Low-Pass Sampling in MPPI Control

• Low-Pass Sampling (LP-MPPI) is a method that filters out high-frequency noise in MPPI, enabling smoother control trajectories aligned with system bandwidth.
• The technique employs digital low-pass filters, like the Butterworth filter, to constrain the noise spectrum, reducing control chattering and actuator wear.
• Empirical evaluations show LP-MPPI can boost performance by up to 41% over traditional MPPI in both simulated benchmarks and physical systems.

Low-Pass Sampling (LP-MPPI) refers to a class of methods within sampling-based Model Predictive Path Integral (MPPI) control that suppress high-frequency variations during action sampling to achieve smoother, more effective control sequences. The LP-MPPI paradigm directly addresses intrinsic limitations of classical MPPI, namely the injection of high-frequency (white) noise that impedes efficient exploration, increases control signal chattering, and accelerates actuator wear. By selectively eliminating frequency components above a tunable cutoff, LP-MPPI methods tightly couple control trajectory generation with the natural bandwidth limitations of real-world robotic platforms, leading to improved performance and robustness across a variety of dynamic systems (Kicki, 13 Mar 2025, Vlahov et al., 2024).

1. Theoretical Motivation and Standard MPPI Overview

Standard MPPI generates control sequences by adding i.i.d. Gaussian noise to a nominal open-loop trajectory, rolling out the perturbed inputs through system dynamics, and forming a weighted average update based on cost (Kicki, 13 Mar 2025). The control trajectory at timestep $t$ is denoted $U = [u_t,u_{t+1},\dots,u_{t+H}]$, and $N$ sampled sequences are constructed as

$$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$

Each sequence $U^i$ is evaluated using model-based rollout, and costs $J_i$ are accumulated. Importance weights are then computed as

$$w_i = \frac{\exp(-\lambda J_i)}{\sum_j \exp(-\lambda J_j)}$$

where $\lambda$ is an inverse temperature. The nominal $U$ is updated as the weighted sum of sampled sequences.

The principal issue arises from time-uncorrelated perturbations: such perturbations implement white noise, distributing energy uniformly across all frequencies. In systems with actuators that dampen high frequencies, these perturbations induce control signals that are both ineffective for meaningful exploration and damaging to hardware.

2. Low-Pass Sampling: Mathematical Formulation

LP-MPPI replaces the i.i.d. noise with low-pass filtered perturbations, ensuring the injected noise energy is concentrated within a prescribed frequency band. Perturbations $\epsilon^i$ are sampled as usual, but are then passed through a digital low-pass filter. In the frequency-domain,

$U = [u_t,u_{t+1},\dots,u_{t+H}]$0

where $U = [u_t,u_{t+1},\dots,u_{t+H}]$1 is a frequency response (commonly a Butterworth filter of order $U = [u_t,u_{t+1},\dots,u_{t+H}]$2) defined by

$U = [u_t,u_{t+1},\dots,u_{t+H}]$3

with cutoff $U = [u_t,u_{t+1},\dots,u_{t+H}]$4.

In practice, discrete-time filters are implemented either recursively via the bilinear transform or by difference equations, e.g. for a first-order filter: $U = [u_t,u_{t+1},\dots,u_{t+H}]$5 These filters are applied per-sample, per-trajectory before rollout.

3. LP-MPPI Algorithmic Structure

The LP-MPPI algorithm incorporates low-pass filtering directly into the sampling process. The high-level steps are as follows (Kicki, 13 Mar 2025):

1. Sample $U = [u_t,u_{t+1},\dots,u_{t+H}]$6 Gaussian control perturbations $U = [u_t,u_{t+1},\dots,u_{t+H}]$7.
2. Filter each perturbation in the time domain: $U = [u_t,u_{t+1},\dots,u_{t+H}]$8.
3. Generate perturbed control sequences: $U = [u_t,u_{t+1},\dots,u_{t+H}]$9.
4. Rollout through model $N$0, accumulating total cost $N$1.
5. Weight samples and update $N$2 with the weighted average as in standard MPPI.
6. Apply $N$3, and shift $N$4 for the next iteration.

The primary innovation is the explicit control over exploration bandwidth via the selection of $N$5 and filter order $N$6. The Butterworth family is recommended for maximal passband flatness (no ripple) (Kicki, 13 Mar 2025).

4. Properties and Analysis

Exploration-Exploitation Control

Through $N$7, the maximum frequency at which exploration occurs is directly tunable. Frequencies $N$8 are passed with minimal attenuation, supporting slow, persistent control signal deviations, while $N$9 are suppressed, reducing useless high-frequency wiggles that do not meaningfully affect the system's cost.

Smoothness and Actuator Wear

Low-pass filtered trajectories exhibit significantly reduced mean-squared second derivative (MSSD) compared with traditional MPPI, and deviate less from off-line smoothed trajectories such as those computed by Savitzky–Golay filters. Reduction in control chattering directly mitigates actuator wear.

Computational Complexity

The digital filter operation is $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$0 per rollout and introduces minimal extra computational burden. Overhead is approximately 2–3% for lightweight models (e.g., F1TENTH race car) and negligible ($$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$1) for models where rollouts dominate cost (e.g., MuJoCo Ant) (Kicki, 13 Mar 2025).

5. Empirical Results

LP-MPPI demonstrates robust improvements across simulated and physical benchmarks (Kicki, 13 Mar 2025):

• Gymnasium Environments (Hopper-v5, Ant-v5, HalfCheetah-v5): LP-MPPI achieves $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$2 higher average reward compared to MPPI, ColoredMPPI, SMPPI, and SCP-MPPI, with greater gains on long time horizons and low rollout count (i.e., better sample efficiency).
• Simulated Quadruped Locomotion (Unitree Go2, MAB Silver Badger): When integrated into Dial-MPC (“LP-Dial-MPC”), LP-MPPI yields 24% and 41% improvement over Dial-MPC baseline for Go2 and Badger, respectively, and outperforms colored-noise variants by a factor of two.
• F1TENTH Autonomous Racing (Physical System): LP-MPPI obtains the largest median track coverage in all tested conditions. For 30 s trials at 30 Hz, it achieves a virtual $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$3 s (∼0.3 m) lap-time improvement over the closest baseline in every regime.

6. Related Methods: Frequency-Shaping and Alternatives

Previous efforts to emphasize low-frequency sampling in MPPI include colored noise generation via spectral shaping (Vlahov et al., 2024). These methods draw frequency-domain samples with power spectral density $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$4 and obtain time-domain perturbations via inverse DFT, enabling control over the noise’s frequency structure through exponent $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$5. This approach successfully increases control smoothness and improves exploration, as shown empirically in off-road vehicle and quadrotor experiments, though the LP-MPPI formulation with filter-based suppression of all components above a strict cutoff delivers more direct and interpretable control over the sampled trajectory's spectrum.

Other relevant paradigms, including those in sparse analog signal recovery, leverage periodic mixing and low-pass filtering to reconstruct signals at sub-Nyquist rates (0901.3950, 0907.3576), but these are architecturally and algorithmically distinct from the trajectory-level filtering employed in LP-MPPI.

7. Implementation, Tuning, and Practical Guidance

• Cutoff Frequency ($$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$6): Should match the actuation bandwidth of the underlying system. Too low a cutoff hampers exploration; too high reverts toward standard MPPI chattering.
• Filter Order ($$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$7): Typically $$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$8–$$\epsilon^i_{t:t+H} \sim \mathcal N(0, \Sigma), \qquad U^i = U + \epsilon^i$$9; higher order yields sharper roll-off but can increase phase delay.
• Other Hyperparameters: Temperature ($U^i$0), rollout count ($U^i$1), horizon ($U^i$2), and covariance ($U^i$3) are optimized as in standard MPPI.
• Filter Selection: Butterworth filters are maximally flat in their passband and do not bias energy inside $U^i$4.

LP-MPPI modifies only the sampling step and is otherwise compatible with any system model, cost function, or MPPI controller structure.

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In conclusion, LP-MPPI is a principled extension of sampling-based model predictive control that integrates frequency-domain knowledge into the trajectory generation process, providing practical gains in exploration efficiency, control smoothness, and hardware robustness across simulated and physical domains (Kicki, 13 Mar 2025, Vlahov et al., 2024).