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Cautious NMPC with Gaussian Process Dynamics for Autonomous Miniature Race Cars

Lukas Hewing, Alexander Liniger, Melanie N. Zeilinger

arXiv:1711.06586v2eess.SYcs.LGcs.SY

TL;DR

Autonomous racing demands accurate dynamics models, but modeling near tire-adhesion limits is challenging and learning-based methods can violate critical constraints during learning. This paper uses GP dynamics learning with uncertainty-aware NMPC and sparse approximations for real-time control. Simulations report improved lap time and safety after one example lap, alongside real-time capability and process-noise tolerance.

  • Problem

    Modeling racing dynamics near tire-adhesion limits is challenging, while learning-based control can have poor transient accuracy and violate critical constraints.

    Source: §I · p. 1

  • Method

    The controller learns dynamics with a GP, incorporates residual model uncertainty into cautious NMPC, and uses a dynamically sparse GP approximation.

    Source: §I · p. 1

  • Results

    The GP-based approaches significantly improve lap time and safety after learning from just one example lap.

    Source: §VI · p. 8

  • Takeaways & Limitations

    Simulations indicate that the sparse GP-based controller is real-time capable and tolerates process noise better than the initial baseline controller.

    Source: B. Results · p. 7

Abstract

from arXiv · show

This paper presents an adaptive high performance control method for autonomous miniature race cars. Racing dynamics are notoriously hard to model from first principles, which is addressed by means of a cautious nonlinear model predictive control (NMPC) approach that learns to improve its dynamics model from data and safely increases racing performance. The approach makes use of a Gaussian Process (GP) and takes residual model uncertainty into account through a chance constrained formulation. We present a sparse GP approximation with dynamically adjusting inducing inputs, enabling a real-time implementable controller. The formulation is demonstrated in simulations, which show significant improvement with respect to both lap time and constraint satisfaction compared to an NMPC without model learning.

I. INTRODUCTION

The paper combines GP-based dynamics learning with uncertainty-aware NMPC for miniature-car racing, aiming to improve performance while maintaining constraint satisfaction. It reports improved racing performance and constraint satisfaction at 30 ms sampling times, with computation on par with NMPC without model learning.

I. INTRODUCTION

  • GP regression learns dynamics from measurements, and its residual uncertainty is integrated into cautious NMPC with chance constraints on states.The formulation permits specifying a minimum probability of satisfying critical constraints, such as track boundaries.Source: §I · p. 1
  • Dynamically selected inducing inputs approximate the GP locally along an approximate state-action trajectory, supporting real-time control despite GP evaluation costs that grow with data.The paper presents this as a response to the computational demands of fast-sampled racing systems.Source: §I · p. 1, B. Gaussian Process Regression · p. 2
  • At 30 ms sampling times, the controller improves racing performance and constraint satisfaction with computation on par with NMPC without model learning.The introduction also reports robustness to process noise.Source: §I · p. 2

A. Notation

  • The notation defines vector and matrix concatenation, element and column indexing, distributions, norms, diagonal matrices, and gradients.The paper denotes a normal distribution with mean μ and variance Σ as N(μ, Σ).Source: A. Notation · p. 2

B. Gaussian Process Regression

  • GP regression infers dynamics from data using a kernel; the paper uses a squared-exponential kernel with a positive diagonal length-scale matrix.The paper notes that other differentiable kernels can also be used.Source: B. Gaussian Process Regression · p. 2

C. Sparse Gaussian Processes

The sparse GP approximation uses inducing inputs to make GP-based prediction more computationally tractable. The approximation conditions training data and test points on inducing targets, and its evaluation cost depends on the number of inducing points rather than the original data count.

C. Sparse Gaussian Processes

  • Sparse GP approximations use inducing targets at inducing inputs and assume training data and test points are conditionally independent given those targets.The approximation is formed by using an inducing conditional distribution to approximate the joint distribution.Source: C. Sparse Gaussian Processes · p. 2
  • The paper uses the Fully Independent Training Conditional approximation to reduce GP computational complexity.Inducing inputs are selected according to the approach described in the preceding passage.Source: C. Sparse Gaussian Processes · p. 2
  • With M-tilde inducing points, predictive mean and variance evaluation costs are O(ndnz M-tilde) and O(ndnz M-tilde^2), respectively.Precomputation makes evaluation complexity independent of the number of original data points.Source: C. Sparse Gaussian Processes · p. 2

III. RACE CAR MODELING

The race-car model combines nominal first-principles dynamics with learned unmodeled dynamics and process noise. Model mismatch and noise are assumed to affect only the velocity states, and the system is discretized for MPC.

A. Car Dynamics

  • The nominal car dynamics use a bicycle model with nonlinear tire forces, while an additional term represents unmodeled dynamics.The model also includes additive Gaussian white noise to account for mismatch from inaccurate parameters and limited model fidelity.Source: A. Car Dynamics · p. 3
  • Model uncertainty and process noise are assumed to affect only the longitudinal velocity, lateral velocity, and yaw-rate states.The first three states follow kinematic relationships in the nominal model.Source: A. Car Dynamics · p. 3
  • The system is discretized with the Euler forward scheme for MPC, and the uncertain dynamics function and process noise are inferred from measurement data.The passage specifies normally distributed, independent process noise.Source: A. Car Dynamics · p. 3

B. Race Track and Constraints

The track is represented by a spline centerline and fixed width, with the car constrained to remain within the track boundaries. Steering angle and motor duty cycle are also bounded.

B. Race Track and Constraints

  • A piecewise cubic spline parameterized by path length defines the centerline and its position and orientation; the car's position is projected onto that centerline for the track constraint.The track constraint uses r as half the track width.Source: B. Race Track and Constraints · p. 3
  • The input constraints limit steering angle to a maximum and require motor duty cycle to remain between zero and one.Source: B. Race Track and Constraints · p. 3

IV. LEARNING-BASED CONTROLLER DESIGN

The controller learns a stochastic residual dynamics model from measurements and combines it with contouring control in an expected-cost NMPC formulation. It uses chance constraints for track limits, while the 30 ms sampling setting motivates computational approximations.

A. Model Learning

  • Training data represent deviations from the nominal system model, from which GP regression infers the residual dynamics function.The resulting state used by the predictive controller is represented as a stochastic distribution.Source: A. Model Learning · pp. 3–4

B. Contouring Control

  • The contouring formulation maximizes track progress over the horizon while penalizing contouring and lag errors and large changes in control inputs and progress.An incremental progress variable is linked to the car’s projected position on the track centerline.Source: B. Contouring Control · p. 4

IV. LEARNING-BASED CONTROLLER DESIGN

  • The controller integrates a learned stochastic GP dynamics model into a finite-horizon NMPC problem that minimizes expected cost and imposes track chance constraints.The design sequence introduces model learning, contouring control, and approximations to make the learning-based controller real-time feasible.Source: §IV · p. 3, B. Contouring Control · p. 4

B. Contouring Control

  • At about 30 ms sampling times, the stochastic optimization is computationally demanding because state distributions generally become non-Gaussian after the first prediction step.The following subsections introduce approximations to reduce complexity while retaining the benefits of learning.Source: B. Contouring Control · p. 4

C. Approximate Uncertainty Propagation

The controller approximates propagated state distributions as Gaussian so it can represent uncertainty through tractable mean and variance dynamics. A first-order Taylor approximation provides a computationally cheap propagation procedure described as sufficiently accurate.

C. Approximate Uncertainty Propagation

  • Because GP uncertainty propagation makes later state distributions non-Gaussian, the controller approximates each predicted state distribution as Gaussian.This approximation supports tractable propagation of state uncertainty through the prediction horizon.Source: C. Approximate Uncertainty Propagation · p. 4
  • A first-order Taylor expansion yields the mean and variance dynamics, offering a computationally cheap procedure of sufficient accuracy.A sigma-point transform is also mentioned as an alternative for deriving Gaussian-distribution dynamics.Source: C. Approximate Uncertainty Propagation · p. 4

D. Simplified Chance Constraints

The method approximates probabilistic track constraints with deterministic bounds on the state mean and variance. The bound uses a confidence-region approximation based on the largest variance direction.

D. Simplified Chance Constraints

  • For Gaussian states, chance constraints can be approximated by deterministic constraints on the state mean and variance.The method uses a confidence-region bound to limit the probability of violating the track constraint.Source: D. Simplified Chance Constraints · pp. 4–5
  • The confidence-region bound uses the chi-squared quantile and the covariance’s maximum eigenvalue to tighten the allowable mean-state region.The resulting confidence region is an outer approximation oriented along the direction of largest variance.Source: D. Simplified Chance Constraints · pp. 4–5
  • Figure 2 shows the car’s planned mean trajectory with active chance constraints, and depicts a 1-σ confidence level perpendicular to the car’s mean orientation.Source: Figure 2 · p. 5
  • The chance-constraint bound and objective require variance dynamics, motivating a further approximation to reduce computational cost.The paper describes the tightening as similar to robust-control constraint tightening, with its amount related to an approximate confidence region around the mean state.Source: D. Simplified Chance Constraints · p. 5

E. Time-Varying Approximation of Variance Dynamics

The controller reduces computation by approximating variance along a reference state-action trajectory and choosing sparse-GP inducing inputs locally around an MPC trajectory. These approximations target fast evaluation while retaining close agreement between full and sparse GP predictions along the predicted path.

E. Time-Varying Approximation of Variance Dynamics

  • The method evaluates variance along an approximate state-action trajectory, trading system-description accuracy for lower computational complexity.The reference trajectory can be chosen for tracking or obtained by shifting a prior MPC solution.Source: E. Time-Varying Approximation of… · p. 5
  • Precomputed variance along the approximate trajectory is independent of optimization variables and is substituted into the chance constraints approximately.The variance can be computed before the state measurement becomes available at each sampling time.Source: E. Time-Varying Approximation of… · p. 5

F. Dynamic Sparse GP

  • Sparse-GP evaluation can remain prohibitive for fast applications with high-dimensional state-input spaces despite reducing computation with few inducing inputs.This motivates selecting inducing inputs locally at each sampling time.Source: F. Dynamic Sparse GP · p. 5
  • Local inducing inputs let the sparse GP approximation focus computation near the state-action region of interest along an MPC trajectory.The approach exploits the expectation that the relevant region at each sampling time lies close to a known trajectory.Source: F. Dynamic Sparse GP · p. 5
  • Full and dynamically sparse GP approximations are in close correspondence along the predicted system trajectory in the illustrated double-integrator example.The example compares the current prediction with the previous prediction used for local approximation.Source: F. Dynamic Sparse GP · p. 5
  • Receding-horizon updates can add or remove individual inducing points with rank-1 updates, allowing efficient updates to the corresponding Cholesky factorizations.The reformulation avoids inversion of the large matrix Qa.Source: F. Dynamic Sparse GP · p. 5

G. Resulting Control Formulation for Autonomous Racing

The integrated approximations turn the GP-based racing controller into a deterministic nonlinear program while retaining approximate uncertainty and simplified chance constraints. The section specifies the sparse-GP setup and simulation conditions used to evaluate the formulation.

G. Resulting Control Formulation for Autonomous Racing

  • Figure 3 compares full-GP and dynamic-sparse posterior variance contours, including their variances along the planned trajectory.The figure distinguishes the current planned trajectory from the previous trajectory used for approximation and marks inducing points.Source: Figure 3 · p. 6
  • Reducing learned dynamics to the GP mean, with approximate variance dynamics and simplified chance constraints, yields a deterministic nonlinear program of moderate dimension.Source: G. Resulting Control Formulation for… · pp. 5–6
  • The cost is quadratically approximated around the shifted prior solution, while constraint evaluation fixes Θᵢ using that previous solution.This lets the track spline be evaluated separately from the optimization procedure.Source: G. Resulting Control Formulation for… · p. 6

A. Simulation Setup

  • The race simulation perturbs nominal dynamics parameters by up to ±15% and compares full-GP and sparse-GP controllers with nominal-model and true-model controllers.The no-process-noise trajectory figure shows the baseline, reference, and sparse GP-based controller.Source: §V · p. 6, Figure 5 · p. 7
  • The dynamic sparse GP uses 10 inducing inputs placed with exponentially decaying density along the previous solution, emphasizing current and near-future states.Source: A. Simulation Setup · p. 6
  • Soft constraints use nonnegative slack variables, and chance-constraint tightening is applied only to the first 15 prediction steps.Source: A. Simulation Setup · p. 6
  • One baseline-controlled lap supplies 350 GP training data points; simulations use a 30 ms sampling time on a 2.6 GHz i7-5600 laptop with 12 GB RAM.Source: A. Simulation Setup · p. 6
  • During a race lap, the sparse GP’s yaw-rate error prediction corresponds well to the learned dynamics error, and its uncertainty matches residual model uncertainty and process noise.The plotted GP uses 10 inducing inputs; the shaded region is a 2-σ confidence interval including noise.Source: A. Simulation Setup · p. 6, Figure 4 · p. 6

B. Results

In simulations, sparse-GP control improved on the nominal-model baseline while approaching true-model performance and maintaining constraint satisfaction. It tolerated process noise and met the reported real-time solve-time threshold; the conclusion reports gains after one example lap.

B. Results

  • The sparse GP controller performs close to the full GP in lap time and constraint satisfaction, and significantly outperforms the baseline; the full GP matches the reference lap-time performance.The reported comparison covers simulations without process noise.Source: Table I · p. 7, B. Results · p. 7
  • Under process noise, the baseline shows severe constraint violations and loses track in eight runs that are removed as outliers, while the other formulations tolerate noise well.The reference is slightly faster than the GP formulations but has higher constraint violations; chance-constraint violation probability tunes the aggression-safety trade-off.Source: B. Results · p. 7
  • 99.9% of sparse-GP solve times are below 25.6 ms, within the 30 ms sampling time; sparse-GP updating adds 2.5 ms on average but does not delay input application.Source: B. Results · p. 7
  • The authors report significant performance improvement while maintaining safety and approaching the true-model reference, alongside real-time implementation and better tolerance of process noise than the baseline.Source: B. Results · p. 7

VI. CONCLUSION

  • Simulations show that GP-based approaches significantly improve lap time and safety after learning from one example lap.Source: §VI · p. 8

APPENDIX I UNCERTAINTY PROPAGATION FOR NONLINEAR SYSTEMS

The passage outlines uncertainty propagation using expectation and variance identities, then invokes first-order expansions around the mean state for approximation.

APPENDIX I UNCERTAINTY PROPAGATION FOR NONLINEAR SYSTEMS

  • The derivation uses the law of iterated expectation and the law of total variance to obtain the propagation expressions.Source: Appendix I · p. 8
  • The mean dynamics expression combines the nominal dynamics f(xᵢ, uᵢ) with the GP mean correction B_dµ_d(xᵢ, uᵢ).Source: Appendix I · p. 8
  • The passage approximates f, µ_d, and Σ_d by first-order expansions around the mean state.Source: Appendix I · p. 8
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