Barrier-Rate Guided MPPI

PaperTitle

BR-MPPI: Barrier-Rate Guided MPPI for Enforcing Multiple Inequality Constraints with Learned Signed Distance Fields

Author

Hardik Parwana¹ *, Taekyung Kim¹ *, Kehan Long², Bardh Hoxha³, Hideki Okamoto³, Georgios Fainekos³, Dimitra Panagou¹

Note

(* equal contribution)

Affiliation

¹ Department of Robotics, University of Michigan; ² Department of Electrical and Computer Engineering, University of California, San Diego (UCSD); ³ Toyota Research Institute of North America (TRINA)

Venue

Paper

https://arxiv.org/abs/2506.07325

2 more properties

BR-MPPI: Uses the same 1,000-sample budget to find a trajectory through the narrow passage.

BR-MPPI incorporates CBF-like constraints directly into MPPI sampling. It augments the state with adaptive barrier-rate variables and projects sampled controls onto the corresponding equality constraints before rollout, allowing the constraint evolution to be optimized together with the robot motion.

Across five robot models, BR-MPPI achieves 87–100% success with no unsafe trials over 500 benchmark runs using analytic barriers. Learned signed distance fields reduce mean command computation time by 2.10–2.52× in the three evaluated models.

Paper

Code

Video

Why guide the samples?

Vanilla MPPI samples control sequences without explicitly enforcing inequality constraints. Collision penalties can discourage unsafe trajectories, but useful samples may become sparse in constrained environments. Applying a CBF filter only to the selected action enforces a local condition without changing how trajectories are sampled over the planning horizon.

Can we incorporate multiple safety constraints into MPPI sampling without repeated constrained optimization?

BR-MPPI vs Vanilla MPPI

BR-MPPI and vanilla MPPI under the same 1,000-sample budget.

Both comparisons use the same 1,000-sample budget. For the dynamic unicycle, BR-MPPI passes through the narrow gap while vanilla MPPI takes a detour; for the hexagonal robot, BR-MPPI reaches the goal while vanilla MPPI remains near its initial position. In the dynamic-unicycle case, increasing vanilla MPPI to 20,000 samples produces a path similar to BR-MPPI.

How BR-MPPI works

The key step in BR-MPPI is to replace each barrier inequality with a parameterized equality and optimize its rate together with the control input sequences.

For each constraint

h_i(x)\geq 0

, BR-MPPI introduces an adaptive rate

\alpha_{i,t}

and uses the barrier equality

h_i(x_{t+1})-h_i(x_t)=-\alpha_{i,t}\,h_i(x_t).

When

h_i(x)\geq 0

, a negative

\alpha_{i,t}

increases the barrier value and moves the predicted state away from the corresponding boundary, while a positive value allows the barrier value to decrease. Each constraint has its own rate, allowing MPPI to adapt the barrier evolution along the sampled trajectory.

BR-MPPI: augment the state, project sampled inputs onto the barrier equalities, roll out the dynamics, and update the sampling distribution.

Augment and sample: Introduce one barrier-rate state per constraint and sample robot pseudo-inputs together with rate increments.

Project jointly, then roll out: At each rollout step, jointly project the sampled pseudo-inputs onto the barrier equalities, then propagate the resulting state and rate dynamics.

Score and update: Evaluate the trajectory cost, including goal, control, and barrier-rate terms, and apply MPPI’s cost-weighted update to the nominal pseudo-input sequence.

Simulation: Five Robot Models

We compare BR-MPPI with MPPI-CBF, Shield-MPPI, SC-MPPI, and GS-MPPI across five robot models. The enlarged panel shows BR-MPPI, while the smaller panels show the four safety-oriented MPPI baselines. Following-camera views emphasize local motion and body clearance, with an inset retaining the global route.

Single Integrator

Point-mass navigation through a dense obstacle field.

Single integrator: BR-MPPI and four safety-oriented MPPI baselines.

Unicycle

Nonholonomic navigation requires the heading and forward velocity to jointly determine the feasible path through obstacles.

Unicycle: BR-MPPI and four safety-oriented MPPI baselines.

Dynamic Unicycle & Live Barrier Rates

The traces below the BR-MPPI rollout show a few applied barrier-rate values

\alpha_{i,t}

over time. Colors correspond to individual obstacle constraints, illustrating how different barrier rates are adapted as the robot moves through the environment.

Dynamic unicycle: trajectory comparison with synchronized BR-MPPI barrier rates.

Planar Quadrotor

The planner handles second-order translational and rotational dynamics while navigating close to obstacles.

Planar quadrotor: BR-MPPI and four safety-oriented MPPI baselines.

Mobile Arm

The mobile base and two articulated arms are planned together, making whole-body obstacle clearance part of the constraint evaluation.

Mobile arm: BR-MPPI and four safety-oriented MPPI baselines.

Benchmark Results

We evaluate 100 trials per method and robot model, for 2,500 trials across five controllers and five robot models. With analytic barriers, BR-MPPI succeeds in 475 of 500 trials and has no unsafe outcomes; the remaining trials terminate in timeout or deadlock.

Benchmark results across five robot models: success rate, unsafe rate, and mean command computation time.

BR-MPPI achieves 95%, 87%, 100%, 98%, and 95% success for the single integrator, unicycle, dynamic unicycle, planar quadrotor, and mobile arm, respectively, with zero unsafe trials across all five models.

Mean command computation time ranges from 26.39 to 145.27 ms. BR-MPPI is slower than MPPI-CBF and GS-MPPI, but faster than SC-MPPI, across all five models.

Learned Robot Geometry for Faster Planning

A neural signed distance field (SDF) represents the distance to the robot body from its geometric description. This provides a compact body-distance model for evaluating obstacle clearance against point-cloud obstacles. Training uses a distance-regression loss together with an Eikonal loss on the spatial gradient.

Analytic and learned signed-distance barriers within BR-MPPI.

Replacing the analytic representation with the learned SDF reduces mean command computation time by 2.10× for the unicycle, 2.52× for the dynamic unicycle, and 2.22× for the planar quadrotor, without retuning BR-MPPI. The effect on navigation performance is model dependent.

Hardware Experiment: Constrained Quadrotor Navigation

We implement BR-MPPI on a quadrotor fitted with a 0.28 × 0.56 m rectangular footprint. The controller plans planar position and heading using Vicon localization, while each obstacle is represented by a 30-point cloud. The task requires the vehicle to translate and rotate through a constrained passage while accounting for its full footprint.

Synchronized hardware and planner views of one BR-MPPI traversal.

Three synchronized views: overhead camera, side camera, and planner visualization. The cyan rectangle marks the quadrotor footprint in the camera views, while the planner shows the modeled footprint, sampled trajectories, optimized trajectory, and goal.

Repeated Hardware Trials

Four additional BR-MPPI hardware trials in the same obstacle course.

These trials show repeated navigation through the constrained passage with different approach motions, complementing the synchronized example above.

BibTeX

@inproceedings{parwana2026brmppi,
  title = {{BR-MPPI}: Barrier-Rate Guided MPPI for Enforcing Multiple
           Inequality Constraints with Learned Signed Distance Fields},
  author = {Parwana, Hardik and Kim, Taekyung and Long, Kehan and
            Hoxha, Bardh and Okamoto, Hideki and Fainekos, Georgios and
            Panagou, Dimitra},
  booktitle = {arXiv preprint arXiv:2506.07325},
  shorttitle = {BR-MPPI},
  year = {2026}
}
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