**3.3. Model validation**

Model validation is possibly the most important step in the modelling process. The model and the optimised parameters obtained from the optimization process were validated in terms of consistency and the prediction error. Two different approaches of model validity are conducted. Different sets of experimental conditions are usually required to define the domain of a model's intended applicability. Therefore, in the first validation process, the model is validated with a different set of data than the training data. To avoid any change in the actual plant, the validation data is obtained in the same experimental arrangement but with a different initial knee joint angle. The responses of the optimised model and new experimental data are shown in Figure 10. It is noted that the two agree closely with one another.

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 333

**Figure 9.** 3D Fuzzy surface of normalised viscoelastic moment

332 Viscoelasticity – From Theory to Biological Applications

**3.1. Optimised value of anthropometric inertia parameters** 

parameters been validated through repeated process of optimisation.

Foot Mass 0.95 kg Shank Mass 3.5 kg Moment of Inertia 0.35823Nm2 Position of COM of Foot 0.035 m Position of COM of Shank 0.22 m

**Table 4.** Optimised Value ofAnthropometric Inertia Parameters

**3.2. Optimised Fuzzy Model as Viscoelasticity** 

near extended knee joint.

**3.3. Model validation** 

another.

Parameter Optimised Value

The optimized value of the anthropometric inertia parameters such masses, moment of inertia about COM and positions of COM along the segmentals length of the subject's lower limb are tabulated in Table 4. The accuracy of these optimised anthropometric inertia

The vicoelasticity is represented by a fuzzy model. The GA optimization is used to optimise 48 parameters including 30 associated with the membership function, 15 weights associated with the minimized fuzzy rules and 3 scaling factors of fuzzy model. The scaling factor for the normalization and denormalization of two inputs and output were 0.0091247, 0.0053982 and 35 respectively. The fuzzy model takes into account the nonlinear component of passive viscoelasticity. A three-dimensional plot that represents the mapping from knee angle and knee angular velocity to viscoelastic moment is shown in Figure 9. This surface plot shows the normalised viscoelasticity (unit less) changes as a function of the normalised knee angle (unit less) and normalised velocity (unit less). The presence of the non-linearities in the viscoelasticity can be noted on this uneven surface shape from both angles. The valley for knee angle between 0.5 and 1 gives high viscoelasticity could be due to the high stiffness

Model validation is possibly the most important step in the modelling process. The model and the optimised parameters obtained from the optimization process were validated in terms of consistency and the prediction error. Two different approaches of model validity are conducted. Different sets of experimental conditions are usually required to define the domain of a model's intended applicability. Therefore, in the first validation process, the model is validated with a different set of data than the training data. To avoid any change in the actual plant, the validation data is obtained in the same experimental arrangement but with a different initial knee joint angle. The responses of the optimised model and new experimental data are shown in Figure 10. It is noted that the two agree closely with one

**Figure 10.** Result of first validation

Second, the optimised segmental masses are validated by repeating the same optimization process for further four times. The five sets of the optimised parameters emanating from five different runs of the GA routines are shown in Table 5. The results of the optimised anthropometric inertia parameters from different simulation runs exhibit acceptable repeatability with only a slight difference between each other with small standard deviation. Therefore, it can be concluded that the optimised masses obtained are valid.

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 335

anthropometric inertia parameters of the lower limb such as foot mass, shank mass, moment of inertia about COM and positions of COM along the segmental length of the lower limb have been optimized between the given ranges to obtain the accurate value of the equations of motion. These optimizations have been performed simultaneously using GA with the objective to minimize the error between prediction model and the experimental data. Each person has a unique composition of muscle tissue in their body therefore this passive

*Department of Automatic Control and System Engineering, University of Sheffield, United Kingdom* 

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**Author details** 

**6. References** 


**Table 5.** Five sets of the optimised parameters
