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Provably Safe Trajectory Generation for Manipulators Under Motion and Environmental Uncertainties

Fei Meng, Zijiang Yang, Xinyu Mao, Haobo Liang, Max Q. -H. Meng

TL;DR

This article integrates a rigid manipulator deep stochastic Koopman operator (RM-DeSKO) model to robustly predict the robot's state distribution under motion uncertainty, and introduces an efficient, hierarchical verification method that combines parallelizable physics simulations with sum-of-squares programming as a filter for fine-grained, formal certification of collision risk.

Abstract

Robot manipulators operating in uncertain and non-convex environments present significant challenges for safe and optimal motion planning. Existing methods often struggle to provide efficient and formally certified collision risk guarantees, particularly when dealing with complex geometries and non-Gaussian uncertainties. This article proposes a novel risk-bounded motion planning framework to address this unmet need. Our approach integrates a rigid manipulator deep stochastic Koopman operator (RM-DeSKO) model to robustly predict the robot's state distribution under motion uncertainty. We then introduce an efficient, hierarchical verification method that combines parallelizable physics simulations with sum-of-squares (SOS) programming as a filter for fine-grained, formal certification of collision risk. This method is embedded within a Model Predictive Path Integral (MPPI) controller that uniquely utilizes binary collision information from SOS decomposition to improve its policy. The effectiveness of the proposed framework is validated on two typical robot manipulators through extensive simulations and real-world experiments, including a challenging human-robot collaboration scenario, demonstrating sim-to-real transfer of the learned model and its ability to generate safe and efficient trajectories in complex, uncertain settings.

Provably Safe Trajectory Generation for Manipulators Under Motion and Environmental Uncertainties

TL;DR

This article integrates a rigid manipulator deep stochastic Koopman operator (RM-DeSKO) model to robustly predict the robot's state distribution under motion uncertainty, and introduces an efficient, hierarchical verification method that combines parallelizable physics simulations with sum-of-squares programming as a filter for fine-grained, formal certification of collision risk.

Abstract

Robot manipulators operating in uncertain and non-convex environments present significant challenges for safe and optimal motion planning. Existing methods often struggle to provide efficient and formally certified collision risk guarantees, particularly when dealing with complex geometries and non-Gaussian uncertainties. This article proposes a novel risk-bounded motion planning framework to address this unmet need. Our approach integrates a rigid manipulator deep stochastic Koopman operator (RM-DeSKO) model to robustly predict the robot's state distribution under motion uncertainty. We then introduce an efficient, hierarchical verification method that combines parallelizable physics simulations with sum-of-squares (SOS) programming as a filter for fine-grained, formal certification of collision risk. This method is embedded within a Model Predictive Path Integral (MPPI) controller that uniquely utilizes binary collision information from SOS decomposition to improve its policy. The effectiveness of the proposed framework is validated on two typical robot manipulators through extensive simulations and real-world experiments, including a challenging human-robot collaboration scenario, demonstrating sim-to-real transfer of the learned model and its ability to generate safe and efficient trajectories in complex, uncertain settings.
Paper Structure (14 sections, 1 theorem, 16 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 16 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Risk Contours Map
  4. Problem Statement
  5. Receding-Horizon Risk-Bounded Trajectory Generation for Manipulators Under Motion and Environmental Uncertainties
  6. Background on Sampling-based MPC for Manipulators
  7. Deep Stochastic Koopman Operator Model for Uncertain Rigid Manipulator to Propagate in MPPI
  8. Hierarchical Collision Risk Verification
  9. Experiments
  10. Data Collection and Implementation Details
  11. Comparison Results of State Prediction Performance
  12. Simulation Experiments for Motion Planning Problem
  13. Sim-to-Real Transferred Physical Experiments
  14. Conclusion

Key Result

Theorem 1

jasour2021RealTime A given tube $\mathcal{E}(\mathcal{L}(t))=\left\{\bm{p}\in\mathbb{R}^3:(\bm{p}-\mathcal{L}(t))^TQ(\bm{p}-\mathcal{L}(t))\leq1\right\}$ along a polynomial trajectory $\mathcal{L}(t)$ satisfies the constraint eq:prob_cons over the time horizon $t \in\left[t_0, t_f\right]$ if the pol where $Q\in \mathbb{R}^{3\times3}$ is a given positive definite matrix, $\hat{\bm{p}}_0 \in \mathbb

Figures (6)

  • Figure 1: Risk-bounded motion planning framework for a manipulator under environmental and motion uncertainties. It combines a RM-DeSKO neural network model predicting states $X_{N,H}$ and costs $\hat{C}_{N,H}$, with a MPPI refining control inputs $u^*$. A supervisory logic enforces environment safety by computing contact force-based collision costs in simulation and checking bounded collision risk for the next predicted state before execution. Noisy state $x_t$ closes the perception-action loop. The new cost function $\hat{c}(x_t, u^*)$ quickly guides stochastic optimization while ensuring risk constraints.
  • Figure 2: Schematic diagram of proposed forward dynamics propagation in MPPI under RM-DeSKO model with key equations for $N$ sampled trajectories over a horizon $H$.
  • Figure 3: Ellipsoidal body description example of a robot arm to be verified for collision risk.
  • Figure 4: Comparison results of the $log10$ of maximum (top) and mean (bottom) prediction errors for a robot arm under state uncertainty across the planning horizon.
  • Figure 5: Snapshots of the planned trajectories by the MPPI control using four kinds of nonlinear dynamical systems, i.e., baseline, Transformer, LSTM, and Ours (from top to bottom). The arm tries to reach the goal (green ball) while keeping the probability of collision with uncertain nonconvex obstacles (red heart-shaped) guaranteed. Only our method succeeds in time because of the correct guiding rollouts (cyan) benefited from our neural network dynamical model and collision risk verification.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1