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Optimization-Based Autonomous Racing of 1:43 Scale RC Cars

Alexander Liniger, Alexander Domahidi, Manfred Morari

arXiv:1711.07300v1math.OCcs.ROeess.SY

TL;DR

The paper addresses real-time autonomous racing near vehicle stability limits, where fast dynamics and obstacle avoidance constrain computational complexity. It develops two optimization-based receding-horizon controllers and demonstrates experimental racing performance, including effective overtaking around static obstacles.

  • Problem

    Autonomous racing requires control near stability limits, real-time obstacle-aware path planning, and low computational complexity for embedded platforms.

  • Method

    Two receding-horizon optimization controllers maximize track progress subject to track and obstacle constraints, using hierarchical planning or a unified contouring-control formulation with locally approximated convex subproblems.

  • Results

    8.9–9.2 s lap times were achieved by MPCC, compared with 9.5–9.8 s for HRHC across three laps.

  • Takeaways & Limitations

    Progress-optimal corridor planning enables fast and safe overtaking maneuvers around static obstacles.

  • Takeaways & Limitations

    The demonstrated overtaking work focuses on static obstacles because predictions of dynamically moving opponents are unavailable.

Abstract

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This paper describes autonomous racing of RC race cars based on mathematical optimization. Using a dynamical model of the vehicle, control inputs are computed by receding horizon based controllers, where the objective is to maximize progress on the track subject to the requirement of staying on the track and avoiding opponents. Two different control formulations are presented. The first controller employs a two-level structure, consisting of a path planner and a nonlinear model predictive controller (NMPC) for tracking. The second controller combines both tasks in one nonlinear optimization problem (NLP) following the ideas of contouring control. Linear time varying models obtained by linearization are used to build local approximations of the control NLPs in the form of convex quadratic programs (QPs) at each sampling time. The resulting QPs have a typical MPC structure and can be solved in the range of milliseconds by recent structure exploiting solvers, which is key to the real-time feasibility of the overall control scheme. Obstacle avoidance is incorporated by means of a high-level corridor planner based on dynamic programming, which generates convex constraints for the controllers according to the current position of opponents and the track layout. The control performance is investigated experimentally using 1:43 scale RC race cars, driven at speeds of more than 3 m/s and in operating regions with saturated rear tire forces (drifting). The algorithms run at 50 Hz sampling rate on embedded computing platforms, demonstrating the real-time feasibility and high performance of optimization-based approaches for autonomous racing.

I. INTRODUCTION

The paper develops optimization-based autonomous racing controllers for nonlinear, near-limit driving under tight real-time and embedded-computing constraints. It presents hierarchical and integrated formulations that maximize track progress while incorporating track and obstacle constraints.

  • Motivation: Autonomous racing requires control near vehicle stability limits, real-time obstacle-aware path planning, and low computational complexity for fast dynamics and embedded platforms.Sampling times must be only a few tens of milliseconds at most.
  • Approach: The proposed approach uses a nonlinear bicycle model with nonlinear tire forces and requires only a track map and a dynamical car model.The experiments use 1:43 scale Kyosho dnano cars reaching more than 3 m/s and operating near friction limits.
  • Control formulations: The paper presents a hierarchical path-planning and NMPC-tracking scheme and a single MPCC formulation combining path generation and tracking.Both formulations maximize progress measured by the vehicle’s projection onto the track center line.
  • Real-time optimization: Linearized LTV models produce tractable convex optimization subproblems, while embedded interior-point solvers support real-time implementation.The resulting optimization problems are solved at each sampling time using FORCES-generated solvers.
  • Obstacle avoidance: Obstacle avoidance is enabled by adapting corridor constraints to opponents, with progress optimization producing automatically planned overtaking maneuvers.Non-convex left-versus-right overtaking choices are handled by high-level corridor planning based on dynamic programming.
  • Contribution: The work addresses limitations of prior racing approaches by combining dynamically feasible optimization with real-time execution and obstacle avoidance.Prior methods include higher-level planners, one-level obstacle penalties, mixed-integer formulations, and sampling-based trajectory generation.

C. Outline

The paper outlines a nonlinear bicycle model for the RC cars and the notation used throughout. The model represents planar vehicle motion, tire forces, motor and resistance effects, and behavior near saturated tire-force limits.

  • Organization: The paper is organized around the vehicle model, hierarchical and combined control approaches, obstacle avoidance, and experimental evaluation.Section II covers the dynamical model, Section III covers control and obstacle avoidance, and Section IV covers the testbed.
  • A. Bicycle Model: The bicycle model treats the car as one rigid body with mass m and inertia I_z, considering only in-plane motion while neglecting pitch, roll, and load changes.Because the car is rear-wheel driven without active brakes, the front-wheel longitudinal force is neglected.
  • A. Bicycle Model: The model’s kinematic states are inertial position X, Y and heading ϕ, while its kinetic states are longitudinal velocity v_x, lateral velocity v_y, and yaw rate ω.The kinetic quantities are expressed in a body-fixed frame centered at the center of gravity.
  • A. Bicycle Model: The inputs are motor PWM duty cycle d and steering angle δ; subscripts x and y denote longitudinal and lateral quantities, while f and r denote front and rear tires.The parameters l_f and l_r give distances from the center of gravity to the front and rear wheels.
  • A. Bicycle Model: Lateral tire forces use a simplified Pacejka model, whose parameters B, C, and D determine the semi-empirical friction-curve shape.The model is described as a good approximation to measured friction curves in practice.
  • A. Bicycle Model: The identified model includes rear-wheel combined-slip effects and can represent saturated or near-saturated tire forces at the car’s handling limits.Wheel-speed measurement is unavailable, preventing use of combined-slip models that require it.

B. Stationary Velocity Analysis

The stationary-velocity analysis identifies steady motions across steering angles and forward speeds, including normal, understeering, and drifting regimes. These regimes differ in tire saturation, stability, and attainable yaw behavior.

  • Stationary velocity computation: Stationary velocities are computed by solving the zero-acceleration equations for lateral velocity and yaw rate across forward velocities and steering angles.The analysis considers constant forward velocity and steering angle, with lateral velocity and yaw rate as unknowns.
  • Velocity regimes: Figure 2 compares stationary velocities at ¯vx = 1.5 m/s and ¯vx = 2 m/s as functions of steering angle ¯δ.
  • Velocity regimes: The model separates normal driving, understeering, and oversteering regions according to lateral velocity, yaw-rate behavior, and tire-force saturation.The oversteering region corresponds to drifting with high lateral velocity.
  • Stability and drift: The drifting region is unstable, whereas the normal-driving and understeering regions are stable; drifting can also involve counter-steering.
  • Speed effects: As speed increases, the nonsaturated steering region narrows and the maximum yaw rate decreases.
  • Geometric interpretation: Every stationary velocity corresponds to uniform circular motion with constant radius and curvature, allowing the model to represent drifting around a circle.

III. AUTONOMOUS RACING CONTROL

The autonomous racing control architecture uses optimization-based receding-horizon planning and control built on a vehicle model. The hierarchical planner generates feasible progress-maximizing trajectories, including drift trajectories, while enforcing track constraints.

  • Control formulations: The paper presents two optimization-based racing formulations: a hierarchical two-level controller and a one-level contouring-control formulation.
  • Hierarchical control: The hierarchical controller repeatedly selects a progress-optimal feasible trajectory over N sampling times and tracks it with a soft-constrained MPC controller.This receding-horizon scheme is called the Hierarchical Receding Horizon Controller.
  • Path planning: The path planner grids stationary velocities from the nonlinear model across longitudinal velocities and steering angles, including normal and drifting points.
  • Path planning: Candidate trajectories are generated by integrating stationary points, producing a countable set of constant-turning-radius trajectories over the horizon.
  • Path-planning objective: The planner maximizes projected track progress while requiring candidate positions to remain inside the track and planned velocity to remain reasonably close to the measured velocity.
  • Path-planning solution: The integer path-planning problem is solved by enumerating about one hundred trajectories against local convex track approximations, and it can select drifting when drift offers greatest progress.
  • Limitations: Path-planner performance depends on horizon length, and its constant velocity over the horizon can cause problems in complicated curve combinations such as chicanes.Long horizons can create infeasible circular trajectories, while short horizons recognize curves too late.

2) Model Predictive Reference Tracking.:

The reference-tracking controller uses a linearized, soft-constrained MPC formulation to track the path planner’s simplified reference trajectory while retaining computational tractability.

  • Reference tracking: The reference trajectory cannot be tracked by directly applying the stationary-point controls because it was generated under simplifying assumptions.
  • MPC formulation: The MPC objective penalizes deviation from the reference trajectory with a quadratic cost on states and controls, plus penalties for soft constraints.
  • MPC constraints: The optimization enforces the initial state, linearized dynamics, soft track-related inequalities, state bounds, and input bounds.
  • Trajectory linearization: The model is linearized around the entire reference trajectory rather than one operating point because position and orientation can vary significantly across the horizon.
  • Real-time solution: The formulation is a convex QP whose parameters include the initial state, trajectory-varying dynamics matrices, and changing halfspace constraints, enabling tailored embedded C-code generation.

B. Model Predictive Contouring Control

The model predictive contouring controller combines path planning and tracking in one nonlinear optimization problem. It maximizes progress along a reference path while using local convex QP approximations for real-time solution.

  • Contouring control: The MPCC formulation maximizes traveled distance along the reference path while assigning low weight to contouring error, producing trajectories similar to expert-driven racing lines.
  • Unified optimization: Path planning and path tracking are combined into one nonlinear optimization problem approximated by local convex QPs at each sampling time.
  • Track constraints: The track is approximated by halfspaces generated for horizon points from nearby center-line points and their projections onto the track borders.
  • Real-time solution: The resulting QPs use multistage structure exploited by FORCES to obtain solution times in the range of a few milliseconds.
  • Reference path: The reference path is represented by third-order splines parameterized by arc length, providing interpolated position and tangent angle values.This interpolation is more accurate than the piecewise-linear parameterization used for the hierarchical controller.
  • Error measures: Contouring error measures orthogonal distance from the reference path, while lag error measures the quality of the controller’s approximate projection along that path.
  • Error approximation: The controller replaces the projection operator with an independent approximate projection and uses position-based linearized contouring and lag errors.

3) MPCC problem.:

The MPCC formulates autonomous racing as a finite-horizon optimization that balances track progress against contouring accuracy while enforcing vehicle, track, and input constraints. An approximate path projection makes the formulation suitable for real-time implementation.

  • MPCC trades off maximum progress and low contouring error over a horizon of N sampling times subject to model, track, and input constraints.The progress and path-following objectives are weighted by γ and q_c, respectively.
  • The vehicle position is determined by the nonlinear dynamics, while the path parameter is the orthogonal projection onto the reference path.The contouring error measures deviation from the reference path, and θ_P,k denotes the associated projection parameter.
  • Corridor constraints contain the vehicle within the allowed racing corridor, and state and input bounds enforce physically admissible values.The corridor is represented through parallel half-space constraints.
  • The exact projection creates a bi-level NLP that is too complex for real-time solution, so MPCC replaces it with an approximate projection θ_A,k.The approximation introduces progress dynamics through θ_A,k+1 = θ_A,k + v_k/T_s and uses a lag-error cost to control approximation quality.
  • Progress bounds and input-rate penalties constrain spurious solutions and encourage smooth controls that avoid amplifying unmodeled dynamics.The projected velocity v_k is bounded, and the lag-error weight q_l is chosen high to strengthen coupling between progress and vehicle dynamics.

4) Solving the MPCC problem.:

The MPCC NLP is solved through successive local convex QP approximations formed by linearizing the nonlinear model and error terms. Soft corridor constraints, real-time iteration, and a structure-exploiting solver support embedded implementation.

  • Local convex QPs are built at each sampling time by linearizing the nonlinear terms of the MPCC optimal control problem.The QP uses a linear time-varying approximation of the vehicle dynamics and contouring and lag errors.
  • The linearized prediction model has dynamics x_k+1 = A_k x_k + B_k u_k + g_k with fixed initial state and progress parameter.The initial conditions are x_0 = x and θ_A,0 = θ.
  • State, progress, input, and projected-velocity bounds remain enforced in the local QP.The progress parameter and projected velocity are bounded alongside the physical state and control inputs.
  • Obstacle avoidance is represented by a corridor derived from the overtaking planner, with the corridor shown as the resulting feasible region after dynamic programming.The planner’s path and corridor are passed to the HRHC or MPCC controller.
  • Soft constraints use slack variables and a high infinity-norm penalty to recover hard-constraint behavior whenever possible.The slack vector is s_k ∈ R^2 and its penalty is weighted by q.
  • The re-linearization scheme performs one sequential-QP step per time step, and FORCES solves the lifted multistage QP.Lifting introduces a copy of the previous input at each stage to accommodate input-rate costs.

C. Obstacle Avoidance

Obstacle avoidance is handled by adapting a feasible racing corridor online to opponent positions. A dynamic-programming planner resolves the non-convex overtaking-side decisions, then supplies corridor constraints to either controller.

  • Choosing whether to overtake each opponent on the left or right is combinatorial and generally produces a non-convex corridor-planning problem.The difficulty increases when more than one opponent is involved.
  • A high-level dynamic-programming planner solves a shortest-path problem on a spatial-temporal grid to choose overtaking sides.The grid is built from the previous lower-level path or QP solution, depending on whether HRHC or MPCC is used.
  • The dynamic-programming result produces adapted corridor borders that are passed to the HRHC or MPCC controller.The corridor is issued after the planner computes an optimal path around prohibited grid points.
  • The two racing controllers maximize progress within a time horizon while directly incorporating track and obstacle constraints.Their main distinction is separate path planning plus tracking for HRHC versus one contouring-control optimal-control formulation for MPCC.
  • Online corridor adaptation incorporates obstacle avoidance by changing optimization constraints according to the current positions of opponents.The corridor is represented using two parallel affine inequalities, or slabs, for feasible positions at each time.
  • Both algorithms compensate fixed control delay by simulating the nonlinear model forward before solving the next planning and control problem.The forward simulation uses second-order Runge-Kutta integration initialized by the Kalman-filter state estimate.

IV. RESULTS

The paper evaluates both controllers experimentally on a 1:43-scale RC-car testbed, with implementation and closed-loop performance reported in the following results section.

  • Performance evaluation uses 1:43 scale RC race cars in an experimental testbed.The passage introduces the setup, implementation, and closed-loop performance as the subjects of evaluation.

A. Experimental Setup

Experiments use 1:43 Kyosho dnano cars on a tracked 18.43 m circuit, with embedded implementations of HRHC and MPCC controllers. MPCC achieves faster single-car laps, while its longer-horizon planning avoids several HRHC trajectory limitations but introduces computational and robustness trade-offs.

  • Platform: The testbed uses 1:43 Kyosho dnano cars capable of speeds over 3 m/s, corresponding to about 465 km/h when upscaled.The cars include front and rear suspension and a rear axle differential.
  • Platform: The 18.43 m track is observed by a 100 Hz infrared camera system with below 4 mm accuracy, while an EKF estimates vehicle state.Reflective markers identify cars and measure their positions and orientations.
  • Single-car performance: 9.5 to 9.8 s HRHC laps are slower than 8.9 to 9.2 s MPCC laps across three single-car laps.Figure 8 encodes velocity profiles by color for the driven trajectories.
  • HRHC limitations: The HRHC path planner’s constant-radius, constant-velocity horizon limits performance through curve combinations such as the chicane.Its short lookahead can produce poor positioning, extra braking, and occasional border contact.
  • MPCC performance: The MPCC’s longer horizon and independently optimized velocities plan better trajectories through complicated curves, reaching at least 1 m/s where HRHC falls to nearly 0.5 m/s.The MPCC trajectory is closer to an ideal line in a least-curvature sense, especially through the chicane.
  • MPCC limitations: MPCC has disadvantages from center-line tracking and contouring-error penalties, which can produce unnecessary center-line movements and S-shaped trajectories.The HRHC can drive some sections straight and therefore gain a speed benefit.
  • Trade-offs: MPCC improves trajectory planning but costs about five times more computation than HRHC and is more sensitive to measurement errors, model drift, and unexpected skidding.HRHC performance could improve with multiple velocities per horizon, but its path-planning complexity would grow exponentially.

D. Racing with multiple cars

Both controllers support progress-maximizing obstacle avoidance using planned corridors and demonstrate overtaking experiments with static obstacles. MPCC plans around obstacle groups earlier, whereas HRHC’s shorter horizon adapts incrementally and both controllers meet soft real-time operation at 50 Hz.

  • Obstacle-avoidance objective: Both controllers optimize progress while planning paths around opponents, enabling fast and safe overtaking maneuvers.The current experiments focus on static obstacles; moving-obstacle overtaking would require behavior prediction.
  • Experimental scenario: The obstacle-avoidance experiments use driven trajectories with static obstacles and visualize three time-point snapshots of the corresponding planned corridors.Figures 9 and 10 show the trajectories and detailed prediction-horizon views.
  • Corridor planning: The high-level corridor planner issues constraints to the controllers according to the obstacle situation and track layout.The corridor is adjusted so obstacle avoidance remains part of the optimization problem.
  • Multiple-car racing: MPCC plans around all obstacles before reaching the first car, while HRHC initially avoids only the first group and discovers a route around the remaining cars later.MPCC’s longer horizon provides earlier planning but can become problematic when model mismatch makes the planned path too optimistic.
  • Computation: Direct controller-time comparison is constrained because equal horizons are infeasible for HRHC without combinatorial multi-segment path planning, while MPCC cannot robustly use HRHC’s short horizon.The different horizon requirements prevent a straightforward like-for-like timing comparison.
  • Computation: 20 ms sampling was missed in 0.07% of HRHC instants and 4.4% of MPCC instants across three laps, while both schemes operated at 50 Hz in soft real time.The reported counts are one HRHC instant and 60 MPCC instants.
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