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A Sequential Quadratic Programming Approach to the Solution of Open-Loop Generalized Nash Equilibria for Autonomous Racing

Edward L. Zhu, Francesco Borrelli

TL;DR

A sequentialquadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration and is locally convergent and central to the effectiveness is a non-monotonic line search method and a novel merit function for SQP step acceptance which helps to improve solver convergence beyond the local neighborhood of a GNE.

Abstract

Dynamic games can be an effective approach for modeling interactive behavior between multiple competitive agents in autonomous racing and they provide a theoretical framework for simultaneous prediction and control in such scenarios. In this work, we propose DG-SQP, a numerical method for the solution of local generalized Nash equilibria (GNE) for open-loop general-sum dynamic games for agents with nonlinear dynamics and constraints. In particular, we formulate a sequential quadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration. The three key elements of the method are a non-monotonic line search for solving the associated KKT equations, a merit function to handle zero sum costs, and a decaying regularization scheme for SQP step selection. We show that our method achieves linear convergence in the neighborhood of local GNE and demonstrate the effectiveness of the approach in the context of head-to-head car racing, where we show significant improvement in solver success rate when comparing against the state-of-the-art PATH solver for dynamic games. An implementation of our solver can be found at https://github.com/zhu-edward/DGSQP.

A Sequential Quadratic Programming Approach to the Solution of Open-Loop Generalized Nash Equilibria for Autonomous Racing

TL;DR

A sequentialquadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration and is locally convergent and central to the effectiveness is a non-monotonic line search method and a novel merit function for SQP step acceptance which helps to improve solver convergence beyond the local neighborhood of a GNE.

Abstract

Dynamic games can be an effective approach for modeling interactive behavior between multiple competitive agents in autonomous racing and they provide a theoretical framework for simultaneous prediction and control in such scenarios. In this work, we propose DG-SQP, a numerical method for the solution of local generalized Nash equilibria (GNE) for open-loop general-sum dynamic games for agents with nonlinear dynamics and constraints. In particular, we formulate a sequential quadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration. The three key elements of the method are a non-monotonic line search for solving the associated KKT equations, a merit function to handle zero sum costs, and a decaying regularization scheme for SQP step selection. We show that our method achieves linear convergence in the neighborhood of local GNE and demonstrate the effectiveness of the approach in the context of head-to-head car racing, where we show significant improvement in solver success rate when comparing against the state-of-the-art PATH solver for dynamic games. An implementation of our solver can be found at https://github.com/zhu-edward/DGSQP.
Paper Structure (28 sections, 1 theorem, 68 equations, 10 figures, 2 tables, 1 algorithm)

This paper contains 28 sections, 1 theorem, 68 equations, 10 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Numerical Solvers for Nash Equilibria of Dynamic Games
  4. Game-Theoretic Methods in Autonomous Racing
  5. Problem Formulation
  6. Generalized Nash Equilibrium
  7. An SQP Approach to Dynamic Games
  8. Local Behavior of Dynamic Game SQP
  9. Numerical Robustness of Dynamic Game SQP
  10. A Novel Merit Function
  11. A Non-Monotone Line Search Strategy
  12. A Decaying Regularization Scheme
  13. The Dynamic Game SQP Algorithm
  14. Frenet-Frame Games for Autonomous Racing
  15. Numerical Study
  16. ...and 13 more sections

Key Result

Theorem 1

Consider the dynamic game defined by eq:dynamic_game. Let Assumptions asm:compact_set_differentiability and asm:optimality hold. Then there exist positive constants $\epsilon_1$ and $\epsilon_2$ such that if and $\lambda_0 = -(\bar{G}_0 \bar{G}_0^\top)^{-1}\bar{G}_0 h_0$, then the sequence $(\mathbf{u}_q, \lambda_q)$ generated by the SQP procedure eq:sqp_step and eq:sqp_qp converges linearly to $

Figures (10)

  • Figure 1: (a) Illustration of the contouring and lag errors $e_c$ and $e_l$. (b) An example of a poor approximation $\bar{s}'$ where $e_l(p,\bar{s})=e_l(p,\bar{s}')=0$, but $\bar{s}' \neq s(p)$.
  • Figure 2: Example head-to-head racing open-loop GNE solutions with $N=25$ for the L-shaped track and 1/10th scale RC car (left) and the first hairpin turn of the Austin F1 track and a full scale race car (right). The red trajectories show pre-computed race lines which are used to warm start Scenarios 2 and 3 of our numerical study. In both plots, the cars are traveling in the counter-clockwise direction.
  • Figure 3: DG-SQP success rates for $\hat{\Gamma}_\text{kin}$ over different regularization weight ($x$-axis) and decay rate ($y$-axis) settings.
  • Figure 4: Results of the solver comparison study on the L-shaped track with the kinematic bicycle model from Scenario 1. Each point corresponds to the average of the sampled initial $x$-$y$ positions for the two agents. $\circ$, $\times$ denote successful and failed trials respectively. The number in the top right of each plot shows the number of successful GNE solves out of the 500 sampled inital conditions.
  • Figure 5: Results of the solver comparison study on the L-shaped track with the dynamic bicycle model from Scenario 2.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof