Simulated Reduced Movement

Motor Control, 2001, 2, 136-150 ® 2001 Human Kinetics Publishers, Inc.

Theory of Motion Planning

Planning Reaching and Grasping Movements: Simulating Reduced Movement Capabilities in Spastic Hemiparesis Ruud G.J. Meulenbroek, David A. Rosenbaum, and Jonathan Vaughan In this paper we describe how a theory of posture-based motion planning recently applied to human grasping may contribute to the understanding of grasping pathology. The theory is implemented as a computer model rendered as a stick-figure animation capable of generating realistic multi joint grasping movements. As shown here, the model can also be used to simulate grasping movements whose kinematics resemble those of grasps performed by people with spastic hemiparesis. The simulations demonstrate effects of: (a) reduced ranges of motion of arm joints on the size of the reachable workspace, (b) awkward starting postures on the time course of the hand closing around an object, (c) increased costs of joint rotations on movement time, and (d) addition of noise to biphasic joint rotations on the low-velocity phase of wrist transport. Key Words : motor planning, computational model, posture, reaching, grasp-

ing, prehension, hemiparesis

I ntroduction

Theories of human motor control should provide a framework for describing and understanding both normal and atypical movements. The present article focuses on one such theory, which was developed originally to account for normal multijoint movements in tasks such as reaching, moving around obstacles, handwriting, and grasping. A brief summary of the theory is given in the next section and then applications of the theory to pathological grasping are presented. R.G.J. Meulenbroek is with the Nijmegen Institute for Cognition and Information,

P.O. Box 9104, 6500 HE Nijmegen, The Netherlands. D.A. Rosenbaum is with the Depart-

ment of Psychology at Pennsylvania State University, University Park, PA 16802-3408. J. Vaughan is with the Department of Psychology at Hamilton College, Clinton, NY 13323. The article is based on a talk presented at the International Workshop on Studies and Models of Human Prehension; Contributions to Understanding Grasping Pathology, Werkenrode, Groesbeek, The Netherlands, January 13, 2000. 136

137

Because outlines of the theory have been described in one of the preceding articles (Rosenbaum et al., this issue) and its details are specified elsewhere (Rosenbaum et al., 1995, 1999, in press), we cover only its main assumptions here. One of the theory's core assumptions is that motion planning involves time-limited, multiple task constraint satisfaction. The actor is assumed to construct a task-specific constraint hierarchy. For example, an actor may decide to make a spatially accurate and slow movement rather than a fast and inaccurate movement. Avoiding collisions with intermediate obstacles also may or may not be part of the constraint hierarchy depending on whether the actor is in a china shop or on a football field. Based on the constraint hierarchy, a goal posture is selected as the postural target for movement. Such a goal posture is found, according to the theory, by means of two processes. The first is evaluation of candidate goal postures in a stored posture base, which consists` of representations of adopted final body positions that may have been adopted in similar or different functional contexts. From this stored posture base, the stored posture that is most suitable for the task at hand, according to the constraint hierarchy, is chosen as the currently most promising stored posture. If additional planning time is available, other possible goal postures near the most suitable goal posture (in posture space) are evaluated. These postures are generated in a gradually widening shell, such that the more planning ti me available, the wider the shells become. The less time available for planning (and so the narrower the shell), the higher the chance of an ill chosen goal posture. When planning time is over, the most suitable of all evaluated postures is defined as the goal posture. Once the goal posture is found, a transition from the starting posture to the goal posture is planned and evaluated. The default transition is based on the principle that all joints start and stop moving simultaneously according to bell-shaped joint angular velocity profiles. This simplified movement-execution principle yields smooth, straight-line movements in joint space, which is an approximation to what is actually seen in biological motion (Hogan & Flash, 1987; Uno et al., 1989). The duration of the transition depends on the total cost of the postural transition, where each joint's angular displacement is weighted by its "expense" factor, and the total cost is assumed to be the sum of the weighted angular displacements. Because higher expense factors correspond in the model to more costly joints, and more costly joints also, by definition, prefer longer times than less costly joints for the same angular displacement, the more costly the joints involved in a movement, the longer the movement will take. Moving smoothly and in a straight-line through joint space does not guarantee avoidance of collisions with intermediate obstacles or even with the object to be grasped (e.g., if the back of the hand rather than the palm of the hand comes in contact with the to-be-grasped object). If a collision is anticipated during movement planning, additional planning may help select a collision-free transition between the starting and goal posture. During this extra motion planning, the model tries to find a detour through joint space that satisfies the collision-avoidance constraint. The way this detour is sought is not by shaping the planned movement directly but instead by searching for a suitable via posture that governs the nature of a movement that is added to the main movement from the starting posture to the goal posture. This added movement proceeds from the starting posture to the via



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posture and back to the starting posture. It adds no net excursion to the transition between the starting and goal posture but instead, when combined with the default straight-line movement in joint space between the starting and goal posture, yields a movement through joint space that is curved rather than straight. The magnitude and direction of the curvature depends on where the via posture is located relative to the starting and goal postures. The via posture is found through a search process similar to the one used for finding the goal posture. (For more details, see

Meulenbroek et al., 1996, Rosenbaum et al., in press, and Vaughan et al., this issue). The rationale for this method of obstacle avoidance can be illustrated as follows. Imagine standing in front of a ball hanging from the ceiling at chest height. A single step forward would result in a collision, but combining the step forward with a lateral swing of the trunk could prevent the chest from hitting the ball. While the forward step is made, the lateral swing of the trunk could help the body circumvent the obstacle. The lateral swing would take the body from its starting posture to a via posture and back while the body proceeds forward from its starting posture to its goal posture. The farther the via posture is from the starting posture in terms of hip angle, the larger the lateral sway will be. For a summary of the behavioral and neurophysiological basis of this technique, see Rosenbaum et al. (1999) and Vaughan et al. (1998).

Applying the Theory to Grasping

The theory described above has been implemented in a computer model rendered as a stick-figure animation capable of generating multijoint, planar grasping movements. The stick-figure has an artificial arm with nine degrees-of-freedom (df) (see Figure 1 and Table 1). Four df represent the proximal joints of the arm: the shoulder, the elbow, and two wrist joints, one connecting the forearm to the index

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finger and the other connecting the forearm to the thumb. Five other df represent the distal joints of the index finger (n = 3) and the thumb (n = 2). It should be noted that it may not seem realistic to presume two df for modeling planar wrist movements, but our model applies to, in principle, any combination of df, and the two df for the wrist were used to model the thumb and finger movements separately. Moreover, various linkages between the joints (e.g., the three joints in the index finger forming a single degree-of-freedom and those in the thumb forming another degree-of-freedom) were applied, mainly to speed up the simulations, but the results were comparable in these situations. The hierarchy of task constraints used to model the grasping movements consisted of: (a) collision avoidance-apart from the final moment of contact with the to-be-grasped object, collisions with the object had to be avoided; (2) spatial accuracy-at the moment of movement completion, the finger tip and thumb tip had to touch the object at opposite sides of the circular object; and (c) movementcost minimization-the total cost of movement summed over all joints, each of

which was given its own weight of contributing to this cost needed to be as low as possible (cf. Vaughan et al., 1998). With these constraints, grasping movements

were simulated as follows. An object was positioned in the workplane such that it was within reach of the stick figure (see Figure 1). The stick figure's reach was determined by the minimum angles and the angular ranges of motion of the stick figure's joints (see Table 1). Given the initial starting posture of the stick figure, a candidate goal Table 1 Simulation Parameters (Constants)

Joint

Label

Minimum angle (°)

Elbow

Elbw

0

Wrist to Thumb

WrsT

Shoulder Wrist to Finger

Metacarpophalangeal joint

Proximal (1st) joint of Index finger

t Figure 1- Stick figure capable of performing multijoint planar grasping movements. A typical starting posture used in the modeling is shown. The object to be grasped (solid circle) is located at 135 ° relative to the point of contact between finger tip and thumb tip at a distance of 20 cm; the object size is 2 cm in diameter. Length of forearm: 18 cm.

139

Distal joint (2nd) of Index finger Proximal (1st) joint of Thumb Distal (2nd) joint of Thumb

Range of motion (°)

Expense factor

160

1.2

-60

170

WrsF

-60

120

1.2

Mtcp

0

35

.2

Shld

Indx1

Indx2 Thmb1

Thmb2

50

0 0 0 0

-35

40 45

-40 -45

1.2

1.2

.2 .2 .2 .2

Note. The labels in the second column are used in Figures 2 and 6 to identify the joints whose kinematics are shown. The Minimum Angle() of a joint is defined by the angle between the linear extension of the proximal limb segment and the distal limb segment connected to it, where positive angles correspond to counterclockwise segment rotations. The Expense Factor reflects the relative cost of rotating a joint through one degree; here, rotating the proximal joints is six times as expensive as rotating the distal joints (see Rosenbaum et al., 1995, in press). Number of postures in memory: 200; Grain: 0.01 (the spatial resolution of the search for suitable goal and via postures, expressed as a fraction of the Range of Motion of a joint); Deadline: 20 (the maximum number of searches for more suitable goal or via postures in ever-widening shells around the candidate goal or via posture); Instructed Movement Time: 50 (arbitrary units).



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posture for the hand was sought such that the aperture of the hand matched the object size. The search took place in a posture base of 200 stored postures. Once the hand goal posture was found, a candidate goal posture for the arm was sought that would bring the hand to the object location. Finally, when a suitable goal posture was found, a trajectory through joint space was sought in the combined arm and hand posture base. If internal simulation of the default, straight-line movement in joint space between starting and goal posture indicated that a collision would result, a via posture was sought. Each search process was subjected to the three constraints specified above (i.e., the collision avoidance, spatial accuracy, and movement-cost constraints). Moreover, each search process was limited to 20 search cycles, both for the goal-posture search and then for the via-posture search. Successive searches took place in a gradually larger region of the posture base, where "gradually larger" meant steps of 1% of the range of motion of the joints constituting the dimensions of the posture base. Further specifications of the simulation can be found in Rosenbaum et al. (in press).

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Applying the Theory to Grasping Pathology

In the remainder of this paper we focus on possible applications of our model to the understanding of grasping pathology. In particular, we focus on simulations of grasping movements performed by people with spastic hemiparesis, although we are also interested in other clinical groups. What is presented is not meant to prove the validity of our model. Tests of the power of our model to account for typical

Accounting for Typical Grasping Kinematics

Figure 2 shows a time-lapse image of a typical simulated grasping movement. Figure 2A shows the corresponding excursions of the proximal joints. Figure 2B shows the excursions of the distal joints. The joint excursions shown in 2A and 2B are normalized with respect to the starting posture (i.e., the joint angles at the start of the movement were set to zero). Figure 2C shows the resulting aperture-time function, from which it can be seen that the aperture overshoot peaked at approximately 60% of the movement time. Figure 2B shows that this aperture overshoot was mainly due to the biphasic movement of the index finger. Thus, extensions of the joints of the index finger until the maximum aperture was reached were followed by flexion of these joints until completion of the grasping movement. Figure 2D shows the tangential velocity of the wrist. These and other simulations from our model yielded two of the components of typical grasping kinematics that have been reported in human performance studies. First, there was a wider opening of the hand than the size of the object. Because this over-widening of the hand arose in our simulations from the need to avoid collisions until movement completion, it is possible that this same constraint accounts for the over-spreading of the hand during actual grasping behavior. It should be noted that the collision-avoidance constraint in our model is a generalization of the possible use of a safety margin in prehension, as has been suggested by Wing, Turton, and Fraser (1986). Second, the moment of maximum aperture occurs during the second half of the grasping movement (Figure 2C). This feature of the simulated grasp arose from the combined effects of the obstacle-avoidance constraint and movement-smoothness constraint, suggesting that human grasping, which shows this same sort of delay, stems from the same constraints. The grasping movement shown in Figure 2 did not contain a low-velocity phase of the wrist but in many cases our model does generate such a low-velocity phase. This evanescence of the low-velocity phase reflects what is found in human performance studies. Sometimes it is seen in human prehension and sometimes it is not (Wallace & Weeks, 1988). Our model provides an account of when the lowvelocity phase should appear and when it should not (Rosenbaum et al., in press).

A

B

Proximal Joint Excursions

m lime (% Movement Time)

C

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m

m m a, 7U Time (% Movement Time) M

Distal Joint Excursions

,m

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n,

,m

Figure 2 - Typical simulated grasping movement. Top: Time-lapse image of grasping movement. A: Proximal joint excursions; B: Distal joint excursions; C: Aperture-time function; D: Tangential wrist-velocity function.



Meulenbroek, Rosenbaum, and Vaughan

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human grasping movements have been reported elsewhere (Rosenbaum et al., 1995, 1998, in press; Vaughan et al., 1998).

Size of the Reachable Workspace The first way our model may contribute to the understanding of pathological grasping is by using it to quantify geometric features of the reachable workspace. Figure Range of Motion of Shoulder and Elbow

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3 shows an example. Given a particular set of ranges of motion for the joints of the stick figure, the position, shape, and size of the reachable workspace can be assessed computationally. Suppose the stick figure can adopt ranges of motion of the shoulder and elbow that are set to 160° (and the ranges of motion of the remaining seven joints are set to the values specified in Table 1). The open circles in the topright panel of Figure 3 correspond to the finger tip positions of 100 adoptable but random joint configurations. When the ranges of motion of the shoulder and elbow decreased to 80°, the result, as seen in the middle-right panel of Figure 3, was a considerable reduction in the size of the reachable workspace. The effects of a further 50% reduction of the functional ranges of motion of the shoulder and elbow appear in the bottom-right panel of Figure 3. Clinical assessments of the ranges of motion of joints in the arm of a subject with spastic hemiparesis could be used to set the parameters of our model so it realistically reflects reduced motion capabilities. In this way our model could serve as a tool to quantify the implications of such limitations for the size of the reachable workspace. Additionally, the position of the workspace relative to the position of the subject could be assessed in detail for any individual who has to cope with reduced ranges of joint-motion. Furthermore, the present example suggests a possible explanation for the pronounced trunk rotations that are generally observed in grasping movements performed by subjects with spastic hemiparesis (see e.g., Steenbergen et al., in press). Placing an object outside the reachable workspace requires the recruitment of additional joints (such as the hip joint) to grasp the object successfully. A more detailed analysis of the functional features of people's workspace-for example, in terms of comfort associated with certain postures adopted at various positions in the workspace-can be found in Schillings et al. (2000).

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Effects of Suboptimal Initial Hand Postures

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Oo

°

o°

40 deg (60-100) ooo

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Figure 3 - Effects of reducing the ranges of joint motion for the shoulder and elbow (left column) on the size of the reachable workspace (right column).

Figure 4 shows a simulated grasping movement that started with a "normal" initial hand posture (left-hand column) or with a grasping movement that started with a sub-optimal or "awkward" initial hand posture (right-hand column). The awkward starting posture was characterized by a heavily extended wrist joint and strongly flexed joints in the index finger. Such a combination of an extension and flexion synergy is characteristic of people with spastic hemiparesis. In addition to the awkward starting posture, we also increased the joint-expense factors in this example. As mentioned earlier, the expense factors in our model are weights that determine costs of joint rotations. These factors were increased to mimic the large amount of effort people with spastic hemiparesis claim to experience during motion. The middle-right panel of Figure 4 shows a time-lapse image of the grasping movement that started with the awkward starting posture. The corresponding aperture-time function is shown in the bottom-right panel of Figure 4. For both grasping movements, the "normal" one shown on the left and the "awkward" one shown on the right, the object location and object size were identical. The most important result of this pair of simulations concerns the aperturetime functions, which appear in the bottom panels of Figure 4. The aperture-time function in the bottom-right panel is for the awkward-start case and looks strikingly similar to aperture-time functions reported by Saling et al. (1998) for grasping objects that are smaller than the initial aperture at the start of movement. In this



Meulenbroek, Rosenbaum , and

Vaughan

atypical situation, one can observe that during the initial part of the grasping movements, the aperture decreases substantially. Following this initial hand closure, the aperture increases to a maximum attained at about 60-80% of the movement time. After the aperture has peaked, it starts to decrease until it finally matches the object

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145

size. The situation shown here is not meant to match exactly the observations made by Saling et al., but the similarity of the results is remarkable, since the present simulation also started with an initial aperture that was larger than the object size. The reason our model also generated an initial decrease in aperture is geometrical: The index finger of the stick figure's starting posture in the present simulation was strongly flexed at the start of the movement, and so the index finger had to extend during the movement in order to get around the object. As it extended, its tip first moved towards the thumb tip, thus yielding an aperture decrease. Once the tip of the index finger had passed beyond the thumb tip, the index finger started moving away from the thumb, thus inducing the aperture increase. Near the object, however, the obstacle-avoidance related aperture overshoot still occurred. In sum, the atypical grasping movement shown on the right side of Figure 4, which was simulated with an awkward starting posture and increased joint-expense factors, shows the same aperture-time function as the one shown by Saling et al. (1998) for initial apertures larger than the object size. Consequently, the biphasic pattern observed by Saling et al. need not be uniquely ascribed to the conditions in which they observed those kinematics. Such complex aperture-time, or other kinematic, functions may also be observed in grasping movements performed by subjects with spastic hemiparesis, but our simulation demonstrates that these complex, atypical kinematics may simply be due to awkward starting postures in these subjects. Effects of Differentially Increased Costs of Joint Rotations A key parameter of our model, the expense factor of any given joint, allows us to investigate the effects of that parameter on movement. Insofar as the parameter corresponds to organismic constraints on movement, manipulation of the parameter can shed light on how these constraints affect motor performance.

750

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100 0

Time (% Movement Time) '6

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Figure 4 -Effects of a sub-optimal starting posture. Left column: Simulated grasping movement starting with a "normal" initial hand posture. Right column: Simulated grasping movement starting with a sub-optimal, awkward initial hand posture characterized by over-extension of the wrist and an over-flexion of the index finger. Note the complex aperture-time function in the bottom-right panel.

Figure 5 - Effects of increasing the joints' expense factors. A: Time-lapse image of grasping movement simulated with expense-factor combination specified in Table 1. B: Time-lapse image of grasping movement simulated with the expense factors of all joints set to 1.2. As a result of making the distal joint rotations as costly as the proximal joint rotations, the wrist does not extend during the slow movement but instead flexes (B) more than in the "normal" grasping movement (A).







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Meulenbroek, Rosenbaum, and Vaughan

Figure 5A shows a grasping movement modeled with a set of joint-expense factors that permitted flexible motions of the distal joints of the hand. Figure 5B, by contrast, shows a grasping movement in which the expense factors of all the joints were increased by a factor of five. Increasing the expense factors made it harder, as it were, for the stick figure to perform the grasping movement. The movement in Figure 5B required more "effort" than the movement shown in Figure 5A. The effort reduction algorithm in our model accounts for a relationship between effort and movement time: Movements requiring much effort should take more time to complete than movement requiring little effort (cf. Rosenbaum et al., 1995). This difference is illustrated in Figures 5A and 5B. The low-expense movement, displayed in Figure 5A, takes much less time to complete than the highexpense movement, displayed in Figure 5B. The duration difference is reflected in the higher spatial density of the intermediate postures between the starting and goal postures in Figure 5B. It is well known that grasping movements by people with spastic hemiparesis are slower than grasping movements performed by control subjects, which is consistent with this thinking.

,,

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of motor output as being the cause for the cascading of series of submovements that gradually reduce the distance between the start and the end of a movement. The three models just described were all developed to explain Fitts' law (1954), the logarithmic relationship between the ratio of movement amplitude to target width on the one hand, and movement time on the other. Even though the validity of features of these models have been questioned, they provide likely ex-

Effects of Adding Tremor to Distal Joint Rotations The final example of how our model may contribute to the understanding of grasping pathology concerns the search for a possible explanation of a phenomenon that has frequently been reported in studies of grasping movements by people with spastic hemiparesis (Levin, 1996; Roby-Brami et al., 1997; Steenbergen et al., 2000; Trombly, 1992). In these studies it has often been observed that tangential wrist velocity-time functions are highly asymmetric; the deceleration phase takes considerably more time than the acceleration phase. Furthermore, during the deceleration phase, multiple peaks in the tangential wrist-velocity function can be observed. There are several likely explanations for these kinematic patterns. According to one, the iterative-correction model of Crossman and Goodeve (1983), multiple peaks in the decelerative part of the tangential wrist-velocity profile are due to the fact that a single reaching movement may actually consist of a series of submovements, each of which reduces the remaining movement error. According to the model, the submovements are controlled on the basis of visual and/or proprioceptive feedback. An alternative account, the impulse-variability model developed by Schmidt et al. (1979), suggests that similar processes might be responsible for the multiple peaks in the decelerative part of the tangential wrist velocity function. This model, however, attributes these peaks to error in the initially planned and executed main movement. According to the impulse-variability model, movement error is scaled to the amount of force that is required to perform the movement. When large forces are needed, large errors tend to occur unless such errors are reduced either by means of additional visuomotor processing or by constraining the endpoint variability through an enhanced level of cocontraction of antagonistic muscle groups. Especially in the context of accounting for characteristics of movements performed by people with spastic hemiparesis, Schmidt's (1979) model seems particularly relevant given the lowered stretch-reflex thresholds that have been reported as the principal cause of the uncontrollable, high-force movement impulses that are frequently observed in such people (Katz & Rymer, 1989). A third model relevant in this context is the optimized initial-impulse model developed by Meyer et al. (1988). This model also emphasizes the inherent unpredictability

Proximal Joint Excursions

Shld

Distal Joint Excursions Thmb2

Wrst

Wrst I ndxl Thmbl l ndx2 Mtcp

Elbw Tangential Wrist Velocity

. - _; - ---

-

------- ------ --- --'t

Time

-------------- ---- -------------------- ------------------------------------------------------------------ -- -------------------------------- ---------------Time

Figure 6 - Effects of adding tremor to the rotations of the distal joints in the hand. Note the tremulous tangential wrist-velocity function (bottom-left panel) that is frequently observed in grasping movements performed by people with spastic hemiparesis.

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Meulenbroek, Rosenbaum, and Vaughan

planations for positively skewed, multiple peaked tangential velocity profiles of an end effector. The explanations they suggest are based on presumed effects of continuous sensorimotor processes or inherent stochastic features of force generation. Our model cannot account for such processes because it focuses on motion planning processes involved in multijoint grasping movements and because it is purely kinematic. Moreover, our model does not generate multiple peaks in the tangential wrist-velocity function during movement deceleration. Nonetheless, it can be used to explore alternative causes of this phenomenon because it enables us to implement artificial deficits at various levels of movement planning or execution and examine their consequences. Figure 6 shows what happens when we added noise to our model. In this example, we added a linear damped, high-frequency (8 Hz) tremor to the joint

rotations. Tremor is a characteristic symptom of Parkinson's disease and cerebellar disorders. Whereas in Parkinson's disease tremor may sometimes disappear

during motion, in cerebellar disorders motion may exacerbate the tremor. Here we added tremor to joint rotations to investigate its consequences on grasping kinematics. Rather than adding the tremor to the rotations of the joints indiscriminately, we added the tremor to the obstacle-avoidance component of the grasping movements only-that is, to the biphasic movements from the starting angle to the via angle and back to the starting angle. The reason why tremor was not added to the main movement from the starting to the goal posture was that this allowed us to be certain about the achievement of the desired goal position. Adding tremor to the

extra back-and-forth movement associated with the via posture let us evaluate the consequences of tremor on movement alone. Figure 6 shows that by adding tremor to the obstacle-avoidance component of the grasping movement, an asymmetrical, multi-peaked tangential wrist-velocity function was obtained. The example shown in Figure 6 suggests, then, that multiple-peaked tangential wrist velocity functions of the sort frequently observed in grasping movements performed by people with spastic hemiparesis may be due to i mproper control of back-and-forth movements superimposed on main movements.

Conclusions

As we said at the beginning of this article, we believe that models of human motor control should provide an accurate account of motor disorders and their typical symptoms. As we have tried to show here, our model can be used for this purpose. First, it can be used as an assessment tool to investigate the consequences of reduced joint mobility due to disease or trauma. Second, it can be informative in analyzing the effects of increasing costs associated with joint rotations on movement kinematics, both in workspace coordinates and in joint-space coordinates. Indeed, in presuming that there is an important role of costs associated with joint rotations in explaining which joint rotations will be realized for a particular movement, our model also provides an opportunity to investigate the consequences of specific organismic constraints such as increased joint-stiffness levels. Such increased joint-stiffness levels are, of course, a symptom not restricted to spastic hemiparesis. Modeling increased joint-stiffness levels-mimicked in our kinematic model by increasing the joints' expense factors-induced prolonged movement ti mes and rigid postural changes. Finally, we demonstrated how our model could be used to examine the effects of tremor.

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