1Computational and Biological Learning Lab, Department of Engineering, University of Cambridge, Cambridge,
2Department of Experimental Psychology, University of Cambridge, Cambridge, United Kingdom
3Bernstein Center for Computational Neuroscience, Albert-Ludwigs Universitaet Freiburg, Freiburg, Germany
Humans regularly interact with objects with internal degrees of freedom from carrying a cup of coffee to folding a shirt. While objects with no internal degrees of freedom can be regarded as a fixed extension of our limbs, non-rigid objects pose a more complex control problem. In recent years stochastic optimal feedback control has emerged as a framework for human motor coordination. Optimal control has been used to explain average movement trajectories as well as trial-by-trial variability in a wide range of motor behaviours, such as obstacle avoidance and bimanual coordination. In this study, we investigated whether the optimal control framework can be extended to object manipulation with internal degrees of freedom and whether it can account for the complex behaviour necessary to control such objects. We used a virtual reality set-up together with a vBOT robotic interface to simulate the dynamics of a virtual object attached to the subject’s hand (for details see Koerding et al., 2004). Subjects started with both hand and object aligned in the starting position and were required to move both the hand and object to the target position within a certain time window that was reduced during training down to 0.8 - 1.2 s. As a prototypical object with internal degrees of freedom, the object was simulated as a damped mass, attached to the hand by a spring. However, we created six different complex dynamic objects by introducing anisotropies for the mass, viscosity and spring constant matrices (such as x-y dependencies and a velocity-dependent rotational force field applied to the mass). Subjects (n = 6) learned to control the six different mass-spring objects until they achieved 25% correct trials. Positional data and forces were recorded at 1000 Hz and the last 25 successful trials were analysed. We used an optimal control model based on the model proposed by Todorov & Jordan (2002) and included the dynamics of the different mass-spring-damper systems. Our optimal control model predicted complex hand trajectories such as loops and s-shaped curves, which deviate substantially from the straight hand paths seen during normal reaching movements. Experimental performance of subjects was well fit explaining 83 - 97% of the variance in all conditions. The results suggest that the framework of optimal control can be extended to manipulation of objects with internal degrees of freedom and underline the generality of the optimal control framework as a theory of motor coordination.