Internal models in the control of posture

Transcription

1 PERGAMON Neural Networks 12 (1999) Neural Networks Internal models in the control of posture P.G. Morasso a,b, *, L. Baratto b, R. Capra b, G. Spada b a Department of Informatics, University of Genova, Systems and Telecommunication, Via Opera Pia 13, I Genoa, Italy b Center of Bioengineering, Rehabilitation Hospital la Colletta, Arenzano, Italy Received 14 December 1998; accepted 27 April 1999 Abstract The aim of the paper is to investigate the application of control schemes based on internal models to the stabilization of the standing posture. The computational complexities of the control problems are analyzed, showing that muscle stiffness alone is insufficient to carry out the task. The paper also re-visits the concept of the cerebellum as a Smith s predictor Elsevier Science Ltd. All rights reserved. Keywords: Motor control; Standing posture; Muscle stiffness; Smith s predictor; Internal models; Cerebellum 1. Introduction Theorists and modelers of motor control have investigated a number of different motor paradigms, usually involving fast and coordinated movements with many degrees of freedom, as in eye, hand, and speech movements. In this way, the main computational problems (planning, coordinate transformation, dynamics, redundancy etc.) have been put into focus and the issue has been raised about the need, for the CNS, to build some kind of internal models in order to carry out the necessary computations. However, little attention has been given to posture, perhaps assuming that this paradigm is only marginal because it is characterized in normal conditions by the (virtual) absence of movement. In this paper we intend to show that this is not correct and the control of posture implies the same kind of subtle interactions that make motor control complex and interesting. We should also emphasize that the interest is not limited to the purely intellectual domain because postural stability is so important in the activities of everyday life and is one of the first skills to be deteriorated as a consequence of aging. It has been shown indeed that one of the main requirements for autonomy is postural stability and its deficiency is the main cause of falls and ultimately bone ruptures (Coogler, 1992; Morasso, Baratto, Capra & Spada, 1998). The complexity and uniqueness of postural control is determined by a number of reasons: The mechanical plant or controlled object is intrinsically unstable. The controlled variable (the Center of Mass: COM) is not directly measurable. The available sensory receptors operate quite close to their absolute threshold of sensibility. Stability is critically dependent upon delays in the reafferent signals. 1 There is a dynamic coupling with other motor subsystems, typically the arm/manipulation controller, which is not relevant so much for accuracy as it is for stability itself. The standing posture is usually conceived merely as the initial state of walking or running and thus is assumed to be simpler. However, it is easy to show that it is just the other way around. Standing requires a continuous stabilization process that is fatiguing, although requires a small amount of external work, whereas walking and running are discontinuous control processes where a significant part of the cycle is purely passive (e.g. the swing phase in walking or the flying phase in running). Moreover, as in cycling, there is an intrinsic dynamic stabilization, unknown to the CNS, which is enhanced by speed; in fact, the exploitation of dynamic effects can also be an effective trick in the rehabilitation of hemiplegic and other motor-impaired subjects. Muscle fatigue in power walking or running is less than in * Corresponding author. Tel.: ; fax: address: morasso@dist.unige.it (P.G. Morasso) 1 Delays in feedback control schemes are always dangerous as regards stability, as they decrease the phase and gain margin of the controlled system. The problem is clearly worse if the plant is intrinsically unstable and the purpose of the feedback control mechanism is to stabilize it in the first place /99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 1174 P.G. Morasso et al. / Neural Networks 12 (1999) of the body: f V mg ˆ mz 0 3 f H ˆ my Fig. 1. Mechanics of quiet standing in the A/P direction. y, z, u: horizontal, vertical and angular position of the COM; u: horizontal position of the COP; f: ground reaction force; t ankle : total ankle torque due to all the muscles operating around the ankle; h, d, m: constant parameters. standing or very slow walking, although a significant amount of external work is involved. 2. The plant It is a fundamental law of physics that the motion of any body standing on a surface (multi-articulated or not) is determined by two external forces: the weight force w, applied to the center of mass (COM), and the ground reaction force f, applied to the center of pressure (COP). We can write the pair of canonical equations: f w ˆ d m Tv 1 dt f COM COP ˆ d I Tv dt where m T is the total mass of the body, I T its inertia tensor, v the velocity of the COM and v the angular velocity. Eq. (1) applies to any conceivable circumstance, including jumping, running, diving etc. If we now narrow our attention to posture control during quiet standing (see Fig. 1), we can make several simplifying assumptions. First of all, we consider that the feet are fixed and the inertial parameters are approximately constant. As a consequence Eq. (1) can be split into a static equilibrium equation for the feet: f H d f V u t ankle ˆ 0 2 t ankle is the torque applied by the ankle muscles, f H and f V are the horizontal and vertical components of the ground reaction, d is the vertical displacement of the ankle from the support surface, and u is the horizontal displacement of the COP from the ankle) and a dynamic equation for the rest t ankle mgy ˆ d dt I _ a u I a u mh 2 y k s h ˆ mhk s y y and z are the horizontal and vertical positions of the COM with respect to the ankle; u is the angle with respect to the vertical line; m is the weight of the body excluding the feet, I a is the corresponding moment of inertia with respect to the ankle joint, h is the distance of the COM from the ankle, and k s is a shape factor that depends on the distribution of mass in the body. In particular, k s ˆ 1 if all the mass were concentrated in the COM (in that case I ˆ mh 2 ) and k s ˆ 1:33 if it were distributed in a uniform way along a thin rod (in that case I ˆ 4=3mh 2 ): for the human body k s is closer to the latter estimate than the former one. There are three approximations in the model, that are well satisfied for sway in quiet standing: (i) the vertical acceleration of the baricenter is negligible; (ii) the moment of inertia is constant and (iii) the angular acceleration is proportional to the horizontal acceleration of the COM. These approximations are compatible with the inverted pendulum model but are not restricted to it. We can now combine Eqs. (2) and (3), obtaining the following sway equation: g g y ˆ y u ˆ y u 4 hk s d h e where h e ˆ k s h d is the effective distance of the COM from the ankle. In other words, in the two main directions of sway (antero-posterior and medio-lateral) the COM COP difference is proportional to the acceleration of the COM in the same direction. The purpose of the posture control system is to keep the position of the COM as close as possible to a reference position (typically 1 2 cm in front of the ankle joint). The position of the COP is directly affected by the activation of ankle muscles: plantar flexors shift the COP forward, whereas dorsal flexors shift the COP backward. In the language of systems theory we can write the following transfer function of the mechanical plant: Y s U s ˆ g=h e s 2 5 g=h e where Y(s) is the transform of the output (the COM) and U(s) is the transform of the input (the COP). Clearly this is an unstable system, as one of the roots is positive, and we need some kind of feedback mechanism to stabilize it. 3. Control It is easy to see that stabilization of the plant can be achieved by means of a simple PD (proportional derivative) control block. The transfer function of the controller and the corresponding characteristic equation of the closed

3 P.G. Morasso et al. / Neural Networks 12 (1999) in principle, either by the intrinsic mechanical properties of the muscles or by a suitable brain-based computational mechanism. In the case of trajectory formation the controversy is still open and we are not going to weigh the pros and cons of either solution (Morasso & Sanguineti, 1997); on the contrary, in the case of sway it is easy to show that there is no chance for the muscle properties alone to achieve stabilization, although a recent paper (Winter, Patla, Prince, Ishac & Gielo-Perczak, 1998) defends a stiffness control paradigm Why stiffness stabilization of sway is not possible Fig. 2. Real sway trajectories. COP (thin line, measured via force platform); COM (thick line, reconstructed by integrating sway Eq. (4)). loop system can be written as follows: U s ˆK p sk d s 2 g=h e K d s g=h e K p 1 ˆ0: The stability condition (roots of the characteristic equation with negative real part) implies that the proportional gain must be strong enough: K p 1: In this case, the system is characterized by a damped oscillatory response to disturbances with the natural frequency f n ˆ 1 2p g K p 1 =h e Š 1=2 and the damping coefficient z ˆ K d =2 g= K p 1 =h e Š 1=2 : Hence, K p is critical for stability and K d for damping. The undamped natural frequency of sway movements, estimated by different means, is of the order of 0.5 Hz, thus yielding the following estimate: K p 2: The damping coefficient appears to be rather low, of the order of z ˆ 0:2 : this yields K d 0:13: The problem then is to find out a control mechanism that is functionally equivalent to the PD controller above. As in the case of fast arm movements, the problem can be solved, Fig. 3. Simulated sway trajectories: the plant is given by Eq. (4) and the control by Eq. (6). The system is persistently perturbed by a noise source (white noise impulsive, quasi-periodic noise). COP (thin line); COM (thick line). Horizontal axis: 20 s. Vertical scale: arbitrary units. 6 We can observe that the PD stabilization scheme could be obtained through the mechanical properties of the ankle muscles that are indeed equivalent to an instantaneous proportional (elastic: K e ) and derivative (viscous: K v ) feedback: t ankle K e u K v _u : If we insert this control law into the third part of Eq. (3) we get the following equation: I a u K v _u K e mgh u ˆ 0: Laplace-transforming this equation and comparing it with the second part of Eq. (6), we get the following expression for the ankle stiffness K e (due to the whole set of muscles operating around the ankle): K e ˆ mgh I a K p 1 g=h e : 8 For m ˆ 80 kg; h ˆ 1m; in limit case of marginal stability K p ˆ 1 we have K e ˆ mgh ˆ 785 N m=rad: However, it must be K p 2; in order to fit the observed values of natural frequency and damping coefficient, and thus assuming that h e ˆ 1:2m; I a ˆ 100 kg m 2 we get K e 1602 N m=rad: Is this figure compatible, at least as an order of magnitude, with available joint stiffness measurements and known muscle properties? Joint stiffness values have been estimated by Flash and Gurevich (1997) for the shoulder joint to vary in the range N m/rad for various levels of bias torque, while performing normal movements. Direct measurements of ankle stiffness by means of a special kind of ergometer have been performed by Hof (1998) showing a range of N m/rad, with a large bias torque (100 N m). In both cases we are still far away from the required values computed above. Similar calculations can be performed at the muscle level and, again, we remain quite behind the necessary level for stabilization, even assuming the highest possible levels of stiffness and a very generous moment arm. Summing up, the required control law u t ˆ K p y t K d _y t cannot come alone from the instantaneous mechanical feedback due to intrinsic properties of the ankle muscles but requires a real feedback mechanism. As K p 1 and K d 0 it means that for each time instant the controller must guarantee that the COP is beyond the COM and this margin of advance must be modulated by the sign of the sway speed. In other words, the time profile of the COP must oscillate back and forth around the smoother COM profile. 7

4 1176 P.G. Morasso et al. / Neural Networks 12 (1999) Fig. 4. General scheme of the control with a Smith s predictor. Fig. 2 shows the two curves in a real sway movement and Fig. 3 the simulated behavior of the system with K p ˆ 2:0; K d ˆ 0:13; disturbed by a random noise. However, for the biological plausibility of the solution there are two critical points: The variable y is not directly measurable. The overall stability is very sensitive to the delay in the feedback Stabilization by means of internal models Although we know the form of the solution for the controller block, implementing it in biological hardware is not trivial for the reasons we already mentioned: (further) instability due to delays in the feedback loops and problems in the estimates of the state variables. For solving this kind of task, as for other biological motor control issues, the computational scheme known as Smith s predictor is a good candidate. In the following we discuss in which way the main idea, initially proposed by Smith (1959) for the stabilization of industrial plants and later applied to biological control (Miall, Weir, Wolpert & Stein, 1993; Wolpert & Kawato, 1998), can be specialized for the control of sway. Fig. 4 shows the general form of a Smith s predictor for motor control, as proposed by Miall et al. (1993). The Controlled Object is driven by the motor commands generated by a simple motor controller (e.g. a PD block) with the help of an internal Forward Dynamic Model. It is assumed that this model has been trained to imitate the controlled object and thus, if fed with an efference copy of the motor command, it will generate a prediction of the real trajectory 2 With f n ˆ 0:5 Hz and z ˆ 0:2 the phase margin is 12, provided that the delay in the feedback loop is null. A delay of 64 ms is sufficient to send the margin to zero and make the system unstable. that can be fed back to the input of the controller. As a consequence, the internal recurrent system (motor controller forward dynamic model) operates as an inverse dynamic mechanism and is capable, in ideal conditions, to force the controlled object to follow the desired trajectory, compensating its intrinsic dynamics. However, inaccuracies in the model and/or unaccounted disturbances will generate a discrepancy between the desired and real trajectory that can be used to modulate the input of the controller. The problem is that in order to be effective the comparison must be performed between two signals (the measured and internally predicted values of the COM) that are not timeshifted with each other. On this purpose, a Forward Output Model is needed that transforms the output of the forward dynamic model into the estimated reafferences (taking into account the delays and transformations performed by the sensory system), thus producing the appropriate corollary discharge for the controller. In this way, the required dynamic performance can be achieved in spite of significant delays in the feedback loop. In the sway control system (Fig. 5) we distinguish between a high-dimensional and low-dimensional computational layer. The former one is related, on the motor side, to the highly redundant set of degrees of freedom and muscles directly or indirectly involved in the active stabilization of posture and, on the sensory side, to the multi-modal flow of reafferences that need to be fused and integrated in order to be effective. The latter computational layer implements the core of the Smith s predictor control scheme, acting on the effective controlled variable, the center of mass y Sensory aspects in the high-dimensional computational layer The multi-modal sensory system that feeds the controller includes the following receptors:

5 P.G. Morasso et al. / Neural Networks 12 (1999) Fig. 5. Application of the Smith s predictor control scheme to the stabilization of standing posture. u: COP; y: COM; r: reference posture (desired value of the COM); apa; anticipatory postural adjustment. joint receptors, particularly those of the ankle and hip joints that are known to be more affected by sway movements; muscle spindles, again with particular reference to those acting around the ankle and hip joints; vestibular receptors, both in the statocyst-macula and the canal-crista; visual static and motion detectors in the retina; cutaneous and muscle receptors in the foot sole. Vestibular receptors share with vision the same body platform (the head) but have quite different transduction characteristics. The effect of vision is usually small because the Romberg coefficient (the ratio between the sway path measure in the open vs closed eyes condition) in normal subjects is close to 1. However, this is not true in a variety of pathological conditions, particularly peripheral neuropathies that affect the other sensory channels (Baratto, Morasso, Spada, Capra, Menozzi & Comaschi, 1996; Simoneau, Derr, Ulbrecht, Becker & Cavanagh, 1996). Hence, vision seems to play a sort of backup role, also taking into account its relative slowness. The activation thresholds of vestibular receptors (Benson, Spencer & Stott, 1986) are of the same order of magnitude of sway acceleration and, hence, it is likely that only the high-frequency parts of sway are detected by this channel during quiet standing. Vision, of course, has complementary characteristics because its nature is well suited for low-frequency, largeamplitude components. Also the activation thresholds of joint receptors and muscle spindles (Fitzpatrick & McCloskey, 1993) appear to be in the same range of sway parameters. All together, visual, vestibular, joint and muscle receptors can only provide information about the dynamic components of the COM (speed and acceleration), not its position in relation with the foot and/or the COP. The last potentially relevant sensory channel, provided by the plantar cutaneous and muscle receptors, is usually overlooked. Similarly, the horizontal component of the ground reaction force is ignored by most researcher as it is small with respect to the weight force. However, this small force is well detectable by the plantar receptors and is proportional to the COM COP difference, in agreement with Eq. (4) as it is proportional to the horizontal acceleration of the COM. In the A/P direction, for example, if the force is directed forward it means that the COP is behind the COM and the other way around if the force is directed backward. Thus, also the sign of this force is directly relevant from the control point of view. The vertical and horizontal components of the ground reaction forces are sensed by plantar tactile receptors and it is possible to hypothesize that the sensory process is carried out in a segregated way. The normal component is slowly changing around a big average value; the weighted average of the stress detected by each element is directly related to the position of the COP, by definition. Slowly or moderately fast adapting receptors with small receptive fields like the Ruffini and Meissner terminations have ideal transduction features for this kind of task. On the contrary, the horizontal component of the ground reaction oscillates much more rapidly around zero and it is not important to localize it. In this case, the Pacinian corpuscles seem to have the right transduction features as they are very fast adapting and have a large receptive field. From the functional point of view, the importance of the plantar

6 1178 P.G. Morasso et al. / Neural Networks 12 (1999) Fig. 6. Storage and prediction of time sequences. tactile sensors in the control of sway is reflected in the sensitivity of sway patterns to the compliance of the support surface and the destabilizing effect of compliant materials. Summing up, a plausible scheme for the sensory aspects in the high-dimensional computational layer of the sway control system ( Sensory Fusion in Fig. 5) can be characterized as follows: the Meissner and Ruffini receptors provide an estimate of the variable u; the Pacinian receptors code the difference y u; the combination of the two codes allows the brain to have an internal representation of the unobserved variable y; the first time derivative of this variable, _y; can be estimated by some kind of distributed fusion of the other sensory receptors (visual, joint, and muscle), thus completing the internal representation of the postural state vector x ˆ y; _yš T : All these computations are probably performed in a distributed way, in terms of population codes Motor aspects in the high-dimensional computational layer Eqs. (4) or (5) are good approximate models of the essential dynamic aspects of body sway, with u as the control and y the controlled variable. In this case, the ankle joint is the key element of the system and the other articulations are virtually frozen. In reality, the Controlled Object of Fig. 5 is obviously more complex and the complexity re-surfaces when unusual conditions occur. For example, the system can loose control of u if the support base is not fixed but free to oscillate in the A/P direction or is very small, as in a ballet dancer standing on shoe-tips or an acrobat standing on stilts. In this case, the person cannot adopt the usual control paradigm, centered on the ankle joint, also known as joint strategy. The only alternative is to push up the level of control, involving more distal districts and degrees of freedom, with respect to the ground surface: for example the hip (from this the so called hip strategy, typically used by elderly people when they start experiencing balance problems) but also more distal parts of the body (arms or head) for experienced users. In the latter case Eq. (4) is still approximately valid but u is not the control variable any more and this equation must be coupled with the equations of other degrees of freedom where the alternative control variables are located. In Fig. 5, the Synergy Formation block refers to the ankle strategy: it transforms the motor command u into the corresponding patterns of activation of the ankle muscles and, at the same time, sets the appropriate levels of stiffness of the other joints in order to effectively freeze them out. In general, however, this part of the model should be considered as just one module in a modular control architecture (Wolpert & Kawato, 1998) Low-dimensional computational layer and cerebellar circuitry Fig. 5 shows the structure of the layer for the particular module considered in this paper: stabilization of sway in

7 P.G. Morasso et al. / Neural Networks 12 (1999) quiet standing. The desired trajectory in this case is null, the Forward Dynamic Model is represented by Eqs. (4) or (5), and the Motor Controller has a PD structure. These two blocks play the same roles of the analogous modules in Fig. 4 but are so simple that do not require complicated learning and computational architectures. The basic problem is to compensate the delay of the sensory reafferences in order to achieve stability: this means to align in time the afferent sequence x sens with the internal representation of the state vector x int thus producing a well synchronized corollary discharge or sensory discrepancy vector Dx: The former sequence is intrinsically delayed with respect to the latter for the time taken by the transduction and sensory fusion processes. Two solutions are possible in principle: 1. Dx ˆ extrapolation=anticipation of x sens delayed x int Š: the internal representation of the state vector is delayed before being compared with the sensory reafferences and the result is then processed by a prediction or extrapolation block that compensates by anticipating the effect of delay; 2. Dx ˆ extrapolation=anticipation of x sens x int : in this case internally represented state vector is directly compared with the anticipated version of the sensory reafferences. In fact the two mechanisms are functionally equivalent because it is the estimate of the delay that allows the correct setting of the extrapolation/anticipation parameters. In general, delaying and anticipating are complementary operations that apply to sequences and require, in order to be implemented, some kind of time buffer or memory of sequence fragments. Cerebellar circuitry ideally suits such requirements, as discussed in the review paper by Braitenberg, Heck and Sultan (1997). The old argument that the beams of parallel lines on the cerebellar cortex might behave as delay lines was abandoned in the 1950s because the pure propagation of action potentials on the cerebellar surface (about 10 cm) is too quick to store any significant sequence fragment. However, recent experiments illustrate the possibility of a more sophisticated mechanism based on the build-up and propagation of tidal waves along the beams of parallel fibers. This kind of collective phenomenon resonates with the arrival of specific sequences on the mossy fibers and is propagated in the medio-lateral direction on the cerebellar cortex at a relatively low speed (it takes about 200 ms in the human cerebellar cortex to cross the area). On its way, the wave intersects the gigantic flat dendritic trees of Purkinje cells, thus reading out stacks of synaptic weights. In this way, a basic module of sequence-in sequence-out transformation is made available for various kinds of control mechanisms that involve sequences as basic primitives. In particular, Fig. 6 shows a possible scheme for the anticipation of the corollary discharge, in order to align it in time with the internal state vector produced by the Forward Dynamic Model. Thetime buffer of the figure is related to tidal wave mechanism. The delay to be recovered by anticipation is of the order of ms and thus is consistent with a capacity of the time buffer of about 200 ms. In the learning phase, a part of the buffer (the recent past) is associated with another part (the remote past) by means of some kind of self-supervised learning algorithm. For example, we can store in the time buffer time series of the y variable and we can use a multi-layer perceptron for carrying out a prediction/anticipation of such variable in the following way: the input layer of the network is driven by a portion of the time buffer (the remote past) and the output is compared with the remaining part of the buffer (the recent past). Back propagation training is then used in order to predict a portion of the time series from a preceding portion. In the recall phase, the same network can be applied directly to the input (delayed) sequence, thus producing an output anticipated sequence that compensates for the delay. The PD controller operates on the internally generated estimates of the COM trajectory, modulated by the well synchronized corollary discharge. The output of the postural controller should be combined with other control commands based on anticipation (apa: anticipatory postural adjustments) in order to feed the postural Synergy Formation block (Fig. 5). Indeed most voluntary movements, such as reaching and grasping, induce inertial reactions that operate as internally generated disturbances for posture; their effect is known to be compensated in an anticipatory fashion (Gahery & Massion, 1981). Thus, the input of the Synergy Formation block is the natural meeting point for the different anticipatory control processes: the postural one proper and the other ones that have a dynamic coupling with it. The underlying rationale, in all cases, is that only anticipation allows the brain to stabilize the intrinsically unstable human inverted pendulum in spite of the damaging delays in the sensory feedback. 4. Discussion We have formulated the notion that the stabilization of posture is a complex control problem that requires a critical fusion of multi-modal sensory data and the compensation of transmission/processing delays. Ultimately, the effective compensation of the delays in the control loop is apparent in the phase lock between the controlled variable, COM, and the control variable, COP, as illustrated in Figs. 2 and 3. We also showed that muscle stiffness alone in not sufficient to carry out a complete stabilization in normal conditions. However, in the response to strong unexpected perturbations muscle stiffness is the only fast reaction mechanism available because the reported latencies in muscle activation (Nashner, 1982) are close to 100 ms, even ignoring the delay between tetanic fusion and force build-up. In fact muscle stiffness is subtracted from the virtual (negative) stiffness due to gravity and thus tends to decrease the natural frequency of oscillation, giving more time for intervention to the planner/controller at the higher level. Many questions

8 1180 P.G. Morasso et al. / Neural Networks 12 (1999) remain open and worth pursuing, as regards a deeper understanding of the cerebellar mechanisms of anticipation in normal subjects and cerebellar patients. Acknowledgements This work was partly supported by ISS and CNR. References Baratto, L., Morasso, P., Spada, G., Capra, R., Menozzi, F., & Comaschi, M.A. (1996). A new pattern of evaluation of postural stability and fall risk in neuropathic diabetic patients. Proceedings of the First World Congress on Prevention of Diabetes and its Complications. Copenhagen, Denmark. Benson, A. J., Spencer, M. B., & Stott, J. R. (1986). Thresholds for detection of the direction of whole body linear movement in the horizontal plane. Aviation Space Environmental Medicine, 57, Braitenberg, V., Heck, D., & Sultan, F. (1997). The detection and generation of sequences as a key to cerebellar function: experiments and theory. Behavior and Brain Science, 20, Coogler, C. E. (1992). Falls and imbalance. Rehabilitation Management, April/May, Fitzpatrick, R., & McCloskey, D. I. (1993). Proprioceptive, visual and vestibular thresholds for the perception of sway during standing in humans. Journal of Physiology, 478, Flash, T., & Gurevich, I. (1997). Models of motor adaptation and impedance control in human arm movements. In P. Morasso & V. Sanguineti, Self organization, cortical maps and motor control, (pp ). Amsterdam: North-Holland. Gahery, Y., & Massion, J. (1981). Coordination between posture and movement. Trends in Neuroscience, 4, Hof, A. L. (1998). In vivo measurement of the series elasticity release curve of human triceps surae muscle. Journal of Biomechanics, 31, Miall, R. C., Weir, D. J., Wolpert, D. M., & Stein, J. F. (1993). Is the cerebellum a Smith s predictor? Journal of Motor Behavior, 25, Morasso, P., & Sanguineti, V. (1997). From cortical maps to the control of muscles. In P. Morasso & V. Sanguineti, Self organization, cortical maps and motor control, (pp ). Amsterdam: North-Holland. Morasso, P., Baratto, L., Capra, R., & Spada, G. (1998). Preventing the risk of falling in elderly people. In I. P. Porrero & E. Ballabio, Improving the quality of life of the European citizen, (pp ). Ios Press. Nashner, L. M. (1982). Adaptations of human movement to altered environments. Trends in Neuroscience, 5, Simoneau, G. G., Derr, J. A., Ulbrecht, J. S., Becker, M. B., & Cavanagh, P. R. (1996). Diabetic sensory neuropathy effect on ankle joint movement perception. Archives in Physical and Medical Rehabilitation, 77, Smith, O. J. M. (1959). A controller to overcome dead time. ISA Journal, 6, Winter, D. A., Patla, A. E., Prince, F., Ishac, M., & Gielo-Perczak, K. (1998). Stiffness control of balance in quiet standing. Journal of Neurophysiology, 80, Wolpert, D. M., & Kawato, M. (1998). Internal models of the cerebellum. Trends in Cognitive Science, 2,