Redundancy, self-motion, and motor control

Abstract

Outside the laboratory, human movement typically involves redundant effector systems. How the nervous system selects among the task-equivalent solutions may provide insights into how movement is controlled. We propose a process model of movement generation that accounts for the kinematics of goal-directed pointing movements performed with a redundant arm. The key element is a neuronal dynamics that generates a virtual joint trajectory. This dynamics receives input from a neuronal timer that paces end-effector motion along its path. Within this dynamics, virtual joint velocity vectors that move the end effector are dynamically decoupled from velocity vectors that do not. Moreover, the sensed real joint configuration is coupled back into this neuronal dynamics, updating the virtual trajectory so that it yields to task-equivalent deviations from the dynamic movement plan. Experimental data from participants who perform in the same task setting as the model are compared in detail to the model predictions. We discover that joint velocities contain a substantial amount of self-motion that does not move the end effector. This is caused by the low impedance of muscle joint systems and by coupling among muscle joint systems due to multiarticulatory muscles. Back-coupling amplifies the induced control errors. We establish a link between the amount of self-motion and how curved the end-effector path is. We show that models in which an inverse dynamics cancels interaction torques predict too little self-motion and too straight end-effector paths.

Figure 1

Structure of the model (white boxes with solid borders and arrows) with rough mapping onto involved neuronal structures (gray boxes with dashed borders). The conceptually innovative part is the neuronal dynamics of the virtual joint trajectory, which transforms a virtual end-effector velocity into a virtual joint trajectory and receives back-coupling from the effector level. The inner structure of this dynamics is characterized by decoupling between task-relevant and task-irrelevant combinations of joint velocities.

Figure 2

(Left) Schematic view from above of a participant moving his or her four-joint arm in a horizontal plane. The task consists of moving the pointer tip from a start location (Start1, Start2, Start3) to a target location (Target1 or Target2). Two arm configurations a and b are shown that lead to the same end-effector position. (Right) The set of joint configurations leading to an identical position of the pointer tip forms a two-dimensional manifold (UCM) in the four-dimensional joint space. The figure shows a one-dimensional cut through that manifold within the three dimensions of the original joint space. The linear subspace that is tangent to this manifold at a given joint configuration is spanned by a basis vector that forms one column of the matrix, E (in the full four-dimensional space, a second basis vector forms the second column of E).

Figure 3

(Left) Schematic illustration of how a joint velocity vector can be decomposed into its components in the null-space and range-space of the Jacobian. Only three dimensions are shown. (Right) How the uncontrolled manifold (UCM) in the space of virtual joint angles, λ, structures the vector field of the neuronal dynamics of the joint configuration is illustrated schematically: a weak vector field within the UCM provides little stabilization of the virtual joint configurations that are redundant with respect to the planned end-effector position. In contrast, the vector field outside the UCM provides strong restoring forces pushing the system toward the UCM.

Figure 4

(Top) The left-most three panels show the end-effector paths of the three participants (S1, S2, and S3), for the four different movements. The thin lines are the real end-effector paths in different trials recorded in experiment. The straight lines link the starting position (filled circles, right panel) to the target location (filled squares, right panel) as a guide to the eye. The movements are labeled in the right-most panel, which shows the associated paths generated by the model. (Bottom) End-effector paths from the same three participants and the model for the four conditions are listed in Table 1. Movements 1 and 2, as well as 4 and 5, share initial and target end-effector position but differ in initial joint configuration. The end-effector path curvature depends only very slightly on the starting effector configuration in both experiments and the model.

Figure 5

Left and middle panels show the four joint trajectories (numbered from proximal to distal) as solid lines and the associated virtual joint trajectories as dotted lines for movement 1. The associated end-effector trajectories are shown in the right panel. These data are generated by simulating the model under two conditions differing by the amount of internal joint motion. The left-most panel is based on the reference parameter set (ṡ = 0, s(0) = 0). The simulation shown in the middle has an additional acceleration inserted into the null-space of the end effector (ṡ = 5, s(0) = 0, in equation 2.9).

Figure 6

The evolution in time of end-effector velocity (length of the end-effector velocity vector) is depicted for movement 1. The left panel shows multiple trials from one participant in the experiment. The crosses mark the initial rise and the terminal decrease of the velocity at 1% and 3% of peak velocity, respectively. These event times are used to time-warp trajectories and compute the mean end-effector velocity (see section 3). The velocity profiles generated from the model are shown in the right panel.

Figure 7

The component of the end-effector velocity along the straight line from the starting position to the target is shown as a function of time (solid line) together with the component orthogonal to that direction (dotted line). The left-most three panels show results from three participants, and the right-most shows model simulations. The top row refers to movement 1, the bottom row to movement 4 (compare to Figure 4).

Figure 8

The total amount of self-motion is shown as a function of time during the movement (solid line). This is the length of the joint velocity vector in the null-space of the Jacobian. The dashed line is the length of the orthogonal component of joint velocity lying in range-space. The left three panels of each row are mean results for the three participants (S1, S2, and S3). The right panel of each row shows results generated from the model at the reference parameter set.

Figure 9

Causes of self-motion in the model are analyzed. The amounts of self-motion (solid line) and range-space motion (dashed line) are plotted as time series for movement 3, computed on the left from the real joint velocities and on the right from the virtual joint velocities. The top row is generated at the reference parameter set. In the simulations shown in the middle row, back-coupling from the real to the virtual joint trajectory dynamics was set to zero (ṡ = 0 with initial condition s(0) = 0). In the bottom row, the back-coupling is reinstated, but the muscle impedance is increased by a factor of 10. Results are very similar for the other movements.

Figure 10

Virtual (thin) and real (thick) end-effector paths are shown from simulations in which back-coupling was set to zero (A) and in which the impedance of all muscles was increased tenfold (B). In the first case, self-motion persists (see Figures 9C and 9D), and the end-effector paths are curved (compare to Figure 4), while in the second case, self-motion is cancelled (see Figures 9E and 9F) and the end-effector paths are straight. (C) Results of a simulation are shown, in which an inverse dynamics was emulated by compensating for the interaction torques. This reduces self-motion (see Figure 11B) and makes end-effector paths too straight compared to experiment (see Figure 4). (D) Results of a simulation are shown in which this form of inverse dynamics was combined with eliminating all multiarticular muscles. This strongly reduces self-motion (see Figure 11D) and makes end-effector paths perfectly straight.

Figure 11

Self-motion (solid) and range-space motion (dashed) for a number of different simulations varying aspects of the control level. (A) Coriolis and centrifugal interaction torques set to zero. (B) Inverse dynamics emulated by adding torques to the right side of the biomechanical dynamics that exactly cancel all interaction torques. (C) Multiarticular muscles eliminated by setting off-diagonal elements of muscle joint model to zero and the diagonal elements to 10. (D) Conditions B and C combined.

Figure 12

(Top) End-effector paths in three dimensions obtained from three participants in a pointing task performed with 10 degrees of freedom (thin lines reflect different trials). (Bottom) Range-space and self-motion as a function of time observed while these participants performed the pointing movements (mean across trials). Both components are normalized to the number of dimensions of the respective subspaces.