Identifiable Decomposition of Submovements in Human Hand Trajectories

1 Introduction

• Sub-ID aims to connect computational movement primitives used in robotics with physiologically grounded submovements for human motor-control research.Submovements have been observed across diverse movement types and used to track neuromotor recovery more finely than endpoint outcomes.Source: §1 · p. 1
• Prior approaches either become intractable beyond roughly ten primitives, constrain the search with unmeasured assumptions, or inherit biases while fixing sampling rate and spatial dimension.The described categories include exact search, search-space-reduction heuristics or fixed libraries, and learning from algorithm-generated labels.Source: §1 · p. 1
• Without an identifiability criterion, decompositions may reflect assumptions introduced for tractability, and algorithm failure cannot be separated from mathematical impossibility.Near-collinear basis functions can yield non-unique solutions, while a spurious fit can still have very small residual error.Source: §1 · pp. 1–2
• Sub-ID combines a spatiotemporal kernel-correlation criterion with adaptive-ridge regularisation, steering optimisation toward distinguishable primitive pairs.Spatially divergent directions can preserve amplitude identifiability despite temporal overlap, while onset detection remains governed by temporal overlap.Source: §1 · p. 2, Figure 1 · p. 2

2.1 Parameter recovery on synthetic data (unimodal) distributions

• Sub-ID reproduces ground-truth duration, amplitude, basis-count, and refractory-period distributions across K=3, 8, and 15, while baselines shift, overfit, or fail.At K=15, Sub-ID’s maximum kernel correlation is 0.45, with only 3.00% of overlaps above 0.8.Source: Figure 2 · p. 3, §2.1 · p. 4

2.2 Generalisation to heterogeneous (bimodal) distributions

• At K=15, Sub-ID preserves the separation of bimodal duration, amplitude, and refractory-period distributions, with basis counts varying between 14 and 16.Recovered modes do not merge, and overlaps above 0.8 are rare.Source: Figure 3 · p. 5, §2.2 · p. 6

2.3 Graceful failure in the non-identifiable regime

• Under extreme overlap at K=15, Sub-ID returns fewer primitives as correlation rises toward 1, surfacing non-identifiability rather than a spurious full-K decomposition.The adaptive ridge penalty grows with Gram-matrix conditioning and favors low-correlation solutions.Source: Figure 3 · p. 5, §2.3 · p. 6

2.4 Real Data: 2D long and short horizon tasks

• On 2D PushT, Sub-ID extracts 14.62 primitives with mean overlap 0.0549 and only 2.17% critical overlaps; SSSUMO extracts 49.8 primitives with maximum overlap 1.0.Sub-ID maintains millimetric reconstruction precision on Letters and PushT, while baseline methods show overfitting, underfitting, or degradation on long sequences.Source: §2.4 · pp. 6–7, Figure 4 · p. 7

2.5 Real Data: 3D long and short horizon tasks

• On Moving and PushT 3D, Sub-ID’s relative position error is 0.95% and 2.12%, while SSSUMO’s is 14.22% and 28.64%, respectively.On PushT 3D, Sub-ID has mean overlap 0.0520 and 5.97% critical interactions; Scattershot was excluded because it fails to converge on long, higher-dimensional tasks.Source: §2.5 · pp. 7–8, Table 1 · p. 8

3 Discussion

• Temporal aliasing is an information-theoretic limit: closely placed primitives can become observationally equivalent, while kernel correlation and Gram-matrix conditioning diagnose recoverability.Source: §3 · p. 8
• Heuristic peak detection restricts the search and adaptive ridge discourages collinear primitives; these designed biases reduce exponential optimization cost, with failure observable as ρi,j approaches 1.Source: §3 · pp. 8–9
• Sub-ID raises the temporal ceiling for decomposing longer actions with many superimposed submovements, creating potential to study human actions with robot imitation-learning frameworks.Source: §3 · p. 9

3.1 Limitations

• Because the real-data diagnostic uses estimated directions, cancelling bases could yield falsely reassuring low correlation; until validated against synthetic ground truth, it is necessary but not sufficient evidence of identifiability.Source: §3.1 · p. 9
• SSSUMO was evaluated as released, so whether retraining on Sub-ID-generated labels would narrow the reported gap remains an open fairness question.Source: §3.1 · p. 9

3.2 Future Work

• Timing and duration come from the coordinate-invariant speed trace, while amplitude is fit in position to limit long-horizon drift, at the cost of a Cartesian final fit whose appropriate frame remains open.Source: §3.2 · p. 9
• Under white velocity-command noise, position noise is Brownian with variance growing over time; Sub-ID’s position fit removes drift but is not maximum-likelihood under that noise model.A joint model of measurement and motor noise is proposed as future work.Source: §3.2 · pp. 9–10

3.3 Conclusion

• Sub-ID recovers ground-truth parameter distributions where exact search is intractable and scales to real 3D, long-horizon movements that existing methods cannot decompose without drift or over-segmentation.Source: §3.3 · p. 10

4 Methods

• Sub-ID represents planar or 3D trajectories as temporal integrations of scaled basis functions, with spatial scales encoding displacement and basis parameters encoding onset, duration, and shape.The method seeks parameter and scale vectors whose combination decomposes trajectories in both velocity and spatial domains.Source: §4 · p. 10

4.1 Basis functions description

• The method is basis-agnostic and demonstrates flexibility with three-parameter Minimum Jerk and five-parameter bounded Log-normal functions.The cited prior work identifies Minimum Jerk with broad voluntary movements [4] and support-bounded Log-normal functions as providing greater descriptive flexibility [51].Source: §4.1 · p. 10

4.2 Step-by-Step Submovement Decomposition Algorithm

• The algorithm initializes primitive timing from coordinate-invariant tangential speed, but recovers spatial scales by fitting position, making the global objective coordinate-dependent.The speed-based stages are invariant to workspace geometry and reference frame; joint-angle or object-centered applications require reformulating the global objective.Source: §4.2 · p. 12

Grow (speed): Greedy Residual Refinement

• Greedy residual refinement iteratively adds lower-amplitude primitives to explain speed-profile features missed by peak-based initialization, then recalculates amplitudes until the residual meets its stopping threshold.Candidates are selected around the largest residual deviation subject to anti-nesting and positivity constraints.Source: Grow (speed): Greedy Residual Refinement · p. 13
• Adaptive ridge regularization activates when temporal-basis multicollinearity threatens stability, shrinking redundant amplitudes and avoiding post-hoc primitive merging.The penalty is zero when the basis functions are sufficiently independent and is chosen to keep the condition number at or below the empirically set threshold of 100.Source: Grow (speed): Greedy Residual Refinement · p. 14

4.3 Identifiability and Spatiotemporal Kernel Correlation

• Submovement identifiability depends on distinguishability: spatial orthogonality makes total correlation zero despite temporal overlap, while aligned directions recover temporal correlation.As basis functions become collinear, the Gram-matrix condition number diverges; under Sub-ID’s linear-Gaussian model, amplitude variance scales with that conditioning, and perfect correlation makes recovery impossible.Source: §4.3 · pp. 14–15
• Sub-ID sets its adaptive ridge parameter analytically from the Gram matrix to favor well-conditioned decompositions, but formal identifiability analysis covers amplitude estimation with temporal parameters fixed.The diagnostic evaluates the Gram matrix’s condition number against a maximum threshold; joint identifiability of onset times, durations, and asymmetries is not addressed formally.Source: §4.3 · pp. 15–16

Synthetic Data Experiments

• Synthetic tests used human-derived duration, amplitude, and refractory-period distributions across K = 3, 8, and 15, with unimodal and bimodal parameter modes.Trajectories had known basis parameters and sequential primitives with high temporal overlap; bimodal parameters combined nominal values with scales modified by 50%.Source: Synthetic Data Experiments · pp. 16–17, §4.4 · p. 16
• When refractory periods were much shorter than durations, temporal aliasing made adjacent primitives nearly linearly dependent, preventing unique decomposition without non-physiological constraints.In that limit, two highly collinear primitives and one larger primitive can produce identical velocity profiles.Source: Synthetic Data Experiments · p. 17

Real Data Experiments

• Real-data protocols covered short-horizon public 2D and 3D tasks and custom long-horizon planar and spatial PushT tasks.Short tasks included handwriting and object moving; the planar PushT task required continuous navigation around the block, while spatial PushT permitted lifting and added Z-axis motion.Source: Real Data Experiments · pp. 17–18

Statistical Analysis and Evaluation Metrics

• Evaluation metrics assessed spatial and velocity reconstruction fidelity, primitive rate and temporal overlap, and spatiotemporal kernel correlation as an indicator of ill-conditioning.Position error measured trajectory RMSE; velocity metrics captured explained variance and absolute error, while overlap metrics included mean, maximum, and the share above 0.8.Source: Statistical Analysis and Evaluation… · p. 18

4.5 Implementation and Adaptation of Comparative Methods

• The comparison evaluated Sub-ID against Scattershot, Gowda et al., and SSSUMO using official baseline repositories, standardized boundary conditions, and the same hardware.The stated setup used an MSI Katana GF66 laptop with an Intel Core i7 processor.Source: §4.5 · p. 18

Scattershot (Jason Friedman implementation)

• Scattershot stopped upon reaching 2% velocity reconstruction error or a basis-function cap, and selected the lowest-error result from 20 inference runs per trajectory.The cap prevented indefinite searches on complex trajectories; the 20-run configuration was chosen to give the implementation more opportunities to find a low-error representation.Source: Scattershot (Jason Friedman… · p. 19

Search-Space Reduction (Gowda implementation); SSSUMO

• Gowda et al.’s search-space reduction was generalized to N dimensions with constrained iterative fitting, while SSSUMO used pretrained models on data resampled at 60 Hz.Gowda’s method used ridge regression for spatial amplitudes and SLSQP for onset and duration, with extraction stopping at the 2% error target or negligible marginal improvement; SSSUMO did not allow a dynamic reconstruction-error threshold.Source: Search-Space Reduction (Gowda… · p. 19, SSSUMO · p. 19

Additional information

• The Sub-ID algorithm is listed with its online repository.https://github.com/AdrianPrados/Sub-IDSource: Additional information · p. 22
• The Character Trajectories Data Set is listed from the UCI Machine Learning Repository.https://archive.ics.uci.edu/dataset/175/character+trajectoriesSource: Additional information · p. 22
• The Scattershot algorithm is listed with Jason Friedman's official repository.https://github.com/JasonFriedman/submovementsSource: Additional information · p. 22
• The Search-Space Reduction method is listed with a repository based on Gowda et al.'s implementation.https://github.com/sgowda/decompose_submovementsSource: Additional information · p. 22
• The SSSUMO method is listed with its repository.https://github.com/dolphin-in-a-coma/sssumoSource: Additional information · p. 22

S1.1 Detect (speed): Heuristic Peak-Based Initialization

• Initialization detects peaks above 5% of maximum tangential velocity, estimates durations from peak widths, and solves nonnegative weights for the preliminary basis.The peak-based pass returns initial primitive parameters and scaling weights.Source: S1.1 Detect (speed): Heuristic… · p. 24

S1.2 Grow (speed): Greedy Residual Refinement

• Greedy refinement repeatedly targets the largest residual, tests candidate duration and skewness, applies proximity and anti-nesting filters, then updates weights with nonnegative least squares.It continues while residual error exceeds tolerance and the active primitive count remains below the maximum.Source: S1.2 Grow (speed): Greedy Residual… · p. 25

S1.3 Human-Grounded Synthetic Dataset Generation

• Synthetic trajectories sample onset, duration, overlap latency, amplitude, and spatial direction from human-analysis reference parameters.The generation procedure supports unimodal or bimodal distributions and outputs onset, offset, and spatial-scale parameters.Source: S1.3 Human-Grounded Synthetic Dataset… · p. 26
• For K = 3, 8, and 15, reconstructed synthetic velocity profiles closely track ground truth; at K = 3, recovered submovements practically coincide with the generating primitives.The example decompositions show the method’s recovery against known synthetic ground truth.Source: S1.3 Human-Grounded Synthetic Dataset… · p. 26, Figure 1 · p. 26

S2 Comparative Analysis of Basis Functions: Minimum Jerk vs. Lognormal Profiles

• Sub-ID uses the same iterative extraction flow with Minimum Jerk and Lognormal profiles, allowing the basis choice to vary with task and user needs.Minimum Jerk is parameterized by two temporal variables, whereas Lognormal profiles can model asymmetry through skewness and optionally include kurtosis.Source: S2 Comparative Analysis of Basis… · p. 27

A. Dimensionality of the Search Space (Grid Search Complexity):

• The initialization search expands from one dimension for Minimum Jerk to two for standard Lognormal and three when kurtosis is optimized, increasing cost while yielding a higher-fidelity initial fit.The Lognormal search spans duration and bounded skewness, with kurtosis adding another dimension.Source: A. Dimensionality of the Search Space… · p. 27

B. Multicollinearity and Sensitivity to Regularization:

• Minimum Jerk bases are rigid and symmetric, tending to produce a well-conditioned, largely orthogonal basis matrix that tolerates a higher Ridge penalty such as α = 0.1.Their shape depends on start and end times, so multiple bases are rarely placed in the exact same temporal window.Source: B. Multicollinearity and Sensitivity… · p. 27
• Lognormal bases can stack within the same time window, making the basis matrix ill-conditioned; adaptive Ridge detects this and typically selects α ≈ 5×10^-5.Without controlled regularization, co-located bases can receive large opposing amplitudes; a stronger penalty may flatten the signal.Source: B. Multicollinearity and Sensitivity… · p. 27
• 2.7 s for Minimum Jerk versus 4.0 s for Lognormal on the same 3D task; both achieved approximately 2×10^-4 position-fit RMSE.The comparison used K = 9 Minimum Jerk bases and K = 10 Lognormal bases.Source: B. Multicollinearity and Sensitivity… · p. 28, Figure 3 · p. 29

S3 Real Data Acquisition setup

• Eight Prime x22 cameras recorded the tasks through ROS1 at 100–120 Hz, with the data interpolated to a uniform 100 Hz frequency.The motion-capture system was used for the Long-Horizon 2D and 3D PushT datasets.Source: S3 Real Data Acquisition setup · p. 29, Figure 4 · p. 30
• The recordings captured tool end-effector position and the T-shaped block’s full pose, with multiple block markers used to reduce positional loss during 3D occlusions.The setup mounted the motion-capture system on the tool and block rather than on participants.Source: S3 Real Data Acquisition setup · p. 29
• Data came from two participants not involved in development; records contained motion trajectories without identifying information or a global spatial reference.The paper reports that no personal data were collected.Source: S3 Real Data Acquisition setup · p. 29

• 119.3 ms synchrony error, versus 151.4 ms for the Ground Truth, suggests the optimizer did not add high-frequency noise or mathematical artifacts to fit the data.Source: S4.1 Bimodal distribution metrics · p. 30, Table 1 · p. 30
• 0.0492 s separation at r = 0.90 and 0.0352 s at r = 0.95 correspond to a maximum spatiotemporal correlation of ρ = 0.6595.Source: S4.1 Bimodal distribution metrics · p. 30
• 0.171 mean overlap remains below the Ground Truth reference of 0.201, while critical interactions (> 0.8 overlap) are 2.90%, comparable to the unimodal case.Source: S4.1 Bimodal distribution metrics · p. 30

• The failure-case metrics show overall reconstruction degradation relative to the Ground Truth, with severe artifacts and kinematic distortions introduced to force the fit.Source: S4.2 Failure case metrics · p. 30, Table 2 · p. 31
• 0.534 mean overlap, up from the 0.251 reference, accompanies a rise in critical interactions (> 0.8 overlap) to 33.90%.Source: S4.2 Failure case metrics · p. 30
• When theoretical spatiotemporal resolution limits are exceeded, the inverse problem becomes inherently ill-conditioned and exact recovery of latent parameters is impossible.Source: S4.2 Failure case metrics · pp. 30–31

S5 Mathematical Details: Identifiability Analysis

• Table 3 summarizes the statistical analysis and evaluation metrics used to assess kinematic fidelity and structural identifiability.Source: Table 3 · p. 31, S5 Mathematical Details:… · p. 31

S5.1 Kernel Correlation and Identifiability Limit

• For collinear submovements with the same duration and shape, temporal correlation depends on their onset shift; near-unit correlation makes the decomposition ill-conditioned, motivating a threshold for minimum resolvable separation.Source: S5.1 Kernel Correlation and… · p. 31

S5.2 Scale Invariance for Minimum-Jerk Kernels

• Correlation depends only on ξ = ∆t/T, so the minimum resolvable onset separation scales linearly with duration; longer primitives require a larger gap to remain identifiable.Source: S5.2 Scale Invariance for Minimum-Jerk… · p. 32

S5.3 Spatiotemporal Generalization

• When primitive directions are orthogonal, their spatiotemporal correlation is zero regardless of temporal overlap, extending identifiability beyond the one-dimensional temporal bound.Source: S5.3 Spatiotemporal Generalization · p. 32

S5.4 Algorithmic Implementation in Multi-Dimensional Spaces

• Sub-ID initializes timing from scalar speed, builds a shared temporal basis, then estimates spatial directions and amplitudes with independent ridge regressions for each coordinate.The approach is described as computationally efficient and implicitly captures the spatial angles governing identifiability.Source: S5.4 Algorithmic Implementation in… · p. 32