This paper investigates deterministic optimal control using Model Predictive Path Integral (MPPI) control, a sampling-based and derivative-free framework well suited for systems with complex dynamics and nonsmooth objectives. In deterministic MPPI, the temperature must be driven to zero to recover the true optimum, yet the design of an effective cooling schedule remains a fundamental challenge. Existing methods typically rely on predefined open-loop schedules, which limit the efficiency and robustness of the algorithm. To overcome this limitation, we propose an Information-Theoretic Adaptive Cooling (ITAC) framework that uses the Shannon entropy of the importance weights as an online feedback signal to regulate the temperature. The proposed mechanism adapts the cooling rate to the current sampling state, enabling fast progress when the weights are diffuse and cautious cooling when they become concentrated. We prove asymptotic convergence of the resulting scheme to the deterministic optimum, and further derive a critical entropy threshold that leads to a smooth barrier against premature weight collapse. Experiments on nonsmooth signal temporal logic motion-planning tasks show that ITAC improves sampling efficiency and achieves substantially faster convergence than state-of-the-art baselines without sacrificing the derivative-free nature of MPPI.
Sampling-based model predictive control methods like MPPI and CEM are essential for real-time control of nonlinear robotic systems, particularly where discontinuous dynamics preclude gradient-based optimization. However, these methods derive from information-theoretic objectives that are agnostic to the geometry of the control problem, leading to pathological behaviors such as mode-averaging when the cost landscape is complex. We present OT-MPC, a sampling-based algorithm that overcomes these limitations through an entropy-regularized optimal transport formulation. By computing an optimal coupling between candidate control sequences and low-cost proposals, OT-MPC refines candidates toward nearby promising samples while coordinating updates across the ensemble to maintain coverage of the solution space. We derive closed-form, gradient-free updates via the Sinkhorn algorithm, enabling real-time performance. Experiments on navigation, manipulation, and locomotion tasks demonstrate improved success rates over existing methods.
Vincent Pacelli, Akash Ratheesh, Evangelos A. Theodorou
This paper proposes a novel method that extends the Model Predictive Path Integral (MPPI) method with motion primitives for additional structured sampling, which enhances the convergence towards a globally optimal solution. By evaluating motion primitives and perturbed control sequences in a real-time sampling-based optimization loop, this work addresses the limitations of the path planning capabilities of sampling-based controllers. The algorithm is implemented on a quadcopter simulator and tested on an obstacle field navigation task. It is demonstrated that the proposed approach enhances exploration of the control space while maintaining the fast, reactive behavior required for real-time control.
Classical proportional--integral--derivative (PID) control remains widely used in industrial control systems, while model predictive control (MPC) is actively studied to achieve higher performance for systems with nonlinear dynamics. Model predictive path integral (MPPI) control is a sampling-based MPC method that optimizes control inputs without gradient calculations and can handle non-differentiable models and objective functions. However, conventional MPPI directly samples control-input sequences, which can produce large temporal input increments and causes the optimization dimension to grow with the prediction horizon. This study proposes MPPI--PID control, which uses MPPI to optimize PID gains online instead of directly optimizing the control-input sequences. By replacing high-dimensional input-sequence optimization with low-dimensional gain-space optimization while retaining the PID structure, the proposed formulation improves sampling efficiency and promotes smoother control inputs. Theoretical analyses are provided for a unified path-integral update, the relation between optimization dimension and effective sample size, and the temporal correlation of input perturbations induced by the PID structure. The method is evaluated on a learning-based path following of a mini forklift using a residual-learning dynamics model that combines a physical model and a neural network identified from real-machine driving data. Numerical results show that MPPI--PID improves tracking performance over fixed-gain PID, yields smaller input increments than conventional MPPI, and maintains favorable performance under reduced sampling budgets.