_{\text{plant}} - \DelayPhaseLag > 0\]

Symbol	Definition	Unit	Notes
$\ZohTransportDelay$	ZOH Transport Delay	$\text{s}$	Equivalent continuous delay introduced by discrete Zero-Order Hold reconstruction ($\SamplingPeriod / 2$).
$\SamplingPeriod, \SamplingRate$	Sampling Period and Rate	$\text{s}, \text{Hz}$	Discrete controller execution period and frequency ($\SamplingRate = 1/\SamplingPeriod$).
$\EnforcerComputeTime$	Enforcer Compute Time	$\text{s}$	Time the enforcer takes to evaluate its constraints within one control period; with $\CommandTransportTime$ it adds to the ZOH delay.
$\PureTransportDelay$	Pure Transport Delay	$\text{s}$	A fixed delay in a feedback or communication channel, contributing phase lag $\AngularFrequency \PureTransportDelay$ at frequency $\AngularFrequency$.
$\CrossoverFrequency$	Open-Loop Crossover Frequency	$\text{rad/s}$	Frequency at which open-loop gain $\lvert \OpenLoopTransferFunction(\ImaginaryUnit\CrossoverFrequency) \rvert = 1$ ($0\text{ dB}$).
$\DelayPhaseLag$	Phase Lag Degradation	$\text{rad}$ / $\text{deg}$	Phase erosion at crossover ($\DelayPhaseLag \approx 28.65^\circ \cdot (\CrossoverFrequency / \SamplingRate)$ from the hold alone) depleting closed-loop stability margin.

Actuator electrical and thermal limits

Actuators are constrained by copper winding heating, DC bus rail droop, and magnetic saturation. Sustainable torque output is governed by the coupled electromechanical balance, written per joint with scalar torque:

\[\DcBusVoltage = \PhaseInductance \frac{d\TorqueProducingCurrent}{d\TimeInstant} + \ElectricalResistance(\WindingTemperature)\, \TorqueProducingCurrent + \MotorBackEmfConstant \RotorAngularSpeed, \qquad \StatorThermalCapacitance \frac{d\WindingTemperature}{d\TimeInstant} = \TorqueProducingCurrent^2 \ElectricalResistance(\WindingTemperature) - \frac{\WindingTemperature - \AmbientTemperature}{\ThermalResistance}, \qquad \CommandedJointTorque = \GearboxTransmissionEfficiency \GearReductionRatio \MotorTorqueConstant \TorqueProducingCurrent\]

Symbol	Definition	Unit	Notes
$\MotorTorqueConstant, \MotorBackEmfConstant$	Torque and Back-EMF Constants	$\text{N}\cdot\text{m/A}, \text{V}\cdot\text{s/rad}$	Motor electromechanical coupling constants linking winding current to rotor shaft torque and rotor speed to back-EMF voltage ($\MotorTorqueConstant \approx \MotorBackEmfConstant$ in SI units).
$\TorqueProducingCurrent$	Torque-Producing Current	$\text{A}$	Quadrature-axis current of a field-oriented drive, $\MotorTorque = \MotorTorqueConstant \TorqueProducingCurrent$. Bare $\Glyph{I}$ is never a current, because the series core reserves it for arithmetic intensity.
$\MotorPhaseRmsCurrent$	Motor Phase RMS Current	$\text{A}$	RMS phase current driving the stator windings of a brushless PM motor; an uppercase $\Glyph{I}$ with a subscript names an RMS value or rating.
$\InstantaneousPhaseCurrent$	Instantaneous Phase Current	$\text{A}$	Signed instantaneous electrical phase current, positive from inverter node into winding; distinct from RMS phase current and torque-producing dq current.
$\ContinuousCurrent, \PeakCurrent$	Continuous / Peak Current	$\text{A}$	RMS current ratings bounded by winding insulation class and inverter MOSFET dissipation.
$\PhaseInductance, \PhaseResistance$	Phase Inductance and Resistance	$\text{H}, \Glyph{\Omega}$	Stator winding inductance and DC resistance; they set $\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance$.
$\ElectricalResistance(\WindingTemperature)$	Temperature-Dependent Resistance	$\Glyph{\Omega}$	Stator copper resistance at winding temperature $\WindingTemperature$: $\ElectricalResistance(\WindingTemperature) = \ReferenceWindingResistance [1 + \CopperTemperatureCoefficient(\WindingTemperature - \CopperReferenceTemperature)]$ with $\CopperTemperatureCoefficient = 0.00393\text{ K}^{-1}$.
$\ThermalResistance$	Thermal Resistance	$\text{K/W}$	Lumped thermal resistance from stator windings to the ambient environment; path segments carry a second subscript ($\JunctionAmbientThermalResistance$, $\JunctionCaseThermalResistance$, $\WindingCoreThermalResistance$, $\CoreAmbientThermalResistance$) and add in series.
$\StatorThermalCapacitance$	Stator Thermal Capacitance	$\text{J/K}$	Lumped heat capacity of the motor core, windings, and potting compound.
$\ElectricalTimeConstant$	Electrical Time Constant	$\text{s}$	Winding current rise time constant $\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance$; current reaches $1 - e^{-1}$ of its target in $\ElectricalTimeConstant$.
$\ThermalTimeConstant$	Thermal Time Constant	$\text{s}$	Actuator heating characteristic time constant ($\ThermalTimeConstant = \ThermalResistance \StatorThermalCapacitance$, typically $60\text{--}300\text{ s}$); a single stage takes a second subscript ($\CopperThermalTimeConstant$, $\CaseThermalTimeConstant$).
$\WindingTemperature, \AmbientTemperature$	Winding and Ambient Temperature	$\text{K}$ or $^\circ\text{C}$	The thermal model’s state and boundary temperatures; the other temperatures follow the subscript rule of the disambiguation table ($\JunctionTemperature$, $\CaseTemperature$, $\WindingTemperatureLimit$, $\EquilibriumWindingTemperature$, $\CopperReferenceTemperature$).
$\JouleanTemperatureRise$	Joulean Temperature Rise	$\text{K}$	Thermal rise above ambient; $\JouleanTemperatureRise = \WindingLossPower \ThermalResistance$ gives the steady-state rise for constant loss power in the lumped model. Transient rise follows the thermal balance; never a time interval.
$\ActuatorDutyCycle$	Actuator Duty Cycle	$[0, 1]$	Ratio of active pulse duration to cycle period ($\ActuatorOnTime / (\ActuatorOnTime + \ActuatorOffTime)$) preventing thermal runaway.

Shielded hazards and the exposure bound

Catastrophic failure rates in embodied systems cannot be certified through stochastic testing alone. For a defined hazard class $\HazardClassIndex$, verified interception coverage and independently bounded residual faults constrain the system’s hazard rate. Under a constant-rate Poisson exposure model, zero-failure exposure follows the Rule of Three:

\[\SystemHazardRate \le \HazardousProposalRate (1 - \ShieldInterceptionCoverage) + \HardwareCommonCauseRate \le \TargetHazardRate, \qquad \FailureObservationInterval \ge \frac{-\ln \HypothesisTestSignificanceLevel}{\TargetHazardRate}\]

Here $\FailureObservationInterval$ is the zero-incident operating exposure, in hours. For independent trials with a common success probability, per-trial campaigns use the same bound in probability form, $\ZeroFailureTrialCount \ge \ln\HypothesisTestSignificanceLevel/\ln \TargetTrialSuccessProbability \approx 3/(1 - \TargetTrialSuccessProbability)$.

Symbol	Definition	Unit	Notes
$\SystemHazardRate$	System Hazard Rate	$\text{h}^{-1}$	Certified dangerous failure rate of the integrated cyber-physical machine for hazard class $\HazardClassIndex$.
$\HazardousProposalRate$	Hazardous Proposal Rate	$\text{h}^{-1}$	Arrival rate of unshielded hazardous action proposals the policy emits before the permission layer, measured against a counterfactual oracle.
$\ShieldInterceptionCoverage$	Shield Interception Coverage	Dimensionless	Verified conditional probability that the shield intercepts a hazardous proposal of class $\HazardClassIndex$, evaluated over the states in which the policy errs.
$\HardwareCommonCauseRate$	Hardware and Common-Cause Rate	$\text{h}^{-1}$	Independently bounded hardware-fault and common-cause hazard rate the shield cannot intercept; the floor $\SystemHazardRate$ reaches when $\ShieldInterceptionCoverage = 1$.
$\TargetHazardRate$	Target Hazard Rate	$\text{h}^{-1}$	Tolerable dangerous failure rate assigned to the safety function by risk assessment, in the band its governing standard defines.
$\DiagnosticCoverage$	Diagnostic Coverage	Dimensionless	Rate-weighted fraction of hardware faults whose hazardous effect is mitigated, $\UnmitigatedHardwareFaultRate = \RawHardwareFaultRate(1 - \DiagnosticCoverage)$; the hardware counterpart of $\ShieldInterceptionCoverage$, which covers policy proposals.
$\TrialFailureProbability, \TargetTrialSuccessProbability$	Per-Trial Probabilities	Dimensionless	Per-trial failure probability, and the per-trial success probability a campaign must demonstrate; trial counts follow $\ZeroFailureTrialCount \ge \ln\HypothesisTestSignificanceLevel/\ln \TargetTrialSuccessProbability$.
$\HypothesisTestSignificanceLevel$	Statistical Significance Level	Dimensionless	Type I error threshold ($\HypothesisTestSignificanceLevel = 1 - \text{CL}$, e.g., $\HypothesisTestSignificanceLevel=0.05$ for 95% confidence where $-\ln(0.05) \approx 3.0$).
$\StatisticalFalseNegativeRate$	Type II Error Allowance	Dimensionless	Probability of rejecting an acceptable candidate ($1-\StatisticalFalseNegativeRate$ is the power); with $\HypothesisTestSignificanceLevel$ it sets the SPRT boundaries $\SprtUpperBoundary$ and $\SprtLowerBoundary$. Diffusion schedules use indexed $\Glyph{\beta_s}$; a KL weight $\Glyph{\beta}$ stays local to its objective.
$\ZeroFailureTrialCount$	Zero-Failure Trial Count	Trials	Required number of incident-free trials to substantiate $\TargetTrialSuccessProbability$; for exposure measured in hours, the bound is on $\FailureObservationInterval$ instead.
$\SequentialLogLikelihoodRatio$	Sequential Log-Likelihood Ratio	Dimensionless	After $\ObservedTrialFailureCount$ failures in $\SequentialTrialCount$ trials, $\SequentialLogLikelihoodRatio = \ObservedTrialFailureCount\ln(\HypothesizedTrialFailureProbability_1/\HypothesizedTrialFailureProbability_0) + (\SequentialTrialCount-\ObservedTrialFailureCount)\ln\frac{1-\HypothesizedTrialFailureProbability_1}{1-\HypothesizedTrialFailureProbability_0}$ for per-trial failure probabilities $\HypothesizedTrialFailureProbability_0$ (unacceptable) and $\HypothesizedTrialFailureProbability_1$ (acceptable); SPRT stops when it leaves $(\SprtLowerBoundary, \SprtUpperBoundary)$.
$\SprtUpperBoundary, \SprtLowerBoundary$	SPRT Boundaries	Dimensionless	Upper and lower stopping thresholds on $\SequentialLogLikelihoodRatio$, $\SprtUpperBoundary \approx \ln((1-\StatisticalFalseNegativeRate)/\HypothesisTestSignificanceLevel)$ and $\SprtLowerBoundary \approx \ln(\StatisticalFalseNegativeRate/(1-\HypothesisTestSignificanceLevel))$; never bare $\Glyph{A}$ and $\Glyph{B}$.
$\AssuranceClaim$	Assurance Claim	Proposition	Top-level safety claim of an assurance argument. Under an independent-premise probability model, its confidence $\Pr(\AssuranceClaim)$ is the product of the premise confidences for a conjunctive argument, and one minus the product of the legs’ failure probabilities for a multi-legged one.

The embodied energy corollary

Untethered mobile manipulators and autonomous robots operate under strict battery energy reserves and peak thermal dissipation ceilings. Total mission energy (distinct from the ML workload’s $\Glyph{E_{\text{total}}}$) splits across mechanical actuation work, edge neural computation, and auxiliary sensor payloads, while the compute energy expended by edge inference system-on-chip (SoC) devices decomposes into data movement and arithmetic operations according to the core energy corollary:

\[\MissionEnergy = \Glyph{E_{\text{act}}} + \Glyph{E_{\text{total}}} + \Glyph{E_{\text{aux}}}, \qquad \Glyph{E_{\text{total}}} \approx \DataVolume \times \Glyph{E_{\text{move}}} + \Operations \times \Glyph{E_{\text{compute}}}\]

For constant loss power in the single-node thermal model, resistive heating in stator windings $\WindingLossPower \approx \sum \TorqueProducingCurrent^2 \ElectricalResistance(\WindingTemperature)$ sets the steady-state rise $\JouleanTemperatureRise = \WindingLossPower \ThermalResistance$. Transient heating during shorter acceleration or contact bursts follows the coupled first-order thermal balance above.

Plant State, Actuation, and Kinematics

The symbols below describe the mechanical configuration, velocities, forces, and inertia of articulated and mobile mechanisms. Units for inertia and generalized loads specialize to revolute coordinates; mixed revolute–prismatic plants require entrywise units consistent with their generalized coordinates.

Symbol	Definition	Unit	Notes
$\PhysicalWorldState_{t} \in \WorldStateSpace$	Physical World State	Mixed	Ground-truth continuous state of the machine, environment, and physical contacts at time $\TimeInstant$, and the state of the decision process in the learning chapters; italic, because it mixes types.
$\MechanicalPlantState$	Mechanical Plant State	Mixed: $\text{rad}$ or $\text{m}$ for positions; $\text{rad/s}$ or $\text{m/s}$ for velocities	Generalized state vector $\MechanicalPlantState = [\GeneralizedCoordinates^\top, \GeneralizedVelocities^\top]^\top \in \mathbb{R}^{2\GeneralizedCoordinateCount}$ containing positions and velocities.
$\GeneralizedCoordinates \in \mathbb{R}^\GeneralizedCoordinateCount$	Generalized Coordinates	$\text{rad}$ or $\text{m}$	Degrees of freedom describing mechanical articulation (joint angles and prismatic link extensions); entries $\JointCoordinate_i$.
$\GeneralizedVelocities \in \mathbb{R}^\GeneralizedCoordinateCount$	Generalized Velocities	$\text{rad/s}$ or $\text{m/s}$	First time derivative of generalized coordinates.
$\GeneralizedAccelerations \in \mathbb{R}^\GeneralizedCoordinateCount$	Generalized Accelerations	$\text{rad/s}^2$ or $\text{m/s}^2$	Second time derivative of generalized coordinates.
$\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount$	Actuator Torque Vector	$\text{N}\cdot\text{m}$ or $\text{N}$	Generalized forces and torques delivered by actuators at mechanical joints; a subscript names a source, as in the contact torque $\Glyph{\JointTorque_c}$.
$\CommandedJointTorque$	Commanded Joint Torque	$\text{N}\cdot\text{m}$	Scalar torque commanded at one joint in single-joint derivations such as the kinetic barrier; bounds and disturbances keep a subscript ($\JointTorqueLimit$, $\Glyph{\tau_{\text{bias}}}$).
$\GeneralizedInertiaMatrix(\GeneralizedCoordinates)$	Generalized Inertia Matrix	$\text{kg}\cdot\text{m}^2$	Symmetric positive-definite mass/inertia matrix of the articulated multibody plant.
$\CoriolisCentrifugalMatrix(\GeneralizedCoordinates, \GeneralizedVelocities)$	Coriolis / Centrifugal Matrix	$\text{N}\cdot\text{m}\cdot\text{s/rad}$	Matrix satisfying $\Glyph{\dot{\mathbf{M}}} - 2\CoriolisCentrifugalMatrix$ skew-symmetry (passivity property).
$\GravitationalTorqueVector(\GeneralizedCoordinates)$	Gravitational Torque Vector	$\text{N}\cdot\text{m}$	Generalized gravitational load acting on links as a function of configuration; a single joint’s share is $\Glyph{\tau_g}$.
$\ManipulatorJacobian(\GeneralizedCoordinates)$	Manipulator Jacobian	Mixed	Linear mapping $\Glyph{\mathbf{v}_e} = \ManipulatorJacobian(\GeneralizedCoordinates)\GeneralizedVelocities$ from joint rates to operational task-space velocity.
$\YoshikawaManipulability(\GeneralizedCoordinates)$	Yoshikawa Manipulability	Depends on task coordinates and scaling	Singularity proximity metric $\YoshikawaManipulability(\GeneralizedCoordinates) = \sqrt{\det(\ManipulatorJacobian\ManipulatorJacobian^\top)}$ vanishing at kinematic singular configurations.
$\BackEmfVoltage$	Back-EMF Voltage	$\text{V}$	Opposing voltage $\BackEmfVoltage = \MotorBackEmfConstant \RotorAngularSpeed$ developed across stator coils by rotor rotation.
$\RotorInertia, \LoadInertia$	Rotor and Load Inertia	$\text{kg}\cdot\text{m}^2$	Moment of inertia of motor rotor shaft and coupled mechanical load.
$\ReflectedRotorInertia$	Reflected Rotor Inertia	$\text{kg}\cdot\text{m}^2$	Effective inertia $\ReflectedRotorInertia = \GearReductionRatio^2 \RotorInertia$ amplified quadratically by gearbox ratio.
$\EffectiveJointInertia$	Effective Joint Inertia	$\text{kg}\cdot\text{m}^2$	Total inertia seen at one joint, link and payload plus $\ReflectedRotorInertia$; the inertia in single-joint torque and acceleration budgets.
$\GearReductionRatio$	Gear Reduction Ratio	Dimensionless	Transmission reduction ratio ($\GearReductionRatio = \RotorAngularSpeed/\JointAngularSpeed_j \gg 1$) between motor rotor and joint output shaft; never bare $\Glyph{N}$.
$\InertiaMatchingRatio$	Optimal Inertia-Matching Ratio	Dimensionless	Ratio $\InertiaMatchingRatio = \sqrt{\LoadInertia / \RotorInertia}$ maximizing load angular acceleration under peak torque.
$\GearboxTransmissionEfficiency$	Transmission Efficiency	Dimensionless	Mechanical power transmission efficiency ($0 < \GearboxTransmissionEfficiency \le 1$); never bare $\LearningRate$, which is the learning rate.
$\PeakTorque, \HoldingTorque$	Continuous / Peak / Holding Torque	$\text{N}\cdot\text{m}$	Thermally continuous holding torque $\HoldingTorque$ and transient peak acceleration torque.
$\Mass$	Mass	$\text{kg}$	Physical body or payload mass, lowercase italic; effective and reduced impact masses carry subscripts ($\EffectiveImpactMass$, $\ReducedImpactMass$).
$\Gravity$	Gravitational Acceleration	$\text{m/s}^2$	Scalar magnitude of gravitational acceleration; the vector $\GravityVector$ and the configuration-dependent torque $\GravitationalTorqueVector(\GeneralizedCoordinates)$ are bold.
$\LinearMomentum$	Linear Momentum	$\text{kg}\cdot\text{m/s}$	Scalar momentum $\Mass \BodySpeed$ of a moving body; distinct from the bold position $\CartesianPosition$ and from a distribution $\ProbabilityFunction(\cdot)$.
$\WheelRadius$	Wheel Radius	$\text{m}$	Rolling radius of a driven wheel, which converts wheel torque to floor force and speed to wheel rate ($\Glyph{\omega} = \BodySpeed/\WheelRadius$); the barrier relative degree $\BarrierRelativeDegree$ appears only in barrier analysis.
$\AngularSpeed$	Angular Speed	$\text{rad/s}$	Magnitude of a rigid body’s angular velocity $\AngularVelocity$; not the crossover frequency $\Glyph{\omega_c}$.
$\BaseForwardSpeed$	Base Forward Speed	$\text{m/s}$	Forward linear speed of the mobile base along its heading, the speed input of the unicycle model; not the speed at the instant a hazard is detected.
$\BaseHeadingRate$	Base Heading Rate	$\text{rad/s}$	Rate of change of a mobile base’s heading angle, the turn input of the unicycle model.
$\BaseLateralVelocity$	Base Lateral Velocity	$\text{m/s}$	Lateral velocity of a mobile base in the plane.
$\BaseLongitudinalVelocity$	Base Longitudinal Velocity	$\text{m/s}$	Forward velocity of a mobile base in the plane.
$\BodySpeed$	Body Speed	$\text{m/s}$	Magnitude of the linear velocity of a moving body or machine; the scalar speed, with the vector velocity bold.
$\CommandedJointTorqueSlew$	Commanded Joint Torque Slew	$\text{N}\cdot\text{m/s}$	Time derivative of the torque at a single joint; for constant effective inertia it equals $\EffectiveJointInertia$ times the joint jerk.
$\ContinuousTorque$	Continuous Torque	$\text{N}\cdot\text{m}$	Largest motor torque a winding can sustain indefinitely at its temperature limit under the stated cooling; set by heat rejection, and at the engineering ceiling it gives the installed rating.
$\DirectAxisCurrent$	Direct-Axis Current	$\text{A}$	Current of a field-oriented drive along the rotor field, which produces no torque and is normally regulated to zero; not the torque-producing quadrature current.
$\ElectricalAngularSpeed$	Electrical Angular Speed	$\text{rad/s}$	Rotor speed times the pole-pair count; the electrical frequency of the drive, not the mechanical shaft speed.
$\ElectricalRotorAngle$	Electrical Rotor Angle	$\text{rad}$	Mechanical rotor angle times the pole-pair count, the angle a field-oriented drive uses to rotate between stator and rotor frames.
$\EncoderAngle$	Encoder Angle	$\text{rad}$	Joint angle an encoder reports; a sample index appears as a subscript, as in ${\EncoderAngle}_{k}$.
$\ForwardKinematicsMap$	Forward Kinematics Map	Function	Map from joint positions to the positions of points on the robot’s links in the base frame.
$\GeneralizedCoordinateCount$	Generalized Coordinate Count	Dimensionless	Number of generalized coordinates of a mechanism, its degrees of freedom (the joints of a serial arm); the plant state has twice this many entries.
$\HeadingAngle$	Heading Angle	$\text{rad}$	Heading of a planar mobile base in the unicycle model, the direction its forward speed points.
$\ConveyorSpeed$	Conveyor Speed	$\text{m/s}$	Speed of the belt that carries a workpiece past the pick station; not the bound on a takeaway conveyor, $\ConveyorSpeedBound$.
$\DrivenWheelLoadFraction$	Driven-Wheel Load Fraction	Dimensionless	Fraction of a vehicle’s total weight carried by its driven wheels; it scales the traction ceiling on braking deceleration.
$\JointCoordinate$	Joint Coordinate	$\text{rad}$ or $\text{m}$	One entry of the generalized coordinates $\GeneralizedCoordinates$, a single joint’s angle or extension.
$\JointPosition$	Joint Position	$\text{rad}$ or $\text{m}$	Position of a single joint along its axis, an angle for a revolute joint and a length for a prismatic one; its time derivatives are $\JointVelocity$ and $\JointAcceleration$.
$\JointAngle$	Joint Angle	$\text{rad}$	Angle of one revolute joint, a scalar; the joint-space vector is $\GeneralizedCoordinates$.
$\JointVelocity$	Joint Velocity	$\text{rad/s}$ or $\text{m/s}$	Velocity of one joint, a scalar; the vector of all joints is $\GeneralizedVelocities$.
$\JointAngularSpeed$	Joint Angular Speed	$\text{rad/s}$	Angular speed of a single arm joint; a subscript zero marks the initial value.
$\JointAcceleration$	Joint Acceleration	$\text{rad/s}^2$ or $\text{m/s}^2$	Acceleration of one joint, a scalar; the vector of all joints is $\GeneralizedAccelerations$.
$\JointJerk$	Joint Jerk	$\text{rad/s}^3$	Jerk of one joint’s trajectory, the third time derivative of its position; a scalar for one joint.
$\JointCount$	Joint Count	Dimensionless	Number of joints of a manipulator, the length of its joint-position vector; it shares its glyph with the zero-failure trial count of the validation chapters.
$\JointIndex$	Joint Index	Index	Position of one joint axis of a manipulator, from one to the number of joints.
$\MotorTorque$	Motor Torque	$\text{N}\cdot\text{m}$	Torque a motor produces at its shaft before the transmission, the torque constant times the torque-producing current; the joint output is $\JointOutputTorque$.
$\JointOutputTorque$	Joint Output Torque	$\text{N}\cdot\text{m}$	Torque a joint delivers at the transmission output: the transmission efficiency $\GearboxTransmissionEfficiency$ times the gear ratio $\GearReductionRatio$ times the motor torque $\MotorTorque$.
$\JointFrictionTorque$	Joint Friction Torque	$\text{N}\cdot\text{m}$	Total friction torque of a joint, referred to its output link.
$\JointTorqueLimit$	Joint Torque Limit	$\text{N}\cdot\text{m}$	Largest torque the actuator of one joint can deliver, indexed by joint; it bounds $\CommandedJointTorque$ at the same joint and is not the transient burst rating.
$\JointTorqueUpperLimit$	Joint Torque Upper Limit	$\text{N}\cdot\text{m}$	Per-joint upper bounds on the torque the actuators can deliver, as a vector; the vector counterpart of the scalar $\JointTorqueLimit$.
$\JointTorqueSlew$	Joint Torque Slew	$\text{N}\cdot\text{m/s}$	Time derivative of the joint torque vector; it couples to jerk through the derivative of the manipulator equation.
$\JointVelocityLimit$	Joint Velocity Limit	$\text{rad/s}$ or $\text{m/s}$	Per-joint bounds on the magnitude of the generalized velocities; a planned trajectory and every admitted command must stay within them.
$\JointTrackingError$	Joint Tracking Error	$\text{rad}$ or $\text{m}$	Vector of differences between the desired and the measured joint positions; not the state estimation error, which shares the glyph.
$\LeverArmLength$	Lever Arm Length	$\text{m}$	Distance from a rotating joint to the point it carries, which converts a joint rotation into tool travel; a scalar length.
$\JacobianSingularNumber$	Jacobian Singular Value	Mixed	Singular value of the manipulator Jacobian $\ManipulatorJacobian$, the tool speed a unit joint rate produces along one singular direction; units follow the Jacobian rows.
$\InverseKinematicsDamping$	Inverse Kinematics Damping	Mixed	Damping length that damped-least-squares inverse kinematics adds to bound the joint step near a singularity; it enters squared, and along a singular direction the gain never exceeds one over twice its value. The ridge-regression analogue of weight decay.
$\PermanentMagnetFluxLinkage$	Permanent-Magnet Flux Linkage	$\text{V}\cdot\text{s}$	Magnetic flux from the rotor magnets that links each stator winding; with the pole-pair count it sets the torque constant.
$\PlantDynamicsFunction$	Plant Dynamics Function	Function	Vector field of a plant’s equations of motion, giving the state rate from the state, the control input and the plant parameters; not the unforced drift part of control-affine dynamics, which shares its glyph.
$\PlantParameters$	Plant Parameters	Record	Physical and hardware properties of the machine in one trial or rollout, such as tolerances, wear, payload mass, state of charge and calibration; not the mass $\Glyph{m}$ of a body.
$\PlantStateRate$	Plant State Rate	Dimensionless per unit time	Time derivative of the mechanical plant state; the decorated form of the plant state, and not the Cartesian velocity of the tool, which shares its glyph.
$\PlantStateSpace$	Plant State Space	Set	Set of all states the mechanical plant can occupy, positions together with velocities.
$\RotorAngularSpeed$	Rotor Angular Speed	$\text{rad/s}$	Angular speed of a motor rotor about its axis; it sets the back-EMF, and on a direct-drive hub motor it equals the wheel’s angular speed.
$\RotorPolePairCount$	Rotor Pole-Pair Count	Dimensionless	Number of magnetic pole pairs on a motor rotor; multiplying the mechanical rotor speed by it gives the electrical frequency of the drive.
$\QuadratureAxisVoltage$	Quadrature-Axis Voltage	$\text{V}$	Voltage the inverter of a field-oriented drive applies along the quadrature axis, the axis that produces torque.
$\TiltAngle$	Tilt Angle	$\text{rad}$	Angular deflection of a body from vertical; zero when the body is upright.
$\TorqueRateLimit$	Torque Rate Limit	$\text{N}\cdot\text{m/s}$	Largest rate of change of joint output torque a drive accepts. Divided by the effective joint inertia it bounds joint jerk.
$\TransmissionTorsionalStiffness$	Transmission Torsional Stiffness	$\text{N}\cdot\text{m/rad}$	Torsional stiffness of the transmission between rotor and output link, the compliance that couples motor to load.
$\YawRate$	Yaw Rate	$\text{rad/s}$	Angular velocity of a body about its vertical axis, one component of the body angular velocity $\AngularVelocity$.

Contact Mechanics and Centroidal Dynamics

Contact turns a free-moving body into a constrained one. The symbols here describe the forces a contact can transmit, the friction cone that bounds them, and the centroidal quantities that sum them over the whole robot.

Symbol	Definition	Unit	Notes
$\ActuationSelectionMatrix \in \mathbb{R}^{\GeneralizedCoordinateCount \times (\GeneralizedCoordinateCount+6)}$	Actuation Selection Matrix	Matrix	Matrix $[\ZeroMatrix, \IdentityMatrix]$ projecting generalized forces onto actuated joints in floating-base systems.
$\ContactPointForce_{k} \in \mathbb{R}^3$	Contact Force Vector	$\text{N}$	Physical contact force exerted by environment on contact patch $\Glyph{k}$ ($\ContactPointForce_{k} = [\TangentialContactShear^\top, \NormalContactForce]^\top$); always indexed by the contact.
$\FingerNormalForce$	Finger normal force	N	Normal force one finger of a grasp presses against the object at its contact.
$\FingerTangentialForce$	Finger tangential force	N	Tangential friction force one finger of a grasp carries at its contact, bounded by the friction cone times $\FingerNormalForce$.
${\ContactPointJacobian}_k(\GeneralizedCoordinates)$	Contact Point Jacobian	Mixed	Mapping joint velocities to contact patch Cartesian linear velocity: $\Glyph{\dot{\mathbf{p}}}_k = {\ContactPointJacobian}_k(\GeneralizedCoordinates) \GeneralizedVelocities$.
$\NormalContactForce$	Normal Contact Force	$\text{N}$	Unilateral compressive force ($\NormalContactForce = \ContactPointForce_{k}^\top \ContactNormal_k \ge 0$) perpendicular to contact plane.
$\TangentialContactShear$	Tangential Contact Shear	$\text{N}$	Friction traction force tangential to contact plane ($\TangentialContactShear = \ContactPointForce_{k} - \NormalContactForce \ContactNormal_k$).
$\CoulombFrictionCoefficient$	Coulomb Friction Coefficient	Dimensionless	Static/kinetic coefficient bounding friction cone: $\lVert \TangentialContactShear \rVert \le \CoulombFrictionCoefficient \NormalContactForce$.
$\MassCenterPosition \in \mathbb{R}^3$	Center of Mass (CoM)	$\text{m}$	Spatial coordinates of the total articulated body mass centroid ($\MassCenterPosition = \frac{1}{\Mass} \sum \Mass_i \CartesianPosition_i$).
$\CentroidalLinearMomentum$	Centroidal Linear Momentum	$\text{kg}\cdot\text{m/s}$	Total plant linear momentum ($\CentroidalLinearMomentum = \Mass \Glyph{\dot{\mathbf{c}}}$), governed by $\CentroidalLinearMomentumDerivative = \sum \ContactPointForce_{k} + \Mass \GravityVector$, with gravitational acceleration $\GravityVector$.
$\CentroidalAngularMomentum$	Centroidal Angular Momentum	$\text{kg}\cdot\text{m}^2/\text{s}$	Angular momentum about CoM: $\CentroidalAngularMomentumDerivative = \sum (\CartesianPosition_k - \MassCenterPosition) \times \ContactPointForce_{k} + \ContactPointTorque_k$.
$\ZeroMomentPoint \in \mathbb{R}^2$	Zero Moment Point (ZMP)	$\text{m}$	Ground-plane point where net horizontal tipping moment vanishes; must reside in $\text{Conv}(\{\CartesianPosition_k\})$.
$\RestitutionCoefficient \in [0, 1]$	Coefficient of Restitution	Dimensionless	Kinematic velocity rebound ratio during non-smooth impact ($\JointVelocity^+ = -\RestitutionCoefficient \JointVelocity^-$).
$\ContactStiffness, \HertzianContactStiffness$	Contact Stiffness	$\text{N/m}$	Elastic stiffness of a structural contact, and of the human body region $\Glyph{H}$ in an impact (with its effective mass $\BodyRegionMass$), in the form ISO/TS 15066 states.
$\EffectiveImpactMass$	Effective Impact Mass	$\text{kg}$	Apparent normal mass at a specified contact point and configuration. For a single frictionless normal impact, it is the reciprocal of the positive scalar Delassus coefficient formed from the contact normal Jacobian and generalized inertia matrix (Rigid Body Contact Mechanics).
$\ReducedImpactMass$	Reduced Impact Mass	$\text{kg}$	Two-body reduced mass of a collision, $\ReducedImpactMass = (1/\EffectiveImpactMass + 1/\BodyRegionMass)^{-1}$, which sets the transferred energy.
$\UnbrakedImpactForce$	Unbraked Impact Force	$\text{N}$	Transient collision force peak ($\UnbrakedImpactForce = \SpeedBound \sqrt{\HertzianContactStiffness \ReducedImpactMass}$) governed by ISO/TS 15066.
$\ArmStoppingDeceleration$	Arm Stopping Deceleration	$\text{m/s}^2$	Deceleration the arm must achieve after braking onset to stop within the latch force limit.
$\BodyRegionMass$	Body Region Mass	$\text{kg}$	Effective mass of a human body region, here the hand, in the ISO/TS 15066 body model.
$\BrakeCommandForce$	Brake Command Contact Force	$\text{N}$	Contact force against the latch at the instant the brake command issues, after the contact loop responds; distinct from a commanded traction force.
$\BrakingOnsetContactForce$	Braking Onset Contact Force	$\text{N}$	Contact force reached against the latch plate at physical braking onset.
$\CentroidalAngularMomentumDerivative$	Centroidal Angular Momentum Derivative	$\text{N}\cdot\text{m}$	Time derivative of the centroidal angular momentum, the net external moment about the center of mass: contact-force moments plus the contact torques.
$\CentroidalLinearMomentumDerivative$	Centroidal Linear Momentum Derivative	$\text{N}$	Time derivative of the centroidal linear momentum, the net external force on the robot: the sum of contact forces plus mass times gravity.
$\ContactApproachSpeed$	Contact Approach Speed	$\text{m/s}$	Speed at which the robot approaches and strikes a contact surface; contact force rises at stiffness times this speed.
$\ContactDirectionVelocity$	Contact Direction Velocity	$\text{m/s}$	Velocity of the robot’s contact point along the contact direction.
$\ContactForce$	Contact Force	$\text{N}$	Force between the robot and the environment at a contact, along the contact direction; the per-point vector is $\ContactPointForce_{k}$.
$\ContactLoopResponseTime$	Contact Loop Response Time	$\text{s}$	Time the contact-control loop takes to respond after its tripwire fires.
$\ContactNormal$	Contact Normal	Dimensionless	Unit vector normal to the contact plane at one contact, pointing into the robot; indexed by contact.
$\ContactNormalPosition$	Contact Normal Position	$\text{m}$	Position of a tool or link along the contact normal, whose change from first contact is the contact penetration in the spring contact model.
$\ContactPointIndex$	Contact Point Index	Index	Position of one contact point among the contacts of a grasp.
$\ContactPointOffset$	Contact Point Offset	$\text{m}$	Position of a grasp contact relative to the reference point of the grasped object, the moment arm of its force; indexed by contact.
$\ContactPointPosition$	Contact Point Position	$\text{m}$	Position of the robot’s contact point along the contact direction.
$\ContactPointTorque$	Contact Point Torque	$\text{N}\cdot\text{m}$	Moment the environment exerts on the robot at one contact patch, the torque part of the contact wrench; indexed by contact. Not the joint torque vector $\JointTorque$, which shares the glyph.
$\ContactTripwireForce$	Contact Tripwire Force	$\text{N}$	Contact force at which the permission path ends an attempt on the machine, set below the structural damage limit.
$\ContactWrench$	Contact Wrench	Varies	Force and moment the environment applies at a contact, stacked into one vector; force rows in newtons, moment rows in newton-meters. Privileged in simulation.
$\ContactWrenchCeiling$	Contact Wrench Ceiling	Varies	Largest contact force and torque an intent lease permits at the tool, stacked as a wrench; units mix newtons and newton-meters.
$\DesiredContactPosition$	Desired Contact Position	$\text{m}$	Setpoint position of the robot’s contact point along the contact direction.
$\DesiredContactVelocity$	Desired Contact Velocity	$\text{m/s}$	Setpoint velocity of the robot’s contact point along the contact direction.
$\EffectiveSeriesStiffness$	Effective Series Stiffness	$\text{N/m}$	Stiffness of a contact whose two sides deflect in series, the reciprocal of the sum of their compliances; the softer side dominates.
$\ExternalWrench$	External Wrench	Mixed	Force and moment the environment applies at the end effector, in newtons for the force and newton-meters for the moments; it maps into joint torques through the transposed manipulator Jacobian.
$\ExternalContactTorque$	External Contact Torque Vector	$\text{N}\cdot\text{m}$	Joint torques produced by external contact forces acting on the manipulator.
$\ExternalJointTorque$	External Joint Torque	$\text{N}\cdot\text{m}$	Torque the environment applies at one joint’s output; the scalar counterpart of the external contact torque vector $\ExternalContactTorque$.
$\FrictionCoefficientFloor$	Friction Coefficient Floor	Dimensionless	Lowest friction coefficient a braking or contact premise assumes the surface provides; an unobserved oil spill can invalidate it.
$\GraspForceCeiling$	Grasp Force Ceiling	$\text{N}$	Largest grasp contact force the object tolerates without crushing or deforming.
$\GraspForceFloor$	Grasp Force Floor	$\text{N}$	Smallest grasp contact force that holds the object against slip; the grasp must stay between this floor and $\GraspForceCeiling$.
$\GraspMap$	Grasp Map	Matrix	Linear map from contact forces to the wrench they exert on the grasped object; indexed by contact when one contact is meant; not the Kalman gain.
$\GravityCenterHeight$	Center-of-Gravity Height	$\text{m}$	Height of a vehicle’s center of gravity above the floor; it sets the tip-over ceiling on deceleration.
$\GravityVector$	Gravity Vector	$\text{m/s}^2$	Gravitational acceleration as a vector pointing down with the magnitude of surface gravity; the scalar is the gravitational acceleration, and it is not the vector of gravitational joint torques.
$\ImpedanceControlledPosition$	Impedance-Controlled Position	$\text{m}$	Measured position of the point an impedance controller regulates; its deviation from $\ImpedanceEquilibriumPosition$, times the stiffness, is the spring term.
$\ImpedanceControlledVelocity$	Impedance-Controlled Velocity	$\text{m/s}$	Measured velocity of the regulated point, the time derivative of $\ImpedanceControlledPosition$.
$\ImpedanceEquilibriumPosition$	Impedance Equilibrium Position	$\text{m}$	Virtual rest point of the emulated spring that an impedance controller pulls the controlled point toward.
$\ImpedanceEquilibriumVelocity$	Impedance Equilibrium Velocity	$\text{m/s}$	Velocity of the virtual equilibrium; its difference from $\ImpedanceControlledVelocity$, times the damping, is the damper term.
$\KineticEnergy$	Kinetic Energy	$\text{J}$	Energy a moving body carries, one half its mass times its speed squared (plus rotational terms where stated); not the standard’s limit on energy transferred to a person.
$\MassCenterHeight$	Center-of-Mass Height	$\text{m}$	Height of a body’s center of mass above its support or contact point; it sets the natural frequency of the linear inverted pendulum, $\sqrt{\Gravity/\MassCenterHeight}$, and is not a camera-frame coordinate.
$\PenetrationDepth$	Penetration Depth	$\text{m}$	Depth by which one body penetrates or compresses another at a contact, measured along the contact normal; penalty and compliant contact models compute normal force from it. Shares its glyph with the Huber threshold of the training chapters.
$\PenetrationRate$	Penetration Rate	$\text{m/s}$	Time rate of change of the penetration depth $\PenetrationDepth$ at a contact; zero at the instant of maximum compression.
$\LatchApproachSpeed$	Latch Approach Speed	$\text{m/s}$	Speed at which the arm presses into the cage-door latch plate; the enforcement and release chapters use the same latch scenario.
$\LatchForceLimit$	Latch Force Limit	$\text{N}$	Structural force limit of the door latch the arm pushes against.
$\LatchPlateStiffness$	Latch Plate Stiffness	$\text{N/m}$	Spring stiffness of the cage-door latch plate the arm compresses.
$\LeadingSupportDistance$	Leading Support Distance	$\text{m}$	Horizontal distance from a vehicle’s center of gravity forward to its leading ground support; it shrinks when an arm extends forward.
$\IntrusionDistance$	Intrusion Distance	$\text{m}$	Distance a body part can intrude past the sensing field before detection; a term of the ISO/TS 15066 protective separation distance.
$\PersonApproachTravel$	Person Approach Travel	$\text{m}$	Distance a person travels toward the robot during the robot’s reaction and stopping time; a term of the ISO/TS 15066 protective separation distance.
$\PersonPositionUncertainty$	Person Position Uncertainty	$\text{m}$	Position uncertainty of the person from the sensing system; a term of the ISO/TS 15066 protective separation distance.
$\KnownComponentStopDistance$	Known-Component Stop Distance	$\text{m}$	Stopping travel that the known components of the stopping budget add up to at a given speed on a dry floor: delay travel, braking, jerk ramps and fixed overheads; the clear distance minus it is the spare the stopping claim holds in reserve.
$\PreImpactNormalVelocity$	Pre-Impact Normal Velocity	$\text{m/s}$	Normal contact velocity just before an impact, negative while the bodies approach; the restitution law links it to the post-impact velocity.
$\PostImpactNormalVelocity$	Post-Impact Normal Velocity	$\text{m/s}$	Normal contact velocity just after an impact, equal to minus the restitution coefficient times the pre-impact normal velocity.
$\RobotStructuralStiffness$	Robot Structural Stiffness	$\text{N/m}$	Stiffness of the manipulator side of a contact, its pads, linkages and joint elasticity acting as one series spring; it combines in series with the stiffness of the struck body.
$\SlipAccelerationBound$	Slip Acceleration Bound	$\text{m/s}^2$	Bound on the residual acceleration of a tracked object slipping relative to its carrier; it widens the evidence horizon of a target estimate.

Geometry, Transformations, and Perception

Every measurement a robot uses arrives in some frame, and every command leaves in another. These symbols name the frames, the rigid transforms between them, and the camera models that project the world onto a sensor.

Symbol	Definition	Unit	Notes
$\SpecialEuclideanGroup$	Special Euclidean Group	Lie Group	Group of 3D rigid-body spatial transformations (translations and rotations: $\mathbb{R}^3 \rtimes \SpecialOrthogonalGroup$).
$\SpecialOrthogonalGroup$	Special Orthogonal Group	Lie Group	Group of $3 \times 3$ proper orthogonal rotation matrices ($\DirectionalCosineMatrix^\top \DirectionalCosineMatrix = \IdentityMatrix, \det(\DirectionalCosineMatrix) = +1$).
$\WorldReferenceFrame, \BodyReferenceFrame, \Glyph{\mathcal{F}_E}$	Coordinate Reference Frames	Reference	Fixed World frame ($\WorldReferenceFrame$), mobile robot Base/body frame ($\BodyReferenceFrame$), and End-effector frame ($\Glyph{\mathcal{F}_E}$).
$\CartesianPosition \in \mathbb{R}^3$	Cartesian 3D Position	$\text{m}$	Position vector $[\Glyph{x}, \Glyph{y}, \Glyph{z}]^\top$ expressing point coordinates in a stated frame.
$\RigidTransformTranslation \in \mathbb{R}^3$	Rigid-Transform Translation	$\text{m}$	Translation part of a rigid transform between two frames (the offset of one frame’s origin in the other).
$\DirectionalCosineMatrix \in \SpecialOrthogonalGroup$	Directional Cosine Matrix	Matrix	$3 \times 3$ orthonormal matrix parameterizing spatial attitude and orientation.
$\UnitQuaternion \in \Glyph{\mathbb{H}}$	Unit Quaternion	Dimensionless	Hypercomplex orientation vector $[\Glyph{w}, \Glyph{x}, \Glyph{y}, \Glyph{z}]^\top$ satisfying $\lVert \UnitQuaternion \rVert = 1$.
$\Pose \in \SpecialEuclideanGroup$	Pose	Matrix	Rigid-body pose of a named frame, such as a grasp, the tool center point, or a target ($\Glyph{\Pose_{\text{grasp}}}$, $\Glyph{\Pose_{\text{TCP}}}$); the transform between two frames is $\HomogeneousTransform$ below.
$\HomogeneousTransform \in \SpecialEuclideanGroup$	Homogeneous Transform	Matrix	$4 \times 4$ matrix $\begin{bmatrix} \TransformRotationBlock & \TransformTranslationBlock \\ \ZeroVector^\top & 1 \end{bmatrix}$ mapping coordinates from frame $\Glyph{B}$ into frame $\Glyph{A}$.
$\AngularVelocity, \AngularAcceleration \in \mathbb{R}^3$	Angular Velocity & Acceleration	$\text{rad/s}, \text{rad/s}^2$	Spatial angular velocity and angular acceleration vectors of a rigid body.
$\SensorLeverArm \in \mathbb{R}^3$	Sensor Lever-Arm Vector	$\text{m}$	Displacement vector from body rotation origin $\Glyph{O}$ to displaced sensor location $\Glyph{P}$; its length, or a link’s lever arm, is the scalar $\LeverArmLength$.
$\FundamentalMatrix, \EssentialMatrix$	Fundamental and Essential Matrices	Matrix	Two-view epipolar constraint $\HomogeneousPixelCoordinates_2^\top \FundamentalMatrix\HomogeneousPixelCoordinates_1 = 0$ with $\EssentialMatrix = [\RigidTransformTranslation]_\times \DirectionalCosineMatrix$; the subscript keeps it distinct from the transition matrix $\StateTransitionMatrix$.
$\CameraIntrinsicCalibrationMatrix \in \mathbb{R}^{3 \times 3}$	Pinhole Camera Intrinsics	Pixels	Calibration matrix $\begin{bmatrix} \HorizontalFocalLength & 0 & \PrincipalPointColumn \\ 0 & \VerticalFocalLength & \PrincipalPointRow \\ 0 & 0 & 1 \end{bmatrix}$ parameterizing focal lengths and principal center.
$\CameraIntrinsicInverseTranspose$	Inverse-transpose intrinsic matrix	Matrix	Transpose of the inverse of the camera intrinsic matrix $\CameraIntrinsicCalibrationMatrix$, which carries an essential matrix into the pixel-coordinate fundamental matrix.
$\CameraFramePoint = [\CameraFrameHorizontalCoordinate, \CameraFrameVerticalCoordinate, \CameraFrameDepth]^\top$	Camera-Frame 3D Coordinate	$\text{m}$	Metric coordinates of a spatial point in the optical camera frame ($\CameraFrameDepth$ is optical depth).
$[\PixelColumnCoordinate, \PixelRowCoordinate]^\top$, $\HomogeneousPixelCoordinates$	Image Plane Pixel Coordinates	Pixels	Discrete sensor pixel coordinates ($\PixelColumnCoordinate \sim \HorizontalFocalLength \frac{\CameraFrameHorizontalCoordinate}{\CameraFrameDepth} + \PrincipalPointColumn$, $\PixelRowCoordinate \sim \VerticalFocalLength \frac{\CameraFrameVerticalCoordinate}{\CameraFrameDepth} + \PrincipalPointRow$), italic as scalars; the homogeneous image point is $\HomogeneousPixelCoordinates = [\PixelColumnCoordinate, \PixelRowCoordinate, 1]^\top$, never bold $\Glyph{\mathbf{u}}$.
$\ExposureDuration$	Exposure Duration	$\text{ms}$	Integration time of one camera frame; the frame’s timestamp belongs at its midpoint, $\ExposureDuration/2$ after exposure starts.
$\BeliefInvalidationHorizon$	Belief Invalidation Horizon	$\text{ms}$	Time since the last supporting evidence after which a belief can no longer be shown to lie inside its task tolerance; a belief expires at $\TimeInstant_0 + \BeliefInvalidationHorizon$.
$\FreeSpaceDepth$	Free-Space Depth	$\text{m}$	Distance from a hull’s surface to the nearest cell that is occupied, unknown, or expired, read from an ESDF built with all three treated as obstacles; the budget motion may spend there (Volumetric metric maps and distance fields).
$\RollingShutterRowReadout$	Rolling Shutter Row Readout	$\MicroUnitPrefix\text{s}$	Exposure time delta between adjacent CMOS sensor scanlines; row $\SensorRowIndex$ starts at $\TimeInstant(\SensorRowIndex) = \FrameExposureStartTime + \SensorRowIndex\,\RollingShutterRowReadout$.
$\RollingShutterDistortion(\SensorRowIndex)$	Rolling Shutter Distortion	$\text{m}$	Kinematic skew displacement $\RollingShutterDistortion = \Velocity \cdot \SensorRowIndex \cdot \RollingShutterRowReadout$ induced by high-velocity platform motion.
$\Glyph{\text{tsdf}}(\VoxelCenterPosition)$	Truncated Signed Distance	$[-1, +1]$	Normalized volumetric distance $\Glyph{\text{tsdf}}(\VoxelCenterPosition) = \text{clamp}(\Glyph{d}(\VoxelCenterPosition)/\TsdfTruncationBand, -1, 1)$ of the voxel centered at $\VoxelCenterPosition$ to the nearest surface.
$\TsdfTruncationBand$	TSDF Truncation Band	$\text{m}$	Metric thickness of the spatial surface interface band stored in voxel memory.
$\TsdfAccumulatedDistance_t(\VoxelCenterPosition), \TsdfAccumulatedWeight_t(\VoxelCenterPosition)$	TSDF Accumulated Volume	Mixed	Running weighted signed distance and accumulated visibility weight at the voxel centered at $\VoxelCenterPosition$.
$\InitialTrackingError, \PositionErrorCeiling$	Initial and Ceiling Error	$\text{m}$	Position error of an estimate when its evidence is fresh, and the largest error the task tolerates; $\Glyph{E}_0$ and $\Glyph{E_{\max}}$ stay energies.
$\PositionalTaskTolerance$	Positional Task Tolerance	$\text{m}$	Radius within which a target estimate must lie for the task to succeed; paired with the orientation tolerance $\OrientationTolerance$.
$\GoalSet$	Geometric Goal Set	Subset of $\SpecialEuclideanGroup$	Admissible terminal poses within the positional and angular tolerances of the task; the spatial component of an intent lease $\IntentLease$.
$\TargetTravelDistance$	Distance to Target	$\text{m}$	Straight-line distance from the measured end-effector position to the admitted target, $\lVert \TargetPosition - \CartesianPosition_0 \rVert$, in the intent-lease reachability test; distinct from the hidden width $\HiddenDimension$.
$\AttitudeError$	Attitude Error	$\text{rad}$	Angular error in an estimated attitude, which leaks gravity into the horizontal acceleration.
$\AzimuthAngle$	Azimuth Angle	$\text{rad}$	Angle of a direction measured in the horizontal plane about the vertical axis; the book may state values in degrees.
$\BaseCameraTransform$	Base-to-Camera Transform	Matrix	Rigid transform mapping camera coordinates into the robot base frame.
$\BodyFramePoint$	Body-Frame Point	$\text{m}$	Position of a perceived point expressed in the robot body frame, the camera-frame point carried through the extrinsics.
$\BodyFramePointCovariance$	Body-Frame Point Covariance	$\text{m}^2$	Covariance of a perceived point’s position error in the robot body frame, the spatial covariance an observation contract carries; rotated camera-frame covariance plus the extrinsic term, not the covariance of a state estimator.
$\CameraExtrinsicRotation$	Camera Extrinsic Rotation	Dimensionless	Rotation part of a camera’s extrinsic calibration, taking camera-frame vectors into the body frame; the rotation block of $\CameraExtrinsicTransform$.
$\CameraExtrinsicTransform$	Camera Extrinsic Transform	Matrix	Rigid transform of a camera’s extrinsic calibration, mapping camera-frame coordinates into the body frame; mixed rotation and translation blocks.
$\CameraFramePointCovariance$	Camera-Frame Point Covariance	$\text{m}^2$	Covariance of a point’s position error in the camera’s optical frame, after unprojection of its pixel and depth measurement; rotated into the body frame by $\CameraExtrinsicRotation$.
$\CartesianXCoordinate$	Cartesian X Coordinate	$\text{m}$	First Cartesian component of a three-dimensional position.
$\CartesianYCoordinate$	Cartesian Y Coordinate	$\text{m}$	Second Cartesian component of a three-dimensional position.
$\CartesianZCoordinate$	Cartesian Z Coordinate	$\text{m}$	Third Cartesian component of a three-dimensional position.
$\ClearanceInset$	Clearance Inset	$\text{m}$	Separation a consumer holds between an estimated obstacle position and the machine’s envelope so the true obstacle stays the required clearance away, except with a stated tail probability; a field of the observation contract.
$\DepthStandardDeviation$	Depth Standard Deviation	$\text{m}$	Standard deviation of the depth a stereo or range sensor reports for one point; for stereo it grows with the square of the range.
$\DisparityStandardDeviation$	Disparity Standard Deviation	pixels	Standard deviation of the disparity a stereo matcher measures; it feeds the stereo depth term $\DepthStandardDeviation$.
$\ElevationAngle$	Elevation Angle	$\text{rad}$	Angle of a direction above the horizontal plane; not the encoder parameters of a variational autoencoder, which render the same glyph.
$\ExtrinsicCalibrationCovariance$	Extrinsic Calibration Covariance	$\text{m}^2$	Covariance of a sensor’s calibrated mounting extrinsic.
$\ExtrinsicMisalignmentAngle$	Extrinsic Misalignment Angle	$\text{rad}$	Angular error of a sensor’s extrinsic calibration relative to its true mounting.
$\FocalLength$	Focal Length	pixels	Focal length of a pinhole camera in pixels, the scale from normalized image coordinates to pixels; a rectified pair shares one value.
$\HorizontalFocalLength$	Horizontal Focal Length	pixels	Camera focal length in pixels along the horizontal image axis, one entry of the intrinsics.
$\FrameExposureStartTime$	Frame Exposure Start Time	$\text{s}$	Instant at which the first row of a rolling-shutter frame begins its exposure.
$\FrameReadoutTime$	Frame Readout Time	$\text{s}$	Time a camera takes to read its pixels out after exposure.
$\EvidenceCaptureInstant$	Evidence Capture Instant	$\text{s}$	Instant, on a monotonic clock, at which the sensor evidence behind an observation or proposal was captured; observation age and proposal validity are measured from it.
$\EvidenceEpoch$	Evidence Epoch	$\text{s}$	Capture instant of the most recent measurement that constrained a stored belief; republishing or predicting never advances it.
$\FreshEvidenceTime$	Fresh Evidence Time	$\text{s}$	Capture time of the fresh physical evidence that renews a belief’s evidence epoch $\EvidenceEpoch$.
$\CurrentTime$	Current Time	$\text{s}$	Wall-clock time at which a consumer evaluates a stored record; evidence age is the current time minus $\EvidenceEpoch$.
$\DispatchObservationAge$	Dispatch Observation Age	$\text{s}$	Elapsed time from the sensor exposure midpoint to dispatch of the processed observation. Current consumer evidence age additionally includes elapsed time after dispatch.
$\FreeConfigurationSpace$	Free Configuration Space	Set	Configurations at which the robot touches no obstacle, the configuration space minus the obstacle set.
$\InverseDepth$	Inverse Depth	$\text{1/m}$	Reciprocal of a point’s depth along the optical axis, the parameterization that lets a landmark at infinity enter an estimator at zero; linear in disparity.
$\ObstacleSeparationDistance$	Obstacle Separation Distance	$\text{m}$	Distance between the machine and the nearest obstacle or boundary at a given time, the signal a clearance predicate scores; written as a function of time.
$\OccupancyLogOdds$	Occupancy Log-Odds	Dimensionless	Natural logarithm of the odds of a voxel’s occupancy probability, the quantity an occupancy map stores per cell; indexed by step, with the prior at step zero.
$\MeasuredTrackedPosition$	Measured Tracked Position	$\text{m}$	Measured Cartesian position of the tracked point, which the tracking error compares against the reference.
$\LastMeasurementTime$	Last Measurement Time	$\text{s}$	Capture time of the most recent measurement that updated a track or map cell.
$\InitialVelocityStandardDeviation$	Initial Velocity Standard Deviation	$\text{m/s}$	Standard deviation of the velocity error of an estimate, here at the onset of occlusion.
$\PixelStandardDeviation$	Pixel Standard Deviation	Pixels	Standard deviation of a measured image position, in pixels; the image-plane noise that propagates into depth and position error.
$\PrincipalPointColumn$	Principal Point Column	Pixels	Horizontal pixel coordinate of the principal point, where the optical axis meets the image.
$\PrincipalPointRow$	Principal Point Row	Pixels	Vertical pixel coordinate of the principal point, where the optical axis meets the image.
$\StereoBaseline$	Stereo Baseline	$\text{m}$	Distance between the optical centers of a stereo camera pair; depth is focal length times baseline over disparity.
$\StereoDisparity$	Stereo Disparity	Pixels	Horizontal pixel offset between the images of one scene point in the two views of a rectified stereo pair; not a travel distance or a sightline range, which share the glyph.
$\TargetRange$	Target Range	$\text{m}$	Distance from a camera to the target it observes, measured along the viewing direction; stereo depth error grows with its square.
$\TargetPosition$	Target Position	$\text{m}$	Position of the target an end effector is commanded toward, in the robot base frame; a decorated form of the Cartesian position.
$\PositionErrorBound$	Position Error Bound	$\text{m}$	Worst-case position error of a target estimate as a function of the time elapsed since its evidence was captured; it grows while the target is occluded.
$\PositionErrorEnvelope$	Position Error Envelope	$\text{m}$	Conservative bound on the position error of a target estimate along one coordinate axis of the base frame; written with the axis index.
$\RangeObservation$	Range Observation	Record	One range observation a sensor returns at a time step; indexed by step, with the history as the sequence of them.
$\SpatialMemoryMap$	Spatial Memory Map	Set	Occupancy map in spatial memory that stores the obstacle geometry around the machine; not a decision-process tuple.
$\TransformRotationBlock$	Transform Rotation Block	Matrix	Rotation block of the homogeneous transform from frame B to frame A, its upper-left three-by-three part.
$\TransformTranslationBlock$	Transform Translation Block	$\text{m}$	Translation block of the homogeneous transform from frame B to frame A, the position of the origin of frame B expressed in frame A.
$\UnprojectionJacobian$	Unprojection Jacobian	Mixed	Jacobian of a camera-frame point with respect to its pixel coordinates and depth; it carries measurement covariance into Cartesian covariance. Not the manipulator Jacobian $\ManipulatorJacobian$.
$\VerticalFocalLength$	Vertical Focal Length	Pixels	Camera focal length expressed in pixels along the vertical image axis.
$\WorldBodyTransform$	World-from-Body Transform	Matrix	Rigid transform giving the pose of the vehicle body frame in the world frame; it maps body coordinates into world coordinates.
$\WorldFramePoint$	World-Frame Point	$\text{m}$	Position of an observed point expressed in the world frame; compare $\CameraFramePoint$, the same point in the camera frame.
$\PerceptionDepthCovariance$	Perception Depth Covariance	$\text{m}^2$	Covariance of the three-dimensional position error of a point lifted from stereo, lidar, or depth data; a matrix, not the scalar depth variance.
$\VacuumLightSpeed$	Speed of Light in Vacuum	$\text{m/s}$	About $3 \times 10^8$ m/s, the upper limit on how fast a signal crosses a chip or link; on-chip wires carry signals at a fraction of it.

Control, Observers, and Safety Filters

The control symbols follow state-space convention. A plant has a state, an input, and a measured output; an observer estimates the state from that output; and a safety filter, such as a control barrier function solved as a quadratic program, edits the input before it reaches the actuators.

Symbol	Definition	Unit	Notes
$\StateSpaceSystemMatrix, \StateSpaceInputMatrix$	Linear State-Space Matrices	Matrix	Continuous LTI plant: $\PlantStateRate = \StateSpaceSystemMatrix\MechanicalPlantState + \StateSpaceInputMatrix\ControlInput + \ProcessNoiseVector$, observed output $\MeasurementVector = \MeasurementMatrix\MechanicalPlantState + \MeasurementNoiseVector$ through the measurement map $\MeasurementMatrix$ below.
$\ProcessNoiseVector(\TimeInstant), \MeasurementNoiseVector(\TimeInstant)$	Process and Measurement Noise	Vector	Zero-mean Gaussian noise vectors with covariance matrices $\ProcessNoiseCovariance \succeq 0$ and $\MeasurementNoiseCovariance \succ 0$; bold $\DirectionalCosineMatrix$ is reserved for rotation.
$\StateTransitionMatrix, \MeasurementMatrix$	Transition and Measurement Maps	Matrix	Discrete state transition $\PredictedStateEstimate = \StateTransitionMatrix\PreviousUpdatedStateEstimate$ and measurement map $\MeasurementVector_k = \MeasurementMatrix\MechanicalPlantState_k + \MeasurementNoiseVector$ of a Kalman filter.
$\KalmanGain_{k}$	Kalman Gain	Matrix	Weight on the innovation $\MeasurementInnovation_k = \MeasurementVector_k - \MeasurementMatrix\PredictedStateEstimate$ in the update step; bold $\CameraIntrinsicCalibrationMatrix$ is reserved for camera intrinsics.
$\EstimationErrorCovariance \in \mathbb{R}^{\StateDimension \times \StateDimension}$	Estimation Error Covariance	Matrix	Symmetric positive-definite matrix $\mathbb{E}[(\MechanicalPlantState - \StateEstimate)(\MechanicalPlantState - \StateEstimate)^\top]$ ($\EstimationErrorCovariance(\TimeInstant)$ continuous, $\UpdatedErrorCovariance$ discrete) evolving via matrix Riccati equation; $\Glyph{\mathbf{P}_{\text{est}}}$ where an LQR cost matrix is also in scope.
$\ObservabilityMatrix, \NonlinearObservationSpace$	Observability Matrices	Matrix	Kalman observability matrix $[\MeasurementMatrix^\top, (\MeasurementMatrix\StateSpaceSystemMatrix)^\top, \dots]^\top$ and nonlinear Lie matrix $[\DerivativeOperator\NonlinearOutputFunction, d\DriftLieDerivativeOperator\NonlinearOutputFunction, \dots]^\top$; calligraphic $\mathcal{O}(\cdot)$ stays asymptotic cost. The dual controllability matrix is $\ControllabilityMatrix$.
$\ControllabilityMatrix$	Controllability Matrix	Mixed	Kalman block matrix formed from the input matrix and its products with powers of the system matrix; full row rank characterizes unconstrained LTI controllability.
$\SafetyBoundaryManifold$	Safety Boundary Manifold	Boundary	Zero level set $\{\MechanicalPlantState \mid \BarrierFunction(\MechanicalPlantState) = 0\}$, the boundary of the safe set $\ForwardInvariantSafeSet$, separating nominal operation from irreversible hazard.
$\BarrierFunction(\MechanicalPlantState), \BarrierFunction$	Control Barrier Function (CBF)	Scalar	Continuously differentiable function whose 0-superlevel set is the safe set $\ForwardInvariantSafeSet$. Written $\BarrierFunction(\MechanicalPlantState)$ on the plant state, and as the function head $\BarrierFunction$ when it takes other arguments or is differentiated ($\BarrierFunctionRate$, $\DriftLieDerivativeOperator \BarrierFunction$, $\BarrierFunction(\Glyph{\theta_{\text{clear}}}, \JointAngularSpeed)$).
$\ForwardInvariantSafeSet \subset \mathbb{R}^{2\GeneralizedCoordinateCount}$	Forward-Invariant Safe Set	Set	Superlevel set $\ForwardInvariantSafeSet = \{\MechanicalPlantState \mid \BarrierFunction(\MechanicalPlantState) \ge 0\}$ of plant states where safety invariants are strictly maintained; a qualifier names a variant ($\Glyph{\ForwardinvariantSafeSet}_{\text{contact}}$, $\Glyph{\ForwardinvariantSafeSet}_{\text{discrete}}$).
$\ControlAffineDriftField(\MechanicalPlantState), \ControlAffineInputField(\MechanicalPlantState)$	Control-Affine Drift and Input Fields	Vector, Matrix	The control-affine model $\PlantStateRate = \ControlAffineDriftField(\MechanicalPlantState) + \ControlAffineInputField(\MechanicalPlantState)\ControlInput$ of barrier analysis; these letters hold only inside that model, where the gravity torque $\GravitationalTorqueVector(\GeneralizedCoordinates)$ and contact forces $\ContactPointForce_{k}$ do not appear.
$\LieDerivativeAlongDrift(\MechanicalPlantState)$	Lie Derivative along Drift	Barrier units/s	Directional gradient derivative $\nabla \BarrierFunction(\MechanicalPlantState) \ControlAffineDriftField(\MechanicalPlantState)$ along unforced plant dynamics; its units are those of the barrier divided by time.
$\LieDerivativeAlongControl(\MechanicalPlantState)$	Lie Derivative along Control	Vector	Directional control effectiveness gradient $\nabla \BarrierFunction(\MechanicalPlantState) \ControlAffineInputField(\MechanicalPlantState)$.
$\BarrierRelativeDegree$	Barrier Relative Degree	Integer	Number of differentiations required before control input $\ControlInput$ appears ($\InputLieDerivativeOperator \DriftLieDerivativeOperator^{\BarrierRelativeDegree-1} \BarrierFunction \neq 0$).
$\KineticInsetBarrierFunction(\MechanicalPlantState)$	Kinetic Inset Barrier Function	Scalar	Relative-degree-1 barrier: $\KineticInsetBarrierFunction(\MechanicalPlantState) = \BarrierFunction(\MechanicalPlantState) - \BodySpeed^2 / (2 \CredibleBrakingDeceleration)$ incorporating stopping distance.
$\ClassKFunction(\cdot), \ClassKFunction$	Class-$\ClassKFunctionSet$ Rate Function	Scalar function	Extended class-$\ClassKFunctionSet$ function in the barrier condition $\BarrierFunctionRate \ge -\ClassKFunction(\BarrierFunction)$, written as the function head $\ClassKFunction$ when applied; the book uses the linear choice $\ClassKFunction(\BarrierFunction) = \LinearBarrierDecayRate \BarrierFunction$.
$\LinearBarrierDecayRate$	Linear Barrier Decay Rate	$\text{s}^{-1}$	Coefficient of the linear class-$\ClassKFunctionSet$ choice, $\BarrierFunctionRate + \LinearBarrierDecayRate \BarrierFunction \ge 0$; bare $\DiscountFactor$ is the reinforcement-learning discount factor.
$\NominalControlCommand \in \AdmissibleControlSet$	Nominal Control Command	$\text{N}\cdot\text{m}$ / $\text{rad}$	Unverified control candidate proposed across the proposal boundary by the learned policy.
$\CertifiedControlAction \in \AdmissibleControlSet$	Certified Control Action	$\text{N}\cdot\text{m}$ / $\text{rad}$	Privileged, safety-filtered control command resolved by the real-time active-set CBF-QP.
$\ReferencePosition$	Reference Position	$\text{m}$	Commanded Cartesian position reference the plant tracks, $\ReferencePosition(\TimeInstant)$; the feedback tracker bounds the tracked point’s position error by $\lVert \CartesianPosition(\TimeInstant) - \ReferencePosition(\TimeInstant) \rVert \le \TrackingErrorBound$, with both positions expressed in the same frame. Whole-body clearance requires a declared output map or metric that bounds swept-volume displacement. Elsewhere bold $\Glyph{\mathbf{r}}$ is local (a moment arm, ellipsoid semi-axes).
$\BumplessTransferBlend(\BumplessTransferProgress)$	Bumpless Transfer Blend	$[0, 1]$	$\Glyph{C}^2$ quintic polynomial $6\BumplessTransferProgress^5 - 15\BumplessTransferProgress^4 + 10\BumplessTransferProgress^3$ of normalized time $\BumplessTransferProgress \in [0, 1]$, ramping between prior and candidate controllers.
$\BumplessTransferBlendSlope$	Blend slope	Dimensionless	Derivative of the bumpless-transfer blend weight $\BumplessTransferBlend$ with respect to normalized progress.
$\BlendingTransitionWindow$	Blending Transition Window	$\text{s}$	Duration of bumpless transfer; for two fixed, latched acceleration commands, bounded by $\Glyph{\tau_{\text{blend}, \min}} = 1.875 \lVert \FiniteChange \ControlInput \rVert / \JerkLimit$. Torque or position commands require their own compatible command-rate bounds.
$\PidProportionalGain, \PidDerivativeGain, \PidIntegralGain$	PID / Feedback Gains	Mixed	Proportional, derivative, and integral gains for joint-level or operational-space error stabilization.
$\LqrFeedbackGain, \LqrStateWeightMatrix, \LqrControlWeightMatrix$	LQR Gain and Weights	Matrix	State-feedback gain $\CertifiedControlAction = -\LqrFeedbackGain\MechanicalPlantState$ and the state and control weights of the quadratic cost; the subscripts keep them distinct from intrinsics, process noise, and rotation.
$\MpcPredictionHorizon$	MPC Prediction Horizon	Control steps	Number of predicted discrete control steps in the finite-horizon MPC formulation; distinct from physical-trial count and learned action-chunk horizon.
$\VirtualDampingMatrix, \Glyph{\mathbf{K}_{\text{virt}}}$	Virtual Damping & Stiffness Matrices	Matrix	Target compliance matrices in impedance control: $\Glyph{\boldsymbol{\tau}_{\text{safe}}} = -\VirtualDampingMatrix \GeneralizedVelocities - \Glyph{\mathbf{K}_{\text{virt}}} \FiniteChange\GeneralizedCoordinates + \GravitationalTorqueVector(\GeneralizedCoordinates)$.
$\AccelerationNoiseSpectralDensity$	Acceleration Noise Spectral Density	$\text{m}^2/\text{s}^3$	Continuous power spectral density of the zero-mean white acceleration noise per axis in a constant-velocity motion model; sets how fast the predicted position uncertainty grows.
$\AdmissibleDisturbanceSet$	Admissible Disturbance Set	Set	Set of disturbance values a falsification search may inject; the search is bounded to it.
$\AdmittedControlCommand$	Admitted Control Command	Per interface	Control command the permission path applies to the actuators after admitting, clipping, or replacing a policy proposal; indexed by step. Not $\NominalControlCommand$ (before the filter) or $\CertifiedControlAction$ (the filter’s projection).
$\AngularFrequency$	Angular Frequency	$\text{rad/s}$	Frequency of a sinusoidal signal, the free variable of a frequency response; evaluated at the crossover frequency when a delay’s phase lag is charged to the phase margin.
$\BackwardEulerTransitionMatrix$	Backward Euler Transition Matrix	Matrix	Discrete-time transition matrix that backward (implicit) Euler integration produces, the inverse of the identity minus the system matrix times the step; unconditionally stable but numerically dissipative.
$\BarrierConstraintBound$	Barrier Constraint Bound	Varies	Bound on the right side of one affine control-barrier constraint, one entry of $\Glyph{\mathbf{b}_{\text{cbf}}}$; its unit follows the barrier function rate.
$\BarrierConstraintNormal$	Barrier Constraint Normal	Varies	Normal vector of a single affine control-barrier constraint on the control input, equal to $-\LieDerivativeAlongControl$ transposed; one row of $\Glyph{\mathbf{A}_{\text{cbf}}}$.
$\BarrierFunctionRate$	Barrier Function Rate	$\text{m/s}$	Time derivative of the control barrier function along the closed-loop dynamics, the rate in the barrier condition; a dot is not an allowed decoration on $\BarrierFunction$, so it has its own entry.
$\BarrierProjectionMultiplier$	Barrier Projection Multiplier	Varies	Lagrange multiplier of the barrier half-space constraint in the CBF-QP projection; starred at the optimum, where it is zero when the nominal command is already safe.
$\BeliefErrorBound$	Belief Error Bound	Varies	Upper bound on the discrepancy between a stored belief and the true state after a given elapsed time without new evidence; the unit is that of the stored quantity (meters for a position, radians for an angle).
$\BumplessTransferStartInstant$	Bumpless Transfer Start Instant	$\text{s}$	Instant at which a bumpless-transfer blend begins, the zero of its normalized progress; other chapters use the same letters for exposure or sweep starts, different quantities.
$\ChannelCommand$	Channel Command	Varies	Scalar command of one actuator channel, such as the base acceleration or one joint torque; the unit is the channel’s command unit. Scalar per-channel form of $\ControlCommand$.
$\ClassKFunctionSet$	Class-K Function Set	Set	Class of continuous strictly increasing functions that are zero at zero; named in “extended class-K function.”
$\ControlCommand$	Control Command	Per interface	Control command vector sent toward the actuators, whatever its source or stage; indices name instances. The stages are $\NominalControlCommand$ and $\CertifiedControlAction$.
$\ControlInput$	Control Input	Per interface	Control input applied to the plant, the command a controller or safety filter sends to the actuators; units depend on the command interface (torques, setpoints).
$\ControlInputDimension$	Control Input Dimension	Count	Number of components of the control input vector; not the mass $\Mass$.
$\ControlInputLowerLimit$	Control Input Lower Limit	Per interface	Per-component lower limit of the control input the actuators can deliver.
$\ControlInputSet$	Control Input Set	Set	Set of control inputs the controller may choose from, a subset of the input space; the actuator box and the barrier-feasible subset are narrower sets inside it.
$\ControlInputUpperLimit$	Control Input Upper Limit	Vector	Per-component upper limit of the control input the actuators can deliver.
$\ControlStepIndex$	Control Step Index	Dimensionless	Position of one discrete control step, or setpoint, in a robot’s control timeline; a subscript on per-step quantities, bare in expressions such as a clipped next step.
$\DerivativeGainMatrix$	Derivative Gain Matrix	Matrix	Derivative feedback gains acting on the tracking error rate in joint-space PD or PID control; its diagonal entries are the per-joint derivative gains.
$\DifferentiabilityClass$	Differentiability Class	Set	Functions whose derivatives up to the order written as the superscript exist and are continuous; $\DifferentiabilityClass^2$ means continuous position, velocity, and acceleration of a blended trajectory.
$\DiscretePlantDynamics$	Discrete Plant Dynamics	Function	Discrete-time map from the current state and applied command to the next state; the deterministic counterpart of the plant’s transition kernel.
$\DriftLieDerivativeOperator$	Drift Lie Derivative Operator	Operator	Rate of change of a scalar function of the state under the unforced dynamics; it is applied repeatedly for higher-order derivatives.
$\DriftSpeedBound$	Drift Speed Bound	$\text{m/s}$	Bound on the unobserved speed at which a target drifts relative to its last measured position; not the bound on the state-estimate drift, $\DriftVelocityBound$.
$\DriftVelocityBound$	Drift Velocity Bound	$\text{m/s}$	Bound on the unmodeled velocity at which an unobserved estimate drifts from the true state.
$\ConveyorSpeedBound$	Conveyor Speed Bound	$\text{m/s}$	Largest speed at which the takeaway conveyor carries a target; the bound used when sizing a catch window, not the belt speed $\ConveyorSpeed$ of a single pick.
$\EstimatedPlantState$	Estimated Plant State	Mixed	Estimate of the mechanical plant state reconstructed from sensor observations; its error covariance is a separate symbol.
$\EstimatedPosition$	Estimated Position	$\text{m}$	Position of a point that a filter propagates under its nominal model.
$\EstimatedVelocity$	Estimated Velocity	$\text{m/s}$	Velocity of a body that a filter propagates under its nominal model.
$\FiniteTimeHorizon$	Finite Time Horizon	$\text{s}$	Length of the finite window from time zero to the horizon over which an optimization, rollout, test, or temporal-logic operator is evaluated; not the wall-clock time to complete a unit of work.
$\ForwardEulerTransitionMatrix$	Forward Euler Transition Matrix	Matrix	Discrete-time transition matrix that forward Euler integration produces, identity plus the system matrix times the step; its spectral radius exceeds one for an undamped oscillator at any step size.
$\HermiteBasisFunction$	Hermite Basis Function	Function	Cubic Hermite basis polynomial that weights an endpoint position or velocity in spline interpolation; indexed by two digits.
$\ImaginaryUnit$	Imaginary Unit	Dimensionless	Square root of minus one, written $\ImaginaryUnit$ by the engineering convention rather than $\Glyph{i}$.
$\InductiveVoltage$	Inductive Voltage	$\text{V}$	Voltage across a coil, the inductance times the rate of change of its current.
$\InequalityConstraintFunction$	Inequality Constraint Function	Varies	Constraint function of an optimization problem, whose nonpositive values define the feasible set; it appears in the interior-point barrier.
$\InputLieDerivativeOperator$	Input Lie Derivative Operator	Operator	Lie derivative along the input matrix field, mapping a scalar function of the state to its sensitivity to the control input.
$\IntegrationTimeInstant$	Integration Time Instant	$\text{s}$	Time instant running over an integration interval, the dummy variable when the interval ends at the time $\TimeInstant$; it keeps the integrand’s time distinct from the upper limit.
$\IntegrationDummyTime$	Integration Dummy Time	$\text{s}$	Dummy time variable of integration inside a time integral, distinct from its limits; a primed time would be a decorated quantity, so this entry names the bound variable.
$\LaplaceVariable$	Laplace Variable	$\text{1/s}$	Complex frequency variable of the Laplace transform, standing for differentiation in time; read as the imaginary unit times the angular frequency for sinusoidal motion.
$\LargestEigenvalue$	Largest Eigenvalue	Varies	Largest eigenvalue of a symmetric matrix, applied to that matrix; its unit is that of the matrix entries.
$\LqrCostToGoMatrix$	LQR Cost-to-Go Matrix	Varies	Symmetric solution of the regulator algebraic Riccati equation (the discrete one in discrete-time LQR), whose quadratic form is the optimal LQR cost-to-go; shares its glyph with the estimation error covariance.
$\MpcControlSequence$	MPC Control Sequence	Varies	Sequence of control inputs over a model-predictive-control horizon, one per step, that the optimizer chooses; only the first is applied before the next re-solve.
$\NonlinearOutputFunction$	Nonlinear Output Function	Varies	Output function of a nonlinear state-space model, mapping the state to the measurement vector $\MeasurementVector$; not the barrier function nor the bias forces of the floating-base equations.
$\OpenLoopTransferFunction$	Open-Loop Transfer Function	Dimensionless	Open-loop transfer function of a feedback loop, evaluated along the imaginary axis for frequency response; never the loss.
$\PidProportionalGainMatrix$	PID Proportional Gain Matrix	Varies	Proportional gain matrix of multi-joint PD or PID feedback, the stiffness acting on the position error.
$\MeasurementVector$	Measurement Vector	Varies	Measurement vector a sensor or detector reports at one step.
$\MeasurementInnovation$	Measurement Innovation	Varies	Difference between a measurement and the measurement the predicted state implies; its normalized square is the NIS.
$\MeasurementNoiseCovariance$	Measurement Noise Covariance	Varies	Covariance of the noise on a sensor measurement vector, such as a pixel position with depth or a Kalman filter measurement; units are those of the measurement squared.
$\PathDelayAsymmetry$	Path Delay Asymmetry	$\text{s}$	Difference between the forward and reverse one-way delays of a clock-synchronization exchange, which biases the estimated clock offset by half its value; two-way time transfer assumes it is zero.
$\PredictedStateEstimate$	Predicted State Estimate	Vector	State estimate at step $\Glyph{k}$ predicted from measurements through the previous step, before the measurement update; the step index is part of the symbol.
$\PredictedErrorCovariance$	Predicted Error Covariance	Matrix	Error covariance at step $\Glyph{k}$ predicted from measurements through the previous step, before the update; the conditioned form of the estimation error covariance.
$\PreviousUpdatedStateEstimate$	Previous Updated State Estimate	Vector	State estimate at the previous step after that step’s update, the input to the next prediction.
$\PreviousUpdatedErrorCovariance$	Previous Updated Error Covariance	Matrix	Error covariance at the previous step after that step’s update, the input to the next prediction.
$\ProcessNoiseCovariance$	Process Noise Covariance	Matrix	Covariance of the process noise vector; the predict step adds it to the error covariance.
$\StateEstimate$	State Estimate	Vector	Estimate of the state vector that a recursive estimator maintains; conditioned forms with the step index baked in have their own symbols.
$\ProportionalGainMatrix$	Proportional Gain Matrix	Matrix	Matrix of proportional feedback gains acting on the tracking error vector in joint-space PD or PID control; its diagonal entries are the per-joint proportional gains, and it is not the camera calibration matrix.
$\SafeControlSet$	Safe Control Set	Set	Set of control inputs that satisfy every active barrier constraint and the actuator limits at the current state; it is empty when saturation leaves no command that keeps the barrier condition, which selects a fallback.
$\SafeStateConstraintSet$	Safe State Constraint Set	Set	Set of plant states that satisfy the state constraints of a model predictive control problem, such as joint limits and obstacle margins; polyhedral, and not the barrier safe set.
$\SafeSeparationDistance$	Safe Separation Distance	$\text{m}$	Smallest separation from an obstacle that a clearance requirement allows.
$\SignalTrajectory$	Signal Trajectory	Function	Trajectory of a system’s real-valued signals over continuous time, the trace a temporal logic formula is evaluated on; two trajectories are told apart by subscripts.
$\StlFormula$	Signal Temporal Logic Formula	Record	Signal temporal logic formula a signal must satisfy, such as a safety invariant or a reachability deadline; subscripts index subformulas.
$\StlPredicateFunction$	Signal Temporal Logic Predicate Function	Function	Real-valued function of a signal whose sign decides an atomic temporal-logic predicate; its unit is the physical unit of that predicate’s margin, and it is indexed per predicate.
$\StlRobustness$	Signal Temporal Logic Robustness	Function	Signed margin by which a signal satisfies (positive) or violates (negative) a temporal logic formula at a time; it carries the units of the atomic predicates when they agree and is dimensionless after a declared normalization. Not a correlation coefficient.
$\SmallChange$	Small Change	Operator	Small change, increment or error in the quantity it prefixes; the companion of the finite change for first-order changes.
$\SmallPerturbation$	Small Perturbation	Operator	Small change or error in the quantity it prefixes, treated to first order, as in a perturbation of a signal or a time; not a labeled localization bound that also renders $\Glyph{\delta}$.
$\PolynomialTrajectoryCoefficient$	Polynomial Trajectory Coefficient	Varies	Coefficient of one power of time in a polynomial joint trajectory segment, indexed by the power it multiplies; its unit is the joint unit divided by seconds to that power, and it is not an acceleration.
$\SetpointSampleIndex$	Setpoint Sample Index	Index	Position of one discrete setpoint sample in a sampled trajectory.
$\SetpointStepInterval$	Setpoint Step Interval	$\text{s}$	Time between consecutive setpoints of an action chunk, the spacing at which the controller consumes them; the reciprocal of the action execution rate.
$\StateSpaceDimension$	State Space Dimension	Dimensionless	Number of continuous coordinates in the task-relevant state.
$\StopCompletionInstant$	Stop Completion Instant	$\text{s}$	Instant, on the permission clock, at which the matched stop of an admitted trajectory record reaches rest; the commitment instant plus the stopping duration.
$\StoppingDeadline$	Stopping Deadline	$\text{s}$	Latest instant at which authority can be revoked and a worst-case stop still ends within the verified clearance.
$\PolicyLipschitzBound$	Policy Lipschitz Bound	Varies	Lipschitz bound of a learned policy over a stated operating domain, the largest ratio of action change to state change; it gives no guarantee unless established for that domain, and times the sensor error bound it bounds the action error.
$\TaskToleranceSemiAxes$	Task Tolerance Semi-Axes	$\text{m}$	Principal semi-axes of the ellipsoid within which terminal position error must lie; the vector of $\TaskToleranceSemiAxis$ values.
$\TaskToleranceSemiAxis$	Task Tolerance Semi-Axis	$\text{m}$	Semi-axis of the task tolerance ellipsoid along one coordinate axis, written with the axis index.
$\TerminalInvariantSet$	Terminal Invariant Set	Set	Plant states from which an invariant fallback control law keeps the system safe indefinitely, the required endpoint of a receding horizon. On an embodied system a validated stop that reaches rest within verified clearance realizes it.
$\TrackingErrorVector$	Tracking Error Vector	Vector	Commanded reference state minus measured state of a feedback controller, acted on by $\ProportionalGainMatrix$. Not the error feedback residual, which shares the glyph.
$\TrackingErrorRate$	Tracking Error Rate	Vector	Time derivative of the tracking error vector, the velocity error of a feedback controller; multiplied by $\DerivativeGainMatrix$ in PD control, so a step in it produces a torque step.
$\UnitImpulseFunction$	Unit Impulse Function	$\text{s}^{-1}$	Dirac delta, the idealized unit impulse of zero width and unit area; with time as its argument it carries inverse seconds.
$\UpdatedErrorCovariance$	Updated Error Covariance	Matrix	Error covariance at a step after incorporating that step’s measurement, the posterior counterpart of $\EstimationErrorCovariance$.
$\UpdatedStateEstimate$	Updated State Estimate	Vector	State estimate at a step after incorporating that step’s measurement, the posterior estimate of a Kalman filter.
$\DerivativeOperator$	Derivative Operator	Operator	Differential operator of an ordinary derivative, written over the differential in a Leibniz quotient. Prefer writing the derivative so the $\Glyph{d}$ sits before a variable.
$\FiniteChange$	Finite Change	Operator	Finite change in the quantity it prefixes, between two states, configurations, or points in time. Differs from the increment fused into named entries such as $\JouleanTemperatureRise$.
$\PathCoordinate$	Path Coordinate	Dimensionless	Scalar coordinate along a geometric path, from zero at its start to one at its end, that a timing law maps to time; not a Laplace variable.

Learned Policies for Robotics

Learned policies propose actions rather than issue them. The symbols below cover how those policies are trained and sampled, including diffusion denoising, action chunking, and vision-language-action models, and how their imitation and reinforcement-learning objectives are written.

Symbol	Definition	Unit	Notes
$\Policy$	Policy	Distribution	Deep policy network parameterized by weights $\ModelParameters$ (italic), mapping observation tensors to action distributions.
$\MultimodalObservation_{t}$	Multimodal Observation	Mixed	Synchronized tokenized sensor packet (RGB images, depth latents, joint encoders, tactile arrays).
$\CognitivePolicyAction_{t} \in \ActionSpace$	Cognitive Policy Action	Normalized	Unprivileged action emitted by neural policy $\Policy$ (normalized joint delta, target pose, or gripper command).
$\EnsemblingWindowLength$	Ensembling Window Length	Time steps	Number of overlapping past chunks blended in receding horizon execution ($\EnsemblingWindowLength \le \ActionChunkHorizon$).
$\DecoderEvaluationCount$	Decoder Evaluation Count	Count	Dependent forward passes of a generative action head per chunk (denoising or ODE-solver steps); distinct from the ensembling window $\EnsemblingWindowLength$ and the step index $\DiffusionStepIndex$.
$\LatentStyleVariable \in \mathbb{R}^{\LatentDimension}$	Latent Style / State Embedding	Dimensionless	Compressed semantic representation or CVAE latent style variable in ACT policies.
$\DiffusionCumulativeSchedule_{k}$	Diffusion Schedule Parameter	$[0, 1]$	Cumulative noise variance schedule parameter $\DiffusionCumulativeSchedule_{k} = \prod_{\Glyph{s}=1}^{\DiffusionStepIndex} (1 - \Glyph{\beta}_{\Glyph{s}})$ at diffusion step $\DiffusionStepIndex$.
$\DiffusionGaussianNoise \sim \GaussianDistribution(\ZeroVector, \IdentityMatrix)$	Diffusion Gaussian Noise	Tensor	Standard normal Gaussian noise corrupted onto clean action trajectories.
$\NoisePredictionNetwork(\MultimodalObservation_{t}, \NoisedActionChunk_t, \DiffusionStepIndex)$	Noise Prediction Network	Tensor	Neural network predicting added noise in Denoising Diffusion Probabilistic Models (DDPM).
$\NoisedActionChunk$	Noised Action Chunk	Tensor	Action chunk after $\DiffusionStepIndex$ forward-diffusion steps, $\sqrt{\DiffusionCumulativeSchedule_{k}}\,\ActionChunk^0 + \sqrt{1 - \DiffusionCumulativeSchedule_{k}}\,\DiffusionGaussianNoise$; the denoising iterate the reverse process refines.
$\DiffusionMatchingLoss(\ModelParameters)$	Diffusion Matching Loss	Scalar	Mean-squared error objective: $\mathbb{E}[\lVert \DiffusionGaussianNoise - \NoisePredictionNetwork(\MultimodalObservation_{t}, \NoisedActionChunk_t, \DiffusionStepIndex) \rVert^2]$ optimized across denoising steps.
$\ActVariationalObjective(\ModelParameters, \CvaeEncoderParameters)$	ACT Variational Objective	Scalar	CVAE objective combining trajectory reconstruction $\Glyph{\mathcal{L}_{\text{recon}}}$ with latent KL-regularization $\LatentKlWeight \KlDivergence$.
$\CvaeRecognitionEncoder(\LatentStyleVariable \mid \MultimodalObservation, \ActionChunk)$	CVAE Recognition Encoder	Distribution	Variational encoder in Action Chunking with Transformers parameterizing latent distribution $\GaussianDistribution(\GaussianMeanVector, \GaussianCovarianceMatrix)$; $\CvaeEncoderParameters$ here is the encoder’s parameter set.
$\DivergenceInstant$	Divergence Instant	$\text{s}$	Time at which the autonomous policy’s rollout begins to leave its supported states before a takeover; the start of the interval excluded from imitation targets.
$\TrajectoryDivergenceResidual$	Trajectory Divergence Residual	$\text{m}$	Signed lateral gripper-position error between measured and planned kinematics in the declared fixture frame for the latch example.
$\DivergenceTolerance$	Divergence Tolerance	$\text{m}$	Positive threshold for the declared trajectory-divergence residual; expressed in that residual’s physical unit.
$\Reward$	Reward	Scalar	Per-step reward $\Reward(\PhysicalWorldState, \CognitivePolicyAction)$ that policy search maximizes; the stage cost of control is its negative. The barrier relative degree $\BarrierRelativeDegree$ also renders $\BarrierRelativeDegree$ but appears only in barrier analysis.
$\ResidualGate$	Residual Gate	$\{0, 1\}$	Deterministic, independently evaluated gate $\ResidualGate(\PhysicalWorldState)$ that admits a learned residual correction only inside the certified contact region and is zero elsewhere.
$\TrialCount$	Trial Count	Trials	Number of physical trials (rollouts) in a learning or evaluation campaign, which sets its wall-clock cost; distinct from the zero-failure trial count $\Glyph{n}$.
$\SingleStepImitationError$	Single-Step Imitation Error	$[0, 1]$	Bounded probability of an unforced policy error during an individual execution step; bold $\DiffusionGaussianNoise$ is diffusion noise, the conformal miscoverage rate is $\ConformalMiscoverageRate$, and other error bounds carry a subscript ($\TrackingErrorBound$).
$\BehaviorCloningHorizon$	Behavioral-Cloning Horizon	Control steps	Episode length in control steps over which imitation error compounds; not the action-chunk horizon $\ActionChunkHorizon$, not a duration $\Glyph{T}$, and not the token sequence length $\SequenceLength$.
$\CovariateDriftBound$	Compounding Covariate-Drift Bound	Task cost	Worst-case excess cost of fixed-archive behavioral cloning, which grows quadratically over a rollout of $\BehaviorCloningHorizon$ control steps with per-step error $\SingleStepImitationError$.
$\VitPatchProjectionWeights, \Glyph{\mathbf{E}_{\text{pos}}}$	ViT Projection & Positional Bias	Matrix	Convolutional/linear patch projection matrix and spatial coordinate position embedding tensors.
$\MultimodalFusedTokenStream$	Multimodal Fused Token Stream	Tensor	Concatenated token sequence $[\VisionTokenEmbeddings, \TextTokenEmbeddings, \ProprioceptiveTokenEmbeddings]$ ingested by embodied transformer backbones.
$\KvCacheMemory$	KV-Cache Memory	bytes	Autoregressive key-value cache memory footprint at the embodied batch size $\BatchSize = 1$ ($2 \times \LayerCount \times \HiddenDimension \times \SequenceLength \times \ElementStorageSize$ with full multi-head attention); the core’s quantity.
$\IntentLease$	Intent Lease	Tuple	$(\GoalSet, [\LeaseAdmissionInstant, \IntentLeaseExpiry], \ContactWrenchCeiling)$: the admitted goal set, its validity interval, and its effort ceilings; it names a target, never a path. Calligraphic $\LossFunction$ stays the loss.
$\IntentLeaseExpiry$	Intent-Lease Expiry	$\text{s}$	$\IntentLeaseExpiry = \min(\TimeInstant_0 + \EvidenceHorizon,\, \LeaseAdmissionInstant + \IntentAdministrativeLease,\, \Glyph{t_{\text{deadline,stop}}})$, where $\Glyph{t_{\text{deadline,stop}}}$ is the instant by which a stop must begin; see the lease table above.
$\EvidenceHorizon$	Evidence Horizon	$\text{ms}$	Time for which a target estimate stays within its task tolerance without fresh evidence.
$\ConformalPredictionSet$	Conformal Prediction Set	Set	Calibrated set of referent candidates that cannot be excluded at miscoverage rate $\ConformalMiscoverageRate$; a singleton admits the referent, and a larger set refuses. It shares the glyph of the safe set $\ForwardInvariantSafeSet$, but the two never appear together.
$\ConformalMiscoverageRate$	Conformal Miscoverage Rate	Probability	Allowed marginal probability that the conformal prediction set misses the true referent under the calibration procedure’s exchangeability assumptions; coverage is at least $1 - \ConformalMiscoverageRate$.
$\ExternalWait$	External Wait	Seconds	Time the machine spends waiting for a remote operator’s response during a handover or intervention.
$\AggregatedSurrogateLoss$	Aggregated Surrogate Loss	Dimensionless	Best-in-class surrogate imitation loss on the states the learner visited, aggregated over the rounds so far; differs from the error measured on expert-visited states.
$\AggregationLearnedPolicy$	Aggregation Learned Policy	Function	Policy trained in one round of corrective dataset aggregation, one of the sequence the bound selects from; carries a round index in the appendix.
$\ChunkExecutionHorizon$	Chunk Execution Horizon	Count	Number of actions of each predicted chunk the controller executes before the policy is queried again; at most the chunk horizon.
$\ChunkStepIndex$	Chunk Step Index	Index	Position of one step within an action chunk, counted from zero.
$\ConservativePenaltyWeight$	Conservative Penalty Weight	Dimensionless	Weight of the conservative penalty that pushes down values of unsupported actions in conservative Q-learning.
$\ContinuousActionSpace$	Continuous Action Space	Set	Set of continuous actions a robot policy may emit, a subset of real space of the action dimension.
$\ContinuousActionVector$	Continuous Action Vector	Varies	Continuous action vector of the robot’s decision process, written without a step index: the argument of a policy, value function, reward, or action distribution, or an action recorded from a demonstrator. A star marks the demonstrated action; it lives in $\ContinuousActionSpace$.
$\CriticActionReturnEstimate$	Critic Action-Value Estimate	Function	Return a learned critic predicts for a state and an action; the action counterpart of the critic’s state-value estimate.
$\DecoderPassTime$	Decoder Pass Time	$\text{s}$	Time of one dependent forward pass of a generative action head, one denoising or ODE-solver step; summed over the decoder evaluations in the proposer inference time.
$\DemonstrationActionDistribution$	Demonstration Action Distribution	Distribution	Conditional distribution of the actions demonstrators took given an observation, the target a behavioral cloning decoder models; its mean minimizes squared error and can fall between modes (mode averaging).
$\DemonstrationDataset$	Demonstration Dataset	Set	Recorded observation-action pairs that an imitation loss averages over.
$\EmpiricalTimestepAverage$	Empirical Timestep Average	Operator	Sample average over the timesteps of a batch of rollouts, the estimate of an expectation.
$\EnsemblingDecayRate$	Ensembling Decay Rate	Dimensionless	Decay factor per chunk index of exponential temporal-ensembling weights; its size sets how fast newer chunks enter the blend.
$\ImitationLearnedPolicy$	Imitation-Learned Policy	Function	Policy learned from expert demonstrations or expert-labeled states, compared against the expert policy; indexed by aggregation round in DAgger.
$\LatentKlWeight$	Latent KL Weight	Dimensionless	Weight on the KL divergence between the latent posterior and prior in a conditional variational autoencoder objective; not the RLHF reference penalty coefficient.
$\LatentPosteriorDistribution$	Latent Posterior Distribution	Distribution	Distribution over the latent style variable that a conditional VAE encoder infers from a demonstration chunk and its observation; compared with $\Glyph{p}$ in the KL term.
$\MixtureComponentCount$	Mixture Component Count	Dimensionless	Number of components of a mixture distribution.
$\MixtureComponentIndex$	Mixture Component Index	Index	Position of one component of a mixture distribution, from one to the component count $\MixtureComponentCount$; it indexes the weight, mean and covariance of a mixture density network.
$\MixtureComponentMean$	Mixture Component Mean	Varies	Mean vector of one component of a Gaussian mixture, output by a mixture density network for the given input.
$\MixtureComponentCovariance$	Mixture Component Covariance	Varies	Covariance matrix of one component of a Gaussian mixture, output by a mixture density network for the given input.
$\ObservationEncodingTime$	Observation Encoding Time	$\text{s}$	Time to encode one observation, visual backbone and proprioception, once per policy proposal; the first term of the proposal latency.
$\ObservationHistoryWindow$	Observation History Window	Varies	Sliding window of the robot’s most recent onboard observations, from the history length before the current step up to the current step; one row per observation step.
$\ObservationHistoryLength$	Observation history length	Steps	Number of past steps the observation history window holds before the current one.
$\RobotAction$	Robot Action	Vector	Action vector a learned robot policy emits, written without a step index as the argument of a policy, value function or action distribution; the same quantity as the cognitive policy action, which carries the step index.
$\RobotObservation$	Robot Observation	Vector	Observation vector a robot policy receives from its sensors, written without a step index.
$\RequestedActionTap$	Requested Action Tap	Record	First stream of the four-stream action tap, the raw policy or operator request before kinematic mapping.
$\SuccessorActionVector$	Successor Action Vector	Vector	Continuous action vector taken in the successor state of a transition, the next action a Bellman backup evaluates.
$\ReturnStepOffset$	Return Step Offset	Index	Number of control steps after the current step at which a reward arrives, the summation index of a discounted return.
$\RolloutHorizon$	Rollout Horizon	Dimensionless	Number of control steps in one executed rollout, the index of its final state; a step count, not a duration.
$\RolloutPolicyParameters$	Rollout Policy Parameters	Vector	Parameters of the policy that collected the current batch of rollouts, frozen while the updated parameters are optimized; the denominator policy of the probability ratio.
$\PpoClippedSurrogateObjective$	PPO Clipped Surrogate Objective	Dimensionless	Clipped surrogate objective that proximal policy optimization maximizes, the empirical mean of the smaller of the ratio-weighted advantage and its clipped counterpart; a whole-batch objective, not the per-token term of the language model variant.
$\PrivilegedSimulatorState$	Privileged Simulator State	Record	Ground-truth simulator quantities such as friction and contact wrenches that a deployed policy cannot measure; a subset of the physical world state exposed by the simulator.
$\SimulationParameterVector$	Simulation Parameter Vector	Varies	Physical parameters that define one simulated environment: link masses, centers of mass, friction, joint damping, transport delays and sensor noise; sampled in domain randomization, with the nominal point marked by a zero subscript.
$\ProprioceptiveTokenEmbeddings$	Proprioceptive Token Embeddings	Vector	Token embeddings of the robot’s joint and force state fed to an embodied transformer; one block of the fused multimodal token stream.
$\StaticDataDistribution$	Static Data Distribution	Distribution	Fixed data distribution from which a static benchmark draws its held-out examples.
$\TextTokenEmbeddings$	Text Token Embeddings	Vector	Token embeddings of the language instruction fed to an embodied transformer, one block of the fused token stream.
$\VisionTokenEmbeddings$	Vision Token Embeddings	Vector	Token embeddings an embodied transformer computes from camera images, one block of the fused token stream.
$\VitPatchSize$	Patch Size	Pixels	Side length of one square image patch a vision transformer tokenizes, typically 14 or 16 pixels.
$\VitPatchTokenEmbedding$	Patch Token Embedding	Vector	Latent token embedding a vision transformer forms from one image patch.
$\CriticReturnEstimate$	Critic Return Estimate	Dimensionless	Remaining-return estimate a learned critic network predicts for an interaction history; the baseline of PPO, which GRPO replaces with the group mean reward.
$\DecisionProcessState$	Decision Process State	Record	Complete environment state of a decision process, including interaction history and variables the agent cannot observe. A record, not a number.
$\DeviationDisadvantage$	Deviation Disadvantage	Dimensionless	Largest extra expected cost of one deviation from the reference action followed by reference recovery, at any state the learner can reach; an assumption about the task, in normalized cost units.
$\DiscountFactor$	Discount Factor	Dimensionless	Factor by which a reward arriving one step later is weighted in the return, between zero and one; one means undiscounted.
$\EnvironmentTransitionDistribution$	Environment Transition Distribution	Distribution	Distribution the environment induces over the next state given the current state and the action taken.
$\EstimatedHistoryAdvantage$	Estimated History Advantage	Dimensionless	Estimate of how much an action taken after a given interaction history beats the policy value at that history.
$\ExpectedHistoryActionReturn$	Expected History-Action Return	Function	Expected remaining return under a policy after taking a given action at a given interaction history and then following the policy.
$\ExpectedRemainingReturn$	Expected Remaining Return	Function	Expected remaining return under a policy from a given interaction history at a decision step; written with the policy as a superscript.
$\ExpectedReturnObjective$	Expected Return Objective	Function	Expected return of a policy as a function of its parameters, the objective reinforcement learning maximizes. Not the imitation cost sum $\ExpectedTaskCostSum$.
$\ExpectedTaskCostSum$	Expected Task Cost Sum	Function	Expected sum over the horizon of the normalized immediate task costs a policy incurs; the imitation bounds compare it for a learner and the reference. Not monetary.

Embedded Real-Time Systems, Power, and Hardware Safety

Response-time analysis keeps the Liu–Layland letters of its literature, but only inside that model and only with a task index $\Glyph{i}$, $\Glyph{j}$, or $\Glyph{k}$: $\WorstCaseExecutionTime_i$ (worst-case execution time), $\TaskBlockingTime_i$ (blocking), $\TaskInterferenceTime_i$ (interference), $\Glyph{J}_i$ (release jitter), $\TaskPeriod_i$ (period), $\WorstCaseResponseTime_i$ (worst-case response time), and $\TaskRelativeDeadline_i$ (relative deadline, which is $\CompletionDeadline$ of task $\Glyph{i}$ in the series’ terms), as in $\WorstCaseResponseTime_i = \WorstCaseExecutionTime_i + \TaskBlockingTime_i + \TaskInterferenceTime_i \le \TaskRelativeDeadline_i$. Outside that model these letters keep their table meanings.

Symbol	Definition	Unit	Notes
$\PowerRailInductance, \PowerRailResistance$	Power Rail Inductance / ESR	$\MicroUnitPrefix\text{H}, \text{m}\OhmUnit$	Parasitic inductance and equivalent series resistance of the power harness feeding the inverter; with $\DcBusBulkCapacitance$ they set the droop $\Glyph{\Delta V_{\text{droop}}}$ under a current step.
$\DcBusBulkCapacitance$	DC Bus Bulk Capacitance	$\MicroUnitPrefix\text{F}$	Inverter decoupling capacitor bank buffering transient inductive voltage collapse.
$\MemoryBusQueueDelay$	Memory Bus Queue Delay	$\text{ms}$	Contention latency $\sum_{\Glyph{k}} {\DataVolume}_{\Glyph{k}} / \Glyph{\text{BW}_{\text{peak}}} + \Glyph{N_{\text{conflict}}} \BankConflictPenalty$ across shared SoC crossbars.
$\TelemetryFreshnessBound$	Telemetry Freshness Bound	$\text{ms}$	Maximum allowable age of sensor telemetry before an interlock trips: $\TelemetryFreshnessBound \le \PositionErrorCeiling / \SpeedBound$.
$\ClockOffset$	Clock Offset	$\MicroUnitPrefix\text{s}$	Offset of one clock from another, estimated from a timestamp exchange as $\ClockOffset = [(\TimeInstant_2 - \TimeInstant_1) - (\TimeInstant_4 - \TimeInstant_3)]/2$; its one-way path delay is $\PtpPropagationDelay$.
$\TimingJitter$	Timing Jitter	$\MicroUnitPrefix\text{s}$	Standard deviation of a nominally periodic event’s timing about its schedule.
$\SynchronizationBlackout$	Synchronization Blackout	$\text{s}$	Interval during which a node receives no synchronization message and free-runs on its own clock.
$\AcquisitionEpoch$	Acquisition Epoch	$\text{s}$	Instant a sensor transduced one observation, such as a camera frame's exposure midpoint or an encoder latch; the evidence epoch of a belief is the latest acquisition epoch among the measurements that constrained it.
$\ActuatorOffTime$	Actuator Recovery Time	$\text{s}$	Recovery duration between actuator bursts in one duty cycle; the burst itself is $\ActuatorOnTime$.
$\ActuatorOnTime$	Actuator Burst Time	$\text{s}$	Duration an actuator runs at burst power in one duty cycle; pairs with the recovery time $\ActuatorOffTime$.
$\AisleTravelSpeed$	Aisle Travel Speed	$\text{m/s}$	Speed the mobile base is set to travel in a warehouse aisle, chosen below the derived dry-floor speed ceiling; it is the speed at detection in the aisle stopping budget.
$\ArrivalTimestamp$	Arrival Timestamp	$\text{s}$	Instant a sensor sample arrives in host memory, when its direct memory access transfer completes; later than the capture timestamp by exposure, readout, transport, and transfer.
$\AvailableBrakingDistance$	Available Braking Distance	$\text{m}$	Clear distance minus the localization bound and the protective clearance margin; the distance braking and blind travel may use.
$\AvailableDeceleration$	Available Deceleration	$\text{m/s}^2$	Deceleration a machine actually has, the smaller of its credible braking deceleration and the traction limit, friction coefficient times $\Gravity$.
$\BankConflictPenalty$	Bank Conflict Penalty	$\text{s}$	Time one memory bank or row conflict adds to a queued transfer, about a precharge plus an activation.
$\BlindTravelDistance$	Blind Travel Distance	$\text{m}$	Distance a machine travels during the pre-brake delay, before retarding force begins: speed at detection times that delay.
$\BrakeChopperResistance$	Brake Chopper Resistance	$\OhmUnit$	Resistance of the absorption resistor a dynamic brake chopper switches across the bus; must dissipate the peak regenerated power at the clamp voltage.
$\BrakingDistance$	Braking Distance	$\text{m}$	Distance a machine travels from brake onset to rest, speed squared over twice the credible deceleration; excludes travel during the pre-brake delay.
$\CaptureTimestamp$	Capture Timestamp	$\text{s}$	Midpoint of the sensor’s exposure or firing window, the instant the observation describes; the letter t carrying index zero.
$\CaseTemperature$	Case Temperature	$^\circ\text{C}$	Temperature of a motor or package case.
$\CaseThermalTimeConstant$	Case Thermal Time Constant	$\text{s}$	Thermal time constant of the motor case stage of a multi-node motor thermal model.
$\ChopperClampVoltage$	Chopper Clamp Voltage	$\text{V}$	Bus voltage at which a dynamic brake chopper switches its absorption resistor across the bus; above the full-charge voltage and below the converter and switch ratings.
$\ClockConversionBound$	Clock Conversion Bound	$\text{s}$	Bound on the error of converting a sensor’s capture timestamp to the clock of the safety microcontroller that hosts the permission path; measured from offset tails, path asymmetry, and drift, and added to the observation age at dispatch.
$\ClockConversionIntercept$	Clock Conversion Intercept	$\text{s}$	Intercept of the affine mapping that converts a capture-clock time to the safety microcontroller’s clock.
$\ClockConversionSlope$	Clock Conversion Slope	Dimensionless	Slope of the affine mapping that converts a capture-clock time to the safety microcontroller’s clock; near one, the drift between the clocks.
$\ConsoleCommunicationTime$	Console Communication Time	$\text{s}$	Time the operator console and link add to a takeover: video transport to the operator plus the returning protocol exchange; shares its glyph with the gradient communication time of training, a different quantity.
$\ConstantHazardRate$	Constant Hazard Rate	$\text{h}^{-1}$	Constant rate of hazardous failures per operating hour of a machine, the rate a failure-free test exposure bounds from above; assumes the flat middle of the bathtub curve.
$\ConverterLockoutVoltage$	Converter Lockout Voltage	$\text{V}$	Input voltage at which a DC-DC converter shuts off (undervoltage lockout); bus droop below it resets the computation the converter supplies.
$\CopperReferenceTemperature$	Copper Reference Temperature	°C	Temperature at which a winding has its reference resistance, the base of the copper temperature coefficient.
$\CopperTemperatureCoefficient$	Copper Temperature Coefficient	$\text{K}^{-1}$	Fractional rise in a copper winding’s resistance per kelvin; about 0.00393 per kelvin for annealed copper at 20 °C.
$\CopperThermalTimeConstant$	Copper Thermal Time Constant	$\text{s}$	Thermal time constant of the winding copper stage in a multi-node motor thermal model.
$\CoreAmbientThermalResistance$	Core-to-Ambient Thermal Resistance	$\text{K/W}$	Convective thermal resistance from a motor’s stator core to the ambient air; it rises when an enclosure removes airflow.
$\EquilibriumWindingTemperature$	Equilibrium Winding Temperature	°C	Temperature a winding approaches asymptotically under a constant loss power: ambient plus loss power times the thermal resistance in the first-order model.
$\DcBusVoltage$	DC Bus Voltage	$\text{V}$	Voltage of the DC bus that feeds a motor inverter.
$\DcRailVoltage$	DC Rail Voltage	$\text{V}$	Voltage of the shared DC power rail that feeds the compute and the motor inverters; the bus of $\DcBusVoltage$ is the inverter’s side of it.
$\DroopedSupplyVoltage$	Drooped Supply Voltage	$\text{V}$	Supply voltage a core sees at the bottom of a droop event.
$\FinalBusVoltage$	Final Bus Voltage	$\text{V}$	DC bus voltage at the end of a regenerative stop when the bus capacitance alone absorbs the returned energy; a screening endpoint, since exceeding a component rating means the stop needs another energy path.
$\GatePropagationDelay$	Gate Propagation Delay	$\text{s}$	Propagation delay of a logic gate as a function of its supply voltage; the alpha-power law makes it grow as the supply approaches the transistor threshold voltage.
$\HeatDissipationPower$	Heat Dissipation Power	$\text{W}$	Power the hardware at one signal handoff dissipates as heat, the energy-and-heat limit of the handoff matrix; from under a watt on the safety microcontroller to tens of watts on the proposer accelerators.
$\DeadlineInstant$	Deadline Instant	$\text{s}$	Clock instant by which an observation or action must be complete.
$\FallbackDeadline$	Fallback Deadline	$\text{s}$	Latest permissible wait after a handover request before the fallback must begin, from the stopping inequality at the current speed and clearance; it clips the handover timeout ceiling $\HandoverTimeoutCeiling$.
$\HandoverTimeoutCeiling$	Handover Timeout Ceiling	$\text{s}$	Configured protocol ceiling on the handover timeout, which the clearance-derived $\FallbackDeadline$ may shorten but never extend.
$\EnforcedCommandTap$	Enforced Command Tap	Record	Third stream of the four-stream action tap, the command the permission check admitted.
$\InitialBusVoltage$	Initial Bus Voltage	$\text{V}$	DC bus voltage at the moment a regenerative deceleration begins; the worst case is a full pack, when the battery refuses charge.
$\NominalSupplyVoltage$	Nominal Supply Voltage	$\text{V}$	Nominal regulated supply voltage of a core rail.
$\InverterCurrentLimit$	Inverter Current Limit	$\text{A}$	Largest current an inverter bridge lets flow, set to protect its transistors and the stator.
$\LoadCurrent$	Load Current	$\text{A}$	Current the loads draw from a bus node.
$\JunctionTemperature$	Junction Temperature	$\text{°C}$	Temperature of a silicon die at its transistor junctions; written as a function of time when it relaxes after a power step. A temperature always carries a subscript, since the bare letter is time.
$\JunctionCaseThermalResistance$	Junction-to-Case Thermal Resistance	$\text{K/W}$	Thermal resistance from a silicon junction to the case of its package.
$\JunctionAmbientThermalResistance$	Junction-to-Ambient Thermal Resistance	$\text{K/W}$	Lumped thermal resistance from a silicon junction to the ambient air, the sum of the junction-to-case, interface-material and chassis paths; not the motor stator-to-ambient path.
$\OhmUnit$	Ohm Unit	Unit	SI unit of electrical resistance, written upright after a prefix as in milliohms.
$\PeriodicTaskCount$	Periodic Task Count	Dimensionless	Number of periodic tasks in a real-time task set; it sets the Liu and Layland utilization bound, which falls toward the natural logarithm of two as the count grows.
$\MicrocontrollerClockTime$	Microcontroller Clock Time	$\text{s}$	Time read on the safety microcontroller’s monotonic clock.
$\MappedCommandTap$	Mapped Command Tap	Record	Second stream of the four-stream action tap, the kinematically mapped command before it crosses to the safety microcontroller.
$\MeasuredResponseTap$	Measured Response Tap	Record	Fourth stream of the four-stream action tap, the plant response measured by joint encoders and current shunts.
$\LeaseAdmissionInstant$	Lease Admission Instant	$\text{s}$	Instant at which the permission path admits an intent lease, the start of its validity interval.
$\NominalPlanExpiry$	Nominal Plan Expiry	$\text{s}$	Instant, on the permission clock, at which the nominal setpoints of an admitted trajectory record end; never executed when the matched stop begins at commitment.
$\ObservationAgeBudget$	Observation Age Budget	$\text{s}$	Configured upper bound on observation age above which a consumer refuses a state.
$\MaximumObservationAge$	Maximum Observation Age	$\text{s}$	Largest observation age a consumer tolerates before acting on the observation is no longer safe or stable; the phase-margin ceiling in the control chapters and the freshness threshold in the memory and machine-learning chapters.
$\MissionDuration$	Mission Duration	$\text{s}$	Duration of one mission of a mobile robot, the unit of operation over which a per-mission failure probability is allocated.
$\MissionEnergy$	Mission Energy	$\text{J}$	Total energy a machine draws to complete one physical mission; distinct from the machine-learning workload’s total energy.
$\OperatorSensoryTime$	Operator Sensory Time	$\text{s}$	Time an operator needs to perceive a takeover stimulus, the perceptual stage of a simple reaction (Model Human Processor perceptual cycle).
$\OperatorCognitiveTime$	Operator Cognitive Time	$\text{s}$	Time an operator needs to decide on a response to a takeover stimulus, the cognitive stage of a simple reaction (Model Human Processor cognitive cycle).
$\OperatorMotorTime$	Operator Motor Time	$\text{s}$	Time an operator needs to move the controls once a response is chosen, the motor stage of a simple reaction; not the electrical time of a motor.
$\OperatorStabilizationInstant$	Operator Stabilization Instant	$\text{s}$	Time after a takeover at which the operator’s corrections settle into a steady return toward the nominal corridor.
$\OrientationTolerance$	Orientation Tolerance	$\text{rad}$	Largest angular orientation error a task accepts.
$\ObservedCycleCount$	Observed Cycle Count	Dimensionless	Number of operating cycles, each one observation of the monitored quantity, over which exceedances are counted.
$\ObservedTrialFailureCount$	Observed Trial Failure Count	Dimensionless	Number of failures observed among the trials of a test campaign, fixed-sample or sequential.
$\OperationalTestDistribution$	Operational Test Distribution	Distribution	Declared joint distribution of trial conditions an evaluation campaign samples, the population its success probability describes; each stratum of a stratified test matrix is its own distribution, and it is not a dataset.
$\PreTriggerWindowDuration$	Pre-Trigger Window Duration	$\text{s}$	Length of the ring-buffer window before an intervention trigger that the logger freezes and keeps; sized from the out-of-loop takeover latency and rounded up to whole seconds.
$\PrebrakeDelayCeiling$	Pre-Brake Delay Ceiling	$\text{s}$	Largest pre-brake delay that still lets the machine stop within its clearance: clearance minus margins and braking distance, divided by speed; the ceiling the lease rule subtracts the loop, transport, actuator and observation times from.
$\StopDecisionComputeBound$	Stop Decision Compute Bound	$\text{s}$	Bound on the stop-decision computation and any still-required input transfer that a hazard-triggered stop spends inside the pre-brake delay; a budget term, not operations over a sustained rate.
$\SupervisoryTakeoverLatency$	Supervisory Takeover Latency	$\text{s}$	Total time from a takeover demand to the first checked human command: the human sensory, cognitive and motor times plus console transport and handshake; the generic form of the in-loop and out-of-loop design values.
$\RemainingExecutionLease$	Remaining Execution Lease	$\text{s}$	Time left on the active execution lease, the lease length minus the time elapsed since its last accepted renewal; a field of the per-tick evidence tuple.
$\ReplacementLatency$	Replacement Latency	$\text{s}$	Time from the capture of the observation a candidate trajectory rests on to that candidate’s admission into the execution queue; a latency, not a period, and not the loss that bare $\Glyph{L}$ names.
$\SignalHandoff$	Signal Handoff	Record	One of the five canonical signal handoffs of the machine, from sensing to perception through enforcer to actuator, at which interface limits are budgeted; written with its position.
$\SynchronizationErrorBound$	Synchronization Error Bound	$\text{s}$	Bound on the error of converting a timestamp from one node’s clock to another’s; the clock-conversion error of the observation contract, not an estimated clock offset.
$\PtpPropagationDelay$	PTP Propagation Delay	$\text{s}$	One-way network path delay between two clocks, estimated from a symmetric two-way Precision Time Protocol exchange.
$\RealTimeTaskIndex$	Real-Time Task Index	Index	Position of one periodic task in a real-time task set, the index of its period, deadline and execution time; written bare when the text names the task.
$\TaskBlockingTime$	Task Blocking Time	$\text{s}$	Longest time a real-time task can wait for a lower-priority task to release a shared resource; written with its task index, and under the priority ceiling protocol at most one lower-priority critical section.
$\ReadStartSequenceCount$	Read-Start Sequence Count	Dimensionless	Value of a sequence-lock counter that a reader loads before copying the shared slot; the read is valid only if it is even and equals the end count.
$\ReadEndSequenceCount$	Read-End Sequence Count	Dimensionless	Value of a sequence-lock counter that a reader loads after copying the shared slot; a change from the start count means a writer preempted the copy and the snapshot is torn.
$\RowActivationDelay$	Row Activation Delay	$\text{s}$	Minimum time from activating a DRAM row to the first column access to it, the row-to-column delay (JEDEC tRCD).
$\RowPrechargeTime$	Row Precharge Time	$\text{s}$	Minimum time a DRAM bank needs to close its open row by precharge before another row can be activated (JEDEC tRP); a row conflict costs it plus the row activation delay.
$\RawHardwareFaultRate$	Raw Hardware Fault Rate	$\text{h}^{-1}$	Rate of hazardous hardware faults before any diagnostic mitigates them, the sum of the component fault rates; usually quoted in failures in time.
$\RatedFatigueLife$	Rated Fatigue Life	Dimensionless	Cumulative load-cycle count at which 10% of a population of identical components is expected to have failed; the subscript ten is part of the symbol.
$\SourceCurrent$	Source Current	$\text{A}$	Current arriving at a bus node through the source path of battery, harness and switches; it differs from the load current during a transient, and the bus capacitance supplies the difference.
$\SourceVoltage$	Source Voltage	$\text{V}$	Unloaded voltage of the supply that feeds a bus node through the source path; minus the drop across the rail resistance and inductance it gives the bus voltage.
$\ReferenceWindingResistance$	Reference Winding Resistance	$\Glyph{\Omega}$	Winding resistance at the reference temperature, the base of copper’s linear resistance law.
$\StatorCoreTemperature$	Stator Core Temperature	$\text{K}$ or $^\circ\text{C}$	Temperature of the stator core in a two-node motor thermal model.
$\ProtectiveSeparationDistance$	Protective Separation Distance	$\text{m}$	Minimum distance a speed-and-separation monitor must keep between a person and a robot; the sum of the human travel, robot reaction travel, robot stopping travel, intrusion and the two uncertainty terms of ISO/TS 15066.
$\RobotReactionTravel$	Robot Reaction Travel	$\text{m}$	Distance a robot travels during its reaction time before stopping begins; a term of the protective separation distance.
$\RobotStoppingTravel$	Robot Stopping Travel	$\text{m}$	Distance a robot travels while stopping; a term of the protective separation distance.
$\RobotPositionUncertainty$	Robot Position Uncertainty	$\text{m}$	Position uncertainty of the robot; a term of the protective separation distance.
$\SequentialTrialCount$	Sequential Trial Count	Dimensionless	Number of trials a sequential test has observed so far, the index of its running log-likelihood ratio.
$\TailRiskObservation$	Tail-Risk Observation	Varies	Monitored physical quantity whose rare large values a tail model describes, such as a contact load or a control-loop latency, treated as a random variable; in the unit of the quantity monitored, and indexed for one independent observation.
$\TailRiskObservationLevel$	Tail-Risk Observation Level	Varies	Level of the monitored quantity at which its distribution or density is evaluated, or one observed value of it; the lower cutoff of a power-law tail is marked by a zero subscript.
$\TailShapeParameter$	Tail Shape Parameter	Dimensionless	Shape parameter of an extreme value tail model, positive for a power-law tail, zero for an exponential tail and negative for a bounded tail; shared by the generalized extreme value and generalized Pareto models.
$\TailSupportUpperEndpoint$	Tail Support Upper Endpoint	Varies	Right endpoint of the support of the monitored quantity’s distribution, possibly infinite; in the unit of the monitored quantity.
$\TargetExceedanceProbability$	Target Exceedance Probability	Dimensionless	Per-sample probability of a catastrophic exceedance that a design margin is sized against; a probability per sampled cycle, not a rate per operating hour.
$\TaskInterferenceTime$	Task Interference Time	$\text{s}$	Processor time that higher-priority tasks and interrupt handlers take from a real-time task within its response window. Written with its task index; it depends on the response time itself, so it is found by a fixed-point recurrence.
$\TaskPeriod$	Task Period	$\text{s}$	Interval between successive releases of a periodic real-time task, written with its task index.
$\TaskRelativeDeadline$	Task Relative Deadline	$\text{s}$	Longest time after its release by which a job of a real-time task must complete, written with its task index. The series-wide deadline is $\CompletionDeadline$.
$\TaskSetUtilization$	Task Set Utilization	Dimensionless	Share of processor time a set of periodic real-time tasks demands, the sum over tasks of $\WorstCaseExecutionTime/\TaskPeriod$.
$\WorstCaseExecutionTime$	Worst-Case Execution Time	$\text{s}$	Worst-case execution time of one real-time task in response-time analysis; used only inside that analysis and only with a task index.
$\WorstCaseResponseTime$	Worst-Case Response Time	$\text{s}$	Longest time from a real-time task’s release to its completion; used only with a task index.
$\TrackingStartInstant$	Tracking Start Instant	$\text{s}$	Instant, on the permission clock, at which nominal tracking of an admitted trajectory record begins; the first of the record’s four deadlines.
$\TimestampSkew$	Timestamp Skew	$\text{s}$	Skew between the time stamps of two records or clocks. The book writes it with the finite-change prefix, $\FiniteChange\,\TimestampSkew$.
$\TrackLastUpdateTime$	Track Last Update Time	$\text{s}$	Time of the most recent observation associated with a track.
$\CurrentWallClockTime$	Current Wall-Clock Time	$\text{s}$	Wall-clock time at which a monitor evaluates staleness; a point in time, like $\TimeInstant$, fixed at the moment of the check.
$\FailureObservationInterval$	Failure Observation Interval	$\text{s}$	Length of the time interval over which failures are counted under the Poisson failure model; a generic window, not the length of one job.
$\ComponentFailureRate$	Component Failure Rate	$\text{h}^{-1}$	Rate at which one component fails; the system-wide rate is the component count times this rate under independent, identical components.
$\UnmitigatedHardwareFaultRate$	Unmitigated Hardware Fault Rate	$\text{h}^{-1}$	Rate of hardware faults whose hazardous effect no diagnostic mitigates, the rate compared with the target hazard rate.
$\WatchdogEarliestServiceTime$	Watchdog Earliest Service Time	$\text{s}$	Time after the last reload before which a windowed watchdog rejects a service attempt as too early, the lower end of its service window. With $\WatchdogLatestServiceTime$ it bounds the valid window.
$\WatchdogLatestServiceTime$	Watchdog Latest Service Time	$\text{s}$	Time after the last reload by which a windowed watchdog must be serviced, past which it trips the safe state.
$\ThrottleTripTemperature$	Throttle Trip Temperature	°C	Junction temperature at or above which a hardware thermal governor reduces the processor clock; paired with a lower clear temperature for hysteresis.
$\TransistorThresholdVoltage$	Transistor Threshold Voltage	$\text{V}$	Gate-source voltage at which a transistor begins to conduct. The headroom between supply and threshold sets gate delay in the alpha-power law.
$\VelocitySaturationExponent$	Velocity Saturation Exponent	Dimensionless	Technology-dependent exponent of the alpha-power law, by which drain current grows with gate overdrive above threshold; it falls from 2 for a long channel toward 1 as carrier velocity saturates.
$\WindingCoreThermalResistance$	Winding-to-Core Thermal Resistance	$\text{K/W}$	Conductive thermal resistance from a motor winding to its stator core; in series with the core-to-ambient resistance it makes the total $\ThermalResistance$.
$\WindingCurrent$	Winding Current	$\text{A}$	Electric current flowing through a motor winding, which heats it at the rate of current squared times resistance.
$\ConductorCurrent$	Conductor Current	$\text{A}$	Electrical current flowing through a power-distribution conductor; resistive loss is its square times the conductor resistance.
$\WindingLossPower$	Winding Loss Power	$\text{W}$	Resistive power dissipated in the motor windings.
$\WindingTemperatureLimit$	Winding Temperature Limit	°C	Largest winding temperature the insulation class and derating policy allow.
$\MicroUnitPrefix$	Micro Unit Prefix	Prefix	SI prefix for one millionth, written before a unit such as seconds; it only scales the unit after it and is never a quantity, a service rate, or a mean.
$\SupplyVoltage$	Supply Voltage	$\text{V}$	Voltage that powers a digital circuit and that voltage and frequency scaling lowers; dynamic power scales with $\SupplyVoltage^2$.
$\ParasiticLoopInductance$	Parasitic Loop Inductance	$\text{H}$	Unintended inductance of the current loop that feeds an accelerator core rail; it turns current slew into voltage droop.

Deep Learning Foundations and Edge Serving

Embodied robotic platforms increasingly run large multimodal foundation models, such as vision-language-action (VLA) backbones and cross-attention policy decoders, alongside deterministic real-time control stacks. Serving these models on an embedded edge computer requires tracking both the neural architecture dimensions and the memory footprint those dimensions occupy:

Symbol	Definition	Unit	Notes
$\AttentionHeadDimension$	Attention Head Dimension	Integer	Per-head hidden dimension in attention layers.
$\HiddenDimension$	Hidden Dimension	Integer	Hidden feature (model) width $\HiddenDimension$ of a transformer layer or representation; the distance to a target, also $\Glyph{d}$, is $\TargetTravelDistance$ and appears only in the intent-lease reachability test.
$\ProbabilityFunction(\cdot)$	Distribution	Distribution	Probability mass or density $\ProbabilityFunction(\Glyph{x})$, conditional $\ProbabilityFunction(\Glyph{y} \mid \Glyph{x})$, or an observation model $\ProbabilityFunction(\MultimodalObservation \mid \PhysicalWorldState)$; lowercase avoids the collision with $\Glyph{P}$ (parameter count).
$\Pr(\Glyph{E})$	Event Probability	Dimensionless	Probability of an event $\Glyph{E}$, as in $\Pr(\text{error by step } \Glyph{t})$; use $\ProbabilityFunction(\cdot)$ for distributions.
$\LayerInputWidth, \LayerOutputWidth$	Layer Input and Output Widths	Integer	Number of input and output features of a dense layer, which holds $\LayerInputWidth \LayerOutputWidth$ weights.
$\LearningRate$	Learning Rate	Scalar	Step size for the optimizer; never bare for hardware efficiency, which is $\EfficiencyHw$.
$\LossFunction$	Loss Function	Scalar	The objective minimized during training; a subscript or superscript names the objective ($\Glyph{\LossFunction_{\text{MSE}}}$, $\Glyph{\LossFunction_{\text{chunk}}}$, $\Glyph{\LossFunction^{\text{CLIP}}}$).
$\ModelParameters$	Model Parameters	Vector	Trainable parameter vector or weights $\ModelParameters$ of a learned model, such as a policy.
$\LayerCount$	Number of Layers	Integer	Total number of layers in a network.
$\SequenceLength$	Sequence Length	tokens	Input context or sequence length $\SequenceLength$ in tokens. The behavioral-cloning horizon $\BehaviorCloningHorizon$ also renders $\Glyph{S}$ but counts control steps; the two never share an equation.
$\WeightMemory$	Weight Memory	bytes	Memory occupied by model weights ($\Parameters \times \text{bytes-per-parameter}$).
$\ElementStorageSize$	Element Storage Size	bytes/element	Bytes per stored tensor element at the deployed precision.
$\ArithmeticIntensity$	Arithmetic Intensity	FLOP/byte	Operations per byte moved from off-chip memory, $\Operations/\DataVolume$; it decides whether an evaluation is bound by memory or by compute.
$\EffectiveSampleSize$	Effective Sample Size	Dimensionless	Number of independent observations that carry the same information about a mean as a serially correlated stream.
$\EnvironmentParameters$	Environment Parameters	Record	External physical variables of one trial or rollout, such as floor friction, illuminance, obstacle geometry, and air temperature; written non-bold, so a record rather than a vector.
$\EvaluationPredicate$	Evaluation Predicate	Function	Binary predicate that maps a realized trial trajectory to its task outcome, zero or one.
$\ExceedanceReturnLevel$	Exceedance Return Level	Varies	Level of the monitored quantity exceeded with the target probability per sampled cycle; its subscript is that exceedance probability.
$\ExceedanceThreshold$	Exceedance Threshold	Varies	High threshold on a performance metric above which peaks-over-threshold modeling fits exceedances; it carries the metric’s units (meters for a distance overshoot).
$\GeneralizedParetoDistribution$	Generalized Pareto Distribution	Function	Cumulative distribution function of the excess over a high threshold, parameterized by a scale and a tail shape.
$\GpdScaleParameter$	Generalized Pareto Scale	Varies	Scale parameter of the generalized Pareto distribution fitted to exceedances over a threshold.
$\FailureUpperConfidenceLimit$	Failure Upper Confidence Limit	Dimensionless	One-sided upper confidence limit on the per-trial failure probability from a finite campaign.
$\HypothesizedTrialFailureProbability$	Hypothesized Trial Failure Probability	Dimensionless	Per-trial failure probability a sequential probability ratio test specifies under one of its two hypotheses, indexed 0 for the unacceptable null and 1 for the acceptable alternative.
$\HazardClassIndex$	Hazard Class Index	Dimensionless	Index of one defined hazard class.
$\FleetRobotCount$	Fleet Robot Count	Dimensionless	Number of robots in a deployed fleet; not the accelerator count of a data center fleet.
$\GaussianCovarianceMatrix$	Gaussian Covariance Matrix	Matrix	Covariance matrix of a multivariate Gaussian; entries carry the products of the component units.
$\GaussianMeanVector$	Gaussian Mean Vector	Vector	Mean vector of a multivariate Gaussian, paired with $\GaussianCovarianceMatrix$.
$\PrefillOperations$	Prefill Operations	$\text{FLOP}$	Arithmetic operations one evaluation’s prefill executes over all of its input tokens; the compute-bound phase of a fresh evaluation, and an instance of the operation count.
$\ProposerEvaluationRate$	Proposer Evaluation Rate	$\text{Hz}$	Rate of fresh proposer evaluations when every evaluation streams the weights.
$\SoftmaxFunction$	Softmax Function	Function	Function that maps a vector of logits to a probability vector that sums to one; normalization by construction, not calibrated confidence, and not the sigmoid.
$\ModelInput$	Model Input	Record	Input of one example, the data a model receives to produce a prediction; its label partner is $\Glyph{\mathbf{y}}$. Not a scalar feature $\Glyph{x_i}$ of a neuron.
$\ModelPrediction$	Model Prediction	Dimensionless	Scalar prediction a model outputs for one example, compared with its label by the loss; the vector of class probabilities is $\Glyph{\hat{\mathbf{y}}}$.
$\LogitVector$	Logit Vector	Vector	Unnormalized score for every class that a classifier outputs before softmax; its entries are $\OutputLogit$.
$\OutputLogit$	Output Logit	Dimensionless	Raw score a model outputs for one class before the softmax, indexed by class as $\Glyph{z_i}$.
$\ValidationLoss$	Validation Loss	Scalar	Loss a trained model incurs on held-out validation data; the outer objective of neural architecture search, never the training loss.
$\WeightMatrix$	Weight Matrix	Matrix	Matrix of learned weights one layer multiplies its input by, or the kernel tensor of a convolution layer; a subscript names the layer, as in $\Glyph{\mathbf{W}_1}$.
$\FunctionJacobian$	Function Jacobian	Matrix	Matrix of partial derivatives of a differentiated function’s outputs with respect to its inputs; not the manipulator Jacobian of a robot arm.
$\ZeroMatrix$	Zero Matrix or Vector	Matrix	Matrix or vector whose entries are all zero; bold, so it is not the scalar zero.
$\ImageHeight$	Image Height	pixels	Number of pixel rows in an image or its segmentation mask; shares its glyph with horizons but counts rows.
$\ImageWidth$	Image Width	pixels	Number of pixel columns in an image or its segmentation mask.
$\IndicatorFunction$	Indicator Function	Function	Function equal to one when its condition holds and zero otherwise.
$\KlDivergence$	KL Divergence	nats or bits	Expected extra code length from encoding one distribution with a code built for another; asymmetric, so $\Glyph{\KlDivergence(p \lVert q) \ne \KlDivergence(q \lVert p)}$, and distillation uses the teacher-to-student order.
$\ShannonEntropy$	Shannon Entropy	bits	Average uncertainty of a probability distribution, the expected negative log probability of its outcomes; the log base sets the unit, base two for bits and natural log for nats.
$\StatisticalHypothesis$	Statistical Hypothesis	Record	A hypothesis about a population: $\Glyph{H_0}$ the null, $\Glyph{H_1}$ the alternative, and $\Glyph{H_{(i)}}$ the hypothesis with the $\Glyph{i}$-th smallest p-value. A record, not a number.
$\SignificanceCriticalQuantile$	Significance Critical Quantile	Dimensionless	Standard-normal quantile at one minus half the significance level, the two-sided critical value of a release test; about 1.96 at a significance level of 0.05.
$\NextDecisionProcessState$	Next Decision-Process State	Record	State a decision process moves to after an action is taken in the current state, as in the transition law $\Glyph{p(s^\prime \mid s, a)}$; the successor of the current state.
$\PolicyObservedState$	Policy Observed State	Record	State or observation a policy conditions its choice of action on, the conditioning argument in $\Glyph{\pi_\theta(a \mid s)}$.
$\PolicyAction$	Policy Action	Record	Action a policy chooses given the state it observes, the chosen argument in $\Glyph{\pi_\theta(a \mid s)}$; agentic rollouts use their own rollout-action symbol.
$\PolicyProbabilityRatio$	Policy Probability Ratio	Dimensionless	Ratio of a sampled token’s probability under the current policy to its probability under the behavior policy that generated it; clipped to an interval of half-width $\RatioClippingWidth$ around one.
$\RatioClippingWidth$	Ratio Clipping Width	Dimensionless	Half-width of the interval around one to which the clipped surrogate limits the probability ratio; between zero and one.
$\PolicyHistoryActionReturn$	Policy History-Action Return	Dimensionless	Expected remaining return under the current policy after taking a given action at a given interaction history and then following that policy, written $\Glyph{Q^{\pi}(h_k, a_k)}$.
$\PolicyRemainingReturn$	Policy Remaining Return	Dimensionless	Expected remaining return under the current policy from a given interaction history, written $\Glyph{V^{\pi}(h_k)}$; the baseline the advantage subtracts.
$\PolicyHistoryAdvantage$	Policy History Advantage	Dimensionless	Return of a given action at a given interaction history minus the remaining return at that history, $\Glyph{\PolicyHistoryAdvantage(h_k, a_k) = \PolicyHistoryActionReturn(h_k, a_k) - \PolicyRemainingReturn(h_k)}$.
$\StateDistribution$	State Distribution	Distribution	Average distribution of states a policy visits over the horizon; the per-step form carries a superscript, $\Glyph{d_\pi^t}$.
$\ReferenceStateDistribution$	Reference State Distribution	Distribution	Average distribution of states the reference policy visits over the horizon; the reference-policy counterpart of $\StateDistribution$.
$\Trajectory$	Trajectory	Record	Ordered record of the states, actions, and observations of one delegated task, $\Glyph{\tau = (s_0, a_0, o_1, \dots, s_H)}$; a record, not a number.
$\ClassCount$	Class Count	Count	Number of classes a classifier distinguishes, or vocabulary entries it scores; the length of its softmax output.
$\ExampleLabel$	Example Label	Record	Label paired with one input, treated as a random variable; a bold form is the label written as a vector.

Latin matrices and vectors are set in bold ($\Glyph{\mathbf{W}}$, $\Glyph{\mathbf{x}}$); scalars, dimensions, indices, and individual matrix entries stay italic ($\Glyph{N}$, $\Glyph{d}$, $\Glyph{i}$, $\Glyph{W}_{ij}$). The generic parameter symbol $\Glyph{\theta}$ is the one conventional exception and stays italic. Local matrix algebra follows the same bold rule: matrix operands are bold ($\Glyph{\mathbf{A}}$, $\Glyph{\mathbf{B}}$, $\Glyph{\mathbf{C}}$), as in general matrix multiply (GEMM) $\Glyph{\mathbf{C}}=\Glyph{\alpha} \Glyph{\mathbf{A}}\Glyph{\mathbf{B}}+\Glyph{\beta} \Glyph{\mathbf{C}}$. The bold form marks a matrix operand and is distinct from the italic scalar batch size $\BatchSize$, which remains batch size in scalar contexts. A GEMM’s scalar dimensions are $\Glyph{M}$, $\Glyph{N}$, and $\Glyph{K}$ ($\Glyph{\mathbf{A}}$ is $\Glyph{M} \times \Glyph{K}$, $\Glyph{\mathbf{B}}$ is $\Glyph{K} \times \Glyph{N}$). Asymptotic cost is written $\mathcal{O}(\cdot)$, which keeps it distinct from the operation count $\Operations$.

Evaluation and Statistical Evidence

A physical trial is expensive, so the evaluation chapters count carefully. These symbols name the trials themselves, the samples a metric is computed over, and the threshold exceedances that tail estimates are built from.

Symbol	Definition	Unit	Notes
$\CalibrationExampleCount$	Calibration Example Count	Count	Number of held-out calibration examples, and so of nonconformity scores, used to set a conformal prediction threshold. Not the quantization calibration set.
$\EvaluationSampleCount$	Evaluation Sample Count	Count	Number of examples in the evaluation set over which an accuracy or calibration metric is computed; it sets the sampling error. Not the training set size $\Glyph{D}$.
$\DiscreteRandomVariable$	Discrete Random Variable	Record	Discrete random variable, the uncertain quantity whose outcomes a distribution describes; uppercase for the variable, lowercase for one of its outcomes.
$\TrialCondition$	Trial Condition	Record	Condition of one physical trial: initial plant state, environment parameters, plant parameters, computational configuration, and termination protocol; drawn from the operational test distribution.
$\TrialIndex$	Trial Index	Index	Position of one physical trial within a learning campaign, counted from one.
$\TrialSuccessProbability$	Trial Success Probability	Dimensionless	True probability that one trial succeeds under the declared evaluation protocol and test distribution.
$\ThresholdExceedanceCount$	Threshold Exceedance Count	Count	Number of trials whose performance metric exceeds the threshold.
$\ThresholdExcess$	Threshold Excess	Varies	Amount by which the monitored quantity exceeds the threshold, one recorded exceedance; it carries the unit of the monitored quantity. Its random-variable form is $\ThresholdExcessVariable$.
$\ThresholdExcessVariable$	Threshold Excess Variable	Varies	Excess of the monitored quantity over the threshold, conditional on exceeding it, treated as a random variable.

Units and Precision

Physical units: This book uses SI (metric) units throughout, including meters, kilograms, seconds, watts, and °C, consistent with standard engineering and scientific practice. Where source data was originally reported in imperial units, the book converts to SI and notes the original values parenthetically. A space always separates the number from the unit (for example, 100 ms, 2 TB/s).
Data and memory: This book uses decimal SI prefixes only: KB = $10^3$ bytes, MB = $10^6$, GB = $10^9$, TB = $10^{12}$. Binary-prefixed units do not appear in prose; model sizes, activation footprints, throughputs, and transfer volumes are reported in decimal units (for example, 2 TB/s, 102 MB). Hardware memory capacities are the one exception: they keep the vendor’s nameplate figure and label. A device modeled here with a binary 80 GB capacity holds $80 \times 2^{30}$ bytes; usable capacity and allocation headroom remain separate.
Compute: The notation distinguishes operation counts from rates. Total work uses FLOPs (for example, GFLOPs, TFLOPs), while throughput uses FLOP/s with decimal prefixes (for example, GFLOP/s, TFLOP/s).
- 1 TFLOP = $10^{12}$ FLOPs
- 1 TFLOP/s = $10^{12}$ FLOPs per second
- Arithmetic intensity conventionally uses FLOP/byte as a unit ratio (floating-point operations per byte moved). This is a ratio unit, not a total-work symbol or throughput symbol.
Power: Power carries a subscript ($\Glyph{P_{\text{avg}}}$, $\Glyph{P_{\text{device}}}$, $\Glyph{P_{\text{cluster}}}$); bare $\Glyph{P}$ remains parameter count.
Latency percentiles: Written p50, p95, and p99 in upright type, inside math as well as in prose.
Currency: Dollar amounts use the dollar sign ($); unless otherwise noted, dollar-denominated costs are U.S. dollars (USD).
Precision:
- FP64: Double precision (8 bytes)
- FP32: Single precision (4 bytes)
- TF32: TensorFloat-32 (19-bit Tensor Core compute mode; inputs and storage remain FP32)
- FP16: Half precision (2 bytes, standard range)
- BF16: Brain float (2 bytes, wide dynamic range)
- FP8: Quarter precision (1 byte, E4M3 or E5M2 format)
- FP4: 4-bit floating-point format
- INT8: 8-bit integer (1 byte)
- INT4: 4-bit integer; lower integer precisions follow the same uppercase INTn pattern (for example, INT3, INT2)

Quick Reference: Resolving Collisions

Throughout this book, two single-letter symbols recur in both machine learning equations and hardware performance budgets. The most frequent symbol collisions across the machine learning, control, and embedded systems literatures are resolved as follows:

Symbol	ML Meaning	Systems Meaning	Book Convention
$\BatchSize$	Batch Size	Bandwidth	Batch Size. Use $\Bandwidth$ for bandwidth.
$\Parameters$	Parameters	Peak FLOP/s	Parameters. Use $\PeakRate$ for peak rate.

As a general principle, ML conventions take precedence for single letters, while systems concepts get subscripts or multi-letter symbols. This preserves compatibility with the ML literature while distinguishing the robotics and embedded systems meanings used in this book.

Symbol	Canonical Convention	Colliding Meaning	Disambiguation Rule
\(\Glyph{\theta}\)	Learned model (policy) parameters, as in \(\Policy\)	Joint angle, trajectory polynomial \(\Glyph{\theta}(\Glyph{t})\), heading, azimuth, attitude or mounting error, clock offset, or thermal resistance \(\Glyph{\theta_{JA}}\)	Bare \(\Glyph{\theta}\) is always the parameter collection of a learned model, set italic (never bold), as the series core defines it. A geometric angle always carries a subscript that names it: \(\JointAngle\) for one joint (\(\Glyph{\theta_{\text{motor}}}\) and \(\Glyph{\theta_{\text{link}}}\) on either side of a reducer, \(\EncoderAngle\) for an encoder reading), \(\HeadingAngle\) for planar heading, \(\AzimuthAngle\) for azimuth, \(\AttitudeError\) for attitude error, \(\ExtrinsicMisalignmentAngle\) for extrinsic misalignment, \(\TiltAngle\) for tilt from vertical, \(\OrientationTolerance\) for an orientation tolerance, and \(\ElectricalRotorAngle\) for the electrical rotor angle; increments keep the subscript (\(\FiniteChange\JointAngle\), \(\SmallChange\EncoderAngle\)). Joint configurations are the generalized coordinates \(\GeneralizedCoordinates \in \mathbb{R}^\GeneralizedCoordinateCount\) with entries \(\JointCoordinate_i\), and a joint trajectory polynomial is \(\JointCoordinate(\TimeInstant) = \sum \PolynomialTrajectoryCoefficient_k \TimeInstant^{\Glyph{k}}\) with boundary values \(\JointCoordinate_0, \JointCoordinate_1\) (never \(\Glyph{\theta}(\Glyph{t})\)). A clock offset is \(\ClockOffset\) (in seconds). Thermal resistance is \(\ThermalResistance\) or \(\JunctionAmbientThermalResistance\) (in \(\text{K/W}\)), never \(\Glyph{\theta_{JA}}\).
\(\Glyph{\mathbf{a}}\) vs. \(\NominalControlCommand\) vs. \(\CertifiedControlAction\) vs. \(\JointTorque\)	Four-stage authority chain (\(\CognitivePolicyAction_{t} \to \NominalControlCommand \to \CertifiedControlAction \to \JointTorque\))	Generic “action” symbol colliding with deceleration \(\CredibleBrakingDeceleration\) or a barrier constraint row	Disambiguate execution authority: \(\CognitivePolicyAction_{t}\) is the unprivileged policy action (dimensionless/normalized); \(\NominalControlCommand\) is the proposed plant control setpoint; \(\CertifiedControlAction \in \AdmissibleControlSet\) is the certified, CBF-filtered control command; \(\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount\) is realized physical actuator torque. Control commands are bold \(\ControlCommand\) (with time index \(\ControlCommand_t\)). Credible linear deceleration is \(\CredibleBrakingDeceleration\) (\(\text{m/s}^2\)); stacked barrier constraints are \(\Glyph{\mathbf{A}_{\text{cbf}}}(\MechanicalPlantState)\ControlCommand \le \Glyph{\mathbf{b}_{\text{cbf}}}(\MechanicalPlantState)\), and a single constraint row is \(\BarrierConstraintNormal(\MechanicalPlantState)^\top\ControlCommand \le \BarrierConstraintBound(\MechanicalPlantState)\).
\(\JointTorque\) vs. \(\Glyph{\tau}\)	Actuator torque vector \(\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount\) (\(\text{N}\cdot\text{m}\) or \(\text{N}\))	Time constants (\(\ElectricalTimeConstant\), \(\ThermalTimeConstant\)), delays (\(\PrebrakeDelay\), \(\ZohTransportDelay\)), observation age, lease durations, or real-time task names	Bare or bold \(\Glyph{\tau}\) is always torque (\(\JointTorque \in \mathbb{R}^\GeneralizedCoordinateCount\), scalar joint torque \(\CommandedJointTorque_i\), component torques such as \(\MotorTorque\), \(\HoldingTorque\), \(\Glyph{\tau_{\text{bias}}}\)), with \(\MotorTorque = \MotorTorqueConstant \TorqueProducingCurrent\) at the motor. A subscripted Greek \(\Glyph{\tau}\) marks a time only for these named constants and delays: \(\PrebrakeDelay\) (pre-brake delay, excluding \(\DispatchAdmissionInterval\)) and its ceiling \(\PrebrakeDelayCeiling\), \(\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance\) (electrical), \(\ThermalTimeConstant = \ThermalResistance \StatorThermalCapacitance\) (thermal, with stage constants such as \(\CopperThermalTimeConstant\) and \(\CaseThermalTimeConstant\)), \(\Glyph{\tau_{\text{fall}}}\) (balance divergence), \(\ZohTransportDelay\) (sample-and-hold delay), \(\EvidenceHorizon\) (evidence horizon), and \(\BlendingTransitionWindow\) (authority blend window). Every other interval uses Latin \(\Glyph{T}\) or \(\Glyph{t}\): observation age is \(\ObservationAge\), never \(\Glyph{\tau}\); leases are \(\ChunkLease\), \(\ManualLease\), and \(\IntentAdministrativeLease\); compute and fieldbus times are \(\EnforcerComputeTime\) and \(\CommandTransportTime\); a pure transport delay is \(\PureTransportDelay\). A real-time task is named “task \(\Glyph{i}\)”, never \(\Glyph{\tau}_i\).
\(\MechanicalPlantState\) vs. \(\Glyph{s}\) vs. \(\CartesianPosition\) vs. \(\VitImagePatch\)	Mechanical plant state \(\MechanicalPlantState = [\GeneralizedCoordinates^\top, \GeneralizedVelocities^\top]^\top \in \mathbb{R}^{2\GeneralizedCoordinateCount}\)	World state \(\Glyph{s} \in \Glyph{\mathcal{S}}\), 3D position \(\CartesianPosition = [\CartesianXCoordinate, \CartesianYCoordinate, \CartesianZCoordinate]^\top\), image point, or ViT image patch \(\VitImagePatch\)	Continuous mechanical plant state is bold vector \(\MechanicalPlantState \in \mathbb{R}^{2\GeneralizedCoordinateCount}\). Ground-truth latent world state is italic \(\PhysicalWorldState_{t}\) (a mixed-type state, never bold). 3D Cartesian position is vector \(\CartesianPosition \in \mathbb{R}^3\), including a camera-frame point \(\CameraFramePoint\) and a voxel center \(\VoxelCenterPosition\); the translation of a rigid transform is \(\RigidTransformTranslation\). An image point in homogeneous pixel coordinates is \(\HomogeneousPixelCoordinates = [\PixelColumnCoordinate, \PixelRowCoordinate, 1]^\top\), never \(\Glyph{\mathbf{x}}\) or \(\Glyph{\mathbf{u}}\). Vision transformer patch embeddings use explicitly subscripted \(\VitImagePatch\) or latent feature \(\VitPatchTokenEmbedding\).
\(\Parameters\)	Model parameter count \(\Parameters\) (series core)	Power (\(\WindingLossPower, \Glyph{P_{\text{peak}}}\)), estimation covariance \(\EstimationErrorCovariance\), failure probability or rate, tail-latency percentile, or 3D point \(\CartesianPosition\)	Use \(\Parameters\) for parameter count. Electrical/mechanical power always carries a subscript (\(\WindingLossPower, \Glyph{P_{\text{peak}}}, \HeatDissipationPower\), in Watts); the estimator’s error covariance is bold matrix \(\EstimationErrorCovariance\), and a generic Gaussian covariance is \(\GaussianCovarianceMatrix\); a per-trial probability is lowercase italic \(\Glyph{p}\) (\(\TrialFailureProbability, \TargetTrialSuccessProbability\)), a rate per hour is \(\Glyph{\lambda}\) (\(\SystemHazardRate, \TargetHazardRate\)), and an event probability is \(\Pr(\cdot)\); percentiles are upright p50, p95, and p99 (never \(\Glyph{P}_{99}\)); 3D Cartesian position is bold lowercase \(\CartesianPosition = [\CartesianXCoordinate, \CartesianYCoordinate, \CartesianZCoordinate]^\top\).
\(\BatchSize\)	Batch size \(\BatchSize\) (series core)	Bandwidth (\(\Bandwidth\)), Body frame (\(\BodyReferenceFrame\)), bus transfer size, SPRT boundary, or input matrix \(\StateSpaceInputMatrix\)	Batch size is \(\BatchSize\); sustained bandwidth is \(\Bandwidth\) (per the single-device decomposition), with a subscript naming the path (\(\SustainedMemoryBandwidth\), \(\Glyph{\Bandwidth_{\text{chirp}}}\)); coordinate reference frames use calligraphic script (\(\WorldReferenceFrame, \BodyReferenceFrame, \Glyph{\mathcal{F}_E}\)); the bytes of bus transaction \(\Glyph{k}\) are \({\DataVolume}_{\Glyph{k}}\); SPRT boundaries are \(\SprtUpperBoundary\) and \(\SprtLowerBoundary\); state-space control input matrix is bold capital \(\StateSpaceInputMatrix \in \mathbb{R}^{\StateDimension \times \ControlInputDimension}\).
\(\BumplessTransferBlend\)	Bumpless transfer blending function \(\BumplessTransferBlend(\BumplessTransferProgress) \in [0, 1]\)	CBF decay (\(\LinearBarrierDecayRate\)), significance (\(\HypothesisTestSignificanceLevel\)), angular acceleration (\(\AngularAcceleration\)), copper tempco (\(\CopperTemperatureCoefficient\)), or noise schedule (\(\DiffusionCumulativeSchedule_{k}\))	Authority transfer uses scalar function \(\BumplessTransferBlend(\BumplessTransferProgress) \in [0, 1]\) of normalized time \(\BumplessTransferProgress \in [0, 1]\); barrier linear decay uses \(\LinearBarrierDecayRate > 0\); statistical confidence significance uses \(\HypothesisTestSignificanceLevel = 1 - \text{CL}\); angular acceleration is bold vector \(\AngularAcceleration \in \mathbb{R}^3\) (\(\text{rad/s}^2\)); copper resistance tempco is \(\CopperTemperatureCoefficient\); diffusion noise schedules use indexed \(\DiffusionCumulativeSchedule_{k}\).
\(\CoriolisCentrifugalMatrix\)	Coriolis matrix \(\CoriolisCentrifugalMatrix(\GeneralizedCoordinates, \GeneralizedVelocities) \in \mathbb{R}^{\GeneralizedCoordinateCount \times \GeneralizedCoordinateCount}\)	Safe set \(\ForwardInvariantSafeSet\), conformal prediction set, assurance claim, state-space output matrix, controllability matrix, capacitance (\(\StatorThermalCapacitance, \DcBusBulkCapacitance\)), CoM \(\MassCenterPosition\), or shield coverage \(\ShieldInterceptionCoverage\)	Coriolis/centrifugal forces use matrix function \(\CoriolisCentrifugalMatrix(\GeneralizedCoordinates, \GeneralizedVelocities)\), always written with its arguments; forward-invariant safe sets use calligraphic \(\ForwardInvariantSafeSet \subset \mathbb{R}^{2\GeneralizedCoordinateCount}\), defined on the plant state, and the conformal prediction set \(\ConformalPredictionSet\) of target grounding shares that glyph but never appears beside a safe set; the top-level claim of an assurance argument is italic \(\AssuranceClaim\); the state-space output map is \(\MeasurementMatrix\) (the same map as the Kalman measurement map), never \(\Glyph{\mathbf{C}}\); the controllability matrix is \(\ControllabilityMatrix\); electrical/thermal capacitance uses italic \(\Glyph{C}\) with a subscript; center of mass is bold vector \(\MassCenterPosition \in \mathbb{R}^3\); safety shield coverage of hazard class \(\HazardClassIndex\) uses \(\ShieldInterceptionCoverage \in [0, 1]\).
\(\ActionChunk\)	Action chunk \(\ActionChunk_t\), \(\Glyph{\mathbf{A}_{t:t+H}}\), and its denoising iterate \(\ActionChunk_t^\DiffusionStepIndex\)	Continuous LTI state matrix, barrier constraint matrix	Bold \(\ActionChunk\) with a time or denoising index is the action chunk. The continuous LTI state matrix is \(\StateSpaceSystemMatrix\) (its discretizations \(\ForwardEulerTransitionMatrix\) and \(\BackwardEulerTransitionMatrix\), the discrete Kalman transition \(\StateTransitionMatrix\)); stacked barrier constraints use \(\Glyph{\mathbf{A}_{\text{cbf}}}\).
\(\Velocity\)	Velocity (\(\Velocity\), end-effector velocity \(\Glyph{\mathbf{v}_e}\))	Measurement noise, voxel	Bold \(\Velocity\) is a velocity. Measurement noise is \(\MeasurementNoiseVector\) with covariance \(\MeasurementNoiseCovariance\) (bold \(\Glyph{\mathbf{R}}\) is reserved for rotation); a voxel is addressed by its center \(\VoxelCenterPosition\).
\(\CameraIntrinsicCalibrationMatrix\)	Camera intrinsic calibration matrix \(\CameraIntrinsicCalibrationMatrix \in \mathbb{R}^{3 \times 3}\)	Contact stiffness (\(\ContactStiffness\)), control gains (\(\PidProportionalGain, \PidDerivativeGain\), LQR gain), motor constants (\(\MotorTorqueConstant, \MotorBackEmfConstant\)), ensembling window \(\EnsemblingWindowLength\), or diffusion step \(\DiffusionStepIndex\)	Camera intrinsics use bold matrix \(\CameraIntrinsicCalibrationMatrix\); contact stiffness is \(\ContactStiffness\), with body-region stiffness \(\HertzianContactStiffness\) (\(\text{N/m}\)); feedback control gains use capitalized scalars \(\PidProportionalGain, \PidDerivativeGain\), and a state-feedback gain matrix is \(\LqrFeedbackGain\); motor torque/back-EMF constants use \(\MotorTorqueConstant, \MotorBackEmfConstant\) (per joint \({\MotorTorqueConstant}_{i}\)); temporal ensembling window length uses integer \(\EnsemblingWindowLength\); the number of denoising or solver evaluations per chunk is \(\DecoderEvaluationCount\), never \(\Glyph{K}\); diffusion noise steps use lowercase index \(\DiffusionStepIndex\); the Kalman gain is \(\KalmanGain_{k}\).
\(\GeneralizedInertiaMatrix\)	Articulated inertia matrix \(\GeneralizedInertiaMatrix(\GeneralizedCoordinates) \in \mathbb{R}^{\GeneralizedCoordinateCount \times \GeneralizedCoordinateCount}\)	Mass (\(\Mass, \EffectiveImpactMass\)), memory footprint (\(\WeightMemory, \KvCacheMemory\)), or fleet size (\(\FleetRobotCount\))	Articulated generalized mass/inertia uses bold matrix \(\GeneralizedInertiaMatrix(\GeneralizedCoordinates)\); physical body mass is lowercase italic \(\Mass\) (\(\text{kg}\)); effective impact mass is \(\EffectiveImpactMass\); hardware memory storage uses capital \(\Glyph{M}\) with descriptive subscript (in bytes); robot fleet size uses \(\FleetRobotCount\).
\(\Time\)	Time (series core)	Temperature	Bare \(\Glyph{T}\) is always time. A temperature always carries a subscript that names where it is measured: \(\WindingTemperature\) (winding), \(\StatorCoreTemperature\), \(\JunctionTemperature\) (junction), \(\CaseTemperature\), \(\AmbientTemperature\) (ambient), \(\WindingTemperatureLimit\) (rated limit), \(\ThrottleTripTemperature\), \(\EquilibriumWindingTemperature\) (equilibrium), and \(\CopperReferenceTemperature\) (the reference temperature of \(\CopperTemperatureCoefficient\)). A temperature rise is \(\JouleanTemperatureRise\) (in \(\text{K}\)); a time interval is \(\FiniteChange \TimeInstant\), never \(\Glyph{\Delta} \Glyph{T}\).
SIL (abbreviation)	Safety integrity level (IEC 61508 family), a grade of a safety function	Software-in-the-loop, the first rung of the verification ladder	Spell out at first use in each chapter. “SIL 2” and similar graded forms always mean safety integrity level; the verification ladder writes software-in-the-loop (SIL) only where the ladder’s abbreviations MIL, SIL, PIL, and HIL appear together, and never beside a safety integrity level.

Symbol	Definition	Unit	Notes
\(\PrebrakeDelay\)	Pre-Brake Delay	\(\text{ms}\)	Time from capture of the frame that first shows a hazard to the onset of retarding force, excluding \(\DispatchAdmissionInterval\); the machine travels \(\SpeedAtDetection\,(\PrebrakeDelay + \DispatchAdmissionInterval)\) before braking begins. Its allowed ceiling is \(\PrebrakeDelayCeiling\).
\(\ObservationAge\)	Observation Age	\(\text{ms}\)	Age of the state the permission path acts on, from the exposure midpoint at the sensor to the decision, which enters \(\PrebrakeDelay\). Its perception-side terms are the exposure midpoint \(\tfrac{1}{2}\ExposureDuration\), sensor readout, transfer, and perception processing; a bound on it is \(\Glyph{t_{\text{age},\max}}\).
\(\ChunkLease\)	Chunk Lease	\(\text{ms}\)	Longest interval an admitted chunk may keep driving the actuators without renewal, which enters \(\PrebrakeDelay\); chosen, then checked against the ceiling the stopping budget derives.
\(\ChunkRenewalPeriod\)	Chunk Renewal Period	\(\text{ms}\)	Interval between successful admissions of the chunk policy; with \(\RenewalJitterAllowance\) it sets the floor of the lease rule, \(\ChunkRenewalPeriod + \RenewalJitterAllowance < \ChunkLease\) (equation 4.4).
\(\RenewalJitterAllowance\)	Renewal-Jitter Allowance	\(\text{ms}\)	Allowance for variation in the interval between successful admissions; chosen, then checked against measured renewal intervals.
\(\PermissionLoopPeriod\)	Permission-Loop Period	\(\text{ms}\)	Period of the permission loop, which enters \(\PrebrakeDelay\) because a hazard may wait up to one period before it is checked.
\(\CommandTransportTime\)	Command Transport Time	\(\text{ms}\)	Fieldbus delivery time from the permission path to the drive, which enters \(\PrebrakeDelay\).
\(\ActuatorResponseTime\)	Actuator Response Time	\(\text{ms}\)	Time from a delivered stop command to retarding force, including gate-drive and current rise, which enters \(\PrebrakeDelay\).
\(\DetectionInterval\)	Detection Interval	\(\text{ms}\)	Recognition or expiry-detection time after the decision-start epoch, not already charged to \(\ObservationAge\); a term of the lease-expiry and hazard-detection forms of \(\PrebrakeDelay\).
\(\FallbackHold\)	Fallback Hold	\(\text{ms}\)	Optional, separately authorized hold or fault-confirmation interval before the stop command reaches the drive; zero when the stop path has none.
\(\ProposerInferenceTime\)	Proposer Inference Time	\(\text{ms}\)	Time for one complete evaluation of the learned proposer (prefill and decode); spent above the proposal boundary and outside the permission loop’s period.
\(\DispatchAdmissionInterval\)	Dispatch-to-Admission Interval	\(\text{ms}\)	Elapsed time from perception dispatch to the admission of the candidate it conditions; evidence keeps aging through it. Not a term of \(\PrebrakeDelay\); its travel \(\SpeedAtDetection\,\DispatchAdmissionInterval\) is charged against the stopping budget’s timing allowance.
\(\TakeoverResponseTime, \DisengagedTakeoverResponseTime\)	Takeover Response Time	\(\text{ms}, \text{s}\)	Human response plus console transport for an in-loop (\(\TakeoverResponseTime\)) and an out-of-loop (\(\DisengagedTakeoverResponseTime\)) takeover; it adds to \(\PrebrakeDelay\) while the machine waits for the person.
\(\CompletionDeadline\)	Deadline	\(\text{ms}, \text{s}\)	A duration budget a computation or handover must meet (the series symbol); qualifiers name the instance, as in \(\FallbackDeadline\), the latest permissible wait before the fallback must begin. A deadline stated as a clock instant is lowercase, \(\DeadlineInstant\) or \(\Glyph{t_{\text{deadline,stop}}}\).
\(\HandoverTimeout\)	Handover Timeout	\(\text{s}\)	Timer an arbiter sets for an unacknowledged handover, \(\HandoverTimeout = \max(0, \min(\HandoverTimeoutCeiling, \FallbackDeadline))\), where \(\HandoverTimeoutCeiling\) is the protocol ceiling.
\(\CommitmentInstant\)	Commitment Instant	\(\text{s}\)	Instant after which a receding-horizon plan’s prefix is committed and can no longer be replaced by a new proposal.
\(\HazardOnset\)	Hazard Onset	\(\text{s}\)	Instant at which a hazard first becomes physically present, the reference from which detection and stopping intervals are measured.

Symbol	Definition	Unit	Notes
\(\StoppingDistance\)	Stopping Distance	\(\text{m}\)	Distance required to bring the plant to rest from speed \(\SpeedAtDetection\).
\(\StoppingDuration\)	Stopping Duration	\(\text{s}\)	Elapsed time from braking onset to standstill in the protective-separation calculation; distinct from controller sampling period.
\(\ClearDistance\)	Clear Distance	\(\text{m}\)	Clear distance from the machine to the boundary it must not cross; the subscript keeps it distinct from dataset size \(\Glyph{D}\).
\(\SpeedAtDetection\)	Speed at Detection	\(\text{m/s}\)	Speed of the machine at the moment the hazard is detected; in stopping, pre-brake, and impact algebra, the scalar travel or approach speed \(\SpeedAtDetection\).
\(\SpeedBound\)	Speed Bound	\(\text{m/s}\)	Largest speed the machine, a tracked object, or an approaching body is credited with in a budget; the bounding speed in \(\TelemetryFreshnessBound \le \PositionErrorCeiling/\SpeedBound\).
\(\CredibleBrakingDeceleration\)	Credible Deceleration	\(\text{m/s}^2\)	Validated low tail of the deceleration the machine achieves under its stated conditions, not the datasheet peak.
\(\ProfileAccelerationBound\)	Profile Acceleration Bound	\(\text{m/s}^2\)	Largest acceleration or deceleration a planned or interpolated profile may command; for the base’s resident stop it is set to \(\CredibleBrakingDeceleration\).
\(\NetBrakingDeceleration\)	Net Braking Deceleration	\(\text{m/s}^2, \text{rad/s}^2\)	Linear or angular deceleration left after a gravity or load bias torque \(\Glyph{\tau_{\text{bias}}}\) is subtracted from the actuator’s braking capacity, in the direction of travel.
\(\ResidentStopDistance(\SpeedAtDetection)\)	Resident Stop Distance	\(\text{m}\)	Travel of the resident stop from speed \(\SpeedAtDetection\): \(\SpeedAtDetection^2/(2\CredibleBrakingDeceleration)\) for constant-deceleration braking; for the jerk-limited stop of equation 11.17, that term plus the jerk ramps’ \(\SpeedAtDetection\,\CredibleBrakingDeceleration/(2\JerkLimit)\) once \(\SpeedAtDetection \ge \CredibleBrakingDeceleration^2/\JerkLimit\).
\(\LocalizationBound\)	Localization Bound	\(\text{m}\)	Bound on localization and state-estimation error.
\(\ProtectiveClearanceMargin\)	Protective Clearance	\(\text{m}\)	Fixed, nonzero protective clearance between the machine envelope and the boundary.
\(\TrackingErrorBound\)	Tracking-Error Bound	\(\text{m}\)	Bound on the deviation between the commanded and the executed trajectory, stated by the enforcement record.
\(\HumanWalkingSpeedBound\)	Human Walking-Speed Bound	\(\text{m/s}\)	Walking-speed bound for a person approaching the machine, used in the form its safety standard states.
\(\JerkLimit\)	Jerk Limit	\(\text{m/s}^3\)	Maximum rate of change of deceleration; sets the jerk ramps of the resident stop in \(\ResidentStopDistance(\SpeedAtDetection)\) (equation 11.17) and bounds the bumpless-transfer window \(\BlendingTransitionWindow\).

Symbol	Definition	Unit	Notes
\(\ZohTransportDelay\)	ZOH Transport Delay	\(\text{s}\)	Equivalent continuous delay introduced by discrete Zero-Order Hold reconstruction (\(\SamplingPeriod / 2\)).
\(\SamplingPeriod, \SamplingRate\)	Sampling Period and Rate	\(\text{s}, \text{Hz}\)	Discrete controller execution period and frequency (\(\SamplingRate = 1/\SamplingPeriod\)).
\(\EnforcerComputeTime\)	Enforcer Compute Time	\(\text{s}\)	Time the enforcer takes to evaluate its constraints within one control period; with \(\CommandTransportTime\) it adds to the ZOH delay.
\(\PureTransportDelay\)	Pure Transport Delay	\(\text{s}\)	A fixed delay in a feedback or communication channel, contributing phase lag \(\AngularFrequency \PureTransportDelay\) at frequency \(\AngularFrequency\).
\(\CrossoverFrequency\)	Open-Loop Crossover Frequency	\(\text{rad/s}\)	Frequency at which open-loop gain \(\lvert \OpenLoopTransferFunction(\ImaginaryUnit\CrossoverFrequency) \rvert = 1\) (\(0\text{ dB}\)).
\(\DelayPhaseLag\)	Phase Lag Degradation	\(\text{rad}\) / \(\text{deg}\)	Phase erosion at crossover (\(\DelayPhaseLag \approx 28.65^\circ \cdot (\CrossoverFrequency / \SamplingRate)\) from the hold alone) depleting closed-loop stability margin.

Symbol	Definition	Unit	Notes
\(\MotorTorqueConstant, \MotorBackEmfConstant\)	Torque and Back-EMF Constants	\(\text{N}\cdot\text{m/A}, \text{V}\cdot\text{s/rad}\)	Motor electromechanical coupling constants linking winding current to rotor shaft torque and rotor speed to back-EMF voltage (\(\MotorTorqueConstant \approx \MotorBackEmfConstant\) in SI units).
\(\TorqueProducingCurrent\)	Torque-Producing Current	\(\text{A}\)	Quadrature-axis current of a field-oriented drive, \(\MotorTorque = \MotorTorqueConstant \TorqueProducingCurrent\). Bare \(\Glyph{I}\) is never a current, because the series core reserves it for arithmetic intensity.
\(\MotorPhaseRmsCurrent\)	Motor Phase RMS Current	\(\text{A}\)	RMS phase current driving the stator windings of a brushless PM motor; an uppercase \(\Glyph{I}\) with a subscript names an RMS value or rating.
\(\InstantaneousPhaseCurrent\)	Instantaneous Phase Current	\(\text{A}\)	Signed instantaneous electrical phase current, positive from inverter node into winding; distinct from RMS phase current and torque-producing dq current.
\(\ContinuousCurrent, \PeakCurrent\)	Continuous / Peak Current	\(\text{A}\)	RMS current ratings bounded by winding insulation class and inverter MOSFET dissipation.
\(\PhaseInductance, \PhaseResistance\)	Phase Inductance and Resistance	\(\text{H}, \Glyph{\Omega}\)	Stator winding inductance and DC resistance; they set \(\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance\).
\(\ElectricalResistance(\WindingTemperature)\)	Temperature-Dependent Resistance	\(\Glyph{\Omega}\)	Stator copper resistance at winding temperature \(\WindingTemperature\): \(\ElectricalResistance(\WindingTemperature) = \ReferenceWindingResistance [1 + \CopperTemperatureCoefficient(\WindingTemperature - \CopperReferenceTemperature)]\) with \(\CopperTemperatureCoefficient = 0.00393\text{ K}^{-1}\).
\(\ThermalResistance\)	Thermal Resistance	\(\text{K/W}\)	Lumped thermal resistance from stator windings to the ambient environment; path segments carry a second subscript (\(\JunctionAmbientThermalResistance\), \(\JunctionCaseThermalResistance\), \(\WindingCoreThermalResistance\), \(\CoreAmbientThermalResistance\)) and add in series.
\(\StatorThermalCapacitance\)	Stator Thermal Capacitance	\(\text{J/K}\)	Lumped heat capacity of the motor core, windings, and potting compound.
\(\ElectricalTimeConstant\)	Electrical Time Constant	\(\text{s}\)	Winding current rise time constant \(\ElectricalTimeConstant = \PhaseInductance/\PhaseResistance\); current reaches \(1 - e^{-1}\) of its target in \(\ElectricalTimeConstant\).
\(\ThermalTimeConstant\)	Thermal Time Constant	\(\text{s}\)	Actuator heating characteristic time constant (\(\ThermalTimeConstant = \ThermalResistance \StatorThermalCapacitance\), typically \(60\text{--}300\text{ s}\)); a single stage takes a second subscript (\(\CopperThermalTimeConstant\), \(\CaseThermalTimeConstant\)).
\(\WindingTemperature, \AmbientTemperature\)	Winding and Ambient Temperature	\(\text{K}\) or \(^\circ\text{C}\)	The thermal model’s state and boundary temperatures; the other temperatures follow the subscript rule of the disambiguation table (\(\JunctionTemperature\), \(\CaseTemperature\), \(\WindingTemperatureLimit\), \(\EquilibriumWindingTemperature\), \(\CopperReferenceTemperature\)).
\(\JouleanTemperatureRise\)	Joulean Temperature Rise	\(\text{K}\)	Thermal rise above ambient; \(\JouleanTemperatureRise = \WindingLossPower \ThermalResistance\) gives the steady-state rise for constant loss power in the lumped model. Transient rise follows the thermal balance; never a time interval.
\(\ActuatorDutyCycle\)	Actuator Duty Cycle	\([0, 1]\)	Ratio of active pulse duration to cycle period (\(\ActuatorOnTime / (\ActuatorOnTime + \ActuatorOffTime)\)) preventing thermal runaway.