ROBOTICS
4.3 ENGINEERING APPLICATIONS
4.3.2 Muscle Activation Modeling for Human Limb Prosthesis and
4.3.2.6 Application to Muscle Dynamic System Parameter Identification
In this section the adaptive method is coupled with the UKF developed by Julier and Uhlmann [65] and used to estimate the parameters of a bilinear model of a binary muscle-activation system that belongs to the Weiner type of block-oriented models. The details of the adaptive Kal-man filter implementation may be found in the work of Vepa and Zhahir [83].
The adaptive UKF is applied to the problem of identification of a typical flexor–extensor muscle pair. A typical set of parameters was assumed for the muscle–limb dynamics and the
(4.11) Lt(t)=Lt0
1+
I
i=1
cie−t/τi
1+
I
i=1
ci
.
(4.12) dLt
dt = −
Lt−Lt0 1−
1−τσ
τε
τε +wt,
state response simulated by numerical integra-tion of the equaintegra-tions in MATLAB. The assumed set of parameters is listed in Table 4.1. Table 4.2 lists the initial values of the states, that were estimated by the adaptive UKF estimator. Of these states, the last seven are assumed to be constants. All the equations are assembled and expressed in discrete form to facilitate the appli-cation of the adaptive UKF.
In the first instance, the UKF was imple-mented without any adaptation of the process noise or measurement noise covariance matri-ces. The first activation inputs used to determine the limb response over a 10-second timeframe corresponding to 10,000 time steps are obtained from real measured EMG data available on the Web [84]. The subject did a protocol of 30 seconds water cycling in a 100 bpm cadence. The second TABLE 4.2 Typical initial values assumed for the muscle and limb dynamic states.
State Description (Units) Initial Value
qf Flexor muscle activation state 0
qe Extensor muscle activation state 0
α Limb rotation angle (rad) 0.2
ω Limb rotation rate (rad/s) 0
Ltf Flexor muscle tendon length (m) 0.1
Lte Extensor muscle tendon length (m) 0.1
df Flexor muscle force moment arm (m) 0.3
de Extensor muscle force moment arm (m) 0.25
dp Proximal distance of muscle origin to joint (m) 0.2 dd Distal distance of muscle insertion point to joint (m) 0.3
Fmax Maximum muscle force (N) 0.01
Ltf0 Flexor muscle initial tendon length (m) 0.1
Lte0 Extensor muscle initial tendon length (m) 0.1
TABLE 4.1 Definitions and typical values assumed for the muscle and limb parameters.
Parameter Description (Units) Value
β Fraction of activation state 0
τact Activation time constant (s) 0.015
Be Coefficient of viscous damping in limb (Kg.m2.s) 1 × 10−04
Ic Moment of inertia of limb (Kg.m2) 2 × 10−05
τε Relaxation time for constant strain (s) 0.1
τσ Relaxation time for constant stress (s) 0.077
rte Ratio of final to initial extensor tendon length 1.3
mlc Limb unbalance (N.m) 0.005
αs Reference limb rotation angle π/2
100 4. BIOMIMETIC ROBOTICS
activation input was scaled white noise. The estimated and simulated responses of the two activation states, the limb rotation, and the rota-tion rate are shown in Figure 4.9.
The case of the adaptive UKF-based state esti-mation and parameter identification, where the process noise covariance is continuously updated, was also considered. A key difference
between the results obtained by the use of the UKF and the adaptive UKF is in the estimates of the initial length of the tendon. For this reason a nonlinear model of the tendon length dynam-ics was also employed. The advantage of using the nonlinear model is that the tendon could not only be allowed to be slack in its rest state, but one is also allowed to incorporate nonlinearities
0 2000 4000 6000 8000 10000
−0.05 0 0.05
Act. State(1)
Muscle States Compared
0 2000 4000 6000 8000 10000
−0.06
−0.04
−0.02 0 0.02
Act. State(2)
0 2000 4000 6000 8000 10000
−0.5 0 0.5
Limb rotation
Simulated Estmtd.
0 2000 4000 6000 8000 10000
−4
−2 0 2
No. of time steps
Rotation rate
Simulated Estmtd.
Simulated Estmtd.
Simulated Estmtd.
Simulated Estmtd.
Simulated Estmtd.
FIGURE 4.9 UKF estimated and simulated responses of the two activation states, the limb rotation, and the limb rotation rate.
such as dead band and hysteresis. However, both tendons cannot be allowed to be slack simultaneously if the limb is to maintain its ori-entation. When the nonlinear model was used only for the tendon that was less stressed, the results resembled the case of the UKF without adaptation of the process covariance.
In this section the successful application of the UKF and the adaptive UKF to the identifica-tion of model parameters of a typical nonlinear muscle-limb dynamic system is illustrated. It has been shown by example that UKF-based and adaptive UKF–based state estimation and parameter identification are an option that is particularly well suited for the dynamic systems associated with muscle-limb interactions.
Although the generic case of a limb actuated by a pair of antagonistic muscles was considered, the identification method could be fine-tuned and applied to any specific pair of antagonistic muscles actuating a particular limb. The method is currently being employed to design nonlinear control laws that can be used to control smart prosthetic limbs.
An adaptive UKF is used to estimate the unmeasurable state variables and kinetic param-eters of the muscle-limb model. Although the UKF has a simple structure, the tuning of these estimators is a relatively difficult task. The use of the adaptive approach eliminates the need for the tuning of the covariance parameters of the UKF estimator. However, the estimates of the process covariance matrices obtained tively can vary widely depending on the adap-tation scheme adopted. For this reason, the adaptive algorithm is recommended to be used only in the initial stages as a tuning method.
One of the features of the parameter-identi-fication problem considered here is the pres-ence of internal parameters in the definition of the applied moment. These internal parame-ters, such as the distances to the two muscle distal and proximal anchor points from the center of rotation, dd and dp, constitute a set of weakly identifiable parameters since they may
in some cases be inaccessible for observation.
Although formal identifiability analysis of the model has not been carried out, the problem of weakly identifiable parameters has been dealt with by the use of optimal state estimation of nonlinear dynamic systems. Moreover, physi-cally meaningful, coupled models of the inter-nal tendons’ length dynamics facilitate the identification of the weakly identifiable parameters.