## Abstract

When humans rotate their wrist in flexion-extension, radial-ulnar deviation, and combinations, the resulting paths (like the path of a laser pointer on a screen) exhibit a distinctive pattern of curvature. In this report we show that the passive stiffness of the wrist is sufficient to account for this pattern. Simulating the dynamics of wrist rotations using a demonstrably realistic model under a variety of conditions, we show that wrist stiffness can explain all characteristics of the observed pattern of curvature. We also provide evidence against other possible causes. We further demonstrate that the phenomenon is robust against variations in human wrist parameters (inertia, damping, and stiffness) and choice of model inputs. Our findings explain two previously observed phenomena: why faster wrist rotations exhibit more curvature and why path curvature rotates with pronation-supination of the forearm. Our results imply that, as in reaching, path straightness is a goal in the planning and control of wrist rotations. This requires humans to predict and compensate for wrist dynamics, but, unlike reaching, nonlinear inertial coupling (e.g., Coriolis acceleration) is insignificant. The dominant term to be compensated is wrist stiffness.

- kinematics
- dynamics

wrist rotations are essential for proper upper limb function but have received little attention, despite the fact that common disorders such as stroke often result in significant wrist impairment. To improve the assistance and rehabilitation of wrist function requires a more thorough understanding of wrist rotations, in terms of both biomechanics and neural control. Wrist rotation paths exhibit a distinctive pattern of curvature (Charles 2008; Charles and Hogan 2010; Hoffman and Strick 1999), but the origin of that curvature is unclear. This report reveals the probable cause of this curvature pattern and what it implies about how the nervous system controls wrist rotations.

During wrist rotations, the hand draws paths on a roughly spherical surface surrounding the wrist joint. These spherical paths could a priori be straight, like great arcs on a globe, or curved. We recently characterized the spatial characteristics of wrist paths between a central target (in neutral wrist position) and peripheral targets requiring flexion-extension (FE), radial-ulnar deviation (RUD), or a combination (Charles and Hogan 2010). We found that while individual wrist rotations show substantial variability, on average wrist rotation paths show an intriguing pattern of curvature: In general, *1*) outbound and inbound paths curve to opposite sides of a straight line (Fig. 1); *2*) movements in the same direction but to opposite targets curve to the same side (for example, moves from an extended wrist position to neutral position and moves from neutral position to a flexed position, both of which involve pure flexion, curve toward radial deviation—see Fig. 1); and *3*) fast movements are more curved than comfortably paced movements.

In addition, from a study by Kakei and colleagues (Kakei et al. 1999), we learn that *4*) the pattern of curvature rotates with pronation-supination (PS) of the forearm (Fig. 1 of Kakei et al. 1999).

What is the origin of this pattern of path curvature? Is it intentional or unintentional? Is it neurally controlled or simply a consequence of wrist biomechanics? In this study, we simulate various aspects of wrist dynamics and demonstrate that the observed pattern of curvature can be fully explained by the passive stiffness of the wrist joint. We also provide evidence against other possible causes, making stiffness the most likely candidate. Finally, we discuss what the observed pattern of curvature and its likely cause imply about how the nervous system controls wrist rotations.

## METHODS

To investigate the cause of the observed pattern of curvature, we simulated wrist dynamics under a variety of conditions, varying the contributions of inertia, damping, stiffness, gravity, and input torque to determine which of these parameters (if any) could cause path curvature. This section describes the model and parameters used to simulate wrist dynamics.

### Model of Wrist Rotation Dynamics

#### Equations of motion.

The pattern of curvature was observed in wrist rotations of moderate size (±15°). The dynamics of wrist rotations of this size are well-approximated^{1} by a set of two linear, coupled equations that can be written in matrix form as (Charles and Hogan 2011):
where β represents wrist orientation in FE (β is positive in flexion) and γ represents wrist orientation in RUD (γ is positive in ulnar deviation). The matrices *I*, *B*, and *K* represent passive tissue properties: *I* is the inertia tensor of the hand about the wrist joint, and *B* and *K* are the passive damping and stiffness tensors of the wrist joint, respectively (due to the damping and stiffness of associated muscles and ligaments). *M*_{β} and *M*_{γ} represent the torque generated by active muscle contraction^{2} in FE and RUD, respectively (*M*_{β} is positive toward flexion, and *M*_{γ} is positive toward ulnar deviation). The last term accounts for the gravitational torque pulling the hand into ulnar deviation (*m* is the mass of the hand, *g* is the gravitational acceleration, and *r* is the distance from the wrist joint to the center of mass of the hand).^{3}

Because wrist inertia is well-approximated by a diagonal matrix, the equations of motion are coupled through the off-diagonal terms in damping and stiffness (*B*_{βγ} and *K*_{βγ}). In other words, the torque required to rotate the wrist in FE (*M*_{β}) depends not only on movement in FE (β, β̇, and β̈) but also on movement in RUD (γ and γ̇), and vice versa (Charles and Hogan 2011).

#### Solutions to equations of motion.

By choosing the torques applied to the wrist (*M*_{β} and *M*_{γ}) and solving *Eq. 1* for wrist orientation (β and γ), we determined the effect of inertia, damping, stiffness, gravity, and input torque on path shape. We solved *Eq. 1* with a numerical solver (ode45 in MATLAB) for convenience.

Some of the observed pattern of curvature is observed in the step response of a simplified model of wrist rotation dynamics in which the degrees of freedom (DOF) of the wrist are not coupled (*B*_{βγ} = *K*_{βγ} = 0):

### Model Parameters

To investigate the cause of the path curvature, we first simulated wrist rotation paths, using our best estimates of the actual values of the model parameters, and then investigated how changing various parameters changed path shape. The parameter values described in this section represent our best estimates of the actual values. The parameter values used in subsequent simulations are described in the context of each simulation in results and are summarized for convenience in Table 1.

#### Inertia, mass, and moment arm.

The values for *m* (mass of the hand), *r* (distance from the wrist joint to the center of mass of the hand), and *I*_{β} and *I*_{γ} (moments of inertia of the hand about the wrist joint) used in this article represent average values from six young, healthy individuals (3 men, 3 women; age range 19–28) who participated in a prior study (Charles and Hogan 2010). The values were calculated from measurements of link lengths and the regression equations of de Leva (1996).

The mean values for *m* and *r* among the six subjects were *m* = 0.37 kg (range 0.28–0.46 kg) and *r* = 0.060 m (range 0.047–0.071 m). For our simulations, we used *m* = 0.37 kg and *r* = 0.06 m. The mean inertial values were *I*_{β} = 0.0020 kgm^{2} (range 0.0010–0.0033 kgm^{2}) and *I*_{γ} = 0.0022 kgm^{2} (range 0.0011–0.0038 kgm^{2}), and the ratio *I*_{γ}/*I*_{β} (discussed later) was on average 1.12 (range 1.10–1.15). For our simulations, we used *I*_{β} = *I*_{γ} = 0.002 kgm^{2} (making the off-diagonal elements of the inertia tensor equal to zero).

Most of the analysis in this report deals with wrist rotations of the hand alone (i.e., the hand is not grasping any objects). In one part of results, however, we compare our simulations to data from an experiment by Hoffman and Strick in which subjects were asked to make wrist movements while grasping an apparatus that increased the inertia of the wrist rotations (Hoffman and Strick 1999). The inertia of the apparatus in RUD is reported to be 0.0025 kgm^{2}, and we estimated the inertia in FE to be 0.005 kgm^{2}, so the total inertia (hand + apparatus) used in those simulations was *I*_{β} = 0.007 kgm^{2} and *I*_{γ} = 0.0045 kgm^{2}.

#### Stiffness.

Several past studies have measured the passive stiffness of the wrist in FE (Axelson and Hagbarth 2001; De Serres and Milner 1991; Gielen and Houk 1984; Leger and Milner 2000), with values generally ranging from ∼0.5 Nm/rad (Gielen and Houk 1984) to 3 Nm/rad (De Serres and Milner 1991). The passive stiffness in RUD was recently measured to be 1.5 Nm/rad (Rijnveld and Krebs 2007). However, *Eq. 1* models wrist rotations in combinations of FE and RUD and therefore requires the passive stiffness in combinations of FE and RUD (i.e., the complete stiffness tensor *K*). This passive stiffness tensor was recently measured by Domenico Formica in our lab (unpublished data) on 10 young subjects (7 men, 3 women) using a wrist rehabilitation robot (Krebs et al. 2007). Subjects relaxed their upper limb while the robot slowly (<11°/s) rotated the wrist in 24 equally spaced directions involving FE, RUD, and combinations (17° in each direction). For each direction, stiffness was calculated as the slope of the force-displacement relationship in that direction. The four elements of the stiffness tensor were computed from an ellipse fit to the stiffness values in each direction. For the 10 subjects in that study, the mean stiffness tensor was (mean ± standard deviation):

It is convenient to represent stiffness (and damping and inertia) graphically as an ellipse, where the ellipse represents the torque produced by a unit displacement (unit velocity or acceleration for damping and inertia, respectively) rotated about the origin (Mussa-Ivaldi et al. 1985). Each ellipse is characterized by its size, shape, and orientation, which can be computed from its underlying tensor (Dolan et al. 1993). Considering the stiffness ellipse in Fig. 2*A*, two important properties of passive wrist stiffness are immediately apparent. First, the stiffness of the wrist is anisotropic, with greater stiffness in RUD than in FE. This was true for all 10 subjects; the ratio of the major to minor ellipse axes was 1.58 ± 0.385 (range from 1.14 to 2.43). Second, the directions of greatest and least stiffness are not aligned with the directions of RUD and FE. Instead, the major and minor axes are rotated (in the direction of pronation) by 21.2 ± 9.20° (range from 8.8° to 37°) with respect to the RUD and FE axes. It is this misalignment that makes *K*_{βγ} nonzero and couples the DOF through stiffness.

The diagonal stiffness tensor *K*_{p} required for *Eq. 2* was obtained by rotating *K* to lie along the FE and RUD axes (Fig. 2*B*):

#### Damping.

*Equation 1* also requires *B*, the tensor of passive damping coefficients in FE, RUD, and combinations. Unfortunately, measurements of passive damping throughout FE and RUD are not available in the literature. The only published measurement of passive damping in the wrist of which we are aware is for FE only: As part of their investigation of active damping in the wrist, Gielen and Houk measured passive damping in four subjects in FE alone and found values of 0.02–0.03 Nms/rad (Gielen and Houk 1984). When combined with our estimates of inertia and stiffness above, this value of damping gives a low damping ratio (0.25 in FE) and much more overshoot than we observed in experiment. To match what we observed in experiment, we chose the damping coefficient in FE to be 0.064 Nms/rad (resulting in a damping ratio of 0.63).

To estimate the damping in combinations of FE and RUD (and not just in FE), we assumed that the damping ellipse has the same shape and orientation as the stiffness ellipse. We based this assumption on the facts that *1*) passive stiffness and damping in this range of motion (i.e., away from the limits) are thought to be caused by the same phenomenon (stretching of relaxed muscles and tendons) and *2*) several studies of the impedance of the shoulder and elbow have shown that, despite some minor differences, the stiffness and damping ellipses of the arm are similar, particularly in terms of orientation (Dolan et al. 1993; Perreault et al. 2004; Tsuji et al. 1995). Therefore, for this study, we assumed a damping tensor proportional to the stiffness tensor, with *B* = (0.05 s)·*K*:

Importantly, we show in results that the main features of the pattern of curvature are independent of damping. The diagonal damping tensor *B*_{p} required for *Eq. 2* was obtained by rotating *B* to lie along the FE and RUD axes (Fig. 2*B*):

## RESULTS

Results are presented in the following order. First, we demonstrate that the model represented by *Eq. 1* reproduced the observed pattern of curvature. Second, we illustrate that it is the stiffness that causes the simulated paths to curve. Third, we show that the other model elements (inertia, damping, gravity, and input torques) are unlikely causes of the observed pattern.

### A Simple Model Recreates Observed Pattern of Curvature

The simple mass-spring-damper model of the wrist (*Eq. 1*) is able to recreate the pattern of curvature in all directions (compare Fig. 1, *A* and *B*). Outbound and inbound paths curve to opposite sides. Movements in the same direction curve to the same side as observed in experiment (for example, movements from ulnar deviation to the central target and movements from the central target to radial deviation, both of which involve pure radial deviation, curve toward flexion).

### Effect of Stiffness

The effect of stiffness is best illustrated in several steps (Fig. 3).

Isotropic stiffness produces paths that are perfectly straight, with coincident outbound and inbound paths (Fig. 3*A*).

Anisotropic (but uncoupled) stiffness produces paths that are straight in the cardinal directions (in pure FE or RUD) but curved in the diagonal directions (in combinations of FE and RUD), with outbound and inbound paths curving to opposite sides (Fig. 3*B*). This phenomenon is explained in Fig. 4.

Anisotropic and coupled stiffness produces paths that are curved in cardinal as well as diagonal directions (Fig. 3*C*). The effect of tilting the stiffness ellipse is to couple the DOF, causing curved paths along all axes but two: the principal axes of the stiffness ellipse (Fig. 5). When the orientation of the stiffness ellipse is aligned with a movement direction (i.e., at tilt angles of 0°, 45°, 90°,…), paths in that direction are straight (Fig. 5, *B* and *F*). At a tilt of 22.5°, the curvature is evenly distributed between the eight movement directions (Fig. 5*D*). The mean stiffness tilt measured in 10 subjects was 21° (methods), explaining why the magnitude of the observed curvature is similar in all four axes (Fig. 1*A*). Note that the direction of curvature (i.e., the side to which the path deviates) in our simulations (Fig. 5) matches the observed direction of curvature (Fig. 1*A*) under certain conditions. If the stiffness ellipse is tilted by <45° in either direction (clockwise or counterclockwise), the simulation matches the observation in the diagonal directions. If the stiffness ellipse is tilted counterclockwise by >0° but <90°, the simulation matches the observation in the cardinal directions. It follows that simulation and observation match in all target directions as long as the stiffness ellipse is tilted counterclockwise by >0° and <45°. In the 10 subjects measured for stiffness, the tilt ranged between 9° and 37° (counterclockwise), consistent with our simulations.

Because tilting the stiffness ellipse by <45° in either direction does not change the pattern of curvature in the diagonal directions (but simply modulates how much curvature is present), we can continue our analysis of path curvature by focusing on diagonal paths in the presence of unrotated stiffness and damping (with zero tilt angle), resulting in uncoupled equations of motion (*Eq. 2*) and greatly simplifying the analysis.

### Initial Path Direction

The observed paths curve to one side of a straight line connecting start and end points and do not generally cross this straight line until the very end, if at all (Fig. 1*A*). Therefore, the direction of curvature (i.e., the side to which the path veers) is largely determined by the path's initial direction. As mentioned above, it is sufficient to investigate the diagonal paths in an uncoupled system (*Eq. 2*). If the input torques are step inputs, the initial direction of the resulting diagonal paths can be shown to be (appendix):
*K*_{γ} > *K*_{β}), stiffness causes the initial slope to lean more toward RUD than FE, as observed above.

### Effect of Inertia

The initial path direction—and therefore the direction to which the path veers—is inversely proportional to the ratio of inertias (*Eq. 3*). A greater inertia in one direction will cause a relatively greater acceleration in the other direction, causing the path to accelerate initially more in the other direction. We calculated *I*_{β} and *I*_{γ} in six young, healthy individuals (3 men, 3 women; age range 19–28), using measurements of link lengths and the regression equations of de Leva (1996) (methods). For these six subjects, the ratio *I*_{γ}/*I*_{β} was on average 1.12 (range 1.10 to 1.15). The fact that *I*_{γ} is slightly larger than *I*_{β} would cause the path to start out toward FE, but this slight anisotropy in inertia is not large enough to overcome the greater anisotropy in stiffness (where the ratio of principal axes is 1.58 with a range from 1.14 to 2.43) (methods), creating a mean initial slope that points toward RUD (dγ/dβ = 1.58/1.12 = 1.41). Note that this phenomenon is quite robust: To neutralize the slope would require *I*_{γ}/*I*_{β} to be 1.58, i.e., our estimate of *I*_{γ}/*I*_{β} would have to be off by 41%. To reverse the phenomenon and cause paths that veer in the opposite direction (by the same amount) would require *I*_{γ}/*I*_{β} to be 2.23, i.e., our estimate of *I*_{γ}/*I*_{β} would have to be off by 100%.

### Effect of Damping

The pattern of path curvature can be reproduced by anisotropic and coupled stiffness, even if the damping is isotropic (Fig. 3*C*). Therefore, anisotropic damping is not necessary to reproduce the observed pattern. In fact, the initial slope of the path, which determines the side to which the path veers, is completely independent of damping (*Eq. 3*). Although damping does not affect the direction of curvature, it does affect the amount of curvature, as demonstrated in Fig. 3*D* and Fig. 6.

Damping also affects the amount of overshoot by modifying the damping ratio, which is a function of inertia and stiffness as well. For an uncoupled system, the damping ratio in the FE direction is:
*B*) shows that increasing the inertia to account for the apparatus (methods) produces overshoot similar to that observed by Hoffman and Strick (Fig. 7*A*).

### Effect of Gravity

The initial path direction (and therefore the side to which the path curves) is completely independent of gravity (*Eq. 3*). The effect of gravity is simply to require a greater input torque in RUD (a constant torque) to overcome gravity. Ignoring gravity (and reducing the input torque in RUD accordingly) produces the exact same kinematics and path curvature, so gravity has no effect on the path curvature of our simulations.

### Effect of Input Torque

For simplicity, the simulations up until this point have assumed that the torque inputs were step functions, but many other physiologically plausible torque inputs produce a similar pattern of curvature. In principle any trajectory, including a straight line, could be generated with an internal model of the dynamics equivalent to *Eq. 1*: Given any specification of β(*t*) and γ(*t*), differentiate and substitute to find *M*_{β}(*t*) and *M*_{γ}(*t*), the so-called inverse dynamics computations. The observations we are attempting to reproduce indicate that the biological controller does something much simpler, sending a similar time course of activation to all muscles that produces the movement. A priori, the only constraint on the torque inputs is that their steady-state values be such that the path will end at the desired target. The final orientation produced by any input can be found by setting the derivatives in *Eq. 2* to zero:
_{ss} = γ_{ss}. Combining with *Eq. 4* gives
*Eq. 6*, *M*_{β}(*t*) and *M*_{γ}(*t*) are scaled and shifted versions of each other. Many such inputs cause path curvature remarkably similar to that produced by step inputs, as illustrated in Fig. 8. Note that *Eq. 6* does not represent a necessary condition: Many other, more complicated input torques (which do not satisfy *Eq. 6*) will produce a similar pattern of curvature.

## DISCUSSION

When humans rotate the wrist in combinations of FE and RUD, the resulting paths (like those produced on a screen while using a laser pointer) exhibit a distinctive systematic pattern of curvature. In this report we showed that the stiffness of the wrist joint is sufficient to account for this pattern of curvature. We showed that with experimentally derived parameters a relatively simple model of wrist dynamics reproduces the observed pattern of curvature. Varying the model parameters, we showed that only the passive stiffness of the wrist, because it is anisotropic and misaligned with respect to the anatomical axes of the wrist, can fully account for the observation that paths curve to opposite sides of a straight line between targets, depending on the direction of movement.

### Why Are Fast Wrist Rotations More Curved than Comfortably Paced Motions?

We propose that faster movements have greater path curvature because the neuromuscular system has insufficient time to straighten the path. To understand this hypothesis, consider the following two findings.

First, in an investigation of muscle activity associated with wrist rotations, Hoffman and Strick found that humans made wrist movements by modulating muscle activity in either of two spatiotemporal patterns, termed amplitude graded and temporally shifted (Hoffman and Strick 1999). Amplitude-graded muscle activity is very prevalent and consists of two bursts of muscle activity (an early agonist burst and a late antagonist burst). Because the wrist joint has many muscles acting on both DOF, adjusting the amplitude of the bursts changes the direction of the resulting movement. Temporally shifted muscle activity is less prevalent and is characterized by a single burst of muscle activity occurring after the normal agonist burst but before the normal antagonist burst. Importantly, temporally shifted muscle activity occurred in muscles that pulled in a direction perpendicular to the direction of the movement.

Second, Hoffman and Strick found that monkeys' wrist movements displayed amplitude-graded muscle activity but not temporally shifted muscle activity, and that monkeys' wrist paths were much more curved than humans' wrist paths, leading them to hypothesize that temporally shifted muscle activity functions to reduce the amount of path curvature (Hoffman and Strick 1999). However, the questions of why and how temporally shifted muscle activity functions to straighten paths remained unanswered. If, as the work reported here indicates, stiffness is the origin of path curvature, it can explain why and how temporally shifted muscle activity could reduce path curvature. In an anisotropic stiffness field, a path that starts out toward its intended target will veer off-target (Fig. 4) unless an appropriate force is applied perpendicular to the direction of travel and toward the higher stiffness to “keep the hand on track” toward the target. Such a force could be supplied by temporally shifted muscle activity, which Hoffman and Strick found to occur *1*) not at the beginning or end but during the movement and *2*) in directions perpendicular to the movement direction. In essence, temporally shifted muscle activity would act to keep the path from “slipping down” the stiffness gradient perpendicular to the movement direction (Charles 2008). Thus the present finding that path curvature is likely caused by stiffness also provides a reasonable mechanism by which temporally shifted muscle activity could serve to reduce path curvature, as hypothesized by Hoffman and Strick.

We therefore propose that faster movements have greater path curvature because the neuromuscular system has insufficient time to straighten the path. As movement duration decreases, the gap between the agonist and antagonist bursts of amplitude-graded muscle activity decreases, limiting the opportunity to insert a burst of temporally shifted muscle activity and thereby reducing the ability of temporally shifted muscle activity to straighten the path. With limited temporally shifted muscle activity, fast movements must rely almost entirely on the agonist and antagonist bursts of amplitude-graded muscle activity. In the presence of stiffness anisotropy, the only way for amplitude-graded muscle activity to produce a path that ends on-target is to aim off-target (toward the stiffer direction) and rely on the stiffness gradient to curve the path toward the target.

### Why Does the Pattern of Curvature Rotate with Pronation-Supination of the Forearm?

Wrist stiffness reflects the combined elastic properties of the muscles and ligaments of the radiocarpal and intercarpal joints. Because the wrist resides distal to the forearm, its ligamentous structure rotates with PS of the forearm. Wrist muscles rotate with PS of the forearm as well [though not as much as the wrist joint itself (Kakei et al. 1999)]. Therefore, as the forearm rotates in PS, the stiffness ellipse of the wrist (anisotropic and misaligned) rotates as well and can explain why the pattern of curvature changes with rotation of the forearm.

### Robust Simulation

The inertial and stiffness parameters used in our simulations (methods) represent our best estimates of the actual physical parameters. For mass, moment arm, and inertia, we used the average values from six young, healthy individuals who participated in a prior study. For stiffness, we used average values measured on 10 young, healthy subjects. Damping was chosen to match the amount of overshoot between our simulations and our observed data (methods).

More importantly, the results presented in this report are insensitive to modest inaccuracies in model parameters. For example, we showed that the direction of curvature (i.e., the side to which a path deviates) is independent of damping (*Eq. 3*) and therefore unaffected by inaccuracies in the value of damping used in our simulations. While inertia and stiffness vary greatly between subjects (because of variations in body size), the direction of curvature depends on ratios (of inertia and stiffness in the 2 DOF) that remain quite similar between subjects of different body size. For example, while subjects' inertia in RUD and FE ranged between 0.001 kgm^{2} and 0.0038 kgm^{2} (380% variation) the ratio of inertias only ranged between 1.10 and 1.15 (<5% variation).

The direction of curvature depends on the orientation of the stiffness ellipse but remains the same as long as the orientation of the stiffness ellipse is within a relatively large range (45°). Furthermore, the mean orientation measured on 10 subjects (methods) was right in the middle of this range (21°), i.e., far from where the direction of curvature would change, and the orientation of all 10 subjects was within the range (9–37°).

Interestingly, according to *Eq. 3*, inertial anisotropy affects initial path direction as much as stiffness anisotropy, even though inertial effects are small compared with stiffness effects in wrist rotation dynamics in general (Charles and Hogan 2011). *Equation 3* is not specific to the wrist but holds for any 2-DOF joint whose dynamics can be approximated as linear and uncoupled (we decoupled the dynamics of the wrist by rotating to the principal axes of the stiffness and damping ellipses—the inertia ellipse is a circle and unaffected by rotation because the inertia tensor is symmetrical). Thus, while proximal joints rotate larger inertias and are generally dominated by inertial effects and distal joints rotate smaller inertias and are generally dominated by stiffness effects, the initial path direction would be the same if the ratio of anisotropies were the same and if the dynamics were linear and uncoupled. However, the dynamics of proximal joints are generally highly coupled through inertial interaction torques (Hollerbach and Flash 1982), so *Eq. 3* does not generally apply to proximal joints.

### Active Stiffness

A limitation of this study is that it relies on measurements of passive wrist stiffness (stiffness measured in the nominal absence of muscle activation) even though the stiffness present during movements is a combination of passive stiffness and active stiffness—muscle activation is always accompanied by an increase in net stiffness that may be due to intrinsic muscle mechanics and/or local spinal reflex pathways (Hu et al. 2011). Unfortunately, measurements of wrist stiffness during movement are not available. However, this study demonstrates the passive dynamics that the neuromuscular controller must deal with (through the production of muscle force and active stiffness) and that the passive dynamics themselves (i.e., in the absence of correcting control) predispose paths to curve. In addition, there is evidence from studies of shoulder-elbow movements indicating that while the size of the stiffness ellipse increases during movement (because of the contribution of active stiffness), the shape and orientation of the stiffness ellipse tend to stay relatively unchanged (Mussa-Ivaldi et al. 1985). In this study we showed that it is the shape and orientation (i.e., the anisotropy and misalignment) of the stiffness ellipse—not its size—that can explain the observed pattern of curvature.

### Other Candidate Causes of Path Curvature

We showed that wrist passive stiffness is sufficient to explain the pattern of path curvature observed in wrist rotations. We have also shown quantitatively that inertia, damping, and gravity are unable to account for the curvature pattern. However, there remain other factors that might, in principle, cause path curvature: kinematic, neuromuscular, and feedback-related effects.

#### Candidate kinematic causes.

The 2-DOF motion of the wrist can be approximated as a universal joint, resulting in nonlinear equations of motion that could potentially cause path curvature. However, the pattern of path curvature was observed for moderately sized wrist rotations (±15°), and we have previously shown that for wrist rotations of this size the difference between the nonlinear equations of motion of a universal joint and the linear equations used in this study (*Eqs. 1* and *2*) is negligible (Charles and Hogan 2011).

Wrist paths lie on a roughly spherical surface surrounding the wrist joint, but the pattern of path curvature discussed in this study is generally illustrated on paths that have been projected onto a plane (as in Fig. 1), and this nonlinear projection could, in principle, cause paths to appear curved. However, we have previously shown that the difference between the curvature of the actual, spherical paths and the curvature of their planar projections is negligible for the moderately sized wrist rotations (±15°) in which the pattern of path curvature was observed (Charles and Hogan 2010).

Finally, the kinematics of the individual carpal bones, which produce wrist motion, are complex and could, a priori, cause path curvature. However, such kinematic constraints are unlikely to cause different path curvature in outbound and inbound directions. Furthermore, it is very unlikely that the complex interaction of carpal bones would produce a pattern of path curvature that varies with movement direction as regularly and smoothly as the experimentally observed pattern (Fig. 1).

#### Candidate neuromuscular causes.

Path curvature could also result if muscles pulling in different directions pulled at slightly different times or rates. For example, if the muscles pulling in RUD acted more quickly than the muscles pulling in FE (either because of a difference in the timing of neural signals or because of a difference in muscular dynamics), a movement to a target in radial deviation and flexion might start out toward radial deviation and then curve toward flexion. The inbound path would start out toward ulnar deviation and then curve toward extension, producing an oppositely curved path, as observed. While this example makes it appear that differences in the timing of muscle activity might be the cause of the observed pattern of curvature, there are at least two pieces of evidence against this hypothesis.

First, while Hoffman and Strick did observe differences in the timing of muscle activity, the “temporally shifted” (i.e., delayed) muscle activity did not act in the right direction but rather acted perpendicular to the intended movement direction (see above). For example, for movements involving flexion and some radial deviation, which curved first toward radial deviation and then toward flexion, the temporally shifted muscle activity occurred in extensor carpi radialis longus, an extensor muscle that acts perpendicular to the intended movement direction (and even contributes to radial deviation).

Second, the strongest evidence comes from a study by Hoffman and Strick in which they recorded the path produced by individual stimulation of five wrist muscles (Hoffman and Strick 1999). Upon stimulation of each muscle separately, the hand traced a path that was initially straight but invariably ended up curving toward flexion or extension. In other words, while it is possible that the path curvature could be caused by differences in muscular or neural timing between muscles, stimulation of individual muscles alone produces marked path curvature that matches the experimental observation. Veering toward flexion or extension is fully consistent with stiffness anisotropy: As the hand increases its deviation from neutral position, the stiffness gradient perpendicular to the movement direction will cause it to turn away from the stiffer direction (roughly RUD) and toward the less stiff direction (roughly FE), which is the “path of least resistance.”

The second argument also indicates that the observed pattern is not caused by differences in strength between muscles acting in different directions. In summary, the available evidence indicates that the observed pattern of path curvature is not due to differences in timing or strength between muscles acting in different directions but due to passive wrist stiffness.

#### Candidate feedback-related cause.

Does visual feedback play a role in the generation or modification of path curvature? In our experiments, the pattern of path curvature was most prominent when movements were performed as fast as possible. Such movements were often composed of an initial, high-speed, high-amplitude movement, which placed the hand close to the target, followed by one or several later, low-speed, low-amplitude movements, which brought the hand to rest within the target boundary. Importantly, the observed pattern of curvature stems from the initial, high-speed, high-amplitude movements, not the later, low-speed, low-amplitude movements. For fast wrist movements, where path curvature was observed most prominently, these initial, high-speed, high-amplitude movements were often completed within 200 ms, making it very unlikely that visual feedback played any role in the generation or modification of the path curvature. Interestingly, none of the subjects who participated in the observations of path shape (Charles and Hogan 2010) was aware of the curvature in his/her wrist movements.

### Implications

The findings in this report have four important implications.

First, the fact that humans can make on-target movements in the presence of anistropic and misaligned stiffness implies that the neuromuscular system can predict and compensate for this stiffness. Both strategies discussed above—using temporally shifted muscle activity (for slow movements) and aiming off-target and relying on the stiffness to curve the path toward the target (for fast movements)—rely on the ability to predict the dynamic behavior of the system. The strategy for fast movements (aiming off-target), in particular, relies on prediction (rather than feedback correction) because the aiming occurs before movement onset. This ability to predict and compensate for stiffness during wrist rotations is similar to the demonstrated ability to compensate for inertial interaction torques during reaching movements (Hollerbach and Flash 1982). However, the dynamics are completely different. In reaching, inertial coupling torques are prominent; the terms depending on velocity are comparable to those depending on acceleration. Unimpaired humans learn to compensate for them, but patients with cerebellar damage exhibit incomplete compensation (Bastian et al. 1996). In contrast, in wrist rotations inertial coupling is irrelevantly small. Instead, the coupling to be compensated arises from wrist stiffness. Whether patients with neurological disorders are unable to compensate for wrist stiffness is not known.

Second, Hoffman and Strick's hypothesis that temporally shifted muscle activity functions to straighten paths, strengthened by our explanation of why and how temporally shifted muscle activity could straighten paths, implies that humans care (at least to some degree) that wrist paths be straight. This is consistent with many past studies of reaching movements that demonstrated that path straightness is a predominant factor (among others) in planning and controlling discrete movements.

Third, if temporally shifted muscle activity acts to straighten paths, then the fact that paths are not perfectly straight implies that the observed curvature is either not perceived or not worth reducing. Wrist rotations, especially fast ones, exhibit a relatively large amount of variability in path shape (Charles and Hogan 2010), and it is possible that the path curvature is not perceived amid the noise in path shape. In addition, the path of the hand during wrist rotations (and therefore the path curvature) spans a relatively small angle in the visual field (e.g., compared with arm movements) and may simply go unnoticed. Indeed, of the subjects who participated in the observations of path shape (Charles and Hogan 2010), none was aware of the curvature in his/her wrist movements. If the curvature is perceived, it is possible that further reduction (beyond what is accomplished through temporally shifted muscle activity) is not worth the effort. After all, if straightness of path is a factor in the control of wrist rotations, it is certainly not the only factor.

Fourth, because passive stiffness plays a substantial role in wrist rotations, one might expect a pattern of muscle activity slightly different from the usual triphasic burst pattern; in movements from the neutral position outward, stiffness could (at least partially) replace the role of antagonist muscles in braking movement and maintaining posture. In the study by Hoffman and Strick (1999), antagonist activity does indeed appear smaller than agonist activity, as expected. Also as expected, agonist activity is present during the third phase of the triphasic pattern (to counteract the restoring effect of stiffness), while antagonist activity is small or absent during that phase (since stiffness can fulfill that role). This is different from normal arm movements, which generally occur far away from the limit of their range of motion, where passive stiffness is relatively low. However, similar behavior has been observed when arm movements are made to the end of their range of motion (Berardelli et al. 1996), where passive stiffness increases in relative importance.

We have illustrated the effects of stiffness on path curvature for movements from or to neutral position and have seen that the stiffness gradient causes curvature in all directions except along the principal axes of the stiffness ellipse. However, we would expect wrist stiffness to have a path-curving effect for movements between any two points within the range of motion of the wrist, as long as the points do not both lie on the same principal axis (in which case we would expect a straight path).

### Conclusion

We have shown that a simple mass-spring-damper model of the wrist can recreate the distinctive pattern of path curvature observed during wrist rotations and that it is the stiffness of the wrist—because of its anisotropic and rotated nature—that causes this pattern. Other possible causes (including kinematic, neuromuscular, and feedback-related causes as well as other mechanical causes) were unable to account fully for the observed pattern. Our results imply that in wrist rotations as in reaching humans attempt to make straight paths, and that they predict and compensate for anisotropic wrist stiffness, not inertial coupling, to do so.

## GRANTS

This study was funded in part by the New York State Center of Research Excellence, Contract CO19772; the Eric P. and Evelyn E. Newman Fund; the Whitaker Foundation Graduate Fellowship; and the Harvard-MIT Division of Health Sciences and Technology MEMP Graduate Fellowship.

## DISCLOSURES

S. K. Charles has no conflict of interest. N. Hogan holds equity in the company that distributes the robot used to derive the stiffness measures applied here.

## AUTHOR CONTRIBUTIONS

Author contributions: S.K.C. and N.H. conception and design of research; S.K.C. performed experiments; S.K.C. analyzed data; S.K.C. and N.H. interpreted results of experiments; S.K.C. prepared figures; S.K.C. drafted manuscript; S.K.C. and N.H. edited and revised manuscript; S.K.C. and N.H. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors thank Donna Hoffman and Peter Strick for thought-provoking discussions and the use of their data.

## APPENDIX: DERIVATION OF INITIAL PATH DIRECTION

The step response of the decoupled set of equations (*Eq. 2*) has a well-known analytical solution (Nise 2000). In general, the step response of a second-order system can take on different forms depending on the level of damping in the system. Wrist rotations are underdamped (Lakie et al. 1984; Sinkjaer and Hayashi 1989), so the step response in FE is
*A*_{β} is the magnitude of the step input in *M*_{β},
_{γ} and ζ_{γ} defined the same as for β(*t*).

The initial path slope is given by
*t* = 0,

In order for the path to end up at a diagonal target, the steady-state response in the two DOF must be equal:

## Footnotes

↵1 Modeling wrist dynamics as a set of linear, coupled equations of motion approximates a more anatomically accurate model of the wrist as a universal joint with nonintersecting axes. For wrist rotations of ±15°, the approximation error is extremely small, only 0.80 ± 0.60% (mean ± SD) of the maximum torque (averaged over wrist rotations of various speeds and directions).

↵2 These active torques are functions of displacement and its derivatives (as well as being functions of neural activation) because they include (in addition to pure torque) torques due to changes in stiffness and damping associated with muscle contraction.

↵3 The pattern of curvature described in the introduction and depicted in Fig. 1

*A*was observed with the forearm in the horizontal plane, midway between pronation and supination, with gravity acting in the direction of ulnar deviation.

- Copyright © 2012 the American Physiological Society