Race Car Problem

Mathematical Formulation

Problem Statement

Find the optimal throttle control \(\theta(t)\) that minimizes the lap time:

\[J = t_f\]

Subject to the dynamic constraints:

\[\frac{d\text{pos}}{dt} = \text{speed}\]

\[\frac{d\text{speed}}{dt} = \theta - \text{speed}\]

And the speed limit constraint:

\[\text{speed}(t) \leq 1 - \frac{1}{2}\sin(2\pi \cdot \text{pos}(t))\]

Boundary Conditions

• Initial conditions: \(\text{pos}(0) = 0\), \(\text{speed}(0) = 0\)

• Final conditions: \(\text{pos}(t_f) = 1.0\), \(\text{speed}(t_f) = \text{free}\)

• Control bounds: \(0 \leq \theta \leq 1\)

Physical Parameters

• Track length: \(L = 1.0\) (normalized)

• Maximum throttle: \(\theta_{\max} = 1.0\)

State Variables

• \(\text{pos}(t)\): Position along track (normalized, 0 to 1)

• \(\text{speed}(t)\): Vehicle speed

Control Variable

• \(\theta(t)\): Throttle input (0 = no throttle, 1 = full throttle)

Notes

The speed limit varies sinusoidally along the track, representing curves where the car must slow down and straights where higher speeds are allowed.

Running This Example

cd examples/race_car
python race_car.py

Code Implementation

examples/race_car/race_car.py

import casadi as ca
import numpy as np

import maptor as mtor


# Problem setup
problem = mtor.Problem("Car Race")
phase = problem.set_phase(1)

# Variables
t = phase.time(initial=0.0)
pos = phase.state("position", initial=0.0, final=1.0)
speed = phase.state("speed", initial=0.0)
throttle = phase.control("throttle", boundary=(0.0, 1.0))

# Dynamics
phase.dynamics({pos: speed, speed: throttle - speed})

# Constraints
speed_limit = 1 - ca.sin(2 * ca.pi * pos) / 2
phase.path_constraints(speed <= speed_limit)

# Objective
problem.minimize(t.final)

# Mesh and guess
phase.mesh([8, 8, 8], np.array([-1.0, -0.3, 0.3, 1.0]))
phase.guess(terminal_time=2.0)  # MUCH CLEANER!

# Solve
solution = mtor.solve_adaptive(
    problem,
    min_polynomial_degree=4,
    max_polynomial_degree=8,
    nlp_options={"ipopt.print_level": 0},
)

# Results
if solution.status["success"]:
    print(f"Lap time: {solution.status['objective']:.6f}")
    solution.plot()
else:
    print(f"Failed: {solution.status['message']}")