Abstract

It is well known that planar reaching movements of the human shoulder and elbow joints have invariant features: roughly straight hand paths and bell-shaped velocity profiles. The optimal control models with the criteria of smoothness or precision, which determine a unique movement pattern, predict such features of hand trajectories. In this letter on expanding the research on simple arm reaching movements, we examine whether the smoothness criteria can be applied to whole-body reaching movements with many degrees of freedom. Determining a suitable joint trajectory in the whole-body reaching movement corresponds to the optimization problem with constraints, since body balance must be maintained during a motion task. First, we measured human joint trajectories and ground reaction forces during whole-body reaching movements, and confirmed that subjects formed similar movements with common characteristics in the trajectories of the hand position and body center of mass. Second, we calculated the optimal trajectories according to the criteria of torque and muscle-tension smoothness. While the minimum torque change trajectories were not consistent with the experimental data, the minimum muscle-tension change model was able to predict the stereotyped features of the measured trajectories. To explore the dominant effects of the extension from the torque change to the muscle-tension change, we introduced a weighted torque change cost function. Considering the maximum voluntary contraction (MVC) force of the muscle as the weighting factor of each joint torque, we formulated the weighted torque change cost as a simplified version of the minimum muscle-tension change cost. The trajectories owing to the minimum weighted torque change criterion also showed qualitative agreement with the common features of the measured data. Proper estimation of the MVC forces in the body joints is essential to reproduce human whole-body movements according to the minimum muscle-tension change criterion.

1  Introduction

An arm reaching movement shows slightly curved path and bell-shaped velocity profiles with respect to the hand trajectory (Soechting & Lacquaniti, 1981; Morasso, 1981; Abend, Bizzi, & Morasso, 1982). The optimal control principle is based on a hypothesis in which the features of the reaching movement are a consequence of minimizing the effort to perform a task or maximizing the task performance. The optimization criterion determining the unique trajectory to reach a target from an infinite number of patterns is one of the central problems in motor control of human movements. Much research has focused on investigating the planning and control mechanisms of human voluntary arm movements.

Flash and Hogan (1985) proposed the minimum jerk model, which implies that the hand trajectory of a reaching movement is specified so that the hand jerk (derivative of the hand acceleration) is minimized. Although a minimum jerk trajectory shows features of a bell-shaped velocity profile, the hand path becomes an exact straight line. Uno, Kawato, and Suzuki (1989) proposed the minimum torque change model, which deals with the nonlinear dynamics of the arm. The minimum torque change model shows better prediction of sightly curved trajectories; however, the torque change cost is not directly provided in the sensory information from proprioception. To improve the physiological adequacy, Dornay, Uno, Kawato, and Suzuki (1996) considered a muscle-tension change cost that could be provided in the sensory feedback of the muscle tension. Soechting and Flanders (1998) proposed a minimum muscle-force change model to consider the contribution of bi-articular muscles in arm reaching movements. The criteria mentioned above explicitly require smoothness of the trajectory. Alternatively, Harris and Wolpert (1998) proposed a criterion directly related to task performance: a minimum end point variance criterion, which takes the contamination of signal-dependent noise into account in the motor command of the muscles. By requiring precision of the end point, minimization of error due to the signal-dependent noise leads to finer motor command and a smoother trajectory.

Most models for motor planning were adopted for simple two-link arm movements in a horizontal plane, and a few studies addressed the motor planning of reaching movements in a bipedal stance (whole-body reaching). The planning of a whole-body reaching task involves the problems of redundant joint degrees of freedom and maintenance of postural balance. Although the whole-body reaching task is more complex than the horizontal arm reaching task, similar features of a slightly curved hand path and bell-shaped velocity profile were observed (Pozzo, Mcintyre, & Papaxanthis, 1998; Patron, Stapley, & Pozzo, 2005). Tagliabue, Pedrocchi, Pozzo, and Ferrigno (2008) proposed an optimization algorithm for whole-body tasks and investigated the optimal criteria of kinematic smoothness for the hand jerk and postural balance for the displacement of body center of mass (body COM) or center of pressure (COP). Although the trajectory reproduced by each criterion was not consistent with the measured trajectory, the combined criteria of smoothness and balance with appropriate weighting predicted a similar trajectory, indicating that both smoothness and balance maintenance should be considered in motor planning. Patron et al. (2005) showed the effects of gravity on whole-body reaching trajectories. In microgravity environments, the movement duration increased and displacements of the ankle and knee joints were smaller while conserving the temporal coordination of trajectories between body COM and hand position. The gravity dependence of the reaching motion implies that the motor planning may deal with the dynamics of the musculoskeletal system. The optimization criteria related to the musculoskeletal dynamics were supported by previous studies on sit-to-stand tasks (Pandy, Garner, & Anderson, 1995; Kuželički, Žefran, Burger, & Bajd, 2005; Yamasaki, Kambara, & Koike, 2011) and arm raising movements (Ferry, Martin, Termoz, Côté, & Prince, 2004).

In this study, we addressed the optimization criteria in relation to body dynamics for the whole-body reaching movement. Although one of the dominant hypotheses of motion planning is the minimization of end point variability, optimization under a stochastic system is still a complicated problem. Furthermore, for the whole-body reaching task, redundant joint degrees of freedom and inequality constraints from balance maintenance make it more difficult to calculate the optimal trajectory. Therefore, in this study, we examined the criterion of smoothness associated with the musculoskeletal dynamics. We examined not only the minimum torque change criterion, which predicts the horizontal arm reaching movement, but also the minimum muscle-tension change criterion, which is physiologically more adequate. Although both models predicted the common properties of the planar arm reaching movement in previous studies (Dornay et al., 1996), we found remarkable differences in their predictions for the whole-body reaching movement. To clarify the dominant factors of the differences of the two criteria, we simplify the muscle-tension change cost to a weighted torque change cost, where the weight of the joint torque change can be specified by the moment arm and the physiological cross-sectional area (PCSA) of the related muscles. We compared the trajectories of the hand, joint angle, body COM, and COP as predicted by the models with the measured trajectories.

2  Methods

2.1  Data Acquisition and Analysis of Whole-Body Reaching Movements

To examine the trajectories predicted by the optimization models, we measured whole-body reaching movements. Four young male subjects (165–170 cm height and 55–65 kg weight) agreed to participate in the experiments. The experiments were approved by the Nagoya University ethical review board.

Figure 1A shows the experimental setup for the measurement of the whole-body reaching movements. Six infrared LED markers, which were measured by a three-dimensional position measurement device (OPTOTRAK Certus, Northern Digital) at 100 Hz, were attached to the body parts of interest on the right side: ankle, knee, hip, shoulder, elbow, and hand. The ground reaction force was simultaneously measured by a force plate (9286B, Kistler). The subjects stood on the force plate and maintained an initial posture with their elbow joints flexed in a 90 degree position. They were instructed to move their hands, with bilateral symmetry, to a target as soon as possible. They were also instructed not to stand on tiptoe while reaching. Following an auditory cue, the subjects performed a whole-body reaching movement to the target. We used a coordinate system where the origin was the position of the ankle joint, and the positive directions of the horizontal axis x and the vertical axis z were the anterior direction and the superior direction, respectively. We examined two target positions in relation to the ankle joint: a near target (0.6 m, 0.1 m) and a far target (0.8 m, 0.2 m). The subjects practiced reaching a few times before performing 20 trials for each target position.

Figure 1:

Experimental setup (A) and model definition (B) for measurement and analysis of whole-body reaching movements.

Figure 1:

Experimental setup (A) and model definition (B) for measurement and analysis of whole-body reaching movements.

The measured joint positions and the COP were low-pass-filtered with a cutoff frequency of 5 Hz. The joint angle, angular velocity, and angular acceleration were calculated from the measured position data. The start times of the movements were specified when the sum-squared angular velocity of all joints was over a threshold (0.1 rad/s). Alternatively, the end time was specified when the velocity of the body COM, the velocity of the hand, and the angular velocity of the joints were sufficiently small because the postural fluctuation around the end state is not as small as the initial state. We modeled the human body as a five-link model of the musculoskeletal system in the saggital plane (see Figure 1B, detailed in section 2.2). The body COM was calculated from the measured position with the segment mass and the position of the segment COM according to the method by Winter (2005). To calculate the mean trajectories of the positions and angles for the data of 20 trials, the movement duration of each trajectory was time-normalized as the mean duration by the cubic spline method. The mean velocities of the hand, body COM, and joint angle were calculated by the numerical derivative of the mean position and joint angle data.

2.2  Kinematics and Dynamics of Musculoskeletal System

Figure 1B shows a five-link musculoskeletal model of the human body in the saggital plane, consisting of the shank, thigh, trunk and head, upper arm, and forearm. We define a state variable representing the state of the body by the angles of body segments related to a horizontal axis:
formula
2.1
The joint angle can be related to the state variable as
formula
2.2
where is a coordinate transformation matrix and is a bias term. An equation of motion for the five-link model can be represented by
formula
2.3
where is the inertia term, is the Coriolis and centrifugal term, and is the gravity term. and are the joint torque and joint viscous moment, respectively. is a diagonal viscous matrix (Nakano et al., 1999; Pozzo, Ouamer, & Gentil, 2001). The relationship between the joint torque and muscle force is represented by the following equation,
formula
2.4
where is a moment arm of the jth muscle with respect to the ith joint. Considering the different maximum voluntary contraction (MVC) force of each muscle, we normalize the muscle force by PCSA as . In this study, we consider 25 muscles related to the joint movements in the model. The biomechanical and anatomical parameter values used for the trajectory optimization are summarized in the appendix.
The positions of the hand and the body COM are given by the state variable with the kinematics equations. The postural balance can be evaluated by the position of the COP (Winter, 1995, 2005). We evaluated the postural stability with the COP position xCOP obtained from the equilibrium conditions of the ground reaction force and the moment around the ankle joint:
formula
2.5
where zankle is the height of the ankle joint and px and pz are the amount of linear momenta of the body segments in the horizontal and vertical directions, respectively. M is the body mass and g is the gravitational acceleration.

2.3  Optimization Criteria

Modeling the motor planning with optimal control theory is a computational approach to explore the determinant of the stereotyped pattern of human movements. In this study, we focus on the smoothness of the movement in consideration of the musculoskeletal dynamics. The minimum torque change model (Uno et al., 1989) is a typical model that precisely reproduces horizontal arm-reaching trajectories. The cost function of the torque change is defined by
formula
2.6
where tf is the movement duration and N is the number of joints. In this model, the torque change of each joint is equally evaluated even though the gravitational and inertial moments of the leg joints are much greater than those of the arm joints.
We examined another criterion related to the musculoskeletal dynamics: the muscle-tension change model. The contribution of each muscle to the movement depends on the MVC force of the muscle. Since MVC force is related to muscle size, we consider the muscle force normalized by PCSA Fj, (L is the number of muscles contributing to the movement execution). The cost function of the muscle-tension change is defined as
formula
2.7
Our cost function is different from the other muscle-level criteria proposed in previous studies of arm reaching movements (Dornay et al., 1996; Soechting & Flanders, 1998), where MVC force or PCSA values of muscles were not considered.

The different prediction of trajectories between the criteria may result from the contributions of muscle interactions with agonist, antagonist, and biarticular muscles and nonuniform muscle parameters (PCSA and moment arms). To clarify the dominant factors of the different prediction, we simplify the muscle-tension change cost function to a weighted torque change cost function with some assumptions.

First, we regard each muscle group of a flexor and extensor of a joint as a monocular muscle. We let the tension per unit cross-sectional area of the flexor and extensor of joint i be , . We approximate the cost function, equation 2.7, as
formula
2.8
The second assumption is that the moment arm and the amount of PCSA are comparable between the flexor and extensor muscles. Then the torque of joint i can be represented by the following equation,
formula
2.9
where ri is a typical value for the moment arm and is the amount of PCSA of the muscle groups in relation to the joint i. The third assumption is that the cross-product term of the flexor and extensor tension change is negligibly small compared with the squared term of each tension change. Then the following approximation is applicable:
formula
2.10
Substituting the approximation, equation 2.10, into the cost function, equation 2.8, we obtained the following weighted torque change cost function:
formula
2.11
formula
2.12

We categorized the extensor and flexor groups of the joints according to the definition of leg muscles (Ward, Eng, Smallwood,& Lieber, 2009) and elbow muscles (Kawakami et al., 1994). For the shoulder joint, we categorized them based on data of a moment arm (Kuechle et al., 2000). The parameter values of the moment arm, the PCSA, and the weight in the cost function CW for each joint are summarized in Table 1. The weights of the cost function of the leg joints are much smaller than those of the shoulder and elbow joints. These differences result from the differences between the PCSA of leg muscles and arm muscles. Based on this approximation, minimization of muscle-tension change may lead to a smaller torque change of the arm joint and a greater torque change of the leg muscle than the original torque change criterion. It should be noted that the PCSA of plantarflexors is greater than that of dorsiflexors, which does not correspond to the second assumption. We specified the weight value of the ankle joint from the parameter values of the plantarflexors because the plantarflexor torque is dominant during the whole-body reaching movement in our study.

Table 1:
Moment Arm ri, PCSA , and Weight of the Cost Function CW in Equations 2.11 and 2.12.
AnkleKneeHipShoulderElbow
PFDFEFEFEFEF
Moment arm (cm) 4.4 −3.6 −3.8 5.4 −7.0 9.1 −4.1 2.7 −2.2 3.8 
PCSA (227 19.0 171 132 110.3 61.6 30.7 14.2 17.2 16.8 
Weight (9.9 5.2 5.3 420.4 995.2 
AnkleKneeHipShoulderElbow
PFDFEFEFEFEF
Moment arm (cm) 4.4 −3.6 −3.8 5.4 −7.0 9.1 −4.1 2.7 −2.2 3.8 
PCSA (227 19.0 171 132 110.3 61.6 30.7 14.2 17.2 16.8 
Weight (9.9 5.2 5.3 420.4 995.2 

Note: PF, DF, F, and E denote plantarflexor, dorsiflexor, flexor, and extensor, respectively.

2.4  Trajectory Optimization

In general optimal control problems, the time series values of joint toque are the optimization variables. However, it is difficult to solve the problem due to the large number of variables and constraints. To reduce the number of variables, the trajectory of the ith joint angle is represented with a polynomial function:
formula
2.13
Then the optimization variables can be represented in the following matrix:
formula
2.14
We choose the 11th-order polynomial functions for the whole-body reaching movements.
From the joint trajectory determined by , the time series of joint torque can be calculated in accordance with the inverse dynamics based on equation 2.3. The torque change cost, equation 2.6, and the weighted torque change cost, equation 2.11, are evaluated from the numerical derivative of the joint torque. To evaluate the muscle tension change cost, equation 2.7, the muscle-tension change is determined by minimizing it for each time t,
formula
2.15
subject to
formula
2.16
formula
2.17

The equality constraints, equation 2.16, are the equation between the muscle force change and joint torque change, which corresponds to the derivation of equation 2.4. The inequality constraints, equation 2.17, are nonnegative constraints of muscle force.

The optimization variables in matrix are determined by solving the following optimization problem,
formula
2.18
subject to
formula
2.19
formula
2.20
The function represents a cost function. The equality constraint, equation 2.19, represents the conditions that the angle, angular velocity, and angular acceleration of the start and end points are consistent with those of the measured data. The inequality constraint, equation 2.20, is the balance condition that the COP is located within the base of support with a safety margin. tl represents the discretized time within the sampling time. When the sampling time is sufficiently small, the whole COP trajectory satisfies the balance condition. In this study, the upper and lower limits of the COP, in relation to the horizontal ankle position, were  m and m, respectively. The sampling time was 0.01 s. The optimization problems of the muscle-tension change and the joint trajectories were solved by the quadprog and fmincon functions, respectively, in the optimization toolbox of Matlab (Mathworks).

3  Results

3.1  Measured Whole-Body Reaching Movements

Figure 2 shows stick figures, hand paths, and body COM paths of 20 trials during the reaching movements. The motion patterns showed common features for the different subjects and the different target positions: a slightly curved shape of hand position and body COM similar to the previous study (Pozzo et al., 1998; Patron et al., 2005).

Figure 2:

Stick figures of initial and terminal postures, hand paths, and body COM paths of measured whole-body reaching movements for 20 trials. The upper and lower columns of the figures show the results of whole-body reaching toward the near and far targets, respectively. The solid lines and dashed lines indicate the paths of hand and body COM, respectively.

Figure 2:

Stick figures of initial and terminal postures, hand paths, and body COM paths of measured whole-body reaching movements for 20 trials. The upper and lower columns of the figures show the results of whole-body reaching toward the near and far targets, respectively. The solid lines and dashed lines indicate the paths of hand and body COM, respectively.

Figure 3 shows the mean time profiles of the hand velocity, body COM velocity, COP position, and joint angles, where the horizontal axes denote the percentage of movement duration. The hand velocity and body COM velocity showed bell-shaped profiles among the subjects, which were indicated as the invariant features in previous observations (Pozzo et al., 1998; Patron et al., 2005). The COP position moved backward in the early phase and forward in the middle and late phases. On the other hand, the joint angular velocity profiles, especially the ankle, knee, and elbow velocity profiles, showed subject-dependent patterns. Figure 4 shows the time profiles of 20 trials for a typical subject (subject 1, reaching to the near target). The profiles show stereotyped patterns among the trials. The horizontal line indicates the boundary of the base of support. The COP position moved within the base of support with a sufficient margin.

Figure 3:

Mean trajectories for 20 trials of the measured whole-body reaching movements to the near target. The positive directions of the ankle, knee, hip, shoulder, and elbow joints are defined as plantarflexion, flexion, extension, flexion, and flexion, respectively.

Figure 3:

Mean trajectories for 20 trials of the measured whole-body reaching movements to the near target. The positive directions of the ankle, knee, hip, shoulder, and elbow joints are defined as plantarflexion, flexion, extension, flexion, and flexion, respectively.

Figure 4:

The profiles of 20 trials of the measured whole-body reaching movements (subject 1, near target).

Figure 4:

The profiles of 20 trials of the measured whole-body reaching movements (subject 1, near target).

3.2  Simulated Movements of Optimal Control Models

The results presented here were calculated with the initial and terminal angle, angular velocity, and angular acceleration data obtained from subject 1. These data were used for the equality constraints, equation 2.19, in the optimization. We obtained similar results with the data from the other subjects.

Figure 5 shows the stick pictures predicted by the three optimization models. The upper and lower figures correspond to the reaching movements for the near and far targets, respectively. The minimum torque change trajectories show strongly curved hand paths, which are inconsistent with the typical human movement. The minimum weighted torque change and muscle-tension change models predict more straight hand paths like the measured human movement. The paths of the body COM are comparable among the models, showing paths similar to the measured movement. Similar to the results for the near target, the minimum torque change trajectory for the far target also showed a strongly curved hand path, and the others showed linear hand paths. Figure 6 shows the time profiles of the hand velocity, body COM velocity, COP position, and joint angles for the near target. Similar to the hand paths in Figure 5, the hand velocity shows a clear difference between the models. The minimum torque change trajectory showed a two-peak velocity profile, while the other two models predicted bell-shaped profiles. The movements of the shoulder and elbow joints resulted in the difference in the hand path and velocity. The curved trajectories close to the body were the consequence of the earlier extension of the elbow and the late flexion of the shoulder and elbow. The greater extension and flexion velocities of the elbow would induce the two peaks of hand velocity. The different predictions were also found in the profiles of the ankle angular velocity and the COP movement. Because of the smaller dorsiflexion velocity of the ankle of the minimum muscle-tension change model, the forward displacement of the COP position was more delayed and overshot before the end; these results are not consistent with the measured human data.

Figure 5:

The hand paths (solid lines) and body COM paths (dashed lines) calculated by the three optimization models. Upper and lower figures show the results for reaching to the near and far targets, respectively. The stick pictures are the postures at start and end times.

Figure 5:

The hand paths (solid lines) and body COM paths (dashed lines) calculated by the three optimization models. Upper and lower figures show the results for reaching to the near and far targets, respectively. The stick pictures are the postures at start and end times.

Figure 6:

Time profiles of the hand velocity, body COM velocity, COP position, and angular velocity of joints calculated by the three optimization models. The solid, dashed, and dash-dotted lines correspond to the minimum muscle-tension change trajectory, minimum weighted torque change trajectory, and minimum torque change trajectory, respectively.

Figure 6:

Time profiles of the hand velocity, body COM velocity, COP position, and angular velocity of joints calculated by the three optimization models. The solid, dashed, and dash-dotted lines correspond to the minimum muscle-tension change trajectory, minimum weighted torque change trajectory, and minimum torque change trajectory, respectively.

Comparison of the minimum muscle-tension change trajectory with the measured trajectories in Figure 4 shows that the velocity profiles of the hand and body COM have good qualitative agreement. The peak timings of the hand and body COM velocities are around 0.4 s; this is almost consistent with the measured data. The peak value of the hand velocity was lower and that of the COP displacement greater than the measured trajectories. Regarding the joint profiles, the knee and hip joint profiles showed good agreement, whereas some peak velocities of the ankle, shoulder, and elbow joints showed inconsistencies.

4  Discussion

In this study, we investigated the motion criteria to determine human whole-body movements. Our primary finding is that the stereotyped features of the whole-body reaching movement can be predicted qualitatively with the minimum muscle-tension change criterion rather than the minimum torque change criterion. In the planar two-link arm movement, although the minimum torque change model reproduces human arm reaching movements well (Uno et al., 1989; Nakano et al., 1999), a few previous studies indicated discrepancies in the slow and fast reaching movements (Engelbrecht & Fernández, 1997; Klein Breteler, Meulenbroek, & Gielen, 2002). Expanding the two-link arm reaching movement to the redundant whole-body reaching movement, we found the other obvious discrepancies of the minimum torque change trajectories, which showed strongly curved hand paths and two-peak velocity profiles. The minimum muscle-tension change trajectories were similar to the actual human whole-body reaching movements. Clear adequacy of the criterion of smoothness in the muscle level rather than the torque level was shown by not only the physiological aspect (Dornay et al., 1996) but also the similarity of motion patterns owing to the expansion of the whole-body reaching movements. It should be emphasized that in our optimization model, no parameters for data fitting are required to reproduce the complex movement patterns with redundant joint degrees of freedom and balance maintenance. In addition, the weighted torque change cost as a simplified muscle-tension change cost also predicted the invariant features of human data. In the simplification, only PCSA and the moment arms are considered for the weighting factors of the torque change. The greater weight of arm joints with smaller muscles leads to a limited torque change of arm joints, which may make the hand path be slightly curved and nearly linear. Since greater inertial and gravitational moments act on the leg joints, the torque changes of leg joints become larger than those of arm joints. Equivalent evaluation of the torque change of leg joints and arm joints may lead to incorrect arm movement. Therefore, the previous models of muscle-level smoothness (Dornay et al., 1996; Soechting & Flanders, 1998), which did not consider PCSA, would result in an inaccurate prediction of whole-body reaching as well as minimum torque change model.

However, the minimum torque change criterion well predicts the features of the horizontal arm reaching movement, so the minimum muscle-tension change criterion cannot deviate far from that model. It will in fact perform the same because the values of weight of shoulder and elbow are comparable. The minimum muscle-tension change criterion for movements of the shoulder and elbow joints becomes equivalent to the minimum torque change criterion. Therefore, the minimum muscle-tension change model is a generalized formulation of a criterion for the smoothness of human whole-body movements. We confirmed that the weighted torque change trajectories with different ratios of weight (0.5–2.0) were invariant and consistent with the measured trajectories in the horizontal arm reaching movements.

We introduced some assumptions to simplify the muscle-tension change cost as the weighted torque change cost. It seems that the interactions of mono- and biarticular muscles related to the first assumption do not contribute much to the trajectory formation at least, although the stiffening of the joints by the co-contraction of the muscles may contribute to the stabilization of body COM and the accurate target pointing in the movement execution. We evaluated the cross-terms for the assumption of equation 2.10 with a minimum muscle-tension trajectory. The sum of the squared cross-terms was about 10% of that of the residual squared terms. Even with the rough approximations of the muscle-tension change criterion, the minimum weighted torque change model still predicts the features of the whole-body reaching movement without any data-fitting parameters, suggesting that the cost evaluation dealing with the volume or MVC force of muscles is crucial for whole-body movements. The parameter values of the muscles from a number of references were not derived through a consistent methodology. In particular, the shoulder joint has a complex structure between the muscle force and the joint motion. Although the accuracy of the parameter values of each muscle is limited, there is no doubt regarding the greater PCSA of the leg muscles than that of the arm muscles.

The model did not provide perfect predictions of the actual human movement patterns, for example, the peak velocities of the hand, body COM, and ankle, shoulder, and elbow joints. Of course, incomplete representation of the movement criterion is a possible cause of the inconsistency, but there are other possibilities. To formulate the whole-body movement by optimization, a large number of biomechanical and anatomical parameters for the musculoskeletal model were specified on the basis of the references. The differences in the musculoskeletal parameters between the model and subjects may cause the inconsistent results of the joint trajectories because the joint angular velocity profiles were subject specific in contrast to the invariant characteristics of the bell-shaped velocity profiles of the hand and body COM. In addition, our model is still incomplete with respect to the dimensions of joints and muscles. The larger COP displacement in the model prediction might be caused by the lack of degrees of freedom in the trunk segment. The trunk segment consisted of the pelvis, spinal bones, and head. Since the trunk segment was bent during the whole-body reaching movement, our model might have overestimated the displacement of the COP position. In addition, the COP displacement of the minimum muscle-tension change trajectory showed a profile different from that of the measured trajectory, which may have resulted from our formulation of the balance maintenance in the optimization.

In this study, the whole-body reaching movement is formulated on the basis of the optimal control principle. We compared the optimal trajectories determined by each single criterion, while it is possible that the central nervous system (CNS) deals with a multiobjective optimization (Tagliabue et al., 2008; Berret, Chiovetto, Nori, & Pozzo, 2011). However, if the performance index is represented by the weighted sum of different criteria, an alternative open question arises: How does the CNS determine the weights of the costs to plan the movement? In contrast to the optimal control approaches, Morasso, Casadio, and Zenzeri (2009) proposed a hypothesis of neural control for whole-body reaching movements on the basis of an equilibrium point hypothesis. The model can formulate the human-like whole-body reaching movement by a time-varying force field. It is beneficial that the complex calculation of the nonlinear dynamics of the human body is not required in this model. However, there are control parameters whose values should be appropriately chosen, such as the virtual stiffness of the hand, the damping component of the body COM velocity, and the admittance matrix specifying each joint velocity. So far, there are many possible mechanisms to explain human whole-body movements, and how the CNS solves the redundancy of movement patterns is still an open question.

One of the dominant hypotheses describing the origin of the smoothness is the consequence of maximization of precision under the signal-dependent noise (Harris & Wolpert, 1998). However, the minimum end point variance criterion was examined for only planar movement of a two-link arm model, and underlying computational difficulties of the stochastic optimal control prevent the extension to whole-body reaching. However, it would be meaningful if the smoothness of muscle tension could be interpreted by the minimum end point variance model. The signal-dependent noise was adopted for the simplified muscle models involving only the filtering effect of a second-order system (van der Helm & Rozendaal, 2000; Winter, 2005) and ignored the MVC force, which depends on muscle volume (Harris & Wolpert, 1998; van Beers, Haggard, & Wolpert, 2004; Todorov & Jordan, 2002). Therefore, it can be supposed that the model minimizing the variability with the simplified muscle model may predict incorrect whole-body reaching movements similar to the original minimum torque change model. A more precise muscle model and muscle-specific signal-dependent noise should be taken into account. The linear scale between the signal and noise results from the size-orderly recruitment of the motor unit (Jones, Hamilton, & Wolpert, 2002). Hamilton, Jones, and Wolpert (2004) showed that the rate of the standard deviation for mean torque of the joints depended on the maximum voluntary torque and the number of the motor units of the relevant muscles. The muscle-specific property of the signal-dependent noise may provide an explanation for the smoothness criterion of the muscle tension CF. Our future work will focus on the prediction of whole-body reaching through minimization of end-point variability under the muscle-specific noise.

Appendix: Biomechanical and Anatomical Parameters in a Musculoskeletal Model

Table 2 shows the biomechanical parameters used for the presented results of trajectory optimization. These parameters were specified for the typical subject 1. The values of the inertia and segment mass were derived by the body mass, and the lengths of the body segments and the positions of the segment COM were calculated from the measured segment length of subject 1 (Winter, 2005). The joint viscosities of the shoulder and elbow joints were specified as 1.76 and 2.13 Nms/rad, respectively, according to Nakano et al. (1999). The values of the ankle, knee, and hip joints were 11, 11, and 17 Nms/rad, respectively, according to Pozzo et al. (2001).

Table 2:
Biomechanical Parameter Values for Trajectory Optimization.
ShankThighTrunk and HeadUpper ArmForearm
Mass (kg) 5.86 12.6 36.4 3.53 2.77 
Length (m) 0.37 0.37 0.45 0.22 0.37 
COM (m) 0.21 0.21 0.23 0.10 0.25 
Inertia (kg0.33 0.73 3.77 0.05 0.26 
ShankThighTrunk and HeadUpper ArmForearm
Mass (kg) 5.86 12.6 36.4 3.53 2.77 
Length (m) 0.37 0.37 0.45 0.22 0.37 
COM (m) 0.21 0.21 0.23 0.10 0.25 
Inertia (kg0.33 0.73 3.77 0.05 0.26 

Table 3 shows the PCSA and moment arms of each muscle. The PCSA values of the leg muscles were determined by Brand, Pedersen, and Friederich (1986); Narici, Landoni, and Minetti (1992); Fukunaga, Roy, Shellock, Hodgson, and Edgerton (1996); Ward et al. (2009). The PCSA of the arm muscles was determined by Ikai and Fukunaga (1968); An, Hui, Morrey, & Linscheid, and Chao (1981); Veeger, van der Helm, van der Woude, Pronk, and Rozendal (1991); Kawakami et al. (1994); Happee and van der Helm (1995); Holzbaur, Murray, and Delp (2002). The moment arms of the leg muscles were given by Ogihara and Yamazaki (2001); Spoor and Van Leeuwen (1992); Fukunaga et al. (1996); Krevolin, Pandy, and Pearce (2004); Németh and Ohlsén (1985); Delp, Hess, Hungerford, and Jones (1999); and Rugg, Gergor, Mandelbaum, and Chiu (1990). The moment arms of the arm muscles were given by Kawakami et al. (1994); Murray, Delp, and Buchanan (1995); Kuechle et al. (2000); Holzbaur et al. (2002); and Bassett, Browne, Morrey, and An (1990).

Table 3:
PCSA and Moment Arms of Muscles.
MusclesPCSA (cm)Moment Arm (cm)
Soleus 162.0 4.3 (A) 
Tibialis anterior 19.0 −3.6 (A) 
Lateral head of gastrocnemius 17.4 4.5 (A), 3.3 (K) 
Medial head of gastrocnemius 47.5 4.5 (A), 3.3 (K) 
Vastus medialis 43.7 −4.0 (K) 
Vastus lateralis 49.8 −4.0 (K) 
Vastus intermedius 49.4 −4.0 (K) 
Biceps femoris short head 6.6 3.5 (K) 
Semimtendinosus 8.9 4.5 (K), −7.9 (H) 
Semimembranosus 32.4 3.5 (K), −7.9 (H) 
Biceps femoris long head 19.3 3.5 (K), −6.7 (H) 
Rectus femoris 28.2 −3.2 (K), 4.9 (H) 
Abductor magnus 23.0 −5.0 (H) 
Goluteus maximus 26.8 −7.8 (H) 
Iliopsoas 33.3 13.2 (H) 
Latissimus dorsi 8.1 −5.4 (S) 
Deltoideus pars scapularis 16.6 −2.2 (S) 
Deltoideus pars clavicularis 8.1 2.4 (S) 
Long head of triceps 6.2 −4.9 (S), −2.2 (E) 
Short head of biceps brachii 2.6 3.4 (S), 3.9 (E) 
Long head of biceps brachii 3.5 2.4 (S), 3.9 (E) 
Lateral head of triceps 5.3 −2.2 (E) 
Medial head of triceps 5.3 −2.2 (E) 
Brachioradialis 2.2 5.2 (E) 
Brachialis 8.9 2.0 (E) 
MusclesPCSA (cm)Moment Arm (cm)
Soleus 162.0 4.3 (A) 
Tibialis anterior 19.0 −3.6 (A) 
Lateral head of gastrocnemius 17.4 4.5 (A), 3.3 (K) 
Medial head of gastrocnemius 47.5 4.5 (A), 3.3 (K) 
Vastus medialis 43.7 −4.0 (K) 
Vastus lateralis 49.8 −4.0 (K) 
Vastus intermedius 49.4 −4.0 (K) 
Biceps femoris short head 6.6 3.5 (K) 
Semimtendinosus 8.9 4.5 (K), −7.9 (H) 
Semimembranosus 32.4 3.5 (K), −7.9 (H) 
Biceps femoris long head 19.3 3.5 (K), −6.7 (H) 
Rectus femoris 28.2 −3.2 (K), 4.9 (H) 
Abductor magnus 23.0 −5.0 (H) 
Goluteus maximus 26.8 −7.8 (H) 
Iliopsoas 33.3 13.2 (H) 
Latissimus dorsi 8.1 −5.4 (S) 
Deltoideus pars scapularis 16.6 −2.2 (S) 
Deltoideus pars clavicularis 8.1 2.4 (S) 
Long head of triceps 6.2 −4.9 (S), −2.2 (E) 
Short head of biceps brachii 2.6 3.4 (S), 3.9 (E) 
Long head of biceps brachii 3.5 2.4 (S), 3.9 (E) 
Lateral head of triceps 5.3 −2.2 (E) 
Medial head of triceps 5.3 −2.2 (E) 
Brachioradialis 2.2 5.2 (E) 
Brachialis 8.9 2.0 (E) 

Notes: Each letter following the value of moment arm indicates the relative joint: A: ankle, K: knee, H: hip, S: shoulder, and E: elbow. The sign of the moment arm indicates the direction of rotation induced by the muscle contraction.

Acknowledgments

This work was supported by Grant-in-Aid for Scientific Research (B) no. 26289129 and no. 26280101.

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