Model Predictive Optimized Path Integral Strategies

Abstract: We generalize the derivation of model predictive path integral control (MPPI) to allow for a single joint distribution across controls in the control sequence. This reformation allows for the implementation of adaptive importance sampling (AIS) algorithms into the original importance sampling step while still maintaining the benefits of MPPI such as working with arbitrary system dynamics and cost functions. The benefit of optimizing the proposal distribution by integrating AIS at each control step is demonstrated in simulated environments including controlling multiple cars around a track. The new algorithm is more sample efficient than MPPI, achieving better performance with fewer samples. This performance disparity grows as the dimension of the action space increases. Results from simulations suggest the new algorithm can be used as an anytime algorithm, increasing the value of control at each iteration versus relying on a large set of samples.

• The paper introduces a joint distribution approach for control sequences using adaptive importance sampling (AIS) to improve sample efficiency.
• The methodology reformulates traditional MPPI by iteratively adapting the proposal distribution based on full trajectory feedback.
• Empirical results demonstrate significant performance gains in high-dimensional, real-time autonomous control tasks.

Model Predictive Optimized Path Integral Strategies: A Technical Essay

Introduction

Trajectory optimization for nonlinear dynamical systems, especially those encountered in autonomous robotics and vehicle control, remains a central problem in control theory. Model Predictive Control (MPC) permits the handling of nonlinear dynamics via on-line optimization, but issues of sample efficiency and distributional adaptivity persist in sampling-based trajectory optimization frameworks. Model Predictive Path Integral (MPPI) control leverages importance sampling to estimate optimal control sequences under general cost functions and system dynamics. The inherent decoupling of proposal distributions at each time step, however, impairs sample efficiency, particularly in high-dimensional action spaces.

The work "Model Predictive Optimized Path Integral Strategies" (2203.16633) generalizes MPPI to view control sequences as samples from a joint distribution across the entire horizon. This enables adaptive importance sampling (AIS) updates over the full control trajectory, significantly improving sample efficiency and performance. The approach preserves MPPI’s universality with respect to system dynamics and cost function structure, while integrating advanced AIS strategies for proposal distribution refinement.

Generalized Formulation and Adaptive Importance Sampling

The authors reformulate the sampling scheme: instead of sampling independent control actions at each time step as in classical MPPI, controls are treated as a single vector over the time horizon, sampled from a joint Gaussian proposal. The joint covariance matrix captures inter-temporal correlations, allowing iterative proposal adaptation using AIS algorithms. Importance sampling weights are evaluated using trajectory costs, with control sequences updated via weighted averages.

Let $U \in \mathbb{R}^{mT}$ denote the control sequence, sampled from $V \sim \mathcal{N}(U, \Sigma)$, where $\Sigma$ is block-diagonal (or more general) over the time horizon. The optimal control rule under this information-theoretic reformulation remains the weighted average over sampled control sequences using normalized importance weights, but adaptation now exploits trajectory-level feedback.

The integration of AIS affords the following benefits:

• Joint proposal adaptation: The AIS procedure updates both the mean and covariance of the control sequence proposal distribution, accommodating correlations between control actions across the time horizon.
• Anytime algorithmic structure: MPOPI can refine its control distribution iteratively, improving control value with additional computation, unlike the batch MPPI paradigm.
• Decoupled control cost and trajectory weights: Modifications to the base distribution and normalization strategies facilitate stable importance weighting, which is critical for numerical reliability.

No theoretical optimality guarantees from MPPI are lost in this generalization. The algorithm is laid out to allow classical constraints and smoothing to be handled via model function augmentation, and the control update procedure is adapted for AIS.

Computational Implications

The classical strong parallelization of MPPI—propagating $M$ trajectories in parallel—conflicts with iterative AIS refinement in MPOPI, which operates on $K$ samples over $L$ sequential AIS iterations. While parallel dynamics propagation is preserved within each AIS iteration, overall parallel capability is reduced as $L$ increases, in exchange for marked gains in sample efficiency.

This trade-off becomes especially favorable in action spaces of larger dimension, where classical MPPI becomes increasingly sample-inefficient. MPOPI thus presents a paradigm shift for sample-limited, real-time trajectory optimization in constrained computational environments.

Empirical and Numerical Results

Simulations highlighted in the paper indicate strong numerical improvements:

• Sample efficiency: For fixed computational budgets (sample count), MPOPI consistently outperforms MPPI in trajectory cost minimization, with the advantage scaling with action dimensionality.
• Performance gains: In multi-agent vehicle scenarios (e.g., car racing tasks), MPOPI produces trajectories of lower cost and increased safety with fewer samples than classical MPPI.

The capacity of MPOPI to operate as an anytime control strategy—improving with longer execution—is highlighted empirically, suggesting practical applicability for systems with variable computational resources.

Theoretical and Practical Implications

From a theoretical standpoint, the generalization to joint trajectory proposals via AIS connects sampling-based control with broader stochastic optimization and sequential Monte Carlo literature. Allowing for arbitrary covariance adaptation opens avenues for exploring non-Gaussian and multi-modal proposals, further increasing robustness in disturbed or stochastic environments.

Practically, these results suggest significant improvements in real-time autonomous control domains, such as autonomous racing, cooperative robotics, and dynamic vehicle maneuvers, especially where adversarial disturbances and high-dimensional planning are involved. The joint-distribution approach anticipates further research in constrained control, reinforcement learning policy adaptation, and hybrid model-based/model-free control integration.

Conclusion

The MPOPI algorithm proposed in "Model Predictive Optimized Path Integral Strategies" (2203.16633) represents a significant formal advancement in sampling-based MPC. By viewing the control sequence as a joint distribution and leveraging AIS techniques, MPOPI overcomes sample efficiency and adaptability limitations inherent in classical MPPI. Empirical and theoretical results indicate robust improvements, especially as action space dimension increases. The approach opens several avenues for advanced optimal control, robust real-time planning, and further integration of stochastic optimization strategies in AI-driven dynamical systems.