Interaction-Stiffness-Guided Basis Allocation in Dynamic Movement Primitives for Efficient Skill Transfer

Chan Xu; Silu Chen; Dehao Wang; Xiyu Chen; Dexin Jiang; Chi Zhang; Guilin Yang; Chenguang Yang

cs.ROOct 1, 2026

Interaction-Stiffness-Guided Basis Allocation in Dynamic Movement Primitives for Efficient Skill Transfer

Authors: Chan Xu, Silu Chen, Dehao Wang, Xiyu Chen, Dexin Jiang, Chi Zhang, Guilin Yang, Chenguang Yang, +1 more

Organizations: Zhejiang Key Laboratory of Precision Actuation and Intelligent Robotics, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China · University of Chinese Academy of Sciences, Beijing 100049, China · Department of Computing, The Hong Kong Polytechnic University, HKSAR, China

Abstract

Dynamic Movement Primitives (DMPs) provide a compact and stable formulation for trajectory representation and generalization in robot skill learning. However, their predefined basis layout limits the allocation of approximation capacity according to stage-dependent precision requirements. To address this issue, this article proposes Stage-Criticality-Guided Dynamic Movement Primitives (SC-DMPs) with adaptive basis allocation for precision-critical skill learning. Operator-robot interaction stiffness and a trajectory-consistency cue derived from cross-demonstration task-space variability are integrated to construct a stage-criticality index. Guided by this index, basis centers are redistributed in normalized time through inverse cumulative criticality and mapped to the canonical phase domain, while their bandwidths are refined to adjust local approximation support. This enables denser and more flexible representation at high-criticality stages while retaining sparser allocation elsewhere. Experiments on handwriting trajectories and three real-robot tasks show that the inferred criticality is concentrated in geometrically demanding and task-constrained regions. Comparisons with DMPs, ProMPs, ProDMP, GP-MP, and KMP demonstrate improved trajectory reproduction, endpoint generalization, and task-critical accuracy while retaining a compact model and the stable structure of classical DMPs.

Figures & tables

Symbol	Description
$M$ , $T$ , $D$	Number of demonstrations, number of aligned time steps, and task-space dimension, respectively.
$t_{i}$	Normalized time at the $i$ -th aligned time step.
$\bm{P}_{i,m}$ , $\bm{K}_{i,m}$ , $\bm{Q}_{i,m}$	Task-space position, axis-wise interaction stiffness, and their concatenated motion-state vector.
$N_{\rm G}$ , $\pi_{j}$ , $\bm{\mu}_{j}$ , $\bm{\Lambda}_{j}$	Number of Gaussian components and the weight, mean, and covariance of the $j$ -th component.
$\gamma_{j}(t)$ , $\bm{\Lambda}_{Qt,j}$ , $\bm{\Lambda}_{QQ,j}$	Posterior responsibility and covariance blocks used in GMR conditioning.
$\bar{\bm{P}}(t)$ , $\bar{\bm{K}}(t)$ , $\bm{\sigma}_{P}^{2}(t)$	GMR reference trajectory, stiffness profile, and task-space variability.

TABLE I: Summary of main symbols.

Fig. 1: Overall framework of the proposed SC-DMPs. Operator-robot interaction stiffness and cross-demonstration task-space variability are used to estimate a stage-criticality index, which guides basis-center redistribution and adaptive bandwidth design for precision-critical skill reproduction.

Fig. 2: Experimental platform for interaction stiffness identification.

Fig. 3: Handwriting letter data acquisition. (a)-(c) present the template, the trajectory with stiffness ellipsoids, and the visualized stage-criticality index for letter “Z”. (d)-(f) show the corresponding results for letter “M”.

Fig. 4: Learning errors of SC-DMPs and DMPs for “Z” under 5, 10, 15, 20, 25, and 30 basis functions on a logarithmic scale. (a) RMSE (mm). (b) W-RMSE (mm). (c) RMSE-Geo (mm). (d) MaxE (mm).

Fig. 5: Learning errors of SC-DMPs and DMPs for “M” under 5, 10, 15, 20, 25, and 30 basis functions on a logarithmic scale. (a) RMSE (mm). (b) W-RMSE (mm). (c) RMSE-Geo (mm). (d) MaxE (mm).

Errors	Mean reduction	95% confidence interval	Adjusted $p$
RMSE	21.42% ± 10.11%	[13.99%, 25.65%]	0.0103
W-RMSE	21.92% ± 10.27%	[13.99%, 26.08%]	0.0103
RMSE-Geo	32.50% ± 13.89%	[23.55%, 38.91%]	0.0020
MaxE	24.67% ± 20.16%	[8.57%, 32.56%]	0.0122

TABLE II: Statistical analysis of SC-DMPs versus DMPs across different basis-function settings.

Fig. 6: Center distributions under 15 and 30 basis functions, where color intensity increases with density. (a) and (c) correspond to “Z”. (b) and (d) correspond to “M”.

Fig. 7: Comparison of learning performance between SC-DMPs and DMPs. (a) and (b) correspond to “Z” with 15 and 30 basis functions. (c) and (d) correspond to “M” with 15 and 30 basis functions.

Fig. 8: Ablation results of basis representation with 20 basis functions. (a) RMSE (mm). (b) RMSE-Geo (mm). (c) MaxE (mm).

Fig. 9: Ablation results of stage-criticality construction with 20 basis functions. (a) RMSE (mm). (b) RMSE-Geo (mm). (c) MaxE (mm).

Letter	Refinement method	Temporal-variation reduction (%)	Critical-stage retention (%)	Construction time (s)
“Z”	Prior-only	0.00	100.00	$<0.001$
	Prior-smoothed	3.24	99.00	0.008
	STR-Net-refined	20.91	91.50	1.258
“M”	Prior-only	0.00	100.00	$<0.001$
	Prior-smoothed	3.21	100.00	$<0.001$
	STR-Net-refined	34.92	85.00	1.147

TABLE III: Stage-criticality refinement characteristics and construction time.

Fig. 10: Endpoint-generalization errors for “Z” under varying numbers of basis functions. (a) NSE. (b) W-RMSE (mm). (c) RMSE-Geo (mm).

Fig. 11: Endpoint-generalization errors for “M” under varying numbers of basis functions. (a) NSE. (b) W-RMSE (mm). (c) RMSE-Geo (mm).

Fig. 12: Endpoint-generalization of DMPs and SC-DMPs on “Z” with 20 basis functions. (a) Translation. (b) Start-point change. (c) Goal-point change. (d) Start-point and goal-point changes.

Fig. 13: Spatial-scaling generalization of DMPs and SC-DMPs on “Z” with 20 basis functions. (a) 0.5

\times

. (b) 2

\times

. (c) 3

\times

. (d) 5

\times

Letter	Scale	NSE		W-RMSE (mm)		RMSE-Geo (mm)
		DMPs	SC-DMPs	DMPs	SC-DMPs	DMPs	SC-DMPs
“Z”	$0.5\times$	0.0055	0.0048	0.9063	0.7465	1.3508	1.0256
	$2\times$	0.0055	0.0048	3.6251	2.9858	5.4032	4.1025
	$3\times$	0.0055	0.0048	5.4377	4.4787	8.1048	6.1537
	$5\times$	0.0055	0.0048	9.0628	7.4645	13.5080	10.2562
Reduction		12.99%		17.64%		24.07%

TABLE IV: Errors under spatial-scaling generalization.

Letter	Method	Learning performance		Endpoint generalization		Computational efficiency
		RMSE (mm)	RMSE-Geo (mm)	NSE	RMSE-Geo (mm)	Model size (kB)	Adaptation time (ms)
“Z”	DMPs	0.7936	1.2318	0.0027	1.3303	1.00	4.77
	ProMPs	0.6136	0.8083	0.8927	68.3981	29.08	0.75
	ProDMP	0.6396	0.9263	0.1004	32.1902	0.99	1.66
	GP-MP	1.1117	1.3649	0.2599	68.2537	9.65	89.86
	KMP	20.6595	16.8022	0.1696	63.0821	65.64	938.29

TABLE V: Comparison of learning performance, endpoint generalization, and computational efficiency on letter trajectories.

Parameter	Tested settings	Default
$\beta_{1}/\beta_{2}$	0.50/1.50, 0.75/1.25, 1.00/1.00, 1.25/0.75, 1.50/0.50	1.00/1.00
GMM components	14, 18, 22, 26, 30	22
Latent dimension (LD)	32, 64, 128	64
Kernel size (KS)	3, 5, 7	5
Bandwidth candidate (BW)	Narrow: {0.75:0.05:1.25} Normal: {0.50:0.10:1.50} Wide: {0.25:0.15:1.75}	Normal

TABLE VI: Tested hyperparameter settings and default values.

Fig. 14: Hyperparameter sensitivity of SC-DMPs. (a) “Z”. (b) “M”.

Fig. 15: Convergence and computational time of bandwidth optimization. (a) “Z”. (b) “M”.

Fig. 16: Effect of demonstration number on the RMSE-Geo of DMPs and SC-DMPs. (a) “Z”. (b) “M”.

Fig. 17: Experimental setups and data visualizations for plug insertion and removal, sequential dipping, and curved-surface wiping. (a) Platform. (b) Demonstrated trajectory and stiffness ellipsoids. (c) Normalized variance. (d) Normalized stiffness. (e) Stage criticality index.

Fig. 18: Trajectory-learning error comparison on three robotic tasks. (a) RMSE (mm). (b) RMSE-TCR (mm). (c) MaxE-TCR (mm).

Task	Refinement	RMSE	RMSE-TCR	MaxE-TCR
Task 1	Prior-only	0.2709	0.7714	2.0715
	Prior-smoothed	0.2709	0.7716	2.0719
	STR-Net-refined	0.2666	0.7409	1.9526
Task 2	Prior-only	0.6737	0.9580	2.5812
	Prior-smoothed	0.6736	0.9578	2.5807
	STR-Net-refined	0.6709	0.9743	2.6242

TABLE VII: STR-Net ablation on robot tasks. All errors are in mm

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Open paper AskPDF arXiv

Expert demonstrations are widely assumed to be the gold standard for robot imitation learning. Yet for fine-grained manipulation such as insertion, stacking, and alignment, we uncover a counterintuitive failure mode: fluent demonstrations can be poor teachers. A skilled teleoperator compresses the decisive moments of alignment and recovery into a brief temporal window, leaving the policy flooded with redundant free-space motion and starved of supervision exactly where precision determines success. We address this bottleneck at two levels. At the data level, slowing down near alignment and resampling critical segments both help, yet the gain comes mainly from broadening the coverage of recovery states the policy must learn, not from reweighting frames it already has. Such data-side fixes, however, leave the policy's per-frame view untouched: a single image still maps directly to an action, and the local motion that governs correction stays implicit. We therefore turn to the representation level and introduce STAIR (\textbf{S}patio-\textbf{T}emporal feature \textbf{A}s an \textbf{I}nterface for \textbf{R}obot learning), a compact dynamic feature that bridges the vision-language model and the action expert, distilling the short-horizon motion already recorded in each trajectory into dense, motion-aware supervision. Trained on fluent data alone, STAIR recovers most of the deliberate-demonstration gain (

50.0

62.2\%

overall, approaching the

64.4\%

of deliberate demonstrations). These results call for a more pedagogical view of robot data, optimized for machine learnability rather than human efficiency alone.

Abstract

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