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Spacetime Splines for Adaptive Robot Motions

Boxuan Zhang, Ante Marić, Sylvain Calinon

arXiv:2610.29839v1cs.RO

TL;DR

Robot demonstrations vary in duration, local timing, and completeness, motivating a representation that considers geometry and timing together. The paper models them in a shared continuous spline and jointly estimates path coverage and phase. Simulations report robust recovery under demonstration-set variation and adaptation to varying continuity.

  • Problem

    Robot demonstrations can differ in duration, local timing, recorded paths, and completeness, while motion adaptation must consider both spatial and temporal information.

    Source: §I · p. 1

  • Method

    A spacetime spline jointly represents shared path geometry and demonstration-specific phase, with alternating optimization estimating path intervals in partial demonstrations.

    Source: §I · p. 1

  • Results

    Simulations show robust recovery under demonstration-set variation and adaptation to varying path continuity; versus DTW+spline, accuracy degrades more slowly as inter-demonstration spatial variation increases.

    Source: A. Simulation Experiments · p. 5, §V · p. 6

  • Takeaways & Limitations

    The formulation illustrates online retiming and synchronization in an assistive robotics task involving human–robot collaboration and visual tracking loss.

    Source: §I · p. 1, §V · p. 6

Abstract

from arXiv · show

Movement primitives commonly represent robot mo- tions through basis functions indexed by time or phase. Inspired by spacetime representations employed in physics, we propose a spline-based formulation in which spatial geometry and temporal progression are represented within a common set of continuous variables. The proposed method jointly estimates a shared Bezier curve, while allowing spatial and temporal model complexity to be adapted independently (e.g., demonstration-specific phase modulation). As use case, we consider an alternating optimization process that simultaneously estimates path intervals observed in partial demonstrations. Simulation experiments demonstrate accurate joint recovery, refinement through modifications in model complexity and continuity, and improved robustness to variance in the demonstration set. Finally, we demonstrate online retiming and synchronization capabilities in a patient-turning task assisted by a robot, evaluated with a physical manikin.

I. INTRODUCTION

Robot demonstrations can differ in duration, local timing, recorded paths, and coverage, so adapting motions requires treating spatial and temporal information together. The paper proposes a shared spacetime spline with independently adaptable geometry and phase, estimates coverage in partial demonstrations, and evaluates the approach in simulations and a patient-turning task.

  • Demonstrations may vary in duration, local speed, recorded paths, sampling rates, and how much of the motion they cover.Source: §I · p. 1
  • A continuous spacetime spline jointly represents shared path geometry and motion timing while allowing spatial and temporal adaptation independently.The phase component can be adapted for individual demonstrations while preserving the shared spatial path.Source: §I · p. 1
  • Alternating optimization adapts spatial control points and temporal coordinates to estimate path intervals in partial demonstrations.Source: §I · p. 1
  • Simulations evaluate recovery and adaptive optimization, and a physical-manikin patient-turning task demonstrates synchronization under changing human timing and temporary interruptions.Source: §I · pp. 1–2

II. RELATED WORK

The related work spans spline fitting, temporal alignment and geometry–phase modeling, and learning from partial demonstrations. This paper addresses spatial and temporal model complexity together, retaining phase and coverage as explicit parts of a shared motion representation.

A. Spline-Based Motion Representation and Adaptive Fitting

B. Temporal Alignment and Geometry–Phase Modeling

  • The method jointly estimates a shared geometric spline and demonstration-specific monotone phase functions, keeping phase explicit for later constraints or modification.Source: B. Temporal Alignment and… · p. 2

C. Learning from Partial Demonstrations

III. METHODOLOGY

The method represents demonstrations in normalized recording time using a shared spatial map and demonstration-specific progression along that map. Partial demonstrations have estimated coverage intervals, while a complete anchor resolves a common reparameterization ambiguity.

A. Spacetime Motion Encoding

  • Using normalized recording time, the encoding pairs a shared spatial map with a demonstration-specific progression function over a coverage interval.Source: A. Spacetime Motion Encoding · p. 2
  • Complete demonstrations cover [0, 1], while partial coverage is estimated jointly with spatial and temporal parameters; one complete anchor fixes the common monotone reparameterization ambiguity.Source: A. Spacetime Motion Encoding · p. 3

B. Coordinate-Specific Spline Parameterization

The parameterization uses cubic splines for both the shared spatial path and demonstration-specific timing profiles. Endpoint, monotonicity, and optional derivative constraints govern continuity and progression.

B. Coordinate-Specific Spline Parameterization

  • The shared spatial map uses cubic Bezier segments with control points and shared endpoints for C0 continuity, with selected G1 junctions aligning tangent directions while allowing different magnitudes.Source: B. Coordinate-Specific Spline… · p. 3
  • Demonstration-specific temporal profiles use piecewise-cubic segments, with endpoint and shared-segment constraints preventing reversal while allowing pause-like progression and C0 continuity.Source: B. Coordinate-Specific Spline… · p. 3
  • Further continuity constraints can be imposed by matching derivatives at internal knots.Source: B. Coordinate-Specific Spline… · p. 3

C. Alternating Spacetime Optimization

For a fixed spline structure, the method minimizes mean squared reconstruction error with curvature and coverage regularization. It alternates temporal and spatial parameter updates while enforcing constraints on temporal blocks.

C. Alternating Spacetime Optimization

  • Coverage intervals for partial demonstrations are parameterized by α_i and β_i, with a minimum coverage length Δ_min.The coverage parameterization uses the sigmoid σ(x) = (1 + exp(−x))^−1.Source: C. Alternating Spacetime Optimization · p. 3
  • For a fixed spline structure, the optimization minimizes mean squared reconstruction error while regularizing curvature and coverage intervals.Source: C. Alternating Spacetime Optimization · p. 3
  • Block-coordinate optimization updates non-anchor temporal blocks in parallel with the spatial block fixed, then updates the spatial parameters with temporal blocks fixed.Temporal blocks enforce monotonicity, endpoints, coverage, and selected continuity constraints; spatial updates use regularized linear least squares for all-C0 junctions and constrained nonlinear least squares with G1 junctions, alternating until relative change in L falls below a threshold.Source: C. Alternating Spacetime Optimization · p. 3

D. Adaptive Coordinate Refinement

The refinement procedure proposes splits on high-error segments, tests spatial junctions for G1 continuity, and accepts structural changes only when their score improves sufficiently. The section also describes temporal-coordinate profiles for partial traversal and online retiming.

D. Adaptive Coordinate Refinement

  • Candidate splits are initialized with De Casteljau subdivision on segments with the largest mean reconstruction errors, while retaining the encoded curves.Source: D. Adaptive Coordinate Refinement · p. 3
  • Spatial junctions meeting the angle and handle-ratio thresholds are reoptimized under G1 continuity and retained if the increase in (13) stays within a threshold.Source: D. Adaptive Coordinate Refinement · p. 3
  • Table I reports recovery on complete and partial demonstrations as mean ± standard deviation over 30 runs, with lower values better.DMP and ProMP are evaluated only on complete demonstrations because their standard formulations do not infer unknown coverage intervals.Source: Table I · p. 4
  • A candidate structure is accepted only if its score decreases by more than ε_struct; when both split candidates are admissible, the larger score reduction is selected.Refinement stops when no split is accepted or the target reconstruction error is reached.Source: D. Adaptive Coordinate Refinement · p. 4
  • Linear temporal profiles allow nominal or partial traversal, while nonlinear profiles support retiming and pause-like progression, including online adaptation in the assistive-task experiment.Source: D. Adaptive Coordinate Refinement · p. 4

A. Simulation Experiments

The simulations test joint recovery of shared geometry, phase, and partial coverage, then assess robustness, continuity selection, and adaptive optimization. Results show the method degrades more slowly under increasing demonstration variation, selects continuity with lower errors, and benefits from joint adaptive fitting.

Joint recovery and robustness (Q1)

  • The joint-recovery evaluation uses six LASA shapes, seven demonstrations per shape, and five trials per shape for each setting.Partial trials expose only a sub-interval; minimum coverage span is 0.12, with at most 30 optimization iterations and tolerance 10^-7.Source: A. Simulation Experiments · p. 4

Alternating optimization and adaptive complexity (Q3)

  • Adaptive geometry complexity selects 3.89 ± 1.45 segments; mean runtimes for 2, A, 4, 6, and 8 segments are 0.58, 21.0, 18.7, 121, and 689 s, respectively.The figure compares adaptive selection with fixed segment counts and illustrates an accuracy–runtime trade-off.Source: Figure 3 · p. 5, A. Simulation Experiments · p. 5

Joint recovery and robustness (Q1)

  • With no privileged clean reference, our method degrades more slowly than DTW+spline as spatial perturbation amplitude increases, although DTW+spline is more accurate at low perturbation.The clean-reference regime deliberately favors reference-based alignment.Source: A. Simulation Experiments · pp. 4–5, Figure 2 · p. 4

Structural preservation and adaptive continuity (Q2)

  • Adaptive continuity achieves the lowest position and tangent errors, avoiding All-C0 tangent inconsistency and globally enforced G1 over-smoothing.Ground-truth phase is supplied to isolate geometric continuity.Source: A. Simulation Experiments · p. 5

Alternating optimization and adaptive complexity (Q3)

  • Removing alternating updates or fixing coverage or phase substantially degrades joint recovery, supporting joint optimization of geometry, phase, and coverage.Table II reports an ablation over 15 runs; lower values are better.Source: A. Simulation Experiments · p. 5, Table II · p. 5
  • For 32 demonstrations, median fitting time falls from 88.5 s with serial finite differences to 5.81 s with analytic gradients and 3.04 s with parallel updates.Per-demonstration updates take 16.9 ms median and 29.8 ms at the 95th percentile.Source: A. Simulation Experiments · p. 5

B. Robot Experiments

A physical-manikin patient-turning task tests online synchronization using a learned human-to-robot motion correspondence. The robot follows the human during tracking and holds its configuration during temporary tracking loss, resuming after recovery.

B. Robot Experiments

  • Kinesthetic guidance establishes correspondence between human motion progress and robot motion during patient-turning demonstrations.An RGB camera tracks the human hand at approximately 15 Hz.Source: B. Robot Experiments · p. 6
  • A monotone estimated human phase provides the robot-phase reference, preventing temporal regression under visual noise or ambiguous local matches.Robot commands update at approximately 100 Hz.Source: B. Robot Experiments · p. 6
  • During temporary hand-tracking loss, the robot holds its current configuration and resumes execution after tracking returns.With continuous tracking, the robot progresses synchronously with its human partner.Source: B. Robot Experiments · p. 6

V. CONCLUSION

The conclusion presents spacetime splines as a way to learn shared paths, demonstration-specific timing, and coverage for complete and partial demonstrations. Simulations show robust recovery and adaptation to path continuity, while a patient-turning task illustrates online retiming under tracking loss.

V. CONCLUSION

  • The formulation jointly estimates a shared geometric path, demonstration-specific temporal profiles, and coverage intervals in one augmented state space.It permits coordinate-specific continuity constraints.Source: §V · p. 6
  • Simulations demonstrate robust recovery under demonstration-set variation and adaptation to varying continuity in demonstrated paths.Source: §V · p. 6
  • A patient-turning assistive robotics experiment illustrates online retiming and adaptation to visual tracking loss; future work will explore collaborative tasks with multiple users and robots.Source: §V · p. 6
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