Feedback-MPPI: Fast Sampling-Based MPC

• Feedback-MPPI is a sampling-based model predictive controller that integrates rollout differentiation to compute local linear feedback gains for continuous correction.
• It employs a two-rate architecture where the costly stochastic optimization is executed at a lower frequency while a fast inner-loop applies immediate linear adjustments.
• Experimental evaluations on quadrupeds and quadrotors highlight enhanced tracking accuracy, stability, and efficiency compared to standard MPPI approaches.

Searching arXiv for papers on Feedback-MPPI and closely related MPPI feedback variants. Feedback-MPPI, abbreviated F-MPPI, is a sampling-based model predictive controller that augments standard Model Predictive Path Integral control with a local linear feedback law computed from the sensitivity of the MPPI solution to the current state. In the specific formulation introduced in “Feedback-MPPI: Fast Sampling-Based MPC via Rollout Differentiation -- Adios low-level controllers,” the purpose of this augmentation is to permit rapid closed-loop corrections around the current state without requiring full re-optimization at each timestep, thereby retaining MPPI’s flexibility for nonlinear dynamics and non-convex costs while adding a Riccati-like inner-loop feedback mechanism (Belvedere et al., 17 Jun 2025). In a broader usage within the MPPI literature, “feedback” can also denote the receding-horizon closed-loop character of MPPI itself; however, F-MPPI names a more specific architecture in which rollout differentiation supplies an explicit local gain matrix rather than relying only on repeated horizon replanning (Leon et al., 2023).

1. Standard MPPI background

F-MPPI is defined relative to the standard MPPI formulation for finite-horizon optimal control. The problem is written as

$$\begin{aligned} \theta^\ast = \arg\min \;& \ell_N(x_N) + \sum_{i=0}^{N-1} \ell_i(x_i,u_i) \ \text{s.t. } & x_0 = \hat x, \ & x_{i+1} = f(x_i,u_i), \ & u_i = \pi_\theta(x_i,t_i). \end{aligned}$$

Here $x_i$ is the state, $u_i$ is the control, $\hat x$ is the current measured state, $\ell_i$ are running costs, $\ell_N$ is a terminal cost, $\theta$ are sampled control parameters, and $\pi_\theta$ is a parameterized input trajectory policy. The policy parametrization may use direct control sampling, splines, or other low-dimensional parameterizations (Belvedere et al., 17 Jun 2025).

MPPI samples around a nominal parameter vector $\bar\theta$,

$$\theta^k = \bar\theta + \Delta^k, \qquad \theta^k \sim \mathcal N(\bar\theta,\Sigma),$$

rolls each sample through the nonlinear dynamics, and evaluates

$x_i$0

The resulting parameter update is a weighted average of perturbations,

$x_i$1

with importance weights

$x_i$2

The control actually applied is

$x_i$3

In standard MPPI, this $x_i$4 is applied directly and typically held constant until the next optimization cycle. That coupling between optimization frequency and actuation frequency is the limitation F-MPPI is designed to relax (Belvedere et al., 17 Jun 2025).

2. Defining feature: local feedback around the MPPI solution

The central addition in F-MPPI is a local linear approximation of the optimal MPPI policy with respect to the initial state,

$x_i$5

This matrix is used as a feedback gain between MPPI solves: $x_i$6 where $x_i$7 is a local setpoint associated with the MPPI solution. The formulation does not simply reuse $x_i$8 as the setpoint. Instead, the setpoint is continuously updated by linearly interpolating between $x_i$9 and $u_i$0 so that inner-loop tracking is smoother (Belvedere et al., 17 Jun 2025).

This yields a two-rate architecture. The outer loop runs standard MPPI at a moderate frequency and produces a nominal sequence and control action. The inner loop runs at a higher rate and applies the linear feedback correction continuously until the next MPPI update. The paper’s description is explicit: the expensive stochastic horizon optimization is performed less often, while the high-bandwidth stabilization is handled by the linear feedback correction. This suggests a division of labor in which MPPI remains the mechanism for global nonlinear and non-convex trajectory optimization, whereas the gain $u_i$1 supplies local corrective authority around the current trajectory (Belvedere et al., 17 Jun 2025).

The intended consequence is high-frequency closed-loop action without re-solving the entire sampling problem every timestep. The paper presents this as especially relevant for fast systems where planning at $u_i$2 Hz is insufficient but actuation at $u_i$3–$u_i$4 Hz is required (Belvedere et al., 17 Jun 2025).

3. Rollout differentiation and gain derivation

The feedback gain is obtained by differentiating the MPPI solution with respect to the initial state $u_i$5. Because

$u_i$6

the chain rule gives

$u_i$7

The derivative of the MPPI parameter update follows from

$u_i$8

which yields

$u_i$9

The key sensitivity identity is the derivative of the normalized importance weights: $\hat x$0 Substitution into the chain rule produces the general gain expression

$\hat x$1

For direct zero-order sampling with

$\hat x$2

the parametrization Jacobian reduces to a selector, and the concrete implementation becomes

$\hat x$3

The gain therefore depends on rollout cost gradients with respect to the initial state rather than on a separate backward dynamic-programming pass (Belvedere et al., 17 Jun 2025).

The paper characterizes this construction as Riccati-inspired rather than Riccati-derived. In a toy validation on a double integrator, it compares MPPI-derived gains to the infinite-horizon LQR gain obtained from the discrete algebraic Riccati equation with $\hat x$4 and $\hat x$5, reporting

$\hat x$6

As the number of samples increases, the MPPI gain converges toward the LQR gain; even with fewer samples, the resulting closed loop remains stable. A plausible implication is that the differentiable sampling law can recover familiar local feedback structure in regimes where linear-quadratic approximations are meaningful, while still being embedded in a nonlinear sampling-based MPC stack (Belvedere et al., 17 Jun 2025).

4. Computational structure, differentiability, and control-theoretic interpretation

F-MPPI changes the computational profile of MPPI without replacing the MPPI optimizer itself. The gain computation requires differentiating rollout costs $\hat x$7, but this operation is parallelizable because each rollout is independent. The reported implementation uses JAX, runs on CUDA-compatible GPUs, computes rollouts and gradients in parallel, and exploits automatic differentiation. The paper states that the gain computation increases runtime by about $\hat x$8–$\hat x$9 relative to plain MPPI, depending on horizon and sample count, but remains practical because it does not require a new full optimization at the inner-loop frequency (Belvedere et al., 17 Jun 2025).

The framework also imposes a differentiability requirement. Because the gain uses derivatives of rollout costs, local differentiability is needed. The paper states that this is not a serious limitation in robotics because many models are piecewise smooth. It also notes a more specific caveat for constraints represented by hard indicator penalties: such terms are locally constant and therefore do not affect the gain. A smoother barrier formulation is identified as a better alternative, but this was left for future work (Belvedere et al., 17 Jun 2025).

These points delimit the method’s position within control theory. F-MPPI is not a fully gradient-based nonlinear MPC solver, and it does not explicitly solve a Riccati equation. It is instead a sampling-based MPC method supplemented by a local sensitivity-based linear correction. That distinction matters because the feedback law inherits both the strengths and limitations of the rollout distribution and cost model. This suggests that the controller’s local behavior is strongly shaped by the quality of trajectory-cost derivatives, even though the nominal plan is still produced by importance-weighted stochastic sampling (Belvedere et al., 17 Jun 2025).

5. Experimental platforms and reported performance

The paper validates F-MPPI on two robotic platforms: a quadrupedal robot in simulation and a quadrotor in simulation and on hardware. In the quadruped study, the platform is a Unitree Aliengo simulated in MuJoCo with a simplified Single Rigid Body Dynamics model and control inputs given by ground reaction forces. The task is to track desired linear velocity references between $\ell_i$0 and $\ell_i$1 m/s over randomly generated rough terrain while withstanding random disturbances of $\ell_i$2 Nm. The cost is

$\ell_i$3

The experiments use $\ell_i$4 samples, horizon $\ell_i$5, $\ell_i$6 s, F-MPPI at $\ell_i$7 Hz, and baseline MPPI at $\ell_i$8, $\ell_i$9, and $\ell_N$0 Hz. The reported result is that F-MPPI at $\ell_N$1 Hz remains competitive with, and in some metrics better than, MPPI running at $\ell_N$2 Hz, while the inner-loop feedback keeps the ground reaction forces updated at $\ell_N$3 Hz (Belvedere et al., 17 Jun 2025).

In the quadrotor study, the platform is a MikroKopter quadrotor with onboard compute on an Nvidia Jetson Orin NX, state estimation via Telekyb3 and an Unscented Kalman Filter, and simulation in Gazebo. The task is to reach a goal in $\ell_N$4D and, in simulation, navigate around circular obstacles. Rotor speeds $\ell_N$5 are sampled with cubic splines. The main configuration uses horizon $\ell_N$6, $\ell_N$7 samples, F-MPPI planning at $\ell_N$8 Hz, and inner-loop updates at $\ell_N$9 Hz. The stage cost includes

$\theta$0

together with

$\theta$1

and obstacle avoidance is represented by

$\theta$2

The reported outcomes distinguish simulation and real flight. In simulation, the robot successfully navigates through obstacles and reaches the goal without getting stuck in local minima. In real flight, F-MPPI outperforms standard MPPI even when MPPI runs at the highest feasible hardware frequency of $\theta$3 Hz. The paper reports post-transient RMSE values of $\theta$4 m in $\theta$5 and $\theta$6 m in $\theta$7 for F-MPPI, versus $\theta$8 m in $\theta$9 and $\pi_\theta$0 m in $\pi_\theta$1 for standard MPPI, corresponding to a $\pi_\theta$2 reduction in $\pi_\theta$3-RMSE and a $\pi_\theta$4 reduction in $\pi_\theta$5-RMSE in the experiment figure caption (Belvedere et al., 17 Jun 2025).

6. Position within the MPPI literature

The term “feedback” has more than one use in the MPPI literature, and the distinction is important. Some papers explicitly state that standard MPPI is already a feedback controller in the model predictive control sense because the state is re-estimated and the optimization rerun at each control step. That interpretation appears in output-sampled MPPI and in safety-oriented MPPI-DBaS, where the feedback character is identified with receding-horizon closed-loop replanning rather than with an added inner-loop gain law (Leon et al., 2023). A related control-theoretic treatment further describes MPPI as a sample-based receding-horizon control method whose feedback behavior is induced by repeated recomputation from the current measured state, and analyzes deterministic MPPI suboptimality as exploration variance is annealed to zero (Homburger et al., 28 Feb 2025).

Other recent MPPI variants introduce explicit corrective layers, but of different types. Manifold-Constrained MPPI is described as a two-level feedback control architecture in which latent-space MPPI produces a near-feasible reference and an execution-level quadratic program corrects residual manifold mismatch in a single solve rather than through iterative projection (Lee et al., 24 May 2026). RL-guided whole-body MPPI places MPPI on top of a pretrained RL locomotion policy as an add-on feedback controller, with the RL policy used as a sampling prior and MPPI continuously correcting that prior online through modular cost terms (Seo et al., 23 Jun 2026). CSC-MPPI is presented as a constrained, feedback-oriented extension that first shifts sampled controls into the feasible region through a primal-dual gradient method and then uses DBSCAN to select representative control inputs instead of taking a global weighted average (Park et al., 19 Jun 2025).

The resulting landscape can be summarized succinctly:

Method	Feedback mechanism	Distinctive feature
F-MPPI	Local linear gain from rollout differentiation	High-rate inner-loop correction (Belvedere et al., 17 Jun 2025)
MC-MPPI	Execution-level QP correction	Hard manifold-constraint enforcement (Lee et al., 24 May 2026)
RGB	MPPI as add-on feedback above RL prior	Modular objective correction without retraining (Seo et al., 23 Jun 2026)
CSC-MPPI	Receding-horizon control with feasible-sample correction and cluster selection	Primal-dual constraint handling and DBSCAN (Park et al., 19 Jun 2025)

A common misconception is therefore that “Feedback-MPPI” names any closed-loop MPPI controller. The literature shows two usages. One is generic, referring to MPPI’s ordinary receding-horizon feedback structure. The other, and more specific, refers to F-MPPI as a controller that differentiates the MPPI solution to obtain an explicit state-feedback gain (Belvedere et al., 17 Jun 2025).

7. Significance, scope, and limitations

F-MPPI’s contribution is best understood as a feedback layer on top of MPPI rather than as a new stochastic optimizer. It keeps the standard MPPI sampling rule, importance weights, and nominal-control update, and it adds a local linear gain derived from the state sensitivity of the sampled solution. This allows high-frequency stabilization and tracking between expensive MPPI updates while preserving the flexibility of MPPI for nonlinear, non-convex, and constrained robotic problems (Belvedere et al., 17 Jun 2025).

The method’s scope is bounded by the assumptions of differentiable rollout costs and the quality of the local linear approximation. Because indicator-style obstacle penalties do not contribute to the gain, the inner-loop correction can be partially blind to hard nonsmooth constraints. This suggests that F-MPPI is particularly well matched to settings where a smooth local correction around the nominal MPPI trajectory is valuable, but where the nominal planner still carries responsibility for global obstacle avoidance and long-horizon nonlinear behavior. The empirical results on a quadruped and a quadrotor indicate improved smoothness, stability, and tracking accuracy under those conditions (Belvedere et al., 17 Jun 2025).

Within the broader MPPI literature, Feedback-MPPI marks a shift from viewing sampling-based MPC purely as repeated open-loop sequence optimization toward treating the sampled solution itself as an object that can be differentiated and turned into a fast corrective controller. That is the sense in which the method bridges sampling-based planning and Riccati-style feedback without abandoning the stochastic rollout framework that defines MPPI (Belvedere et al., 17 Jun 2025).