LP-MPPI: Low-Pass Filtering for Efficient Model Predictive Path Integral Control

Technical Summary: LP-MPPI: Low-Pass Filtering for Efficient Model Predictive Path Integral Control

Problem Statement
Model Predictive Path Integral (MPPI) control is a robust, sampling-based approach for real-time motion planning, valued for its ability to handle arbitrary dynamics and cost functions without requiring differentiability. However, standard MPPI relies on sampling control trajectory perturbations from a white noise distribution (Gaussian with independent elements). This results in a lack of temporal correlation within trajectories, effectively introducing high-frequency noise. In robotic systems, where dynamics naturally dampen high frequencies, these components do not contribute to finding optimal controls. Instead, they cause "chattering" in the applied control signals, leading to actuator wear and inefficient search processes. Existing solutions, such as spline interpolation or input lifting, offer only coarse or indirect control over the frequency spectrum. Other approaches using colored noise (e.g., in iCEM or specific MPPI variants) dampen signal components proportionally to the inverse of their frequency, which can overly bias the search toward the lowest frequencies, potentially missing optimal behaviors that require moderate frequency variations (e.g., in legged locomotion).

Methodology
The authors propose Low-Pass Model Predictive Path Integral Control (LP-MPPI), which integrates a low-pass filter directly into the MPPI sampling process.

• Core Mechanism: Unlike standard MPPI, which samples white noise $\epsilon$ and adds it to the nominal control $U$, LP-MPPI samples white noise and then passes it through a low-pass filter (specifically a Butterworth filter) to generate filtered perturbations $\epsilon_{LP}$. The control sequence is then updated as $U + \epsilon_{LP}$.
• Frequency Shaping: This approach introduces temporal correlation to the perturbations, effectively shaping the frequency spectrum of the search distribution. It biases the search toward low-frequency signals while allowing the user to explicitly define the passband.
• Parameters: The method introduces two interpretable physical parameters:
  1. Cutoff Frequency ($f_c$): Controls the width of the passband, determining the range of frequencies with the highest power density in the perturbations.
  2. Filter Order ($o_{LPF}$): Controls the roll-off slope in the stopband, determining how aggressively high-frequency components are attenuated.
• Implementation: The filtering is applied to the noise sequences before simulation, not to the final control signal. This ensures that the applied control signal remains reactive (the first control input can still change rapidly) while the trajectory search is biased toward smooth, plausible paths. The computational cost is negligible, as filtering requires only a small number of multiplications proportional to the horizon length.

Key Contributions

1. Algorithmic Innovation: The proposal of LP-MPPI, which shapes the frequency spectrum of control trajectory perturbations to eliminate detrimental high-frequency noise without sacrificing search efficiency.
2. Interpretability and Tuning: The method provides direct, intuitive control over the frequency spectrum via physically meaningful parameters ($f_c$ and $o_{LPF}$), unlike colored noise approaches which offer less flexible biasing.
3. Comprehensive Evaluation: A thorough experimental analysis across three distinct domains:
   • Gymnasium Environments: Hopper-v5, Ant-v5, and HalfCheetah-v5.
   • Simulated Quadruped Locomotion: Unitree Go2 and MAB Silver Badger robots using a diffusion-based MPC framework (Dial-MPC).
   • Real-World Application: F1TENTH autonomous racing on a physical oval track.

Experimental Results
The paper reports consistent performance improvements of LP-MPPI over state-of-the-art MPPI variants (including original MPPI, SMPPI, SCP-MPPI, and ColoredMPPI):

• Gymnasium Environments: LP-MPPI outperformed baselines across a wide range of horizon lengths ($H$) and rollout counts ($N$). On average, it improved results by 24% compared to baselines. It demonstrated particularly high sample efficiency, showing larger gains with fewer rollouts.
• Control Smoothness: Quantitative metrics (Mean Squared Second Derivative and Mean Savitzky-Golay Filter Deviation) confirmed that LP-MPPI generates significantly smoother control signals, reducing chattering while maintaining high rewards.
• Quadruped Locomotion: When integrated with Dial-MPC (creating LP-Dial-MPC), the method achieved an average performance gain of 32% over the standard Dial-MPC and significantly outperformed a ColoredMPPI variant (which yielded rewards roughly half as high).
• Real-World Racing: In F1TENTH autonomous racing, LP-MPPI achieved the highest median distances covered in 30-second trials, outperforming all baselines. It maintained real-time performance on a standard CPU (Intel Core i5-12500H) with negligible computational overhead (approx. 2.4% increase in F1TENTH, and even a slight decrease in simulation time for Ant-v5, likely due to simplified physics interactions).

Significance and Claims
The authors claim that LP-MPPI represents a simple yet powerful modification to the MPPI algorithm. Its significance lies in:

• Efficiency: It improves the search for optimal controls by focusing the exploration on the admissible frequency range where optimal behaviors typically reside, rather than wasting samples on high-frequency noise that robotic dynamics filter out.
• Practicality: It reduces actuator wear by smoothing control signals and is easy to implement with minimal computational overhead, making it suitable for real-time edge devices.
• Adaptability: The method is highly adaptable to various robotic systems through intuitive parameter tuning (cutoff frequency and filter order) and can be seamlessly integrated with other sampling-based MPC approaches, as demonstrated with Dial-MPC.

The paper concludes that LP-MPPI offers a superior balance between search efficiency, control smoothness, and computational cost, making it a strong candidate for real-time robotic control tasks requiring high-performance trajectory optimization.