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Safety-Critical Control of Euler-Lagrange Systems Subject to Multiple Obstacles and Velocity Constraints

Zhi Liu, Si Wu, Tengfei Liu, Zhong-Ping Jiang

TL;DR

This work tackles safety-critical control for Euler-Lagrange systems subject to multiple obstacles and velocity constraints by presenting a cascade inner-outer loop. The outer-loop QP-based law generates a Lipschitz velocity reference $v^*$ that respects obstacle constraints defined by $h_i(q)=|q_{oi}-q|-d_{si}\ge0$ and the velocity bound $|\dot{q}|\le \bar{v}$, while the inner-loop velocity-tracking law leverages the EL energy structure via $u=N(q,\dot{q})+C(q,\dot{q})v^*-k_D\tilde{v}$ with $\tilde{v}=\dot{q}-v^*$ to ensure convergence. A key contribution is the feasible-set reshaping using a positive basis, which guarantees Lipschitz continuity of $v^*$ even with multiple obstacles, and a distance-condition on obstacles ensuring feasibility. The paper also introduces a reduced-neighborhood variant to cut computation while preserving safety, and proves safety via a max-type Lyapunov function. Numerical simulations and experiments on planar and 6-DOF manipulators validate obstacle avoidance and velocity-limit compliance under uncertain dynamics.

Abstract

This paper studies the safety-critical control problem for Euler-Lagrange (EL) systems subject to multiple ball obstacles and velocity constraints in accordance with affordable velocity ranges. A key strategy is to exploit the underlying inner-outer-loop structure for the design of a new cascade controller for the class of EL systems. In particular, the outer-loop controller is developed based on quadratic programming (QP) to avoid ball obstacles and generate velocity reference signals fulfilling the velocity limitation. Taking full advantage of the conservation-of-energy property, a nonlinear velocity-tracking controller is designed to form the inner loop. One major difficulty is caused by the possible non-Lipschitz continuity of the standard QP algorithm when there are multiple constraints. To solve this problem, we propose a refined QP algorithm with the feasible set reshaped by an appropriately chosen positive basis such that the feasibility is retained while the resulting outer-loop controller is locally Lipschitz. It is proved that the constraint-satisfaction problem is solvable as long as the ball obstacles satisfy a mild distance condition. The proposed design is validated by numerical simulation and an experiment based on a $2$-link planar manipulator.

Safety-Critical Control of Euler-Lagrange Systems Subject to Multiple Obstacles and Velocity Constraints

TL;DR

This work tackles safety-critical control for Euler-Lagrange systems subject to multiple obstacles and velocity constraints by presenting a cascade inner-outer loop. The outer-loop QP-based law generates a Lipschitz velocity reference that respects obstacle constraints defined by and the velocity bound , while the inner-loop velocity-tracking law leverages the EL energy structure via with to ensure convergence. A key contribution is the feasible-set reshaping using a positive basis, which guarantees Lipschitz continuity of even with multiple obstacles, and a distance-condition on obstacles ensuring feasibility. The paper also introduces a reduced-neighborhood variant to cut computation while preserving safety, and proves safety via a max-type Lyapunov function. Numerical simulations and experiments on planar and 6-DOF manipulators validate obstacle avoidance and velocity-limit compliance under uncertain dynamics.

Abstract

This paper studies the safety-critical control problem for Euler-Lagrange (EL) systems subject to multiple ball obstacles and velocity constraints in accordance with affordable velocity ranges. A key strategy is to exploit the underlying inner-outer-loop structure for the design of a new cascade controller for the class of EL systems. In particular, the outer-loop controller is developed based on quadratic programming (QP) to avoid ball obstacles and generate velocity reference signals fulfilling the velocity limitation. Taking full advantage of the conservation-of-energy property, a nonlinear velocity-tracking controller is designed to form the inner loop. One major difficulty is caused by the possible non-Lipschitz continuity of the standard QP algorithm when there are multiple constraints. To solve this problem, we propose a refined QP algorithm with the feasible set reshaped by an appropriately chosen positive basis such that the feasibility is retained while the resulting outer-loop controller is locally Lipschitz. It is proved that the constraint-satisfaction problem is solvable as long as the ball obstacles satisfy a mild distance condition. The proposed design is validated by numerical simulation and an experiment based on a -link planar manipulator.
Paper Structure (12 sections, 5 theorems, 93 equations, 12 figures, 1 table)

This paper contains 12 sections, 5 theorems, 93 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Cascade Controller Design
  4. QP-based Control Law to Handle Position Constraints
  5. Velocity-Tracking Control Law
  6. Synthesis of the Inner-Outer-Loop Controller
  7. Refined Control Law with a Possibly Reduced Neighborhood
  8. Numerical and Experimental Validations
  9. A Safety-Critical Control Scenario and Controller Design
  10. Numerical Simulation
  11. Experimental Results
  12. Conclusions

Key Result

Proposition 1

Consider the kinematics part of the EL system EL: where $v^*\in\mathbb{R}^n$ is the solution of the QP problem QPvirtualcontrolllaw1--QPvirtualcontrolllaw3 and $\tilde{v}\in\mathbb{R}^n$ represents the velocity-tracking error. Suppose that Assumptions assumption.vc and assumption.balldistance are satisfied. Under the conditions that with $\bar{v}$ defined right after barv, $d_a$ and $d_b$ define

Figures (12)

  • Figure 1: The proposed cascade control structure to handle constraints.
  • Figure 2: The basic idea of feasible-set reshaping.
  • Figure 3: A scenario of safety-critical control of a $2$-link robotic manipulator.
  • Figure 4: Trajectories of the velocity reference signal $v^*$ in the numerical simulation.
  • Figure 5: The motion of the $2$-link robotic manipulator in the workspace.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Proposition 3
  • Theorem 2