Model Predictive Path Integral (MPPI)

Problem Statement

MPPI addresses nonlinear trajectory tracking without local linearization by sampling control perturbations and weighting trajectories by cost. It is effective in regimes with nonconvex costs and uncertain dynamics.

Model and Formulation

For control sequence U, MPPI computes update:

$$ \Delta u_t = \frac{\sum_{k=1}^{K} \exp\left(-\frac{1}{\lambda}S_k\right)\epsilon_{k,t}}{\sum_{k=1}^{K} \exp\left(-\frac{1}{\lambda}S_k\right)} $$

where S_k is rollout cost and \epsilon_{k,t} is sampled perturbation at time t.

Algorithm Procedure

1. Sample K noisy control sequences around nominal controls.
2. Roll out dynamics and compute trajectory costs.
3. Compute importance-weighted control correction.
4. Shift horizon and repeat at each control cycle.

Tuning Guidance

• Increase sample count K for better solution quality.
• Lower temperature \lambda sharpens elite trajectory selection.
• Match exploration covariance to expected disturbance magnitudes.

Failure Modes and Diagnostics

• Insufficient samples lead to high-variance control updates.
• Overly aggressive exploration destabilizes near-hover behavior.
• Large horizon with slow hardware can violate realtime deadlines.

Implementation and Execution

python -m uav_sim.simulations.trajectory_tracking.mppi

Evidence

References

• Williams et al., Information Theoretic MPC for Model-Based Reinforcement Learning (2017)
• Theodorou et al., Policy Improvement with Path Integrals (2010)

Related Algorithms

• Nonlinear MPC
• Feedback Linearisation