Performance Enhanced Risk-Aware MPPI using Gaussian Process for Robust Mobile Robot Control

This paper presents a performance enhanced Risk-Aware Model Predictive Path Integral (RA‑MPPI) control using Gaussian-process (GP). To this end, the real system is run with arbitrary control input and the resulting data is saved. Based on the nominal model of the system and the collected data, uncertainty affecting the state transition is estimated by a Jacobian-based method offline. Then, the used control input and the estimated uncertainty are used to train the Gaussian process which generates the mean and variance of the uncertainty. Online, the gap between the nominal and real dynamics is compensated by the trained GP. In other words, the nominal model is refined using the estimated noise mean, while the estimated noise variance is directly incorporated into the RA‑MPPI framework, eliminating the need to manually tune the disturbance covariance parameter. Validation is conducted through simulation experiments in Gazebo using a F1TENTH car with a bicycle model navigating a track with and without obstacles. Also, real‑world performance is further demonstrated on the F1TENTH car platform driving the same track under similar conditions. Across both simulation and real‑world scenarios, the proposed method demonstrates improved safety and trajectory tracking performance compared to baseline MPPI and RA‑MPPI techniques in terms of lap times and the number of crash‑free lap completions. Videos of the F1TENTH car simulations, experiments, and similar Gazebo simulation results using a Clearpath Jackal with a differential drive model are available at the project webpage: https://gp-ramppi.github.io.

1. Collect data in open space. Drive the robot in open space manually or with a simple controller (e.g., baseline MPPI) at speeds/trajectories similar to the target task and log pairs \( (u_k, x_{k+1}) \) to form the dataset \( \mathcal{D}=\{(u_k,x_{k+1})\}_{k=0}^{N-1} \).
2. Estimate actual input and lumped uncertainty (Jacobian-based). For each sample \(k\), initialize \(u_{\mathrm{act},k}^{(0)} = u_k\) and iterate:
   1. \(\hat{x}_{k+1}^{(i)} = f(x_k,\;u_{\mathrm{act},k}^{(i)})\)
   2. \(\Delta x_k^{(i)} = x_{k+1} - \hat{x}_{k+1}^{(i)}\)
   3. \(\delta u^{(i)} = J\!\big(x_k,u_{\mathrm{act},k}^{(i)}\big)^{\dagger}\,\Delta x_k^{(i)}\)
   4. \(u_{\mathrm{act},k}^{(i+1)} = u_{\mathrm{act},k}^{(i)} + \alpha\,\delta u^{(i)},\ (0<\alpha\le 1)\)
   where \(J=\partial f/\partial u\) and \(^{\dagger}\) denotes the pseudo-inverse. After convergence, compute the lumped input-channel uncertainty \( \hat{w}_k = u_k - \hat{u}_{\mathrm{act},k} \), and build \( \mathcal{D}_w=\{(u_k,\hat{w}_k)\}_{k=0}^{N-1} \).
3. Train Gaussian Processes per input channel. Fit \(n_u\) independent GPs (RBF kernel) on \( \mathcal{D}_w \) to learn \( u \mapsto w \). Hyperparameters (e.g., \( \sigma_f, \ell \)) are optimized via negative log marginal likelihood minimization. The trained models return, for any input \(u\), a mean \( f_{\mu}(u) \) and variance \( f_{\sigma^2}(u) \) of the uncertainty: \( f_{\mathcal{GP}}(u) \sim \mathcal{N}\!\big(f_{\mu}(u),\,f_{\sigma^2}(u)\big) \).