Autonomous Racing using Learning Model Predictive Control

I. INTRODUCTION

• Long-horizon lap-time planning must limit speed before curves, yet receding-horizon control does not guarantee an optimal closed-loop trajectory.Real-time path generation for racing in real-world conditions remains challenging.Source: §I · p. 1
• Earlier autonomous-racing work includes layered and single-layer MPC, tracking and lap-time MPC, and ILC tracking of an offline aggressive trajectory,,,.The cited ILC approach was experimentally tested on a full-size vehicle.Source: §I · p. 1
• The paper formulates minimum-lap-time racing as reference-free iterative control and couples Learning MPC with system identification to address model mismatch.The approach uses previous iterations, with a relaxation intended to reduce computational burden and meet real-time requirements.Source: §I · p. 1
• At each MPC time step, the proposed identification method estimates dynamics from previous-iteration input-output data near the current state.This is motivated by the difficulty of modeling lateral dynamics near handling limits and the possible computational demands of online LPV estimation.Source: §I · p. 1

A. Vehicle Model

• The vehicle is represented in a curvilinear abscissa frame, with longitudinal progress, lateral and heading errors, velocities, and yaw rate as state quantities.The inputs are longitudinal acceleration and steering angle.Source: A. Vehicle Model · p. 2, Figure 1 · p. 2

B. Minimum Time Problem

• The controller's objective is to minimize the time to reach the finish line at s_target.Source: B. Minimum Time Problem · p. 2
• The optimization is constrained by vehicle dynamics, the initial condition, and state and input constraints, with the road boundaries represented by a convex state set.Source: B. Minimum Time Problem · p. 2
• The set L contains points beyond the finish line, and the formulation assumes a controller can keep the vehicle on the road after crossing it.This assumption treats L as control invariant: from every state in L, an input can keep the next state in L.Source: B. Minimum Time Problem · p. 2

III. LMPC PRELIMINARIES

• At each iteration, Learning MPC plans a trajectory from the shared starting point to the invariant set, using iteration data to improve performance toward a local optimum.The constrained infinite-horizon problem is difficult to solve, especially in real time.Source: §III · p. 2

A. Sampled Safe Set

• Every iteration starts from the same initial state x_S, and the sampled trajectories include the states and inputs realized during those iterations.Source: A. Sampled Safe Set · p. 2
• The sampled safe set SS_j collects state trajectories from successful iterations up to iteration j.Source: A. Sampled Safe Set · p. 2

B. Iteration Cost

• The cost-to-go along the realized closed-loop trajectory is defined using the problem's stage cost, and the iteration cost is its value at the starting time.Source: B. Iteration Cost · p. 3
• The iteration cost at the start quantifies controller performance for that iteration.Source: B. Iteration Cost · p. 3
• Q_j assigns each state in the sampled safe set its minimum cost-to-go across the trajectories in that set.Source: B. Iteration Cost · p. 3

IV. SYSTEM IDENTIFICATION

• The dynamics estimate combines a known system component with an online-identified function whose entries use state-dependent features and estimated parameters.The unknown function is identified using least mean squares.Source: §IV · p. 3
• The data selection uses the trajectory point closest to the current state and measurements from recent steps and nearby portions of prior safe-set trajectories.The measurement vector draws on the last n̄ steps and data around the selected time index in the last n_j+1 trajectories.Source: §IV · p. 3
• The identification procedure uses N̄ = n̄+1+n_j(n̄+n+1) points, with n̄, n_j, and n chosen by the designer.Source: §IV · p. 3

A. LMPC Formulation

• LMPC solves a finite-horizon constrained optimal-control problem at each time step as an approximation to the infinite-horizon problem.Source: A. LMPC Formulation · p. 4
• The optimization enforces system dynamics, the initial condition, state and input constraints, and a terminal state in the previous iteration’s safe set.Source: A. LMPC Formulation · p. 4
• Only the first element of the optimal control sequence is applied, after which the problem is solved again using the new state.This repeated update is a receding-horizon strategy.Source: A. LMPC Formulation · p. 4

B. LMPC Relaxation

• The sampled safe set makes the LMPC a nonlinear mixed-integer program that may be computationally challenging in real time.Source: B. LMPC Relaxation · p. 4
• A local approximation of the prior trajectory uses a fifth-order polynomial, and the relaxed finite-horizon problem is repeatedly solved as the state updates.Source: B. LMPC Relaxation · pp. 4–5
• The prior-iteration cost-to-go is approximated along its trajectory, with the approximation computed through a least-mean-square problem.Source: B. LMPC Relaxation · pp. 4–5
• The relaxation reformulates LMPC with time-varying approximations of the prior safe set and cost-to-go to obtain a computationally tractable problem.Source: B. LMPC Relaxation · pp. 4–5

VI. SIMULATION RESULTS

• The simulation uses CarSim vehicle and tire models identified from experimental passenger-vehicle data and runs closed-loop on a dSpace MicroAutobox II at 100 ms sampling time.The LMPC horizon is N = 10.Source: §VI · p. 5
• The controller cuts the curve with a constant-radius trajectory and saturates the road constraint in the curve.The paper identifies this behavior as optimal for minimum-time racing problems.Source: §VI · p. 5, Figure 3 · p. 6
• The velocity profile slows before the corner and accelerates after its midpoint, while braking until the midpoint to prevent drifting off track.The profile is described as consistent with racing-driver performance.Source: §VI · p. 6, Figure 4 · p. 6
• Travelling time decreases at each iteration until the LMPC converges.The controller successfully decreases the time from the starting point to the finish line.Source: Figure 5 · p. 6, §VI · p. 6
• The distance-based identification technique estimates the vehicle dynamics within an acceptable tolerance against the actual system state.Figure 7 compares one-step prediction error with the actual state at the 45th iteration.Source: §VI · p. 6, Figure 7 · p. 7
• The controller saturates the handling capability of the left front and rear tires, identifying and operating at the vehicle’s performance limit.The simulation tests the control logic on a single corner of a race track.Source: §VI · pp. 5, 7, Figure 8 · p. 7

• The proposed learning nonlinear MPC uses information from previous laps to improve closed-loop performance over iterations.Source: §VII · p. 7
• A time-varying approximation safe set and terminal cost are learned from previous iterations.Source: §VII · p. 7
• Distance-based system identification is coupled with LMPC to support operation at the vehicle’s handling limit.CarSim simulations test the control logic and show vehicle-dynamics identification and operation close to that limit.Source: §VII · p. 7

VIII. APPENDIX I

• The known dynamics for identification are assumed to be a kinematic model in the error reference frame.Source: §VIII · p. 8
• The linear regressor is used to identify longitudinal and lateral vehicle dynamics.Its features are chosen based on the dynamic bicycle model.Source: §VIII · p. 8
• The model describes path curvature and vehicle lateral-distance and heading-angle errors, alongside longitudinal velocity, lateral velocity, and yaw rate.Source: §VIII · p. 8