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

I. INTRODUCTION

• Precision-oriented learning requires global trajectory fidelity and local accuracy at less error-tolerant stages, while preset DMP bases limit stage-wise adaptability –.Controller-based improvements may require extra information, interaction, computation, or specialized design without directly addressing encoding errors –.Source: §I · p. 1
• Existing representation refinements use trajectory structure or informativeness, but may miss task-imposed tolerances, as in geometrically simple yet precision-demanding peg-in-hole insertion –.Basis-center redistribution moves local support without increasing basis count or changing stable DMP dynamics.Source: §I · pp. 1–2
• Operator-robot interaction stiffness can reveal regulation of positional deviations beyond trajectory geometry, while cross-demonstration consistency provides a complementary cue –,,.Interaction stiffness is available across kinesthetic-teaching stages, unlike contact-stiffness estimates that can be unreliable outside sustained contact –.Source: §I · p. 2
• SC-DMPs combine stiffness and trajectory variability to guide basis-center redistribution and bandwidth adaptation, with experiments reporting improved reproduction, generalization, and task-critical accuracy.The reported comparisons include DMPs, ProMPs, ProDMP, GP-MP, and KMP, while retaining a compact model.Source: §I · p. 2

II. PROBABILISTIC MODELING OF DEMONSTRATIONS

• The demonstration pipeline aligns M trajectories with dynamic time warping, normalizes their common time axis to [0, 1], and represents the resulting dataset.The aligned sequence has length T.Source: §II · p. 2
• A Gaussian mixture model represents the joint temporal, task-space motion, and interaction-stiffness distribution, with component weights summing to one.The model partitions each component’s mean and covariance by time, position, and stiffness.Source: §II · p. 3
• Gaussian mixture regression infers conditional distributions and mixture expectations for the reference trajectory and stiffness profile.Cross-demonstration task-space variability is characterized using position covariance and variance across dimensions; lower variance indicates stronger consistency.Source: §II · p. 3
• Interaction stiffness and task-space consistency are treated as complementary task-informed cues for estimating stage-dependent reproduction requirements and guiding basis allocation.The paper associates higher stiffness with stronger active regulation and lower variance with stronger cross-demonstration consistency.Source: §II · p. 3

III. STAGE-CRITICALITY-GUIDED ADAPTIVE DMPS

• The section constructs a stage-criticality index from interaction stiffness and task-space trajectory variability.The index guides adaptive basis allocation in the canonical phase domain.Source: §III · p. 4
• The index links the two cues to adaptive basis allocation in the canonical phase domain.Source: §III · p. 4
• The SC-DMP framework is presented as the method for using this index to guide allocation.Source: §III · p. 4

• The dimension-wise prior weights normalized interaction stiffness and inverse task-space variability to estimate stage criticality.The paper treats high stiffness and low variability as indicators of task-critical phases.Source: A. Physically Motivated Stage… · p. 4
• STR-Net refines the physical prior with temporal context to mitigate fluctuations from local estimation errors, misalignment, and measurement noise.Its Huber and temporal-smoothness terms regularize the prior while preserving coherent criticality intervals.Source: A. Physically Motivated Stage… · p. 4
• The refined dimension-wise scores are averaged into a scalar index, where larger values indicate stronger stage-wise reproduction-fidelity requirements.Source: A. Physically Motivated Stage… · p. 4

• Equal accumulated criticality between adjacent centers gives higher-criticality regions denser basis support in normalized time.The method partitions cumulative criticality uniformly and inverts that profile to locate centers.Source: B. Stage-Criticality-Guided Basis… · pp. 4–5
• Criticality-guided centers are mapped from normalized time into the canonical phase domain, preserving the allocation while matching DMP phase evolution.Source: B. Stage-Criticality-Guided Basis… · p. 5
• Under constant stage criticality, the allocation reduces to uniformly spaced centers and thus standard DMP center placement.Source: B. Stage-Criticality-Guided Basis… · p. 5

• Adjacent-center spacing sets reference bandwidths, which are scaled to adjust basis-function support and overlap.Source: C. Adaptive Bandwidth Design · p. 5
• Cyclic coordinate descent evaluates bandwidth candidates one scaling factor at a time because overlapping bases couple the reconstruction cost.A candidate update is retained only when it reduces the objective by more than δ.Source: C. Adaptive Bandwidth Design · p. 6
• The finite search terminates at a coordinate-wise local minimum within tolerance δ when a sweep accepts no update, but global optimality is not guaranteed.Source: C. Adaptive Bandwidth Design · p. 6

A. Experimental Setup

• Robot and arm motion are recorded alongside four muscle sEMG signals to estimate operator–robot interaction stiffness.Robot joint and body-position data are recorded at approximately 100 Hz, while sEMG is acquired at 2000 Hz; signals are processed and timestamp-aligned before stiffness estimation.Source: A. Experimental Setup · p. 6, Figure 2 · p. 6
• The evaluation uses RMSE, W-RMSE, RMSE-Geo, MaxE, and NSE to assess trajectory learning and generalization.W-RMSE weights errors by inferred stage-criticality, RMSE-Geo evaluates regions in the top 20% of reference-trajectory curvature, and NSE is used for varying start or target configurations.Source: A. Experimental Setup · p. 6
• The reported NSE definition follows the setup passage introducing that metric.The supplied setup passage introduces NSE but does not include its defining expression.Source: A. Experimental Setup · p. 6

B. Handwriting Letter Experiment

• SC-DMPs and DMPs are compared on “Z” learning errors across 5, 10, 15, 20, 25, and 30 basis functions.The reported metrics are RMSE, W-RMSE, RMSE-Geo, and MaxE.Source: Figure 4 · p. 7, 1) Learning Performance and… · p. 7
• Twelve demonstrations per letter provide marker trajectories and corresponding interaction-stiffness estimates for the “Z” and “M” templates.Reference trajectories and stage-criticality profiles are derived from the demonstrations.Source: B. Handwriting Letter Experiment · pp. 6–7, Figure 3 · p. 7
• For both letters, stage criticality peaks around turning regions, where small deviations can noticeably distort the letter shape.The figure presents each letter's template, trajectory with stiffness ellipsoids, and visualized criticality index.Source: B. Handwriting Letter Experiment · p. 7, Figure 3 · p. 7

1) Learning Performance and Basis-Allocation Analysis:

• SC-DMPs reduce mean errors by 21.42%–32.50% relative to DMPs across 12 letter-basis settings, with all adjusted Wilcoxon tests significant.They achieve lower errors in 45 of 48 letter-basis-metric comparisons, and all 95% confidence intervals exclude zero.Source: Table II · p. 7, 1) Learning Performance and… · p. 7
• SC-DMPs place denser centers in high-criticality regions and sparser centers elsewhere, improving trajectory matches around bends and turns.The visualized center distributions compare 15 and 30 basis functions for both letters.Source: 1) Learning Performance and… · p. 7, Figure 6 · p. 7

2) Ablation Analysis:

• Bandwidth refinement and criticality-guided centers each reduce errors, while their combined benefit depends on the demonstrated letter’s geometry.Full SC-DMPs perform best on “M”; DMPs-BW has the lowest errors on “Z”.Source: 2) Ablation Analysis · pp. 7–8, Figure 8 · p. 8
• Stiffness-only and variability-only cues are not consistently reliable across letters; their fusion is more balanced, while STR-Net refinement trades critical-stage retention for stronger temporal-variation reduction.Prior smoothing reduces temporal variation by about 3.2% while retaining at least 99% of prior-defined critical stages; STR-Net reduces it by 20.91% for “Z” and 34.92% for “M”, retaining 91.50% and 85.00%, respectively.Source: 2) Ablation Analysis · p. 8

3) Generalization Performance Analysis:

• 34 of 36 endpoint-generalization comparisons favored SC-DMPs; with 5 basis functions, error reductions ranged from 21.86% to 35.22%, and with 20 or more, RMSE-Geo fell 23.74%-47.32%.For “Z” with 20 basis functions, the generalized trajectories more closely follow locally demanding segments.Source: 3) Generalization Performance Analysis · pp. 8–9, Figure 12 · p. 9
• Across 0.5×–5× spatial scaling, SC-DMPs preserved relative improvements over DMPs for both letters, while W-RMSE and RMSE-Geo increased proportionally with trajectory size.The models learned with 20 basis functions were kept fixed during scaling.Source: 3) Generalization Performance Analysis · p. 9, Figure 13 · p. 9, Table IV · p. 9

4) Comparison with Recent Methods:

• SC-DMPs achieved the lowest learning and endpoint-generalization errors for both letters, with approximately 1 kB of online storage and adaptation time close to classical DMPs.The comparison covers trajectory errors, endpoint generalization, model size, and adaptation time.Source: 4) Comparison with Recent Methods · p. 9, Table V · p. 10

5) Sensitivity and Stability Analysis:

• All tested hyperparameter settings retained positive RMSE-Geo reductions; default settings reduced error by 45.64% for “Z” and 24.32% for “M”.Kernel-size settings varied least, with standard deviations of 0.22 and 0.25 percentage points for “Z” and “M”.Source: 5) Sensitivity and Stability Analysis · p. 10, Figure 14 · p. 10, Table VI · p. 10
• Bandwidth optimization terminated monotonically in all 18 runs after 3-6 sweeps and approximately 0.42-0.65 s, reducing the objective by 6.26%-35.24%.The finite coordinate-descent procedure guarantees a coordinate-wise solution, not global optimality.Source: 5) Sensitivity and Stability Analysis · p. 10, Figure 15 · p. 10
• SC-DMPs had lower RMSE-Geo than DMPs for every tested demonstration count; with two demonstrations, reductions were 30.86%±8.87% for “Z” and 25.62%±9.75% for “M”.The tested counts were 2, 4, 6, 8, 10, and 12 demonstrations.Source: 5) Sensitivity and Stability Analysis · p. 10, Figure 16 · p. 10

C. Real-Robot Experiments

• The evaluation used plug insertion and removal, sequential dipping, and curved-surface wiping, with four kinesthetic demonstrations per task and a GMR reference trajectory.Demonstrations were resampled to 1000 points.Source: C. Real-Robot Experiments · p. 10, Figure 17 · p. 11
• 5.83%-17.51% lower RMSE and 19.93%-23.86% lower RMSE-TCR than DMPs; SC-DMPs also had the lowest MaxE-TCR, 6.56%-42.01% below DMPs.ProMPs attained lower mean trajectory errors in several comparisons, but higher MaxE-TCR in every task.Source: Figure 18 · p. 11, C. Real-Robot Experiments · p. 11
• High criticality concentrated around plug-socket interaction, tube insertion-withdrawal, and wiping contact, while less constrained transfer segments received lower criticality.The profiles combine independently normalized trajectory variance and interaction stiffness.Source: C. Real-Robot Experiments · p. 11, Figure 17 · p. 11
• Prior smoothing produced nearly unchanged errors without consistent improvement; STR-Net reduced across-task mean RMSE, RMSE-TCR, and MaxE-TCR by 1.38%, 2.43%, and 4.74% versus prior-only.The comparison varied the stiffness-variability prior, its moving-average refinement, and the STR-Net-refined variant while keeping other settings unchanged.Source: C. Real-Robot Experiments · p. 11, Table VII · p. 11

V. CONCLUSION

• SC-DMPs combine interaction stiffness and cross-demonstration trajectory variability in a stage-criticality index to redistribute basis centers and refine bandwidths.The conclusion reports improved reproduction, generalization, and task-critical accuracy while retaining the compact and stable DMP formulation.Source: §V · p. 12
• The formulation targets precision-oriented skills whose critical stages are reflected in interaction stiffness and trajectory consistency; low-stiffness critical stages may require complementary cues.The conclusion names force, vision, object-dynamics, and contextual cues as possible additions.Source: §V · p. 12