Representation and Control of the Task Space in Humans and Humanoid Robots

From a robotics perspective, the human body is a complex, high degree-of-freedom motor system, with a plethora of sensory inputs, motor outputs, and multiple endeffectors. The challenge for the nervous system is to synergistically coordinate motion and forces from collective muscles, limbs, and joints, to skillfully and efficiently achieve tasks such as reaching, grasping, and walking. The complexity of planning and controlling such skills is daunting, particularly as most of the motions we execute are redundant (having an infinite set of possible motor inputs that can achieve the same motor output; Figure 6.1). How the nervous system chooses its particular motor solution, even for the simplest of tasks, is still an active topic in neuroscience research. Roboticists, however, commonly work with redundant systems, and have developed computational methods for reducing complexity. Given a model of a redundant robotic system, with specified end-effector(s), techniques exist for decomposing the robot’s kinematics and dynamics into the task space (the lower-dimensional subspace of motion and forces directly relevant for task achievement). Orthogonal to the task space is the null space (the subspace of task-irrelevant motion and forces). These decomposition methods enable the planning and control of tasks in a simplified, reduced dimensional space, while still allowing for exploitation of redundancy for achieving secondary tasks or absorbing disturbances.

In neuroscience, there has been previous study of how humans plan and coordinate movement within a task space. As early as 1930, Lashley recognized the importance of kinematic redundancy for motor equivalence (the ability to achieve the same motor function with multiple but unequal movement) [10]. However, it was Bernstein who first formalized the degrees-of-freedom problem: how does the nervous system cope with the indeterminate mapping from goals to actions [4]? Bernstein postulated that the nervous system is capable of functionally “freezing” or tightly coupling joints, a hypothesis that still holds today, particularly in the study of motor learning. When examining eye movement and gaze, Donder observed a specific and fixed eye torsion for every gaze angle, and thus proposed that the nervous system may have a fixed one-to-one mapping from task variables to joint variables (i.e., Donders law). Limb movements, however, have been shown to violate this relatively simple law [28]. In more recent work, Scholz and Schöner demonstrated via repeated sit-to-stand movements that humans consistently controlled their center of mass, as opposed to task-irrelvant and highly variable hand motion [21]. Lockhar and Ting show how a task-level center-of-mass acceleration can be used to predict synergies of muscle activation in the legs of balancing cats [11]. Todorov and Jordan suggest the minimum intervention principle (motor output errors are only corrected when they interfere with task performance) [30].

The goal of this chapter is to bring to light some of the current robotics methodology in task space and redundant control, and to demonstrate their applications toward the modeling of human movement control and redundancy resolution. We first describe, primarily from the robotics point of view, the theory and methodology for representing and controlling task-space motion and forces. We show how these modeling techniques are a useful tool for probing the nervous system’s strategies for motor control, learning, and adaptation. We first introduce the notation and methodology for task-space control in robotics, including inverse kinematics and operational space control. Next, we show evidence of task space control and learning from an experiment in human reaching. We propose a controller for such a movement that exploits an internal model of task-space dynamics, while allowing the disturbance to alter redundant motion. Finally we describe and present solutions to some of the challenges of extending these models from single limbs into whole-body domains, where underactuation and external contacts must be considered.