J Neurophysiol 95: 2898-2912, 2006.
First published January 25, 2006; doi:10.1152/jn.00983.2005
0022-3077/06 $8.00
Is Equilibrium Point Control Feasible for Fast Goal-Directed Single-Joint Movements?
Dinant A. Kistemaker,
Arthur (Knoek) J. Van Soest and
Maarten F. Bobbert
Institute for Fundamental and Clinical Human Movement Sciences, Vrije Universiteit, Amsterdam, The Netherlands
Submitted 19 September 2005;
accepted in final form 22 January 2006
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ABSTRACT
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Several types of equilibrium point (EP) controllers have been proposed for the control of posture and movement. EP controllers are appealing from a computational perspective because they do not require solving the "inverse dynamic problem" (i.e., computation of the torques required to move a system along a desired trajectory). It has been argued that EP controllers are not capable of controlling fast single-joint movements. To refute this statement, several extensions have been proposed, although these have been tested using models in which only the tendon compliance, force–length–velocity relation, and mechanical interaction between tendon and contractile element were not adequately represented. In the present study, fast elbow-joint movements were measured and an attempt was made to reproduce these using a realistic musculoskeletal model of the human arm. Three types of EP controllers were evaluated: an open-loop 
-controller, a closed-loop 
-controller, and a hybrid open- and closed-loop controller. For each controller we considered a continuous version and a version in which the control signals were sent out intermittently. Only the intermittent hybrid EP controller was capable of generating movements that were as fast as those of the subjects. As a result of the nonlinear muscle properties, the hybrid EP controller requires a more detailed representation of static muscle properties than generally assumed in the context of EP control. In sum, this study shows that fast single-joint movements can be realized without explicitly solving the inverse dynamics problem, but in a less straightforward manner than implied by proponents of conventional EP controllers.
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INTRODUCTION
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It has been widely acknowledged that the viscoelastic properties of muscles facilitate the control of posture (e.g., Milner 2002
) and movement (e.g., Brown and Loeb 2000; Van Soest and Bobbert 1993). Equilibrium point (EP) controllers form an important class of control models that exploit these viscoelastic properties. In an EP controlled system, voluntary movements are generated by changing EPs. Various EP controllers have been proposed in the literature, which differ in the way EPs are defined (see McIntyre and Bizzi 1993), but have in common that there is no need to calculate the torques required to move a system along a desired trajectory by solving the "inverse dynamics problem."
Two classes of EP controllers have received a lot of attention in the literature: the -model and the -model. According to the -model (e.g., Bizzi and Abend 1983; Hogan 1984), an EP is set by defining the "rest length" and the stiffness of the muscles crossing a joint, both of which are determined by the -motoneuron activity (McIntyre and Bizzi 1993). Although the -model does not need information about the dynamical properties of the musculoskeletal system, it does require information about its static properties; it presupposes a representation of the relation between -motoneuron activity and the resulting equilibrium position. Other types of EP controllers recognize that -motoneuron activity depends not only on supraspinal input but also on low level spinal loops. For example, at a given supraspinal input, -motoneuron activity, and consequently muscle force, depends on afference from muscle spindles. This prompted Merton (1953) to suggest that low-level spinal loops could form the basis for EP control. In his "servo"-controller, 
-motoneuron activity prescribes the desired lengths of muscle spindles, that is, the lengths that they would have if the system were in the desired EP. As long as actual spindle length deviates from the desired length, spindles produce afference that drives -motoneuron activity, causing the system to accelerate in the direction of the EP. However, this type of control is not supported by experiments because at movement initiation - and -motor units become active simultaneously (Vallbo 1970). Another EP controller, in which -motoneuron activity depends on feedback, is the so-called -model (e.g., Feldman 1986). In abstract terms, the -model assumes that an EP can be set by defining the threshold () of the stretch reflex of all muscles involved. At least in the original version, this controller needs only a representation (or "map") that relates joint angles to muscle lengths to generate control signals.
To improve the performance of EP controllers, two extensions have been proposed in the literature. First, a cocontraction command was added to increase stiffness, resulting in a faster movement toward the desired EP (e.g., Gribble et al. 1998). However, as was already argued by Gottlieb (1998b), whereas this would be straightforward to implement if the muscles in the effector system had identical invariant angle–torque characteristics, it remains to be shown whether an adequate cocontraction command can be easily generated when realistic muscle models are used. Second, velocity feedback was added to improve the damping characteristics of the system (e.g., Feldman 1986), and it has been advocated to include a contraction velocity reference signal and feed back the difference between desired and actual contraction velocity (de Lussanet et al. 2002; McIntyre and Bizzi 1993 rather than the actual velocity.
It has been argued that EP control is not feasible for fast single-joint movements (e.g., Schaal 2002; Schweighofer et al. 1998) or, alternatively, that EP controllers need complex EP trajectories to control fast single-joint movements (e.g., Bellomo and Inbar 1997; Hogan 1984; Latash and Gottlieb 1991; Popescu et al. 2003). However, both claims, as well as attempts to refute them (e.g., Gribble and Ostry; 2000; Gribble et al. 1998; St-Onge et al. 1997), were based on results of simulations with models of the musculoskeletal system in which the viscoelastic properties of muscles were inadequately represented or not at all. When testing EP controllers with musculoskeletal models, it is of paramount importance to use a realistic model of the muscles. For example, the maximally attainable movement speed depends on muscle properties such as the force–velocity relationship. It has been argued (Gribble et al. 1998; but see Nakano et al. 1999) that the complex and nonmonotonic nature of (virtual) EP trajectories derived on the basis of comparison of simulation results with experimental results (Gomi and Kawato 1996; Latash and Gottlieb 1991) was attributed to oversimplification of the force–velocity relationship of the modeled muscles. Furthermore, most EP controllers have been tested with models that ignored the series compliance of muscle; especially in pennate muscles, with relatively short muscle fibers, this series compliance may be large (the compliance resides not only in tendons, but also in aponeuroses); we will use "tendon compliance" as shorthand for the total series elasticity. In a musculoskeletal model that does not incorporate tendon compliance, EP controllers can control joint angle by setting the appropriate muscle lengths (i.e., the length of the muscle–tendon complex as a whole) using feedback from muscle spindles (e.g., Gribble et al. 1998; Latash and Gottlieb 1991). In reality, however, muscle spindle feedback is not related to muscle length but to muscle fiber length, and tendon compliance causes the relation between muscle fiber length and total muscle length to depend on muscle force. Furthermore, when tendon compliance is ignored, contraction velocity becomes proportional to angular velocity, which results in physiologically implausible muscle behavior (e.g., Zajac 1989).
Interestingly, both neurophysiological and behavioral studies suggest that humans control their movements intermittently. For example, during voluntary movements of the finger (Evans and Baker 2003; Valbo and Wessberg 1993), wrist (Conway et al. 2004; Kakuda et al. 1999), and elbow (Conway et al. 1997; Doeringer and Hogan 1998), humans typically show 6- to 12-Hz variations in angular velocity. Wessberg and Vallbo (1996) provided evidence that reflex responses during finger movements are too weak to account for the observed intermittent modulation of motor output. It is conceivable that motor neuron pool dynamics play a role in the observed intermittency. However, recent studies of Gross et al. (2002) and Pollok et al. (2005) indicated that finger movements are driven by a cerebello-thalamo-cortical loop with a frequency of about 8–12 Hz. These observations are complemented by a recent study of Conway et al. (2004) in which a low-frequency (<12 Hz) coupling was found between motor cortex EEG and muscle EMG during the execution of fast wrist movements. Combined, these findings suggest that, whereas the feedback components of muscle stimulation are sent out continuously, the open-loop components are sent out intermittently. This leads to the question to what extent this intermittency affects the EP control of fast movements.
The main purpose of this study was to investigate whether EP control using simple EP trajectories is feasible for fast single-joint arm movements. Elbow angle and EMG were measured during fast point-to-point elbow movements. Subsequently, it was attempted to reproduce the experimentally observed movements with a realistic musculoskeletal model of the arm, controlled by three types of EP controllers: an open-loop -controller, a closed-loop -controller, and a hybrid open- and closed-loop EP controller. To investigate the influence of intermittent control, we also performed simulations in which the open-loop components of these EP controllers were sent out intermittently. It will be shown that all EP controllers gain movement speed when control signals are sent out intermittently rather than continuously. Furthermore, it will be shown that an intermittent version of the hybrid EP controller is capable of making the model reproduce the fast point-to-point movements observed in the subjects, from which we will conclude that EP control cannot be dismissed when considering fast point-to-point single-joint movements.
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METHODS
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Experimental protocol
After providing informed consent, six male participants (age 27 ± 2 yr) were seated on a nonrotating chair behind a table with their arm in a horizontal plane through both shoulders. The lower arm was strapped to a light glass fiber tube (0.48 m; 0.345 kg) that was attached to the table with a hinge. The participants were seated such that the flexion–extension axis of the elbow joint was aligned with the axis of the hinge. The shoulder angle was 45° (see Fig. 1, A and B for definitions of elbow and shoulder joint angle). Three blocks made of polystyrene were placed on the table at a distance of 60 cm from the hinge, corresponding to elbow angles of 45, 95, and 145°. A laser pointer was mounted to the end of the tube such that when the tube rotated in the horizontal plane, the laser beam hit the blocks at their centers. Participants were instructed to direct the light of a small laser pointer (13.5 g), mounted on the end of the tube, to the marked middle of one block and then, on an auditory cue, to direct it as fast as possible to the marked middle of another block. Participants performed fast point-to-point movements in six randomized conditions, each consisting of 15 trials: elbow flexion movements Flex1–3 of 100° and Flex1–2 and Flex2–3 of 50°; and extension movements Ext3–1 of 100° and Ext2–1 and Ext3–2 of 50°. Before the 15 trials in each condition, participants practiced until they could move fast to the target with minimal overshoot. Breaks of 5 min were interspersed between conditions.
Data collection and signal processing
Elbow angle time histories were recorded using a potentiometer (Sakae, FCP40A, sample rate 1,000 Hz) and low-pass filtered at a cutoff frequency of 20 Hz using a fourth-order zero-lag Butterworth filter. Myoelectric activity was recorded (TMS Porti, Enschede, The Netherlands, sample rate 1,000 Hz) from m. biceps, m. brachoradialis, m. triceps caput longum, and m. triceps caput laterale, using pairs of surface electrodes (Medi-trace, pallet electrodes, center-to-center electrode distance 2 cm) attached to the skin after standard skin preparation (Hermens et al. 1999). EMG signals were amplified, band-pass filtered (10–500 Hz), and full-wave rectified. For each participant, the six movements with the highest peak angular velocity per condition were selected that did not show more than 10° overshoot of the target.
Musculoskeletal model of the arm
The musculoskeletal model of the arm consisted of three segments, connected by two hinges representing the glenohumeral joint and the elbow joint, respectively (Fig. 1B). To reproduce the experimental data, only elbow flexion–extension movements in the horizontal plane were allowed. The lower arm was actuated by four lumped muscles: a monoarticular elbow flexor (m. brachioradialis, m. brachialis, m. pronator teres, m. extensor carpi radialis), a monoarticular elbow extensor (m. triceps brachii caput laterale, m. triceps brachii caput mediale, m. anconeus, m. extensor carpi ulnaris), a biarticular elbow flexor (m. biceps brachii caput longum and caput breve), and a biarticular elbow extensor (m. triceps brachii caput longum). Muscle-specific parameters (maximal isometric force, tendon slack length, optimum contractile element length, and moment arms; see APPENDIX A) were obtained from the literature (Murray et al. 1995, 2000; Nijhof and Kouwenhoven 2000). The parameter values of the lumped muscles were a weighed sum of the parameter values of their component muscles, with the weight factors depending on the contribution of each component muscle to the total moment. Nonspecific muscle parameters were taken from Van Soest and Bobbert (1993).
The muscles were modeled as Hill-type units (see APPENDIX A, Fig. A1) consisting of a contractile element (CE), a parallel elastic element (PE), and a series elastic element (SE). In previous research, it was shown that the implemented Hill-type muscle model is capable of reproducing the characteristic features of the dynamical behavior of muscles (e.g., Bogert et al. 1998; Winters and Stark 1985; Zajac 1989). Activation dynamics, describing the relation between the excitatory signal of the muscle and active state, was modeled according to Hatze (1981; see also Fig. 3). In line with Hatze's description, the excitatory signal of the model was termed muscle stimulation (STIM).
It was previously shown elsewhere that the resulting muscle model accounts for the experimentally observed length-dependent Ca2+ sensitivity of a muscle and the stimulation dependency of optimum length (Kistemaker 2005). The parameters used were equal for all controllers and kept constant during the simulations. A detailed description of the muscle model is provided in APPENDIX A and all parameter values are listed in Tables A1 and A2 of APPENDIX A. For relevant abbreviations, see Glossary (APPENDIX C).
In a previous study a musculoskeletal model was developed in which isometric elbow angle–torque relations agreed with those reported in a recent study (Kistemaker et al., unpublished observations). Using this model, it was found that stable open-loop EPs could be achieved over the whole range of motion of the elbow joint and stiffness (ranging from 18 to 42 Nm rad–1 depending on elbow-joint angle) could be controlled independently by means of cocontraction. In the present study we also tested the sensitivity of the simulation results to model parameter values determining the activation and contraction dynamics (see APPENDIX B).
A variable step-size ordinary differential equation solver based on the Runge–Kutta (4,5) formula was used to numerically solve the differential equations of the musculoskeletal model. Model and simulations were implemented in MATLAB 6.5 Release 13.
Controllers
An EP trajectory was defined as a "ramp" trajectory from the initial to the final position (see Fig. 2A). Based on the experimental data, desired movement time was estimated to be 0.20 s for the movements of 100° and 0.18 s for the movements of 50°. The stimulation defining an open-loop EP in the absence of external forces will from now on be referred to as STIMopen. As mentioned earlier, three different types of EP controllers were implemented: an open-loop -controller, a closed-loop -controller, and a hybrid open- and closed-loop controller. The muscle stimulation generated by the -controller was only STIMopen. The muscle stimulation generated by the -controller depended solely on the difference in desired () and actual CE length (lCE) and CE contraction velocity (vCE). Feedback of lCE and vCE was assumed to be linear and a 25-ms time delay in the feedback loop was modeled using a fifth-order Padé approximation (Golub and Van Loan 1989). The muscle stimulation that resulted solely from feedback will from now on be referred to as STIMclosed. The muscle stimulation generated by the hybrid EP controller was the sum of STIMopen and STIMclosed. The contraction velocity feedback in this controller depended on the difference between desired (
) and actual vCE. Figure 2 shows the and traces for - and hybrid EP controller. More details on the implementation of the controllers are provided below.
-Controller
When the -controller was used (see Fig. 3A), STIMopen was the only input of the musculoskeletal system
 | (1) |
Not surprisingly, exploration of the model showed that the fastest movements were generated when STIMopen was set to produce the highest low-frequency joint stiffness possible (results not shown). Because our interest was in fast movements, the STIM yielding the highest stiffness was selected. The only exception was STIMopen at the equilibrium starting position, which, in line with the empirical observation that muscle activity is practically absent at initial positions (e.g., Gottlieb 1998a; Ostry and Feldman 2003; Suzuki et al. 2001), was chosen such that it minimized the sum of the individual muscle stimulations.
-Controller
The total muscle stimulation generated by the -controller (STIM) depended on the difference between (activation CE threshold length) and lCE and on vCE (see Fig. 3B)
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where kp and kd are feedback constants that will be optimized (see following text) and 
(=0.025 s) is a short-latency reflex delay. In the present study it was assumed that the muscle spindles provide accurate time-delayed information about lCE and vCE. The expression {x}01 means that values of x > 1 were set to 1 and negative values were set to 0.
To set an EP in terms of values of the muscles around the elbow joint, the relation between elbow angle and CE length of the individual muscles must be known. Because SE length depends on the force delivered by the CE, no one-to-one relation exists between elbow angle and CE lengths. This implies that a choice had to be made with respect to the mapping of CE to . In analogy to the reciprocal command (R) of the -model (e.g., Feldman et al. 1990), was chosen such that it equaled CE length in a desired EP with zero stimulation (see Fig. 2B).
Hybrid EP controller
McIntyre and Bizzi (1993) hypothesized that a combination of the -controller and -controller increases movement speed. They also suggested that feedback on the difference between actual and desired velocity, instead of feedback on actual velocity alone (i.e., linear damping), would enhance the performance of the system. Both suggestions provided grounds for formulating a hybrid EP controller of the form (see Fig. 3C)
 | (3) |
values were derived from the steady-state solution of the muscle model using the equilibrium angle and STIMopen. is the time derivative of (see Fig. 2, E and G) and kp and kd are feedback constants. It was assumed that the CNS is capable of generating a reference signal on basis of the desired trajectory (McIntyre and Bizzi 1993), although direct physiological evidence for this assumption is absent in the literature.
Intermittent controllers
Intermittent control was implemented by updating STIMopen, , and at a frequency of 10 Hz. Feedback was updated continuously. Note that when sending out and intermittently, is no longer the time derivative of . The algorithms defining the intermittent -controller, -controller, and hybrid EP controller were the same as those for the continuous versions (see also Fig. 2).
Optimization of feedback gains
The feedback gains used for the -controller and the hybrid EP controller were optimized to minimize the total sum of the squared differences between simulated and desired movements for all six conditions. Thus for each controller, only one set of feedback gains was used. It is implausible that endpoint oscillations of the actual movement are part of the desired movement. Therefore for the desired movements, minimal jerk trajectories (see Fig. 2B) were used to optimize the feedback gains, rather than the experimental data themselves. Minimal jerk trajectories have often been used to describe the kinematics of fast point-to-point movements and do not show oscillations (e.g., Hogan 1984). The optimal values for the feedback gains were identified using a combination of a grid search and a Nelder–Mead simplex search method (Lagarias et al. 1998).
Kinematic features
To quantify the performance of the controllers, three movement features were extracted from the simulated and experimental data: peak angular velocity (
p), time to peak velocity (tpv), and "movement duration" (tmov). tpv was defined as the time between the instant that the elbow reached 5% of the total distance to be covered and the instant at which p was reached. tmov was defined as the time needed to move the arm from 5 to 95% of the total distance to be covered. Furthermore, the root-mean-squared (RMS) values were calculated between the simulated and (averaged) experimentally observed position traces. RMS values were calculated by first calculating the cross-correlation function between experimental and simulated data. Subsequently, the lag yielding the highest cross-correlation was identified. Then, the RMS value between experimental data and the simulated data, time shifted with the indicated lag, was calculated (see Table 1).
Statistical analysis
Condition-related differences in kinematic parameters were analyzed using a 2 (Direction) x 3 (Trajectory) repeated-measures ANOVA. The factor Direction had levels flexion and extension and the factor Trajectory had levels 45–145, 45–95, and 95–145° (and vice versa). Single-sample t-tests ( = 0.05) were used to test for differences between the features of the simulated and experimental data.
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RESULTS
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Column I of Fig. 4 and columns I and III of Fig. 5 show the recorded elbow angle, angular velocity, and EMG traces for six trials of one representative participant in all six conditions. Before comparing simulation results with experimental data, it is useful to summarize the results of a statistical analysis of the experimental results. For peak angular velocity, a significant main effect of Trajectory [F(1,5) = 246.4; P < 0.001] and a significant Direction x Trajectory interaction [F(2,10) = 5.38; P = 0.026] were found. For time to peak angular velocity, we found a significant main effect of Direction [F(1,5) = 10.15; P = 0.024], a main effect of Trajectory [F(1,5) = 68.4; P < 0.001], and a significant Direction x Trajectory interaction [F(2,10) = 7.56; P = 0.010]. For movement duration we also found a significant main effect of Trajectory [F(1,5) = 154.2; P < 0.001] and a significant Direction x Trajectory interaction [F(2,10) = 4.46; P = 0.041]. Post hoc t-test revealed that each of the dependent variables reached different values in the large-amplitude condition (movements of 100°) than in the two small-amplitude conditions (movements of 50°), whereas the values reached in the small-amplitude conditions were not statistically different. In the large-amplitude condition higher peak angular velocities were reached than in the small-amplitude movements (mean 930 vs. 655°/s), the subjects took longer to arrive at peak velocity (mean 0.074 vs. 0.052 s), and total movement duration was greater (mean 0.128 vs. 0.093 s). For time to peak velocity, the main effect of Direction resulted from a significantly longer time to arrive at peak angular velocity in the flexion condition. The significant Direction x Trajectory interaction was the result of significant differences for peak angular velocity (t = 3.33; P = 0.021) and movement duration (t = –3.62; P = 0.015) between the Flex1–3 and Ext3–1 conditions: in the Flex1–3 condition higher peak angular velocities were reached (mean 975 vs. 886°/s) and movement duration was shorter (mean = 0.118 vs. 0.137 s) than in the Ext3–1 condition. For time to peak velocity, the significant Direction x Trajectory interaction was the result of significant differences between the small-amplitude movements. Peak angular velocity and movement duration did not differ significantly between the small-amplitude conditions, but movement duration was significantly smaller in the Ext2–1 condition than in all other movements of 50°.
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FIG. 5. Experimental data (columns I and III) and simulation results obtained with the intermittent hybrid EP controller (columns II and IV) for conditions Flex1–2 and Flex2–3 (A) and for conditions Ext2–1 and Ext3–2 (B). Top EMG/STIM traces refer to the 2 elbow flexors, bottom traces to the 2 elbow extensors.
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Column II of Fig. 4 shows the time histories of kinematic data and STIM of the -controller for conditions Flex1–3 (Fig. 4A) and Ext3–1 (Fig. 4B). Single-sample t-test showed that all kinematic features produced by the simulations with the continuous and intermittent -controller were significantly different (P < 0.01) from those subtracted from the experimental data (see Table 2). When EPs were set intermittently, the model reacted less sluggishly and movement speed increased, but insufficiently so to match human performance. The overall resemblance between simulated movement using the -controller and experimental data was poor, as indicated by the high RMS values (on average 37.7 and 22.5° for the continuous and intermittent version, respectively; see Table 2).
The movements produced with the -controller (column III of Fig. 4) also failed to match experimental data: all kinematic features were significantly different (P < 0.001) from those observed experimentally. Apart from reaching a significantly lower peak angular velocity (see Table 2) and taking more time, the model exhibited marked endpoint oscillations. The relatively high feedback gains, necessary to minimize the error between the actual trajectory and the minimal jerk trajectory, were the main cause of these oscillations. When additional "penalties" on endpoint oscillations were introduced in the optimization criterion, feedback gains and oscillations decreased, but at the cost of movement speed (results not shown). As with the -controller, the RMS values were high (on average 23.5 and 17.9° for the continuous and intermittent version of the -controller, respectively), indicating a poor resemblance between simulated and experimentally observed movements.
In both Fig. 4 and Fig. 5, Column IV shows the simulation obtained with the hybrid EP controller for all six conditions. This hybrid EP controller was the only controller capable of producing movements at least as fast as those observed experimentally (see Tables 2 and 3 for complete results of statistical comparison). In only one of six conditions (Ext3–2) was peak p in the simulation significantly lower than that observed experimentally. Similarly, in only one of six conditions, the duration of the simulated movement was significantly longer than that observed experimentally (Flex2–3). Finally, peak velocity in the simulated movements was never reached later than in the experimental movements. In fact, tpv was significantly smaller in the simulations in five of six conditions. The close resemblance between movements generated using the intermittent hybrid EP controller and those observed experimentally is reflected by the average RMS value, which was found to be only 2.3° (vs. 3.9° for the continuous EP controller; Table 2). In sum, the intermittent hybrid EP controller was capable of generating point-to-point movements at least as fast as those observed experimentally.
The muscle stimulation patterns generated with the hybrid EP controller were qualitatively similar to the experimentally observed EMG patterns (Figs. 4 and 5). The controller produced the triphasic burst pattern typical of fast point-to-point movements (e.g., Prodoehl et al. 2003; Wachholder and Altenburger 1926; Yamazaki et al. 1993; see also Fig. 4). It can be seen in Fig. 6, which presents the separate contributions of open-loop and feedback components to the total muscle stimulation of the hybrid EP controller, that the triphasic patterns mainly originated from the contraction velocity feedback (see also Bullock and Grossberg 1992). It is still debated whether triphasic EMG bursts result predominantly from central or peripheral processes (e.g., Prodoehl et al. 2003). In line with the position of Bullock and Grossberg (1992), the results obtained in this study with the hybrid EP controller suggest that triphasic bursts need not be preprogrammed.
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FIG. 6. Contribution of STIMopen, feedback of lCE (STIMs) and feedback of vCE (STIMd) to the total muscle stimulation of agonists (column A) and antagonists (column B) generated by the intermittent hybrid EP controller (STIMh) for the Flex1–3 condition. Continuous lines refer to the monoarticular elbow flexor and extensor, dashed lines refer to the biarticular elbow flexor and extensor.
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DISCUSSION
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The purpose of this study was to examine whether EP control is feasible for fast goal-directed single-joint movements. In an attempt to reproduce the experimentally observed fast movements, a realistic model of the human arm was used that was controlled by several types of EP controllers. All controllers generated faster movements when the control signals were sent out intermittently. However, movements as fast as those produced by the subjects could be generated by the musculoskeletal model only when the intermittent hybrid EP controller was used.
The speeds generated by the -controller strongly depended on the local stiffness of the EP that was set. The stiffness in an EP can be seen as the change in steady-state net torque for a given deviation from that EP. Even though STIMopen was chosen such that it maximized joint stiffness, and even though in a previous study it was shown that the maximal stiffness of the model was in agreement with the literature (Kistemaker et al., unpublished observations), maximal movement speed was insufficient to match the experimental data. By sending out the control signals intermittently rather than continuously, deviations of the current position from the (intermittent) target were initially larger, so that greater forces were produced, resulting in faster movements (see Fig. 4). The requirement that at least start and endpoint be set resulted in a minimal modulation frequency of 5 Hz. Movement speed increased when this frequency was used, but stayed well below actually observed speeds.
The -controller was also incapable of generating movements fast enough to match experimental data. This was mainly attributed to the fact that muscle stimulation resulted from feedback only, whereas feedback gains had to be limited to prevent time-lag–related instability and oscillatory behavior. A shorter time lag diminished this effect, but a time lag <25 ms used in this study is physiologically unrealistic (e.g., Funase and Miles 1999). Implementing a reference contraction velocity and intermittency in the -controller enhanced maximal movement speed, but the resulting movement was still not fast enough to match experimental data and, moreover, did not eliminate the oscillations shown in Fig. 4. Other studies (Gribble et al. 1998; St-Onge et al. 1997) used a duration of the EP trajectory that was half that of the intended movement. Additional simulations with the present model indeed showed that the -controller reacted less sluggishly and gained movement speed when the duration of the EP trajectory was halved. However, again, the resulting movement speed stayed well below that observed experimentally and, as a result of the increased velocity, the endpoint oscillations increased in amplitude. Even when both options (i.e., halved duration of EP trajectory and reference feedback) were implemented, the -controller was still incapable of reproducing the experimental data (see APPENDIX B).
Although substantially faster than the movements controlled by both the (intermittent) - and -controller, the movements controlled by the continuous version of the hybrid EP controller were not fast enough to match the experimental data. Maximal movement speed could be increased to about the experimentally observed value by raising the feedback gains, but only at the cost of a large overshoot of the target (results not shown). However, when the control signals of the hybrid EP controller were sent out intermittently, the musculoskeletal model produced movements that were in general as fast as or faster than the fast point-to-point movements observed in our experiments. Additional simulations (APPENDIX B) further showed that the reference velocity adopted in the hybrid EP controller was indispensable to achieve a close match between simulated and experimentally observed results. As an aside we note that, given the main question of this study, it was not attempted to maximize the fit between experiment and simulation. Several possibilities exist to further improve this fit, for example by condition-specific manipulation of the duration of the EP trajectory or by allowing different feedback gains for the different muscles. Similarly, the unrealistically high levels of cocontraction near movement completion could easily be reduced by lowering the level of cocontraction (lowering STIMopen) near the end of the movement (see APPENDIX B).
Some studies used a duration of the EP trajectory that was half the intended duration (Ghafouri and Feldman 2001; St-Onge et al. 1997). The performance of the continuous hybrid EP controller improved when such an EP trajectory was used, but its performance was worse than that of the intermittent EP controller (see APPENDIX B). The model reacted slightly faster, but was incapable of keeping up with this much faster (1,000 instead of 500°/s) EP trajectory. As a consequence, damping arising from feedback started to counteract the movement at the instant the EP trajectory reached the endpoint because from this instant onward the reference velocity equaled zero. Although an experimental study indicated that the shifts in EP during fast point-to-point movements may end approximately near peak velocity (Ghafouri and Feldman 2001), it does not seem directly beneficial for the control of fast movements when the difference between desired and actual velocity is fed back.
The hybrid EP controller, which combined open- and closed-loop components, was not simply a combination of the implemented -controller and -controller. This is because the desired CE length () depends on the total muscle stimulation (recall that, because of tendon compliance, steady-state CE length at a given joint angle depends on stimulation). In other words, the values set by the -controller are not equal to the CE lengths in a desired EP defined by the STIMopen of the -controller. The hybrid EP controller requires an internal representation of the following static properties of the musculoskeletal system: 1) a map relating desired EP to STIMopen, 2) a map relating STIMopen to joint stiffness, and 3) a map relating STIMopen to steady-state CE length. (In essence, the implemented -controller uses the first two maps.) Thus as a result of the complexity of the musculoskeletal model, the hybrid EP model requires a more detailed representation of static properties of the musculoskeletal system than generally used in the context of EP controllers (e.g., Gribble et al. 1998; Latash and Gottlieb 1991; McIntyre and Bizzi 1993). The feedback parameters were optimized to minimize the error between actual and desired movement and, consequently, were implicitly tuned to the parameters determining the dynamics of the musculoskeletal system. Additional simulations showed that the performance of the hybrid EP controller model was not noticeably affected when the inertial parameters of the musculoskeletal model were changed by 10% without reoptimizing the feedback gains (see APPENDIX B). This result suggests that for single-joint movements, EP control is not critically dependent on representation of dynamical parameters of the musculoskeletal system. To conclude, the present study showed that fast single-joint movements of a realistic musculoskeletal model can be adequately controlled without the requirement of solving the "inverse dynamics problem" and without an internal representation of the dynamical properties of the musculoskeletal system, albeit in a less straightforward manner than implied by most proponents of conventional EP controllers.
A major difference between the -controller presented in this study and most -models presented in the literature is that our -controller lacked a cocontraction or coactivation (C) command: a "simple" command that shifts all values to a shorter length (Feldman 1986). Because of the nonlinear behavior of the musculoskeletal system, such a simple C-command could not be derived for the present musculoskeletal model without affecting the equilibrium position (see also Gottlieb 1998b; Windhorst 1994, 1995). However, using the internal representation of the static properties that was required for the hybrid EP controller, cocontraction can be added to the -controller without affecting the equilibrium position. In fact, it can be shown that such a -controller is mathematically equivalent to the hybrid EP controller. Basically, we need to use the same three maps as those used for the hybrid EP controller, but "replace" STIMopen by a cocontraction command (note that this cocontraction command is specific for each muscle involved). As a result of tendon compliance, the steady-state CE lengths in an EP can match the values (in terms of CE lengths) only if the controller is generating exactly the appropriate muscle stimulation at the EP: this may be accomplished by STIMopen in the case of the hybrid EP controller and by a cocontraction command in the case of the -controller (note that at the EP, cocontraction is the only "source" of muscle stimulation). Once a desired EP with a desired stiffness is selected, it can easily be deduced that the cocontraction command (*) for each individual muscle must equal
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This also shows that * is in essence an open-loop component: the cocontraction command is the open-loop muscle stimulation defined in terms of CE length. Because steady-state CE length is completely determined by the cocontraction command no insight is gained by referring to the set of values as the "position" command (note again that, because of tendon compliance, does not relate one to one with joint angle). By substituting by ( + *) in Eq. 2, it can be seen that the modified -controller is mathematically equivalent to the hybrid EP controller. Therefore it can be concluded that when an appropriate cocontraction command is added to the (intermittent) -controller, using the representation of the static properties of the musculoskeletal system mentioned before (e.g., represented in maps), the controller is capable of generating fast point-to-point movements that resemble experimental movements. Interestingly, Feldman et al. (1990) acknowledged the need for maps relating the number of active motoneurons and their firing rates to control variables and kinematic variables; the present study provides a more detailed description of the variables that need to be coded in such maps.
When the EP controllers sent out their control signals in an intermittent rather than continuous fashion, the movements were faster and corresponded better with experimentally observed data (see Fig. 4 and Table 2). Although several studies provided evidence that humans control their movements intermittently rather than continuously (e.g., Craik 1947; Gross et al. 2002; Vallbo and Wesberg 1993; Wessberg and Vallbo 1996), intermittent control at first sight appears to be irreconcilable with the experimental results reported by Bizzi et al. (1984). Based on experiments with deafferented monkeys, Bizzi et al. (1984) showed that the final position is not set immediately at movement onset and proposed a gradual evolution of EPs. In the experiments in question, monkeys were trained to move their invisible arm to a target position. During some trials, a servo-motor was used to move the monkey's arm unnoticed from the initial position to the target position. Because the monkeys were encouraged to move their arm to this very position one would expect the arm to stay put, but instead, when released, it started to move back in the direction of initial position. This result militates against a control scheme in which the endpoint is immediately set at movement onset. Using an estimation of movement time and distance from Fig. 6 in Bizzi et al. (1984), the intermittent versions of our controllers would generate an EP trajectory in steps of 6°. Thus the intermittent versions of the -, - (if extended with the cocontraction command), and hybrid EP controllers would be capable, at least qualitatively, of reproducing Bizzi's data. Furthermore, as suggested by Bizzi and colleagues (1984), it is conceivable that for very fast movements, as studied here, the shifts in EP are more abrupt. Taken together, intermittent control as implemented in the current study does not contradict the results of experimental studies on trajectory formation during arm movement such as those reported by Bizzi et al. (1984).
A consequence of sending out control signals intermittently is that the EP trajectory is no longer identical to the desired trajectory. Other studies also suggested that EP trajectories other than the desired trajectories, such as N-shaped EP trajectories (Hogan 1984; Latash and Gottlieb 1991), underlie (and improve) the control of fast movements. However, the intermittent EP trajectory used in the present study has two advantages over N-shaped EP trajectories: 1) as argued in the INTRODUCTION, it has both a neurophysiological and a behavioral foundation; and 2) although the total trajectory may be different from the desired trajectory, the set points themselves are part of the desired trajectory and therefore do not require additional calculations. In the literature, the existence of intermittent behavior is often seen as an "imperfection" of the structure and functioning of the CNS (e.g., Craig 1947; Hanneton et al. 1997; Miall et al. 1993). Our results, however, suggest that intermittent control is functional in the control of fast point-to-point movements.
In this study we have assumed that the feedback gains did not vary during the movement. Although to our knowledge all studies on EP control have used continuous feedback, several studies suggest that the short-latency reflex contributions to stimulation of muscles are significantly lowered at movement onset and termination (e.g., Shapiro et al. 2002, 2004). Because this could militate against the feasibility of an EP controller (Shapiro et al. 2004), we performed additional simulations in which the contribution of feedback was gated using a Gaussian function such that feedback was absent at the beginning and end of the movement (the gating of feedback did not interact with the open-loop muscle stimulation, as may be noted from Eq. B1 in APPENDIX B). It was found that the hybrid EP controller suffered no performance loss when feedback was gated (see APPENDIX B); the lack of feedback at the beginning of the movement was compensated by the much higher (optimized) feedback gains. Because feedback is absent at the beginning and end of the movement, the high feedback gains used did not result in instability or oscillatory behavior. Thus the hybrid EP controller is still capable of controlling fast single-joint movements when feedback is diminished at movement on- and offset.
All in all, the results of this study refute the claim that EP controllers cannot account for fast single-joint movements (e.g., Schaal 2002; Schweighofer et al. 1998), as well as the claim that EP controllers need complex EP trajectories to control fast single-joint movements (e.g., Bellomo and Inbar 1997; Hogan 1984; Latash and Gottlieb 1991; Popescu et al. 2003). However, they do not refute the claim that EP controllers predict high stiffness during fast single-joint movements (e.g., Gomi and Kawato 1997; Popescu et al. 2003). Given the intricacies of experimental determination of stiffness, we would suggest addressing this issue by comparing experimentally estimated stiffness to that predicted by the EP controlled model when subjected to the same perturbation. This remains a challenge for further research.
Although we showed that EP control is feasible in the context of fast point-to-point single-joint movements in the absence of external forces, the question remains whether the present results are generalizable to the control of movements under various circumstances involving, for example, multijoint movements and gravity. The presented hybrid EP controller is presumably able to generate control signals for multijoint movements, provided that the maps mentioned earlier are expanded for all joints and muscles involved. However, the resemblance between experimental and simulated data of single-joint movements does not guarantee that simulated multijoint movements will match experimental data as well. An important difference between single- and multijoint systems is that, in the latter, stiffness is no longer a single-valued function of STIMopen. This implies that nontrivial assumptions have to be made to select STIMopen. Also, it has been shown that "interaction torques" (i.e., torques that arise at one joint as a result of movements at the other) are taken into account by the CNS, which would be inconsistent with EP control (but see Gribble and Ostry 2000). Furthermore, it has been suggested in the literature that mono- and biarticular muscles have distinct roles during movement (Hof 2001; Van Ingen Schenau 1987), a proposition not taken into account by EP controllers, at least not explicitly.
In this study, the arm was restricted to move in a horizontal plane, so that the movement was not affected by gravity. When the arm is moved outside the horizontal plane, static errors between desired EP and actual position will occur, unless gravity on the arm and on loads possibly carried by the hand is accounted for in the maps used by the EP controller. Errors (or their compensations) will depend on the exact orientation of the arm in the gravitational field, its static properties, and the stiffness induced by both muscles and feedback. When the upper arm of the model was placed vertically and the lower arm horizontally (thereby maximizing the influence of gravity), the deviation from the desired EP was <2° if STIMopen was set such that open-loop stiffness was maximal (in this case about 20 Nm rad–1), and about 5° with the lowest possible amount of cocontraction. When external loads on the hands are considered, however, the deviation from the desired position soon became unacceptably large. Thus in our view, external forces such as gravity should be taken into account by the controller.
In sum, although this study has demonstrated that EP control cannot be dismissed when considering fast point-to-point single-joint movements, it remains to be established whether EP control is feasible when multijoint movements in the presence of external forces are considered.
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APPENDIX A
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The implemented Hill-type muscle model consists of a contractile element (CE), a series elastic element (SE), and a parallel elastic element (PE) as shown schematically in Fig. A1. The mechanical behavior of the muscle is described by two coupled first-order differential equations. One first-order dynamical system describes the activation dynamics and related the muscle stimulation (STIM) to active state (q). The other first-order dynamical system describes the relation between the contraction velocity (vCE) and the length of the contractile element (lCE), depending on q, the force–length–velocity relationship of CE, and the passive forces of SE and PE. The length of the muscle–tendon complex (lMTC) and moment arm (arm) are functions of joint angle. A detailed description of the muscle model is provided below (see also the flowchart of the model presented in Fig. 3; see Glossary in APPENDIX C for the relevant abbreviations).
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fig. A1. Schematic representation of the Hill-type muscle model used in this study. For abbreviations see Glossary (APPENDIX C).
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Activation dynamics
Activation dynamics was modeled according to Hatze (1981; see also Kistemaker et al. 2005) and related muscle stimulation (STIM) to active state (q) in two steps. A first-order dynamical system related the free Ca2+ concentration (relative to its maximum value; rel) to STIM. Subsequently, an algebraic relation described how active state q depends on rel and (by 
) on CE length relative to its optimum (lCE_rel)
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 | (A2) |
with a function of lCE_rel
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where 
, k, c, m, and q0 are constants (see Table A1).
The original equations of Hatze are slightly simplified for clarity. For a graphical representation of the STIM–q relationship as a function of lCE_rel, see Fig. A2C.
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fig A2. A: force–length relationship for parallel elastic element (PE): lPE_rel = lPE/lCE_0. B: force–length relationship of series elastic element (SE): lSE_rel = lSE/lSE_0. C: q(t) for STIM = 0.3 and lCE_rel = 0.6, 0.8, 1.0, 1.4 (longer lCE_rel = higher q). D: isometric force–length relationship for STIM = 0.05, 0.1, 0.15, 0.3, 1.0 (higher STIM = higher force). E: force–velocity relationship for lCE_rel = 1.0 and q = 0.1, 0.4, 0.7, 1.0 (higher q = higher FCE_rel). Note that for q <0.3, maximal shortening velocity scales with q. F: force–velocity relationship for STIM = 0.2 and lCE_rel = 1.0, 1.4, 0.8 (dashed), 0.6 (in order of highest maximal FCE_rel). Note that for lCE_rel >1, maximal shortening velocity scales with Fisom_n. Maximal shortening velocity of the lower force–velocity relationship in F is diminished because at this STIM and lCE_rel, q is <0.3 (see E).
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Contraction dynamics
Contraction dynamics was modeled by relating the contraction velocity (vCE) to lCE
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The contraction velocity was derived from the difference between the isometric force (Fisom), calculated using the force–length relationship, and the actual force to be generated by the CE (FCE). Assuming that the mass of the muscle was negligible with respect to the force it is producing, FCE equaled the difference between the force of SE (FSE) and that of PE (FPE). The concentric (vCE < 0 or FCE < Fisom) and eccentric (vCE > 0 or FCE > Fisom) parts of the force–velocity relationship were modeled separately. The concentric part was described based on the classic Hill equation, which was solved for vCE_rel (the time derivative of lCE_rel)
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with FCE_rel (=FCE/FMAX) and Fisom_n (=Fisom/FMAX). Based on experimental results (Stern 1974), maximal contraction velocity was made dependent on Fisom_n by setting: arel = arelFisom_n when lCE > lCE_opt and arel = arel when lCE 
lCE_opt. Furthermore, based on experimental results of Pertrofsky and Philips (1981), for low values of q, maximal contraction velocity was made dependent on q by setting
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when q < qcrit and brel = brel when q 
qcrit. The values for the constants arel, brel, and qcrit are given in Table A1.
The eccentric part of the force–velocity relationship was modeled using a hyperbola. To prevent numerical problems, a hyperbola with a slightly slanted asymptote was used (for the sake of conciseness, it was not solved here for vCE_rel)
 | (A7) |
The parameters p1, p2, p3, and p4 were calculated using four criteria: 1) the concentric and eccentric curve are continuous; 2) based on Katz (1939), the derivative of FCE_rel with respect to vCE_rel at vCE_rel = 0 of the eccentric curve was twice that of the concentric curve; 3) the asymptote had a value of 1.5qFisom_n at vCE_rel = 0; and 4) an arbitrary small value for the slope of the asymptote. Note that the calculated parameters were functions of Fisom_n such that both parts of the force–velocity relationship depended on Fisom_n and FCE_rel. See Fig. A2, E and F for a graphical representation of the force–velocity relationship at different values of q and lCE_rel.
Normalized isometric force (Fisom_n) was modeled as a second-order polynomial with an optimum at lCE_rel = 1 and two zero-crossings at lCE_rel = 1 ± width
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ABSTRACT
INTRODUCTION
). Skeletal dynamics describes the equations of motion of the skeleton. For abbreviations see Glossary (APPENDIX C).
