| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Laboratory for Intelligent Mechanical Systems, Mechanical Engineering Department, Northwestern University, Evanston, Illinois
Submitted 2 December 2004;
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
Although a great deal of work has studied how motion freedoms [redundancy 1) above] are resolved in unconstrained point-to-point arm motions (e.g., Alexander 1997
; Flash and Hogan 1985; Harris and Wolpert 1998; Kuo 1994; Todorov and Jordan 1998; Uno et al. 1989) and how muscle load-sharing [redundancy 3) above] is resolved in completely constrained single-arm isometric tasks (e.g., Buchanan et al. 1986; Flanders and Soechting 1990; Gomi 2000; van Bolhuis and Gielen 1999), there has been less work on understanding how constraint force freedoms [redundancy 2) above] are resolved in partially constrained tasks. Because applying forces against constraints has no effect on the task, the manner in which force freedoms are resolved provides powerful clues to the organization of the motor control system. For example, although many hypotheses have been proposed to explain experimental unconstrained arm motion data (minimum Cartesian jerk, minimum rate of torque change, minimum metabolic cost, etc.), these hypotheses' differing predictions of constraint forces can be used to determine whether they are applicable also to the case of constrained manipulation. Ideally an organizing principle would be able to explain both unconstrained and constrained arm motions.
Consider, for example, the following experiment. A subject consistently chooses a trajectory (and associated path) solving a particular point-to-point unconstrained reaching task. Now we place a frictionless guide rail exactly along that chosen path and ask the subject to again perform the reaching task several times. Optimization models of the dynamics may predict that the subject will learn to use the new force freedom by applying forces against the rail to further decrease the objective function. In other words, the presence of the rail allows the subject to "optimize" further (Lynch et al. 2000). This is exactly the effect we are interested in: how subjects naturally take advantage of force freedoms.
There are reports of previous studies on natural interaction with smooth constraints. Perhaps most related to the present paper are studies on how subjects turn a crank (Russell and Hogan 1989; Svinin et al. 2003). Russell and Hogan showed that subjects apply significant radial forces (compressing or extending the crank) even though they are workless. Svinin et al. argued that experimental forces and motions during crank-turning can be described by minimization of a weighted combination of changes in hand force and joint torques. Van der Helm and Veeger (1996) studied the related problem of shoulder muscle activation during wheelchair propulsion, and both their shoulder mechanism model (van der Helm 1994) and experimental results showed that subjects apply forces normal to the rim of the wheel. The efficiency of the motion is also discussed. Gomi (1998) showed that the natural stiffness at the hand during motion is altered in the presence of a guiding constraint. Scheidt et al. (2000) studied persistence of motor adaptation during constrained multijoint arm movements. Their decomposition into kinematic and dynamic criteria influencing disadaptation correspond roughly to our decomposition into trajectory and force freedoms.
This paper reports the results of the simplest possible experiment studying how subjects resolve a constraint force redundancy. Each subject was asked to use the right hand to hold stationary a handle being pulled with constant force along a low-friction linear rail (m = 1 motion freedom). The arm was supported in a horizontal plane with the wrist cuffed, so that the arm could be treated as a two-joint shoulder–elbow mechanism. To hold the handle stationary, the subject had to apply an equal and opposite force along the rail, but subjects were also free to apply a force against the constraint, orthogonal to the direction of the pulling force (one force freedom in the horizontal plane because the arm was treated as a two-joint mechanism). Despite the fact that constraint forces increase the magnitude of the total force vector at the hand and have no effect on the task, we found that subjects applied significant constraint forces in a consistent manner dependent on the arm and constraint configuration. We show that the constraint forces can be interpreted in terms of an objective function describing how subjects choose a particular hand force from a family of hand forces that accomplish the task. Without assuming any particular form for the objective function, the data show that its level sets are convex and scale invariant (i.e., the level set shapes are independent of the hand-force magnitude). We derive the level sets, or "isocost" contours, of subjects' objective functions directly from the experimental data. In other words, in contrast to previous work on optimization models that use experimental data to support or invalidate candidate objective functions based on a biomechanical model, we use a new method for directly measuring the level sets of the objective function without assuming any particular form for it. These level sets may be thought of as nonparametric objective functions that act as descriptors and predictors of behavior, independent of any interpretation in terms of biomechanics and neural control. Importantly, the objective functions appear to be independent of the arm configuration when expressed as objective functions on joint torques. We have compared our results to the predictions of several biomechanical models of force generation, and although these results are inconclusive because of uncertainty in subjects' physiological parameters, models based on the sum of muscle tensions and stresses can be effectively ruled out. An objective function that is a simple positive-definite quadratic form on the joint torques appears to fit the data well. If the muscles are springlike (e.g., Mussa-Ivaldi et al. 1985), one interpretation of this objective function is that subjects choose a hand force that satisfies the task while minimizing the potential energy stored in the muscles. Portions of this work have previously appeared in conference form (Pan et al. 2004; Tickel et al. 2002).
|
METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
Fourteen healthy right-handed male subjects (Table 1) were seated in a custom-made high-backed chair with an adjustable seat, to raise or lower the height of the shoulder plane based on the height of the subject. To fix the shoulder location, subjects were restrained by a 4-point harness. The wrist was immobilized by an over-the-counter commercially available wrist cuff, and the subject grasped a vertical handle on a slider on a horizontal low-friction linear rail. The rail is mounted on a lazy Susan turntable, allowing the rail to be rotated 360° in the plane. The orientation of the rail can be fixed at any angle by clamping the turntable. The handle can spin freely about a vertical axis so that no torques at the hand are involved, and a support plate is attached to the handle to support the weight of the forearm (Fig. 1A). This support maintains the arm in a horizontal plane throughout experiments without fatiguing the shoulder.
|
|
Each trial consisted of the subject holding the handle while a weight was hung from the cable, causing a tangential pulling force on the slider. The subject then stabilized the position of the handle at the center of the rail. Forces normal to the rail were then recorded for 1 s and averaged. The weight was then removed from the cable.
For 8 subjects (subjects 1 through 8), the slider handle was located at (0, 45 cm) in a frame fixed to the shoulder, as shown in Fig. 1B. Sixteen angles of the rail were used, evenly spaced at 22.5° intervals. At each of the 16 test angles, 2 different weights were hung from the cable, 0.858 kg (the light weight) and 1.759 kg (the heavy weight). These resulted in tangential forces of 8.4 and 17.3 N, respectively. For each angle and weight, the experiment was repeated 3 times. Therefore, for each handle position, we collected 16 x 2 x 3 = 96 data points. The ordering of the trials was randomized to minimize any history effect in the results. Subjects were told to suspend the weight as naturally and comfortably as possible, and not to excessively cocontract to stiffen the position of the handle. Fatigue was minimized by the short durations of each experiment, and subjects were permitted to take a break at any time. The total testing time for each subject was about 1 h.
For 6 subjects (subjects 9 through 14), the experimental protocol was similar, except only a single weight of 1.2 kg was used, giving 11.8 N tangential force. Each of these 6 subjects was tested at 5 different positions of the hand, as shown in Fig. 9A.
|
Objective functions and isocost contours
The organizing principle governing how subjects apply constraint forces can be expressed as an objective function describing the "cost" of generating a particular force vector at the hand. This objective function may reflect the "effort" involved in applying a particular force, or it may simply reflect the organization of the motor control system. In either case, the role of the objective function is simply to resolve the freedom in the applied constraint force.
An objective function g may be viewed as a mapping from the arm configuration and the force and torque applied at the hand to a nonnegative real number representing the "cost." For ease of discussion, we will assume that the arm configuration is implied and that the hand force can be written as a 2-vector fh = (fhx, fhy) in a Cartesian frame aligned with the shoulder frame (Fig. 1B). Then the objective function g(fh) can be viewed as a function g : 
2 
. (Any objective function on joint torques or muscle tensions uniquely defines an objective function on the hand forces.) Thus the objective function forms a 2-dimensional surface. A weak assumption on the form of g is that the cost increases monotonically as we move outward along any ray from the origin fh = (0, 0). For low to moderate hand forces, this seems intuitively correct: when a subject can solve a task with any hand force 
fh for 
1, the subject is likely to choose approximately the minimum necessary force, 
1. An objective function satisfying this condition defines a bowl in the hand-force space, and there are no local minima except the global minimum at (0, 0). The level sets (curves) of a bowl-shaped cost function are concentric, closed, and star-shaped (can be written as a function of the polar angle) about the origin. We call these level curves isocost contours in the hand-force space.
Two important subclasses of bowl-shaped objective functions are those whose isocost contours are all convex, and those whose isocost contours are scale invariant—each isocost contour is a uniformly scaled version of every other. These properties are shown graphically in Fig. 2. An objective function g(fh) is convex if its Hessian matrix 
is positive definite at all fh, i.e.
 | (1) |
 | (2) |
 |
) satisfying k(1) = 1 and monotonically increasing with . A common example of k() is a power law p (p > 0). If an objective function is both convex and scale invariant, we refer to it as a CSI objective function. These properties of an objective function generalize immediately to hand forces and torques in more than 2 dimensions.
|
 |
|
If the objective function is scale invariant, the direction of the optimal total force fh depends only on the direction of ft, not on its magnitude. In other words, if the required tangential force ft is scaled by , then the optimal normal force is also scaled by . This does not hold for more general objective functions.
Reconstructing isocost contours
The measured force fh applied by a subject in a given trial is decomposed into 2 orthogonal components: a component tangential to the rail denoted ft with value equal and opposite to the pulling force, and a component perpendicular to the rail, denoted fn, which is the object of study. For each rail orientation we define a reference frame such that fh = (fn, ft), where fn and ft are the scalar values of the force against and along the rail, respectively. The +ft-axis is oriented along the rail in the direction the subject must apply a force to prevent motion of the slider, and the +fn-axis is 90° clockwise, as shown in Fig. 1B.
The results for a single subject at a single slider position can be plotted as shown in Fig. 6 in the RESULTS section. There are 2 plots, one corresponding to the light weight and one to the heavy weight. Each plot shows the normal force applied by the subject as a function of the angle of the tangential force ft along the rail.
|
Each point on a normal force plot, as in Fig. 6, indicates a point in the hand force (fhx, fhy) space, at an angle 
relative to the +fhx-axis (Fig. 4). At this point, the direction of the normal force fn is tangent to the isocost contour, as shown in Fig. 3. Therefore, the p data points of the normal force plot give us a set of angles i, i = l p, and a tangent direction 
i associated with each i. Choosing a point at an arbitrary radius r1 (say r1 = 1) along a ray at angle 1 from the origin of the (fhx, fhy) space, follow the tangent angle 1 (i.e., integrate) until 2 is reached. Then using angle 2, integrate until 3 is reached, and so on. (More sophisticated interpolating numerical integration could instead be used.) Continue around angularly until the curve reaches 1 again. If the normal force plot comes from a scale-invariant objective function, the curve will close at 1. The key point is that for scale-invariant objective functions, the tangent direction depends only on the angle of the force fh, not the magnitude ||fh||, and therefore the data do not have to be derived from the same isocost contour to be able to reconstruct an isocost contour.
|
Transforming to joint space
For the two-joint arm of Fig. 1B, isocost contours in the hand-force space are transformed to isocost contours in the joint-torque space by the relation
 | (3) |
= (1, 2)T is a vector of shoulder and elbow torques, and the arm Jacobian J(
) is
 |
Ellipse fitting of isocost contours
In the RESULTS section we see that isocost contours in joint-torque space appear rather elliptical, so we can fit ellipses to the data. We use the general form Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey = 1, where A, B, and C describe the shape of the ellipse, and D and E describe its offset from the origin. The coefficients A, B, C, D, and E can be found by a least-squares fit minimizing 
(1 – Ax2 – 2Bxy – Cy2 – 2Dx – 2Ey)2 over the data points. Defining features of the ellipse are its orientation, given by tan–1 2B/(A – C), and its eccentricity, given by
 |
 |
 |
Biomechanical modeling
Joint torques are caused by a complex set of uniarticular and biarticular muscles crossing both the shoulder and the elbow (An et al. 1981; Meek et al. 1990; Murray et al. 2000; Pigeon et al. 1996; van der Helm 1994; Wood et al. 1989). The torque generated by each muscle is a function of the muscle tension stemming from muscle activation and the joint-angle–dependent moment arms based on the bone attachment points. The maximum tension available from a muscle is roughly a function of the physiological cross-sectional area (PCSA) and muscle stretch (and, in nonisometric settings, the rate of lengthening or shortening).
To simplify the model, we follow van Bolhuis and Gielen (1999) and Gomi (2000) and combine the muscles into 6 muscle groups: shoulder extensor and flexor, elbow extensor and flexor, and biarticular extensor and flexor. We define the muscle tension vector 
= (se, sf, ee, ef, be, bf)T 
6 to capture the tension of each of these groups of muscles. All elements of the vector must be nonnegative, indicating that each muscle group is capable of pulling only. This simplification into muscle groups makes the assumption that all muscles in each group are activated proportionally (van Bolhuis and Gielen 1999). With this model, the joint torques are obtained from the muscle tensions by
 | (4) |
) 2x6 is a matrix of joint-angle–dependent moment arms.
Figure 5 shows a model of the arm with these 6 muscle groups (adapted from Gomi 2000). By Eq. 3, torque arising from shoulder monoarticular muscles causes hand forces along the line of the forearm, torque arising from elbow monoarticular muscles causes hand forces along the line through the shoulder, and biarticular muscles with 1 = 2 generate hand forces parallel to the upper arm.
|
 | (5) |
2 and 6, there is an infinite set of muscle tension vectors that generate a specified fh. Thus, in addition to the freedom to provide any force normal to the rail, the subject has freedom in how to share the load across muscle groups.
We consider the following minimization models as candidates for interpreting experimental isocost contours. Some of the muscle load-sharing models were considered for isometric force generation by Gomi (2000) and van Bolhuis and Gielen (1999).
i i2. For a stationary robot arm with identical DC motors at the shoulder and elbow, this solution minimizes the electrical power to the motors. TW, where W is a positive-definite 2 x 2 symmetric matrix (only 3 unique elements). Isocost contours are ellipses in the joint-torque space, a generalization of the T2 model, where W is the identity matrix and the isocost contour is a circle in joint-torque space. i ik, i {se, sf, ee, ef, be, bf}. The model k = 1 was proposed by Yeo (1976), and Nelson (1983) and Hogan (1984) suggest that metabolic power consumed by a muscle is proportional to the square of muscle force k = 2. i (i/PCSAi)k where PCSAi is the physiological cross-sectional area of muscle i (Crowninshield and Brand 1981). This is a measure of the activation of the muscle. There is some evidence that muscle endurance time is inversely proportional to (/PCSA)3 (Prilutsky et al. 1998). ). These values are lumped parameters obtained from data found in Meek et al. (1990). In these parameters, the moment-arm matrix A is independent of the joint angles. A definitive test of optimization models would, of course, require a method for obtaining the parameters of Table 2 for each subject.
|
The linear models MT1 and MS1 predict activation of only one of the muscle groups for a given task, whereas higher-order models predict greater sharing of the load across the muscle groups. Each of the models predicts a normal force plot that can be compared to experimental data.
To obtain the prediction for model T2, let fn represent the force applied against the constraint, where fn is a unit vector normal to the tangential force ft applied by the subject to resist motion. Then satisfies the equation
 |
To obtain the predictions of models MTk and MSk, k = 1, 2, 3, we solve for the tension vector minimizing the objective function, subject to 0 (all muscles pulling) and
 |
), C code implementing sequential quadratic programming. |
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
|
Intertrial consistency
The intertrial variations in the normal force curves over the 3 trials are small, indicating that resolution of the force freedom is systematic rather than random. A glance at Fig. 6 (subject 1) shows that the envelope of normal forces over the 3 trials is narrow relative to the peak normal forces. For subject 1, the average variation in normal forces applied at a particular constraint angle, as a percentage of the range of average normal forces over all angles, was 16.6% for the light weight and 15.3% for the heavy weight. Other subjects displayed similar behavior. For subjects 1 through 8, the average intertrial variation was 22.5 ± 7.3% for the light weight and 19.4 ± 5.0% for the heavy weight.
Convexity
As described in the METHODS section, a nonconvex objective function implies discontinuities in the normal force plot as a function of the tangential force angle. Although it is impossible to prove the absence of discontinuities in the underlying normal force curves from sampled data, the data in Fig. 7 appear to be indicative of smooth underlying curves. This supports convexity of the underlying objective function.
Scale invariance
For our experiment, the scale-invariance hypothesis can be written
 |
To further test this hypothesis, we pooled all 8 subjects data and performed a 2-way ANOVA with "constraint angle" and "force magnitude" as experimental factors. The results showed no main effect for the "force magnitude" factor on the data (F = 1.143, P = 0.286). There was no evidence of system deviation from the scaling hypothesis across subjects for "constraint angle x force magnitude" interaction (F = 0.586, P = 0.884, adjusted R2 = 0.62). We then performed 2-way ANOVAs for each subject individually (n = 8). The results showed no main effect for the "force magnitude" factor except for one subject. The results also showed that there are some idiosyncratic deviations from the scale-invariance hypothesis within each subject attributed to the "constraint angle x force magnitude" interaction. However, the pooled analysis, along with correlation coefficients for a linear fit to the scaling hypothesis (0.876 ± 0.053), show that the data reasonably support scale invariance for the tangential forces we tested.
Reconstructing isocost contours
The isocost contour reconstruction procedure outlined in the METHODS section results in a closed curve only if the normal force data is zero mean. We can see that the experimental normal force curves are in fact approximately zero mean. In general, the amount of uniform shift (raising or lowering) of a normal force curve to achieve zero mean is small relative to the maximum normal force, and in nearly all cases the shifted curve remains within the 3-trial variability (see, for example, Fig. 6). The ratio of the amount of shift |
f| to the range (from min to max) of the average normal forces for subjects 1 through 8 is 4.3 ± 2.7 percent for the light weight and 3.0 ± 3.0 percent for the heavy weight. This is significant, because although all scale-invariant objective functions predict zero mean normal force curves, this is not true for general non-scale-invariant objective functions. The fact that the experimental data are approximately zero mean is further evidence of the CSI objective function model.
To reconstruct a subject's isocost contour, we first shift the curve of average normal forces by subtracting the mean value of the normal force. This shifts the curve uniformly up or down and produces a zero mean curve. We then apply the numerical integration scheme described in the METHODS section. The results for subjects 1 through 8 at the hand position (0, 45 cm) in the shoulder frame are shown in Fig. 8. Because of the scale-invariance hypothesis, only the shape of the isocost contours is of interest; their sizes are arbitrary.
|
). The long axis of the isocost contour for most subjects passes approximately through the shoulder or leans to pass between the shoulder and the elbow. This is another common feature of stiffness ellipses. A local minimum (maximum) of an isocost contour is defined as a point such that no nearby point on the contour is closer to (further from) the origin, in a Euclidean sense. If ft lies on a local maximum or minimum of an isocost contour, then the predicted normal force is zero. Therefore, every zero crossing in the normal force data predicts a local extremum in the isocost contour at the angle of the vector ft. More specifically, if the slope at the zero crossing is negative (i.e., the normal force changes from positive to negative as the constraint angle increases), it is a local minimum, and otherwise it is a local maximum. Any closed curve containing the origin must attain an equal number of local minima and maxima, with a minimum of one each. Every average normal force curve in our experiments showed 4 zero crossings (Fig. 7), predicting 2 local maxima and 2 local minima in the isocost contours for all subjects.
The key points of the reconstructed isocost contours are that 1) they are independent of any strong biomechanical modeling assumptions apart from CSI and 2) they represent how the constraint force freedoms are used by subjects in solving static manipulation tasks.
Configuration dependency of isocost contours
To investigate the dependency of the isocost contours on arm configuration, we performed similar experiments with subjects 9 through 14 at 5 different hand positions shown in Fig. 9A: (0, 45 cm) as before (labeled CENTER), (0, 55 cm) (labeled FAR), (0, 35 cm) (labeled NEAR), (31.8 cm, 31.8 cm) (labeled RIGHT), and (–31.8 cm, 31.8 cm) (labeled LEFT) in the shoulder frame. LEFT and RIGHT are obtained from CENTER by rotating ±45° about the shoulder. The procedure is the same as with the previous experiments, but only one weight (1.2 kg, tangential force of 11.8 N) is used.
The isocost contours for a typical subject are shown in Fig. 9B. We can see that the isocost contour at RIGHT and LEFT have shapes similar to the isocost contour at CENTER, rotated approximately ±45°. In contrast, changing both the elbow and shoulder angle (FAR and NEAR positions) causes the isocost contour to both change shape and rotate. For example, the isocost contour at FAR becomes more anisotropic because of the greater extension of the elbow. These results suggest that the objective function may be better expressed in the joint-torque space rather than the hand-force space, as discussed next.
Invariance in joint space
To investigate the dependency of the isocost contours on arm configuration, we transform isocost contours in the hand-force space to isocost contours in the joint-torque space (Eq. 3). Figure 10 shows the isocost contours of Fig. 9 expressed in the joint-torque space. We see that the joint-torque isocost contours are nearly constant over the different arm configurations. These results are typical for all subjects.
|
Assuming zero offset from the origin, a torque ellipse can be expressed in the form TW = c, where c > 0 is a constant (i.e., the WT2 model of the METHODS section). A typical W matrix obtained by a least-squares fit to a joint-torque isocost contour is
 |
In Mussa-Ivaldi et al. (1985), stiffness at the hand is also interpreted in terms of a positive-definite stiffness matrix K, invariant to the arm configuration when expressed as a stiffness matrix R in joint space. If 
is the vector of joint angle displacements from the equilibrium point, the potential energy stored in the springlike muscles can be written TR. By the relation = R, this potential energy can be rewritten as TR–1. Thus the inverse of the stiffness matrix R–1 can be used as the W matrix in the WT2 model with the following interpretation: iso-cost contours are contours of constant potential energy, and subjects choose a hand force that minimizes the stored potential energy in the spring-like muscles. The inverse of the stiffness matrix found in Mussa-Ivaldi et al. (1985), normalized so that the top left element is identical to the W matrix given above, is
 |
Biomechanical modeling
The 9 force-generation models described in the METHODS section predict hand-force isocost contours as shown in Fig. 11 using the physiological parameters of Table 2. Note the strong anisotropy of the MSk isocost contours, attributed to the large PCSA of the uniarticular shoulder muscles. For the WT2 isocost contour shown, the positive-definite W matrix is the inverse of the stiffness matrix R–1 of Mussa-Ivaldi et al. (1985), as described above.
|
) and the measured upper and forearm lengths for each individual subject. The results are shown in Fig. 12. The HAND model is not included because it predicts zero normal forces for all experiments, and thus can be discarded as a candidate.
|
2.
A weakness of the approach for the MTk and MSk models is that we are forced to assume physiological parameters, and published parameters demonstrate significant variations. For instance, the parameters used by van Bolhuis and Gielen (1999) are significantly different. Using their parameters, the correlation coefficients are as shown in Fig. 13. With these parameters, the MS3 and MS2 models outperform the MT3 and MT2 models, although the correlation is somewhat less than obtained with the MT2 and MT3 models under Gomi's parameters. One reason the MSk models become more competitive is that the PCSA values of the shoulder uniarticular muscles given by van Bolhuis and Gielen are closer to those of the other muscles, meaning that the isocost contours of the MSk models are more isotropic than those under the Gomi parameters.
|
and van Bolhuis and Gielen (1999). For the MT2 and MT3 models, Gomi's parameters yield isocost contours more closely resembling the experimental data. For the MS2 and MS3 models, the isocost contours obtained using van Bolhuis and Gielen's parameters are superior. This is consistent with the correlation coefficients results.
|
; Gomi 2000; Gribble et al. 1998; Lemay and Crago 1996; van Bolhuis and Gielen 1999), as shown in Table 3. For simplicity, in the Monte Carlo tests we assume all physiological parameters are uniformly distributed within 1SD of the mean. The resulting correlation coefficients are 0.28 ± 0.24 for MT1, 0.43 ± 0.25 for MT2, 0.48 ± 0.24 for MT3, 0.07 ± 0.21 for MS1, 0.24 ± 0.21 for MS2, 0.31 ± 0.21 for MS3, 0.66 ± 0.16 for T2, and 0.75 ± 0.15 for WT2. (Note that the T2 and WT2 models do not use the physiological parameters, so their SDs show only the variation between subjects.) The SDs in the MT and MS models are relatively large, meaning that the model predictions change dramatically according to changes in the physiological parameters.
|
Ci/Pj), where Ci is the correlation coefficient and Pj is the physiological parameter. This sensitivity is a linear estimate of the percentage change in the variable Ci caused by a 1% change in the parameter Pj. The sensitivity index is relatively small for all parameters (the mean value is <1 for most models, and the SD is very small except in the MT1 and MS1 models), which shows that the model predictions are not particularly sensitive to small changes in any particular parameter. The large percentage variations in the PCSA parameters reported in the literature, however, when taken together, lead to significant variations in the model predictions. We also did Monte Carlo tests to compare the predictions with the experimental data for subjects 9 through 14 at 5 different hand positions. The results are similar to the previous results for subjects 1 through 8. Although there is little conclusive that we can say to validate or invalidate particular optimization models (other than the MT1 and MS1 models, which are inferior to their higher-exponent counterparts), the simple WT2 model is competitive for any set of physiological parameters. We return to this in the discussion.
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
Our use of a CSI objective function is as a descriptor and predictor of behavior in constrained tasks. It does not require that we commit to a particular interpretation of it. For instance, it may describe that subjects' natural behavior minimizes some notion of "effort." Although we find this interpretation appealing, the objective function could also describe properties of the motor control system that resist simple interpretation of behavior as minimization of effort. For example, Krylow and Rymer (1997) argue that smooth motions in motor control may arise largely from intrinsic muscle mechanics.
Our observation that objective functions in the hand-force space are CSI is consistent with most existing models for static muscle load-sharing (sum of muscle group stresses or tensions raised to the power of 1). Each of these models defines a CSI objective function in the muscle stress or tension space, and linear mappings (from muscle stress to muscle tension, from muscle tension to joint torques, and from joint torques to hand forces) preserve convexity and scale invariance. Previous research has also found that as the direction of an applied force remains fixed but the magnitude is scaled, the muscle activation and force patterns also scale, further evidence of scale invariance (Buchanan et al. 1986; Flanders and Soechting 1990; Valero-Cuevas et al. 2000). We note that there is recent evidence for an objective function that is the sum of linear and squared terms of the muscle tension (FCT van der Helm, personal communication), which is not scale invariant, but such scaling effects would become more noticeable at larger percentages of maximum force, which were not explored in this study.
Relationship to stiffness ellipses
The approximately elliptical shape of the joint-torque isocost contours, and their approximate invariance to arm configuration, brings to mind the stiffness ellipse description of natural stiffness at the hand (Gomi and Osu 1998; McIntyre et al. 1996; Milner 2002; Mussa-Ivaldi et al. 1985; Perreault et al. 2001). The joint-torque isocost contours of Figure 10 and the correlation coefficients of Figures 12 and 13 also indicate that the 3 unique entries of the symmetric positive-definite weighting matrix W in the WT2 model provide a reasonable low-complexity description of subject behavior. This model better captures the shape of the joint-torque isocost contours (approximately elliptical, not aligned with the 1–2 axes) than the joint-torque circle of the T2 model. The W matrix used in our analysis is the inverse of the joint stiffness matrix from Mussa-Ivaldi et al. (1985), which leads to the interpretation that subjects minimize the potential energy stored in springlike muscles while stabilizing the manipulandum.
It is important to keep in mind, however, that stiffness ellipses and isocost contours are not the same thing; the former express the behavior of the arm in response to brief perturbations, whereas the latter express how subjects actively resolve force freedoms.
Generalizations and assistive guides
The work described in this paper can be extended in at least 2 ways: by extending to partially constrained reaching tasks, similar to the crank-turning work of Russell and Hogan (1989) and Svinin et al. (2003), where the arm dynamics become significant; and by increasing the number of degrees of freedom of the arm and the number of force freedoms to be resolved. We have begun work toward the former extension by designing and building a planar manipulandum that can implement arbitrary constraint curves in the plane (Worsnopp et al. 2004). This manipulandum is superior to traditional robotic manipulanda at enforcing smooth constraints because the constraint is generated by a steerable wheel rolling on a table. For the latter extension, the notions of orthogonal "tangential" and "normal" (workless) force and torque subspaces must be generalized properly, according to the kinetic energy metric of the manipulandum.
One reason we are interested in these generalizations is that an eventual goal of this work is to design constraint surfaces to assist a human in manipulating a load from one configuration to another. Constraint surfaces are passive and inherently safe to interact with, and a properly designed constraint surface or guide rail can improve the ergonomics of a repetitive material handling task. Before tackling this problem, however, we must understand how humans naturally take advantage of the presence of kinematic constraints. The work described in this paper is a step toward that understanding.
|
APPENDIX |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
and the associated tangent direction are derived from the normal force plot, 
is the constraint angle, r is the current radius of the isocost contour, d is the increment of , and dr is the increment of r. We have
 | (A1) |
 | (A2) |
 | (A3) |
) = r(0)
 |
 | (A4) |
 |
 |
 |
 |
 |
= (/2) + (for fn < 0) or = (3/2) + (for fn > 0), so finally we get
 |
|
GRANTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
|
ACKNOWLEDGMENTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
|
FOOTNOTES |
|---|
1 In generalized force spaces representing both forces and torques, "tangential" forces do work on the manipulandum, whereas "normal" forces are workless. Tangential and normal forces are orthogonal by the inertia metric of the manipulandum rather than the standard Euclidean metric. For our system, the inertia metric is equivalent to the Euclidean metric. 
Address for reprint requests and other correspondence: K. M. Lynch, Mechanical Engineering Department, 2145 Sheridan Road, Evanston, IL 60208 (E-mail:kmlynch{at}northwestern.edu)
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
An KN, Hui FC, Morrey BF, Linscheid RL, and Chao EY. Muscles across the elbow joint: a biomechanical analysis. J Biomech 14: 659–669, 1981.[CrossRef][Web of Science][Medline]
Buchanan TS, Almdale DP, Lewis JL, and Rymer WZ. Characteristics of synergic relations during isometric contractions of human elbow muscles. J Neurophysiol 56: 1225–1241, 1986.
Crowninshield RD and Brand RA. A physiologically based criterion of muscle force prediction in locomotion. J Biomech 14: 793–801, 1981.[CrossRef][Web of Science][Medline]
Flanders M and Soechting J. Arm muscle activation for static forces in three-dimensional space. J Neurophysiol 64: 1818–1837, 1990.
Flash T and Hogan N. The coordination of arm movements: an experimentally confirmed mathematical model. J Neurosci 5: 1688–17703, 1985.[Abstract]
Garner BA and Pandy MG. Estimation of musculotendon properties in the human upper limb. Ann Biomed Eng 31: 207–220, 2003.[Medline]
Gomi H. Anisotropic stiffness reduction during constrained multijoint arm movement. Proc 20th Ann Int Conf IEEE Eng Med Biol Soc, 1998, vol. 20, p. 2336–2337.
Gomi H. Minimum principles for muscle-coordination evaluation by the directional preference of muscle activation during multijoint-arm force regulation tasks. Proc IEICE Neurocomput Tech Meeting, 2000, p. 9–16.
Gomi H and Osu R. Task-dependent viscoelasticity of human multijoint arm and its spatial characteristics for interaction with environments. J Neurosci 18: 8965–8978, 1998.
Gribble PL, Ostry DJ, Sanguineti V, and Laboissiere R. Are complex control signals required for human arm movement? J Neurophysiol 79: 1409–1424, 1998.
Harris CM and Wolpert DM. Signal-dependent noise determines motor planning. Nature 394: 780–784, 1998.[CrossRef][Medline]
Hogan N. Adaptive control of mechanical impedance by coactivation of antagonist muscles. IEEE Trans Automat Contr 29: 681–690, 1984.[CrossRef]
Krylow AM and Rymer WZ. Role of intrinsic muscle properties in producing smooth movements. IEEE Trans Biomed Eng 44: 165–176, 1997.[CrossRef][Web of Science][Medline]
Kuo AD. A mechanical analysis of force distribution between redundant, multiple degree-of-freedom actuators in the human: implications for the central nervous system. Hum Mov Sci 13: 635–663, 1994.[CrossRef]
Lawrence C, Zhou JL, and Tits AL. User's Guide for CFSQP, version 2.3, No. 94-16. College Park, MD: Institute for Systems Research, University of Maryland, 1994.
Lemay MA and Crago PE. A dynamical model for simulating movements of the elbow, forearm, and wrist. J Biomech 29: 1319–1330, 1996.[Medline]
Lynch KM, Liu C, Sorensen A, Peshkin MA, Colgate JE, Tickel T, Hannon D, and Shiels K. Motion guides for assisted manipulation. Int J Robot Res 21: 27–43, 2000.[CrossRef]
McIntyre J, Mussa-Ivaldi FA, and Bizzi E. The control of stable arm postures in the multi-joint arm. Exp Brain Res 110: 248–264, 1996.[Web of Science][Medline]
Meek SG, Wood JE, and Jacobsen SC. Model-based, multi-muscle EMG control of upper-extremity prostheses. In: Multiple Muscle Systems: Biomechanics and Movement Organization, edited by Winters JM and Woo SLY. New York: Springer-Verlag, 1990.
Milner TE. Contribution of geometry and joint stiffness to mechanical stability of the human arm. Exp Brain Res 143: 515–519, 2002.[CrossRef][Web of Science][Medline]
Murray WM, Buchanan TS, and Delp SL. The isometric functional capacity of muscles that cross the elbow. J Biomech 33: 943–952, 2000.[CrossRef][Web of Science][Medline]
Mussa-Ivaldi FA, Hogan N, and Bizzi E. Neural, mechanical, and geometric factors subserving arm posture in humans. J Neurosci 5: 2732–2743, 1985.[Abstract]
Nelson WL. Physical principles for economies of skilled movements. Biol Cybern 46: 135–147, 1983.[CrossRef][Web of Science][Medline]
Pan P, Lynch KM, Peshkin MA, and Colgate JE. Static single-arm force generation with kinematic constraints. Proc 2004 IEEE Int Conf on Robotics Autom, 2004, vol. 3, p. 2794–2800.
Perreault EJ, Kirsch RF, and Crago PE. Effects of voluntary force generation on the elastic components of endpoint stiffness. Exp Brain Res 141: 312–323, 2001.[CrossRef][Web of Science][Medline]
Pigeon P, Yahia L, and Feldman AG. Moment arms and lengths of human upper limb muscles as functions of joint angles. J Biomech 29: 1365–1370, 1996.[CrossRef][Web of Science][Medline]
Prilutsky BI, Isaka T, Albrecht AM, and Gregor RJ. Is coordination of two-joint leg muscles during load lifting consistent with the strategy of minimum fatigue? J Biomech 31: 1025–1034, 1998.[CrossRef][Medline]
Russell D and Hogan N. Dealing with constraints: a biomechanical approach. Proc IEEE Conf Eng Med Biol, 1989, p. 892–893.
Scheidt RA, Reinkensmeyer DJ, Conditt MA, Rymer WZ, and Mussa-Ivaldi FA. Persistence of motor adaptation during constrained, multi-joint arm movements. J Neurophysiol 84: 853–862, 2000.
Svinin M, Ohta K, Luo ZW, and Hosoe S. Towards understanding of human movements constrained by the external environment. Proc IEEE/RSJ Int Conf Intell Robots Syst, 2003, p. 155–161.
Tickel T, Hannon D, Lynch KM, Peshkin MA, and Colgate JE. Kinematic constraints for assisted single-arm manipulation. Proc 2002 IEEE Int Conf Robotics Autom, 2002, vol. 2, p. 2034–2041.
Todorov E and Jordan MI. Smoothness maximization along a predefined path accurately predicts the speed profiles of complex arm movements. J Neurophysiol 80: 696–714, 1998.
Uno Y, Kawato M, and Suzuki R. Formation and control of optimal trajectory in human multijoint arm movement: minimum torque change model. Biol Cybern 61: 89–101, 1989.[CrossRef][Web of Science][Medline]
Valero-Cuevas FJ, Towles JD, and Hentz VR. Quantification of fingertip force reduction in the forefinger following simulated paralysis of extensor and intrinsic muscles. J Biomech 33: 1601–1609, 2000.[Medline]
van Bolhuis BM and Gielen C. A comparison of models explaining muscle activation patterns for isometric contractions. Biol Cybern 81: 249–261, 1999.[CrossRef][Web of Science][Medline]
van der Helm FCT. A finite element musculoskeletal model of the shoulder mechanism. J Biomech 25: 551–569, 1994.
van der Helm FCT and Veeger HEJ. Quasi-static analysis of muscle forces in the shoulder mechanism during wheelchair propulsion. J Biomech 29: 39–52, 1996.[Medline]
Wood JE, Meek SG, and Jacobsen SC. Quantitation of human shoulder anatomy for prosthetic arm control—I. Surface modeling. J Biomech 22: 273–292, 1989.[CrossRef][Web of Science][Medline]
Worsnopp T, Peshkin MA, Colgate JE, and Lynch KM. Controlling the apparent inertia of passive human interactive robots. Proc 2004 IEEE Int Conf Robotics Autom, 2004, vol. 2, p. 1179–1184.
Yeo BP. Investigations concerning the principle of minimal total muscular force. J Biomech 9: 413–416, 1976.[Medline]
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
