Implicit learning of object geometry by reducing contact forces and increasing smoothness

Moving our hands smoothly is essential to execute ordinary tasks, such as carrying a glass of water without spilling. Past studies have revealed a natural tendency to generate smooth trajectories when moving the hand from one point to another in free space. Here we provide a new perspective on movement smoothness by showing that smoothness is also enforced when the hand maintains contact with a curved surface. Maximally smooth motions over curved surfaces occur along geodesic lines that depend on fundamental features of the surface, such as its radius and center of curvature. Subjects were requested to execute movements of the hand while in contact with a virtual sphere that they could not see. We found that with practice, subjects tended to move their hand along smooth trajectories, near geodesic pathways joining start to end positions, to reduce contact forces with constrained boundary, variance of contact force, tangential velocity profile error and sum of square jerk along the time span of movement. Furthermore, after practicing movements in a region of the sphere, subjects executed near-geodesic movements, less contact forces, less contact force variance, less tangential velocity profile error and less sum of square jerk in a different region. These findings suggest that the execution of smooth movements while the hand is in contact with a surface is a means for extracting information about the surface’s geometrical features.

💡 Research Summary

The paper investigates how smooth hand movements and the reduction of contact forces while the hand is constrained to a curved surface can serve as a mechanism for implicitly learning the surface’s geometry. Building on the classic minimum‑jerk model of Flash and Hogan, the authors extend the optimization problem to a spherical constraint using Lagrange multipliers. Numerical solutions obtained with MATLAB’s BVP4C reveal that the optimal trajectory lies on the geodesic (the shortest path on a sphere) connecting the start and end points, and that the tangential velocity follows a bell‑shaped profile identical to that observed in free‑space movements.

In the experimental part, 22 right‑handed participants interacted with a haptic device (PHANToM 3.0) that rendered an invisible virtual sphere of 20 cm radius. Participants were blindfolded and asked to move between three memorized targets placed at the vertices of an equilateral triangle intersecting the sphere. Training consisted of 300 repetitions on the right‑hand side of the triangle, while testing involved 60 repetitions on the opposite side. Position data were sampled at 100 Hz and contact forces at 1 kHz.

Performance metrics included average path deviation (APD) from the theoretical geodesic, average contact force (ACF) directed toward the sphere’s centre, contact‑force variance (CFV), velocity‑profile error, and the integral of squared jerk. Early trials showed non‑geodesic, irregular paths and asymmetric velocity profiles. With practice, APD decreased by about 12 % in training and 10 % in testing (p < 0.01), indicating that participants progressively aligned their hand paths with the geodesic. Simultaneously, ACF fell by roughly 29 % during training and 39 % during testing (p < 0.001), reflecting reduced penetration into the virtual boundary. Importantly, χ² analyses demonstrated that changes in APD and ACF were statistically independent, suggesting that smoothness optimization and force minimization are driven by distinct neural control processes that both contribute to surface learning. CFV also showed significant reductions, confirming that participants learned to predict and stabilize the contact dynamics.

The key insight is that, when constrained to a curved surface, humans implicitly acquire a representation of the surface’s curvature and centre by progressively producing minimum‑jerk, geodesic‑aligned movements and by minimizing contact forces. This dual adaptation occurs not only for the practiced region but also generalizes to untrained regions, supporting the hypothesis that smoothness is not merely an efficiency criterion but a learning strategy for extracting geometric information. The findings have implications for haptic rehabilitation, tactile interfaces, and human‑robot interaction, where leveraging the natural tendency toward smooth, low‑force movements could facilitate implicit learning of complex environmental constraints. Future work should explore other surface geometries and multi‑constraint scenarios to broaden our understanding of this sensorimotor learning mechanism.