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Certified Gradient-Based Contact-Rich Manipulation via Smoothing-Error Reachable Tubes

Wei-Chen Li, Glen Chou

TL;DR

This work tackles the difficulty of applying gradient-based trajectory optimization to contact-rich manipulation by introducing a principled smoothing of hybrid contact dynamics and explicit, tight bounds on the resulting model mismatch. A novel differentiable simulator based on convex programs provides informative gradients and κ-gradients, enabling planning despite discontinuities, while a set-valued error model bounds the true dynamics around the smoothed approximation. The authors jointly optimize a nominal κ-smoothed trajectory and an affine feedback policy to produce tube predictions that certify robust constraint satisfaction for the true hybrid system. Experiments on planar pushing, bimanual manipulation, and in-hand dexterous tasks demonstrate improved constraint satisfaction, lower goal errors, and scalable performance, illustrating a pathway to certifiable gradient-based control in contact-rich domains.

Abstract

Gradient-based methods can efficiently optimize controllers using physical priors and differentiable simulators, but contact-rich manipulation remains challenging due to discontinuous or vanishing gradients from hybrid contact dynamics. Smoothing the dynamics yields continuous gradients, but the resulting model mismatch can cause controller failures when executed on real systems. We address this trade-off by planning with smoothed dynamics while explicitly quantifying and compensating for the induced errors, providing formal guarantees of constraint satisfaction and goal reachability on the true hybrid dynamics. Our method smooths both contact dynamics and geometry via a novel differentiable simulator based on convex optimization, which enables us to characterize the discrepancy from the true dynamics as a set-valued deviation. This deviation constrains the optimization of time-varying affine feedback policies through analytical bounds on the system's reachable set, enabling robust constraint satisfaction guarantees for the true closed-loop hybrid dynamics, while relying solely on informative gradients from the smoothed dynamics. We evaluate our method on several contact-rich tasks, including planar pushing, object rotation, and in-hand dexterous manipulation, achieving guaranteed constraint satisfaction with lower safety violation and goal error than baselines. By bridging differentiable physics with set-valued robust control, our method is the first certifiable gradient-based policy synthesis method for contact-rich manipulation.

Certified Gradient-Based Contact-Rich Manipulation via Smoothing-Error Reachable Tubes

TL;DR

This work tackles the difficulty of applying gradient-based trajectory optimization to contact-rich manipulation by introducing a principled smoothing of hybrid contact dynamics and explicit, tight bounds on the resulting model mismatch. A novel differentiable simulator based on convex programs provides informative gradients and κ-gradients, enabling planning despite discontinuities, while a set-valued error model bounds the true dynamics around the smoothed approximation. The authors jointly optimize a nominal κ-smoothed trajectory and an affine feedback policy to produce tube predictions that certify robust constraint satisfaction for the true hybrid system. Experiments on planar pushing, bimanual manipulation, and in-hand dexterous tasks demonstrate improved constraint satisfaction, lower goal errors, and scalable performance, illustrating a pathway to certifiable gradient-based control in contact-rich domains.

Abstract

Gradient-based methods can efficiently optimize controllers using physical priors and differentiable simulators, but contact-rich manipulation remains challenging due to discontinuous or vanishing gradients from hybrid contact dynamics. Smoothing the dynamics yields continuous gradients, but the resulting model mismatch can cause controller failures when executed on real systems. We address this trade-off by planning with smoothed dynamics while explicitly quantifying and compensating for the induced errors, providing formal guarantees of constraint satisfaction and goal reachability on the true hybrid dynamics. Our method smooths both contact dynamics and geometry via a novel differentiable simulator based on convex optimization, which enables us to characterize the discrepancy from the true dynamics as a set-valued deviation. This deviation constrains the optimization of time-varying affine feedback policies through analytical bounds on the system's reachable set, enabling robust constraint satisfaction guarantees for the true closed-loop hybrid dynamics, while relying solely on informative gradients from the smoothed dynamics. We evaluate our method on several contact-rich tasks, including planar pushing, object rotation, and in-hand dexterous manipulation, achieving guaranteed constraint satisfaction with lower safety violation and goal error than baselines. By bridging differentiable physics with set-valued robust control, our method is the first certifiable gradient-based policy synthesis method for contact-rich manipulation.
Paper Structure (27 sections, 4 theorems, 105 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 27 sections, 4 theorems, 105 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Statement
  4. Differentiable Simulator with Contact Smoothing Bounds
  5. Differentiable Simulator
  6. Smoothing of Contact Dynamics
  7. Smoothing of Contact Geometry
  8. Trajectory Optimization and Policy Synthesis
  9. Cost and Constraints
  10. Solver Algorithm
  11. Baseline
  12. Experiment Results and Discussion
  13. Bimanual Planar Bucket Manipulation
  14. Bimanual Planar Box Manipulation
  15. In-hand Cube Reorientation
  16. ...and 12 more sections

Key Result

Theorem 1

The true dynamics $x^+ = f_0(x,u)$ and the smoothed dynamics $x_\kappa^+ = f_\kappa(x,u)$ satisfy where $\lambda_{\kappa,i}$ is the $i$th contact force under the smoothed dynamics and $w_i \in [1,2]$, under the assumption that $J_i P^{-1} J_j^\top = 0$ for all $i \neq j$. Moreover, the bound is tight in the sense that for every $w_i \in \{1,2\}$, there exist $(x,u)$ pairs that satisfy eq:smoothi

Figures (9)

  • Figure 1: We evaluate on a in-hand cube reorientation task. (a) Keyframes from the executed rollout. (b) The closed-loop trajectory stays within the reachable tube for all object DoF, validating the constraint satisfaction guarantees. Guiding planning with these tubes allows the smoothed dynamics to strategically overshoot the target to compensate for model mismatch, ensuring low goal error on the true hybrid dynamics.
  • Figure 2: 1D pusher. (a) System schematic. (b) Discrete-time dynamics $x^+ = f(x,u)$, with $x_o^+$ plotted versus the input $u$.
  • Figure 3: Linearization of the smoothed dynamics $z^+ = f_\kappa(z,v)$ for a 1D pusher as viewed by (a) TO-CTR and (b) our method.
  • Figure 4: Rollouts of the 1D pusher system for pushing the object to reach a goal position (dashed line) using (a) TO-CTR and (b) our method.
  • Figure 5: Rollout of bimanual planar bucket manipulation. (a) Time-lapse. (b) Rollout and predicted tube.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Example 1
  • Remark 1
  • Theorem 2
  • Example 2
  • Lemma 1
  • proof
  • Corollary 1