Tracing curves in the plane: Geometric-invariant learning from human demonstrations

Abstract

The empirical laws governing human-curvilinear movements have been studied using various relationships, including minimum jerk, the 2/3 power law, and the piecewise power law. These laws quantify the speed-curvature relationships of human movements during curve tracing using critical speed and curvature as regressors. In this work, we provide a reservoir computing-based framework that can learn and reproduce human-like movements. Specifically, the geometric invariance of the observations, i.e., lateral distance from the closest point on the curve, instantaneous velocity, and curvature, when viewed from the moving frame of reference, are exploited to train the reservoir system. The artificially produced movements are evaluated using the power law to assess whether they are indistinguishable from their human counterparts. The generalisation capabilities of the trained reservoir to curves that have not been used during training are also shown.

Fig 1. Points on a curve extremizing the distance to a point r.

When these points are interior, the tangent at those points is always perpendicular to the segment joining r to the point itself.

Fig 2. Self-looped configurations of a trained ELM to forecast discrete-time dynamical systems.

Note that Δt in a discrete-time experiment corresponds to the sampling rate.

Fig 3

(a) The figure of eight was provided as the reference curve to the subjects to perform tracing demonstrations on and the training data for the ELM was prepared by selecting the data-points where the speed was non-zero. (b) Omitted regions with the curvature close to zero are marked in green.

Fig 4

S1 tracing data for the figure of eight (the reference curve geometry) showing (a) speed v, (b) curvature κ, (c) Euclidean reparameterisation of tracing data t ↦ p* to yield the pairs {r_t, c(p*)} satisfying (11), and (d) the piecewise linear power law estimated for S1 tracing data {r_t}, yielding critical speed $v_c^{real}$ and curvature ${\kappa }_c^{real}$.

Fig 5. Self-looping ELM: Generalisation of the tracing task is based on the observation set z_t constituting of the signed error evaluated at the current point with respect to the reference geometry c(p*) after the reparameterisation step is done, the Euclidean curvature in moving frame μ′ and the parameter p*—the same structure as the training set data {z˜t} (Eq 19).

The trained ELM predicts the moving frame velocity ${\dot{\mathit{r}}}_t$ that the tracing task requires at time t. The instantaneous spatial frame position of the tracing task is updated to r_{t+ 1} with an additional force field α(c(p*) − r_t) imposed to ensure stability, where α = 0.05 was heuristically chosen.

Fig 6

(a) The ELM self-looping behavior may be observed as going slow at the corners and picking up speed along the straighter segments, plotted against the reference curve geometry and the ELM-induced velocity field around the reference geometry. (b) Piece-wise power law evaluated for ELM data. (c) Comparison of the ELM performance with the human demonstration data based on the piecewise power law.

Fig 7

Tracing performance of ELM trained using demonstration data from S1 on (a) Affine transformed figure of eight, (b) Affine transformed cloverleaves, (c) Ellipse and (d) Spiral.

Fig 8. Standard choices of clustering techniques fail to separate the human and ELM-generated power-laws.

Fig 9. The importance of geometric invariance is demonstrated by the ELM’s failure to generalize even for the same curve, just described with a different speed.

In (a)-(b), the ELM trained using c_ss, successfully traces the figure of eight even at different reparameterisations of n = {1, 2} (see Eq 22). However, in the non-invariant ELM (which was trained using c_pp), in (c)-(d), the generalisation performance declines.