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PccDiffuser: Multi-solution Motion Planning for Continuum Robots

Ke Qiu, Sifan Chen, Si Wang, Rong Xiong, Yue Wang, Haojian Lu

arXiv:2609.09745v1cs.RO

TL;DR

Continuum-robot motion planning must handle multiple feasible whole-body paths for the same target while preserving terminal accuracy and executable timing. PccDiffuser learns a conditional diffusion distribution over configuration-space paths, samples candidates in parallel, and converts selected paths into trajectories under actuator constraints. It achieves high success and efficiency against conventional planners, with parallel sampling making the computational advantage larger for multiple candidates.

  • Problem

    Continuum-robot redundancy creates multiple terminal configurations and whole-body paths, while existing planners generally return one solution and may suffer workspace-quality or inverse-kinematics limitations.

  • Method

    PccDiffuser uses conditional diffusion with PCC exponential-coordinate kinematics, graph-based obstacle encoding, analytical differential-kinematics guidance, parallel candidate sampling, and actuator-constrained time allocation.

  • Results

    PccDiffuser achieved a 91.11% success rate and a 34.0 ms mean runtime per successful path, outperforming the reported conventional benchmarks while preserving the benefit of parallel multi-solution sampling.

  • Takeaways & Limitations

    The framework produces diverse feasible paths in free and cluttered spaces and was validated in simulation and on a three-section tendon-driven continuum robot.

Abstract

from arXiv · show

We present the PccDiffuser, a conditional diffusion framework for continuum robots that learns a multimodal distribution over complete configuration-space paths and samples multiple candidate solutions in parallel, which are subsequently converted into an executable trajectory by time allocation considering actuator constraints. Under the piecewise constant-curvature model, we use exponential co-ordinates to describe the robot kinematics, and use graph neural network to encode a variable number of environment obstacles. Analytical differential kinematics is incorporated in the denoising process to improve terminal accuracy and whole-body clearance. On a mixed test set comprising workspace with zero to four obstacles, PccDiffuser achieved a success rate of 91\%. Compared with existing sampling- and optimisation-based benchmarks, it delivered both a higher success rate and greater computational efficiency, with the latter advantage becoming more substantial when sampling more candidate solutions. Experiments on a three-section tendon-driven continuum robot further demonstrate consecutive planning, multi-solution planning, and whole-body obstacle avoidance.

I. INTRODUCTION

Continuum-robot planning is inherently multimodal because redundant robots can reach the same target through different whole-body paths. PccDiffuser addresses this by generating diverse configuration-space candidates in parallel with conditional diffusion.

  • Continuum robots support applications requiring adaptable motion in cluttered environments.Their compliance and adaptability motivate use in minimally invasive surgery, industrial inspection, maintenance, and human-robot interaction.
  • Kinematic redundancy creates multiple terminal configurations and substantially different whole-body paths for the same target.This multiplicity is useful for obstacle avoidance and runtime objectives but makes planning underconstrained and multimodal.
  • Existing sampling- and optimisation-based planners typically return one solution per run and face workspace or configuration-space limitations.Configuration-space methods may produce undesirable workspace paths, while workspace methods can suffer inverse-kinematics discontinuities or local minima.
  • Diffusion models can learn conditional multimodal distributions and generate several candidates in parallel from different noise samples.This capability motivates applying diffusion-based planning to continuum robots.
  • PccDiffuser generates obstacle-conditioned configuration-space paths, applies analytical kinematic guidance, and time-allocates selected paths under actuator constraints.A graph neural network handles variable obstacle counts, while batched denoising makes parallel candidate exploration practical.

A. Modelling and Planning

The paper models continuum robots with PCC kinematics and uses exponential coordinates to avoid straight-configuration singularities. It combines diffusion-based planning context with analytical differential kinematics for tip-related guidance.

  • Modelling and Planning: The PCC model represents continuum robots as concatenated circular arcs with explicit forward kinematics at lower computational cost than Cosserat-rod models.Even under PCC, one tip position can correspond to multiple terminal configurations and backbone shapes.
  • Modelling and Planning: Workspace and configuration-space RRT methods have complementary weaknesses and normally require repeated searches for multiple feasible motion modes.Workspace methods depend on inverse kinematics, while configuration-space methods receive limited direct task guidance and may yield undesirable workspace paths.
  • Modelling and Planning: Diffusion denoising starts from noisy samples and can explore distinct modes through different initial noise samples.DDIM can shorten sampling and optionally make solutions reproducible through deterministic sampling.
  • Modelling and Planning: The planner separates geometric path generation from subsequent time allocation for trajectory execution.This formulation follows the paper's stated planning decomposition.
  • Non-Singular Kinematics Representation: Exponential coordinates replace curvature-and-bending-plane parameters because the latter are discontinuous and singular at the straight configuration.The robot configuration concatenates variable twist entries across the three sections.
  • Non-Singular Kinematics Representation: The tip transformation is obtained by a product of exponentials, and the translational Jacobian component provides analytical differential-kinematics guidance during sampling.The Jacobian maps configuration changes to tip-position changes used in denoising guidance.

B. Obstacle-Aware Trajectory Planning

Obstacle-aware planning represents obstacles and the robot body geometrically, then separates collision-free path generation from minimum-duration scheduling under actuator velocity constraints.

  • Obstacle Representation: The environment is modeled as a collection of spherical obstacles, while the robot occupies a volume in R3.Each obstacle is specified by a center and positive radius.
  • Planning Formulation: The planning problem seeks a configuration path from an initial configuration to a target tip position.The path is defined over a physical-time interval, with diffusion-step time kept separate from execution time.
  • Path and Time Separation: Geometric path requirements are invariant to time parameterisation, so the planner can generate multiple candidate paths before scheduling execution.Paths are discretised into configurations associated with a later time schedule.
  • Collision Assumption: The method assumes sufficiently close adjacent collision-free configurations imply collision-free interpolated configurations.This assumption supports discrete path checking between sampled configurations.
  • Time Allocation: Each selected path receives a minimum-duration schedule subject to actuation velocity constraints, producing an executable trajectory.Path planning and time allocation together form the complete trajectory planner.

IV. PLANNING WITH CONDITIONAL DIFFUSION

PccDiffuser learns a conditional distribution over complete configuration-space paths, samples multiple solutions in parallel, and allocates time to convert a selected path into a trajectory.

  • PccDiffuser learns a conditional distribution over complete configuration-space paths and draws multiple solutions in parallel.Time allocation converts selected paths into trajectories.

A. Conditional Diffusion Model

The conditional diffusion model learns to denoise configuration-space paths conditioned on the initial configuration, target tip position, and obstacles. Deterministic DDIM sampling generates paths from Gaussian noise over a shortened reverse schedule.

  • Conditional formulation: A feasible configuration-space path is denoised conditionally using the initial configuration, target tip position, and obstacles.The model learns pθ(x_t−1|x_t,c), where c contains these planning conditions.
  • Forward process: The forward process progressively corrupts paths with Gaussian noise according to a diffusion schedule.The cumulative schedule permits direct sampling at arbitrary diffusion steps.
  • Reverse process: The reverse denoiser predicts the noise in the current path rather than reverse mean and covariance independently.Sampling uses deterministic DDIM with η = 0.
  • Reverse process: 50 reverse steps selected from T = 1000 training steps produce a path conditioned on the planning inputs, starting from xT ∼ N(0, I).The shortened schedule is applied from an initial Gaussian sample.
  • Conditioning: The diffusion step, initial configuration, target position, and obstacle graph are embedded into a shared conditioning vector for temporal U-Net denoising.The conditioning vector combines task and environment information with the noisy configuration-space path.

B. Network Architecture

The network combines temporal path processing with embeddings of task conditions and a graph representation of obstacles. This produces a shared context that conditions the denoiser throughout its residual blocks.

  • Denoiser: A one-dimensional temporal U-Net receives the noisy path and a binary channel marking the inpainted initial configuration.The architecture is designed to preserve temporal path structure during denoising.
  • Obstacle encoding: A GNN encodes a variable number of obstacles, and graph pooling converts them into a fixed-dimensional embedding.Obstacle node features include curvature-related quantities and relative centre information in message passing.
  • Conditioning: The obstacle embedding is fused with diffusion-step, initial-configuration, and target-position embeddings into a shared context vector.The resulting context represents the complete planning task.
  • Conditioning: Feature-wise affine transformations inject the shared context into residual blocks at every U-Net resolution.This conditions the denoiser across resolutions while maintaining temporal path structure.

C. Training

Training uses generated PCC configuration-space paths and masked conditional noise prediction, while analytical differential kinematics guides denoising toward terminal accuracy and whole-body clearance.

  • Training data: Training paths use a three-section PCC robot with H = 64 waypoints, bending limit κmax = π/2, and multiple terminal inverse-kinematics solutions.Targets and initial configurations are paired from randomly sampled configurations, with obstacle scenes filtered for collision-free paths.
  • Training objective: The masked loss excludes the initial configuration because it is restored by inpainting rather than noise prediction.The mask is zero at the first configuration and one elsewhere.
  • Optimization: AdamW optimization runs for 100k updates, with an exponential moving average of parameters used for inference.All variables are normalized before training.
  • Differential guidance: Analytical differential kinematics refines terminal task satisfaction during the reverse process using the terminal objective and its gradient.The resulting guidance is subsequently incorporated into the DDIM update.
  • Differential guidance: Backbone points are sampled to identify the deepest penetration, whose displacement from a safe boundary defines configuration-space repulsion.The repulsion uses the Jacobian at selected backbone points and leaves the initial configuration update fixed at zero.
  • Post correction: Post correction moves terminal and penetrating configurations along their respective gradients before the next DDIM update.The corrected final path estimate is substituted into the update.
  • Guided prediction: Guided prediction instead combines terminal and clearance objectives and modifies the predicted noise through an analytically propagated gradient.Automatic differentiation propagates the geometric gradient through the denoiser-dependent estimate.
  • Sampling evolution: The reverse process converts initially unstructured samples into feasible paths over a 50-step DDIM evolution.Snapshots compare starts from a random initial configuration and a straight configuration.

E. Trajectory Generation

Sampled configuration-space paths are mapped into actuator space and time-parameterized under actuator limits to produce executable trajectories while preserving planned geometry.

  • Trajectory conversion: A linear mapping converts each sampled configuration-space path into an actuator-space path.The mapped path is then scheduled subject to actuator velocity and acceleration limits.
  • Time allocation: Minimal execution time is assigned under actuator velocity and acceleration constraints without changing the planned geometry.The resulting trajectory can be spline-interpolated for hardware execution.

V. EXPERIMENTS

The experiments test whether PccDiffuser generates diverse feasible paths across obstacle settings. Results show multiple terminal configurations and collision-free whole-body motions for the same task, including increasingly cluttered scenes.

  • PccDiffuser was evaluated in simulation and real-world experiments for multi-solution generation, guidance effects, planner comparison, and execution.
  • Multi-Solution Planning: The planner generates alternative terminal configurations, distinct collision-free motions, and corresponding 6-dimensional PCC configuration paths for identical cases.
  • Multi-Solution Planning: The experiments include scenes with zero to four ball obstacles and feasible solutions in increasingly cluttered environments.
  • Multi-Solution Planning: Across obstacle-free and obstructed examples, multiple samples reached their targets while preserving distinct configurations and whole-body motions.

B. Ablation Study

The ablation and benchmark studies assess analytical refinement and compare PccDiffuser with sampling- and optimisation-based planners. Post correction offers the best refinement trade-off, while PccDiffuser achieves higher success and lower runtime than the reported benchmarks.

  • Ablation Study: The ablation compares pure diffusion, post correction, and guided prediction using identical initial noise, 50 sampling steps, and 10 candidate paths.Analytical refinement was activated during the final 20 denoising steps.
  • Ablation Study: Post correction achieved the highest success rates of 99.62% in obstacle-free spaces and 91.11% in mixed obstacle spaces.It also reduced mean normalised terminal tip error by 55.6% and 45.4%, respectively.
  • Ablation Study: Post correction increased mean runtime by 15.1% in obstacle-free spaces and 56.8% in mixed obstacle spaces, versus 60.4% and 102.2% for guided prediction.
  • Planner Comparison: The benchmark comparison includes configuration-space RRT/RRT*, workspace RRT/RRT*, and APF on a dataset containing zero to four ball obstacles.Success required meeting terminal tip-error tolerance while keeping every sampled backbone point collision-free.
  • Planner Comparison: 91.11% success exceeded workspace RRT/RRT* and APF at around 70%, while configuration-space RRT/RRT* achieved approximately 65%.
  • Planner Comparison: PccDiffuser had the lowest mean runtime at 34.0 ms per successful path, and parallel denoising kept total sampling time nearly unchanged as candidate count increased.This runtime advantage became more pronounced for multi-solution planning.

D. Real-World Evaluation

Hardware experiments evaluated consecutive planning, multiple solutions for identical targets, and whole-body obstacle avoidance on a three-section tendon-driven continuum robot. The executed trajectories reached targets, remained continuous across replanning, and maintained clearance from obstacles.

  • Evaluation setup: Hardware evaluation tested consecutive planning, multiple solutions for identical targets, and whole-body obstacle avoidance.Multiple configuration-space trajectories were generated, with one selected for execution according to test-time requirements.
  • Consecutive Planning: The robot reached two sequential targets while maintaining its shape during waiting and continuity across replanning boundaries.The second target was conditioned on the terminal configuration from the first motion.
  • Multi-Solution Planning: Two samples reached the same target through visibly different whole-body shapes and well-separated configuration-space trajectories.The results provided alternative motions for identical boundary conditions rather than online perturbations of one nominal path.
  • Obstacle Avoidance Planning: The executed trajectory reached the target without collisions while retaining whole-body clearance from a ball obstacle.Collision checking considered the complete discretised backbone, and denoising evolved dispersed path points into an obstacle-aware solution.
  • Execution: Selected paths were converted into executable trajectories through time allocation under actuator constraints, and hardware experiments validated the planner.The conclusion reports hardware validation after simulation demonstrated diverse solutions in free and cluttered spaces.
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