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Planning Trajectories that Bounce: Reflection Classes for Collision-Tolerant Robots

Subhadeep Koley, Subhrajit Bhattacharya, David Saldaña

arXiv:2609.27145v1cs.RO

TL;DR

Robots with limited maneuverability may spend substantial effort avoiding contact, even when wall interactions could be exploited. The paper represents reflection sequences as distinct search classes and executes selected paths with a contact-aware controller. In the tested simulation, reflecting references had higher completion rates and generally required less effort and time, despite being longer.

  • Problem

    For robots with limited maneuverability, avoiding contact can demand substantial steering effort and time even when surrounding surfaces could be safely exploited.

    Source: §I · p. 1

  • Method

    The planner tracks ordered reflection sequences in an augmented state graph, finds paths within distinct classes, and executes them with a contact-aware sampling controller under robot dynamics.

    Source: §I · p. 2

  • Results

    80.0% of reflecting-reference runs succeeded versus 59.1% for reflection-free runs; among matching successful pairs, median control effort fell 16.6% and traversal time 5.3%, despite 6.9% greater path length.

    Source: §VI · p. 8

  • Takeaways & Limitations

    In the tested map and robot model, planned wall interactions can improve execution efficiency for robots with limited free-space turning authority.

    Source: §VI · p. 8, B. Experiment 2: Executing Reflecting… · p. 8

    The results cover one map and start–goal pair, with perfectly elastic, frictionless contact and no linear drag.

    Source: B. Experiment 2: Executing Reflecting… · p. 8

Abstract

from arXiv · show

Robot navigation methods tend to avoid contact, and consequently search for collision-free trajectories. For robots with high inertia and limited maneuverability, however, avoiding contact can require substantial steering effort and time, even when interactions with surrounding surfaces could be safely exploited. In this paper, we develop a planning method that deliberately uses controlled wall reflections to generate trajectories that can be easier and more efficient to execute than purely collision-free motion. We consider planar navigation in environments where a mobile robot is permitted to bounce off surrounding surfaces. To represent the resulting alternatives, we construct a reflection-augmented state graph in which paths are partitioned into distinct classes according to the sequence of walls used for reflection. This representation enables systematic enumeration of reflection strategies and identification of the lowest-cost path within each class. We show that, although a reflecting path cannot be shorter than the shortest collision-free path, it can reduce execution time and actuation effort by replacing costly changes in heading with controlled environmental interactions. The planned trajectories are executed using a contact-aware sampling-based controller with the robot's full dynamics. In our experiments, we demonstrate that in our simulated test scenario, the best reflecting class can reduce time and control effort. Our results show that controlled contact can provide dynamically advantageous navigation strategies that are excluded by conventional collision-avoidance formulations.

I. INTRODUCTION

The paper studies navigation for robots that can tolerate or exploit contact, where bouncing may avoid costly free-space turns. It represents wall-reflection sequences explicitly so distinct path classes can be searched and compared.

I. INTRODUCTION

  • For robots with limited maneuverability, a wall bounce may require less aggressive steering than a tight free-space turn, motivating contact-tolerant planning.The problem statement considers a bounded, simply connected 2D workspace with obstacles and paths through free space.Source: §I · p. 1, §II · p. 2
  • Unlike approaches where contact sequences emerge as solver by-products, the reflection invariant makes distinct sequences enumerable and optimizable as separate classes.The paper distinguishes this representation from collision-state sampling and mixed-integer contact selection.Source: §I · p. 2
  • The authors hypothesize that environmental contacts can replace actuator-generated direction changes, allowing a longer path to require less actuation and traversal time.This hypothesis targets robots with limited free-space turning authority.Source: §I · p. 2
  • The framework tracks wall-contact sequences with an r-signature, searches an augmented graph for optimal paths within reflection classes, and executes selected paths with a contact-aware controller.In simulation, reflecting paths reduced control effort while achieving comparable execution time across the controller’s speed range.Source: §I · p. 2

II. PROBLEM STATEMENT

  • The workspace is modeled as a bounded planar region containing closed, connected obstacle regions, with robot paths defined as curves in free space.Source: §II · p. 2

A. Workspace Discretisation and the Planning Lattice

The planner discretizes position and heading into a state lattice, then augments it to track reflections and search for reference paths.

A. Workspace Discretisation and the Planning Lattice

B. Robot Model

The paper models a planar bicopter whose thrust acts along one body axis, so changing acceleration direction requires rotating the body. It models wall contact as specular reflection and rejects ambiguous corner hits and successors outside free space.

B. Robot Model

  • The planar bicopter has two parallel thrusters and can accelerate only along its body axis, so changing acceleration direction requires first rotating the body.The robot is represented as a disc for collision purposes.Source: B. Robot Model · p. 3
  • The robot state comprises planar position and velocity, body orientation, and body rate; its controls are the left and right thrusts.The body-axis direction is (cos θ, sin θ).Source: B. Robot Model · p. 3

C. Contact Model

  • A step crossing a wall segment is replaced by specular reflection: its normal displacement component is reversed while its tangential component is preserved.For the axis-aligned walls in the occupancy grid, this flips the step’s x-component at vertical walls or y-component at horizontal walls.Source: C. Contact Model · p. 3
  • The contact model rejects simultaneous vertical and horizontal hits and reflected successors that lie outside free space.Source: C. Contact Model · p. 3

D. Problem Formulation

The paper classifies paths by their ordered wall-reflection sequences, then searches for an optimal representative of each class. Its reflection words record physical contacts without cancellation, and the no-revisit rule bounds their length.

D. Problem Formulation

  • Intentional wall contact can change motion direction without actuator-driven reorientation, potentially reducing control effort and execution time even for a geometrically longer path.Source: D. Problem Formulation · p. 3
  • The planning problem is to enumerate path families by reflection sequence, find each family’s best representative, and test whether the advantage persists under full robot dynamics.Source: D. Problem Formulation · p. 3

III. REFLECTION CLASSES AND PATH ENUMERATION

  • Wall segments are maximal runs of adjacent boundary faces in one direction on the same obstacle component or workspace edge, with runs shorter than ℓ_min excluded.The implementation sets ℓ_min to 3 cells.Source: §III · p. 3
  • Reflection order distinguishes classes: s1s5, s6s1, and s5s1 are distinct, and a path that revisits a wall in its word is rejected.Source: §III · p. 4
  • An r-signature records the wall segments a path reflects from as an ordered word; a path with no reflections has the empty word.Each wall segment is a letter representing one specular reflection event.Source: §III · p. 4
  • Unlike the h-signature’s reduced word, an r-signature has no cancellation because a later reflection does not undo an earlier one.Source: §III · p. 4
  • The no-revisit rule rejects any path that reflects from a wall segment already in its word, bounding word length by the number of segments and ensuring finitely many valid words.Source: §III · p. 4
  • Two paths belong to the same reflection class exactly when their r-signatures are equal.Source: §III · p. 4

A. Concatenation of Reflection Words

The reflection word for a path is extended by concatenating the next path’s word, provided the two portions do not revisit a shared wall segment. In the lattice, a reflected successor appends the contacted wall to its parent’s word.

A. Concatenation of Reflection Words

B. The r-Augmented Lattice

The r-augmented lattice adds the accumulated reflection word to the planning state. Its search preserves that word across free-space steps, appends it at reflections, and records optimal representatives of distinct classes.

B. The r-Augmented Lattice

  • Each r-augmented lattice vertex combines the planning state with the reflection word accumulated so far.The start vertex carries the empty word, and the goal is matched on position alone, with heading treated as a wildcard.Source: B. The r-Augmented Lattice · pp. 4–5
  • Free-space edges leave the word unchanged, while a valid reflected edge appends its wall segment and updates the successor heading.Reflections are attempted only on cardinal and ordinal strides; ambiguous corner hits and successors outside free space are rejected.Source: B. The r-Augmented Lattice · p. 5
  • A reflected edge pays distance cost without a turn penalty, while sustained nonreflecting wall contact increases the distance cost to suppress wall-sliding paths.Source: B. The r-Augmented Lattice · p. 5
  • A* records the first expanded goal vertex for each previously unseen word as that class’s optimal representative, continuing until enough classes are found or the open list is exhausted.The augmented construction has a factor N_θ more vertices than the unlatticed one.Source: B. The r-Augmented Lattice · p. 5

IV. CONTACT-AWARE MPPI FOR EXECUTION

A contact-aware MPPI controller tracks planned paths under full vehicle and wall-contact dynamics. Its contouring formulation separates path deviation from along-path lag while rewarding progress and tightening tracking near the goal.

IV. CONTACT-AWARE MPPI FOR EXECUTION

  • MPPI generates dynamically feasible controls while accounting for underactuated vehicle dynamics and hybrid wall contacts.It tracks each planned path as a geometric reference.Source: §IV · p. 5

A. Contouring formulation

  • The controller samples candidate inputs because sampling admits the discontinuous contact dynamics that a gradient-based solve would not allow.Arc length is a state whose rate the controller chooses online, and progress is rewarded.Source: A. Contouring formulation · p. 5
  • Contour error measures deviation perpendicular to the path, whereas lag error measures displacement along its tangent between the robot and the reference marker.Lag is harmless by itself but must stay small so the marker remains near the robot's position on the path.Source: A. Contouring formulation · p. 5, Figure 6 · p. 5
  • A high contour-weight floor keeps both references close, while a narrow Gaussian bump tightens tracking over the final σg of arc length near the goal.The bump addresses the risk of drifting off the path before arrival when progress is otherwise cheap.Source: A. Contouring formulation · pp. 5–6

B. Contact-aware rollouts

The contact-aware rollouts use sampling-based MPPI to optimize finite-horizon control sequences through simulated impacts. Rollouts resolve wall penetration with reflection dynamics and penalize contacts outside the reference path's reflection class.

B. Contact-aware rollouts

  • At each step, MPPI samples K bounded control sequences, rolls them out to horizon H, and updates the nominal sequence with an exponentially weighted average.The weights depend on rollout costs and the MPPI temperature η; the first updated control is applied for Δt.Source: B. Contact-aware rollouts · p. 6
  • Each rollout integrates impact dynamics, reflecting penetrating displacement and negating the corresponding velocity component with restitution e.The impact resolution is applied per axis when a step would enter an obstacle or leave the workspace.Source: B. Contact-aware rollouts · p. 6
  • Walls are part of the dynamics, but contacts are admissible only when sanctioned by the reference reflection signature; other contacts are penalized.The impact update leaves θ and ω unchanged.Source: B. Contact-aware rollouts · p. 6

V. EXPERIMENTS

The experiments use a graph-search planner and a simulated robot execution setup. The reported enumeration table covers twenty lowest-cost reflection classes across four maps and three grid resolutions.

V. EXPERIMENTS

  • Planning uses a single-threaded DOSL fork on one Intel Core i7-11800H core with 15 GB RAM available under WSL2.Reported planning times are single runs and exclude map parsing and image output.Source: §V · p. 6
  • Execution is simulated in Python with batched PyTorch rollouts on an RTX 3070 laptop GPU, using a robot with bounded thruster forces.The simulated parameters include m = 1.0, I = 0.16, L = 0.55, r = 0.40, bang = 0.30, and blin = 0.Source: §V · p. 6
  • Table I reports wall-clock time to enumerate twenty lowest-cost reflection classes for four maps at three grid resolutions, alongside wall-segment count |S| and expanded states Nexp.|S| includes the four boundary walls.Source: Table I · p. 6

A. Experiment 1: Enumerating Reflection Classes

The experiments enumerate reflection classes and compare the best reflecting reference with a reflection-free reference under simulated execution. In the tested map and robot model, reflecting paths generally use less control effort and time despite being longer.

A. Experiment 1: Enumerating Reflection Classes

  • The planner finds the twenty lowest-cost classes in four maps at 20×20, 50×50, and 100×100 resolutions, using fixed heading, path-length, word-length, and cost parameters.The three resolutions isolate lattice cost from alphabet cost because the obstacle geometry and S remain unchanged.Source: A. Experiment 1: Enumerating… · p. 6
  • Enumeration time rises steeply with grid resolution and wall-segment count: Map 4 has 2.5× Map 1's segments and takes 8× longer at 20×20, widening to 50× at 100×100.The passage attributes the widening gap to the word factor multiplying lattice size, so cost depends jointly on resolution and segment count.Source: A. Experiment 1: Enumerating… · p. 6

B. Experiment 2: Executing Reflecting and Reflection-Free Paths

  • Map 4 is chosen as the demanding execution test because its largest alphabet and narrow corridors force repeated reversals; the comparison pairs its top-ranked class with a same-start, same-goal reflection-free path.At Δmax = 1, the baseline variant discards blocked edges and allows ±45° turns at no cost because no reflection-free path exists otherwise; that path is 49.02 versus 52.38 cells.Source: B. Experiment 2: Executing Reflecting… · pp. 6–7
  • The reflecting reference is the lowest-cost representative of the top-ranked class, while the reflection-free baseline permits ±45° turns at no cost.Figure 7 shows these paths in gold and blue, respectively, for the same start–goal pair in each of four maps.Source: Figure 7 · p. 7
  • The execution comparison varies progress reward µ over eleven settings with ten shared seed identifiers per setting, totaling 220 runs.It measures completion time, integrated squared thrust, and mean path error; direct pairwise reductions use only matching settings and seeds where both references succeed.Source: B. Experiment 2: Executing Reflecting… · p. 7
  • As µ increases, completion time generally falls while control effort rises; reflecting references retain lower median control effort and path error at every tested setting.The controller penalizes illegal contacts and permits contacts according to each reference class; reflecting runs succeed only when every planned vertex is contacted legally and no other contact occurs.Source: B. Experiment 2: Executing Reflecting… · p. 7
  • Across 52 matching pairs, reflecting references reduce control effort 16.6%, traversal time 5.3%, and mean path error 27.4%, despite being 6.9% longer.These results cover one map and start–goal pair with perfectly elastic, frictionless contact (e = 1) and no linear drag (blin = 0).Source: B. Experiment 2: Executing Reflecting… · pp. 7–8, Figure 9 · p. 8
  • At µ = 500, all ten reflecting runs succeed in 11.40 ± 0.33 s, while seven reflection-free runs succeed with a 13.60 s mean.Across the 52 pairs, effort is lower in 51 and traversal time shorter in 43.Source: B. Experiment 2: Executing Reflecting… · p. 7

VI. CONCLUSION

The authors introduce the r-signature and reflection-augmented state lattice, then report that reflecting references improved execution outcomes in the tested map despite being longer. Future work will evaluate additional environments, physical platforms, and contact effects.

VI. CONCLUSION

  • 80.0% vs 59.1% of runs succeeded; among 52 matching successful pairs, reflecting references cut median control effort 16.6% and traversal time 5.3%, despite being 6.9% longer.The comparison used the same contact-aware MPPI controller for both references.Source: §VI · p. 8
  • The r-signature represents ordered wall-reflection sequences, supporting a reflection-augmented state lattice that finds lowest-cost paths in distinct reflection classes.Source: §VI · p. 8
  • Future work will evaluate additional environments and physical collision-tolerant platforms, including friction, inelastic impacts, and contact uncertainty.Source: §VI · p. 8
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