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On the control of biologically and kinematically redundant manipulators C.C.A.M. Gielen *, B.M. van Bolhuis, M. Theeuwen Dept. of Medical Physics and Biophysics, University of Nijmegen, Geert Grooteplein Noord 21, NL 6525 EZ Nijmegen, The Netherlands

Abstract

One of the key problems in motor control concerns the apparent redundancy of muscles and joints. This biological and kinematic redundancy has been an object of study since long. In this paper we will give a review of the various approaches which have been proposed to solve this problem. We will give a comparison of the results of these approaches with special emphasis on recent models, which try to deal with this problem by eliminating the number of degrees of freedom. This reduction will be achieved by imposing constraints, which follow from the biomechanics of the system under study or from some plausible requirements on the behaviour of these systems in various motor tasks. Since this problem is a key issue both in the control of biological and artificial robot manipulators, we will also discuss models from the robotics community, as well as the possible relevance of biological models for robotics.

1. Introduction Most biological limbs have a large flexibility in terms of degrees of freedom due to the relatively large number of joints, and due to the fact that the number of muscles acting across a joint usually exceeds the number of degrees of freedom in that joint. Yet, this same flexibility is one of the most complex problems in understanding motor performance, since a kinematically redundant limb allows a movement to be made by a large * Corresponding

author.

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variety of changes in joint angles and since the same joint rotation can in principle be the result of various muscle activation patterns for muscles acting across that joint. This problem, well known as the “motor equivalence” problem, was clearly formulated by Bernstein (19671, who was also the first person who gave a well-defined suggestion for an approach to a possible solution to this problem. Since then, the control of muscle systems with multiple degrees of freedom has been analysed in terms of muscle force, kinematics, EMG signals and single motor-unit activity, matrix and tensor transformations, and neural network studies (see, e.g., Flanders et al., 1992; Gielen and van Zuylen, 1986; Hasan and Karst, 1989; Lacquaniti, 1992; Hogan, 1985; Mussa-Ivaldi et al., 1988; Nichols, 1989; Pellionisz and Llinas, 1982; Gorinevsky and Connolly, 1994; Winters and Woo, 1990). The problem of coordination of kinematically redundant manipulators and solving the ill-posed computational problems related to motor redundancy have been an important problem in robotics too. The robotic and biological communities have been operating rather independently for a while, but a gradual dissemination of results and approaches can be observed in the last decade. In fact, some ideas from one field, e.g., the idea of generalized inverses (see Klein and Huang, 1983; Pellionisz and Llinas, 1982) have been tested in the other field. In this paper we will discuss some of the interactions between these two communities. Despite the fact that the problem of biological and/or kinematic redundancy is one of the central problems in the two fields of research, there has not been presented an adequate solution yet. For biological limbs it is well accepted that a unique activation pattern is observed for each particular motor act. However, it is not clear which constraints give rise to the unique activation patterns. Moreover, in the literature on behavioural studies there is increasing evidence that in addition to biophysical and anatomical constraints, psychomotor and even cultural factors may contribute to the selection of postures and to the formation of movement trajectories (Hollerbach, 1990a; Hollerbach, 1990b). For robotics, there is no general algorithm yet which prescribes how the number of degrees of freedom can be reduced in a consistent and reproducible way. However, this is not to say that no progress has been made. There is a growing awareness that a solution to the problem has to be found in finding some constraints that give rise to a reduction in the number of degrees of freedom. However, these constraints do not come from control theory, but may have to be found in the biomechanical properties of the manipulator or in task-related constraints, as suggested by Smeets (1994) who demonstrated that optimiz-

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ing the accuracy of force control or position control may give rise to a different activation pattern for mono- and bi-articular muscles. Any constraints following from psychomotor and cultural effects can be incorporated once more detailed insight in these constraints has been gained. The aim of this paper is to give a review of the various models and theories, which have been postulated to describe or to explain the coordination of biologically and kinematically redundant limbs and manipulators. Where appropriate, we will also deal with the self-organizing principles, which may underly the learning process of muscle coordination. We will focus on the advantages and disadvantages of the various approaches in order to conclude with suggestions for further research.

2. Theories on how to deal with kinematic redundancy In this section we will discuss various hypotheses in motor control, which are related to the problem of kinematic or biological redundancy. Most emphasis will be on hypotheses with regard to biological limbs, but where relevant, also ideas from robotics will be reviewed. 2.1. Equilibrium point hypothesis The idea that a limb’s posture is controlled through selection of the agonist and antagonist muscles’ length-tension curves was first proposed by Feldman (1966; see also Feldman, 1986). The notion of an “equilibrium point” controller was born from the observation that in a single-joint system, the length-tension properties of antagonist muscles interact to stabilize the limb at some joint angle. If external perturbations are applied, the limb opposes these perturbations by a restoring force, the amplitude of which increases with displacement. The joint angle corresponding to the equilibrium position can be adjusted by shifting the rest lengths of the muscle pair. In this way any posture can be obtained by selecting the appropriate set of rest lengths. These ideas were supported by neurophysiological findings on the properties of muscles and of reflex actions, which give muscles a spring-like behaviour in which both the rest length as well as the spring constant can be adjusted (see Nichols and Houk, 1976; Houk and Rymer, 1981). It was hypothesized that the setting of the rest length of muscle is the result of a central command, which may include both the (Yand y-activation.

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For multi-joint systems like the arm, these ideas were elaborated in 2 dimensions, in which stiffness for a single joint is replaced by a matrix representing the stiffness in joint space. When the stiffness of single joints in a multi-joint limb is known, the restoring forces to external perturbations at the end effector in world space can be calculated. The stiffness matrix R in joint coordinates 0 is related to the stiffness matrix K in world space coordinates

axi

by the Jacobian matrix J with Jij = ae_ (see Hogan, 1985): I

R = JTEw. The requirement that the orientation of the stiffness ellipse in workspace does not change largely for various positions in space, corresponding to various joint angles (see Mussa-Ivaldi et al., 19851, put a constraint on the contributions of mono- and bi-articular muscles to stiffness. However, this constraint is in itself not sufficient to reduce the number of degrees of freedom due to the multiple muscles in the human arm to a unique activation pattern. Still, the available evidence suggests a unique activation pattern. Moreover, the shape of the stiffness ellipse for a given arm position could not be changed voluntarily; only the size of the ellipse could be changed, presumably by coactivation of synergistic and antagonistic muscles (see Mussa-Ivaldi et al., 1985). This illustrates that the relative activation of muscles is not under voluntary control and that a fixed and consistent set of relative activation is used to control posture. Later, Mussa-Ivaldi et al. (1988) gave a mathematical relation between stiffness in work-space, stiffness in joint space, and muscle stiffness. They pointed out that given the stiffness of muscles, the inverse kinematic problem could be easily solved by a mechanical model of the manipulator provided with springs to simulate the muscles in the following way. By simply moving the tip of the (arbitrarily complicated) manipulator in the direction of the position of the intended target point, the joint angles of this mechanical model automatically find their appropriate values, corresponding to the minimum potential energy stored in the system. However, this model works only when muscle stiffness is given, and as we know the effective stiffness of muscle is not a fixed variable but is flexible, under control of the central nervous system by (Y- and y-activation. Therefore, this model does not really provide a solution to the inverse kinematics problem. It only gives the joint angles when muscle stiffness or (since muscle stiffness is uniquely related to muscle activation) when muscle activation is known, i.e., when the problem has been solved already. Other studies have elaborated on the role of stiffness of a multi-joint

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limb to predict movement trajectories. As explained above, joint stiffnesses determine posture of the limb in a unique way. The idea to generalize the role of stiffness to movements is supported by computer simulations, which demonstrate that the trajectory of the hand during a movement can be obtained by a gradual shift of the equilibrium point (see, e.g., Flash, 1987; Shadmer et al., 1993). These results are consistent with the hypothesis that the Central Nervous System programs a reaching movement by gradually shifting the equilibrium position of the hand from the start position to the target position. This model predicts movement trajectories very well in agreement with observed trajectories. However, since it does not predict stiffness, this model does not provide a solution to the kinematic redundancy problem either. One of the attractive key issues of the equilibrium point model is that of stability. However, as recently pointed out by Dornay et al. (19931, the spring-like behaviour of muscles by itself is not sufficient to guarantee a stable equilibrium point. Limb stability is critically influenced by the geometrical arrangement of muscles and, in particular, by their position-dependent moment arms. Recognizing this fact provided a way to falsify the theory: the equilibrium point hypothesis is a viable control hypothesis if and only if certain specific geometrical constraints are met by the muscles involved in the postural task or movement. Dornay et al. (1993) investigated this in more detail for the monkey arm. Their analysis demonstrated that by keeping the muscle tendons more distant from a joint, their mechanical advantage increases. However, at the same time the stability is substantially decreased. This result suggests that only those muscles should be activated for which the mechanical advantage as a function of joint angle behaves in a particular way, which may depend on the particular motor task. This is consistent with simulations by Smeets (1994) who found that the relative activation of mono- and bi-articular muscles depends on the stability for controlling force and position. The constraints which follow from this analysis certainly deserve to be explored in more detail for various conditions. 2.2. Generalized inverse methods The problem of kinematic redundancy is also an important topic in robotics, where the requirement of flexibility imposes a large number of degrees of freedom. Although any joint configuration corresponds to a unique position of the end effector in work space, the large number of

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degrees of freedom makes that a particular position in work space may correspond to a large number of joint configurations. Since movements are planned in work space, the problem is to find a consistent solution to choose the appropriate set of joint angles. Since the inverse of the mapping from joint space to work space (the well known Jacobian, which was introduced earlier) does not exist for kinematically redundant manipulators, the first attempts to deal with the problem of kinematic redundancy suggested to use a generalized inverse or pseudo-inverse Jf of the Jacobian J. The idea behind a generalized inverse is that there is a unique choice such that there is a unique end position for each set of joint angles and, reversed, such that there is a unique set of joint angles for each position of the end effector. Mathematically, this implies that a generalized inverse satisfies the equations J = JJ+J and J+= J+JJ+. For several reasons (see Klein and Huang, 1983 for a review) the Moore-Penrose inverse was inverse chosen, which is defined by J&,= JT( JJT)- ‘. The Moore-Penrose satisfies the equalities defined above. For each position of the end effector it chooses that set of joint angles which brings the end effector to the desired position with the additional constraint that the sum of squares of changes in joint angles is minimal. However, this approach did not produce results which were in agreement with experimental results on the activation of motor units in human arm muscles during isometric contractions (see Gielen and van Zuylen, 1986; Jongen et al., 1989). Moreover, this particular generalized inverse is not integrable which gives rise to a hysteresis in the sense that the set of joint angles, corresponding to a particular position of the end-effector in work space, depends on the trajectory which brought the manipulator to that position (see Klein and Huang, 1983). This observation is not in agreement with experimental observations on human arm movements, and in addition leaves the joint configuration of robot manipulators difficult to control because of the dependence on trajectory history. The problem that the joint configuration is subject to hysteresis depending on the previous movement trajectory, was addressed recently by Mussa-Ivaldi and Hogan (1991) who proposed a class of integrable generalized inverses which did not have the disadvantage of hysteresis. However, no quantitative comparison between the theoretical results of this study and experimental data has been presented yet. 2.3. Possible constraints for the reduction of degrees of freedom Another, but related, way to solve the problem of redundancy is to look for constraints which reduce the number of degrees of freedom. Several

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approaches (e.g., Penrod et al., 1974; Hatze, 1976; Pedotti et al., 1978; Zajac et al., 1984) have proposed optimization techniques which minimize or maximize some objective function. However, for many movements it is not easy to identify an objective criterion for minimization or maximization and, if one is found, it is still an open question whether the central nervous system uses the optimization of that particular criterion for the coordination of muscles. As an alternative it has been proposed that any constraints should be found in the biomechanical properties of the motor system itself, rather than in theoretical considerations such as minimum energy, minimum fatigue, minimum muscle torque, etc. (see Herzog and Binding, 1994; Nelson, 1983; Dul et al., 1984; Yeo, 1976; Uno, 1989). A recent proposal came from van Ingen Schenau (1989) who took into consideration that translations of the end effector of a limb (e.g. the hand or the foot> are caused by joint rotations. This requires a very tight coupling between rotations in multiple joints in order to generate the desired movement trajectory (cf. Lacquaniti et al., 1987). The transformation from desired movement trajectory to joint rotations appears to be subject to some interesting constraints which were not explicitly recognized until recently. For example, it was found (Bobbert and van Ingen Schenau, 1988) that muscle activation patterns of mono- and bi-articular muscles in man need a very precise timing in order to jump in a vertical direction. Small variations in the relative onset time of activation of muscles of as small as 10 ms lead to a relatively large decrease of height of the jump and to significant deviations from vertical elevation. This is due to the fact that rotational energy related to joint rotations has to be transformed into translational kinetic energy. Realizing the consequences of these constraints it was possible to explain why the extremities of man and animal are not only equipped with muscles which cross over one single joint, but also with muscles which cross over two or more joints (see van Ingen Schenau et al., 1987; van Ingen Schenau, 1989). Related to these findings, there are other conditions in which the use of bi-articular muscles seems preferential above the activation of mono-articular muscles. For some movement directions of the hand (see Fig. 1) the torque in the elbow or shoulder joint may be opposite to the change in joint angle, which is required for the desired movement. In such a case, the contribution of that joint to the work delivered by the hand is negative. If this torque would be generated by a mono-articular muscle, it would imply that this muscle is dissipating energy, rather than contributing positive

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Fig. 1. Schematic view of a subject pushing an object against an external force F,,,. The force exerted by the subject is opposite to F,,,. This requires a flexion torque both in the elbow and in the shoulder. However, the movement requires an extension of the elbow and a flexion of the shoulder. Note, that the torque and change in joint angle have opposite sign in the elbow, not in the shoulder. Therefore, if monoarticular elbow flexors are used in this motor task, they will dissipate energy rather than contribute positive work.

work to the movement. A more efficient activation pattern for this type of movement would be to activate a bi-articular muscle, which contributes to the required joint torques, but which does not change in length as much as a mono-articular muscle due to opposite contributions in length change caused by the changes in the two joint angles. A more detailed analysis can be found in Gielen and van Ingen Schenau (1992). The predictions of this model are qualitatively in agreement with observations by Karst and Hasan (1991) and Sergio and Ostry (1994) who found that the activation of bi-articular muscles, such as the biceps brachii muscle, was greatest when it participated as agonist at the two joints and was less when it served as agonist at one joint and as antagonist at the other. See also Doorenbosch et al. (1994) who found similar results for bi-articular leg muscles in man. In order to investigate whether the activation of mono- and bi-articular muscles was in agreement with the special role attributed to bi-articular muscles by van Ingen Schenau (1989) the activation patterns of arm muscles were studied quantitatively in more detail during isometric contractions and during movements against or assisting an external load. The activation pattern of the mono- and bi-articular muscles appeared to be different. This is illustrated in Fig. 2, which shows the amount of EMG activity for forces exerted in various directions during isometric contractions and during movements against or assisting an external load. For reasons explained before (see Theeuwen et al., 1994a) circles were fitted to the data. Clearly, both the amplitude and orientation of the circles is

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Fig. 2. The amount of EMG activity plotted in polar coordinates for forces in various directions. Filled circles, open circles and crosses refer to the amount of EMG activity for isometric contractions and for movements against or assisting an external load, respectively, for four muscles. Force amplitude at the hand was the same in all conditions; only the direction of force was varied for each of the three conditions. (Reprinted with permission from Multi-Sensory Control of Movement, A. Berthoz (Ed.), Oxford University Press, 1993).

different for the three conditions. These differences appeared to be significant and consistent for all subjects tested (see Theeuwen et al., 1994b). These results indicate that the biological redundancy allows various muscle activation patterns. However, a unique activation pattern is observed for all subjects which may be different for different motor tasks.

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The results on muscle activation patterns in human arm muscles by Theeuwen et al. (1994b) have shown that the suggestion of a particular role for mono- and bi-articular muscles (see van Ingen Schenau, 1989) cannot quantitatively account for all experimental data. Given the activation pattern for isometric contractions, the changes in activation observed for the movements are in the direction as predicted according to the ideas by van Ingen Schenau. However, quantitatively there is a discrepancy. This illustrates that the present ideas on muscle coordination may be in the good direction, but certainly not fully adequate yet to explain the experimental observations. 2.4. Algorithms to deal with kinematically redundant robot manipulators In robotics the functional requirements on an algorithm for the control of kinematically redundant manipulators are often the same as for biological limbs. However, contrary to the biological case, there are no data to which the performance of an algorithm has to be matched. Therefore, any solution, which gives repeatable results will be adequate for robotics. In robotics, it is sufficient to have a solution, which gives repeatable trajectories both in Cartesian work space as well as in joint space. Once a reduction in degrees of freedom has been obtained, the extra degrees of freedom can be used for other behaviour patterns such as obstacle avoidance, singularity avoidance, or other criteria that improve the performance. Basically, most of the work in robotics dealing with the problem of kinematically redundant manipulators is based on generalized inverses. Since the Moore-Penrose generalized inverse gives rise to hysteresis, other types of generalized inverses have been proposed. Shamir and Yomdin (1988) proposed to use a generalized inverse which guarantees repeatability. This is achieved by considering integral surfaces for a distribution in joint space. Basically, this approach is similar to the previously described approach described by Mussa-Ivaldi and Hogan (1991). This type of approach is adequate for the robotics field. Since no quantitative comparison has been made yet with biological data, it is not clear which solution (out of many !) has been adopted by living beings. Some variations on the use of the Moore-Penrose generalized inverse have been proposed. In most cases these studies impose additional constraints in order to resolve the problem of non-repeatability in manipulator control (see, e.g., Maciejewski, 1991; Dubey, 1991). Although these models may be adequate for the control of robot manipulators, their selection of

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Fig. 3. The orientation of an object after two noncollinear rotations depends on the order of rotations. (Reprinted with permission from Muscle activation patterns and joint-angle coordination in multijoint movements, S. Gielen, in: Multi-Sensory Control of Movement, A. Berthoz (Ed.), pp. 293-313.).

just one out of the many possible constraints makes it questionable whether one of these algorithms may be useful to model the control of biologically redundant limbs. 2.5. Reduction of 3-D rotational degrees of freedom in joints The considerations concerning biological limbs mentioned above dealt with the constraints to select the appropriate muscle activation pattern for mono- and bi-articular muscles. However, a similar problem exists for kinematically redundant limbs for which a selection has to be made out of the available degrees of freedom in joints. Without attempting to give a full account of all possible approaches to deal with this problem, an interesting proposal has come by realizing that rotations along 3 orthogonal axes do not commute (Tweed and Vilis, 1987; van Opstal, 1993). This implies that the orientation of a limb depends on the order of two non-collinear rotations for joints with 3 degrees of freedom such as the shoulder, head, and eye. The fact that rotations along axes in 3-D space do not commute is illustrated by Fig. 3, which shows that the orientation of a dice depends on the order of two non-collinear rotations. Therefore, if one imposes the constraint that the orientation of the eye, head or shoulder, when pointing or looking in a particular direction, is always the same, irrespective of any previous rotations, not all rotations should be allowed. It has been shown that a unique orientation for the eye, head and shoulder for each gaze or pointing direction, can be achieved by eliminating one degree of freedom (see Tweed and Vilis, 1987; Helmholtz, 1866) thus allowing only 2 rotational degrees of freedom.

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For further explanation of the reduction of degrees of freedom, let us consider the position of the upper arm due to rotations in the shoulder. The reduction of degrees of freedom is most easily explained, if one considers the position of the upper arm as the result of a single-axis rotation in the shoulder which brings the upper arm from a particular initial reference position to that particular position. This implies that every position of the upper arm can be represented by a rotation vector. It has been shown (Miller et al., 1992; Hore et al., 1992) that there exists a unique reference position, called primary position, such that the rotation vectors which describe the upper arm positions lie in a plane. Similar results were found earlier for the eye (see Helmholtz, 1866; in von Helmboltz, 1962) and head (Theeuwen et al., 1993). This reduction of degrees of freedom is illustrated in Fig. 4, which shows the rotation vectors describing the position of the upperarm, forearm (hand), and head for a subject while pointing to targets at a distance of about 2 meter from the shoulder. The three panels give three orthogonal views on the rotation vectors describing the positions of the arm, hand and head. Clearly 3-D rotation vectors describing the orientation of the upperarm, forearm, and head are lying in a thin slice of the 3-dimensional space containing the rotation vectors. The thickness of the slice can be obtained by fitting a 2-dimensional surface through the rotation vectors and calculating the distance of the rotation vectors to that plane. The standard deviation of the scatter of the data points relative to the plane is about 4 degrees (see Theeuwen et al., 1993; Hore et al., 19921, which is small considering that the shoulder has the capability of rotations over 180 degrees along each rotation axis. Fig. 4 illustrated that only particular rotation vectors are used. The data in Fig. 4 illustrate that the rotation vectors which describe the position and orientation of these limbs are restricted to a plane. This plane is orthogonal to the primary position. Given the fact that the rotation vectors, which describe the orientation of the upper arm, are contained in a 2-D plane, it is not trivial that the rotation vectors, which describe the orientation of the forearm, are contained in a 2-D plane as well, since the orientation of the forearm relative to the upper arm can be changed by supination/pronation. These results have many implications for angular velocity vectors, which bring the limb from a start position to a target position, and for the neuronal circuitry, which may be responsible for the precise control of these joint rotations. For a more detailed and more complete description

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Fig. 4. Orientations of the arm (upper three panels), hand (middle three panels), and head (lower three panels) relative to fixed reference orientations for a task in which subjects were instructed to point to and to fixate imagined targets located throughout space. Data in the left column show front views of the planes, which were fitted to the rotation vectors. The middle and left columns present top and side views. The units along the horizontal and vertical axes are those of rotation vectors, where - 0.75 and 0.75 correspond to approximately -75 and +75 deg (see Theeuwen et al., 1993). Reprinted with permission from Theeuwen et al., 1993.

we refer to (Tweed and Vilis, 1990; Straumann et al., 1991; Miller et al., 1992; Hore et al., 1992). For the present review it suffices to say that these results illustrate that a plausible constraint such as requiring that the orientation of a limb or eye for a particular pointing or looking direction does not depend on previous rotations, gives rise to a reduction of degrees of freedom. Again, this reduction is not sufficient to remove the kinematic redundancy of the arm completely and we stress to say that no satisfactory

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solution is available yet to deal with biological and kinematic redundancy of biological limbs. 2.6. Neural network approaches A completely different approach to solve the problem of inverse kinematics and to obtain a unique muscle activation pattern comes from neural network models. Neural networks are biologically inspired systems with many simplified neuron-like elements, called “neurons”. Most or all neurons are interconnected and the weight of the connections from a neuron to another is specific for each pair of neurons. This weight results from a learning rule which prescribes how the connections between neurons have to be modified in order to obtain the desired behaviour of the neural network (see Hertz et al., 1991). Neurons can be in two states, either they are at rest or they are active. The input to each neuron is given by the state of all other neurons, weighted by the strength of the connection between each pair of neurons. When the total input signal exceeds a particular threshold, the neuron becomes active. For a more detailed review of neural networks, see Hertz et al. (1991) and Gutfreund and Toulouse (1994). Neural networks can be trained to perform any desired mapping from a particular input space to an output space, since most familiar types of neural networks are universal approximators (Hornik et al., 1989). The desired performance is obtained by presenting examples of corresponding input/output pairs. During this training period the strength of the connections in the network is updated according to a particular learning rule such that the output of the network for a particular input will gradually approximate the desired output during the training session. After a while this process converges to the desired network state. At this state the network has learned the desired mapping with the remarkable property of generalization, which implies that the network also performs well on input signals which are in the ball park of the input signals in the training set but which it has never met before. Recently, several papers on applications of Artificial Neural Networks (ANN’s) to the approximation of the inverse kinematics problem have been published. Two types of networks commonly used to solve this problem are the multi-layered feedforward network, trained with back-propagation, and Kohonen’s self-organizing maps. A recent review was presented by Gorinevsky and Connolly (1994) who found that networks with hidden units with radial-basis properties, similar to the receptive fields of neurons,

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demonstrate the best performance. However, the solution for the inverse kinematics problem found by these neural networks depends very much on the stimulus ensemble used to train the network. Moreover, the solutions of the network are not always compared with real data (such as in Cruse and Steinkuhler, 19931, or may not be in agreement with experimental data. The latter was the case in the work by Cruse et al. (19931, who proposed some additional constraints to match the experimental observations. This demonstrates that the present state of the art of neural network research cannot give a good explanation for the experimental observations yet. A neural-network type of approach similar to Kohonen’s topographical maps was proposed by Rosenbaum and colleagues to explain the unique selection of postures. According to their approach the brain has stored a (more or less random) set of standard postures. These postures are thought to be stored topographically in a neural map. This implies that within the neural map neighbouring neurons represent very similar postures. In order to achieve a particular position for the end effector the actual posture is a linear combination of the standard postures, weighted by a coefficient related to the comfort of each posture and inversely proportional to the difference between the desired position of the end effector and the position corresponding to the standard posture (for more details, see Rosenbaum et al., 1995). Smeets and Denier van der Gon (1994) presented a biologically inspired neural network for the development of homonymous and heteronymous projections from muscle receptors to spinal motoneurons in order to obtain a proper reflex coordination of muscles to compensate for external perturbations. This was done for a two-joint limb with 4 mono- and 2 bi-articular muscles. Although this model came up with a fairly realistic activation pattern for reflexes, the model obviously is too simple, as noted by the authors, and some of the simplifications may need some more study, since they may have a large effect on the behaviour of the network. 2.7. Estimating the contribution of muscles to joint torque Since it has not been possible yet to come up with a good theory for dealing with biologically or kinematically redundant manipulators, several authors have followed another approach in which they tried to estimate the contribution of a muscle to joint torque quantitatively. This information is important for a better understanding of human movement coordination,

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and the idea was that detailed quantitative information about the contribution of muscles or subpopulations of motor units might give a hint as to the underlying principles for the coordination of biologically redundant limbs. Jorgenson and Bankov (1971) have estimated the contribution of muscles to joint torque based on the assumption that the activation of muscle is proportional to it’s physiological cross-sectional area. However, it is well known now that human arm muscles have several subpopulations of motor units each with a different activation (see ter Haar Romeny et al., 1982; ter Haar Romeny et al., 1984; van Zuylen et al., 1988; Segal et al., 1991; Segal, 1992). Therefore, only part of the muscle is activated for a particular motor task and, consequently, the assumption made by Jorgenson and Bankov (1971) may not be correct. Recently, it has been shown that it is very hard to estimate the reliability of results from approaches based on assumptions on muscle physiology and on biomechanical parameters since a small error in one of the estimates can have a large effect on the results (see, e.g., Challis and Kerwin, 1994). In a later study Cnockaert et al. (1975) tried to estimate the relative contribution of muscles to joint torque by measuring the EMG activity of muscles. When the relative activation of muscles can be varied, for example by studying the EMG activity in various motor tasks in which each muscle plays a different role, the relative activation of each muscle can be estimated by assuming a constant factor which relates the amount of EMG activity to muscle force. However, the results by Cnockaert et al. have rather large error bounds (about 40%). This is due to the fact that EMG activity recorded with surface electrodes is noisy and may contain cross-talk from other muscles. Although this method seems to work well for relatively simple joints when only a few (antagonistic) muscles are involved, such as the human wrist (see An et al., 1983; Buchanan et al., 1993), it failed for the highly redundant muscle system of the human elbow (see Buchanan, 1986). A possible explanation may be that the procedures used to estimate the contribution of muscles are not robust for noise, such as the inevitable cross-talk from other muscles or the stochastic behaviour of surface EMG, which makes these methods fail for more complex situations. Recently, Theeuwen et al. (1995) have tried to extend the method of Cnockaert et al. to data on motor-unit recruitment behaviour in human arm muscles in isometric contractions. Since the recruitment behaviour of motor units can be determined fairly accurately and since all motor units in a given subpopulation are thought to receive the same activation according to the so-called “size-principle” (Henneman, 19811, the procedure could be

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applied without the problems inherent to surface EMG recordings. The results were consistent for several subjects and the error in the estimates appeared to be about 10%. These results indicate the important role of the (sometimes complex) biomechanical advantage of muscle. For example, m. biceps contributes both to torque in flexion and supination direction. When m. biceps is activated to contribute to a pure supination torque, some other muscle must compensate for the contribution to supination torque by m. biceps. This is done by m.triceps, which mechanically contributes to extension torques only. This explains why the direction of maximal activation of muscle does not always correspond to that for the optimal mechanical advantage (see Jamison and Caldwell, 1993). Moreover, it illustrates that factors other than the physiological cross-sectional area and mechanical advantage determine the activation of muscles in various motor tasks. It has been shown now by various authors (see, e.g., Tax et al., 1989; Tax et al., 1990a; Tax et al., 1990b; Theeuwen et al., 1994b; Kalaska et al., 1989) that the relative activation of muscles may be different in different motor tasks. In the present paper we have interpreted this as evidence for different constraints which act upon the motor system. However, this is not necessarily the only explanation. It is well known that different parts of the central nervous system are involved differently in various types of movements (see, e.g., Kuypers, 1981). Recently, it was shown (see Martin and Ghez, 1993; Martin et al., 1993) that motor cortex and red nucleus are differently involved in single joint tasks in the elbow and in reaching and placing movements. Inactivation of area 47 of motor cortex in cat did not increase reaction time of single joint elbow tasks and produced only modest and inconsistent reductions in response amplitude. In contrast, inactivation in the forelimb representation of red nucleus consistently increased reaction time. During reaching, the same inactivations in motor cortex and red nucleus produced a substantial loss of accuracy. It could be that the different relative activation merely reflects a different anatomical projection from these different parts in the central nervous system to the spinal cord, without a specific functional relevance. Further research will be necessary to clarify this point.

3. Discussion The main message of this paper is that the problem of biological and kinematic redundancy is not solved yet. Some algorithms have been pro-

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posed, which may be suitable for some particular applications, but no general solution is available now. The models based on the equilibrium point hypothesis do not seem to give a solution, since they only work, when the stiffness in all joints is known, i.e., when the muscle activation pattern is known. Using principles and constraints from control theory does not seem to work either. Imposing constraints based on the biomechanical properties of the system under study or based on functional requirements seems to give results in the correct direction. However, a good quantitative comparison between the results of these models and experimental observations is still lacking. This is a necessary step which has to be taken in order to make progress in this field of research. In order to be able to derive constraints from the biomechanical, anatomical and physiological constraints, it is important to have a good understanding of the functional implications of the properties of muscles, receptors, etc. This necessitates modeling approaches of skeletal muscles, such as those described by Winters (19951, Huijing (1995) and Vaal et al. (1995). It is our opinion, that significant progress can be made only in a context where modelling and experiments go hand in hand, guided by detailed and specific hypothesis. It has been shown by many authors (see, e.g., van Zuylen et al., 1988; Tax et al., 1989; Tax et al., 1990a; Tax et al., 1990b) that for each particular motor task the same muscle activation pattern is found for all subjects. Later, Theeuwen et al. showed that a different muscle activation pattern is found for different motor tasks, such as isometric contractions and movement tasks (see Theeuwen et al., 1994b). This indicates that multiple activation patterns are possible indeed. Yet, the observed activation pattern is consistent for each particular motor task for all subjects. Since muscle activation patterns can be modified after surgery, this indicates that the muscle activation pattern is subject to adaptation and is not a hardwired pattern which is once and for all fixed at birth. This clearly suggests some adaptive learning process. Some neural network approaches have attempted to model this learning process. Although these approaches may seem to be speculative, at least they demonstrate that with plausible learning rules (see Smeets and Denier van der Gon, 1994; Gomi and Kawato, 1993) plausible results can be obtained. With regard to neural network approaches, the main problem with models based on neural networks seems to be that neural networks can model almost anything. Therefore, the fact that a neural network can

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model inverse kinematics for a kinematically redundant manipulator, does not learn us much about the biological implementation of the solution used by human beings. Presumably, more specific knowledge about the basic properties of the biological system under study has to be incorporated in the model in order to obtain physiologically relevant results. The model by Smeets and Denier van der Gon (1994) is certainly a first step in that direction.

Acknowledgements

This research was supported by the European Community in the ESPRIT BASIC RESEARCH Program MUCOM, project number 6615. We also gratefully acknowledge the support by the Dutch Foundation for Biophysics (NWO).

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