Data-driven modelling of low-dimensional dynamical structures underlying complex full-body human movement

One of the central challenges in the study of human motor control and learning is the degrees-of-freedom problem. Although the dynamical systems approach (DSA) has provided valuable insights into addressing this issue, its application has largely been confined to cyclic or simplified motor movements. To overcome this limitation, the present study employs neural ordinary differential equations (NODEs) to model the time evolution of non-cyclic full-body movements as a low-dimensional latent dynamical system. Given the temporal complexity full-body kinematic chains, baseball pitching was selected as a representative target movement to examine whether DSA could be extended to more complex, ecologically valid human movements. Results of the verification experiment demonstrated that the time evolution of a complex pitching motion could be accurately predicted (R^2 > 0.45) using the NODE-based dynamical model. Notably, approximately 50% of the variance in the latter half of the pitching motion was explained using only the initial ~8% of the temporal sequence, underscoring how subsequent movement evolves from initial conditions according to ODE-defined dynamics in latent space. These findings indicate the potential to extend the DSA to more complex and ecologically valid forms of human movement.

💡 Research Summary

The paper tackles the longstanding degrees‑of‑freedom (DoF) problem in human motor control by introducing a data‑driven dynamical systems framework that can handle complex, non‑cyclic full‑body movements. Traditional dynamical systems approaches (DSA) have been successful in describing cyclic tasks such as bimanual coordination, but they rely on predefined collective variables and often assume periodicity or stationarity, limiting their applicability to realistic athletic actions. To overcome these constraints, the authors employ Neural Ordinary Differential Equations (NODEs), a recent deep‑learning technique that learns the governing differential equations directly from data, thereby eliminating the need for hand‑crafted state variables.

The experimental domain is baseball pitching, a movement that proceeds sequentially from the lower limbs through the trunk to the upper limbs and exhibits high kinematic redundancy yet remarkable reproducibility. Motion capture data were collected from eight collegiate pitchers (200 Hz, 15 joints, 3‑D positions) without temporal normalization, preserving the natural time course from the onset of the stride to ball release and, in some trials, the follow‑through phase. Each trial was segmented using biomechanical landmarks (maximum knee height and peak wrist velocity) to define a consistent movement window.

The proposed architecture consists of four main components: (1) a Transformer encoder that compresses the high‑dimensional time series into a low‑dimensional latent representation; (2) a latent‑space NODE that treats the initial latent state (derived from the first ~8 % of the motion) as the initial condition of an ODE defined by a multilayer perceptron (MLP) vector field; (3) an MLP decoder that maps the integrated latent trajectory back to the original joint‑space; and (4) a VAE‑style loss combining reconstruction error and a KL‑divergence term to regularize the latent distribution toward a standard normal. The latent space is three‑dimensional, meaning the full 45‑dimensional kinematic data are effectively captured by three collective variables learned directly from the data.

Training employed 10‑fold cross‑validation per pitcher, with Adam optimization, a batch size of 32, and 1,500 epochs. Model hyper‑parameters (e.g., three Transformer layers, eight attention heads, 256 model dimension, three‑layer MLP vector field with 128 hidden units) were chosen to balance expressiveness and computational tractability. Performance was evaluated using frame‑wise root‑mean‑square error (RMSE) and the coefficient of determination (R²) across the predicted time series, compared against a baseline that predicts each frame by the mean of the training data.

Results show that the NODE‑based model achieves an average R² > 0.45 over the entire pitching motion, indicating substantial predictive power despite the high variability inherent in the later phases of the movement. Notably, using only the initial 8 % of the temporal sequence, the model explains roughly 50 % of the variance in the subsequent 92 % of the motion, confirming that the latent ODE flow captures the deterministic evolution of the movement from early initial conditions. The time‑resolved R² curves reveal that prediction accuracy remains robust even in the post‑release phase where inter‑trial variability is greatest, outperforming the statistical baseline by a wide margin.

The authors interpret these findings as empirical support for the DSA hypothesis that complex motor behavior can be reduced to low‑dimensional dynamical laws. By learning the differential equations directly, NODE sidesteps the need for manually defined collective variables, offering a scalable pipeline for other sport‑specific or rehabilitation movements. The study also demonstrates that a small set of latent variables (three in this case) suffices to encode the essential coordination patterns of a full‑body pitch, aligning with theoretical notions of motor synergies and uncontrolled manifolds.

Limitations include the fixed latent dimensionality (three), which may be insufficient for more intricate tasks, and the lack of explicit testing under external perturbations or across a broader population of athletes. Future work could explore adaptive latent dimensionality, incorporate force or EMG data, and assess generalization to other non‑cyclic actions such as gymnastics or dance. Moreover, integrating the learned latent dynamics into real‑time feedback or assistive devices could translate the methodological advances into practical coaching or injury‑prevention tools.

In summary, this paper presents a compelling proof‑of‑concept that Neural ODEs can serve as a powerful, data‑driven bridge between high‑dimensional biomechanical recordings and low‑dimensional dynamical system representations, extending the reach of dynamical systems theory into ecologically valid, full‑body human movements.