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Volume 17, Number 18, Issue of September 15, 1997 pp. 7119-7128
Copyright ©1997 Society for NeuroscienceObstacle Avoidance and a Perturbation Sensitivity Model for Motor Planning
Philip N. Sabes and Michael I. Jordan Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
ABSTRACT
A novel obstacle avoidance paradigm was used to investigate the planning of human reaching movements. We explored whether the CNS plans arm movements based entirely on the visual space kinematics of the movements, or whether the planning process incorporates specific details of the biomechanical plant to optimize the trajectory plan. Participants reached around an obstacle, the tip of which remained fixed in space throughout the experiment. When the obstacle and the start and target locations were rotated about the tip of the obstacle, the visually specified task constraints retained a rotational symmetry. If movements are planned in visual space, as indicated from a variety of studies on planar point-to-point movements, the resulting trajectories should also be rotationally symmetric across trials. However, systematic variations in movement path were observed as the orientation of the obstacle was changed. These path asymmetries can be accounted for by a class of models in which the planner reduces the likelihood of collision with the obstacle by taking into account the anisotropic sensitivity of the arm to external perturbations or uncertainty in joint level control or proprioception. The model that best matches the experimental results uses planning criteria based on the inertial properties of the arm.
Key words: human psychophysics; visuomotor control; motor planning; reaching; obstacle avoidance; optimal control; theoretical model
INTRODUCTION
Many of the motor tasks facing the CNS are characterized by extrinsic (visual space) constraints, which are insufficient to identify a motor response uniquely. When reaching to a visual target, for example, the CNS must convert visual information such as the target position and the location of obstacles in the workspace into one of the infinite possible motor sequences that would attain the goal. One solution, the visual planning model, is to begin by treating the arm as a single point in space, i.e., an end point such as the index finger, and specifying an extrinsic trajectory for that point. This strategy is attractive, because it allows the CNS to plan movement in the space where tasks are typically defined, leaving the kinematic and dynamic details of the biomechanical plant to a subordinate controller. On the other hand, these details could prove useful to the planner, providing information to satisfy the task constraints in a more optimal manner.
The visual planning model derived from observations of invariances in the extrinsic kinematics of pointing movements (Bernstein, 1967
; Morasso, 1981
One reason for the inconclusive nature of these studies is that they have mostly been based on an overly restrictive set of tasks: simple point-to-point reaching movements. In this paper, we explore a more complex task, that of reaching around a visually displayed obstacle. From trial to trial the obstacle tip remained fixed in space, whereas the obstacle, the initial position, and the target were all rotated around the fixed obstacle tip (Fig. 1). As a result, the task constraints were rotationally symmetric across trials.
Fig. 1. Obstacle rotation experiment. A, Position of the participant relative to the virtual image. The triangular obstacle and start and target circles are shown for two different trials, one in black and one in gray. The tip of the obstacle, which remains fixed throughout the experiment, is chosen to lie at a particular location in the participant's joint coordinates,= (
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This obstacle rotation design has a dual purpose. First, we want to determine whether this more complex set of movements displays extrinsic kinematic invariances, i.e., whether these obstacle avoidance trajectories obey the rotational symmetry of their task constraints. Systematic differences in the trajectory as a function of obstacle orientation would suggest that movement planning is not based entirely on the extrinsic coordinate frame but, rather, takes information such as the kinematic or dynamic properties of the arm into account. Second, the obstacle rotation design allows a quantitative analysis of any such trajectory variation, from which we can hope to identify the operative planning criteria.
We show that participants did exhibit systematic variations in movement trajectories as a function of obstacle orientation. To account for these variations, we propose a class of models based on minimizing the sensitivity of the arm, with respect to the obstacle, to position uncertainty or force perturbations. Finally, we show that one of those models, that based on the inertia of the arm, accounts best for the observed data.
MATERIALS AND METHODS
Apparatus. Participants were seated at the virtual visual feedback system shown in Figure 2. Participants wore a strap to ensure that their right shoulder was fixed in space and rested their right arm on a table at shoulder height. Also, the right wrist and index finger were fixed in a fully extended posture. Movements made with that arm were thus constrained to two degree of freedom (shoulder and elbow rotation) planar motions. Finger tip location and joint angles were recorded with a Northern Digital (Waterloo, Ontario, Canada) Optotrak infrared position-monitoring system. Participants wore an infrared marker on the tip of the index finger and a rigid body containing six markers on the upper arm. Experiments began with a calibration procedure in which the positions of the shoulder and elbow with respect to the rigid body were measured, allowing for on-line determination of the shoulder and elbow angles. Participants' view of their arm was blocked by a mirror reflecting a projection screen. A 72 Hz 640 × 480 VGA projector (MediaShow, Sayett Technology) provided visual feedback in the form of a 1-cm-diameter white-filled circle, the virtual image of which followed the position of the index fingertip. Obstacles, starting locations, and targets were similarly displayed.
Fig. 2. Virtual visual feedback system.
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Procedure. Each trial began with a white (start) circle, a blue (target) circle, and a yellow triangular obstacle appearing in the workspace (see Fig. 1). Participants were instructed to move their finger into the start circle and wait for a tone, at which point they were to reach around the obstacle tip to the target circle, making sure to avoid hitting the obstacle with their finger. If the fingertip collided with the obstacle, a low tone was sounded, and the trial was restarted. Otherwise, when the participant's fingertip came to rest in the target circle, a high tone was sounded, and the screen went blank until the next trial. Participants were given no further instructions, except to move naturally and comfortably.
For each experimental session, the location of the obstacle tip was prespecified as a point in joint space ( , corresponding to the orientation of the obstacle with respect to the positive x axis (rightward). If the presentation angle was
Trajectory analysis. Velocities were calculated by simple first differencing of positions. For higher derivatives, the planar positions of the fingertip were fit with cubic smoothing splines, and derivatives were taken analytically from the spline fit. Curvature of movements was calculated using the equation: where v. and a. are the velocity and acceleration, respectively, in the subscripted direction.
Four trajectory landmarks were defined: the near point (NP) or point of closest approach to the obstacle tip; the apex or point of maximal deviation from the straight line path (AP); the location of the local minimum of velocity (VM), if there was one; and the location of the peak of curvature (CP). For each landmark a corresponding angle, , is defined as the difference between the presentation angle and the angle of the landmark from the obstacle tip. Figure 3 illustrates the case of the near point angle,
Fig. 3. Definition of the near point angle,
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A sensitivity model Later we will show that the trajectory near points tended to cluster at opposite poles of the obstacle tip, roughly aligned with the orientation of the forearm. This observation suggests that the planner chose the near point location based, indirectly at least, on the configuration of the arm. What properties of the arm would make one location more desirable for the near point than others? We suggest that the key to answering this question is the notion of the anisotropic sensitivity of the arm. Because the only constraint on the movement, other than the start and target points, is to avoid colliding with the obstacle, it would be desirable to choose a path which minimizes the sensitivity of the arm to uncertainty or perturbations in the direction of the obstacle. We next introduce three definitions of sensitivity: one purely kinematic, one based on the inertial properties of the arm, and one based on its elastic properties. We then show why these directional sensitivities are relevant to trajectory planning.
Kinematic sensitivity: manipulability. The first definition of sensitivity is based solely on the kinematics of the arm. We are interested in how uncertainty in joint angle control or proprioception propagates to uncertainty in the end point position. Assume that the joint controllers (or sensors) are noisy, with independent noise at each joint having variance 2. Then the covariance of the resulting uncertainty in achieved (sensed) Cartesian end point position can be derived as follows:
where dx and d
(1) indicates the matrix transpose. The approximation at the second step follows from the definition of the Jacobian and is valid as long as the uncertainty is sufficiently small. From Equation 1 we see that the matrix
shapes the independent joint noise into anisotropic end point noise. M can thus be thought of as a measure of directional sensitivity; it is more difficult to position or sense accurately along the major eigenvector of M than along its minor eigenvector.
Yoshikawa (1990)
In the present experiment, the arm is constrained to planar two-joint movements, so the Jacobian is well approximated by: where l1 and l2 are the lengths of the upper arm and forearm, respectively. We can thus compute the manipulability matrices from experimental data. These matrices can be displayed as ellipses representing 1 SD of end point noise. Examples are shown as the solid ellipses in Figure 4.
Fig. 4. Sensitivity ellipses at various joint configurations. Solid, Manipulability; dashed, mobility; and dotted, admittance. The absolute magnitude of the ellipse is irrelevant for our purpose; we are concerned only with its shape and how that shape changes over the workspace. The major (minor) axis of the ellipse corresponds to the major (minor) eigenvector.
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We note that the assumption of independent and equal magnitude measurement or control uncertainty at the two joints is simplistic. The existence of muscles (and hence spindle receptors) such as the biceps, which span both the shoulder and elbow, make it clear that uncertainty at the two joints should not be independent. Scott and Loeb (1994)
Inertial sensitivity: mobility. A second measure of sensitivity is based on the instantaneous response of the arm to dynamic perturbations. Following the definition of Hogan (1985) where I(
Direct measurements of the inertia of the arm are not available. Instead, we used a simple model, which treats each segment of the arm as a point mass, m1 and m2, located a fraction a or b along the respective segment length. The resulting inertia matrix is: where c
Elastic sensitivity: admittance. Finally, the importance of the elastic stiffness, or mechanical impedance, of the arm for movement control has long been discussed in the literature (Bizzi et al., 1976
We first define the stiffness of the arm in joint space, R( = Rd
Z determines the Cartesian displacement from the current equilibrium, which will result from a static force perturbation: dx = Zf. Again the eigenvectors are easily interpretable: forces of a given magnitude applied at the end point will result in maximal (minimal) displacement when the force is oriented along the major (minor) eigenvector of Z.
(2)
Although we did not measure the admittance of participants' arms, we were able to estimate values roughly for the joint stiffnesses from data presented by Mussa-Ivaldi et al. (1985) Given these values, the end point admittance Z could be computed for a given participant by Equation 2. This procedure makes a number of simplistic assumptions, such as a linear dependence of R on
Sensitivity and obstacle avoidance. For all three matrices introduced above, the minor eigenvector represents the least sensitive direction, i.e., the one in which the least response to position uncertainty or force perturbations is expected. This relationship is the key to the following discussion. Because the comments apply equally well to all three matrices, they will be referred to collectively as sensitivity matrices.
To understand how a sensitivity matrix relates to movement planning, consider the examples of Figure 5. The top panel shows a possible path around an obstacle, the presentation angle of which is along the x-axis. The sensitivity matrix for that location in the workspace is shown as an ellipse centered at the obstacle tip. Would sensitivity considerations deem this a good path? The region around the obstacle tip is expanded in the right panel, which shows that the line from the obstacle to the near point lies along the minor eigenvector of the sensitivity matrix. Because the arm is most vulnerable to collisions when it is closest to the obstacle, it is desirable for the arm to be relatively insensitive to uncertainty or perturbations along the perpendicular to the path when passing the near point. In this example, that criterion is maximally satisfied, because the path perpendicular is the direction in which the arm is least sensitive. Note that we have drawn the sensitivity matrix for the obstacle tip, not for the actual location of the finger. This simplification is justifiable, because all three sensitivity matrices vary slowly over the workspace. 
Fig. 5. The relationship between a sensitivity matrix and obstacle avoidance planning. Two paths identical up to a 90° rotation are shown. The ellipses represent the sensitivity matrix at the obstacle tip. For further discussion, see Materials and Methods.
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Figure 5, bottom panel, shows an obstacle centered at the same location but rotated 90°. The path displayed here is also the same as above but rotated to achieve the start and target points. The near point now lies along the major eigenvector of the sensitivity matrix. This means that when the fingertip comes closest to the obstacle, the arm is maximally susceptible to uncertainty or perturbations along the direction that will lead to a collision. For this presentation angle, then, the same near point angle is a poor choice.
These considerations can be turned into a simple model of near point placement: the minor eigenvector of the sensitivity matrix represents a preferred axis for the near point. To minimize the risk of collision, the planner chooses the path of the arm to bring the near point closer to this minimally sensitive axis. This idea can be captured formally with the following statistical model of the dependence of the near point angle where
(3) is zero mean, normally distributed noise with SD
and y = x%180°, the "signed modulus," is defined as the y in the interval [
90°,90°] such that x = y + n 180° for some integer n. The two parameters of the model are the preferred axis
and the slope b. The latter is a measure of the strength of the dependence of
To see how the model of Equation 3 relates to the idea of a preferred axis for near point placement, consider the hypothetical data in Figure 6. The top row shows data generated from Equation 3 with the parameters 
Fig. 6. Hypothetical near point data generated from Equation 3. A, C, Near point location with respect to the obstacle tip. B, D, Near point angle as a function of presentation angle. A, B, b = 0.5;
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Figure 6, bottom row, shows a second data set generated from Equation 3, this time with b = 0. Here, there is no dependence of
Given a set of experimental data, we wish to find the values of the preferred axis 2 (McCullagh and Nelder, 1989
RESULTS
Subjects were able to perform the task easily, never colliding with the obstacle on more than one or two trials per session. The mean (SD) movement time across subjects was 736 (118) msec.
If the visual planning model were correct, there should be no systematic variation in trajectory shape as the presentation angle changes. However, participants' paths did not display this rotational symmetry. Figure 7 shows two sets of paths from one participant, rotated into a canonical orientation. The presentation angles for the trials in the two panels were ~90° apart, and there are marked differences between these two sets of movements. Those in the left panel are fairly symmetric, with near points clustering along the line of the obstacle. But when the presentation angle was shifted 90°, the paths became much less symmetric. In particular, near points tended to cluster away from the obstacle tip. Such differences in movement path were characteristic of all participants in the experiment; at some presentation angles paths tended to be symmetric, and at other angles they were more skewed. 
Fig. 7. Sample paths, J. M. at position 1. Black paths are clockwise movements; gray paths are counterclockwise. Circles indicate near points. Presentation angles for trials in the two figures are 90° apart, but the paths have been rotated into a canonical position for comparison. Insets, Actual orientations of the movements. In the left panel, the presentation angles were near the preferred axis.
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The lack of rotational symmetry in obstacle avoidance paths can be seen more clearly by looking at all the landmark locations in an experiment. If planning is performed in Cartesian space, the position of the landmarks relative to the obstacle should be independent of the presentation angle, 
Fig. 8. Landmark locations and angles for P. B. at position 1. From the top, near point, apex, curvature peak, and velocity minimum. Crosses, Clockwise movements; circles, counterclockwise movements.
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Fig. 9. Landmark locations and angles for J. M. at position 2. From the top, near point, apex, curvature peak, and velocity minimum. Crosses, Clockwise movements; circles, counterclockwise movements.
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We can quantify this agreement by fitting the piece-wise linear model of Equation 3 to each data set, yielding estimates of the preferred axis 
Fig. 10. Near point angle regression results. Top, Preferred axis
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Although the piece-wise linear model does account for a significant amount of the variance in near point angles, there are some aspects of the data it does not capture. In particular, the direction of movement has a significant effect on the shape of the path. Consider again Figure 7, right panel, which shows movements with presentation angles near the antipreferred axis. The paths are skewed toward the movement origin, and the near points for the two directions of movement lie in separate clusters, each closer to the respective target. This example is a special case, because the model is discontinuous at 
Fig. 11. Comparison of clockwise (CW) and counterclockwise (CCW) near point placement. These composite plots contain half the data (randomly selected) from each of the 10 experiments. The abscissa represents the presentation angle relative to the preferred axis of each experiment,
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Returning now to the preferred axis regression, we can compare the results with the predictions of the sensitivity models introduced above. A summary of the comparison is shown in Figure 12, in which the preferred axes for the two experimental positions are plotted against each other for each subject. Note that the area of the plot represents the space of possible model predictions. If the preferred axis were independent of location in the workspace, the data would lie on the dashed line representing identity. In fact, the data lie significantly above this line, as do the predictions from all three sensitivity models. This illustrates that the sensitivity model in general is able to capture the dependence of the preferred axis on workspace position qualitatively. The mobility model in particular exhibits good quantitative agreement with the data, especially considering the range of possible predictions. 
Fig. 12. Summary of model predictions and experimental results for near point preferred axes. Both axes of the plot represent preferred near point axis, in degrees, for the respective workspace locations. A 95% confidence ellipse (across participants) is shown for the experimental data. Note that the manipulability and admittance model predictions overlap to a large degree. The dotted line is the identity, x = y.
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DISCUSSION
There are three main points of this paper: (1) we introduced the obstacle rotation paradigm, which provides a means for the systematic study of trajectory planning of obstacle avoidance movements; (2) the experiment revealed a dependence of movement path on presentation angle, ruling out a strict visual planning model; and (3) the nature of the variation is consistent with a sensitivity model of path planning. We discuss each of these points in turn.Obstacle avoidance
One reason researchers are still largely divided over the validity of the visual-planning model is the fact that most relevant studies have been limited to simple point-to-point reaching movements. The class of models that has proven most successful in capturing these experimental data is based on the principle of optimal control. These are essentially models of smoothness or efficiency, defined either extrinsically (Nelson, 1983
Other researchers have investigated reaching under similar conditions. Abend et al. (1982) Systematic path variations
We have argued that the dependence of the landmark locations on the presentation angle rules out a strict visual planning model. However, there are alternate explanations for these data. Let us begin by assuming that the movement variations are truly planned, i.e., they are represented in the central neural command. In this case, there must be some criteria by which the CNS varies the path according to presentation angle. Those criteria could be based on any combination of visual cues or distortion, kinematics of the arm, or dynamics of the arm. The sensitivity models presented in this paper fall into the two latter categories, which are both inconsistent with a visual planning model. What about the possibility that the path asymmetries have a purely perceptual origin? It is known that visual distortions of the workspace can be associated with corresponding distortions in movement path in the case of point-to-point reaching (Wolpert et al., 1994
Finally, it is possible that the results of the obstacle rotation experiment are attributable entirely to low-level dynamic factors and not to a central planning mechanism. This concern is also not addressed by the results of this paper, but see the work of Sabes (1996) The sensitivity model
The three sensitivity models qualitatively capture the main features of interest in the experimental data: the clustering of near points about a preferred axis and the dependence of that axis on workspace location. The mobility model provides the best quantitative match to the experimental data, suggesting that the CNS uses information about the inertia of the arm in planning obstacle avoidance movements. It has been shown that when participants are asked to estimate the location of the tip of a visually occluded object that they are allowed to wield freely, their responses are quite accurate and are predictable given only the eigenstructure of the inertia of the object (Fitzpatrick et al., 1994
The interesting difference between the three sensitivity models is the nature of the information they embody purely kinematic or both kinematic and dynamic, for example
Whatever its exact definition, the sensitivity constraint, in isolation, would be maximally satisfied if the near point always lay at the preferred axis, i.e., if the slope of Equation 3 were unity. Why then do we find a relatively small value of 0.17 for the mean estimated slope in our experiments? First, at presentation angles 90° away from the preferred axis, the model has no preference for direction of the near point angle. This ambivalence is seen in the data as well. Figure 11, A and B, shows large near point angles of both signs at the antipreferred axes, resulting in a downward bias in the estimated slopes. Nonetheless, the same figure shows that the "true" slope is certainly smaller that unity. Why is the sensitivity criterion only partially satisfied? We argue that this is the result of a tradeoff between a set of planning criteria, of which the sensitivity-based collision avoidance scheme is only one. For example, some notion of smoothness is almost surely a consideration in the planning process, and larger near point angles result in less symmetric paths, decreasing the overall smoothness of the movement. The difference between clockwise and counterclockwise movements may reflect another such criterion, one in which there is a preference for paths skewed toward the movement origin (perhaps to allow time for feedback to influence the movement before crossing close to the obstacle).
And finally, we note that these results are not necessarily inconsistent with recent perturbation studies supporting the visual planning model for point-to-point reaching (Flanagan and Rao, 1995
FOOTNOTES
Received March 4, 1997; revised June 26, 1997; accepted July 1, 1997.
This project was supported by a grant from the United States Office of Naval Research. P.N.S. was supported by a training grant from the National Institute of General Medical Sciences. We thank D. M. Wolpert and N. Hogan for many helpful discussions and an anonymous reviewer for helpful suggestions on an earlier draft of this paper.
Correspondence should be addressed to Philip N. Sabes, Salk Institute, Computational Neurobiology Lab, 10010 North Torrey Pines Road, La Jolla, CA 92037.
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