JUMPING FROM STRUCTURE TO CONTROL A Simulation Study of Explosive Movements Ph.D. Thesis of: Arthur J. van Soest Faculty of Human Movement Sciences Vrije Universiteit van der Boechorststraat 9 NL 1081 BT Amsterdam Netherlands email: KNOEK@SARA.NL 22 October 1992 SUMMARY The central question in the area of 'movement control' is how animals and humans organize their movements in order to meet specific environment-related goals (e.g. thread a needle; jump a fence). From a purely scientific point of view, this question is interesting in its own right. In addition, partial answers to this question are applicable in fields like rehabilitation medicine, sports and ergonomics. The actor and the environment, being the essential elements in the central question mentioned, are mutually coupled: on the one hand, we change the environment through our movements (which are based on perceptual cues), and on the other hand we perceive the (changes in the) environment. In this thesis one of these actor-environment couplings is studied. We focus largely on the 'output' side of the actor, i.e. the high energy coupling between actor and environment. Furthermore, we restrict ourselves to a purely mechanical perspective. Specifically, this study is aimed at improving our understanding of the relation between muscle stimulation pattern and structure of the musculo- skeletal system on the one hand, and achievement on the other hand. Many specific questions in this respect cannot be answered in an experimental setting because the independent variables do not lend themselves to experimental manipulation. Therefore, mathematical modelling and simulation have a prominent place in this study. If a reductionistic approach in which the information coupling between actor and environment is largely neglected, is to improve our understanding of the 'real' system, tasks must be considered in which the mechanical properties of the musculo-skeletal system are important and in which the nervous system can be considered as a mere generator of muscle stimulation patterns (henceforth called 'STIM patterns'). This combination of demands is met in 'explosive movement tasks', examples of which are throwing, kicking, jumping and any push-off in running and skating. Muscle properties are expected to be important in these tasks because achievement depends directly on the amount of mechanical work done by the muscles. The nervous system can be considered as an open-loop STIM generator in these tasks because the duration of explosive movements is so short that, due to latencies of neural feedback loops, neural feedback cannot play an important role. From the group of explosive movements, human vertical jumping is the task considered in this thesis. For maximum-height vertical jumping, the goal can be defined in mechanical terms: to maximize the height reached by the body center of mass. As this variable can be calculated from the position and velocity at take-off, only the push- off phase will be considered. Modelling and simulation are the tools used in this study. To investigate the relation between musculo-skeletal properties and STIM pattern on the one hand, and vertical jumping achievement on the other hand, a musculo-skeletal model is needed of which STIM is the input and the resulting movement is the output. In Chapter 2, a software system is described with which the dynamical equations of motion of the skeletal system can be derived. The skeletal system is represented by a 2-dimensional linkage of four rigid segments representing feet, lower legs, upper legs and head-arms-trunk. In Chapter 3, the structure of the Hill-type muscle model used throughout this study is described in detail. Furthermore, an in situ experiment on isolated muscle described in literature is simulated, using parameter values derived from structural data. Simulation results are shown to be similar to experimental data, indicating the adequacy of the muscle model. Indications that the model adequately represents the 'real' system are indispensable because it is not the model that we are interested in, but the real system. An important indication of model adequacy is obtained in Chapter 4. An essential finding reported there is that both kinematical and kinetical aspects of the maximum-height jump that can be achieved with the model closely match experimental data of well-trained subjects instructed to jump as high as possible. This finding supports the assumption that well-trained subjects have 'optimized' their STIM pattern in order to maximize jump height. At the same time, it implies that the model used adequately represents the salient features of the real system that are important in the task studied. Having gained sufficient confidence in the model, two questions concerning the effect of properties of the musculo-skeletal system on jumping achievement are addressed. In Chapter 4, an 'old' idea is tested in a modelling and simulation environment. This idea stemmed from the results of a combined inverse-forward analysis of observed movements. From this analysis it was found that the bi-articular gastrocnemius muscle, delivering a plantarflexing ankle moment and a flexing knee moment, 'transported power' from knee to ankle joint in the final phase of the push-off movement. It was argued that this phenomenon enhances jumping achievement. In other words, the idea was formulated that the bi-articularity of the gastrocnemius muscle contributes to jumping achievement. Obviously, this idea is not amenable to experimental testing in humans. As a substitute, this idea is tested through simulation. First the optimal jump height for the standard model is found using numerical optimization. Next, gastrocnemius is changed into a mono-articular ankle plantarflexor and the optimal jump height for this 'handicapped' model is calculated as well. By comparing jump heights, the specific contribution of bi- articularity is determined. This contribution is found to amount to 1 cm. Thus, although in a quantitative sense the effect is small, the idea outlined above is supported by the simulation results. Furthermore, the optimal STIM pattern for the 'handicapped' model is found to be different from that for the standard model. Apparently, when the control is optimally adapted to potentially detrimental changes in system structure, the effect of such changes can be kept within limits. Finally, the change in jump height is noted to depend critically on values of the model's parameters, specifically on the way in which moment arm of the gastrocnemius at the knee is related to knee joint angle. From this finding it is concluded that, given the present state of knowledge of the structure of the musculo-skeletal system, it is hazardous to use modelling and simulation as the ultimate test of ideas on the role of specific aspects of structure. The question addressed in Chapter 5 concerns muscle characteristics as well, albeit in a more practically relevant direction. The question is what happens to vertical jumping achievement when muscle strength is increased, as is typically achieved through strength training in reality. The purpose of this chapter is to investigate whether changes in muscle strength alone are sufficient to increase vertical jumping achievement, or whether they need to be accompanied by changes in control. Because the control signals are not amenable to direct measurement, let alone to manipulation, a modelling and simulation approach is used. First, the STIM pattern that is optimal for the standard model is applied to a model with strengthened muscles. Despite the larger work potential of the muscles, in this case jump height decreases rather than increases. As expected, when the STIM pattern is optimized for the strong model, jump height is improved. From these findings, the conclusion is drawn that changes in muscle strength, as initiated through strength training, are beneficial in complex tasks like vertical jumping only when the STIM pattern is properly tuned to the new system properties. It is suggested that repeatedly solving the problem posed by the task requirements is the most straightforward way to tune the STIM pattern to the increased strength. The questions addressed in Chapters 6, 7 and 8 concern aspects of control, in relation to characteristics of the musculo-skeletal system. The basic question here is, how humans manage to be successful in open-loop controlling an inverted-pendulum-like skeletal system that is moving at high velocity. Various easily observed phenomena are not explained by existing theory on the control of fast movements. Consequently, modelling and simulation, tools used earlier for testing of a hypothesis, have been used for the generation of new hypotheses in these chapters. Chapter 6 starts from the observation that disturbances that can be assumed to occur during execution of any explosive movement do not lead to disintegration of the movement. Yet, from the fact that the skeletal system is analogous to an inverted pendulum, it would be expected that the movement is sensitive to disturbances. In this chapter, it is investigated if muscle properties, i.e. the force- length-velocity relation of muscle, can explain the observed insensitivity to disturbances. To this end, two types of open loop control signals are applied: STIM and MOM, the latter representing net joint moments. In case of STIM control muscle properties influence the joint moments exerted on the skeleton; in case of MOM control, these moments are directly prescribed. By applying perturbations and comparing the deviations from a reference movement for both types of control, the reduction in the effect of disturbances due to muscle properties can be determined. It is found that the skeletal system is very sensitive to perturbations in case of MOM control; the sensitivity to perturbations is markedly less in case of STIM control. From these results, it is concluded that muscle properties constitute a peripheral feedback system that has the advantage of zero time delay over neural feedback loops. In other words, the visco-elastic type of behavior of muscle that results from its force-length-velocity relation counteracts disturbances to position and velocity without any modification to the neural input of the muscle. In fact, this zero-lag peripheral feedback system reduces the effect of disturbances occurring during human vertical jumping to such a degree that, when disturbances are not too large, adaptation of the muscle stimulation pattern is not necessary for successful execution. In Chapter 7 an explanation is offered for the observation that humans are perfectly able to perform explosive movements (e.g. vertical jumping; hitting an approaching ball) from a wide range of starting positions. In fact, between-trial variability of kinematic parameters may even decrease as the movement unfolds. Due to the short duration of such movements, these observations cannot be attributed to corrections initiated on the basis of neural feedback. The explanation of this observation commonly adhered to is based on the idea of 'general motor programs'. According to this explanation, the values of a number of parameters of a stored 'general motor program' are set in such a way that a well-coordinated movement results. In this chapter, an elegant alternative hypothesis is proposed. This hypothesis is based on 'Control That Works' as opposed to 'Optimal Control'. Specifically, it is shown that, surprisingly, close-to-optimal performance for a wide range of starting positions can be obtained by using one single STIM pattern without any adaptation. This STIM pattern is not optimal for any specific starting position considered. Rather, it results in close-to-optimal jumps for a wide range of starting positions. The success of this strategy again relies heavily on the force-length-velocity relation of muscle. In comparison to other possible explanations of the observations mentioned, the strategy proposed in this chapter is more sparse. In fact, if such a strategy is used in reality, this would in extremo reduce the problem of controlling vertical jumps to learning, storing and retrieving a single muscle stimulation pattern. The questions addressed so far all concern maximal execution of explosive movements. However, in daily life submaximal execution of tasks is more commonly required, for example to jump a fence or to pick an apple from a not-too-high branch. The control problem in preparing for a submaximal vertical jump is to pre-program an appropriate STIM pattern on the basis of desired jump height as specified by perceptual cues (the fence; the apple). In Chapter 8, it is suggested that this control problem may be solved in reality on the basis of scaling of the stored representation of the muscle stimulation pattern. This suggestion is supported by simulation results. First, it is shown that a scaling algorithm exists that allows pre-calculation of net joint moments as a function of time on the basis of desired jump height. This algorithm results in identical movement trajectories at different velocities. Interestingly, 'trajectory invariance' is also found experimentally when subjects are asked to execute the same task submaximally at different velocities. According to this scaling algorithm, the net joint moments must be considered to consist of a 'posture component' that annihilates the effect of gravitational forces, and a 'movement component' that is responsible for the movement. Unfortunately, net joint moments are not the control signal in biological systems; rather, STIM is. In that case muscle properties, helpful as they may be in other respects, complicate the situation. However, it is shown that when only one of the steady-state nonlinearities of the STIM-MOM relation is taken into account, successful scaling can be achieved. Thus, specification of the desired jump height on the basis of perceptual cues can be unequivocally translated to an adequate STIM pattern on the basis of stored posture and movement components that pertain to maximum-height jumping. In the literature support can be found for the notion of separate posture and movement control systems. Finally, it is argued that it is not at all beyond the possibilities of biological neuronal networks to achieve the required transformations.