Why soft obstacles instead of hard “if collision discard rollout”?

Hard rejection creates **discontinuous** costs and brittle gradients for sampling methods—many rollouts die in useless configurations. Smooth penalties keep **grad-like** information in the exponential weights while still exploding cost on deep penetration.

Is this literally a hovercraft dynamics model?

Only as a **cartoon**: real hovercraft have attitude–thrust coupling, fan dynamics, and ground effect. The **double integrator with ‖u‖₂** cap is the standard **holonomic** surrogate used in curricula to isolate **MPC in the plane**.

Why does the robot sometimes scrape obstacles despite MPC?

Soft costs allow **trade-offs**: if **q_pos** dominates or **horizon** is short, the planner may briefly accept small penetration to cut a corner. Raise **q_crash**, lengthen **H**, or increase **R** on control to be more cautious.

Does disabling the rollout fan change the controller?

No—it only hides the **purple** ensemble for performance. The **green** best-path preview and applied **u** are unchanged.

Holonomic 2D Hovercraft MPC (MPPI)

This page extends the sampling-based MPC idea from engineering/mpc-pendulum to a planar holonomic “hovercraft” modeled as a double integrator: ẍ = u_x, ÿ = u_y with a bounded Euclidean thrust ‖(u_x, u_y)‖₂ ≤ u_max. A Model Predictive Path Integral (MPPI) loop draws K noisy open-loop acceleration sequences over a horizon H, rolls each trajectory forward with simple semi-implicit Euler steps plus light velocity damping, and scores rollouts with a quadratic goal attraction to a click-draggable target, velocity regularization, control effort, and soft obstacle costs around fixed circular rocks (exponential “skirt” when outside plus a strong quadratic penalty if the robot disk penetrates the inflated obstacle boundary). The weighted-average control update matches MPPI’s information-theoretic reweighting; the simulator applies the first mean action until the next replan at interval Δt. Purple fan trajectories (optional when K is not huge) visualize the ensemble; the green polyline is the lowest-cost sample path from the last plan. Layout presets create a narrow corridor or an offset pillar field so students can trade horizon, temperature λ, noise σ, and obstacle weights against conservatism and compute budget.

Who it's for: Students who finished the pendulum MPPI demo and want the same sampling intuition on a 2D navigation task before touching full nonholonomic models or QP-based MPC.

Key terms

Holonomic robot
Double integrator
Model Predictive Path Integral (MPPI)
Trajectory sampling
Soft collision constraints
Obstacle avoidance
Warm-started control sequence

How it works

Holonomic 2D “hovercraft” with ẍ = u_x, ÿ = u_y and bounded L2 thrust: MPPI rolls K noisy acceleration trajectories over horizon H, adds obstacle skirts around circular rocks, and reweights the mean control each step — same sampling spirit as engineering/mpc-pendulum, now with goal seeking and collision avoidance in the plane.

Key equations

ẍ = u_x, ÿ = u_y, with ‖(u_x,u_y)‖₂ ≤ u_max — holonomic double integrator (hovercraft abstraction).

MPPI: sample noisy accelerations along the horizon, score rollouts with goal, speed, effort, and obstacle soft penalties, then exponential weights update the mean control sequence.

Frequently asked questions

Why soft obstacles instead of hard “if collision discard rollout”?: Hard rejection creates discontinuous costs and brittle gradients for sampling methods—many rollouts die in useless configurations. Smooth penalties keep grad-like information in the exponential weights while still exploding cost on deep penetration.
Is this literally a hovercraft dynamics model?: Only as a cartoon: real hovercraft have attitude–thrust coupling, fan dynamics, and ground effect. The double integrator with ‖u‖₂ cap is the standard holonomic surrogate used in curricula to isolate MPC in the plane.
Why does the robot sometimes scrape obstacles despite MPC?: Soft costs allow trade-offs: if q_pos dominates or horizon is short, the planner may briefly accept small penetration to cut a corner. Raise q_crash, lengthen H, or increase R on control to be more cautious.
Does disabling the rollout fan change the controller?: No—it only hides the purple ensemble for performance. The green best-path preview and applied u are unchanged.

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