Low Frequency Sampling in Model Predictive Path Integral Control

TL;DR

This work identifies high-frequency chatter as a limitation of Gaussian sampling in Model Predictive Path Integral (MPPI) control and proposes a colored-noise, low-frequency sampling strategy generated in the frequency domain with Hermitian symmetry. The method yields time-correlated, Gaussian samples whose spectral density follows $PSD(f) \propto 1/f^\gamma$, allowing explicit control of smoothness via $\gamma$ while preserving MPPI's update structure, requiring only updates to the mean trajectory through $\mu_t^{k+1}=\Psi(\mu_N^{k+1})$. Empirical results across a hardware off-road platform, a simulated quadrotor, and a double-integrator system show that colored sampling can achieve smoother trajectories, larger effective exploration, and comparable or improved performance, with reduced high-frequency control content and modest computational overhead. These findings suggest colored sampling generalizes Gaussian MPPI to systems with varying bandwidths and delays, offering practical benefits for real-time autonomous control and a path toward integration with advanced MPPI variants and multi-hypothesis strategies.

Abstract

Sampling-based model-predictive controllers have become a powerful optimization tool for planning and control problems in various challenging environments. In this paper, we show how the default choice of uncorrelated Gaussian distributions can be improved upon with the use of a colored noise distribution. Our choice of distribution allows for the emphasis on low frequency control signals, which can result in smoother and more exploratory samples. We use this frequency-based sampling distribution with Model Predictive Path Integral (MPPI) in both hardware and simulation experiments to show better or equal performance on systems with various speeds of input response.

Figures (7)

• Figure 1: (Upper) The off-road vehicle in a desert terrain just before an autonomy test. (Lower) A screenshot of the Flightmare quadrotor simulator
• Figure 2: 300 samples of state trajectories generated from running frequency-based (Left) and Gaussian (Right) control samples through the off-road vehicle dynamics described in \ref{['subsec:hw_platform']}. The frequency-based throttle and steering samples are generated using $\gamma = 4$ and both distributions used the same $\sigma$ set to $0.4$. Descriptions of $\gamma$ and $\sigma$ can be found in \ref{['subsec:colored_noise_def']}. (Lower) Variance in the x direction of the same 300 samples from frequency-based and Gaussian distributions over time. There is an additional Gaussian distribution with $\sigma = 2.0$ to show that increasing the standard deviation does not allow Gaussian sampling to explore as far as colored samples. Colored samples reach further extremes in both the $x$ and $y$ directions due to the reduction of high frequency signals in the control trajectories.
• Figure 3: A comparison of Gaussian and Colored noise with various exponents. As $\gamma$ increases, the samples generated become smoother and can reach tail-end values more consistently.
• Figure 4: Hardware experiments using Gaussian and Colored Sampling on the vehicle. These graphs show the vehicle trying to maneuver in a zig-zag corridor shown in red. The colors along the trajectories indicate the amount of time spent in autonomy before manual override/goal achieved with purple dots indicating the starting points. The Colored samples can go from one end of the corridor to the other while the Gaussian sampling struggles to make more than a single turn even when started in various locations.
• Figure 5: Controls from the vehicle hardware attempts. Picture are the controls from the first eight seconds of the first attempt of each controller starting from the same location. We can see that the colored sampling technique is achieves smoother and larger throttle, brake, and steering commands. Throttle and brake are combined into a single graph with brake values being below the red line.