Movement Planning A Robotic Models

Computationally, the control of a robotic arm by means of a computer and the control of a biological arm by the brain share many similarities. Accordingly, investigators studying the neural control of limb movements have often attempted to use the algorithms that have been used successfully by roboticists as a template for understanding biological motion. In fact, robotic models of movement planning have had considerable influence on models for neural control of movement.

Computer algorithms for movement planning evolve in a serial fashion, with one stage devoted to movement kinematics and a second stage devoted to movement kinetics. Consider the kinematic aspects of movement planning first. In this module, the first step is to define the location of the target of the motion. This specification is said to be in extrinsic (or task) coordinates; for example, the x, y, and z coordinates with reference to some origin. Generally, the extrinsic coordinates do not define a unique configuration of robot or human arms because these arms have more than three degrees of freedom. For example, the human arm has 4 degrees of freedom, counting only the shoulder and elbow, and 7 degrees of freedom if the wrist is included. Therefore, there is not a unique solution; more than one posture is compatible with the target location. A unique solution is imposed by introducing some constraints. For example, it could be desired that the limb posture be as far away as possible from the limits of motion at each of the joints, or it could be desired to limit the amount of motion at each joint.

Two other things also need to be specified in such robotically inspired models before the movement can be constructed: the spatial and the temporal evolution of the movement. Thus, it has been suggested that movements are planned to follow straight lines with a velocity profile that is bell shaped. According to this suggestion, movement planning would involve the following sequence of steps, organized and executed serially: localization of a point in extrinsic space, specification of a rectilinear hand trajectory, and the computation of the joint angles (in intrinsic space) that correspond to each point in extrinsic space. As stated previously, there is not a unique set of joint angles for any given hand position, and a unique solution is obtained by invoking kinematic constraints.

The last step in these models of movement planning involves kinetics. Specifically, necessary joint torques are derived from the desired kinematics (joint angles). This step is commonly referred to as inverse dynamics (Fig. 3) because the forces are derived from motion rather than the opposite, as would be implied by cause and effect. For a biological system, there would be one last step—namely, the partitioning of the torque across the redundant actuators (the motor units), taking into account the musculoskeletal properties described in the previous section.

B. Trajectory Planning of Human Limb Motion

The robotic model has been influential in guiding experimentation and it has also received considerable support. For example, it is generally found that for arm movements from one point in space to another,

Figure 3 Hybrid model of feedforward and feedback control. The desired kinematics of a planned movement are transformed into motor commands (kinetics) through the inverse model. The motor commands are transformed by the musculoskeletal system into the actual movement kinematics. These are sensed by afferents to provide sensory feedback. An efference copy of the motor commands is transformed into the expected kinematics via a forward model. The difference between the expected kinematics and the actual kinematics (provided by sensory feedback) provides an error signal used to update the motor commands.

Figure 3 Hybrid model of feedforward and feedback control. The desired kinematics of a planned movement are transformed into motor commands (kinetics) through the inverse model. The motor commands are transformed by the musculoskeletal system into the actual movement kinematics. These are sensed by afferents to provide sensory feedback. An efference copy of the motor commands is transformed into the expected kinematics via a forward model. The difference between the expected kinematics and the actual kinematics (provided by sensory feedback) provides an error signal used to update the motor commands.

the wrist follows a path that is close to straight and the wrist's speed is well approximated by a bell-shaped profile.

Furthermore, the results of psychophysical studies in humans and electrophysiological studies in nonhuman primates have been interpreted from this perspective. For example, there has been considerable work dealing with errors in pointing to targets in three-dimensional space. In these experimental paradigms, a target is presented visually or proprioceptively (by moving the arm into a given posture), and the subject is instructed to move the arm to the target in the absence of visual guidance. Both constant and variable errors in this type of task are usually very reproducible, and these errors have been interpreted as arising from errors in kinematic coordinate transformations. Specifically, it has been proposed that errors arise because of approximations in transforming target location in extrinsic, eye-centered coordinates into intrinsic, shoulder- or hand-centered coordinates.

Also, single unit recordings from parietal and frontal cortical areas have been related to kinematic parameters defined in different frames of reference. In general, cortical neurons whose activity is related to limb movement are tuned to the direction of movement (a kinematic parameter). Moreover, in some areas, direction is defined in eye-centered coordinates (i.e., a direction relative to the foveal direction of gaze). In other instances, direction appears to be specified in arm-centered coordinates (i.e., a direction relative to the orientation of the arm).

Thus, considerable evidence supports the first stage of the robotically inspired models—a transformation of a specification of target location in extrinsic space (or, more precisely, in retinotopic coordinates) into a specification of target location in joint coordinates. The question then arises: Is there also a kinematic plan of the trajectory? Evidence has been offered in support of such a supposition. Essentially, the velocity profiles of movements are highly repeatable and predictable. Specifically, if movements are relatively straight, bell-shaped (unimodal) velocity profiles are observed. If a curved trajectory is produced intentionally during drawing movements or when a curved path is necessary to avoid an obstacle, there is a precise relation between speed and the trajectory's curvature: The smaller the radius of curvature, the slower the speed. Taken at face value, these observations imply a precise plan of the moment-to-moment evolution of a movement's kinematics.

However, such a kinematic plan of movement does not require that kinematics be determined first without regard to the kinetic requirements. In fact, extensive modeling studies undertaken by Mitsuo Kawato and colleagues led to the opposite conclusion. They began with the observation that wrist trajectories for pointing movements are usually gently curved, and they were able to predict this curvature by assuming that the hand followed a path according to a "minimum torque change'' criterion. Thus, kinetic constraints determined the kinematic plan of movement. Our own studies on limb posture at the end of pointing movements also support this interpretation. When subjects made arm movements from different starting locations, the arm's posture at the target depended on the starting location. This variation in the final posture could be predicted by invoking a kinetic criterion, one related to minimizing energy expenditure during the movement.

Thus, it seems clear that movement trajectory is regulated, even when it does not need to be (such as during pointing movements in the absence of obstacles). However, it is not clear that the observed regularities occur because a particular trajectory is planned explicitly, since they could also occur implicitly from regulation of the movement's kinetics.

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