Residual Wrench Certification and Margin-Aware Control Synthesis for Aerial Physical Interaction

I. INTRODUCTION

• Force–attitude coupling in conventional planar multirotors can restrict pressing, bracing, inspection, or manipulation while preserving pose.Safe interaction also depends on retaining corrective authority under sustained loading.Source: §I · p. 1
• Prior work addresses aerial interaction, constrained allocation, and wrench-capacity measures, but does not jointly certify task-family reserves and local tracking regions with synthesized gains.The cited approaches include tilted and omnidirectional platforms,, allocation methods,, and force-polytope methods for zero-moment force margins.Source: §I · p. 1
• A feasible contact wrench may leave negligible pose-correction authority, while certification must cover a task family and distinguish reserve exhaustion from actual infeasibility.Redundant allocation makes actuator clearance allocation-dependent, and exact wrench geometry can require extensive facet enumeration.Source: §I · p. 1
• The framework combines reserve certificates, slack-maximizing allocation, margin-preserving command projection, and structured gain synthesis for locally certified tracking under stated assumptions.It establishes nominal exponential convergence and robust ultimate boundedness, and uses contact simulations to distinguish reserve thresholds from loss of realizability.Source: §I · p. 1
• “Contact-persistent” refers to actuator-side wrench feasibility during sustained loading; physical persistence additionally requires unilateral, frictional, and contact-moment conditions.Contact-admissible constraints may be represented by intersecting the actuator-generated wrench set with a contact-admissible set.Source: §I · pp. 1–2

II. MODELING AND PROBLEM FORMULATION

• Each of N thrust modules contributes force and moment according to its nonnegative thrust, mounting location, orientation, and reaction-moment parameters.The body-frame actuation wrench is assembled from the module contributions.Source: §II · p. 2
• For a selected morphology, the allocation map M(α) sends thrust vectors in the admissible box Λ to attainable wrenches.The attainable set is W(α) = {w : w = M(α)λ, λ ∈ Λ}.Source: §II · p. 2
• Because force and moment have different units, wrench distances and reserves use fixed scaled, dimensionless wrench coordinates.Interiority and rank are preserved by the nonsingular scaling, and a normalized reserve guarantees feasibility for perturbations within the corresponding reserve ball.Source: §II · p. 2
• Constrained full actuation at an operating wrench requires it to lie inside the attainable set, not merely that the allocation map have rank six.The distance to the boundary also depends on the finite actuator box and operating point.Source: §II · p. 2
• The formulation seeks morphologies that realize every wrench in a compact task family with a prescribed convex reserve, then uses available Euclidean authority for feasible commands and structured gains.The tracking region must provide nominal exponential convergence and robust ultimate boundedness under quantified disturbance bounds.Source: §II · p. 2

• The actuator-interiority certificate uses thrust-bound clearance alone unless contact inequalities are included directly; quasi-static authority comparisons do not certify servo rates, transient singularities, or closed-loop stability.The morphology parameterization distinguishes fixed tilt from admissible time-varying rotor angles.Source: §III · pp. 2, 5
• For a feasible task wrench, the largest Euclidean residual radius equals its distance to the feasible-set boundary; convex reserves yield signed margins whose sign distinguishes interior, boundary, and infeasible wrenches.For a general convex reserve, the margin is the minimum facet slack divided by the reserve support value; ellipsoids provide direction-dependent reserves.Source: §III · p. 3
• An interior actuator allocation certifies a Euclidean reserve of at least its actuator clearance divided by the selected right-inverse norm.Slack-maximizing allocation maximizes the clearance lower bound; positive optimized clearance certifies an interior allocation.Source: §III · p. 3
• A common positive Euclidean reserve for an entire task family exists exactly when its worst-case signed margin is positive; a negative worst-case margin means some task is infeasible.Clipping a negative margin to zero is only a numerical diagnostic, not a nonnegative reserve certificate.Source: §III · pp. 3–4
• Structured gain synthesis maximizes a certified Lyapunov level subject to pole-abscissa and local-coordinate constraints, yielding nominal exponential convergence and, when χ★ < 1, robust invariance and an ultimate bound.The guarantee assumes the task remains in the prescribed family, the disturbance bound holds, and the local model remains valid; the combined commanded wrench remains actuator feasible in the certified ellipsoid.Source: §III · p. 4
• A desired command can be projected onto a tightened feasible set so admissible feedback and additional commanded wrenches preserve the required reserve.The tightened intersection must be nonempty; the stated reserve guarantee concerns commanded wrench demand, while dynamic disturbances are treated separately.Source: §III · p. 5
• The push margin withholds authority in all normalized wrench directions, and a finite direction set is only diagnostic without a covering-error bound.The computations use exact half-space models; fixed rotor directions have allocation rank at most four and cannot provide a six-dimensional residual neighborhood.Source: §III · p. 5

Morphology comparison

• Within the stated task family and actuator limits, the optimized capacity-matched octarotor attains 30.3% larger maximum task margin and 13.3% larger maximum certified push than the optimized hexarotor.At 36° tilt, the capacity-matched octarotor retains 96.9% of the hexarotor’s task margin and 95.4% of its certified push.Source: §IV · p. 6, Figure 1 · p. 6, Table I · p. 6

Residual-authority certificates

• At γ=36°, slack maximization raises O8’s allocation-interiority certificate from 0.072 to 0.105, a 45.3% increase, without enlarging the polytope.The exact point margins for H6, O8, and O8,eq are 0.076, 0.137, and 0.073, respectively.Source: §IV · p. 6, Figure 2 · p. 6

Task-margin maps

• The task-margin maps compare geometry and capacity under equal per-rotor thrust limits across 17,061 force–torque cases per architecture.The sweep spans ρmax∈[0, 0.38] and μmax∈[0, 0.12], with common contours and color limits.Source: §IV · p. 6, Figure 3 · p. 6

Controller synthesis

• At γ=36°, certified levels are c★_6=0.200, c★_8=1.012, and c★_8,eq=0.188; the structured search returns k_p=0.076 and k_d=0.4 for all cases.The synthesis connects reserve-based gains to local tracking guarantees.Source: §IV · p. 6

Closed-loop push sweeps

• At equal installed thrust, VTQ4’s certified push limit is ρ★≈0.696 versus O8’s ρ★≈0.247; at μmax=0.04, the certified push ratio is 2.81.The O8 envelope is limited by a rotor-thrust facet, while VTQ4 is limited by its tilt cone.Source: §IV · p. 7, Figure 4 · p. 7

Beyond-boundary tests

• Beyond the certified boundary, O8 becomes infeasible when rotor bounds activate, while VTQ4 reaches its 36° tilt limit with the predicted 0.096 N force residual.The tests use O8 at ρ=0.280 and VTQ4 at ρ=0.730.Source: §IV · p. 7, Figure 5 · p. 8

V. CONCLUSIONS

• Certified reserves support structured gain synthesis with nominal exponential convergence and robust ultimate boundedness under the stated assumptions.The framework converts residual wrench authority into actuator-feasible local tracking guarantees.Source: §V · p. 8
• Task-dependent margins quantify corrective authority that rank and conditioning cannot certify, while redundant allocation improves lower certificates.Equal-thrust comparisons link morphology to attainable reserves.Source: §V · p. 8
• Contact simulations distinguish exhaustion of a prescribed reserve from actual infeasibility and reproduce predicted thrust and tilt limits.The results support task-specific morphology and control design.Source: §V · p. 8
• Architectural comparisons are confined to the tested task families, and the variable-tilt certificate remains local.Source: §V · p. 8