Hybrid Feedback Sampling for Sample-Efficient Model Predictive Control

Chaoyi Pan; Zeji Yi; John Zhang; Zachary Manchester; Guannan Qu; Guanya Shi

cs.ROAug 19, 2026

Hybrid Feedback Sampling for Sample-Efficient Model Predictive Control

Authors: Chaoyi Pan, Zeji Yi, John Zhang, Zachary Manchester, Guannan Qu, Guanya Shi

Organizations: Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA · Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract

Thanks to its parallelizability and flexibility, sampling-based Model Predictive Control (MPC) has become widely popular for controlling real-world robotic systems. However, for high-dimensional and open-loop unstable dynamical systems, the required number of samples to improve the control sequence will grow exponentially with the horizon, leading to poor sample efficiency and numerical instability. This paper investigates the instability of shooting methods in sampling-based MPC and shows that the optimal sampling proposal distribution can be realized by sampling with an optimized feedback policy. We refer to this algorithm as Feedback Sampling MPC (FS-MPC). FS-MPC involves a hybrid sampling design which balances local and global search based on the system stability and the available computation budget. Our theoretical analysis shows that our hybrid sampling approach achieves faster convergence than standard MPPI and better optimality than standard feedback sampling. Empirically, in diverse contact-rich control tasks like humanoid loco-manipulation and dexterous manipulation, we show that FS-MPC successfully tackles dynamically unstable tasks where standard sample-based approaches struggle, and strictly outperforms feedback policies alone. Finally, we validate our method on humanoid robot locomotion and manipulation tasks in the real world.

Figures & tables

Fig. 1: (a) Motivation of FS-MPC: showing the sampling distributions of FS-MPC (left) and standard MPPI (middle), and the distributions of their samples’ costs (right), where the quadrotor is initialized at the same state. Given the long horizon navigation task with unstable quadrotor dynamics, feedback sampling can stabilize the system while MPPI’s samples diverge. Therefore, the cost distribution of FS-MPC is significantly more optimal. (b) Deploy FS-MPC on the real humanoid robot. When planning over contact-rich and unstable humanoid dynamics, FS-MPC can control the robot with both stability and precision.

Fig. 2: Visualization of shooting control sequence (a), its joint covariance matrix (b) and its linear realization (c). (1) sampling with iLQR feedback

K^{*}

(c1) would lead to stable sampling distribution (b1) which has a rich pattern. (2) Standard MPPI sampling rule (c2) would lead to an isotropic Gaussian distribution (b2), which is not optimal. From the perspective of joint control distribution design, feedback sampling can be viewed as the state space equivalent realization of optimal covariance matrix

\Sigma_{k}^{*}

Fig. 3: Hybrid feedback sampling balances local and global search. (1) Hybrid feedback sampling draws from both local and global proposal distributions, enabling escape from local minima. (2) Standard feedback sampling relies solely on the local distribution, often failing to traverse high-cost barriers.

Fig. 4: Task overview over different controllers. (a) Acrobot: the controller needs to swing up the pendulum from the bottom to the top and then balance it at an unstable equilibrium. (b) Quadrotor: the controller needs to track a desired trajectory and fly through a few gates. The quadrotor is unstable and colliding with a gate can lead to a crash. (c) Quadruped Hill: the controller needs to track desired key points to navigate a hill. Due to the uneven terrain, the quadruped would fall and need to get up. (d) H1-2 PNP: the controller needs to do whole-body control for a 23-DoF humanoid robot to pick up and place a box on the target location. The system involves both an unstable system (the humanoid robot) and long-horizon reasoning over the contact. (e) Allegro: a in-hand reorientation task with allegro hand. The robot needs to reorient the object to a random target orientation. In all tasks, the white line is the sampled trajectory, and the pink line is the planned trajectory.

Controllers	iLQR	MPPI	F-MPPI	FS-MPC
Acrobot	$104.55\pm 0.63$	$117.16\pm 0.20$	$122.12\pm 7.45$	$\mathbf{76.38\pm 15.24}$
Quadrotor	$8.44\pm 0.37$	$13.36\pm 9.77$	$8.78\pm 0.20$	$\mathbf{7.99\pm 0.40}$
Quadruped Hill	$3.39\pm 0.70$	$2.79\pm 0.02$	$2.68\pm 0.17$	$\mathbf{2.51\pm 0.04}$
Allegro	$16.86\pm 2.40$	$12.22\pm 4.84$	$9.27\pm 1.28$	$\mathbf{6.50\pm 0.38}$
H1-2 PNP	$1.72\pm 0.06$	$1.55\pm 0.15$	$2.00\pm 0.43$	$\mathbf{1.06\pm 0.10}$

TABLE I: Cost comparison between controllers with iLQR feedback. From top to bottom: tasks are sorted from lower-dimensional to higher-dimensional, and from contact-free to contact-rich. FS-MPC can consistently outperform MPPI and iLQR.

Tasks	humanoidrun	humanoidstandup
iLQR	-	-
PPO	$-12.34\pm 0.01$	$-24.76\pm 0.01$
MPPI	$-12.50\pm 0.00$	$-27.75\pm 0.47$
F-MPPI (w/ PPO)	$-10.07\pm 0.22$	$-21.14\pm 0.06$
FS-MPC (w/ PPO)	$\mathbf{-15.65\pm 0.10}$	$\mathbf{-29.41\pm 0.55}$

TABLE II: Rollout trajectory cumulative cost on the humanoid running/standing up task.

Fig. 5: Tracking error of iLQR and FS-MPC on real robot.

Methods	MPPI	iLQR	FS-MPC
Tracking Error (m)	Failed	$0.93\pm 1.00$	$0.70\pm 0.66$

TABLE III: Tracking error comparison over different controllers in the real world. MPPI failed due to the high variance of the sampling procedure. FS-MPC can still stabilize the robot thanks to the feedback design.

Fig. 6: FS-MPC on real robot. Upper: H1 humanoid robot walking from one side of the room to the other. Lower: H1 humanoid robot holding a cylinder and put it into a cart. FS-MPC can finish the task while MPPI failed due to unstable sampling.

Symbol	Meaning	Definition
$n$	state dimension
$m$	control dimension
$A_{t}$	Discrete-time system matrix at time $t$	$A_{t}\in\mathbb{R}^{n\times n}$
$B_{t}$	Discrete-time input matrix at time $t$	$B_{t}\in\mathbb{R}^{n\times m}$
$Q_{t}$	state cost matrix at time $t$	$Q_{t}\in\mathbb{R}^{n\times n},Q_{t}\succeq 0$
$R_{t}$	control cost matrix at time $t$	$R_{t}\in\mathbb{R}^{m\times m},R_{t}\succeq 0$

TABLE IV: Notation table

Model Predictive Control

Model Predictive Control optimizes future actions with a system model, repeatedly applying the first action as new information arrives.

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Sampling-Based Control via Entropy-Regularized Optimal Transport

Vincent Pacelli, Akash Ratheesh, Evangelos A. Theodorou

Autonomous Control and Decision Systems Laboratory Georgia Institute of Technology

Robotic Control Nonlinear MPC Model Predictive Control Optimal Control Optimal Transport

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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.

Abstract

Model Predictive Control

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