## Abstract

Determining the principles used to plan and execute movements is a fundamental question in neuroscience research. When humans reach to a target with their hand, they exhibit stereotypical movements that closely follow an optimally smooth trajectory. Even when faced with various perceptual or mechanical perturbations, subjects readily adapt their motor output to preserve this stereotypical trajectory. When humans manipulate *non*-rigid objects, however, they must control the movements of the object as well as the hand. Such tasks impose a fundamentally different control problem than that of moving one's arm alone. Here, we developed a mathematical model for transporting a mass-on-a-spring to a target in an optimally smooth way. We demonstrate that the well-known “minimum-jerk” model for smooth reaching movements cannot accomplish this task. Our model extends the concept of smoothness to allow for the control of non-rigid objects. Although our model makes some predictions that are similar to minimum jerk, it predicts distinctly different optimal trajectories in several specific cases. In particular, when the relative speed of the movement becomes fast enough or when the object stiffness becomes small enough, the model predicts that subjects will transition from a uni-phasic hand motion to a bi-phasic hand motion. We directly tested these predictions in human subjects. Our subjects adopted trajectories that were well-predicted by our model, including all of the predicted transitions between uni- and bi-phasic hand motions. These findings suggest that smoothness of motion is a general principle of movement planning that extends beyond the control of hand trajectories.

## INTRODUCTION

Determining the underlying principles humans use to plan and execute movements is a vital question in motor neuroscience research (Engelbrecht 2001; Gomi and Kawato 1996; Harris and Wolpert 1998; Wolpert and Ghahramani 2000). Much attention has focused on how humans make point-to-point reaching movements with their hands or manipulate simple rigid objects. When adapting to various perceptual and mechanical perturbations, humans eventually return to making stereotypical smooth movements (Bock 1993; Dizio and Lackner 1995; Flanagan and Lolley 2001; Flanagan et al. 1999; Flanagan et al. 2003; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a). These previous experiments artificially manipulated different parameters (e.g., inertia, stiffness, damping, etc.) of the arm. Such manipulations constitute *parametric* perturbations to the control system because they do not alter the fundamental structure of the arm (e.g., the number of joints, muscles, etc.). Therefore they do not alter the number or type of state variables used to characterize the arm's equations of motion. These equations of motion remain effectively unchanged except for the values of one or more of their parameters (e.g., mass, stiffness, damping, etc.). Accordingly, such parametric perturbations do not require any changes in the movement kinematics required to accomplish a given task. The exact same kinematics can be achieved by adjusting the profile of forces and torques being input into the system. This helps explain why humans learn to regain their original movement patterns despite these perturbations.

However, humans often perform tasks involving non-rigid objects, such as a cup of coffee or a briefcase with a hinged handle. In these tasks, the goal is to impose some particular motion on the *object* (e.g., the coffee in the cup) rather than the hand. Manipulating non-rigid objects poses a fundamentally different control problem than that of moving one's arm (Huang et al. 2002; Lynch and Mason 1999; Schaal and Atkeson 1993). New state variables (i.e., the position and velocity of the *object*) are added to the “arm-plus-object” system being controlled. This then fundamentally alters the structure of the governing equations of motion: entirely new equations must be defined for each new state variable. Such tasks therefore constitute *structural* perturbations to the system being controlled. Consequently, limb motions that were optimal for the unperturbed system may no longer work for the perturbed system (Lynch and Mason 1999). Although one could still adjust his input forces and torques accordingly to achieve the previously optimal motion of the hand, this may result in a completely irrelevant or even unwanted motion of the object. Achieving the desired motion of the *object* requires learning to apply the appropriate sequence of input forces *to the object* (Dingwell et al. 2002). Furthermore, humans cannot control the motions of non-rigid objects directly because those objects are not directly acted on by our muscles. The motion of the object can only be controlled indirectly through our interaction with the dynamics of the object itself.

Given these fundamental differences between parametric and structural perturbations and their implications, it is important to determine whether or not the underlying principles believed to be involved in the planning and execution of reaching movements can be extended to tasks involving the purposeful manipulation of complex dynamic objects. It has been argued that to fully validate any theory intended to describe how movements are planned and executed requires that we apply those theories to so-called “critical tests” involving tasks that are substantially different from the tasks for which those theories were originally formulated (Engelbrecht 2001). We believe tasks involving structural perturbations to the control system, like the one described in the present study, may provide just such a critical test. In particular, the present study was conducted to determine if the principle of achieving optimum smoothness remains a viable goal for motion planning when manipulating complex dynamic objects and to determine what kind of hand motions are required to ensure smooth movements of these objects.

Here we formulate a testable hypothesis, optimally smooth transport (OST), whereby the controller derives the motions of the hand that achieve the maximally smooth motion of the transported object, compatible with the constraints of the hand/object interaction. Our OST model is a generalization of the minimum-jerk principle for reaching (Flash and Hogan 1985) where now the optimization criterion is applied to the object rather than the hand. We applied this principle to the task of transporting a mass on a spring to a target. We demonstrate that this task violates the originally proposed minimum-jerk principle in the sense that the task can *not* be achieved by producing either a minimum-jerk movement of the hand or a minimum-jerk movement of the mass. Our extended OST model of smoothness overcomes these limitations in the original theory. The OST model predicts some movement contexts in which the optimal solutions are similar to those of minimum-jerk and other contexts in which the solutions are very different. Subjects adopted trajectories that were well-predicted by our model under all of these different contexts. Our results demonstrate that subjects learned to alter their hand trajectories accordingly under each test condition so as to achieve an optimally smooth motion of the object. These findings suggest that the CNS achieves near optimal smoothness when planning the movements of the endpoint (in this context, the mass) in a general sense.

## METHODS

### Task

The task we studied was that of transporting a mass-on-a-spring attached to the hand from one point to another in the horizontal plane. The equations of motion for this object were 1 where *r*_{H}(*t*) is a vector describing the trajectory of the hand in the workspace. For motions in the horizontal plane, the mass could oscillate in both the *x* and *y* directions, so *r*_{H}(*t*) = [*x*_{H}(*t*), *y*_{H}(*t*)]′. Note that, unlike real physical springs, *Eq. 1* implies that the slack length of the spring is zero: i.e., that the force applied to the object is zero when *r*_{O} = *r*_{H}. Rearranging *Eq. 1* to solve for *r*_{H} demonstrates that for any object trajectory, *r*_{O}, only one unique hand trajectory will bring that desired object trajectory about 2

The task of transporting the mass to the target can be completed successfully by an infinite variety of object trajectories, just as for point-to-point reaching (Flash and Hogan 1985) (see discussion and Fig. 8 for examples). What *Eq. 2* demonstrates is that the objective of the controller in this task is now to choose some desired object trajectory and then to determine the unique hand trajectory that will produce that desired object trajectory. The question addressed in the present study was whether or not humans would plan and execute this task by adopting basic principles similar to those used in point-to-point reaching.

### Optimally smooth object transport

The mathematical model developed for optimal performance in this task is an extension of the minimum-jerk model for point-to-point reaching (Flash and Hogan 1985; Hogan 1984; Nelson 1983). Briefly, minimum-jerk proposes that reaching trajectories are well-described by fifth-order rectilinear polynomial functions in time. The values of the polynomial coefficients are determined by the boundary conditions (BC) imposed by the task. One way to imagine what these BC mean is to consider a simplified model of this task: that of moving a point mass (i.e., the hand). The goal is for the hand to be “at rest” at the beginning and end of the movement. At rest implies zero force, and therefore zero acceleration (by Newton's second law, *F* = *ma*). Thus there are six independent BC for the task of point-to-point reaching governing the position, velocity, and acceleration of the hand at the beginning and end of the movement. A fifth-order polynomial, having six independent constant coefficients, is the lowest-order polynomial that can exactly satisfy these BC. This solution is predicted by solving an optimization problem that minimizes the mean-squared jerk (i.e., third derivative of position) of the hand trajectory, *r*_{H}(*t*), over the whole movement (Flash and Hogan 1985; Hogan 1984). Thus the minimum jerk solution can be interpreted as the ideal solution obtained for a perfect controller manipulating an ideal hand (i.e., a point mass).

Here we extend this principle of optimal smoothness to the task of transporting a mass on a spring to a target. The primary goal of this task is now to control the motion of both the hand *and* the mass, such that both are at rest at the beginning and end of the movement. This then implies zero acceleration of both the hand and the mass at both the beginning and end of the movement. A first logical assumption of how this task could be achieved might be to apply a minimum-jerk trajectory to the object and then compute the corresponding required hand trajectory from *Eq. 2*. For a reaching movement of length *L* made in time *T*, starting at position *r*_{Oi}, the minimum jerk object trajectory would then be (Flash and Hogan 1985) 3

Differentiating *Eq. 3* twice with respect to time, *t*, demonstrates that this trajectory does satisfy the required BC on the velocity and acceleration of the object trajectory at *t* = 0 and *t* = *T*. However, when these expressions are then substituted back into *Eq. 2* to compute the hand trajectory required to achieve this minimum-jerk object trajectory, it is easily shown that at the endpoints of the movement (*t* = 0 and *t* = *T*) and

Thus the required BC on the hand trajectory are *not* satisfied for any real values of *M*_{O} and *K*_{O}. This is shown graphically in Fig. 1 for a movement of *L* = 0.25 m, made in *T* = 1.0 s, from *r*_{Oi} = 0. It is clear that obtaining a minimum-jerk object trajectory in the present task requires a discontinuous (i.e., physiologically impossible) hand trajectory. It can similarly be shown that imposing a minimum-jerk trajectory on the hand path will violate the required BC on the object path. Thus this task clearly violates the minimum-jerk hypothesis as originally proposed by Flash and Hogan (1985). These hand velocity and acceleration terms can *only* go to zero (and thus satisfy the BC) in the two trivial limiting cases that either *M*_{O} → 0 (i.e., when there is no object) or *K*_{O} → ∞ (i.e., if the object becomes infinitely stiff, i.e., rigid). In both of these limiting cases, the optimal solution then reduces to the minimum jerk solution (*Eq. 3*), as described in the preceding text.

This paradox can, however, be resolved by first recognizing that the task is now to control the motion of both the hand *and* the object. Controlling two independent masses would require satisfying 12 BC (6 each for the hand and object). However, the object and hand are not independent because they are dynamically coupled through the relationship prescribed by *Eq. 1*. Differentiating *Eq. 2* twice demonstrates that the required BC on the hand acceleration can be satisfied by imposing corresponding BC on the third and fourth derivatives of the object trajectory, *r*_{O}(*t*). Thus a unique solution for both trajectories is obtained by imposing 10 *independent* BC on the object trajectory and its derivatives. The lowest-order polynomial that will satisfy these BC is a ninth-order polynomial. This solution is predicted by solving the following dynamic optimization problem, which minimizes the mean-squared-*crackle* (i.e., 5th derivative of position) of the object trajectory, *r*_{O}(*t*), over the whole movement 4

We can solve the unconstrained optimization problem (see Flash and Hogan 1985; appendix A) posed by this minimum-crackle cost function by applying the Euler-Poisson equation 5

Substituting the argument of the cost function from *Eq. 4* into *Eq. 5* and taking the necessary derivatives, yields *d*^{10}*r*_{O}/d*t*^{10} = 0, which when integrated 10 times, produces a ninth-order polynomial trajectory for the object, *r*_{O}(*t*). For the task of making a reaching movement of length *L* in time *T*, the required BC for the object trajectory become

Applying these BC to the ninth-order polynomial and solving for the appropriate polynomial coefficients yields the optimally smooth object trajectory 6 where *r*_{Oi} defines the location of the initial starting position. Computing the appropriate derivatives of *Eq. 6* and substituting into *Eq. 2* yields a second ninth-order polynomial solution for the hand trajectory, *r*_{H}(*t*). Note that the object trajectory (*Eq. 6*) depends only on the reaching time (*T*) and distance (*L*) and is independent of the object's mechanical properties (*M*_{O} and *K*_{O}). Only the hand trajectory will be affected by differences in object dynamics.

### OST model predictions

The OST model of optimum performance in the mass-spring manipulation task is independent of the details of the dynamics of the arm itself. As such, the OST model predicts that hand and object movements will be made along straight lines and that optimal movements of the same distance (*L*) and duration (*T*) will exhibit the same velocity profiles regardless of the location in the workspace (i.e., invariance under translation) or the direction of movement (i.e., invariance under rotation). Furthermore, for any given object and movement duration (*T*), hand and object velocities will scale linearly with movement distance. Object velocity profiles will always be smooth and have a uni-phasic shape (i.e., with a single peak; Fig. 2, *B* and *D*). These predictions are similar to those of minimum jerk (Flash and Hogan 1985) and were directly tested in the present experiments.

Unlike minimum jerk, the OST model predicts that, for any given object (*M*_{O} and *K*_{O}) and reaching distance (*L*), velocity profiles for the hand will have a uni-phasic shape for slower movements but a bi-phasic shape (i.e., 2 peaks) for faster movements (Fig. 2*A*). Specifically, the OST model explicitly predicts the critical movement time (*T*_{C}) where these transitions will occur. Because the solutions are symmetrical, we compute the second time derivative of hand velocity, evaluate it at *t* = *T*/2, set the resulting expression equal to zero, and solve for *T*_{C} 7

Thus for any given object (*M*_{O} and *K*_{O}), the model predicts uni-phasic hand velocity profiles for *T* > *T*_{C} and bi-phasic hand velocity profiles *T* < *T*_{C}.

Furthermore, when reaching distance (*L*) and duration (*T*) are held constant, the optimal object trajectory is independent of *M*_{O} and *K*_{O} (Fig. 2*D*). However, the hand trajectory required to achieve optimal object transport depends in a unique way on the resonant frequency of the mass-spring object. Specifically, the OST model predicts uni-phasic hand velocity profiles when transporting objects with higher *f*_{O} and bi-phasic hand velocity profiles when transporting objects with lower *f*_{O} (Fig. 2*C*). The critical resonant frequency (*f*_{C}) where these transitions occur can be computed analytically by inverting the expression obtained in *Eq. 7* 8

Thus for any given movement time, *T*, the OST model predicts uni-phasic hand velocity profiles for objects of *f*_{O} > *f*_{C} and bi-phasic hand velocity profiles for objects of *f*_{O} < *f*_{C}. Furthermore, because hand trajectories are predicted to depend only on *f*_{O}, which depends only on the *ratio* of object mass to stiffness (*K*_{O}/*M*_{O}), the OST model also predicts identical hand kinematics when manipulating objects with different *M*_{O} and *K*_{O} values but the same mass-to-stiffness ratio. The existence of these critical transition points demonstrates that the OST predictions are fundamentally different from those of minimum jerk and also provide a specific critical test for the model.

### Experiments and data analysis

We trained 14 neurologically healthy subjects (7 male and 7 female) who provided written informed consent to transport a virtual mass on a spring attached to their hand quickly to a target and stop them both there. Subjects made these reaching movements in the horizontal plane using a robotic manipulandum (Dingwell et al. 2002) (Fig. 3*A*) that simulated a two-dimensional mass on a spring attached to their hand (Fig. 3, *B* and *C*). Both the subject's arm and the mass-spring object were free to move in both directions in the horizontal plane. Motors on the manipulandum supplied real-time haptic feedback about the forces being exerted on the hand by the virtual object (Fig. 3*A*). A video monitor above the manipulandum provided real-time visual feedback about the positions of both the hand and the object. Because *Eq. 1* assumes a zero rest length for the spring, subjects experienced no forces when the handle and object were both at rest. Reaching movements were always initiated with the hand and object both at rest. Subjects were instructed to bring both their hand *and* the object to the target zone and stop them both there within the required time (±0.2 s) for each condition. Subjects were given audio feedback as to whether their movements were appropriate, too slow, or too fast. No instructions were given concerning what *trajectories* subjects could follow. Subjects were free to adopt any kinematic trajectory they desired as long as they were able to stop in the target zone within the designated time.

On the first day of training, subjects completed a sequence of 1,200 movements, between 10 and 25 cm in length, to pseudorandomly located targets. Movements were made with a simulated object with *M*_{O} = 3.0 kg and *K*_{O} = 120 N/m (i.e., *f*_{O} = 1.0 Hz). These values were chosen because they imposed a nontrivial perturbation on the arm. The time allowed to reach each successive target was slowly decreased from 2.5 to 1.0 s. Movements were completed in six blocks lasting ∼5 min each, where each new target appeared on the screen at the predesignated time and subjects had to make a continuous sequence of movements from each target to the next. Subjects were only instructed to bring their hand and the mass to a stop at the target within the time allowed. They were given no instructions regarding *how* to achieve this. Four of the 14 subjects were unable to achieve the required task speed after the first day and were brought back for a second day of training where this protocol was repeated. After this second day, two subjects were still unable to achieve the required movement speed. This level of proficiency was not unexpected, given the difficulty of the task (Dingwell et al. 2002). Therefore a total of 12 subjects were brought in for a final day of testing.

Subjects completed each of the three experiments (*A, B,* and *C*) on the final day of testing designed to test the three different sets of predictions posed by the OST model. The order of presentation of the experiments and of the conditions within each experiment were randomly assigned to each subject and balanced across subjects and genders to negate any possible order effects (Table 1). During *experiment A,* targets were presented in pseudo-random order. During *experiments B* and *C,* the order of presentation of the conditions was chosen such that each subject had to alternate between conditions where the model predicted uni- and bi-phasic hand movements. *Experiments A* and *B* were conducted with the same simulated object used during the training phase of the experiment. In *experiment C,* subjects sequentially manipulated each of three different objects, each having mechanical parameters different from the originally learned object (Fig. 6*A*). During all three experiments, subjects made individual reaching movements from a specified starting location to a specified target. Subjects started each movement with the hand and object both at rest and then moved to the target location in the specified amount of time. Once the subject reached the target and both their hand and the mass had stopped moving (speed: <0.1 m/s), they were given visual and audible feedback about whether the executed movement was too fast, too slow, or within the specified time limits. Data regarding the positions of both the hand and the simulated mass were recorded from the manipulandum at 100 Hz and analyzed.

For each condition tested, the last 10 hand-velocity profiles executed by each subject were averaged together. Predicted hand-velocity profiles were computed analytically by substituting *Eq. 6* into *Eq. 1*, solving for *r*_{H} and computing its derivative, where *r* was taken to be along the desired movement direction (i.e., from the starting point to the center of the target). Boundary conditions were assumed to be those for an optimal reach [i.e., *r*_{O}(0) = *r*_{H}(0) = 0, *r*_{O}(*T*) = *r*_{H}(*T*) = *L*, and 1st-4th derivatives = 0]. The best model fit for each averaged trajectory was then obtained by allowing the total movement time (*T*) to vary, and selecting that value of *T* (called *T*_{Fit}) where the maximum amount of variance (i.e., the maximum *r*^{2}) was accounted for by the model. It was necessary to vary *T* in fitting the minimum crackle model to the data because while the model assumes an *ideal* controller that can reach the target with exactly zero error, human subjects are not ideal controllers. They reach the target with some nonzero error that they must then correct for. Therefore subjects may attempt to move with an internal “desired” *T* that may be shorter (or longer) than what was asked for. Attempting to achieve a shorter *T* would allow subjects more time to dampen extraneous oscillations at the end of the movement, and thus still complete the task “successfully.” Therefore for each experimental condition, values of *T*_{Fit} and the variance accounted for (VAF) were pooled across subjects and compared. These analyses were focused on the hand motions because it is the hand that is directly controlled by human subjects. Therefore what subjects must learn in this task is the appropriate hand trajectory that will result in some desired object trajectory. Because the object motions were strictly determined by the imposed hand motions (*Eq. 1*), any additional analyses of the object trajectories would be redundant.

## RESULTS

*Experiment A* tested the OST model's predictions that movements of the same distance and duration would be made in a straight line and with the same tangential velocity profiles, regardless of the hand location in the workspace or the direction of the movement (Flash and Hogan 1985). Subjects completed three blocks of 150 movements each of 25-cm reaches to each of six targets, while transporting a virtual 3.0-kg mass attached to a 120 N/m spring (*f*_{O} = 1.0 Hz). Movements of both the hand (Fig. 4*A*) and object (Fig. 4*B*) were nearly straight in all directions. Examination of the hand-velocity profiles (e.g., Fig. 4*C*) demonstrated that *all* 12 subjects exhibited bi-phasic hand velocity profiles as predicted by the OST model, regardless of location in the workspace or the direction of the movement. This finding was substantiated by the analyses of the model fits to these velocity profiles. Fitted movement times (*T*_{Fit}) were similar across the different targets (Fig. 3*D*): mean = 1.037 s, range = 0.79-1.22 s. Across all subjects and directions, the variance accounted for (VAF) by the model (Fig. 3*E*) was high: mean = 89.84%, range = 79.9-98.8%.

*Experiment B* tested the OST model's predictions that for a given object, hand- and object-velocity profiles would scale linearly with movement distance (*L*) and that object-velocity profiles would remain uni-phasic across distance and duration variations (Flash and Hogan 1985), whereas hand-velocity profiles would alternate between uni- and bi-phasic shapes for slower and faster movements, respectively (Fig. 2*A*). Subjects completed four blocks of 75 movements each while transporting a virtual mass-spring object of *M*_{O} = 3.0 kg and *K*_{O} =120 N/m (*T*_{C} = 1.34 s), where each block tested one combination of either slow (*T* = 1.68 s) or fast (*T* = 1.0 s) movements made to either short (*L* = 12.5 cm) or long (*L* = 25.0 cm) target distances. For all conditions, object-velocity profiles (not shown) remained uni-phasic as predicted by our model. All 12 subjects exhibited bi-phasic hand-velocity profiles for fast movements, and all but one subject exhibited uni-phasic hand-velocity profiles for slow movements, independent of the distance moved (Fig. 5*A*). Both hand- and object-velocity profiles scaled linearly with reaching distance in all subjects. One subject, for the slow movements, exhibited a tri-phasic hand-velocity profile (Fig. 5*D*), consisting of an initial fast (bi-phasic) movement, followed by a slower (uni-phasic) movement. Although this strategy was not predicted by the model, it also demonstrates that there were alternative strategies available that subjects could have adopted. All 11 of the remaining subjects followed patterns similar to those shown in Fig. 4*A*. *T*_{Fit} values were larger for the slow movements than for fast movements, as expected, but exhibited no consistent trend with respect to reaching distance (Fig. 5*B*). Neglecting the one exception noted in the preceding text, the OST model predicted the majority of the variance across all subjects and conditions (Fig. 5*C*): mean = 86.6%, range = 66.1-98.8%, although the faster movements tended to exhibit somewhat higher VAF values.

*Experiment C* tested the OST model's predictions that for a given movement duration (*T*) and reaching distance (*L*), optimal object trajectories would be the same, while hand-velocity profiles would be different for objects of different resonant frequency (*f*_{O}; Fig. 2*C*). *Experiment C* also tested the prediction that subjects would exhibit the same hand kinematics when transporting objects with different mass and stiffness values, but the same resonant frequency. Subjects completed three blocks of 150 movements with each of three objects (Fig. 6*A*). All movements were made to the same target (*L* = 25 cm) in the same desired duration (*T* = 1.0 s; *f*_{C} = 1.35 Hz). All subjects exhibited bi-phasic hand-velocity profiles for objects 1 and 3 and uni-phasic hand-velocity profiles for object 2 (Fig. 6*B*). Furthermore, hand-velocity profiles were of similar amplitude for objects 1 and 3. For all conditions, object-velocity profiles remained uni-phasic as predicted by our model (Fig. 6*B*). *T*_{Fit} values were smaller for object 2 than for objects 1 and 3 (Fig. 6*C*). Our model predicted the large majority of the variance across all subjects and objects (Fig. 6*D*): mean VAF = 90.3%, range = 72.6-99.3%, although object 2 yielded higher VAF values.

In all of the preceding results, OST model fits were obtained by allowing *T* in *Eq. 6* to vary until an optimal fit (*T*_{Fit}) was obtained between the data and the model for each case. This method of estimating movement time was very different from the method used during the on-line execution of the task, where movements were considered “finished” when the hand and object velocities remained below a predefined threshold for a specified amount of time (*T*_{Mvt}). Although both of these measures are somewhat arbitrary, it is important to establish that they provided comparable results. We therefore compared the differences between these two measures across all three experiments. Because different experimental conditions required different desired movement times (*T*_{Des}), we first subtracted each measure of movement time from *T*_{Des} to allow valid comparisons across test conditions. We then examined the correlations between these two normalized measures of movement time across the different movement conditions tested in our experiments (Fig. 7). The overall correlation between these two measures (Fig. 7*A*) was significant (*r*^{2} = 46.5%; *P* < 0.001). Although errors in movement time varied between the different conditions tested, the trends in these variations across test conditions were also quite similar for both measures (Fig. 7*B*). In general, *T*_{Mvt} values were almost always greater than *T*_{Fit} values. This was both reasonable and anticipated because *T*_{Mvt} included any additional time that may have been required at the end of each movement to dampen out any additional perturbations that may have occurred during the movement. Thus we are confident that the *T*_{Fit} values we obtained are indeed reasonable estimates of actual movement time.

Based on our results, we can reject the hypothesis that subjects executed this object manipulation task using a minimum-jerk strategy. We reject this hypothesis first on the theoretical grounds that minimum-jerk trajectories cannot satisfy the BC of the task (see Fig. 1). We reject this hypothesis second on the experimental grounds that whereas minimum jerk always predicts uni-phasic hand-velocity profiles, our subjects consistently violated this prediction by generating bi-phasic hand-velocity profiles under a wide variety of conditions.

Furthermore, these results provide very strong support for the alternative hypothesis that subjects adopted a strategy of optimizing the smoothness of the object trajectories, subject to the constraints of the hand/object interaction. The primary movement feature, as predicted by the OST model, that these experiments were designed to test (i.e., uni-phasic vs. bi-phasic hand-velocity profiles) was exhibited in 154 of the 156 total cases tested (12 subjects × 13 conditions). In only two cases (Fig. 5*D*) for only one subject did the model not predict the basic movement pattern observed. All other subjects exhibited the predicted behavior for this condition, and this subject exhibited the predicted behavior for all other conditions. Additionally, and also as predicted by the OST model, all subjects readily switched back and forth between these seemingly different movement strategies when the task conditions were changed accordingly.

## DISCUSSION

In the present work, we demonstrate for the first time how humans adapt to a structural perturbation that alters the basic kinematics between the arm and the moving end effector. This adaptation is similar in some respects to that observed in response to the parametric perturbations imposed by various force fields (Bock 1990, 1993; Dizio and Lackner 1995; Flanagan et al. 1999; Lackner and Dizio 1994; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a). In those contexts, adaptation requires learning of modified relationships among desired motions, motor commands, and sensory consequences and may be facilitated by the acquisition of an internal model of the applied perturbation (Flanagan et al. 2003; Gribble and Scott 2002; Krakauer et al. 1999; Miall and Wolpert 1996; Shadmehr and Mussa-Ivaldi 1994; Thoroughman and Shadmehr 2000; Wolpert et al. 1995b). Our observations can be considered a form of adaptation because our subjects learned a systematic mapping between their motor commands and the motion of the object in extrinsic coordinates. They then used this mapping to generate smooth rectilinear motions of the object according to a simple and experimentally verified kinematic law of motion. This adaptation occurred despite the fact that it required executing hand trajectories that were substantially different from those exhibited during free reaching or when adapting to parametric perturbations (Bock 1990, 1993; Dizio and Lackner 1995; Flanagan et al. 1999; Lackner and Dizio 1994; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a).

The demonstration of adaptation to the structural perturbation generated by the mass-spring object provides insight into the nature of neuromuscular control in several ways. First, it is clear that the capacity of humans to acquire implicit knowledge about the physics of a task and to exploit that knowledge for control extends beyond the control of limb movements alone. Our results complement recent findings related to other object manipulation tasks involving throwing (Dupuy et al. 2000) or catching (McIntyre et al. 2001) balls, bouncing a ball on a racket (Schaal et al. 1996; Sternad et al. 2001), balancing a ball on a beam (Huang et al. 2002), and balancing an inverted pendulum (Mah and Mussa-Ivaldi 2003a,b; Mehta and Schaal 2002). That humans *can* acquire this implicit knowledge of physics is an important initial step in understanding *how* the nervous system processes information so it can effectively implement control strategies using this knowledge. Although this latter question was not directly addressed in the present paper, recent evidence supports the idea that humans acquire and use internal models of an external object's dynamics (Dingwell et al. 2002; Mah and Mussa-Ivaldi 2003a,b; Mehta and Schaal 2002) in much the same way they use an internal model of arm dynamics for controlling reaching (Bhushan and Shadmehr 1999; Dizio and Lackner 1995; Lackner and Dizio 1994; Shadmehr and Mussa-Ivaldi 1994; Thoroughman and Shadmehr 2000; Wolpert and Ghahramani 2000; Wolpert et al. 1995b).

### How much “choice” did subjects have?

The original minimum-jerk model (Flash and Hogan 1985; Hogan 1984; Hogan et al. 1987) demonstrated that reaching movements satisfy a specific smoothness principle. A central question that has been debated since then is whether these properties reflect constraints in the limb control system (Alexander 1997; Harris and Wolpert 1998; Nakano et al. 1999; Uno et al. 1989) or constraints on motion planning (Flash and Hogan 1985; Hogan et al. 1987; Wolpert and Ghahramani 2000; Wolpert et al. 1995a). The task studied in the present experiment involved manipulating an object that imposed its own significant dynamics on the arm and controller and also imposed additional boundary conditions on the task that are not experienced during typical reaching movements. One could then reasonably ask how much these additional constraints limited subjects' choice to acquire appropriate movement solutions.

In general, the mechanics of the mass-spring object and these additional BC do *not* require the object to follow any specific trajectory. Many achievable trajectories exist that subjects did not employ. We give two examples here. First, our OST model states that we need at least a ninth-order polynomial to satisfy all the BC. Satisfying the five BC at *t* = 0 forces the lowest five polynomial coefficients to zero. We solve for the remaining five coefficients to satisfy the five BC at *t* = *T* (*Eq. 6*). However, we could just as easily use the first five terms of any polynomial higher than ninth order, setting all other coefficients to zero. As the polynomial order gets higher, the resulting trajectories become more asymmetrical (Fig. 8*A*). Because the BC for the task are symmetrical about the end-points of the movement, a second family of curves could be defined from the mirror images of those shown in Fig. 8*A* (i.e., starting faster and ending slower). Furthermore, for any of these higher-order polynomials, one could take any five terms (above the 4th-order term), set the remaining coefficients to zero, and solve for a unique solution. Thus we can obtain an infinite family of possible solutions, many of which are physiologically achievable, none of which were exhibited by our subjects.

Furthermore, polynomials provide only one possible family of solutions. Any reasonable basis function can be similarly solved to obtain reasonable solutions that satisfy all the BC. For example, one could define a Fourier series of sine and cosine terms to obtain plausible solutions (Alexander 1997). Figure 8*B* shows several solutions obtained when the following Fourier series was used to define the object trajectory 9 where *r*_{Oi} is the initial starting position (e.g., a movement of length *L*, starting at *r*_{Oi} = -*L*/4 would go from -*L*/4 to 3*L*/4). Varying the starting position (which is arbitrarily chosen by us anyway) is similar to varying the phase shift on the cosine terms. The parameters, *b*_{i}, are then determined by requiring that *Eq. 9* and its derivatives satisfy the required BC, as before. Figure 8*B* shows five physiologically achievable solutions for movements of length *L* = 0.25 m for *r*_{Oi} ∈ {-3*L*/4, -5*L*/8, -*L*/2, -3*L*/8, and -*L*/4}. While none of the solutions shown in Fig. 8 were derived to be “optimal” in any specific way, they are all *physiologically achievable* in that subjects *could* have adopted them within reasonable physiological limits. An infinite number of other such solutions could likewise be derived as intermediaries of those shown and/or from other basis functions (e.g., Gaussians, wavelets, etc.). Therefore simply satisfying the BC does not guarantee that any given solution is optimal. None of the subjects in our experiments adopted these alternative trajectories. These examples therefore demonstrate that the hand (and therefore object) trajectories observed in our experiments represent a definite choice.

There is a second factor that supports the notion that the observed trajectories were not dictated by the task dynamics. Subjects in this study took a much longer time to learn to complete the mass-spring object transport task successfully, especially at faster speeds, than subjects learning to adapt to parametric perturbations like force fields (Flanagan et al. 1999; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a). This extended period of learning also suggests that subjects were not naturally driven by forces beyond their control (i.e., the object's dynamics) into a narrow set of possible solutions. Instead, they actually had to work quite hard to acquire the solutions that they did. We believe there is a logical explanation for this. Unlike point-to-point reaching movements, where only the position of the hand needs to be controlled, subjects did not have either direct actuation or somatosensory information about the object on the end of the spring. The mechanical linkage between the hand and object was therefore quite different from the physiological linkages that exist between limb segments. This connection was entirely passive: there were no muscles connecting the hand to the mass, and the spring stiffness was chosen arbitrarily by us. We believe the additional learning time required for subjects to achieve competence in this experiment came about because subjects needed sufficient practice to construct an “internal model” of the object's dynamics to formulate an appropriate control strategy. We addressed this question in more detail in a previous paper (Dingwell et al. 2002).

### Model “goodness of fit”

In all three experiments, our results were consistent with the primary predictions of the OST model, which predicted the majority of the variance (>70% VAF) observed in nearly all cases. The model fits (like those of any model) were not perfect. In particular, the laterally directed movements (directions 3 and 4) in *experiment A* (Fig. 3, *A* and *B*) were not exactly straight, and the hand-velocity profiles for slow movements in *experiment B* (Fig. 4*A*) were not exactly uni-phasic. The deviations from straightness shown in Fig. 3 were likely due to geometrical constraints (e.g., those imposed by joints and ligaments, etc.) within the human arm itself as reflected by the fact that movements generally tended to curve away from the body. We do not yet know exactly what the additional inflections exhibited toward the beginning of the slow movements (Fig. 4*A*) indicate. However, it should be noted that minimum jerk and its competitors were originally developed primarily to explain the behavior exhibited during fast movements where subjects must rely on preplanned feed forward control. Therefore we suspect that because these were relatively slow movements, these additional inflections may indicate that subjects were not relying entirely on a preplanned feed-forward strategy that attempted to optimize the smoothness of the motion as a whole. It is possible that in this context, they may have been relying more on feedback control than on feed forward control. Obtaining a full account of these inflections and their origins will require further testing, modeling, and/or experiments, which we hope to pursue in the near future.

Likewise, in *experiment C,* the subject's responses across all three objects were not perfectly predicted by the model. Subjects exhibited somewhat faster than required movements with the stiffer object (object 2) and did not exhibit identical trajectories when manipulating the two kinematically equivalent objects (objects 1 and 3). We believe the lower fitted movement times (*T*_{Fit}) and the higher VAF values for object 2 were likely due to the increased stiffness of this object, which made its behavior more like that of a rigid object and thus easier to manipulate. The lower fitted movement times (*T*_{Fit}) for object 3 compared with object 1 were likely due to the increased mass of object 3. A larger mass would impose a larger perturbing force for the same kinematic “error” in the trajectory. Subjects may have been trying to achieve a faster *T*_{Fit} during these trials to have more time to absorb these larger perturbing forces at the end of the movement, while still completing the task “successfully.” These deviations from perfect model fit are consistent with our previous findings that subjects control the kinematics of the object using an internal model that specifies the forces to be exerted by the hand on the object (Dingwell et al. 2002). This suggests that although humans are capable of achieving near optimal kinematic performance, the movements they generate are still subject to the innate dynamic constraints of their own actuators (i.e., their arms).

It should be emphasized here, however, that the OST model was neither intended, nor expected, to explain every detail of these trajectories. Any model, by definition, is an approximation of the real situation that is measured during an experiment. This is especially true for the model we describe here. The OST model is completely independent of the biomechanical and neurophysiological complexities of the human arm and the systems that control it. As such, its predictions must be interpreted as ideal solutions obtained for a perfect controller manipulating an “ideal” hand (i.e., a point mass). The real (human) control system must contend with specific physiological limitations (e.g., limb geometry, strength, rate of force production, noise, etc.) that may prevent it from achieving this ideal performance. How well any model fits the experimental data depends on what features of the data that model is intended to capture. Because we adopted only a very coarse model of our system (i.e., the hand as point mass), we might expect only a very coarse approximation to the movements subjects made. However, as shown in Figs. 4, 5, 6, we believe that the model fits we obtained were in fact quite good given the extreme simplicity of the model. Our results provide strong evidence that subjects strive to, and largely succeed at, achieving optimally smooth performance in the presence of those biomechanical and neurophysiological limitations not accounted for in the OST model.

### Underlying principles versus physiological mechanisms

Finally, it could be argued that the results obtained here might be just as well explained by some other alternative model or theory. Several authors have proposed more biologically based alternative optimization criteria that establish some means whereby rapidly oscillating (i.e., nonsmooth) movements are penalized in their respective cost functions, through minimizing rates of change in joint torques (Gomi and Kawato 1996; Nakano et al. 1999; Uno et al. 1989) or excess neuronal or biomechanical noise (Harris and Wolpert 1998; Todorov and Jordan 2002). Others have argued that the smoothness observed in reaching movements may a byproduct of the natural properties of muscles (Alexander 1997; Karniel and Inbar 1997; Krylow and Rymer 1997). Because all of these alternatives penalize less smooth movements (in one way or another), it is not difficult to imagine that application of these criteria might produce trajectories similar to the minimum crackle trajectories predicted by the OST model. It is possible they might even provide more accurate fits (i.e., higher VAF) to the experimental data than were obtained here.

However, there is a second fundamental and very important difference between the predictions of the OST model and these other alternatives that goes beyond which model best fits the experimental data. That is that the OST predictions make absolutely no assumptions about the nature of the underlying controller (i.e., the arm), whereas all of the proposed alternatives rely intimately on detailed models of the biomechanics and neurophysiology and include all the of the physiological limitations and complexities that living humans have to contend with. The OST solutions therefore establish a theoretical limit describing the best possible solution period (i.e., as executed by an ideal controller), whereas biologically based alternative models provide solutions describing the best solution a person can achieve given their inherent physiological limitations.

The distinction made here is thus a distinction between an *underlying principle* on which movements are planned and executed and the *physiological mechanism* by which these principles are implemented. To move our understanding of motor control forward, we must first establish the underlying principles and then try to find out how those principles are implemented at the biological level. Flash and Hogan (1985) moved the field of study forward by proposing just such a principle and challenging others to determine how that principle was implemented. Here, we present (we believe for the first time) an experimental paradigm that clearly violates Flash and Hogan's minimum-jerk principle as they defined it, and we propose a generalization of their original principle that will allow the field to once again move forward. In this respect, the principle of optimal smoothness, as embodied in the minimum-jerk hypothesis and in the principle of OST, provides a *descriptive* rather than an *explanatory* account for the behaviors observed.

Whether a law is descriptive or explanatory is a very important distinction in science. Take the equivalence between Newton's law of mechanics (F = ma) and Hamilton's principle of least action as an example (Engelbrecht 2001; Marion and Thornton 1995). A stone falling under gravity “finds” the trajectory that minimizes the difference between its kinetic and potential energies. But the stone certainly does not compute this quantity. In our experiments, the situation is the same. We observe that movement trajectories obey a principle of optimum smoothness. However, calculating the trajectories does not necessarily require computing smoothness. Indeed, one criticism of the original minimum-jerk hypothesis is that humans lack the biological capacity to either sense or compute fifth-order derivatives of limb movement (Krylow and Rymer 1997). It is even less likely that the nervous system is computing still higher (9th order) derivatives of the motions of external objects. While laws like Hamilton's principle must be understood as descriptive rather than explanatory, such principles are also extremely powerful because they apply to an infinite variety of systems (Engelbrecht 2001; Marion and Thornton 1995). In the same way, the principle of optimum smoothness is extremely powerful precisely because it is completely independent of the biological details of the controller.

From this perspective, we believe the principle of OST and other proposed alternative models are in fact complementary rather than contradictory: i.e., optimizing smoothness explains the *what*, whereas biologically inspired models explain the *how*. In that sense then, it would be advantageous if one or more of these other alternative models were to produce results similar to the OST model because they would then suggest plausible ways in which this very abstract and nonbiological principle can be implemented at the biological level. Doing this additional modeling work in a sufficiently rigorous manner is beyond the scope of this particular manuscript. However, we hope the work presented here will provide the necessary foundation from which such future efforts will stem.

### Neural control of non-rigid objects

One question that remains unanswered is whether the neural structures involved in controlling non-rigid objects are the same as those involved in controlling reaching movements and manipulating rigid objects. The results of the present study demonstrate that subjects manipulate non-rigid objects using optimally smooth movements, in much the same way they do when executing reaching movements (Flash and Hogan 1985). Previous results from our lab on this task (Dingwell et al. 2002) and on a similar non-rigid object manipulation task (Mah and Mussa-Ivaldi 2003a) further suggest that humans manipulate these objects by learning an internal model of the object's dynamical behavior in much the same way they learn an internal model of their own arm for controlling reaching movements (Flanagan et al. 1999, 2003; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a). Together, these findings imply that there is a great deal of overlap among the control of arm movements, rigid objects, and non-rigid objects. This in turn implies that many of the neural structures involved in the planning and control of the former kinds of movements are also involved in the planning and control of non-rigid objects.

However, there is at least one fundamental difference between non-rigid object manipulation tasks and these other tasks. Because there is no direct muscle activation of the object itself, the CNS cannot acquire direct proprioceptive feedback information about the states (i.e., positions and velocities) of the object. Our previous findings indicate that the internal object model that subjects learn is composed as a specific mapping between hand-object interface forces and the states of both the hand and the object (Dingwell et al. 2002; Mah and Mussa-Ivaldi 2003a). This implies that for subjects to construct such a mapping, they must be able to acquire accurate sensory feedback regarding *both* forces *and* states. Unlike reaching tasks or tasks involving manipulating rigid objects, the state information for non-rigid objects can only come from visual sources. Indeed, anecdotal evidence from our lab suggested that when subjects were given either visual or proprioceptive feedback alone, in the absence of the other, they were incapable of learning to manipulate the mass-spring object successfully. These observations suggest that the CNS should place a greater importance on visual information when learning to manipulate non-rigid objects as compared with rigid objects. Thus the neural structures involved in the visual control of movements are likely to be more active during the learning of such tasks. The non-rigid-object-manipulation task presented in this study allows the complete decoupling hand-object interface forces (generated by the manipulandum motors, which can be switched on or off) from object state information (obtainable *only* through visual feedback). This task and experimental setup therefore provides a unique paradigm from which the relative roles of proprioceptive and visual information in the control of object manipulation tasks can be addressed in future studies.

## Acknowledgments

GRANTS

This work was supported by National Institutes of Health Grant F32-HD-08620-01 awarded to J. B. Dingwell. Additional support was provided by National Institutes of Health Grant NS-35673 and National Science Foundation Grant BES-9900684 awarded to C. D. Mah and F. A. Mussa-Ivaldi.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2004 by the American Physiological Society