Computer Simulation of the Ontogeny of Bipedal Walking. Kazunori Hase1 and Nobutoshi Yamazaki2

Transcription

1 Anthropological Science 106 (4), , 1998 Computer Simulation of the Ontogeny of Bipedal Walking Kazunori Hase1 and Nobutoshi Yamazaki2 1 National Institute of Bioscience and Human-Technology, Tsukuba Faculty of Science and Technology, Keio University, Kouhoku-ku, Yokohama (Accepted November 16, 1998) ôgh ô Abstract ôgs ô Computer simulations of human infant walking, changing from supported walking with weight support to independent bipedal walking, were conducted to identify the biomechanical factors of the neuro-muscular system necessary to acquire bipedal walking. We simulated the autonomous acquisition of independent walking by repeating walking trials and modifying neuronal structures and parameters so that walking efficiency is maximized, and forces supporting the upper body are minimized. The results of simulation showed that reinforced interaction with the system of body dynamics, such as feedback from somatic senses, especially information concerning angular velocities of joints, is essential to the development of walking. Other significant factors for bipedal walking are the development of neuronal feedback about the hip joint position, which determines the posture of the upper body, and improvements in the extensors' ability to produce joint moments. ôgh ô Keywords: ôgs ô supported infant walking, independent infant walking, development, computer simulation, biomechanics Introduction The ontogenetic development of walking in human infants has been extensively studied from various viewpoints from neurophysiology to biomechanics. Findings from these studies provide useful insights into the mechanisms of acquiring general locomotor skills and the evolutionary process for erect bipedal walking in humans. The approach of these studies on infant walking is analytical, based on experimentation and measurements. Most studies, however, are qualitative, and only a few studies have been conducted quantitatively using restricted measurement data, such as surface electromyograms (Okamoto and Goto,1985) and ground reaction forces (Endo and Kimura, 1972; Beck and others, 1981). This is because measurements and experimental procedures which include invasive methods cannot be applied to human infants. Recently studies using computer simulation have overcome the experimental restrictions of movement analysis and allow a new approach to the principles of motor control. Computer simulation has the advantage to provide quantitative data concerning the internal

2 328 Hase K. and Yamazaki N. status of organisms, including the activities of neuronal systems and muscular tension. Taga and others (Taga and others, 1991; Taga, 1995a) have presented mathematical models of walking based on a theory asserting that the interaction between rhythms generated by the neuronal and the musculo-skeletal systems produces autonomous walking movement. They have also simulated newborn stepping and quiet standing in infancy by applying this model (Taga, 1997). The present authors have also simulated walking in various human body models from infants to adults and considered the relationship between the development of body proportions and walking patterns (Yamazaki and others, 1996). However, to our knowledge, no study has been made that simulates the developmental process of autonomously acquired bipedal walking and analyzes the factors in this process. The purpose of this study is to elucidate the conditions for the acquisition of bipedal walking by simulating this process using a mathematical model in a computer. The developmental process examined in this study is from supported walking to independent bipedal walking. Here, supported walking means an incomplete bipedal walking movement with the infant's weight supported partly by additional external forces, such as parental help or a "baby walker" in order to maintain the balance and posture during walking. Independent bipedal walking refers to the complete free bipedal striding performed without supporting forces except those acting between the feet and the ground. In developing the ability to walk, infants perform "supported walking", until their neuro-muscular system is fully developed. Through repeated supported walking movements they finally acquire the ability to walk independently. The transitory stage appears to be important for acquiring the nerve and body dynamics needed for generating mature bipedal walking. This study focuses on the changes in the mechanical properties of the neuronal and the musculo-skeletal systems to identify the biomechanical factors necessary for acquiring independent walking. Model of Bipedal Walking Fundamental Theory Neurophysiological studies on animal locomotion have revealed that the basic rhythm of locomotion is controlled by a rhythm-generation mechanism called the central pattern generator (CPG) (Grillner, 1975). Taga and others (Taga and others, 1991; Taga, 1995a) developed this theory to apply it to human bipedal walking and presented a theory asserting that global entrainment between the neuro-musculo-skeletal system and the environment produces human walking. The neuronal system autonomously produces rhythmic patterns of neural stimuli and the system of body dynamics generates movements according to the rhythm pattern. Information concerning somatic senses, such as information about foot-ground contact and segment angle, is fed back to the neuronal system, and the rhythm pattern of neural stimuli is regenerated based on this information. This theory holds that this interaction between the neuronal system and the system of body dynamics produces movement. Computer simulations have shown that the walking movement generated by this model can be flexibly applied to changes in the environment, such as slopes, and mechanical perturbations that occur during walking (Taga and other, 1991; Taga, 1995b).

3 Simulation of the Ontogeny of Bipedal Walking 329 We constructed a walking model based on this theory. Model of Body Dynamics The system of body dynamics in bipedal walking is modeled by seven rigid segments representing the feet, shanks, thighs, and "upper body" (Figure 1). The "upper body" is the trunk, neck, and both arms and is regarded as one segment to simplify the model. The movements of the model are restricted to the sagittal plane. This is the most basic model representing the characteristics of human bipedal walking (Onyshko and Winter, 1980; Mochon and McMahon,1980; Witte and other, 1991). Each foot is represented by a triangle of appropriate proportions. The posterior point of the triangle represents the heel where the foot-ground contact is made. The anterior point of the triangle represents the head of the first metatarsus where the foot-ground contact ends. The interaction between the foot and ground is modeled as a combination of springs and dampers acting at the two foot-ground contact points (Taga, 1995a). Passive joint structures such as ligaments are modeled as nonlinear, visco-elastic elements (Davy and Audu, 1987). To simplify the model, details of the musculo-skeletal system, such as bi-articular muscles, are omitted, and the output of the neuronal system is assumed to directly induce torque at the joints. Figure 1 Model of body dynamics for bipedal walking.

4 330 Hase K. and Yamazaki N. Table 1 Body parameters used in simulation In this computer simulation, the mechanical conditions of supported walking must be explained through modeling. Therefore, supporting springs and dampers are included as shown in Figure 1. If the heights of the hip and shoulder joints go below certain heights, vertical visco-elastic forces act on the individual joints. Visco-elastic supporting force Fsupport is expressed by the following equation: Fsupport={ ks(z-z0)+csmax(-z, 0) for z<z0 0 otherwise, (1) where ks and cs are the coefficients, z is the height of each joint, and z0 is the prescribed height of which force starts to act. The body parameters used, such as segment mass, were those of 1-year-old infants (Table 1). These parameters were based on literature data (Jensen, 1978, 1989; Sun and Jensen, 1994), and are assumed to be constant while the neuronal system changes. Model of the Neuronal System Neuronal systems change through the ontogenetic development from infant crawling to supported and independent bipedal walking. The following assumptions were made based on findings from studies of the development of infant walking (Thelen and others, 1987a, 1987b; Clark and others, 1988). The model of the neuronal system was constructed according to these assumptions. (a) Since actions such as kicking and step-like activities can be observed even in the early stages of infancy, a rhythm-generation mechanism for walking is considered to be inherited. Therefore, a neural oscillator representing the CPG mechanism is assumed to exist for each joint. The neural oscillators consist of pairs of flexor and extensor neurons and each neuron involves flexion and extension motions of the joints (Brown, 1914). The neurons are linked reciprocally via inhibitory connections so that each neural oscillator can generate rhythm patterns. The connection between flexor and extensor neurons in each joint is assumed to be fixed through the developmental process. The equations for the neural oscillators are given later. (b) Since simple reflexes can be observed even in the early stages of infancy, basic receptors for somatic senses are also inherited. Receptors for somatic senses are thus also assumed to exist at the beginning of the developmental process, and we assume that the

5 Simulation of the Ontogeny of Bipedal Walking 331 somatic senses receive information about the following variables: joint angle; joint angular velocity; and foot-ground contact. These signals are fed back to the neural oscillators. (c) Although changes in neuronal structure are partly determined genetically, most of them are acquired and learned by receiving stimuli from the environment. An infant therefore can be regarded as having the basic components to form a neuronal network at birth, but the connections are not yet developed. The neuronal connections are established by dynamic interactions between the infant and the environment, before it starts walking bipedally. Inhibitory connections with neural oscillators in other joints and feedback connections from sensory receptors to neural oscillators are thus considered to change according to the development of the walking movements, that is, the developmental process can be modeled by a mathematical search process. (d) To accomplish the alternative movements of the left and right legs, the structure of the neuronal system must be bilaterally symmetric and the neural oscillators for the hip have inhibitory connections to each other. (e) If the output of a receptor is connected or fed back to a specific neural oscillator, the same output signal is input to both the flexor and extensor neurons. This innervation, Figure 2 An example of the neuronal system.

6 Ė' 332 Hase K. and Yamazaki N. however, is reciprocal. If one of the connections is inhibitory, then the other connection is excitatory. Figure 2 is an example of a neuronal structure based on the above assumptions. There is a neural oscillator for each joint, and the model has six pairs of neural oscillators in total. There are seven joint angle receptors, seven angular velocity receptors for the joints, and two foot contact receptors. In this figure, broken lines represent fixed connections and their connection weights are constant. Connections shown in this figure are only feedback signals toward the neural oscillators in the right leg, and their symmetrical signals are input for the neural oscillators in the left leg. Moreover, only connections from receptors to the flexor neurons of the hip and ankle joints, and the extensor neuron for the knee joint are drawn in this figure, but the same signals are actually input to the extensor neurons of the hip and ankle joints, and the flexor neuron for the knee joint as well, while their connection states (inhibitory or excitatory) are reversed. Numerical Expression of the Activities of the Neuronal System The structure of the neuronal system is described mathematically as follows: The dynamics of a neuron can be expressed by two differential equations (Matsuoka, 1985): Ėi ui=-ui-Ĉđi+u0+ kĉikfk, (2a) iđi=-đi-yi, (2b) yi=max(ui, 0), (2c) where ui is the inner state of the ith neuron; đi is the state variable representing the fatigue of the i th neuron; yi is the output of the ith neuron; u0 is the constant stimulus; Ė i and Ė'i are time constants; and Ĉ is the fatigue constant. fk represents the sensory signals fed back from receptors as well as input signals from other neural oscillators. We call this the "feedback unit", and will refer below to this as FBU. ĉik represents the connection between the FBU and a neuron. If the kth FBU is connected to the ith neuron, then ĉik becomes 1, if not, it becomes 0. FBUs are mathematically expressed as follows: fk=wksk, sk={ h(y1) c (3a) h(y12) k(ľ1-ľ1) c =k(ľ7-ľ7), (3b) k(ľ1) c k(ľ7) h(rr) h(rl)

7 Simulation of the Ontogeny of Bipedal Walking 333 h(x)={ 1 for x>0 0 otherwise (3c) k(x)={ 1 for x>0-1 otherwise, (3d) where wk is the connection weight of the kth FBU; sk is chosen from the signals as shown in Eq. (3b); y1, c, y12 are the output of neurons; ľ1, c, ľ7 are joint angles (as for the upper body segment, relative angle from the vertical axis); ľ1, c, ľ7 are joint angles in standing posture; ľ1, c, ľ7 are angular velocities at joints; Rr is the ground reaction force of the right leg; and Rl is that of the left leg. These signals sum up to a total of 27, that can be selected as input to sk. h( E) and k( E) gives each signal a discrete value of -1, 0, or 1. There is no physiological meaning in using these functions and they were only used to normalize the strength of feedback signals, that is, these procedures allow us to make quantitative comparisons with the connection weight wk of each FBU and to quantitatively examine differences in the importance of a signal. As shown by Eq. (2) and (3), the neuronal structure can be described mathematically, and the problem of determining the neuronal structure becomes a problem of determining the combinations of FBUs. As described above, the output signal of each neural oscillator multiplied by the conversion coefficient corresponds to the rotational moment that acts on each joint, yielding the following equations: nj=nj,f-nj,e, (4a) nj,f=piyi (i=1, 3, 5, c), (4b) nj,e =piyi (i=2,4,6, c), (4c) where nj is the joint moment of the jth joint; nj,f is the flexion moment of the jth joint; nj,e is the extension moment of the jth joint; and pi is the conversion coefficient. Overview Algorithms for the Development of Walking The following algorithm is introduced to model the developmental process of walking: (1) The model generates walking patterns. Calculation begins with supported walking. (2) Indices to evaluate the quality of walking patterns are calculated based on obtained walking movement. An index, for example, represents the achievement level of independent bipedal walking. The remainder of this paper refers to the indices as evaluative criteria. Specific evaluative criteria are described later. (3) Neuronal structures and parameters are modified so that the results on the evaluative criteria of movement are improved. Modifying the neuronal structures means changing the combinations of FBUs, that is, determining the value of ĉik, and the neuronal parameters modified in the procedure are u0, Ėi, wk, and pi. (4) The above procedures are repeated until independent bipedal walking is accomplished.

8 334 Hase K. and Yamazaki N. In this study, the developmental process of walking is modeled as a search process for the neuronal structures and neuronal parameters. This search process progresses by evaluating some criteria of walking. The effectiveness of an interaction with the environment can be quantitatively evaluated by using criteria based on walking patterns obtained in the simulation, rather than using criteria based on the neuronal structure and parameters themselves. Such a search process can be mathematically considered as an optimization problem if the evaluative criteria for walking are treated as an objective function, and the neuronal structures and neuronal parameters are treated as unknown variables, respectively. However, this paper calls the problem a "search process" for simplification. Neuronal factors which should be determined by the search process include not only the neuronal parameters but also the neuronal structure itself, such as the selection of FBU signals and neural oscillators connected by the FBUs. In this study, a computational technique called genetic algorithms (Goldberg, 1989) is used as a search procedure. However, the basic algorithms were partially adjusted for our purposes. Evaluative Criteria of Walking As explained above, to evaluate the effects of interactions with the environment, we defined the following evaluative criteria of walking, which are based on obtained walking patterns: (a) Dependence on a supporting force: The first evaluative criterion required for a search process is the level of motivation that an infant has to acquire independent walking from the supported-walking stage. This evaluative criterion was expressed by minimizing the supporting force acting on the upper body as follows: C1=1/MgT çfsupportdt min, (5) where M is body weight; g is gravitational acceleration; T is the time required for walking. (b) Step length: This study gives a target for the step length. As an evaluative criterion of walking, minimizing the difference between an obtained step length and a given target step length is considered as follows: C2=(L-L0)2 min, (6) where L is the step length of walking in the simulation and L0 is the target step length. Once such specific targets are given, the model may lose the characteristics of a simulator for the developmental process of autonomous acquirement of walking. However, in the actual development of infant walking, clear demonstrations of walking patterns by the parents and the tendency of infants to mimic the walking patterns of adults are considered to be significant factors for the acquisition of the normal walking pattern. This evaluative criterion is valid if the criterion is regarded as a value obtained by modeling these factors. The target step length L0 was set at 22cm based on empirical data obtained from walking infants (Statham and Murray, 1971).

9 Simulation of the Ontogeny of Bipedal Walking 335 (c) Specific power: Since adult walking patterns are believed to be determined by minimal energy expenditure (Yamazaki and Hase, 1992), improvement of energetic efficiency may also be a critical factor in infant walking patterns. To evaluate the efficiency of level locomotion, an index called specific power is generally used (Gabrielle and Karman, 1950). Specific power is calculated by dividing consumed energy by body weight and distance traveled for normalization. Energy consumed while walking can be given by the mechanical work calculated from joint torques nj and angular velocity ƒæj and by metabolic energy, that is: C3=1/MgD ç( j njƒæj +B)dt min, (7) where D is the distance traveled and B is the metabolic power consumed. (d) Muscle fatigue: Both energy consumption and the effect of muscle fatigue, namely reduction of the ability to produce muscular force, are important indices for evaluating movement. In this study, the level of muscle fatigue was calculated by the following equation and used as an evaluative criterion: C4=1/MgD ç{ j nj/nmaxj 3}1/3dt min, (8) where nmaxj is the maximum joint moment of the jth joint. Crowninshield and Brand (1981) have shown that the maximum endurance time of muscular contraction (that is, the time span during which a muscle can maintain a certain muscular tension) is proportional to the -3rd power of muscle stress. The above equation represents this characteristic expressed by the joint moment. Muscle fatigue was also normalized with body weight and distance traveled, similar to specific power. (e) Combination of evaluative criteria: These four criteria are linearly added, as shown in Eq. (9), to form an overall evaluative criterion for infant walking: C=a1C1+a2C2+a3C3+a4C4 min, (9) where a1, c, a4 are weight coefficients showing the importance of the corresponding criterion. Weight coefficients from a2 to a4 are determined by trial and error, then fixed (that is, a2=5, a3=1, a4=300). However, weight coefficient a1 for dependence on supporting force was designed to begin with 0 at the onset of the search process and increase as the search progresses. The weight coefficient al was determined by the following equation: a1={ 0.001s for s<10, otherwise (10) where s is the number of repeated search calculations. This equation allows the model to be supported at the beginning of the search and eventually forces independent walking according to the progress of the search. Such an operation may be valid as a model of

10 336 Hase K. and Yamazaki N. motivation to acquire independent walking. Conditions for Numerical Simulation This computational algorithm was coded in C language. Differential equations were numerically integrated by the Euler method for the generation of the walking movement. The integration interval for solving the differential equations was 0.5ms. We synthesized a maximum of 10 walking steps in each walking trial. The number of FBUs in the initial stage was set at 30; this quantity included the eight FBUs representing the fixed connections expressed by the broken lines in Figure 2, with the remainder being determined randomly. Parameters for the neural oscillators such as the time constants Ėi, Ė'i, were determined so that they could oscillate at a rate of about 1 s per cycle. Under these conditions, the calculations made in order to find the walking patterns were repeated 15,000 times. We call these repetitions a "search step" and use it as our unit of progress in the search for the optimal combination of factors. Results Changes in Walking Patterns Figure 3 shows the simulated walking pattern. Figures are traced at intervals of 0.1 s. Horizontal broken lines indicate heights where supporting forces were applied. The thick, shaded bands show the periods when supporting forces were actually applied to the upper body. At the beginning of the search process, walking could not be generated without additional supporting forces. Walking was possible only if upward directed supporting forces were applied at the shoulder and the hip joint, and its pattern was asymmetric. After 5,000 steps, a few steps could be made without the supporting force, but later on the maintenance of equilibrium depended on an additional support. The walking pattern became more periodic, symmetrical, and stable than earlier. It became possible to keep the upper body upright. After 10,000 steps, completely independent walking can be performed. The step length of walking, however, was short, the cycle was also short, and the walking pattern was tottering. After 15,000 steps, independent bipedal walking has become more efficient: the step length became longer, the cycles also became longer, and the variation between steps was reduced. The resulting walking patterns showed many characteristics of the walking patterns observed in infants: for example, the extension of the knee joint, which was insufficient during the first half of the stance phase, and foot-ground contact made by the toes, not only by the heels (Statham and Murray, 1971; Susa and Endo, 1981). Changes in Evaluative Criteria Figure 4 shows changes in the evaluative criteria for walking. The horizontal axis indicates the number of search steps. Each line on the figure indicates an evaluative criterion for walking as defined by Eqs. (5)-(8). Each value was normalized with the initial value for the repetitive calculations for searching. When dependence on a supporting force became

11 Simulation of the Ontogeny of Bipedal Walking 337 Figure 3 Simulation results. Broken horizontal lines: height at which supporting forces start to act. Thick shaded bands: periods when supporting forces actually act on the upper body. 0 after 9,000 steps, independent walking was achieved. Interestingly, specific power and muscle fatigue that were small during the phase of supported walking reached maximum levels at around 9,000 steps, that is, after independent walking had begun, then decreased again. Both specific power and muscle fatigue related to the biomechanical load of locomotion are small during supported walking, and reach maximum levels at the transition

12 338 Hase K. and Yamazaki N. Figure 4 Changes in the results of the evaluative criteria for walking. to independent walking. Specific power and muscle fatigue decreased with growing adaptation to this pattern of walking. A similar tendency was observed in the evaluative criteria of step length. Such a tendency of the evaluative criteria to change has not been observed quantitatively in the actual development of human walking, but corresponds well to the results of subjective observations (Alexander and others, 1993). Changes in Neuronal Structure Figure 5 shows the transitions in the neuronal structures during the search process illustrated in Figure 3. The descriptions of this figure are the same as those of Figure 2. As this figure illustrates, the neuronal structure becomes simplified as searching proceeds. Since the FBUs were arranged randomly in the initial stage, the neuronal structure was not functional. However, with the continuation of the search process, connections changed so that it integrated the input from movement receptors. Information concerning the angular velocity of the upper body, for example, began to be fed back strongly to the neural oscillator of the hip joint that controlled the upper body. The number of FBUs, in which the distance between a sensory receptor and neural oscillator is wide, decreased. Examples of such FBUs are the FBU from the sensory receptor of the hip joint to the neural oscillator of the ankle joint and the opposite FBU from the sensory receptor of the ankle joint to the neural oscillator of the hip joint. Finally the network of neurons became simple. These changes in the neuronal structure and its properties were quantitatively analyzed with respect to the value of connection weight wk of the FBUs. Differences in the

13 Simulation of the Ontogeny of Bipedal Walking 339 Figure 5 Changes in the neuronal structure at special search steps.

14 340 Hase K. and Yamazaki N. information fed back to the neuronal system were initially investigated. FBUs are roughly grouped into four types: FBUs from the neural oscillators, FBUs from the angle receptors of joints, FBUs from the angular velocity receptors of joints, and FBUs from the foot contact receptors. The latter three types of FBUs could together be regarded as FBUs from the somatic sense receptors. For each of the four types of FBU groups, the sum of the connection weight wk of the FBUs was calculated for each type of information. In other words, changes in the significance of each FBU group for the development of the walking patterns were investigated (Figure 6). As seen in this figure, the sum of the connection weight associated with the FBU from the angular velocity receptor of joints significantly increased as searching proceeded. However, the weight of the connections associated with the FBUs from the neural oscillators decreased. Secondly, changes in the status of neuronal connections were investigated for each movement in one of the joints. After the sum of the connection weights of the FBUs from the somatic sense receptors (information on angles, angular velocity, and foot-ground contact) were independently calculated for the hip, knee, and ankle joints, the changes were plotted against the number of search steps (Figure 7). As a result, it was found that the changes in the FBU connections of the somatic sense of the hip joint were more marked than in the knee and ankle joints and the degree of the change increased with progress in the searching for optimal locomotion. A significant change in the status of neuronal connections in the hip joint appeared, especially, after about 10,000 steps, that is, the Figure 6 Changes in the connection weights for four FBU groups.

15 n Simulation of the Ontogeny of Bipedal Walking 341 Figure 7 Changes in the connection weights for three FBU groups. transition stage from supported walking to independent walking. This finding indicates that the control system of the hip joint is most important. Thirdly, changes in the ability to produce joint moments were investigated. In the neuronal system of this model, the application of Eqs. (2)-(4) yields joint moments. The ability to produce joint moments is determined by several parameters. Here, the ability to produce a joint moment ni is defined simply by using the following equation: ni=pi(u0+ kĉik fk ), (11) i corresponds to the maximum joint moment produced by the ith neuron when internal fatigue of the neuron is not affected (that is đ2=0), all output from the FBUs act so that the total output from the neural oscillators will increase, and the effect of the time constant Ėi is ignored after sufficient time has passed. The sum of ni was calculated for the flexors and extensors as follows: nflexor=n1+n4+n5, (12a) nextensor=n2+n3+n6, (12b) where nflexor is the sum of ni for the flexors and nextensor is the sum of ni for the extensors. nflexor and nextensor are assumed to represent the ability to flex and to extend the entire lower limb, and changes in these values were plotted against the number of search steps (Figure S). As can be seen from this figure, the ability to produce joint moments increased for both the flexors and extensors with the continuation of searching, but the growth rate in the extensors was more pronounced. Marked changes were noted

16 342 Hase K. and Yamazaki N. Figure 8 Changes in the ability of joint moments for flexors and extensors. in the first half of the searching, that is, during supported walking. These findings indicate that independent walking is required to improve the ability to produce joint moments, especially by the extensors. Discussion Changes in Walking Patterns This simulation selected the supporting forces defined in Eq. (1) for creating supported walking, which work only in a vertical direction. In this case, no horizontal resistance force could be generated. Therefore, the step length of walking becomes longer and walking speed becomes higher in the early stage of the search process when only supported walking is observed. These values decrease when supported walking gradually changes to independent walking. This movement pattern differs from actual human infant walking, but is not necessarily unnatural if treated as simulating supported walking aided by a "baby walker" with casters (that is, a device helping a baby to walk). As described above, the walking pattern after the loss of support represents well the actual walking patterns of human infants. In order to compare the significance of each feedback signal in the neuronal system quantitatively, the FBUs were defined as giving discrete values as seen in Eq. (3). However, changes of the joint moments and the walking patterns observed in this simulation occurred smoothly. This is because the neural oscillators produced smoothly adapting series of output signals against input signals. As for changes in the evaluative criteria on the biomechanical loads of locomotion, such as specific power and muscle fatigue, these values were relatively small during supported walking, reached maximum levels when independent walking was acquired and decreased

17 Simulation of the Ontogeny of Bipedal Walking 343 again after independent walking has become perfect. This tendency is due to reductions in supporting forces and is due to improvements of walking movements. In the early stages, as long as walking depends on supporting forces and the lower extremities are not subject to the entire weight, joint moments and consumed energy are small, resulting in low evaluative criteria. As searching proceeds and walking patterns change to independent walking, the lower limbs gradually take over the body weight without supporting forces. Therefore, joint moments and consumed energy increase and the evaluative criteria temporarily become worse. After walking has got independent, the movement pattern improves and the biomechanical load of locomotion becomes lower, which leads to decreasing values for the evaluative criteria. Changes in Neuronal Structure The neuronal structure obtained from this study is not necessarily a precise imitation of the actual neuronal structure in humans. The bulk of the neuronal structures which control human walking, however, has not yet been neurophysiologically clarified. Taga (1995a) has presented a complex model for bipedal walking, which is connected with the musculo skeletalsystem. When constructing the model, trial and error was used for quite a number of parts of the model by referring to electromyograms and so on. Previously, we have also developed a similar model of walking and applied a search process based on biomechanical indices to minor adjustment for the neuronal parameters, but modification of the basic neuronal structure depended on human trial and error rather than that of a computer (Yamazaki and others, 1996). Compared to previous studies, our approach allows autonomous determination for the values for the neuronal parameters and even of the neuronal structure itself. It is important for the values of neuronal parameters and the neuronal structure to be autonomously determined by imitating the developmental process without human trial and error. Although it is unclear whether details of the neuronal system model presented in this study are valid or not, the "macroscopic" behavior of parameters seen in Figures 6-8 reflects that seen in human development well. Figure 6 shows the importance of velocity feedback on the developmental process of walking. Since reflexes (for example, positive support reactions) were observed in human infants, it was expected that independent walking would be acquired by increasing the significance of the feedback about foot-ground contacts. Simulation results, however, showed little significance of the foot-ground contact. Velocity feedback turned out to be the most important factor in the development of the walking pattern, and is more sensitive to changes in movement than feedback signals such as the joint angle or foot-ground contact. In generating a walking pattern, it would be too late and inappropriate to follow up changes in the system of body dynamics and to control them after the status of the movement, such as foot-ground contact, has changed. Predictive action must be taken before changes in movement occur. Therefore, velocity feedback, which is more predictive to changes in movement, is more important than the signal which is produced at foot-ground contact.

18 344 Hase K. and Yamazaki N. The importance of connections between neural oscillators decreased. The rhythm pattern of the neural stimuli created by connections between neural oscillators is effectively a pattern created merely inside the neuronal system. On the other hand, the rhythm pattern of neural stimuli generated by the feedback from somatic senses is created from interaction between the neuronal system, body dynamics, and the environment. One of the most characteristic traits in the development of walking is the change from movement caused only by the neuronal system to movements caused by the interaction between the sub systemswhich consist of the neuronal system, body dynamics, and the environment. As explained in Figures 7 and 8, a comparison between the joints shows that the importance of the hip joint increases with the increase in walking abilities. Comparing extensors and flexors, the importance of the extensors increases. The hip is used to create the basic reciprocal movements of the lower limbs in walking, that is, walking steps of the right and left legs, and keeps the upper body vertical. Therefore, reinforcement of feedback on the hip joint is of great value for developing independent walking. Moreover, the extensors are antigravity muscles. Improvements in the ability, especially of extensors, to generate joint moments is crucial to developing a walking movement. Since our simulation model did not deal with mechanical muscle properties, such as physiological cross-sectional area, it is unknown whether improvements in the ability to produce joint moments is due to the development of the muscles themselves or to the neuronal system involving muscular stimuli. However, improvements in the ability to produce joint moments is quite important when learning to walk. From what has been said above, we believe that one of the properties in the development of walking is to clarify the role of movement for the body structure, such as the function of each joint and the functions of the flexors and extensors. Significance of Computer Simulation Because experimental studies of the developmental processes of infancy require high effort and imply dangers for the child, and also because the transitions in these developmental processes occur at extremely irregular times among individuals, it is generally difficult to identify trends in experimental data. We propose a synthetic technique with computer simulation as an alternative approach to experimental studies. Computer simulation enables us to standardize conditions and to make quantitative evaluation easier. The abilities of computer simulations are particularly worthful in cases, such as human infants, where an experimental approach is subject to restrictions. It took close to 10,000 repeated calculations to reach fully independent walking in this simulation. Human infants do not actually have to repeat their attempts to walk as many times as this simulation result. In this simulation, initial connections of the neuronal structure were given randomly, which may have delayed the acquisition of the walking pattern. Another reason for this delay may be that the search for a neuronal structures by genetic algorithms, corresponding to the learning of movement, is not very effective. In contrast, we can see from these findings how remarkable the human ability to learn movement is. Kawato and others (1990) have proposed a theoretical model for the ability of the neuronal

19 Simulation of the Ontogeny of Bipedal Walking 345 system to learn movement. Such investigations are popular in the area of neurophysiology, and we would do well to incorporate the findings from neurophysiological simulation studies into our investigations. There are three possible types of developmental factors in the walking of infants: improvements in the ability to control movement due to development of the neuronal system; increases in the ability to produce joint moments that depend on the development of muscles; and changes in body proportions. Among these three factors, changes in body proportions have been demonstrated to be a factor for the development of bipedal locomotion through a computer simulation similar to this study (Yamazaki and others, 1996). In the present study, the main emphasis was placed on the development of the neuro - muscular system, and biomechanical factors of the neuro-muscular system have been identified which influence the acquisition of independent bipedal walking. However, this study deals with a relatively simple model for bipedal walking and does not consider spinal curvature, the bi-articular muscles, and the mechanical properties of muscles and connective tissues, such as visco-elasticity. These elements are also indispensable in perfecting efficient walking (Preuschoft and other, 1988; Witte and other, 1997), but are difficult to analyze through direct measurement. Usage of a computer simulation as in this study would be advantageous in examining these factors. The danger that computer simulations are likely, however, to deviate from actual biological phenomena and to become armchair theory cannot be ruled out. We, therefore, have to consider comparisons with actual biological events to avoid this danger. Conclusion Using computer simulation, we followed up the developmental process from supported walking to independent walking in which human infants autonomously acquire the bipedal walking movement, and attempted to identify the mechanisms of acquiring walking movement from a biomechanical point of view. Basic structures of the neuronal system and body dynamics were modeled to generate walking movement and the developmental or learning process for walking was modeled as a search process for the neuronal structure and parameters. The results of the simulation showed that the neuronal system, which was initially arranged randomly and unable to generate a walking pattern without additional supporting forces, gradually acquired the ability to create independent bipedal walking. Examination of changes in the neuronal system indicated the significance of velocity feedback, feedback involved with the hip joint, and the extensors' ability to produce joint moments during the development of walking. This study implemented computer simulation techniques not frequently used in the study of kinesiology, and the usefulness of such simulation was confirmed. Results obtained from this methodology are quantitative and theoretical. They permit analyses that cannot be made with an experimental approach. The significance of such simulation techniques is expected to increase.

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