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Extending First-order Robotic Motion Planners to Second-order Robot Dynamics

Mayur Sawant, Abdelhamid Tayebi

TL;DR

This work addresses the challenge of using first-order motion planners in robots governed by second-order dynamics. It introduces two control strategies: Dynamic Damping Feedback (DDF), which combines the first-order planner with a distance-weighted damping term when a scalar potential aligned with the planner is available, and Velocity Tracking Feedback (VTF), which enforces convergence of the robot's velocity to the planner without requiring such a potential. The authors prove forward invariance of the free space and almost global asymptotic stability of the target under both schemes, and validate the approaches through simulations in environments with circular, ellipsoidal, and multiple obstacles. The results enable safe navigation and efficient convergence for second-order robots using existing first-order planners, without imposing rigid initial-velocity restrictions.

Abstract

This paper extends first-order motion planners to robots governed by second-order dynamics. Two control schemes are proposed based on the knowledge of a scalar function whose negative gradient aligns with a given first-order motion planner. When such a function is known, the first-order motion planner is combined with a damping velocity vector with a dynamic gain to extend the safety and convergence guarantees of the first-order motion planner to second-order systems. If no such function is available, we propose an alternative control scheme ensuring that the error between the robot's velocity and the first-order motion planner converges to zero. The theoretical developments are supported by simulation results demonstrating the effectiveness of the proposed approaches.

Extending First-order Robotic Motion Planners to Second-order Robot Dynamics

TL;DR

This work addresses the challenge of using first-order motion planners in robots governed by second-order dynamics. It introduces two control strategies: Dynamic Damping Feedback (DDF), which combines the first-order planner with a distance-weighted damping term when a scalar potential aligned with the planner is available, and Velocity Tracking Feedback (VTF), which enforces convergence of the robot's velocity to the planner without requiring such a potential. The authors prove forward invariance of the free space and almost global asymptotic stability of the target under both schemes, and validate the approaches through simulations in environments with circular, ellipsoidal, and multiple obstacles. The results enable safe navigation and efficient convergence for second-order robots using existing first-order planners, without imposing rigid initial-velocity restrictions.

Abstract

This paper extends first-order motion planners to robots governed by second-order dynamics. Two control schemes are proposed based on the knowledge of a scalar function whose negative gradient aligns with a given first-order motion planner. When such a function is known, the first-order motion planner is combined with a damping velocity vector with a dynamic gain to extend the safety and convergence guarantees of the first-order motion planner to second-order systems. If no such function is available, we propose an alternative control scheme ensuring that the error between the robot's velocity and the first-order motion planner converges to zero. The theoretical developments are supported by simulation results demonstrating the effectiveness of the proposed approaches.

Paper Structure

This paper contains 25 sections, 6 theorems, 62 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Distance to a set
  4. Geometric sets
  5. Hyperplane
  6. Problem statement
  7. Feedback control design
  8. Feedback control law for Problem \ref{['problem_with_NF']}
  9. Feedback control law for Problem \ref{['problem_with_VF']}
  10. Simulation results
  11. Conclusion
  12. Proof of Lemma \ref{['lemma:NF']}
  13. Proof of Claim \ref{['lemma:NF_claim1']}
  14. Proof of Claim \ref{['lemma:NF_claim2']}
  15. Proof of Claim \ref{['lemma:NF_claim3']}
  16. ...and 10 more sections

Key Result

Lemma 1

Consider the closed-loop system equation:second_order_system-controller:NF, under Assumptions assumption:conditions_on_unsafe_set and assumption:common. If $d_{\mathbf{x}}(0) > 0$, then the following statements hold:

Figures (10)

  • Figure 1: Block diagram illustrating the adaptation of first order motion planners to second-order dynamics via obstacle-aware damping.
  • Figure 2: Block diagram of the dynamic damping feedback controller \ref{['controller:NF']}.
  • Figure 3: Block diagram of the velocity tracking feedback controller \ref{['controller:VF']}.
  • Figure 4: Robot $\mathbf{x}$-trajectories under the DDF control law (blue curve), the VTF control law (magenta curve), and the fixed damping control law (red curve) which are defined in \ref{['controller:NF']}, \ref{['controller:VF']}, and \ref{['controller:fixed_damping']}, respectively.
  • Figure 5: (a) Robot $\mathbf{x}$-trajectories under the DDF control law and the VTF control law, represented using solid curves and dashed curves, respectively. (b) Path lengths of $\mathbf{x}$-trajectories in meters.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Theorem 1
  • Remark 4
  • Lemma 2
  • Theorem 2
  • Remark 5
  • Lemma 3
  • ...and 2 more