The Stark model, as outlined in [26, 35], forms a good compromise between com- plexity and accuracy, and is firmly established in the literature as a valuable tool for theoretical investigations of the human motor system. As shown in Figure 12 and Table I, it consists of an antagonistic pair of muscles (flexor and extensor), which act symmetrically on their common load (inertia Jp, viscosity Bp and elasticity Kp). Each muscle is constructed from the classic Hill-model: a single contractile element, a parallel viscosity that incorporates the well-known nonlinear force- velocity relationship ((F +a)(v +b) = CONSTANT), and a nonlinear series elasticity (F ∝exp(k.1x)).?
Since the Stark model is highly nonlinear and its mathematical description is not well defined, it is quite difficult to use it for simulations of a complete control system. Some modifications were considered necessary (Table I). These do not al- ter the core of the model, but just expand it for cases in which the original becomes unrealistic.
The original model collapses when the muscle activation (HTL, HTR) appro- aches zero. Instead of considering a minimum muscle coactivation as in [26], we produced a simplified form of the original equations for the case of small activation by ignoring terms that at that point become too small. Such an elimination of terms leads to two possible choices for Fsl andFsr: either set them to zero (due to Equation (A.1)0of Table I), or define them from Equation (A.5). We choose to set it equal to HTL or HTR, which is equivalent to ignoring only the first term of (A.1)0 related to HTL/HTR, that decreases faster than the second. The switching between the two operating regions is smoothened by a sigmoid function.
? In [26, 35] the feedback through the spindles is also modeled. Since this requires an extra couple of neural input signals, we omitted it for simplicity.
Table I. Stark model for the human arm – state equations Modified model:
˙ XL=
Bh(HTL−Fsl) 0.25 HTL+Fsl ãσL+
v+k2(HTLHTL˙ +1)
(1−σL) if HTL>thres_HTL, fx1(θ )−XL
τL if HTL<thres_HTL,
(A.1a)
˙ XR=
Bh(HTR−Fsr)
0.25 HTR+Fsr ãσR+
v+k2(HTRHTR˙ +1)
(1−σR) if HTR>thres_HTR, fx1(θ )−XR
τR if HTR<thres_HTR,
(A.1b)
˙
v=Jp−1(−Bpv−Kpθ+Fe+Fsl−Fsr), (A.2)
θ˙=v, (A.3)
with σL=
1+e−1.5((Bh(HTL−Fsl)/(0.25 HTL+Fsl))−4.5dXlo)−1
, (A.4a)
σR=
1+e−1.5((Bh(HTR−Fsr)/(0.25 HTR+Fsr))−4.5dXro) −1
, (A.4b)
Fsl=max
0, k1(ek2(XL−θ )−1)
, Fsr=max
0, k1(ek2(XR−θ )−1)
, (A.5a,b)
fXL(θ )=fXR(θ )=θ−0.1(θ−Xlo)2, (A.6)
HTL˙ =NL−HTL
T , H˙TR=NR−HTR
T . (A.7)
The original Stark model comprises of Equations (A.1a,b)0 below and Equations (A.2), (A.3), (A.5) and (A.7).
1.25HTL
Bh+ | ˙XL|X˙L=HTL−Fsl, 1.25HTR
Bh+ | ˙XR|X˙R=HTR−Fsr, (A.1a,b)0 where: the subscripts 1 and r denote left and right muscle, θ position and v velocity of the arm; XL and XR internal model variables; Kp, Jp and Bp, passive arm parameters, Bh, T , k1, k2, Xlo, dXlo, Xro, dXro, thres_HTL and thres_HTR are constants; NL and NR the neural input; HTL and HTR activation levels (“hypothetical tension”); Fsl/Fsr the left/right muscle’s force; andFean external force. The original papers also use the notation:
BVL=BL(VL)= 1.25HTL Bh+ | ˙XL|, BVR=BR(VR)= 1.25HTR
Bh+ | ˙XR|.
Figure 13. Stark muscle model. Final arm position for various 1st agonist (AG1) and an- tagonist (ANT) pulses’ heights. The 2nd agonist pulse is calculated to provide “clamping”
[35].
Also, in the original model XL and XR are not updated when the muscle re- ceives no input, unless the non natural hypothesis of a negative neural input holds.
Thus subsequent activations of the muscle lead to increasingly high forces. In the modified scheme,XL/XRis now updated when the muscle is inactive, taking into account the natural movement of the muscle inside a moving arm and mimicking a length-tension curve. This allows us to also drop the “absolute operation” from Equation (A.1)0. The result of these modifications is a smooth and realistic arm trajectory. In Figure 13 the final arm position predicted by the model for various ex- citations is illustrated. Matlab’s Simulink (©The MathWorks Inc., 1984–1994) files for the modified Stark model can be found inhttp://www.robotics.ntua.gr/
∼pplaton.
An inverse of the human arm model is of crucial importance, since it can be used to estimate the neural input when its measurement is obscure, and is needed in simulation. Stark et al. [5] have proposed an iterative inversion method, which however would be not easy to implement on-line. We have developed a family of non-iterative algorithms for the modified model. These are based on first identify- ing the operation region (Table I) for the current sample, and then inverting the local model. Periodical fine tuning of the prediction can be achieved by calculating the model output from the estimated input. Our fastest inverting algorithm requires the measurement of the relative activation of the muscles, which is easily achievable with EMG. This is due to the fact that at the beginning of the antagonist and the sec- ond agonist pulse both muscles are active, so that the inversion problem becomes ill-defined. To raise the ambiguity we have to measure the relative activation, or assume that the force of the muscle that is being switched off is decaying in a predictable (e.g., exponential) way, or resort to iterative methods. In general, our
algorithms are successful for output trajectories previously generated by the model, but often fail when just a target point is specified. This is because not all combi- nations of possible arm forces’ trajectories can be generated by the model. This would not be a problem for an experiment, since the inverse model will use as input a natural trajectory. Partially successful routines that produce a human-like output force were also devised. Because of this problem, for simulations we specifically calculated the 2nd agonist (AG2) pulse so as to stabilize the arm at its current position (when AG2 starts). This is in accord with physiological evidence that it is a “clamping” pulse [35]. To choose suitable 1st agonist (AG1) and antagonist (ANT) pulses, we used a look-up table (Figure 13).
Acknowledgement
This research has been conducted within the framework of the MobiNet EU-TMR Research Network.
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