Spacetime Splines for Adaptive Robot Motions

Boxuan Zhang; Ante Marić; Sylvain Calinon

cs.ROOct 21, 2026

Spacetime Splines for Adaptive Robot Motions

Authors: Boxuan Zhang, Ante Marić, Sylvain Calinon

Organizations: Technical University of Munich (TUM), Munich, Germany. · Idiap Research Institute, Martigny, Switzerland.

Abstract

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.

Figures & tables

Fig. 1: The spacetime spline representation can be used to preserve a shared spatial path geometry while modulating the motion timing by adapting the phase component of the augmented state space. Left: uniform (standard) in $(q_{1},q_{2},\tau)$ space and phase adaptation to provide a minimum-jerk motion. Right: Same movements in the spatial $q_{1}$ – $q_{2}$ plane, showing that the path geometry is preserved under phase adaptation.

Setting	Method	Traj. RMSE	Geom. RMSE	Phase RMSE	Cov. MAE
Complete	Ours	$0.0154\pm 0.0012$	$0.0127\pm 0.0014$	$0.0170\pm 0.0004$	–
	DTW+spline	$\mathbf{0.0105\pm 0.0017}$	$\mathbf{0.0061\pm 0.0031}$	$\mathbf{0.0055\pm 0.0003}$	–
	Linear phase	$0.1368\pm 0.0134$	$0.0523\pm 0.0227$	$0.1728\pm 0.0000$	–
	DMP	$0.1372\pm 0.0136$	$0.0550\pm 0.0202$	–	–
	ProMP	$0.1367\pm 0.0136$	$0.0526\pm 0.0227$	–	–
Partial	Ours	$0.0200\pm 0.0012$	$0.0187\pm 0.0032$	$0.0270\pm 0.0031$	$0.0348\pm 0.0047$

TABLE I: Recovery on complete and partial demonstrations (mean

\pm

standard deviation over 30 runs). Lower is better. DMP and ProMP are evaluated on complete demonstrations only because their standard formulations do not infer unknown coverage intervals.

Fig. 2: Geometry recovery under increasing independent spatial perturbations across demonstrations. The horizontal axis shows the perturbation amplitude $\kappa$ in units of $10^{-2}$ . Without a privileged clean reference, our method degrades more slowly than DTW+spline as the perturbation amplitude increases.

Panel of Figure 4 — Fig. 3: (a) Structural ablation. Position and tangent RMSEs are normalized by the corresponding medians of the all- $C^{0}$ variant. $O$ , $C^{0}$ , and $G^{1}$ denote our method, all- $C^{0}$ , and all- $G^{1}$ , respectively. (b) Adaptive geometry complexity. $A$ denotes the adaptive model, which selects $3.89\pm 1.45$ segments (mean $\pm$ standard deviation); the other labels indicate fixed segment counts. Mean runtimes for $2$ , $A$ , $4$ , $6$ , and $8$ are $0.58$ , $21.0$ , $18.7$ , $121$ , and $689$ s, respectively.

Method	Geom. RMSE	Phase RMSE	Cov. MAE
Ours	$\mathbf{0.0210\pm 0.0017}$	$\mathbf{0.0282\pm 0.0027}$	$\mathbf{0.0347\pm 0.0049}$
One-shot	$0.0541\pm 0.0019$	$0.0687\pm 0.0017$	$0.0813\pm 0.0083$
Fixed coverage	$0.0571\pm 0.0075$	$0.1036\pm 0.0013$	$0.2070\pm 0.0000$
Linear phase	$0.0945\pm 0.0100$	$0.1431\pm 0.0042$	$0.1359\pm 0.0076$

TABLE II: Ablation of the joint optimization (mean

\pm

standard deviation over 15 runs). Lower is better.

Fig. 4: Patient-turning robot assistance experiment. From top to bottom: demonstration collection through guidance; nominal execution under continuous hand tracking; and online adaptation during temporary tracking loss. Figures on the right show camera observations used for phase estimation. During tracking loss, the current phase is retained until visual observation is recovered.

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Modern imitation-learning policies for robot manipulation often represent actions as fixed-resolution action chunks, which are simple and effective but expose limited geometric and temporal structure before execution. This paper studies Spline Policy (SP), a structured representation that replaces action chunks with spline parameters while keeping the policy backbone unchanged. The predicted spline can be decoded as a compact continuous trajectory, queried at different temporal resolutions, constrained or edited in parameter space, and passed to downstream controllers. For quadratic spline outputs, the same representation can also be converted into a state-dependent vector field through an analytical distance-field construction. Under the regularity and projection assumptions of this construction, the induced dynamics do not increase the distance to the generated spline, yielding a principled local corrective mechanism around the predicted motion. The spline output further supports uncertainty propagation from observations to spline parameters, trajectories, and flow fields, and can be combined with classical control mechanisms such as null-space collision avoidance without retraining the policy backbone. We instantiate SP with diffusion, flow-matching, transformer-based, and vision-language-action backbones. Experiments in low-dimensional motion learning, simulated manipulation under matched backbones, dexterous manipulation, and real-robot case studies show that SP remains compatible with modern policy learners while exposing useful motion-structure properties, including compact decoding, temporal resampling, local correction around predicted motions, uncertainty evaluation, and controller compatibility.

Abstract

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