Mujoco-Ackermann-Experiments

Mini repository that contains various ways to calculate the steering angles for Ackermann geometry. The goal is to find the approximation with the least error for MuJoCo cars — specifically, a quartic polynomial that maps the central reference steering angle to a front-wheel steer joint, emitted as a MuJoCo joint-equality polycoef="a0 a1 a2 a3 a4" vector.

Features

• Calculate quartic polycoef coefficients for Ackermann steering in MuJoCo, via two methods:
  • Taylor expansion of the exact geometry around centre (taylor_differentiation.py)
  • Least-squares fit over the steering range (numpy_polyfit.py) — fits over a symmetric range of ±delta_deg (function default ±30°, driven by max_angle_deg from the CLI), sampled at num_samples evenly-spaced points (default 200)
• Print an error table (SSE / RMSE / max error) comparing the methods
• Graphical comparison of how each method fits the exact geometry
• Ideal for automotive engineers, designers, and enthusiasts working on steering design

Installation

The project uses a conda environment named ackermann.

# Create the environment from the spec
conda env create -f environment.yml
conda activate ackermann

Already have the ackermann env? Update it in place instead:

conda env update -f environment.yml --prune

Prefer plain pip? Use requirements.txt (note: Tkinter must come from your Python install — it is not a pip package):

pip install -r requirements.txt

Usage

python main.py [L] [W] [max_angle_deg] [n_samples]

• L — wheelbase (default 1.2)
• W — track width (default 1.0)
• max_angle_deg — steering lock, the +/- range to fit over (default 23)
• n_samples — number of sample points (default 400)

Running it prints the error table and the polycoef strings, then pops up the comparison figure. Example:

python main.py 1.2 1.0 23 400

Why RMSE is the metric that matters most

The error table reports three numbers, but RMSE (root mean square error) is the one to optimize for. Here's the reasoning:

• SSE (sum of squared errors) depends on how many sample points you take. Double n_samples and the SSE roughly doubles, even though the fit hasn't changed at all. That makes it useless for comparing runs with different sampling — it measures the grid as much as the fit.
• Max error captures only a single worst-case point. It's noisy: it swings with where that one peak lands and ignores how the fit behaves everywhere else. A method can have a slightly worse max error but be better across the entire range.
• RMSE is the square root of the average squared error, so it's independent of the sample count (unlike SSE) and it reflects the whole steering range rather than a single point (unlike max error). The squaring also penalizes large deviations more than small ones, which matches what we care about: a fit that's a little off everywhere is fine, but a fit that's badly off anywhere causes visible steering misalignment.

Because RMSE is in the same units as the data (degrees), it also reads directly as "the typical steering error you should expect from this fit" — which is exactly the quantity a MuJoCo car model cares about.

Project layout

File	Responsibility
main.py	CLI entry point: prints the error table and the polycoef strings
graph.py	Renders the comparison figure (curves + error)
src/ackermann.py	Core geometry/error math shared by the table and the plot
src/methods/taylor_differentiation.py	Taylor-expansion coefficients
src/methods/numpy_polyfit.py	Least-squares fit coefficients

Run python main.py from the repository root so the src package resolves.

Dependencies

• Python 3.14
• NumPy 2.4.6
• Matplotlib 3.10.9 (with the Tk backend for the interactive window)

See environment.yml for the conda spec, or requirements.txt for pip.

Citations

Formula for Ackermann steering angles — https://raw.org/book/kinematics/ackerman-steering/