An Axiomatic Resilience Functional for Fault-Tolerant Control of Redundant Manipulators: Representation, Rank Invariance, and Sample Complexity

Comparing fault-tolerant controllers for redundant manipulators requires reducing several incommensurable criteria—tracking degradation, detection latency, reconfiguration time, energy, real-time feasibility, constraint violation—to a single ordering. In practice, this is done with weighted sums whose functional form is never justified and whose weights are never audited, so that reported rankings may be artifacts of the aggregation rule. This paper supplies the missing theory; it proposes no new controller and takes the controllers being compared as given. Faults of eight kinds are written as one perturbation of the rigid-body dynamics, and post-fault task capability is characterized by the Chebyshev radius of the zonotope of attainable task velocities, which is concave in the effectiveness vector and norm-equivalent to the classical locked-joint measure. Six axioms on the aggregation step—normalization, strict monotonicity, homogeneity, continuity, non-compensability and multiplicative compounding—reduce the admissible functionals, within the quasi-arithmetic class, to the weighted geometric mean alone, excluding the weighted arithmetic mean the field currently uses. Three consequences are made computable: the score is Lipschitz with constant one in logarithmic coordinates; the exact ℓ1 distance from the nominal weights to the weight vectors that reverse a pairwise comparison has a closed form; and the effect of misspecifying the normalization constants is bounded by an explicit exponent. Distribution-free bounds give the episodes needed before a difference may be reported: 738 to state one controller’s score to within 0.05 at 95% confidence and 343 to state which of two controllers scores higher under paired sampling. A campaign of 32,000 episodes verifies every bound; shows that the excluded arithmetic mean reverses the reported ranking on three of the five plants; and quantifies how the ranking responds to the detection thresholds, the weights, and the fault distribution.