PaperlayerView exact revision on arXiv

Source-linked AI summary

History-Conditioned Flow Matching for Probabilistic Dynamics of Tendon-Driven Continuum Robots

Hang Yang, Tingcong Liu, Junjie Xiong, Fangju Yang, Ke Wu

arXiv:2609.25658v1cs.RO

TL;DR

Tendon-driven continuum robots have uncertain, partly unobserved mechanics, and their observed motion can depend on prior loading. This paper uses history-conditioned flow matching with a nominal dynamics reference to predict probabilistic configurations. It reports lower Energy Scores and coverage closer to nominal levels than compared baselines in simulation, and lower errors with spread ratios closer to one on two held-out physical inputs.

  • Problem

    Unobserved mechanical states and uncertain parameters limit what nominal deterministic predictions can quantify, while prior loading can make current configuration and command insufficient to characterize subsequent motion.

    Source: §I · p. 1

  • Method

    The framework uses conditional flow matching to predict the next complete joint configuration from motion and actuation histories, with a nominal dynamics reference and kinematics-based supervision of recursive motion.

    Source: §I · p. 1

  • Results

    Across both simulated scenarios, Flow achieves lower Energy Scores and coverage closer to nominal than compared baselines; on two held-out physical inputs, its Energy Scores are 11.91 and 11.42 mm and spread ratios are 1.648 and 1.217.

    Source: §VI · pp. 7–8, C. Results · p. 7

  • Takeaways & Limitations

    The results support history-conditioned probabilistic dynamics prediction that combines motion history with mechanical and geometric structure under incomplete observations.

    Source: §VI · p. 8

Abstract

from arXiv · show

Deterministic dynamics modeling of tendon-driven continuum robots remains challenging owing to uncertainties in material behavior, tendon transmission, friction, and contact. Measured joint configurations and nominal tendon commands do not fully characterize these internal mechanical factors, leaving uncertainty in the subsequent motion. We therefore develop a history-conditioned, physics-informed flow-matching framework for probabilistic dynamics prediction, using motion and actuation histories to predict the distribution of the next complete joint configuration. By recursively sampling next-step configurations under prescribed commands, the model predicts distributions of future whole-body motions. In simulation, scenario-specific models achieve five-second trajectory Energy Scores (lower is better) of 12.05 mm under internal friction variation and 9.29 mm under unobserved actuation disturbances. Relative to the conditional variational autoencoder and diffusion baselines, Flow attains lower Energy Scores and coverage closer to the nominal level in both scenarios. Ablations support history and structural conditioning in both scenarios. On the physical robot, predictions under two tendon-command profiles excluded from training capture the principal motion sequences, with five-second Energy Scores of 11.91 and 11.42 mm, lower than the compared baselines. The predicted-to-measured spread ratios are 1.65 and 1.22 (closer to 1 is better). These results support history-conditioned probabilistic dynamics prediction under incomplete mechanical observations.

I. INTRODUCTION

Tendon-driven continuum robots have uncertain, partly unobserved mechanics, and their observed motion can depend on prior loading. The paper proposes history-conditioned probabilistic prediction and evaluates it for SpiRob, a two-tendon manipulator represented by 23 joints.

I. INTRODUCTION

  • Nominal deterministic predictions do not quantify uncertainty from incomplete mechanical information, while unobserved mechanical effects can preserve prior-loading effects that current configuration and command miss.The relevant effects include friction, contact, elasticity, and tendon-transmission nonlinearities.Source: §I · p. 1
  • The framework predicts a distribution over the next joint configuration from motion and actuation histories, with a nominal dynamics reference and kinematics-based supervision.Its recursive predictions represent future whole-body motion.Source: §I · p. 1
  • Simulations and experiments show lower Energy Scores and higher mean-motion prediction accuracy than the compared probabilistic baselines; ablations support history and structural conditioning.The reported evidence covers both simulated scenarios and physical experiments.Source: §I · p. 1

A. Studied Manipulator and Available Measurements

  • Figure 1 depicts a two-tendon manipulator with 23 moving-link centers and alternative next configurations under the same history and command.The alternatives share a common base and link geometry.Source: Figure 1 · p. 2
  • SpiRob is represented by 23 revolute joints, and tracked markers provide partial observations of its body geometry.The current command is specified separately from the configuration and command history.Source: A. Studied Manipulator and Available… · p. 2

B. Probabilistic Configuration Dynamics

The section formulates next-configuration uncertainty by treating unobserved mechanical state and uncertain parameters as hidden quantities in the transition. It learns the resulting distribution from recorded motions and commands.

B. Probabilistic Configuration Dynamics

  • The hidden state ξ_t collects remaining mechanical state, uncertain parameters, and actuation disturbances over the next sampling interval.These quantities account for elastic and damping effects, tendon transmission, friction, contact, and deviations between nominal and actual actuation.Source: B. Probabilistic Configuration Dynamics · p. 2
  • The transition assumes that ξ_t contains all remaining information needed for the next step, making the next configuration independent of earlier history once q_t, u_t, and ξ_t are given.Available history and command condition the distribution of ξ_t.Source: B. Probabilistic Configuration Dynamics · p. 2
  • Uncertainty in hidden mechanical quantities induces a distribution over possible next configurations, which the model learns jointly across all 23 next-step angles.Forward kinematics maps sampled configurations to whole-body shapes.Source: B. Probabilistic Configuration Dynamics · p. 2

III. METHODOLOGY

The method conditions flow-based configuration sampling on motion history and a nominal physical reference. Recursive rollouts and geometric supervision connect predicted configurations to motion distributions.

III. METHODOLOGY

  • History-conditioned flow matching predicts the next complete joint configuration, using motion history for information beyond the current configuration and command.Kinematics-based losses supervise the resulting motion distributions.Source: §III · p. 2

A. Structural Dynamics Conditioning

  • The nominal dynamics reference supplies known mechanical relationships as a condition for the learned predictor.The reference is integrated for one sampling interval using the latest configurations and command, with joint limits enforced; Flow learns its relation to the observed transition rather than an additive residual.Source: A. Structural Dynamics Conditioning · pp. 2–3
  • The structural balance includes inertia, Coriolis–centrifugal, and effective stiffness matrices, with gravity contributing no generalized joint torque for horizontal motion.The effective stiffness includes adjacent-joint coupling and near-limit stiffening.Source: A. Structural Dynamics Conditioning · p. 3
  • For ideal frictionless routing, virtual work relates tendon torques to tendon-length changes, using a tendon-length Jacobian and command-dependent nominal tensions.The structural reference is one condition among the inputs to next-configuration sampling.Source: A. Structural Dynamics Conditioning · p. 3, Figure 2 · p. 3

B. Conditional Configuration Transition

A recurrent encoder conditions the transition on motion–actuation history, the current command, and a structural reference. A conditional flow learns next-configuration increments and generates samples that are added to the current configuration before joint limits and forward kinematics are applied.

B. Conditional Configuration Transition

  • The recurrent history representation, current command, and structural reference jointly form condition ct for predicting configuration changes affected by unobserved mechanics.The history representation supplies temporal context beyond instantaneous observations.Source: B. Conditional Configuration Transition · p. 3
  • The flow predicts normalized increments using training-set means and positive standard deviations for the configuration increments.The increment target is 23-dimensional.Source: B. Conditional Configuration Transition · p. 3
  • The conditional rectified-flow objective interpolates Gaussian source samples and data targets along flow time, using an endpoint-difference velocity target.Training evaluates the one-step objective without numerical flow integration.Source: B. Conditional Configuration Transition · p. 3
  • Generated samples are inverse-normalized into configuration increments, added to the current configuration, and projected onto joint limits.Forward kinematics then preserves link geometry and connectivity.Source: B. Conditional Configuration Transition · p. 4

C. Geometric Supervision of Recursive Motion

Recursive rollouts expose errors that one-step fitting does not directly supervise. A geometric Energy Score loss evaluates sampled predictions in known forward-kinematics position space using whole-body and other motion distances.

C. Geometric Supervision of Recursive Motion

  • The training loss uses K ≥2 recursively generated rollouts of H transitions, updating each rollout’s history and physical condition.Known forward kinematics maps predictions into body-position space without a learned shape decoder.Source: C. Geometric Supervision of Recursive… · p. 4
  • The body-position distance uses a fixed position scale, and rollout scores are averaged over matched history–continuation records.The motion distance measures backward-difference velocity error across all centers, scaled by sv.Source: C. Geometric Supervision of Recursive… · p. 4
  • The geometric loss combines Energy Score terms for body, end-effector, terminal, and motion distances with nonnegative weights.The end-effector distance uses the last moving-link center, while terminal distance evaluates only the final time.Source: C. Geometric Supervision of Recursive… · p. 4

D. Training and Recursive Prediction

Training progressively extends the rollout window and adds geometric loss, with gradients propagated through sampled transitions and histories but not the structural-model branch. Inference repeatedly predicts configurations under prescribed commands.

1) Training:

  • After one-step training, the rollout window is progressively extended and geometric loss is added with weight λgeo ≥0.The geometric weight is calibrated using training-set gradients, while wr values are prescribed.Source: 1) Training · p. 4
  • Gradients propagate through sampled transitions and histories, excluding the structural-model branch; training data set normalization and geometric scales.The prescribed wr values and calibrated λgeo determine the geometric-loss weighting.Source: 1) Training · p. 4
  • Evaluation metrics and units are presented in Table I, whose arrows indicate lower-is-better or closer-is-better metrics.The supplied text also specifies pointwise error and covariance calculations with a Gaussian reference for S90.Source: Table I · p. 4, 1) Training · p. 4

2) Inference:

  • Inference starts all K trajectories from a common initial history and appends each trajectory’s own predicted configurations under prescribed commands.Training rollouts and inference use 40 Euler steps per transition over τ ∈[0, 1].Source: 2) Inference · p. 4

IV. SIMULATION

The simulation compares probabilistic predictors under internal-friction variation and actuation disturbances, with retrained ablations for history and structural conditioning. It evaluates one-step predictions and five-second recursive predictions against CVAE and diffusion baselines.

1) Simulation Scenarios:

  • The 23-joint, two-tendon SpiRob is simulated under varying internal friction or temporally correlated Gaussian disturbances to both tendon commands.Friction varies across repetitions and links but remains fixed within each execution; the disturbance scenario uses fixed mechanics.Source: 1) Simulation Scenarios · p. 4

IV. SIMULATION

  • Table II covers one-step prediction from measured histories, and Table III covers five-second recursive prediction; the supplied passages do not report their outcomes.The comparison evaluates probabilistic baselines under both simulated scenarios.Source: Table II · p. 4, §IV · p. 4, Table III · p. 5

2) Training and Baselines:

  • Each scenario uses 64 training executions, with complete executions separated into training, development, and test sets.Independent test repeats use the same nominal commands.Source: 2) Training and Baselines · p. 5
  • CVAE and conditional diffusion are the baselines; all predictors use causal history, predict configuration increments, and have approximately 2.2M parameters.Flow and diffusion use 40 Euler and DDIM steps, respectively.Source: 2) Training and Baselines · p. 5

3) Prediction Protocol:

Flow outperforms the probabilistic baselines on five-second Energy Score in both simulated scenarios, while conditioning ablations support the use of history and structural reference. Evaluation uses shared measured histories and recursively sampled trajectories under prescribed commands.

3) Prediction Protocol:

  • For each command, 64 trajectories start from a common 0.5-second history and are evaluated at the first step and over 300 recursive transitions, or five seconds.Recursive predictions use prescribed nominal commands and each trajectory’s own predicted history; results concern repetitions of known commands, with one training seed per model configuration.Source: 3) Prediction Protocol · p. 5

4) Evaluation Metrics:

  • Evaluation groups measures of mean-motion accuracy, distribution agreement, and coverage using RMS body-position distance in millimeters, with H = 1 for one-step queries and H = 300 for recursive forecasts.Source: 4) Evaluation Metrics · p. 5

1) Baseline Comparisons:

  • Over five seconds, Flow’s Energy Score is 12.048 mm under internal friction variation, versus 12.407 mm for diffusion and 15.611 mm for CVAE; under actuation disturbances, it is 9.290 mm, versus 10.377 mm and 15.759 mm, respectively.Under friction variation, spatial coverage is 85.3%, closer to the nominal 90%; under disturbances, coverage is 79.0%, versus 69.6% for diffusion and 28.1% for CVAE.Source: 1) Baseline Comparisons · p. 5

2) History and Structural Conditioning:

  • Shortening history to two observations raises five-second Energy Score from 12.048 to 30.707 mm under friction variation and from 9.290 to 57.666 mm under actuation disturbances; removing the structural reference raises it to 14.968 and 30.920 mm.With full conditioning, spread ratio D is 1.559; the short-history friction model has 90.5% coverage but excessive dispersion (D = 4.100).Source: 2) History and Structural Conditioning · p. 5, Table IV · p. 5

3) Computational Cost:

On an Apple M4 Max, Flow generates 64 next-step states in 21.6 ms and propagates 64 trajectories over five seconds in 4.94 s. The physical setup uses a tendon-driven robot tracked by body-mounted markers.

3) Computational Cost:

  • Flow generates 64 next-step states in 21.6 ms and propagates 64 trajectories over 300 steps (five seconds) in 4.94 s, 27.2% and 40.1% faster than diffusion, respectively.Measurements include conditioning, sampling, and forward kinematics on an Apple M4 Max; CVAE remains faster.Source: 3) Computational Cost · p. 5, Table V · p. 6

A. Experimental Setup

  • SpiRob runs on a horizontal surface with two torque-controlled J60-series tendon actuators, while a fixed camera tracks 18 body-mounted markers.Recordings use five two-channel command profiles from a curled reset target; the structural reference combines nominal inertial properties with fitted stiffness, damping, and effective actuation parameters.Source: A. Experimental Setup · p. 6, Figure 5 · p. 6

B. Experimental Protocol

The physical evaluation holds out two tendon-command profiles and recursively predicts five-second motion from measured history. Flow captures the principal motion sequence and achieves lower marker RMSE and Energy Score than the compared baselines on both inputs.

B. Experimental Protocol

  • The split uses 80 training and 20 validation repeats under three profiles, with 54 test repeats under two held-out profiles; predictors train only on real recordings and use the same test set.Source: B. Experimental Protocol · pp. 6–7
  • For each test input, 64 trajectories start from a shared 0.5-second measured history and advance 100 steps under prescribed commands, updating each trajectory’s history with its own predictions.Source: B. Experimental Protocol · p. 7
  • Metrics use valid marker observations; RMSE measures ensemble-mean error, while RMSE, Energy Score, and coverage are averaged equally over repeats.Distances use a nominal scale of 1.1119 mm/pixel, and spatial coverage uses covariance regularization of (0.556 mm)2I2.Source: B. Experimental Protocol · p. 7

C. Results

  • Under both held-out physical inputs, Flow has the lowest marker RMSE (16.29 and 15.86 mm) and Energy Score (11.91 and 11.42 mm); its spread ratios of 1.648 and 1.217 are closer to one than CVAE’s.CVAE has coverage closer to the nominal 90%, but its spread ratios are 4.717 and 3.780, indicating broader ensembles than measured repeats; Flow follows the principal curling and extension sequence under Test input 2.Source: C. Results · p. 7, Figure 6 · p. 7, Table VI · p. 8

VI. CONCLUSION

  • The conclusion reports lower Energy Scores and coverage closer to nominal for Flow in both simulated scenarios, with ablations supporting history and structural conditioning.For the two held-out physical profiles, it reports lower prediction errors and spread ratios closer to one despite lower pointwise coverage than CVAE.Source: §VI · pp. 7–8

Related papers

Loading 2609.25658v1…