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Model Predictive Control of Tensegrity Robots via Contact-Aware Graph Neural Dynamics Model

Nelson Chen, Patrick Meng, Charles Tang, Angelina Degay, Zachary Brei, Rebecca Kramer-Bottiglio, Kostas E. Bekris, Mridul Aanjaneya

arXiv:2609.08958v1cs.RO

TL;DR

Tensegrity robots are difficult to model and control amid complex contacts and partial observations. This work combines contact-aware GNN dynamics with MPPI and turning primitives; across five MuJoCo tasks, hybrid MPPI consistently outperformed standalone MPPI and A* re-planning.

  • Problem

    Tensegrity dynamics are difficult to model under complex contacts, and prior GNN models assume horizontal-ground contacts rather than explicitly representing non-horizontal surfaces or self-collisions.

    Source: §I · p. 1

  • Method

    The approach adds differentiable contact detection to GNN dynamics, uses the model for MPPI rollouts, and iteratively adds navigation trajectories to training data.

    Source: §I · pp. 1–2, A. Model Predictive Path Integral… · p. 3

  • Results

    Across five MuJoCo tasks, hybrid MPPI consistently outperformed standalone MPPI and A* re-planning; on the incline, it achieved 100% success from iteration one.

    Source: B. Navigation Tasks · p. 7, §VI · p. 8

  • Takeaways & Limitations

    The reported evaluations support contact-aware learned dynamics with hybrid MPPI for tensegrity navigation across simulated terrain and obstacle tasks.

    Source: §I · p. 2, §VI · p. 8

    The approach is restricted to planar surface representations, and its turning primitives are hand-engineered.

    Source: §VI · p. 8

Abstract

from arXiv · show

Tensegrity robots offer lightweight, compliant mobility over challenging terrain but remain difficult to model and control due to complex contact-rich dynamics and partial observability. This work presents a model predictive path integral (MPPI) controller for a three-bar tensegrity robot driven by a learned graph neural network (GNN) dynamics model. This work first extends prior GNN-based models with a differentiable contact detection module. The extension allows the dynamics model to reason over non-horizontal planar terrains, obstacles, as well as self-collisions. Then, the learned dynamics model and the MPPI controller operate in a closed data-collection loop, iteratively improving model accuracy and control performance. This work further introduces a hybrid MPPI strategy that combines MPPI with turning motion primitives to improve maneuverability. Experiments are performed in MuJoCo across five navigation tasks, which include, wall obstacles, inclines, narrow corridors, low-clearance structures, and a composite 3D obstacle course. The experiments demonstrate that the hybrid MPPI controller operating over the learned GNN dynamics model improves predictive accuracy over a flat-ground baseline model and achieves superior navigation performance compared to $A^*$-based re-planning and MPPI-only variants. Results show that the contact-aware learned dynamics combined with the sampling-based model predictive control enable robust tensegrity navigation in complex, contact-rich environments.

I. INTRODUCTION

Tensegrity robots are difficult to model and control because compliant structures interact with complex contacts, while prior approaches have limited runtime dynamics reasoning. This work combines contact-aware GNN prediction, MPPI, iterative data collection, and turning primitives for navigation.

I. INTRODUCTION

  • The compliant rod–cable structure and complex environmental contacts make accurate tensegrity dynamics modeling difficult.Prior work largely avoided system dynamics or relied on approaches with limited locomotion capabilities.Source: §I · p. 1
  • A differentiable contact detector extends GNN dynamics to non-horizontal planar terrain, environmental obstacles, and tensegrity self-collisions.The resulting contact representations become graph node and edge features for contact-dependent prediction.Source: §I · pp. 1–2
  • MPPI uses the learned GNN as its predictive model, and trajectories collected during navigation are added to training data for the next model iteration.The controller is run in a closed loop until task completion or the time limit.Source: §I · p. 2, Figure 2 · p. 2
  • The work adds turning motion primitives to MPPI and evaluates the model and controllers across five MuJoCo navigation scenarios.The scenarios include an unseen composite 3D course; comparisons include A* re-planning, MPPI-only, and hybrid MPPI.Source: §I · p. 2

II. RELATED WORKS

Prior tensegrity navigation methods often avoid runtime dynamics reasoning, while learned GNN models under partial observations assume horizontal-ground contacts. This work extends contact-aware GNN dynamics and integrates them with MPPI for navigation.

II. RELATED WORKS

  • Face-transition planning tiles tensegrity polygonal faces,, while other approaches use precomputed motion primitives [7], or simulation-trained reinforcement learning policies, [5].These methods generally demonstrate limited locomotion capabilities, such as reproducing human-engineered primitive-like behaviors.Source: §II · p. 2
  • MPC repeatedly optimizes control sequences using an internal dynamics model, but tensegrity applications remain challenging because of nonlinear dynamics and complex contacts [11].MPPI evaluates candidate sequences through parallel forward rollouts and is described as robust to local minima.Source: §II · pp. 2–3
  • Differentiable simulators,, can improve system identification, but generally assume full state observability, limiting runtime-control applicability.Earlier analytical simulators,, are typically non-differentiable and sensitive to modeling errors.Source: §II · p. 3
  • Prior GNN tensegrity models, [8] support partial observations but assume horizontal-ground contacts and omit explicit non-horizontal-surface interactions and self-collisions.This work adds differentiable contact detection for planar terrain, workspace obstacles, and internal self-collisions, then integrates the model with MPPI.Source: §II · p. 3

III. ROBOT: THREE BAR TENSEGRITY

The paper uses a three-bar prismatic tensegrity robot whose actuated cables change its shape and center of mass to produce locomotion. Its state representation combines rod-level quantities with a planar pose for navigation.

III. ROBOT: THREE BAR TENSEGRITY

  • The evaluated platform is a three-bar prismatic tensegrity based on the open-source design in [7], with three identical rods, each 33 cm long.Two motors per rod control cable lengths, allowing extension and contraction that shift the center of mass.Source: §III · p. 3
  • Motor encoders estimate cable lengths, while capacitive strain sensors and passive sensing cables provide partial observations of robot configuration.The six short cables are actively actuated; three longer central tendons are passive elastic elements.Source: §III · p. 3
  • The robot state is formed from the three individual rod states, which include position, orientation, and angular velocity.Each rod also has two endpoints and a principal axis.Source: §III · p. 3
  • The robot center of mass is the mean of rod positions, and its principal axis is the normalized mean of rod principal axes.For navigation, the planar principal axis is projected onto the xy-plane and rotated 90° to define heading and an SE(2) pose.Source: §III · p. 3

IV. APPROACH

The approach uses MPPI to evaluate sampled control sequences through the learned dynamics model and combines it with turning primitives. Sampled trajectories are cost-weighted to estimate the optimal control sequence.

A. Model Predictive Path Integral Controller

  • At each control step, MPPI samples candidate control sequences of horizon H and rolls them through a dynamics model from initial state X0 to obtain trajectories and costs.The MPPI controller uses the extended GNN dynamics model.Source: A. Model Predictive Path Integral… · p. 3
  • The hybrid algorithm selects either MPPI or clockwise or counterclockwise turning primitives based on robot state, local costs, and an orientation tolerance.It estimates the local cost-gradient direction and compares its angular difference with the robot heading before choosing the control type.Source: A. Model Predictive Path Integral… · p. 4
  • When the estimated heading difference is within tolerance, the algorithm selects MPPI; otherwise, it selects a turning primitive.The algorithm returns a control type and actions.Source: A. Model Predictive Path Integral… · p. 4
  • The optimal control sequence is estimated as a weighted average of sampled sequences using importance-sampling weights.The weights depend on trajectory costs; the minimum sampled cost is denoted ρ, and β is the inverse-temperature parameter.Source: A. Model Predictive Path Integral… · p. 4

B. Hybrid MPPI and turning motion primitives

The hybrid controller pairs MPPI with turning primitives because MPPI struggles to sample effective turns for the three-bar tensegrity robot. It switches between the two according to the robot’s heading relative to the cost gradient.

B. Hybrid MPPI and turning motion primitives

  • MPPI readily finds forward and backward motion but struggles to sample effective turns, making long horizons and many samples prohibitively expensive.The proposed hybrid controller adds human-engineered turning motion primitives to address this limitation.Source: B. Hybrid MPPI and turning motion… · p. 4
  • When the robot’s heading or its opposite aligns with the cost gradient, MPPI resumes goal-directed locomotion.The controller uses MPPI to exploit forward or backward motion.Source: B. Hybrid MPPI and turning motion… · p. 4
  • When the heading is misaligned with the cost gradient, turning primitives reorient the robot toward the desired direction.MPPI takes over again once alignment is established.Source: B. Hybrid MPPI and turning motion… · p. 4

C. Obstacle-Aware, Wave-Front Cost Function

The obstacle-aware wave-front cost function avoids the misleading estimates that Euclidean distance can produce in cluttered workspaces. It estimates cost-to-go over a graph of collision-free grid cells.

C. Obstacle-Aware, Wave-Front Cost Function

  • Euclidean distance can favor straight-line paths through obstacles, yielding infeasible paths and overly optimistic cost-to-go estimates.The issue arises because the metric ignores obstacle constraints.Source: C. Obstacle-Aware, Wave-Front Cost… · p. 4
  • The wave-front cost function searches a graph whose nodes are collision-free grid cells and whose edges connect adjacent cells.The workspace is discretized into a uniform grid.Source: C. Obstacle-Aware, Wave-Front Cost… · p. 4
  • A shared closed list reuses previously expanded nodes with known optimal cost-to-go, allowing later searches to terminate early.The reuse improves efficiency while propagating obstacle-aware distance estimates outward from the goal.Source: C. Obstacle-Aware, Wave-Front Cost… · p. 4

D. Extended GNN Model for Planar Terrains and Obstacles

The extended GNN adds differentiable contact detection for non-horizontal planar terrain, environmental obstacles, and rod self-collisions. Contact quantities become graph features used to predict dynamics for MPPI rollouts.

D. Extended GNN Model for Planar Terrains and Obstacles

  • For environment and self-collisions, minimum signed distances, surface normals, and relative velocities are computed at the corresponding surface points.Distances are added as node features, while distances, normals, and relative velocities are encoded as contact-edge features.Source: Figure 4 · p. 5, D. Extended GNN Model for Planar… · p. 5
  • The graph neural processing stage predicts future rod-velocity changes, which are applied autoregressively to simulate MPPI rollouts.The graph represents robot state and connected body, cable, and contact edges.Source: D. Extended GNN Model for Planar… · p. 5
  • The differentiable contact module extends the flat-ground GNN to model planar environment contacts and self-collisions between cylindrical rods.It is parallelized and integrated into graph and feature generation.Source: D. Extended GNN Model for Planar… · pp. 4–5
  • Contact edges connect end-cap nodes to nearby planar surfaces and rod pairs when their distances fall below a user-defined threshold.The k smallest distances are added as node features, restricting interactions to local contacts.Source: D. Extended GNN Model for Planar… · p. 5

E. GNN-MPPI Data Collection Loop

The GNN and MPPI controller operate in an iterative training and navigation loop, collecting new task data to update the model. The loop can continue until model and controller performance no longer improves.

E. GNN-MPPI Data Collection Loop

  • Each training phase fits the GNN dynamics model using the current dataset, initially bootstrapped with data from executing only primitives.The trained model is then deployed in the navigation phase.Source: E. GNN-MPPI Data Collection Loop · p. 5
  • During navigation, MPPI executes desired tasks and collects new, diverse data for the next GNN training iteration.The loop combines model training and controller deployment.Source: E. GNN-MPPI Data Collection Loop · p. 5
  • The training and navigation loop can run until the model and MPPI controller no longer improve.Source: E. GNN-MPPI Data Collection Loop · p. 5

V. SIMULATION EXPERIMENTS & RESULTS

The contact-aware GNN outperforms the flat-ground baseline, and prediction errors decrease over successive data-collection iterations. Ablations show that both contact edges and distance-to-contact node features improve accuracy, with the distance features having greater impact.

A. GNN Dynamics Evaluation

  • Model performance is evaluated using rod-length-normalized positional and rotational errors over the MPPI prediction horizon.Errors are averaged across test trajectories after rolling out sampled starting states over the horizon.Source: A. GNN Dynamics Evaluation · p. 6
  • The initial model uses human-designed motion-primitive data; three subsequent iterations add trajectories collected by the current MPPI controller.Source: A. GNN Dynamics Evaluation · p. 6
  • The extended GNN consistently outperforms the flat-ground baseline, while prediction errors decrease across four training and data-collection iterations.Controller-generated trajectories are added to the training data for subsequent iterations.Source: Figure 5 · p. 5, A. GNN Dynamics Evaluation · p. 6
  • Both contact edges and distance-to-contact node features improve prediction accuracy, with distance features having the larger impact.All variants use the iteration 0 dataset to isolate the contact representations’ contributions.Source: A. GNN Dynamics Evaluation · p. 6, Figure 7 · p. 6

B. Navigation Tasks

The controller is evaluated on five navigation tasks, with each task executed 30 times and success rate and completion time recorded. Hybrid MPPI performs differently across courses, while collision-avoidant A* baselines are not evaluated where traversal requires intentional obstacle contact.

Flat obstacle course

  • On the flat obstacle course, Hybrid MPPI reaches 100% success from iteration 1 onward; both A* baselines ultimately reach 100%, with reverse-motion A* fastest.MPPI-only performs inconsistently, while Hybrid MPPI improves rapidly after adding controller-generated data.Source: B. Navigation Tasks · p. 7, Table I · p. 7

Incline course

  • On the 5° incline, Hybrid MPPI achieves 100% success from iteration 1, while both A* baselines fail and MPPI-only never reaches high success rates.A* runs drift in heading during the climb, after which the changed dynamics prevent recovery.Source: B. Navigation Tasks · p. 7, Table I · p. 7

Narrow corner and passageway

  • In the narrow corner and corridor, Hybrid MPPI achieves substantially higher, though imperfect, success rates; collision-avoiding A* cannot traverse the course.The passage is only slightly wider than the robot and requires careful alignment; traversal requires intentional obstacle interaction.Source: B. Navigation Tasks · p. 7, Table I · p. 7

Low-clearance structure

  • At low clearance, both MPPI variants perform similarly by iteration 3, but Hybrid MPPI reaches that performance level one iteration earlier.The opening is below the robot’s resting height, so successful traversal requires temporary compression; nominal-height A* primitives cannot reach the goal.Source: B. Navigation Tasks · pp. 7–8, Table I · p. 7

3D obstacle course

  • On the unseen 3D course, MPPI-only remains at 0% success until the final model iteration, when it reaches 35%; Hybrid MPPI improves rapidly once the model is sufficiently accurate.The course combines inclines, walls, narrow passages, and low-clearance structures.Source: Table I · p. 7, B. Navigation Tasks · p. 8

VI. CONCLUSION

The conclusion presents a contact-aware GNN-MPPI framework for three-bar tensegrity navigation and reports consistent hybrid-MPPI gains across five MuJoCo tasks. It identifies planar surface modeling, hand-engineered turning primitives, and the lack of a real-robot demonstration as boundaries.

VI. CONCLUSION

  • Across five MuJoCo navigation tasks, hybrid MPPI consistently outperformed standalone MPPI and A* re-planning, with performance improving across training iterations.Source: §VI · p. 8
  • The extended GNN uses differentiable contact detection to predict interactions with ramps, walls, low-clearance structures, and rod self-collisions under partial observability.Source: §VI · p. 8
  • Hybrid MPPI supplements learned control with human-engineered turning primitives and an obstacle-aware grid-based wavefront cost function.Source: §VI · p. 8
  • A coupled data-collection and model-training loop iteratively improves the system.Source: §VI · p. 8
  • The current approach is restricted to planar surface representations; fully unstructured terrain and more complex multi-room or outdoor environments remain future directions.Source: §VI · p. 8
  • Turning primitives are hand-engineered, and learning them from data is proposed to improve generality and reduce manual design effort.Source: §VI · p. 8
  • A real three-bar tensegrity demonstration of the iterative modeling and control improvement remains planned future work.Source: §VI · p. 8
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