This paper presents an efficient model predictive path integral (MPPI) control framework for systems with complex nonlinear dynamics. To improve the computational efficiency of classic MPPI while preserving control performance, we replace the nonlinear dynamics used for trajectory propagation with a learned linear deep Koopman operator (DKO) model, enabling faster rollout and more efficient trajectory sampling. The DKO dynamics are learned directly from interaction data, eliminating the need for analytical system models. The resulting controller, termed MPPI-DK, is evaluated in simulation on pendulum balancing and surface vehicle navigation tasks, and validated on hardware through reference-tracking experiments on a quadruped robot. Experimental results demonstrate that MPPI-DK achieves control performance close to MPPI with true dynamics while substantially reducing computational cost, enabling efficient real-time control on robotic platforms.
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.
This paper presents a model-based reinforcement learning (RL) framework for optimal closed-loop control of nonlinear robotic systems. The proposed approach learns linear lifted dynamics through Koopman operator theory and integrates the resulting model into an actor-critic architecture for policy optimization, where the policy represents a parameterized closed-loop controller. To reduce computational cost and mitigate model rollout errors, policy gradients are estimated using one-step predictions of the learned dynamics rather than multi-step propagation. This leads to an online mini-batch policy gradient framework that enables policy improvement from streamed interaction data. The proposed framework is evaluated on several simulated nonlinear control benchmarks and two real-world hardware platforms, including a Kinova Gen3 robotic arm and a Unitree Go1 quadruped. Experimental results demonstrate improved sample efficiency over model-free RL baselines, superior control performance relative to model-based RL baselines, and control performance comparable to classical model-based methods that rely on exact system dynamics.
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.