Actuation Uncertainty

Definition

Actuation uncertainty is the discrepancy between the commanded force/torque and the realized one: the plant is driven by an input the controller never exactly specified. It is distinct from state (estimation) uncertainty, which corrupts what the controller knows, and from parametric uncertainty, which corrupts the model it uses — actuation uncertainty corrupts what the actuators do. In a planning/control stack it appears in three separable questions, and the literature answers them with three different objects:

1. Model — what distribution does a commanded input induce over the realized one? The classical spaceflight answer is the maneuver execution-error covariance: a commanded velocity increment is dispersed by a magnitude error and a pointing error, giving a Gaussian dispersion around the nominal burn (Chioma & Titus 2008, below). The classical robotics/control answer is an additive input disturbance, ${\mathbf{\delta }}_u$, or — where the mechanism is a hard limit rather than noise — a multiplicative deficit, which is the actuator saturation model.
2. Propagation — how does that input distribution become a state distribution? Through covariance propagation along the (linearized) dynamics, the input covariance enters the state covariance through the input map.
3. Budget — what does the planner do with the resulting state dispersion? It tightens the constraints it plans against, so that the nominal trajectory keeps a margin large enough that the dispersed trajectory still satisfies them with the required probability. This is exactly the chance constraint machinery: the tightening margin is the derate. The alternative to a principled tightening is a hand-tuned safety margin, and the covariance-steering literature makes replacing that heuristic its explicit selling point (Kumagai & Oguri 2025, below).

Regime matters. A free-floating base has no base actuators at all, so “actuation uncertainty” there lives entirely in the joint torques and is filtered through momentum conservation (see free-flying vs free-floating). Our free-flying base is fully actuated in 6 DOF, so the base wrench is a decision variable and a noise entry point — the same channel carries the command and the error. Every source below is annotated with the regime it assumes.

Our instantiation. The project’s fragility grid injects actuation uncertainty in the simplest of the forms above: additive zero-mean Gaussian force noise on the base command, the config key uncertainty.actuation.force_std (Inspection/validation/fragility_study.py), swept at 0.5 / 2.0 / 5.0 N against a mean command scale of order 16 N. That is a magnitude choice; this page is the modelling and budgeting complement to it — the precedent for the choice of form, not of size.

Key Equations

Symbols per notation.md. The stacked physical input $\mathbf{u}=[{\mathbf{f}}_b;{\mathbf{\tau }}_b;\mathbf{\tau }]$ (base force, base torque, joint torques — all three individually registered) and the input-error symbol ${\mathbf{\delta }}_u$ with covariance ${\mathbf{\Sigma }}_u$ are not yet in notation.md; they are used here in the generic stochastic-control sense, and the quantile multiplier $\Gamma$ is the same one as in chance constraints (source notation, distinct from the repo’s coordinate transform $\mathbf{\Gamma }$).

Additive execution-error model — the realized input is the commanded input plus a zero-mean draw:

$${\mathbf{u}}_{\text{realized}}=\mathbf{u}+{\mathbf{\delta }}_u{\mathbf{\delta }}_u\sim \mathcal{N}(0,{\mathbf{\Sigma }}_u)$$

This is the form the fragility grid injects (diagonal ${\mathbf{\Sigma }}_u$ on the base force block). The execution-error family (Chioma & Titus 2008) is richer: it makes the dispersion a function of the command itself, with separate magnitude and pointing (direction) contributions, so ${\mathbf{\Sigma }}_u$ is neither constant nor isotropic but scales with, and is oriented by, the commanded impulse. That structure is not reproduced here because the source has not been read in full — see the flag under Source Support.

Propagation into the state — for a linearization with state matrix $\mathbf{A}$ and input map $\mathbf{B}$, the input covariance enters the state covariance additively:

$${\mathbf{\Sigma }}_{k+1}=\mathbf{A}{\mathbf{\Sigma }}_k{\mathbf{A}}^{\top }+\mathbf{B}{\mathbf{\Sigma }}_u{\mathbf{B}}^{\top }$$

The input map $\mathbf{B}$ is where the regime enters: for a fully actuated base it has full row rank on the base block, so base actuation noise is not attenuated by any momentum constraint — it passes straight into the state.

The budget (constraint tightening) — a linear constraint ${\mathbf{a}}^{\top }\mathbf{x}\le b$ imposed with confidence $1-ϵ$ under the propagated $\mathbf{\Sigma }$ becomes the deterministic, tightened constraint

$${\mathbf{a}}^{\top }\mathbf{x}+\Gamma \sqrt{{\mathbf{a}}^{\top }\mathbf{\Sigma }\mathbf{a}}\le b\Gamma ={\Psi }^{-1}(1-ϵ)$$

with ${\Psi }^{-1}$ the standard-normal inverse CDF (chance constraints, full statement and its CVaR inner approximation there — not repeated). The term $\Gamma \sqrt{{\mathbf{a}}^{\top }\mathbf{\Sigma }\mathbf{a}}$ is the derate: the share of it attributable to ${\mathbf{\Sigma }}_u$ is the actuation-uncertainty budget, and it is a computed quantity, not a tuned one.

Source Support

In the corpus (verified bibkeys)

• ekal2021online — free-flying (Astrobee, fully actuated). The corpus’s closest free-flyer-under-uncertainty anchor. It treats inertial-parameter uncertainty rather than actuation uncertainty, but it establishes the coupling that motivates this page: the paper’s own argument is that Astrobee’s severe actuation limit ($u_{\max }\le 0.4$ N) makes an accurate model safety-critical, i.e. a tight actuation budget is what makes every other uncertainty bite. Its EKF + information-aware planner is the template a realized-wrench error model would slot into.
• yao2021adaptive — free-flying space manipulator. The deterministic end of the same axis: the saturation deficit $\Delta \mathbf{\tau }=(\Theta (\mathbf{\tau })-\mathbf{I})\mathbf{\tau }$ is a realized-minus-commanded torque residual, lumped with the disturbance and absorbed adaptively. Bounded-but-unknown, not distributed — see actuator saturation.
• ren2022chance — regime-agnostic. Supplies the SOC reformulation and the risk-allocation idea (${\sum }_k{\pi }_k{ϵ}_k=ϵ$) that the tightening above rests on. The uncertainty it propagates is generic; nothing in it forbids $\mathbf{\Sigma }$ carrying an actuation component.
• nemirovski2006convex — regime-agnostic. Establishes that the CVaR constraint is the tightest convex conservative approximation of the chance constraint, which bounds how much conservatism any tractable actuation budget must pay.
• ono2015chance — spaceflight (EDL/landing, not a manipulator). The space-robotics precedent for a joint chance constraint over a whole trajectory — the shape our inspection-trajectory budget would take.
• akella2024risk — survey; places chance constraints and tail-risk measures in the worst-case / risk-neutral / risk-aware taxonomy that the precedents below populate.
• hakobyan2019risk — CVaR-constrained motion planning; the closest thing the corpus has to a coherent-risk budget, though the uncertainty it constrains is disturbance/obstacle uncertainty, not actuation.

Precedents identified but not yet in the corpus (no bibkey — do not cite as \cite{})

Ranked and verified in notes/actuation_uncertainty_precedents.md (Semantic Scholar metadata; none has a references.bib entry as of 2026-07-11, so each is named by title + DOI here):

• Nakka & Chung 2021, “Trajectory Optimization of Chance-Constrained Nonlinear Stochastic Systems for Motion Planning Under Uncertainty,” IEEE T-RO, DOI 10.1109/TRO.2022.3197072, arXiv:2106.02801 — free-flying (3-DOF robotic-spacecraft testbed). gPC-SCP propagates uncertainty in actuation and physical parameters into a chance constraint. The closest published match to our setting; first in the ingestion queue.
• Kumagai & Oguri 2025, “Robust Cislunar Low-Thrust Trajectory Optimization Under Uncertainties via Sequential Covariance Steering,” JGCD, DOI 10.2514/1.g009092, arXiv:2502.01907 — spacecraft trajectory (no manipulator). Folds maneuver execution errors into chance-constrained covariance steering and argues this replaces heuristic safety margins — the explicit statement of the “computed derate, not tuned margin” position taken above.
• Chioma & Titus 2008, “Expected Maneuver and Maneuver Covariance Models,” AIAA, DOI 10.2514/1.31154 — spacecraft trajectory. The canonical magnitude-plus-pointing execution-error covariance for a commanded $\Delta v$; grounds the vocabulary. Not read in full — hence no formula transcribed above.
• Hou, Shen & Zhuang 2026, “Stochastic Fault-Tolerant Attitude Tracking and Control Allocation of Overactuated Spacecraft,” JGCD, DOI 10.2514/1.g009078 — overactuated spacecraft attitude (no manipulator). Recasts probabilistic actuator constraints as deterministic QCQP equivalents inside control allocation: the derate-under-degradation end of the axis, where the uncertainty is in the fault estimate itself.
• Zhu et al. 2018, “Robust model predictive control for multi-step short range spacecraft rendezvous,” Adv. Space Research, DOI 10.1016/J.ASR.2018.03.037 — rendezvous (thrust-actuated, no manipulator). A worked chance-constrained-MPC example that tightens using the known statistics of thrust uncertainty.
• Lew, Bonalli & Pavone 2020, “Chance-Constrained Sequential Convex Programming for Robust Trajectory Optimization,” ECC, DOI 10.23919/ECC51009.2020.9143595 — uncertain 6-DOF spacecraft. Constraint tightening from propagated uncertainty inside SCP; pairs directly with sequential convex programming.
• Yang, Liu & Gao 2025, “Reliability-Constrained Uncertain Spacecraft Sliding Mode Attitude Tracking Control With Interval Parameters,” IEEE TAES, DOI 10.1109/TAES.2025.3529798 — spacecraft attitude. The non-Gaussian alternative: actuator/parameter uncertainty as interval bounds with a time-varying reliability constraint (cf. interval arithmetic).
• Astrobee platform papers — Barlow et al. 2016 (i-SAIRAS, NTRS 20160007769, no DOI); Smith et al. 2026, IEEE T-FR, DOI 10.1109/TFR.2026.3664810; Bualat et al. 2018, AIAA SpaceOps, DOI 10.2514/6.2018-2517. Free-flying. The actuation-authority anchors for the benchmark vehicle class (impeller + steerable nozzles, per-axis control-force authority) — the hardware side of any realized-wrench error model.

Related Topics

• Chance Constraints — the machinery that consumes $\mathbf{\Sigma }$ and emits the tightened constraint; the budget in point 3 above is a chance constraint whose covariance carries an actuation term.
• Covariance Propagation — the step that turns ${\mathbf{\Sigma }}_u$ into the ${\mathbf{\Sigma }}_k$ the constraint needs.
• Actuator Saturation — the deterministic, bounded-deficit sibling: same realized-minus-commanded residual, hard limit instead of a distribution.
• Conditional Value at Risk — the tail-severity budget the chance constraint cannot express; no precedent applying it to actuation was found (see Open Questions).
• Sequential Convex Programming / Risk Aware MPC — the two optimizers in which the tightening is actually imposed.
• Free Flying vs Free Floating — the regime distinction that decides whether base actuation is a noise entry point at all.
• Parameter Estimation — the adjacent uncertainty channel; ekal2021online shows the two are coupled through the actuation limit.

Open Questions

• No CVaR-on-actuation precedent. The precedent search surfaced chance-constraint and covariance-steering derates but nothing applying a coherent/tail risk measure to actuation uncertainty specifically. If risk phase C wants a CVaR budget on the realized wrench, it may be a genuinely open formulation rather than a citation gap — but the search was one pass, and a dedicated one is owed before that claim is made.
• No realized-force error figure for an Astrobee-class free-flyer. No paper quantifying Astrobee’s realized-versus-commanded force error was located. Astrobee actuates by impeller and steerable nozzle (a ducted fan), not discrete RCS thrusters, so classical minimum-impulse-bit models may not transfer — an additive Gaussian ${\mathbf{\Sigma }}_u$ is defensible as a form but has no measured $\sigma$ behind it for this vehicle class.
• Is the additive model the right one for us? Our injection is a constant-$\sigma$ additive force noise, independent of the command. The execution-error family says the dispersion should scale with the command (magnitude error) and rotate with it (pointing error). Under a helix-tracking profile the base command varies substantially, so command-proportional and constant-$\sigma$ models are not interchangeable; which one the fragility grid should use is unresolved.
• Where does the derate land across subsystems? The tightening term is a scalar per constraint, but our plant splits into the decoupled CoM loop and the coupled base+EE block (subsystem decomposition). Base-force noise enters both. Whether the budget should be allocated per subsystem — and whether the coupled block inflates $\mathbf{\Sigma }$ enough to dominate the tightening — is untested.
• The Gates execution-error model (C. R. Gates, ~1963, JPL TR), the origin of the magnitude+pointing decomposition that Chioma & Titus formalize, was not retrievable via Semantic Scholar and would need a direct NTRS/library pull if the foundational derivation is wanted.