MPPI Control

Real-time whole-body controllers for legged robots typically plan through a fixed nominal model and degrade when the deployed dynamics change. Adaptive methods typically require a model structure that contact dynamics do not provide, or they need offline training for each anticipated condition. We present Look-back and Look-ahead Adaptive Model Predictive Path Integral control (LLA-MPPI). The method converts whole-body adaptation into selection over a bank of GPU-batched contact simulators with different physical or structural parameters. Windowed prediction errors select the simulator that best explains recent motion. A whole-body MPPI planner optimizes controls through the selected model. The framework requires no offline training, and its selected hypotheses are physically interpretable. Across four simulated tasks, it achieves 97.5% success while the strongest baseline reaches 74% and an oracle with the true model reaches 98.5%. Hardware validation on a Unitree Go2 shows the robot walking under a payload added mid-run, walking after one leg is disabled, and pushing a box to its goal while increasing its mass on the fly. Code, videos, and project details are available at: https://lla-control.github.io

This work proposes the L1 Adaptive Model Predictive Path Integral (L1-MPPI). It cascades L1 adaptive control with the Model Predictive Path Integral (MPPI) to improve tracking of high-speed UAV trajectories. Thanks to the L1augmentation, the tracking remains accurate even under model uncertainties and external disturbances, such as an additional payload or a mismatch in the modeled aerodynamic drag. In contrast to existing MPPI approaches for UAV control that do not explicitly model aerodynamic effects, varying payloads, and typically neglect the dynamics of low-level motor controllers, our L1-MPPI approach enhances the dynamic model used in the MPPI by incorporating the low-level flight controller and motor dynamics, as well as an iterative mixing scheme that reflects the approach of the low-level controller. The proposed method demonstrates improved tracking performance in both simulation and the real world, even when the UAV is subjected to an unknown payload. In flight with 35% mass increase, our approach lowers the RMSE by 58.61% with respect to plain MPPI. Compared to the same MPPI using an online mass estimator in place of the L1 augmentation, the RMSE is lower by 38.59%. During the real-world experiments the UAV reaches speeds up to 13.50 m/s and accelerations up to 2.5 g.

Autonomous UAV flight through cluttered and partially unknown environments requires reasoning not only about observed obstacles but also about occluded regions that the sensor cannot observe. We present OA-MPPI, an obstacle- and occlusion-aware extension of Model Predictive Path Integral (MPPI) control for quadrotor flight that accounts for potential moving agents emerging from these regions into the vehicle's path. At every planning step, we extract a 3D occlusion boundary from the online occupancy map and use it to model the regions that hidden agents could reach over the prediction horizon. We penalize trajectories that enter these expanding regions within MPPI rollouts generated using nonlinear quadrotor dynamics and accounting for individual rotor thrust limits. We validate the proposed approach in simulation and hardware flight experiments, with the complete pipeline running onboard the vehicle in real time. Results show increased clearance from occlusion boundaries compared to baseline MPPI in both settings, as well as avoidance of an agent emerging from occlusion in simulation.

Online residual learning can reduce model mismatch in predictive control, but passive data collection may fail to adequately cover states that become important later in the task. Task-agnostic active learning targets uncertain or informative regions, but information acquired in such regions does not necessarily improve task performance. This paper introduces Task-Oriented Information Acquisition (ToIA), an active-learning criterion for model predictive path integral control (MPPI) with online Gaussian process (GP) residual learning. For each sampled control sequence, ToIA estimates how much an observation obtained early in the rollout would reduce predictive uncertainty at later states on the same rollout, and weights this reduction by the rollout's relevance to the task. The score is evaluated over the existing MPPI rollout batch without sampling future observations or re-optimizing control under hypothetical posterior updates. In simulated off-road navigation across held-out maps with heterogeneous terrain, ToIA improved the goal-reaching success rate over passive GP learning by 19.3 and 27.4 percentage points and outperformed task-agnostic active-learning baselines across dense and sparse online-learning intervals. An ablation study indicates that task relevance is particularly important under sparse model updates. The implementation supports online control at 20 Hz on an NVIDIA RTX 2080 Ti.

We present a hierarchical UAV navigation framework that aligns natural-language intent with dynamically feasible flight behaviors in cluttered indoor environments. To bridge the gap between abstract semantics and low-level control, we employ a parallelized ensemble of six behavior-conditioned Model Predictive Path Integral (MPPI) planners. Crucially, by designing mode-specific guiding costs and sampling biases, we induce distinct trajectory modes that converge to unique behavioral means, yielding a compact set of intentionally diverse candidates rather than mere stochastic variations. We project these 3D candidates onto the onboard first-person-view RGB stream, turning language grounding into a visual action selection problem. A pretrained vision--language model (VLM) asynchronously selects the candidate index given the overlaid FPV image and a natural-language prompt, while MPPI replans at 20Hz and a PID-based low-level controller tracks the selected trajectory. We implement the full pipeline in NVIDIA Isaac Sim and on a real-world quadrotor platform equipped with LiDAR and RGB sensing. Experiments in both simulation and real-world flights show semantically meaningful behavior diversity, robust language alignment despite VLM latency, and safe, repeatable flight across all modes, achieving 100% task success in our evaluated scenarios.

Tensegrity robots offer lightweight, compliant mobility over challenging terrain but remain difficult to model and control due to complex contact-rich dynamics and partial observability. This work presents a model predictive path integral (MPPI) controller for a three-bar tensegrity robot driven by a learned graph neural network (GNN) dynamics model. This work first extends prior GNN-based models with a differentiable contact detection module. The extension allows the dynamics model to reason over non-horizontal planar terrains, obstacles, as well as self-collisions. Then, the learned dynamics model and the MPPI controller operate in a closed data-collection loop, iteratively improving model accuracy and control performance. This work further introduces a hybrid MPPI strategy that combines MPPI with turning motion primitives to improve maneuverability. Experiments are performed in MuJoCo across five navigation tasks, which include, wall obstacles, inclines, narrow corridors, low-clearance structures, and a composite 3D obstacle course. The experiments demonstrate that the hybrid MPPI controller operating over the learned GNN dynamics model improves predictive accuracy over a flat-ground baseline model and achieves superior navigation performance compared to $A^*$-based re-planning and MPPI-only variants. Results show that the contact-aware learned dynamics combined with the sampling-based model predictive control enable robust tensegrity navigation in complex, contact-rich environments.

This study proposes a novel Model Predictive Path Integral (MPPI)-based task-space control framework. The proposed framework explicitly solves rigid-body dynamics within a real-time MPC formulation and enforces safety constraints, enabling accurate motion and force control that yields compliant behaviors for safe and effective physical interaction of robotic manipulators in unstructured environments. By leveraging MPPI, the proposed framework efficiently handles nonlinear dynamics that are difficult to solve with conventional MPC approaches in real-time. Furthermore, we develop a torque-sampling-based control architecture that enables efficient exploitation of GPU-based parallelization, resulting in effective compliant and force-aware behaviors. As a result, the proposed framework achieves a solver update rate of over 166 Hz with a 0.18 s prediction horizon, and its performance is validated through real-world experiments on a 7-DoF manipulator.

Combining learned policies with model predictive control can leverage learned task priors while retaining online adaptation to new objectives and constraints, but performance degrades when the policy is out of distribution. In policy-guided model predictive path integral (MPPI) control, a policy-centered warm-start approach centers the sampling distribution on the policy output. When the prior is mismatched, centering the sampling distribution on the policy output restricts exploration around an unsuitable solution and prevents recovery toward the task optimum. We propose Proximal Prior Injection (ProxPI), which retains nominal-centered MPPI sampling and incorporates the policy through a soft proximity cost. This matches the in-distribution performance of existing prior-injection schemes while enabling the optimizer to escape an inaccurate policy and recover vanilla MPPI-level performance. We theoretically show that re-centering on the prior discards the optimizer's correction at every update, whereas nominal-centered sampling retains it and converges to a solution set by both the task cost and the prior, and that this failure is not removed by a larger rollout budget. Simulations and real-robot experiments demonstrate robust performance under both in-distribution and out-of-distribution tasks.

Safe robot navigation in crowded spaces requires planning that accounts for where people will be, not only where they are now. Model Predictive Path Integral (MPPI) control is an effective sampling-based planner, but many implementations encode humans as static point obstacles at their current positions, underestimating risk in dynamic scenes. We propose Predictive Gaussian Interaction Fields (PGIF), a spatiotemporal cost formulation that propagates pedestrian predictions forward over the full planning horizon and encodes them as anisotropic Gaussian repulsive fields aligned with each pedestrian's direction of motion. The forward spread of each field grows with the pedestrian's speed, creating a motion cone danger zone that penalises robot trajectories entering the pedestrian's path of travel more strongly than those approaching from behind. The formulation is closed-form and fully parallelisable across rollouts, adding no measurable computational overhead. Evaluated over 300 randomised crowd scenarios at three density levels, PGIF-MPPI achieves a 0% collision rate at every density level, compared with up to 82% for vanilla MPPI, while maintaining real-time planning performance.

Model Predictive Path Integral (MPPI) control is widely used in manipulation for its gradient-free, parallel handling of non-convex costs. Manipulation tasks, however, often impose constraints that hold throughout the motion: a closed kinematic chain that two grasping arms keep exactly, or joint limits and obstacle clearances that are never crossed. MPPI handles such constraints only through the cost, as soft penalties that hold approximately and fail under a strong task cost. To address this, we propose Projection-Retraction MPPI (PR-MPPI), which enforces the constraints inside the sampled dynamics. At every rollout step, the sampled velocity is projected to satisfy both constraint types: the equality restricts it to a subspace, and each inequality to a half-space within that subspace, so inequality handling never breaks the equality. This projection, however, satisfies the constraints only to first order, and a finite step leaves a small drift off the equality. Therefore, we retract the returned command back onto the constraint to numerical tolerance and independent of task weighting. We validate PR-MPPI on 14-DoF dual-arm systems. In simulation, the returned commands satisfy the closed-chain equality to numerical tolerance through a joint-limit stress test and randomized obstacle avoidance. On real hardware, the arms of a Unitree H1-2 humanoid reactively avoid a moving obstacle. Code and experiment videos are available at https://rcilab.github.io/prmppi.

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.

We study Model Predictive Path Integral (MPPI) control for nonlinear systems with additive process disturbances whose covariance is unknown, spatially varying, and slowly time-varying. A mismatched disturbance covariance produces a persistent penalty in closed-loop stability certificates, while online estimation can reduce this penalty as data are collected. We propose a cell-wise recursive covariance estimator with spatial diffusion and prove a finite-horizon error bound that separates stochastic-approximation error, spatial-smoothing bias, and temporal-drift effects. The diffusion kernel is chosen to be reversible with respect to the stationary visitation measure, making the diffusion operator dissipative in the weighted Lyapunov analysis. We then substitute the resulting covariance estimate into the MPPI sampling distribution and derive an adaptive stability certificate with an explicit learning penalty. The main result is a payoff theorem: after a computable crossover time, the adaptive controller achieves a strictly tighter certified stability bound than any fixed covariance choice whose mismatch exceeds the residual smoothing and drift allowance. Numerical experiments illustrate the estimator convergence and the resulting stability-tightening effect.

Sampling-based model predictive control methods handle nonlinear dynamics and complex cost landscapes through Monte Carlo rollouts, yet typically employ fixed constraint penalties that do not adapt to model-plant mismatch. This paper proposes Residual-Conservative Model Predictive Path Integral Control (RC-MPPI), a sampling-based MPC framework that modulates safety conservatism online using the prediction-execution residual. RC-MPPI combines three coupled mechanisms: residual-dependent constraint tightening, adaptive safety-cost shaping, and residual-adaptive sampling modulation through exploration contraction and temperature relaxation. The temperature adaptation reflects a key insight: when the model is inaccurate, rollout cost evaluations become unreliable, and increasing temperature reduces overcommitment to apparent cost rankings. Under Lipschitz dynamics and sub-Gaussian disturbances, we derive probabilistic bounds on constraint violation and show that the joint effect of the adaptive mechanisms reduces violation probability as the residual grows. A rollout-cost uncertainty analysis further shows that model-plant mismatch perturbs MPPI importance weights in proportion to residual magnitude and inversely with temperature, providing theoretical justification for residual-adaptive temperature relaxation. Simulations on an LTI point-mass system and a planar 2R manipulator show improved safety margin, success rate, and control efficiency compared with vanilla MPPI under significant model-plant mismatch.

With the widespread availability of parallel computing hardware, sampling-based motion planning methods such as Model Predictive Path Integral (MPPI) control have become increasingly powerful for complex nonlinear systems in non-smooth task spaces. However, the sampling and forward-simulation pipeline in MPPI suffers from averaging-induced failure in cluttered environments, where the importance-weighted update averages incompatible rollouts and leads to hesitation or even collision when an obstacle lies directly ahead. This paper proposes Clustering-Embedded MPPI (CE-MPPI), a framework that architecturally resolves the averaging-induced failures inherent in standard MPPI within non-convex environments. Rather than simply mitigating interference, CE-MPPI redefines the control law by integrating a high-fidelity pruning and clustering stage. By leveraging density-based spatial clustering of applications with noise (DBSCAN) alongside a novel geometric direction feature that is extracted from collision-derived reference points, the system isolates feasible trajectory modes from the noise of infeasible rollouts. This is paired with an intelligent selection logic that optimizes for minimum cost in static scenes while actively steering opposite to obstacle flux in dynamic environments. Experiments in 2-D JAX-accelerated simulations show that CE-MPPI alleviates obstacle-front hesitation and avoids persistent coupling with moving obstacles in dynamic scenes. In particular, real-world tests on a 6-DoF UR5e manipulator with CUDA-parallel rollouts in Isaac Gym achieve a 48% reduction in time-to-goal and a 12% shorter end-effector path.

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.

We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizon-dependent stacked cost matrices, making explicit that the finite-sample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevel set over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a process-noise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DARE solution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.

Humanoid robots require whole-body controllers that are both robust and precise in contact-rich environments. While deep reinforcement learning (RL) achieves robust stability, its behavior is tightly coupled to the training objective and command interface, making it difficult to add new feedback objectives without retraining. In this study, we propose an RL guided whole-body model predictive path integral (MPPI) framework that acts as an add-on feedback controller on top of a pretrained RL policy. Instead of using RL policy as the final controller, we use it as a sampling prior that biases MPPI rollouts toward dynamically feasible behaviors. Task objectives are specified through modular MPPI cost terms, and MPPI closes the loop by continuously correcting the RL prior online to satisfy these objectives without retraining the policy. Simulations on a 29-DoF Unitree G1 humanoid in MuJoCo demonstrate stable high-rate control (average 280~Hz). The proposed method improves task-level precision over a pure RL baseline under the same command interface. This is achieved by correcting systematic drift during straight walking and tracking additional whole-body reference signals imposed through the cost.

Robots deployed in the real world must plan motions across diverse scenarios without per-scenario retuning. End-to-end reinforcement learning (RL) can generalize across scenarios but often becomes brittle under distribution shift, reward misspecification, and stochastic interactions. Model predictive path integral (MPPI) control enables strong real-time refinement without gradients, but its performance depends on a well-shaped sampling prior, while manually designing the priors does not scale to multi-scenario deployment. We present HOLO-MPPI (High-level Offline, Low-level Online MPPI), a multi-scenario motion planning framework that combines high-level policy learning with low-level stochastic optimal control. Offline, we learn a high-level policy that proposes scenario-robust plans in an abstract action space, with a learned world model for online rollout. Online, the policy serves as a data-driven prior generator that parameterizes MPPI's sampling distribution conditioned on the current observation and goal. MPPI then optimizes low-level control sequences around this prior in real time to adapt to local disturbances. We instantiate HOLO-MPPI in autonomous driving by designing an effective high-level action space and tailored model architectures. Our evaluation across diverse driving scenarios shows that HOLO-MPPI improves upon MPPI and end-to-end RL baselines while maintaining real-time control.

Chance-constrained Model Predictive Path Integral (MPPI) control is increasingly adopted for navigation in dynamic environments to explicitly bound collision risk. However, these probabilistic guarantees implicitly assume that upstream uncertainties from localization and perception are well-calibrated. In practice, estimators are often miscalibrated, inducing characteristic closed-loop failure modes: overconfidence leads to systematic safety violations, while underconfidence triggers overly conservative freezing or probability dilution. To address this critical gap, our primary contribution is a rigorous evaluation methodology applying proper scoring rules to assess the statistical validity of predicted collision risks during closed-loop execution. Concurrently, Dual-Uncertainty Chance-Constrained Tube MPPI (DUCCT-MPPI) is proposed as a real-time, risk-aware planning architecture. DUCCT-MPPI jointly integrates localization uncertainty via a one-tube Unscented Transform (UT) approximation and dynamic obstacle prediction uncertainty via Monte Carlo aggregation. Through extensive physics-based simulations, the framework demonstrates robust failure-mitigation, seamlessly transitioning to safe, conservative maneuvering without succumbing to functional deadlocks in highly cluttered environments. In highly cluttered environments, DUCCT-MPPI achieves superior robustness, outperforming established Monte Carlo MPPI baselines by nearly 28% in navigation success rate, while simultaneously recording the lowest travel times and minimizing induced social forces. Ultimately, these findings establish that reliable probabilistic safety in autonomous navigation dictates not only expressive risk models but statistically valid uncertainty estimates throughout the entire autonomy stack.

Sampling-based model predictive control methods, such as Model Predictive Path Integral (MPPI), offer derivative-free optimization and robustness in complex robotic systems. However, standard MPPI relies on cost-based soft penalties that cannot guarantee hard-constraint satisfaction, severely limiting its applicability to highly constrained tasks such as closed-chain manipulation. To address this, we propose Manifold-Constrained MPPI (MC-MPPI), a real-time sampling-based control framework that regulates nonlinear equality constraints while preserving the computational advantages of MPPI. The key idea is to decouple the constrained optimal control problem into latent-space planning and execution-level correction. At the planning stage, a Variational Autoencoder (VAE) learns a low-dimensional latent representation of the constraint manifold, enabling MPPI to efficiently generate near-feasible candidate trajectories without per-sample modification. Since this reference enables accurate linearization of the equality constraints, an execution-level Quadratic Programming (QP) controller resolves the residual manifold mismatch in a single solve rather than through iterative projection. Experiments on a 14-DoF closed-chain dual-arm system in both simulation and real-world settings demonstrate 100 Hz planning and 500 Hz execution, equality-residual regulation in static and dynamic environments, and a 95% success rate over 40 randomized hardware trials. Supplementary videos and implementation details are available at https://rcilab.github.io/mcmppi.

Autonomous and safe navigation of tractor-trailer systems requires accurate, real-time collision avoidance and dynamically feasible control, particularly in cluttered and complex agricultural environments. This is challenging due to their articulated, deformable geometries and nonlinear dynamics. Traditional methods oversimplify vehicle geometry or rely on precomputed distance fields that assume a known map, limiting their applicability in dynamic, partially unknown environments. To address these limitations, we propose a geometric neural encoder that provides fast and accurate distance estimates between the full tractor-trailer body and raw LiDAR perception, enabling real-time, map-free geometric reasoning. These learned distances are integrated into a Model Predictive Path Integral (MPPI) controller, allowing the system to incorporate true articulated geometry directly into its cost evaluation and enabling more responsive navigation in challenging agricultural settings. Simulation results demonstrate that the proposed framework generates dynamically feasible and safe trajectories for navigating tractor-trailer systems in cluttered and complex environments.

A central challenge of visual control with model-based reinforcement learning (RL) is reliable long-horizon planning: long rollouts with learned latent dynamics exhibit branching futures and multi-modal action-value distributions. In addition, compounding model errors amplified by visual occlusions make deep imagination brittle. We present ELVIS, a latent model predictive controller (MPC) designed to make long-horizon planning practical. ELVIS plans in a Dreamer-style recurrent state space model (RSSM) and replaces standard unimodal model predictive path integral (MPPI) with a Gaussian-mixture MPPI that maintains multiple coherent hypotheses over long horizons, avoiding mode averaging under branching rollouts. In parallel, ELVIS stabilizes deep imagination with a shared uncertainty-aware lambda-return: an ensemble of latent critics defines an upper-confidence-bound (UCB) score that gates a time-varying lambda, adaptively trading off bootstrapping versus look-ahead to limit compounding error during planning. The same return is used both to train an actor-critic prior from imagined rollouts and to score candidate trajectories inside GMM-MPPI, aligning RL objectives with the planner's long-horizon optimization. On fourteen DeepMind Control Suite visual tasks, ELVIS establishes state-of-the-art performance compared with TD-MPC2 and DreamerV3. Finally, ELVIS transfers zero-shot to a real-world sand-spraying task with severe occlusions, improving surface-quality metrics and demonstrating robustness beyond simulation.

Model Predictive Path Integral (MPPI) control is a powerful sampling-based strategy for nonlinear autonomous systems. However, its performance is often bottlenecked by the fidelity of nominal dynamics. We propose ICODE-MPPI, a robust framework that leverages Input Concomitant Neural Ordinary Differential Equations (ICODEs) to learn and compensate for unmodeled residual dynamics. Unlike discrete-time learners, ICODEs maintain physical consistency and temporal continuity during the MPPI prediction horizon. High-fidelity simulations on complex trajectories demonstrate that ICODE-MPPI achieves up to a 69% reduction in cross-tracking error under persistent disturbances compared to standard MPPI control. Furthermore, our analysis confirms that ICODE-MPPI significantly suppresses control chattering, yielding smoother steering commands and superior robust performance.

Dense, dynamic crowds pose a persistent challenge for autonomous mobile robots. Purely reactive planning methods, such as Model Predictive Path Integral (MPPI) control, often fail to escape local minima in complex scenarios due to their limited prediction horizon. To bridge this gap, we propose Ray-based Task-Oriented Latent Dynamics (RAY-TOLD), a hybrid control architecture that integrates obstacle information into latent dynamics and utilizes the robustness of physics-based MPPI with the long-horizon foresight of reinforcement learning. RAY-TOLD leverages a LiDAR-centric latent dynamics model to encode high-dimensional sensor data into a compact state representation, enabling the learning of a terminal value function and a policy prior. We introduce a policy mixture sampling strategy that augments the MPPI candidate population with trajectories derived from the learned policy, effectively guiding the planner towards the goal while maintaining kinematic feasibility. Extensive tests in a stochastic environment with high-density dynamic obstacles demonstrate that our method outperforms the MPPI baseline, reducing the collision rate. The results confirm that blending short-horizon physics-based rollouts with learned long-horizon intent significantly enhances navigation reliability and safety.

Motion planning for autonomous vehicles often requires satisfying multiple conditionally conflicting specifications. In situations where not all specifications can be met simultaneously, minimum-violation motion planning maintains system operation by minimizing violations of specifications in accordance with their priorities. Signal temporal logic (STL) provides a formal language for rigorously defining these specifications and enables the quantitative evaluation of their violations. However, a total ordering of specifications yields a lexicographic optimization problem, which is typically computationally expensive to solve using standard methods. We address this problem by transforming the multi-objective lexicographic optimization problem into a single-objective scalar optimization problem using non-uniform quantization and bit-shifting. Specifically, we extend a deterministic model predictive path integral (MPPI) solver to efficiently solve optimization problems without quadratic input cost. Additionally, a novel predicate-robustness measure that combines spatial and temporal violations is introduced. Our results show that the proposed method offers an interpretable and scalable solution for lexicographic STL minimum-violation motion planning within a single-objective solver framework.

Many safety-critical control systems operate under latent uncertainty that sensors cannot resolve at decision time. Such uncertainty, arising from unknown physical properties, disturbances, or unobserved geometry, affects dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider systems with measured state and an unknown, time-invariant parameter represented by a belief distribution. We propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) controller that plans under this belief, regularizes performance using Conditional Value-at-Risk (CVaR), and imposes a CVaR constraint on a trajectory safety margin over the horizon. For the exact risk-constrained formulation underlying this controller, we establish three properties: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the objective risk weight tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In contact-rich MuJoCo simulations of vision-guided dexterous stowing, where a manipulator inserts a grasped object into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared with 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces. Project page: https://clintonenwerem.com/belief-cvar-mppi/.

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 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.

This paper introduces a method for Model Predictive Path Integral (MPPI) control that optimizes sample generation towards an optimal trajectory through Stein Variational Gradient Descent (SVGD). MPPI relies upon predictive rollout of trajectories sampled from a distribution of possible actions. Traditionally, these action distributions are assumed to be unimodal and represented as Gaussian. This assumption can lead to suboptimal rollout predictions due to sample deprivation and, in the case of differentiable simulation, sensitivity to noise in the cost gradients. By introducing SVGD updates in between MPPI environment steps, we present Stein-Optimized Predictive Path-Integral Inference (SOPPI), an MPPI/SVGD algorithm that can dynamically update noise distributions during rollouts to better capture action sampling distributions without an excessive increase in computational requirements. We demonstrate the efficacy of SOPPI through experiments on a planar cart-pole, 7-DOF robot arm, a planar bipedal walker, and a Unitree G1 29-DOF Robot. Results indicate improved inference of an underlying multi-modal action distribution compared to other Stein-based MPPI algorithms, with improved system performance and similar or improved efficiency across a range of hyper-parameters.

Model Predictive Path Integral (MPPI) control provides a sampling-based framework for optimal control, while Control Barrier Functions (CBFs) provide a principled means of enforcing safety constraints. We introduce BR-MPPI, which integrates CBF-like conditions into MPPI's control sampling procedure. CBFs impose inequality constraints that bound the rate of change of barrier functions using a class-K function of the barrier value. We instead impose the CBF condition as an equality constraint using a parametric linear class-K function and augment the system state with its parameter. The parameter's time derivative serves as an additional control input optimized by MPPI. We further design a cost function that promotes parameter values consistent with Nagumo's condition at the safe-set boundary, thereby encouraging safety. The resulting multiple state- and control-dependent equality constraints pose a challenge for random control sampling. We address this through state transformations and control projections inspired by manifold path planning that map sampled controls onto the constraint manifold. We also incorporate learned signed distance fields to represent robot geometry and reduce computation time. Simulations demonstrate improved sample efficiency over vanilla MPPI and higher navigation success rates across five robot models compared with safety-oriented MPPI variants. Hardware experiments on a quadrotor further demonstrate the method's ability to navigate constrained environments near safe-set boundaries.