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Belief-Informed Hybrid Control with Almost-Sure Target-Set Convergence

Clinton Enwerem, Saleh Kemal, John S. Baras, Calin Belta

arXiv:2610.07674v1eess.SYcs.ROcs.SYmath.OC

TL;DR

Under parameter uncertainty, informative actions can temporarily move a hybrid system away from its target, complicating regulation. The paper combines bounded-slack expected-decrease constraints with posterior-conditioned lookahead; it proves conditional convergence and reports improved assembly accuracy over reference tracking.

  • Problem

    Informative actions under parameter uncertainty may temporarily move a hybrid system away from its target, while immediate-progress requirements can exclude actions that improve subsequent decisions.

    Source: §I · p. 1

  • Method

    The selector combines a posterior-based expected-decrease constraint with bounded cumulative slack and observation-conditioned two-step lookahead.

    Source: B. Contributions and Outline · p. 2, §IV · p. 3

  • Results

    96.4% lower infinity-norm position error than the reference-tracking baseline was achieved in simulated assembly, and the paper proves almost-sure target-set convergence under its stated conditions.

    Source: 7) Assembly Completion and Accuracy · p. 7, §VI · p. 8

  • Takeaways & Limitations

    At lower friction, posterior conditioning detects an empty admissible set where a fixed-prior alternative admits an action violating the conditional decrease constraint.

    Source: 8) Belief Refinement and Action… · p. 7

Abstract

from arXiv · show

Controlling a hybrid system to a target set under parameter uncertainty can require informative actions that temporarily drive the system state away from the specified set. We propose a belief-informed dual-control algorithm that combines belief-space receding-horizon selection with an expected-decrease constraint on a nonnegative target-set progress function. To accommodate exploratory deviations, a scalar controller state bounds the constraint's cumulative slack. Our algorithm's two-step lookahead selection uses predicted observations to update the parameter belief before evaluating the subsequent action's admissibility. Under distance-comparison bounds, correct conditional prediction, and recursive feasibility, we prove almost-sure target-set convergence at decision times and bound both the sum of expected progress-function values and the expected neighborhood-entry time. Upper confidence bounds extend these convergence guarantees to bounded model samples with summable error probabilities. In planar regulation with an unknown control direction, we verify recursive feasibility: the proposed two-step selection reduces the Euclidean state norm below 0.01 within 11 decisions for either sign, whereas a myopic one-step selection loses admissibility immediately after zero input. In a simulated bimanual assembly task, our algorithm's one-step implementation uses clearance feedback to complete the assembly under nominal friction, yielding a 96.4% lower infinity-norm relative-position error than the reference-tracking baseline at the method's completion time. At lower friction, its posterior conditioning successfully detects an empty admissible set, whereas a fixed-prior alternative admits an action that violates the conditional decrease constraint. Project page: https://clintonenwerem.com/belief-hybrid-control/.

I. INTRODUCTION

The paper frames uncertain hybrid control as a joint regulation-and-identification problem: informative actions may temporarily move the system away from its target. It combines belief-space action selection with bounded slack and posterior-conditioned lookahead, and situates this approach within hybrid stability and belief-space control.

I. INTRODUCTION

  • A probing motion can reveal slip while displacing a part, so immediate-progress requirements may exclude actions that improve later decisions.The motivating assembly task combines continuous motion with contact- and pose-triggered mode transitions under unknown friction.Source: §I · p. 1

A. Related Work

  • Posterior-conditioned lookahead ranks actions by predicted observations, while bounded cumulative slack permits temporary increases in the progress function and hybrid barrier conditions restrict actions for safety.The contribution extends belief-space control by predicting hybrid endpoints through flows and jumps.Source: A. Related Work · p. 2

B. Contributions and Outline

  • The paper’s contributions are bounded-slack, posterior-dependent two-step action selection; conditional convergence guarantees; and demonstrations in planar regulation and bimanual assembly.The convergence result assumes distance comparison, correct prediction, and recursive feasibility; a sampled extension is also provided.Source: B. Contributions and Outline · p. 2

A. Flows, Events, and Feedback Actions

  • The model represents a hybrid state with physical coordinates, controller memory, clocks, and a finite mode, and feedback actions specify control laws, event responses, and stopping rules.Actions are evaluated at decision times after the events assigned to that time.Source: A. Flows, Events, and Feedback Actions · p. 2

B. Parameter Belief and Observations

The controller infers a fixed but uncertain parameter from observed states and interval observations, then uses the posterior to predict action outcomes. The target set is defined over continuous coordinates and mode, with a progress function that vanishes exactly on the set.

B. Parameter Belief and Observations

  • The controller predicts outcomes by integrating the parameter-conditioned endpoint and observation model against the current posterior belief.The belief conditions on observed states, selected actions, and interval observations available before the next action.Source: B. Parameter Belief and Observations · p. 2
  • The posterior over the fixed parameter is updated by Bayes’ rule using the observed endpoint and interval observations, including information in the physical response.For numerical evaluation, the paper specializes to deterministic parameter hypotheses and noiseless likelihoods.Source: B. Parameter Belief and Observations · p. 2

A. Target Set and Progress Function

  • The target set includes both continuous coordinates and mode; its nonnegative progress function is zero exactly on the target set and is bounded above and below by distance comparison functions.The convergence problem is to make the distance to the target set tend to zero almost surely.Source: A. Target Set and Progress Function · p. 2

B. Admissible Actions and Expected Decrease

Admissibility requires predicted expected progress to meet a decrease condition, with bounded cumulative slack permitting limited deviations. Under the stated assumptions, this yields almost-sure convergence and bounds on expected progress and neighborhood-entry time.

B. Admissible Actions and Expected Decrease

  • The slack required for an action is the excess of its posterior expected endpoint value over the desired decrease threshold, and the remaining budget decreases by that amount.The total slack spent is bounded by the initial budget on every trajectory.Source: B. Admissible Actions and Expected… · p. 3
  • Recursive feasibility requires an admissible action with finite conditional expected progress after every reachable history, including after observations and slack updates.The decrease condition applies over the complete action interval, including flows and jumps.Source: B. Admissible Actions and Expected… · p. 3

C. Convergence Under Exact Prediction

  • Under the theorem’s prediction, comparison, and feasibility assumptions, the progress function tends to zero almost surely and in mean, and distance to the target tends to zero almost surely.The result also bounds the sum of expected progress values and expected neighborhood-entry time.Source: C. Convergence Under Exact Prediction · p. 3
  • Expected physical-time neighborhood entry is bounded by the decision-time entry bound plus h, the maximum action duration.The proof uses the nonnegative augmented function formed from progress and remaining slack.Source: C. Convergence Under Exact Prediction · p. 3

IV. BELIEF-INFORMED ACTION SELECTION

The selector uses posterior-conditioned two-step lookahead to account for how first-action observations affect the next action’s cost and admissibility. A finite-sample variant uses upper confidence bounds to guard against underestimated endpoint means.

IV. BELIEF-INFORMED ACTION SELECTION

  • Because the decrease constraint only determines admissibility, the two-step objective evaluates the second-stage cost-to-go after conditioning on each first action’s predicted observation.The selector assigns infinite cost when the second-stage admissible set is empty, then executes the first action and replans after observing its outcome.Source: §IV · p. 3
  • The selector minimizes the two-step objective over admissible first actions, with ties resolved by catalog order, then replans after the observed outcome.The comparison one-step selector minimizes the posterior expected endpoint progress over the admissible set.Source: §IV · p. 3

A. Finite-Sample Action Selection

  • The finite-sample selector uses an upper confidence bound when estimating expected endpoint progress from model samples, avoiding admission through underestimated means.The stated setup assumes bounded progress values and accounts for discrepancy between the model posterior and the current posterior.Source: A. Finite-Sample Action Selection · p. 3

11 Execute ak until its stopping rule gives tk+1

The sampled selector uses confidence bounds to preserve admissibility while accounting for model-sampling error. Under summable error probabilities and recursive feasibility, the convergence guarantee persists with a sampling-error allowance.

12 Observe (sk+1, ok+1)

  • An action is accepted when its upper-confidence-bound violation εk does not exceed the remaining slack Bk, which is then reduced by εk.The posterior-approximation mean error is bounded by Vmaxηk; the confidence construction gives coverage probability at least 1 − δk.Source: §12 · p. 4
  • The sampled selector retains almost-sure convergence and the expected neighborhood-entry-time bound, with Vmax∆ added to the bound's numerator.The guarantee assumes the physical and comparison conditions, the sampling conditions, and an accepted action returned after finitely many samples at every decision.Source: §12 · p. 4
  • For any 0 < ∆ < 1, choosing δk = ∆/[(k + 1)(k + 2)] satisfies the summability condition.The proof conditions on the full computational random stream and uses conditional independence to relate the selected action's mean to the next progress-function value.Source: §12 · p. 4

V. NUMERICAL EXPERIMENTS

The numerical experiments use planar regulation to verify the convergence hypotheses and compare informative two-step selection with one-step selection. They also examine posterior-updating one-step selection in bimanual assembly.

V. NUMERICAL EXPERIMENTS

  • The experiments evaluate conditional means over finite deterministic hypotheses; assembly uses one-step selection to examine withdrawal feedback and posterior-dependent admissibility.The planar experiment verifies the hypotheses of Theorem 1 and compares informative two-step selection with one-step selection.Source: §V · p. 4

A. Planar Regulation with Unknown Control Direction

  • The planar system's zero-input drift preserves radius, and for nonzero state a nonzero-gain action identifies the unknown direction from the observed radius change.The target is the origin and the progress function is the squared Euclidean norm; the test observation identifies the sign before contracting radial feedback is selected.Source: A. Planar Regulation with Unknown… · p. 4, Figure 2 · p. 4
  • Both control directions reach ∥x∥ ≤ 0.01 by decision 11, while the exact solution's expected progress-function sum is approximately 2.59, below the 11.2 upper bound.The exact solution verifies the sum bound, and simulated trajectories reproduce it; the two-step selection first tests and then applies contracting feedback.Source: A. Planar Regulation with Unknown… · p. 5
  • The one- and two-step selectors differ at 55 of 399 prior–slack pairs; in 40 cases, two-step selection tests where one-step selection holds.All 55 differences occur for 0.5 ≤ p ≤ 0.7; at the other 15, hold leaves no admissible next action and test exceeds available slack, so no action has finite two-step cost.Source: A. Planar Regulation with Unknown… · p. 5
  • At p = 0.6 and β = 0.13, one-step selection chooses hold and leaves no admissible next action, while two-step selection chooses the test at cost 0.377 and preserves recursive feasibility along both possible trajectories.The test requires slack 0.125; the observation identifies the contracting gain, which requires zero slack thereafter.Source: A. Planar Regulation with Unknown… · p. 5

B. Bimanual Assembly

The bimanual assembly task is modeled as a hybrid system with free-body motion, contact, and mode transitions. Its target requires specified assembly conditions to hold through a dwell period, alongside completed acquisition and lift flags.

B. Bimanual Assembly

  • The simulated bimanual robot comprises left and right variants of RealHand A7 7-DoF arms, each fitted with a matching RealHand L6 6-DoF robotic hand (Fig. 3).Source: 1) State and Target · p. 5
  • Assembly is simulated with free bodies under gravity and contact, 1 ms plant steps, and unknown friction fixed during execution and shared across contact geometries.The robot state includes configuration and velocity, object poses and twists, reference phase, clocks, and contact outputs; the mode tracks progress and acquisition flags.Source: 1) State and Target · p. 5

1) State and Target:

  • The desired relative pose is p∗ = (−0.0975, 0, 0) m with identity relative orientation, and completion uses residuals normalized by the scales in Table I.The completion predicate requires all listed residuals to be nonpositive; the insertion predicate requires only the position and rotation conditions.Source: 1) State and Target · p. 5, Table I · p. 5
  • The target requires all completion conditions, a 2 s dwell, terminal mode, and housing-acquisition, part-acquisition, and lift flags set.If the completion predicate fails during dwell or terminal mode, the dwell clock resets or the system returns to dwell.Source: 1) State and Target · p. 5
  • Forward modes advance after 40 ms when guards and required contacts hold; a failed required contact for 20 ms triggers recovery to an acquisition mode.The transition logic prioritizes housing recovery and resets transition clocks; transfer and alignment use specified anchors and component-wise tolerances.Source: 2) Supervisory Modes · p. 5

2) Supervisory Modes:

The section describes ordered supervisory modes and a feedback architecture that adjusts waypoint tracking and finger commands using object-pose and contact feedback.

2) Supervisory Modes:

  • Fig. 4 maps selected first actions across prior probability and normalized slack, marking empty admissible sets and first actions without a feasible second decision.The comparison holds the plant, gains, initial state, and expected-decrease constraint fixed; the nominal case is (p, β) = (0.5, 0.12).Source: Figure 4 · p. 6
  • Using joint-limited inverse kinematics, the method generates 92 waypoints from prescribed palm poses and finger postures over a nominal reference duration Tref = 33 s, regularizing toward preceding configurations.Source: 3) Feedback Architecture · p. 6

3) Feedback Architecture:

  • The feedback architecture corrects the right-arm reference using object-pose feedback from lift through unloading.It compensates housing motion during transfer through unloading and bounds the Jacobian correction by joint limits and 0.2 rad.Source: 3) Feedback Architecture · p. 6
  • Clearance feedback translates the right-palm reference by (−40, 0, 20) mm in the housing frame after part–hand load falls below 0.01 N.The correction retains orientation, finger, and left-arm commands; finger offsets for tightening and relaxation do not accumulate.Source: 4) Feedback Actions and Endpoint… · p. 6

4) Feedback Actions and Endpoint Prediction:

Feedback actions are predicted through supervisory transitions, and a mode-rank-plus-residual progress function evaluates their endpoints. The reported assembly outcomes compare selectors and a reference-tracking baseline at a shared completion time.

4) Feedback Actions and Endpoint Prediction:

  • Endpoint prediction composes plant and contact updates, feedback, reference and supervisory updates, and stops at the first supervisory transition or duration limit.The duration limit is 8 s for advancement and clearance and 0.1 s for holding and finger adjustment.Source: 4) Feedback Actions and Endpoint… · pp. 6–7

5) Progress Function:

  • The progress function combines mode rank with a nonnegative mode residual, and its zero set is the target set.The assembly residual ZA measures normalized completion-condition violations, dwell-time deficit, and unset acquisition or lift flags.Source: 5) Progress Function · p. 7

IDENTICAL ENDPOINTS. MINT DENOTES SELECTION; SALMON DENOTES

  • At θ = 0.7, the compared selectors are evaluated at 30.96 s, when the robot reaches the target set under either selector.The reference-tracking baseline retains object-pose feedback, impedance control, and guards, but advances the reference within each mode and holds during dwell.Source: Table V · p. 7, 6) Comparison Protocol · p. 7
  • The outcome table reports final progress of 0 versus 3.94 and position error of 0.188 versus 5.20 mm for the quantity selector and baseline.It also reports rotation error of 0.00194 versus 0.0453 rad and hand clearance of 7.14 versus 0 mm.Source: IDENTICAL ENDPOINTS. MINT DENOTES… · p. 7, Table V · p. 7

6) Comparison Protocol:

At nominal friction, both one-step selectors produce identical assembly trajectories, with friction identified from acquisition observations.

6) Comparison Protocol:

7) Assembly Completion and Accuracy:

At nominal friction, clearance feedback completes withdrawal while preserving the assembled pose; at lower friction, posterior updating detects that no action is admissible. The discussion connects these outcomes to the paper’s bounded-slack, posterior-dependent selection and convergence result.

7) Assembly Completion and Accuracy:

  • 96.4% lower infinity-norm position error than the reference-tracking baseline is measured at the selectors’ completion time, with zero residual hand–housing load.After insertion, clearance feedback drives hand withdrawal while preserving the assembled part pose.Source: 7) Assembly Completion and Accuracy · p. 7
  • At θ = 0.35, posterior updating detects an empty admissible set, while the fixed-prior selector admits tightening that violates the conditional decrease constraint by 0.0543.The posterior slack requirement is 0.1143 against 0.0936 remaining; the fixed-prior realized progress value is approximately 5.7728.Source: 8) Belief Refinement and Action… · p. 7

VI. CONCLUSION

  • The conclusion states almost-sure target-set convergence at decision times under prediction, comparison, and recursive-feasibility conditions, with bounds on summed expected progress values and expected neighborhood-entry time.The discussion attributes the assembly accuracy improvement to clearance feedback after insertion and says posterior conditioning accounts for slack that fixed-prior prediction underestimates at lower friction.Source: §VI · p. 8, C. Discussion · p. 8
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