Biomimetics of human movement

IOP PUBLISHING

BIOINSPIRATION & BIOMIMETICS

doi:10.1088/1748-3182/4/3/033001

Bioinsp. Biomim. 4 (2009) 033001 (7pp)

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Biomimetics of human movement: functional or aesthetic? Christopher M Harris SensoriMotor Laboratory, Centre for Theoretical and Computational Neuroscience, Centre for Robotics and Neural Systems, University of Plymouth, Plymouth, Devon PL4 8AA, UK

Received 16 October 2008 Accepted for publication 15 June 2009 Published 30 June 2009 Online at stacks.iop.org/BB/4/033001 Abstract How should robotic or prosthetic arms be programmed to move? Copying human smooth movements is popular in synthetic systems, but what does this really achieve? We cannot address these biomimetic issues without a deep understanding of why natural movements are so stereotyped. In this article, we distinguish between ‘functional’ and ‘aesthetic’ biomimetics. Functional biomimetics requires insight into the problem that nature has solved and recognition that a similar problem exists in the synthetic system. In aesthetic biomimetics, nature is copied for its own sake and no insight is needed. We examine the popular minimum jerk (MJ) model that has often been used to generate smooth human-like point-to-point movements in synthetic arms. The MJ model was originally justified as maximizing ‘smoothness’; however, it is also the limiting optimal trajectory for a wide range of cost functions for brief movements, including the minimum variance (MV) model, where smoothness is a by-product of optimizing the speed–accuracy trade-off imposed by proportional noise (PN: signal-dependent noise with the standard deviation proportional to mean). PN is unlikely to be dominant in synthetic systems, and the control objectives of natural movements (speed and accuracy) would not be optimized in synthetic systems by human-like movements. Thus, employing MJ or MV controllers in robotic arms is just aesthetic biomimetics. For prosthetic arms, the goal is aesthetic by definition, but it is still crucial to recognize that MV trajectories and PN are deeply embedded in the human motor system. Thus, PN arises at the neural level, as a recruitment strategy of motor units and probably optimizes motor neuron noise. Human reaching is under continuous adaptive control. For prosthetic devices that do not have this natural architecture, natural plasticity would drive the system towards unnatural movements. We propose that a truly neuromorphic system with parallel force generators (muscle fibres) and noisy drivers (motor neurons) would permit plasticity to adapt the control of a prosthetic limb towards human-like movement. (Some figures in this article are in colour only in the electronic version)

(reversible adhesion). He also recognized that the problem was transferable to synthetic (human-made) systems, and that Velcro might be an improvement over zip fasteners. Thus, ‘functional’ biomimetics is not only transferring nature’s solution to a synthetic system, but also recognizing the problem that nature has solved, and recognizing that a similar problem exists in a synthetic system. In contrast, in some types of biomimetics, nature is copied for its own sake. We call this

Introduction When George de Mestral observed how thistle burrs stuck to his clothing, he was inspired by nature’s way of creating reversible adhesion with miniature hooks. He was probably not so interested in the advantage to burdock genes of dispersing seeds via passing furry animals, but he was able to recognize the immediate problem that nature solved 1748-3182/09/033001+07$30.00

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Bioinsp. Biomim. 4 (2009) 033001

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generate the same mean firing rate with many different spike patterns. Yet, neurophysiological experiments have revealed that motor neuron signals are also highly organized and stereotyped. Notably, individual motor neurons fire stochastically with inter-spike interval distributions having low coefficients of variability (Clamann 1969), and motor units are recruited orderly with increasing strength (size principle, Henneman 1967). Thus, in spite of the almost infinite number of ways of generating a specified force, nature is restrained (or constrained), implying a lawful control objective is being optimized as well. In robotics, this neurophysiological stereotypicity may seem irrelevant for conventional motors and controllers. However, a linkage between neurophysiological stereotypicity and kinematic stereotypicity is now emerging.

‘aesthetic’ biomimetics. Aesthetic mimicry is not without value: it is the fundamental goal of prosthetics, but the value lies in mimicking the appearance or function without consideration of the problem that is being solved by nature. Recognizing the distinction between functional and aesthetic biomimetics not only allows for clarity of purpose, but also forces us to address some fundamental questions about the kinds of problems confronting natural and synthetic systems. Nature solves complex real-world problems by maximizing fitness via natural selection. We see the end product of this optimization in biological structure and function, and if we are lucky (as de Mestral was) we may be able to infer nature’s problem that led to the observed solution (the real bio-inspiration). However, we seldom see the details of the problem confronting nature, particularly the constraints. We may believe we have found a novel solution to a problem, but it may not be applicable to synthetic systems because of differing constraints. The mimicry of human movements in robots is an archetypal example. The grace and smoothness of skilled human movements has inspired many in the field of robotics to program their machines to mimic human movement, but why? What problem in nature does ‘smoothness’ of movement solve? Unlike burdock burrs, the answer is not obvious. Does nature’s problem exist in synthetic systems? If so, does it solve a problem in robots? Or, like toy robots, is this aesthetic biomimetics? In this article, we focus on point-to-point arm reaching movements to explore these issues. They are not straightforward, and perhaps surprisingly, our ‘answers’ are only now emerging and depend on the type of noise perturbing the motor system, and the robot’s core electronic and motor architecture.

The minimum jerk trajectory The cost function that has been very influential in the study of human movement is the MJ model, first proposed by Hogan (1984). There is considerable cross over to robotics (a Google search of ‘minimum jerk’ and ‘robotics’ listed 3900 hits). The MJ trajectory is a member of the family of minimum square derivatives (MSD) that maximize ‘smoothness’ by minimizing the integrated square of a high-order derivative (Harris 1998). Consider a one-dimensional movement in which the position of the effector (e.g. arm) at time, t, is denoted by y(t). Assume the task is to move the arm from the origin (y(0) = 0) to an end position y(T) = A, such that the duration of the movement is T. We assume that the arm starts from rest, and ends at rest. There are many feasible trajectories, and we assume that each one yields a positive scalar cost to the organism, J. The MSD family of trajectories minimize the cost functional T k 2 d y J = dt, (1) dt k 0

The degrees of freedom problem and optimal control, cost models

where k is a constant. The optimal trajectory is a polynomial of order 2 k − 1, which can be found by standard variational calculus. The MJ trajectory model corresponds to k = 3, and is a quintic polynomial. For a movement of amplitude A and with initial and final velocity and acceleration set to zero (boundary conditions), the classic MJ trajectory is obtained: 4 5 3 t t t − 15A + 6A . y(t) = 10A T T T

It has long been recognized that limb movement is extremely under-constrained, and a central problem has been understanding how the degrees of freedom (DOF) in a multi-jointed limb are controlled—the ‘degrees-of–freedom problem’ (Bernstein 1967). In spite of the huge potential for different trajectories afforded by so many DOFs, human pointto-point movements tend to be remarkably stereotyped, with straight trajectories and smooth bell-shape velocity profiles (Morasso 1981). Attempts to explain this preference have relied heavily on an optimality approach, in which it is assumed that nature chooses a trajectory because it optimizes some biologically relevant performance index, or cost function (Nelson 1983). The fundamental problem in biology is to find nature’s cost function, and many models have been suggested, including minimum jerk (MJ) model (Hogan 1984, Flash and Hogan 1985), minimum torque change model (Uno et al 1989), minimum variance (MV) model (Harris and Wolpert 1998) and many more (not reviewed here). At the neurophysiological level, there is an explosion of DOFs. For forces below maximum voluntary contraction, the same force could be generated by many different combinations of motor neurons, and each motor neuron could

The velocity profiles of the MJ model are bell-shaped and symmetrical, and independent of duration (dashed lines in figure 1(c)). For rapid movements, the fit to data is impressive, although for longer movements it becomes progressively worse as observed velocity profiles become increasingly skewed. The model can be extended to more dimensions and yields straight trajectories with the MJ profile occurring along the trajectory, similar to empirical observations (Flash and Hogan 1985). From a functional mimetic viewpoint, building MJ trajectories into synthetic systems implies (a) that nature’s problem is maximizing smoothness, and (b) that smoothness is important in synthetic systems—both solved by the MJ model. But, is this really the case? What is special about ‘smoothness’, and what is special about k = 3? 2


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with p(t) denoting the system’s impulse response function (figure 1(a)), or in state-space notation as x˙ (t) = Ax(t) + Bu(t),

(2c)

where x(t) is the 1 × k state vector, u(t)is the scalar control input, A is the k × k state matrix, and B the 1 × k control vector. Now consider a general weighted quadratic cost functional of the form T f (t)u2 (t) dt, (3) J = 0

where the weighting function f (t) is an arbitrary positive finite continuous function with the property that f (t) −−−−→ f0 > 0.

(4)

t→0

Thus, the optimization goal is to minimize the total control input ‘energy’ weighted by the function f (t). Substituting (2a) into (3) and changing the variable τ = t/T , we have 1 J = f (τ T ) 0

ak dk y ak−1 dk−1 y + ak−1 + · · · + a1 y × T k−1 dτ k T k−2 dτ k−1

Figure 1. (A) Simplified lumped motor system. Effector position, y(t), is the output of a linear system (plant) with impulse response function, p(t), driven by the motor command, u(t). (B) Some possible types of wideband noise perturbing the motor command (equation (8)), showing the relationship between instantaneous ¯ The standard deviation σu (t) and mean of motor command u(t). horizontal line shows signal-independent noise, corresponding to z = 0 (e.g. additive zero-mean Gaussian noise). Compressive function shows the effect of renewal noise (RN), z = 0.5, (e.g. a Poisson process). Ascending line shows the effect of proportional noise (PN), z = 1, which has been measured on human isometric force production. (C) (solid lines) The optimal velocity profile for a point-to-point movement that minimizes the output endpoint variance due to PN on the motor command for different movement durations (see the text for details). The plant was modelled as a third-order system with time constants 1.0, 0.1, 0.05 time units (arbitrary). Optimal trajectory for brief movement of duration 0.25 units is quasi-symmetrical, but becomes increasingly skewed with increasing duration. (dashed lines) Minimum jerk velocity profiles for the same durations as above. Note similarity for brief movements (see the text), but velocity profile remains symmetrical for longer movements. All velocity profiles are normalized to unit area.

(5)

which is clearly proportional to the cost function for the MSD family of trajectories. Thus, the MSD family of trajectories also minimize the cost function given by (3) for small T. In particular, for a third-order system (k = 3), the optimal trajectory is the MJ trajectory as duration becomes infinitesimal. We can even consider a time-varying system, where the coefficients a1 , . . . , ak are finite functions of time. Provided, ak (t) −−−−−→ ak (0) > 0, the MSD will still be t→0

optimal. We can also envision many nonlinear systems where the same limit will be reached. Thus the MJ trajectory could be optimal for a completely different cost functional than originally proposed. The optimal solutions have smoothness as a by-product of minimizing input energy in the low-pass system (2) for brief movement durations. Mathematically, we have no way of distinguishing among the infinite number of cost functions that could give rise to MJ. However, as duration increases, lower-order terms (and any nonlinearities) will have an effect and the optimal trajectory will increasingly depart from an MSD trajectory. This is precisely what happens in arm reaching and saccadic eye movements, and indicates that (1) is only a limiting approximation to nature’s cost function. Indeed, without knowing nature’s cost function, mimicking the MJ trajectory can only be aesthetic. We now turn to proportional noise (PN) and the MV model as an alternative biological explanation for smooth movements, which has a cost functional of the form given by (3).

Could the MJ trajectory be optimal for another cost function? If so, then maximizing smoothness per se may not be the control objective of natural movements. Consider a kth-order low-pass (pole only) linear system given by dk y dk−1 y + a + · · · + a1 y = u(t), k−1 dt k dt k−1

dτ.

For brief movements (small T), the higher-order terms in (5) dominate the cost function, and in the limit T k 2 T d y ak f0 f (t)u2 (t) dt −−−−→ = k−1 . dt, (6) t→0 T dt k 0 0

Limiting cost function

ak

2

(2a)

where u(t) is the control input, y(t) is the one-dimensional output trajectory position, and a1 , . . . , ak are finite constants. This system can also be written in the standard form: t y(t) = p(t − t )u(t ) dt , (2b) 0

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Minimum variance model trajectory

Signal-dependent noise

It has been proposed that smoothness is not the control objective of movement control, but the by-product of minimizing the effects of PN on the motor command (Harris 1998, Harris and Wolpert 1998, 2006, Hamilton and Wolpert 2002, Miyamoto et al 2004). Proportional noise is a type of signal-dependent noise (SDN) where the instantaneous standard deviation of signal noise is proportional to the instantaneous signal mean (figure 1(b)). PN has unusual properties when it is transmitted through a linear (low-pass) system. Larger commands carry more noise which increases the output variance and leads to greater endpoint error for point-to-point movements. As a consequence, PN imposes a speed–accuracy trade-off, as moving more quickly requires larger commands and causes more end-point error. Let us return to the task of moving the effector through an amplitude, A, taking time T. Again, we assume that we start at rest and finish at rest but we also assume that the effector must stay at its final position for some period of time, F. Our goal is to minimize the amount of variance over the postmovement period T t T + F . It can be shown that this is equivalent to minimizing the cost functional: T f (t)u2 (t) dt, (7a) J =

The term ‘signal-dependent noise’ is used for stochastic signals where the standard deviation and other moments depend on the signal mean. One type of relationship occurs when the instantaneous standard deviation σu (t) of a stationary signal u(t) is proportional to a power a function of the signal mean: ¯ z, σu = C |u|

where C is a constant (see figure 1(b)). Thus, when z = 0, σu is equal to C and the noise is independent of the signal mean, as typically assumed in additive Gaussian noise. However, when z = 1, σu is proportional to the mean (i.e. PN): ¯ , σu = C |u|

where

T +F

p2 (t − t) dt ,

(9)

and the noise is signal-dependent, as seen in multiplicative noise. The case with z = 0.5 is important as it is the asymptotic relationship for renewal point processes (e.g., Poisson process), which we call ‘renewal noise’ (RN) and C 2 is sometimes called the ‘Fano factor’. Unfortunately, PN and RN can be confused as they are both types of signal-dependent noise. RN is important because the statistics of individual neuronal firing rates can be approximated as a renewal process. However, it is important to distinguish between the statistics of firing rates and inter-spike intervals (ISIs). Consider a stationary renewal process where the ISI distribution has a mean μISI and standard deviation σ ISI. Assume that σ ISI is proportional to μISI with a constant coefficient of variation CISI = σISI /μISI , then it can be shown (Cox and Miller 1965) that over a long counting period, the mean firing rate is u¯ = 1/μISI , and the standard deviation of firing rate is

0

f (t) =

(8)

(7b)

T

¯ 1/2 . σu = CISI |u|

and p(t) is the impulse response function of the system (equation (2b)), (Harris and Wolpert 2006). There is a unique solution (solid lines in figure 1(c)) which provides excellent fits to observed trajectories including the increasing skew with longer durations, when the order of the plant is k = 3 ∼ 4. For small durations the trajectories appear very similar to MJ, as might be expected from the limiting property of (6), but provides a slightly better fit than MJ in the Fourier domain (Harwood et al 1999, Harris and Harwood 2005). The model is also well grounded because PN has been empirically observed in human isometric force generation, at least below 50% maximum voluntary contraction (Schmidt et al 1979, Galganski et al 1993, Enoka et al 1999, Slifkin and Newell 1999, Laidlaw et al 2000, Jones et al 2002, Hamilton et al 2004). Thus, we believe that we have found nature’s problem—moving as quickly and as accurately as possible, but with the speed–accuracy constraint imposed by PN on the motor command. The solution is the MV trajectory, and smoothness emerges as a by-product of this optimization. We are beginning to see MV controllers emerge in robotics (Simmons and Demiris 2004), but what problem does this solve in the robot? Moving quickly and accurately seem reasonable objectives in the robot, although we hardly need to be inspired by nature to value these goals. The synthetic solution, however, depends on the type of noise in the synthetic system. Different kinds of noise lead to different optimal solutions, which we next show using optimal control theory.

(10)

Thus, proportionality on the ISI moments leads to z = 0.5, and does not translate to PN on firing rates. As a wellknown example, if the ISI distribution is exponential, then CISI = 1, corresponding to a Poisson process. Recordings from actual motor neurons indicate a roughly Gaussian ISI distribution with CISI ≈ 0.2—far from exponential (Clamann 1969). The CISI was originally viewed as being constant for any given motor neuron, but recently it has been claimed that CISI decreases with mean firing rate (Moritz et al 2005), indicating that z < 0.5. With such a low CISI, the statistics of firing rate depend strongly on the counting interval and are considerably more complex than the asymptotic relationship (10). Even taking this into account, a single renewal process cannot produce PN (Harris 2002). Optimal control The exponent z modifies the MV cost functional to J = T 2z 0 f (t)|u(t)| dt and leads to different optimal trajectories depending on z. To see this we can use optimal control theory and Pontryagin’s minimum principle (Bryson and Ho 1975). Using state space notation, x˙ (t) = Ax(t) + Bu(t), the MV Hamiltonian is H (x, u, t)) = f (t) |u(t)|2z + ΛT (t) [Ax(t) + Bu(t)] , (11) where ΛT (t) is the transpose of the co-state vector (1 × k vector of time-dependent Lagrange multipliers). From 4


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Pontryagin’s minimum principle, the optimal control u(t)∗ minimizes H (x, u, t)with respect to u. For z > 0.5, u(t)∗ will always be finite and is found by simultaneously solving the ˙ = −∂H /∂x, and when z = 1 (PN) co-state equation: Λ the solution is the MV model (Harris and Wolpert 2006). However, for z 0.5, the minimum of H (x, u, t)is either unbounded and u(t)∗ → ±∞, or u(t)∗ = 0 depending on z and the co-state vector. This is called ‘bang-bang’ or ‘bangoff-bang’ control. This does not fit observed trajectories as we have shown (Harris 1998). Thus PN and the MV model cannot be explained as optimizing RN noise, nor (we deduce) noise arising from an individual motor neuron. In other words, if our robot motor is driven by a command signal which is perturbed by signal-independent noise (z = 0), or renewal noise (z = 0.5) then speed and accuracy are simultaneously maximized by bang-bang control, and there would be no functional point in mimicking the MV (or MJ) trajectory. Should we mimic PN? To answer this, we need to understand where PN originates in the natural system. Stein et al (2005) have shown that the cortical motor command follows the RN rule (z = 0.5), so consequently PN must emerge downstream. There is now evidence that its source lies at the neurophysiological level of motor unit recruitment.

Figure 2. Schematic to illustrate different optimal strategies depending on the intrinsic noise on the motor command. (A) For proportional noise (PN) on the motor command, moving quickly leads to more inaccuracy, due to the larger motor command. It is not possible to fast and accurate (forbidden zone). Minimum variance (MV) trajectories lie on the optimal curve (curve). (B) For additive signal-independent noise on the motor command, accuracy depends only on the movement duration and improved by making the movement as brief as possible. The optimal trajectory is given by bang-bang (BB) control (point).

Motor unit recruitment In biological motor control, muscle force is increased not only by raising firing rates of motor neurons, but also by augmenting the number of motor neurons actively recruited, and it is now well established that as more muscle force is generated stronger motor units are recruited (size principle, Henneman 1967). In an elegant experiment, Jones et al (2002) artificially stimulated human grip force and showed that the standard deviation of grip force remained roughly constant (z = 0). Whereas, when the same range of grip forces was generated voluntarily and naturally, the standard deviation of the noise increased linearly with mean force (z ∼ 1). They concluded that the PN must have originated in the recruitment strategy. Recently, we have shown that PN may itself be an optimal recruitment strategy for maximizing the number of signals that can be transmitted through a network of thresholded noisy binary neurons (Harris 2008). Consider a pool of motor neurons in which each unit is driven by a common motor command but has potentially a different threshold, so that when the command exceeds a unit’s threshold it is switched on and generates a firing pattern with a fixed mean rate and RN noise (10). The problem we have posed is how to organize the units so that the pool can transmit as many different steadystate signals as possible (for a given probability of error per signal). We have shown that the optimal arrangement staggers the units’ thresholds with weakest units recruited first and stronger units recruited for large forces. Remarkably, the size principle emerges and the output noise is PN (Harris 2008).

off occurs because of the properties of PN and would not occur if the motor command were perturbed by constant or renewal noise. Let us put the argument in reverse. We start with a network of units (motor neurons), which are each inherently noisy with adjustable thresholds, and use them to drive a linear system for motor control. After optimizing the network to minimize motor output variance, the end result would be the emergence of PN. In turn, PN tightly constrains what movements are allowed and forbidden (figure 2(a)). Movements cannot be simultaneously fast and accurate, and the optimal trade-off is given by smooth trajectories given by the MV model, which for short durations would be asymptotically close to MJ trajectories (for a thirdorder system). We now have some insight into why human movements are so stereotyped. Our next question is whether a similar problem exists in synthetic systems? Clearly, building robot arms that are fast and accurate are reasonable objectives, but the crucial constraint of PN on the motor command is not typical of synthetic systems. A robot arm with nominal signaldependent additive Gaussian noise would not have a speed– accuracy trade-off. End-point error would be minimized by keeping the duration as small as possible (less time to accumulate variance), and so speed and accuracy would be simultaneously maximized by bang-bang control (sequentially switching the motor command between its maximum and minimum limits) (figure 2(b)).

Discussion Based on current evidence and theory, smooth trajectories optimize a trade-off between speed and accuracy. This trade5


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the natural system. One way is via a truly neuromorphic device with parallel force channels and noisy drivers. This is, in principle, possible, although not yet technically available. Another way is to simulate the natural Hamiltonian in the human–machine interface, so that the human nervous system ‘sees’ a natural limb.

In reality, robot arms and their controllers are more complex, and it may be desirable to limit high derivatives to reduce unwanted oscillations or wear and tear, but this does not lead to the MV trajectories. It is also possible that some degree of PN may emerge in controllers that are based on internal models of the arm dynamics (such as real-time inverse dynamic controllers). However, the Hamiltonians of such systems would need to be calculated case by case. We conclude, therefore, that there is no functional reason for mimicking MV (or MJ) trajectories in conventional robots (unless it can be shown that PN is the dominant noise source). This does not preclude a role for functional biomimetics in robotics. We may be inspired by how nature solves control problems, but when we apply such controllers to conventional robots we would not, in general, expect natural-appearing behaviour.

Conclusion The broad field of artificial devices inspired by nature is growing at a very fast rate, but it is easy to lose sight of what we mean by bio-inspiration and biomimetics. We have distinguished between functional and aesthetic biomimetics. In functional biomimetics we need to look at the biology. As shown here, this can be a difficult task, but it can give us insight into both nature’s underlying problem and its means of solution, which then can truly inspire us in synthetic systems. When we do not know the problem that nature is solving, we are essentially copying nature for its own sake—aesthetically. Aesthetic biomimetics may also be valuable, as in prosthetics, but it is essential to recognize the different engineering goals between functional and aesthetic biomimetics.

Prosthetics and neuromorphic muscle Mimicking the appearance and the behaviour (function) of missing body parts is the goal of prosthetics, and philosophically it is aesthetic rather than functional biomimetics. Clearly building a device that appears natural and moves in a human-like way is valuable, but if we want the device to dovetail into the human nervous system, we must understand the control objectives of the natural system (just as we did for functional biomimetics above). Human limb movement trajectories are under constant adaptive control (motor learning). Experiments with exogenous perturbations (loading, force fields) induce a period of motor learning in which the perturbed trajectories are gradually adapted until a new steady state is reached, often similar to the original steady state (Shadmehr and MussaIvaldi 1994, Brashers-Krug et al 1996). These results can be interpreted as an adaptive controller that modifies motor input to meet a control objective. We would expect this plasticity to also occur in the control of an artificial limb, so that with practice, the user’s adaptive control would (unconsciously) modify the trajectory towards some steady state. In general, this steady state would not appear to be natural, but would reflect the optimal solution given by the Hamiltonian of the system, which depends on the intrinsic cost function of the task, the constraints laid down by the possible motor commands and the architecture of the prosthesis. To illustrate, consider two extreme hypothetical scenarios in a reaching task. First, consider an artificial limb whose overall transfer function (for a simple reaching task) is linear and perturbed with additive Gaussian noise. Adaptive control would attempt to minimize error and duration simultaneously, which would be achieved by bang-bang control (figure 2(b)). In general, this behaviour would not be human-like. Now consider a truly neuromorphic prosthesis with multiple parallel force generators (artificial muscle fibres) and noisy thresholded drivers (artificial motor neurons) that obey the size principle. The user’s adaptive control would seek to minimize error and duration, which are in conflict, and it would find the MV solution, which would be human-like (figure 2(a)). Thus we need to mimic the Hamiltonian of

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