akella2024risk

Risk-Aware Robotics: Tail Risk Measures in Planning, Control, and Verification

Authors: Akella, Dixit, Ahmadi, Lindemann, Chapman, Pappas, Ames, Burdick · Year: 2024 · Venue: Survey/tutorial (IEEE-style, controls/robotics community)
Raw: md

Summary

A tutorial survey arguing that tail risk measures (Value-at-Risk, Conditional-Value-at-Risk, Entropic-Value-at-Risk) borrowed from financial mathematics provide a principled middle ground between the overly conservative worst-case paradigm and the over-optimistic risk-neutral (expectation) paradigm for robotic autonomy under uncertainty. It surveys risk-aware behavior planning ((PO)MDPs), motion planning/control (risk-aware MPC, risk-aware control barrier functions), and risk-aware verification & validation (V&V) using the quantitative robustness semantics of signal temporal logic (STL). The unifying construction is to map a (stochastic) closed-loop trajectory to a scalar cost random variable and then either minimize its tail risk or constrain it.

Key Claims

• CVaR and EVaR are coherent risk measures; VaR is not (it fails subadditivity), which limits VaR’s utility when risks must be jointly assessed (Def. 2; Coherent Risk Measures section).
• The three measures form an ordered chain of upper bounds: VaR_β(X) ≤ CVaR_β(X) ≤ EVaR_β(X), so any of these being ≤ 0 certifies the chance constraint P(X ≤ 0) ≥ 1−β. CVaR is a “loose” and EVaR the “tightest” convex upper bound on VaR.
• Tail risk measures yield convex inner approximations of chance constraints regardless of the uncertainty distribution (Gaussian or not), avoiding the mixed-integer program that sampling-based chance-constrained MPC otherwise requires (Sidebar on MPC).
• For dynamic settings, time-consistent dynamic coherent risk measures are the right object; the existence of a Risk-aware Control Barrier Function (RCBF) implies ρ-safety (Thm. 1), and a finite-time RCBF implies ρ-reachability with an explicit upper bound on the reach time t* (Thm. 2).
• In V&V, two systems can share the same robustness VaR_β (same minimum probability of satisfying an STL spec) yet be distinguished by CVaR: CVaR_β(R₁) ≥ 0 while CVaR_β(R₂) < 0 shows system 1 still satisfies the spec in expectation over the worst 100β% of cases (Motivations for Tail Risk Measures in Verification).
• The β convention used here: β → 1 (or β ≃ 1) is risk-neutral, β → 0 is maximally risk-averse, with lim_{β→0} EVaR_β(X) = ess sup(X).

Method

The survey is built on a measure-theoretic foundation. For a probability space (Ω, F, P) and cost random variable X: Ω → ℝ with CDF F_X, the β-tail is defined (Eq. after Def. 1) as

X_β = F_X^{-1}(1 − β + β·U), U ~ Uniform[0,1],

and a β-tail risk measure ρ satisfies ρ(X) = ρ(X′) whenever X_β =_d X′_β (Def. 1).

Definitions transcribed faithfully:

• VaR: VaR_β(X) := inf{ z ∈ ℝ | F_X(z) ≥ 1−β } = F_X
• CVaR (Rockafellar–Uryasev): CVaR_β(X) := inf_{z∈ℝ} E[ z + (X−z)^+ / β ], with (X−z)^+ = max(0, X−z).
• EVaR: EVaR_β(X) := inf_{z>0} [ z^{-1} ln( E[e^{Xz}] / β ) ], derived via the Chernoff bound, with a dual relative-entropy (distributionally robust) interpretation.
• Coherence (Def. 2): subadditivity, monotonicity, translational invariance, positive homogeneity (the latter two imply convexity). Representation theorem (Def. 3): every coherent ρ(X) = sup_{Q∈𝒬} E_Q(X) over a convex closed ambiguity set 𝒬 with Q ≪ P.

The control/planning model is a generic discrete-time stochastic system x(t+1) = f(x(t), u(t), d(t)) with d(t) ~ ξ(x,u,t). A closed-loop trajectory is a sample of a trajectory-valued random variable Σ(x₀); a cost map C: 𝒮^𝒳 → ℝ produces the scalar X = C(Σ(x₀)) on which ρ acts. Behavior planning is posed as risk-aware (PO)MDPs with nested one-step risk measures over the horizon (Eqs. for J(x₀,π,γ) and π*). Risk-aware MPC replaces the expectation/chance constraint with ρ in the receding-horizon QP; the linear-Gaussian sidebar shows the deterministic reformulation using Φ^{-1}(1−β) tightening, and the risk-aware version replaces it with ρ_β(F_x x − g_x) ≤ 0. Safety-critical control uses RCBFs: h is an RCBF if ρ(h(x(t+1))) ≥ α(h(x(t))) for a convex class-𝒦 α with α(r) < r (Def. 9). V&V uses STL robust semantics ρ^φ(x,t) (signed-distance-based, recursively defined) as the random cost whose tail risk is then assessed.

Regime: This is a domain-general robotics survey (ground/legged/wheeled robots, bipeds, lane-keeping, subterranean traversal). It is not specific to space manipulators and does not address free-flying vs free-floating base dynamics, dynamic coupling, or the generalized Jacobian. The system model x(t+1) = f(x,u,d) is fully actuated and abstract; mapping it to a manipulator regime is left to the reader.

Relevance to thesis

This is the canonical reference for the risk layer that sits atop the nominal free-flying space-manipulator guidance/control stack. Once nominal algorithms are perfected, uncertainty (state estimation, actuator, disturbance) can be folded in via: (i) CVaR/EVaR convex inner-approximations of chance constraints for collision and pointing avoidance during inspection trajectories, distribution-free; (ii) risk-aware MPC for receding-horizon trajectory tracking with tail-risk costs; (iii) RCBFs to certify ρ-safety/ρ-reachability of capture or station-keeping sets under stochastic disturbance; and (iv) STL-robustness tail-risk V&V to verify a candidate controller against timed safety/reachability specs. The ordered bound VaR ≤ CVaR ≤ EVaR gives a tunable conservatism knob (β) the pencil-and-paper reviewer can audit analytically.

Connections

Topics: conditional_value_at_risk · chance_constraints · risk_aware_mpc · control_barrier_functions · signal_temporal_logic

Key Equations / Quotes

“CVaR_β(X) := inf_{z∈ℝ} E[ z + (X−z)^+ / β ]” (Conditional Value-at-Risk section)

“VaR_β(X) ≤ CVaR_β(X) ≤ EVaR_β(X) ≤ 0, ⟹ P(X ≤ 0) ≥ 1−β.” (Entropic Value-at-Risk section)

“Many tail risk measures such as CVaR and EVaR, provide intuitive convex, inner approximations of chance constraints regardless of the uncertainty distribution.” (Risk-Aware MPC sidebar)

RCBF safety condition (Def. 9):

ρ( h(x(t+1)) ) ≥ α( h(x(t)) ), ∀ x(t) ∈ 𝒳, α convex class-𝒦 with α(r) < r.

Finite-time reach bound (Thm. 2):

t* ≤ log( (ε − h(x(0))) / ε ) / log( 1/γ ).

Open Questions

• Risk-aware learning: how to provide guarantees when the uncertainty distribution itself is learned/estimated from finite data (the survey’s “Risk-Aware Learning” open-problem section).
• Risk-awareness with nonstationary and dependent data — most results assume i.i.d. disturbances; correlated/drifting noise is open.
• Computational scalability: coherent-risk POMDP synthesis is limited to “hundreds of states”; risk-level dynamic programs generally cannot guarantee a globally optimal value function (Hau et al. concern).
• V&V sample complexity and compositional verification of risk-aware specifications remain open.