**2. Material and methods**

The passive behaviour of the knee joint depends on the knee joint elastic moment ( *M*<sup>s</sup> ) and the viscous moment ( *M*<sup>d</sup> ). Considering the inertial ( *M*<sup>i</sup> ) and gravitational ( *M*<sup>g</sup> ) moments, a moment balance equation can be written as (Ferrarin and Pedotti, 2000):

$$M\_i = M\_\text{g} + M\_\text{s} + M\_\text{d} \tag{1}$$

or

322 Viscoelasticity – From Theory to Biological Applications

especially during swing phase (Doane and Quesada, 2006).

spastic or normal legs (Lin and Rymer, 1991).

**2. Material and methods** 

Segment mass of human body is an elementary inertial parameter for kinetic analyses of human motion. Many methods exist to estimate body segment properties. In the past, the most popular approach used to estimate segment parameters has been based on data obtained from elderly male cadavers. This database is quite limited in that a small number of cadavers have been studied. Thus, the database for making inertial estimates is not representative of the subjects often under investigation in many exercise and sport biomechanics studies. Dempster (1956) addressed the problem of mass model using cadaveric studies to establish segmental masses expressed as a percentage of total body mass (Winter, 1990). Other techniques have been developed in which inertial properties are directly measured for an individual. Zatsiorsky and Seluyanov (1985) used gamma mass scanning as a means of quantifying mass distribution in analyzing human body segment inertial characteristics. Both of these methods were promising and widely used but gave different measurement of the segment mass since calculated properties can vary drastically depending on the method used. Segment properties can significantly affect such variables,

The pendulum test of Wartenberg is a technique commonly applied to evaluate passive properties in which the leg is allowed to drop from an initially extended position under the influence of gravity and then allowed to oscillate freely (Wartenberg, 1951). The test is very attractive in that it requires no special equipment and is very simple and the oscillatory movements of the lower leg recorded at the knee joint, are captured with electrogoniometers (Bajd and Vodovnik, 1984; Le Cavorzin et al., 2001), uniplanar video-based methods (Jamshidi and Smith, 1998) and 3D motion analysis systems (White et al., 2007). Most of the analyses of the pendulum motion depend on a second-order linear model to extract the elastic and viscous moments from the recorded leg oscillations. However a second order linear model does not provide an adequate description of the motion for either

In this paper a new approach for estimating passive properties of the paraplegic's knee joint based on pendulum test is described. On the basis of these experimental and optimization results, a non-linear fuzzy model is proposed which can be used to estimate the passive viscoelastic knee joint moment as a function of knee angle and knee velocity. The model of a dynamic system of the lower limb is derived using Kane's equations (Josephs and Huston, 2002) with accessibility to estimate of the foot mass, shank mass, moment of inertia about

The passive behaviour of the knee joint depends on the knee joint elastic moment ( *M*<sup>s</sup> ) and the viscous moment ( *M*<sup>d</sup> ). Considering the inertial ( *M*<sup>i</sup> ) and gravitational ( *M*<sup>g</sup> ) moments,

*M*igsd *MMM* (1)

COM and position of COM along the segmental length of the subject.

a moment balance equation can be written as (Ferrarin and Pedotti, 2000):

$$M\_i - M\_g = M\_s + M\_d \tag{2}$$

In this research the *M*<sup>i</sup> *Mg* represented by equation of motion for dynamic model of the lower limb and *M*s d *M* represented by fuzzy model as viscoelasticity. The subject was a 48 year-old T2&T3 incomplete paraplegic male with 20 years post-injury with height = 173cm and weight = 80kg. Informed consent was obtained from the subject.

In this section, first the procedure to perform the pendulum test is presented to get the experimental data. Second, the equations of motion for dynamic model of the lower limb are introduced. Third, the estimations of anthropometric inertia parameters lower limb model are described briefly. Forth, the optimization of fuzzy model as passive viscoelasticity is outlined. Lastly new method for estimation and optimization of passive properties using GA by comparing with experimental data is elaborated. The procedure for estimation of the anthropometric inertia parameters and optimisation of FIS as passive viscoelasticity of the knee joint model are shown in Figure 1.

#### **2.1. Pendulum test**

Pendulum test can be used to evaluate passive properties such as viscosity and elasticity moments of the knee. Viscoelasticity is combination of elasticity and viscosity and represents passive resistances to joint motion associated with the structural properties of the joint tissue and of the muscular-tendon complex. Elasticity can be considered as an intrinsic property of the tissue to resist deformation, while viscosity is related to cohesive forces between adjacent layers of tissues. Both parameters may influence the joint range of motion affecting knee angle (Valle et. al, 2006). A genetic optimization algorithm is used to identify the unknown viscoelasticity by minimizing the error between the data obtained experimentally and from the simulation model. The pendulum test was conducted to measure the passive knee motion of an SCI patient. The subject sat on a chair, which allowed the lower leg to swing freely, while ankle joint was monitored to be at 0°. Reflexive or voluntary activation of muscles acting on the knee occurred during the pendulum test has been monitored to avoid the influence of pendulum movements.

In the pendulum test the knee was slowly extended, by having the experimenter lift it with minimal acceleration at the starting position (1) and then it was released as shown in Figure 3. The knee angle was recorded using electro-goniometer until the final position (2). A Biometric electro-goniometer was used to measure knee movements with sampling time of 0.05s. The electro-goniometer arrangement is shown in Figure 4.

#### **2.2. Equations of motion for dynamic model of the lower limb**

The inertial ( *M*<sup>i</sup> ) and gravitational ( *M*<sup>g</sup> ) moments are represented by mathematical model of a dynamic system of the lower limb based on Kane's equations as follows:-

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 325

**Figure 2.** Limb oscillation during pendulum test (Valle et. al, 2008)

**Figure 3.** Electro-goniometer arrangement for pendulum test

=knee acceleration, 2

*r* = position of COM along the shank, 2

where,

1

1 

=knee velocity, 1

2 2 2

=ankle angle *g* =gravity.

 

*r* = position of COM along the foot, 1

(3)

   

=knee angle,

2 2 1 2 1 11 1 1 2 1 2 11 1 2 2 1 cos cos *M M mq r mg r I mg q mr mq i g*

*m*<sup>1</sup> = shank mass, *m*<sup>2</sup> =foot mass, 1*I* = moment of inertia about COM, 2 *q* = shank length

The lower limb model with the angles sign convention is shown in Figure 4. Anthropometric measurements of length of the lower limb were made as shown in Table 1.

**Figure 1.** Optimization and Estimation Procedure

**Figure 2.** Limb oscillation during pendulum test (Valle et. al, 2008)

**Figure 3.** Electro-goniometer arrangement for pendulum test

$$M\_i - M\_\chi = -m\_2 q\_2 \dot{\theta}\_1^2 r\_2 + m\_1 g \cos \theta\_1 r\_1 - I\_1 \ddot{\theta}\_1 + m\_2 g \cos \theta\_1 q\_2 - m\_1 r\_1^2 \ddot{\theta}\_1 - m\_2 q\_2^2 \ddot{\theta}\_1 \tag{3}$$

where,

324 Viscoelasticity – From Theory to Biological Applications

**Figure 1.** Optimization and Estimation Procedure

*m*<sup>1</sup> = shank mass, *m*<sup>2</sup> =foot mass, 1*I* = moment of inertia about COM, 2 *q* = shank length 1 *r* = position of COM along the shank, 2 *r* = position of COM along the foot, 1 =knee angle, 1 =knee velocity, 1 =knee acceleration, 2 =ankle angle *g* =gravity.

The lower limb model with the angles sign convention is shown in Figure 4. Anthropometric measurements of length of the lower limb were made as shown in Table 1.

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 327

> Dempster's Method

more accurate moment of inertia of the knee, the minimum (0.32 <sup>2</sup> *Nm* ) and maximum (0.58

Vatnsdal et al (2008) addressed the masses and positions of the COM of the lower limb are possible error source in the lower limb model. Dempster (1956) estimated the mass from the total height and weight of the person using standard human dimensions (Winter, 1990). Zatsiorski and Seluyanov (1983) estimated the mass on the basis of regression equations based on statistics obtained from measuring cadavers. However, both of the methods give different measurements for a selected subject as shown in Table 2. Therefore they can potentially introduce large errors, especially while modelling the passive properties, as the pendulum test results may be affected by those parameters. Thus, fine tuning these parameters around the predicted values with an efficient stochastic search algorithm such as GA could lead to good solution with immense potential of extracting accurate subject specific results. GA is used to optimise the foot and shank mass between the two measurements with ±10% tolerance. While the approximated positions of COM along the segment length are assumed as in the centre of the segment length. To get the accurate positions, the position of the COM of shank and foot are optimised between ±10%

Seluyanov's Method

Modelling based on conventional mathematical tools is not well suited for dealing with complex and nonlinear nature systems. By contrast, a fuzzy inference system employing fuzzy 'if–then' rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise quantitative analyses (Dinakaran, 2009). Besides that, the purpose of modeling passive viscoelascity is to be used as a part of the simulation platform and need to integrate with the active properties of the knee joint model for the controller application. Controller such as fuzzy control, neural network does not require the mathematical model of the plant. Therefore fuzzy model are well suited for modeling nonlinear models such as passive viscoelasticity. The passive viscoelastic joint moment is represented as a non-linear function of knee angle and knee angular velocity in a fuzzy model. An automatic Mamdani fuzzy system design method integrates three stages; determines membership functions, the rule-consequent parameters, and scaling factor at the same time. Gaussian membership functions are preferred, because of their continuously differentiable curves and smooth transitions. There are 58 parameters to optimise including

Shank 1.0499kg 1.16kg Foot 3.3973kg 3.72kg

**2.4. Optimization of fuzzy model as passive viscoelasticity** 

<sup>2</sup> *Nm* ) range as reported by M. K. Lebiedowska (2003) is optimized using GA.

*2.3.2. Mass and position of COM of shank and foot* 

Segment Mass Zatsiorski and

**Table 2.** Mass of patient using different methods

tolerances.

**Figure 4.** Lower limb model


**Table 1.** Anthropometric data of subject

## **2.3. Estimation of anthropometric inertia parameters**

The lower limb dynamics is complex and less well defined because the form is not composed of simple geometrical shapes. Indeed, even the location of the mass center of elemental parts is imprecise, and comprehensive analyses of joint kinematics are extremely difficult (Josephs and Huston, 2002). To quantify limb dynamics, accurate estimates are needed of anthropometric inertia parameters (mass, COM location, and moments of inertia). Therefore these equations have accessed to optimise mass, moment of inertia about COM and position of COM along the segmental length of the subject using GA tuning method to fit the experimental data.

## *2.3.1. Moments of Inertia*

Moments of inertia are fundamental parameters describing the mass distribution of body segments which enter into all computations involving segmental rotations. Methods based on the geometrical segment body models and appropriate anthropometric measures were used in measuring the moment of inertia of various segments in cadavers and in living subject (Dempster, 1955; Hatze, 1980; Jensen, 1986; Schneider and Zernicke, 1992). Different procedures were applied to identify the segment moment of inertia from the dynamic equation of motion. There is no unique way of calculating the moment of inertia To obtain more accurate moment of inertia of the knee, the minimum (0.32 <sup>2</sup> *Nm* ) and maximum (0.58 <sup>2</sup> *Nm* ) range as reported by M. K. Lebiedowska (2003) is optimized using GA.

#### *2.3.2. Mass and position of COM of shank and foot*

326 Viscoelasticity – From Theory to Biological Applications

**Figure 4.** Lower limb model

fit the experimental data.

*2.3.1. Moments of Inertia* 

**Table 1.** Anthropometric data of subject

**2.3. Estimation of anthropometric inertia parameters** 

<sup>2</sup> *<sup>q</sup>* <sup>0</sup>

1*r*

Segment Length (m) Shank 0.4256 Foot 0.0675 Approximated position of COM of shank 0.2128 Approximated position of COM of soot 0.03375

The lower limb dynamics is complex and less well defined because the form is not composed of simple geometrical shapes. Indeed, even the location of the mass center of elemental parts is imprecise, and comprehensive analyses of joint kinematics are extremely difficult (Josephs and Huston, 2002). To quantify limb dynamics, accurate estimates are needed of anthropometric inertia parameters (mass, COM location, and moments of inertia). Therefore these equations have accessed to optimise mass, moment of inertia about COM and position of COM along the segmental length of the subject using GA tuning method to

Moments of inertia are fundamental parameters describing the mass distribution of body segments which enter into all computations involving segmental rotations. Methods based on the geometrical segment body models and appropriate anthropometric measures were used in measuring the moment of inertia of various segments in cadavers and in living subject (Dempster, 1955; Hatze, 1980; Jensen, 1986; Schneider and Zernicke, 1992). Different procedures were applied to identify the segment moment of inertia from the dynamic equation of motion. There is no unique way of calculating the moment of inertia To obtain

1

> <sup>2</sup>

2*r*

Vatnsdal et al (2008) addressed the masses and positions of the COM of the lower limb are possible error source in the lower limb model. Dempster (1956) estimated the mass from the total height and weight of the person using standard human dimensions (Winter, 1990). Zatsiorski and Seluyanov (1983) estimated the mass on the basis of regression equations based on statistics obtained from measuring cadavers. However, both of the methods give different measurements for a selected subject as shown in Table 2. Therefore they can potentially introduce large errors, especially while modelling the passive properties, as the pendulum test results may be affected by those parameters. Thus, fine tuning these parameters around the predicted values with an efficient stochastic search algorithm such as GA could lead to good solution with immense potential of extracting accurate subject specific results. GA is used to optimise the foot and shank mass between the two measurements with ±10% tolerance. While the approximated positions of COM along the segment length are assumed as in the centre of the segment length. To get the accurate positions, the position of the COM of shank and foot are optimised between ±10% tolerances.


**Table 2.** Mass of patient using different methods

#### **2.4. Optimization of fuzzy model as passive viscoelasticity**

Modelling based on conventional mathematical tools is not well suited for dealing with complex and nonlinear nature systems. By contrast, a fuzzy inference system employing fuzzy 'if–then' rules can model the qualitative aspects of human knowledge and reasoning processes without employing precise quantitative analyses (Dinakaran, 2009). Besides that, the purpose of modeling passive viscoelascity is to be used as a part of the simulation platform and need to integrate with the active properties of the knee joint model for the controller application. Controller such as fuzzy control, neural network does not require the mathematical model of the plant. Therefore fuzzy model are well suited for modeling nonlinear models such as passive viscoelasticity. The passive viscoelastic joint moment is represented as a non-linear function of knee angle and knee angular velocity in a fuzzy model. An automatic Mamdani fuzzy system design method integrates three stages; determines membership functions, the rule-consequent parameters, and scaling factor at the same time. Gaussian membership functions are preferred, because of their continuously differentiable curves and smooth transitions. There are 58 parameters to optimise including 30 parameters that determine the centre and width of the 15 Gaussian membership functions, 25 weights associated with the fuzzy rules, and 3 scaling factors for the normalization and denormalization of 2 inputs and 1 output of fuzzy model respectively. The weights of the 25 fuzzy rules have been optimised between 0 and 1. The rules are identified based on expert knowledge that refers to relationship between knee angle and velocity and passive torque. Table 3 summarizes the rule base for the fuzzy model encompassing possible AND combinations of the input fuzzy values.

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 329

). These scaling factors influence the performance of the

*et yt yt* () () () ˆ (4)

(5)

defuzzified crisp output from the normalised universe of the model output into an actual

model and optimise simultaneously using GA. The ranges of the scaling factors are set as

of the experimental data.

of the experimental data.

The GA is based on natural selection and population genetics theory (Goldberg, 1989). This evolutionary algorithm is chosen to estimate passive knee joint properties of paraplegic because the search space is large and complex. The advantage of the GA approach is robust, searches many points simultaneously, and is able to avoid local optima that the traditional

GA is used to estimate a dynamic characterization of passive viscoelasticity of the knee joint using fuzzy model and estimation of the anthropometric inertia parameters such as foot mass, shank mass, moment of inertia about COM and positions of COM along the segmentals length of the lower limb as shown in Figure 6. The goal of GA optimization process is to minimize the error between the knee angle obtained experimentally and and

where *y*( )*t* is the experimental data and *y*ˆ( )*t* is the estimated current output of knee angle. The 'goodness of fit' of the identified model is determined using the objective function by

1

The GA optimization procedure is shown in Figure 7 and implemented in MATLAB with GA Toolbox. First, an initial population of individuals is generated. Each individual corresponds to a chromosome, which is a set of specific genes from the biological point of view. Then, the performance of each member of the population is assessed through an objective function imposed by the problem. This fires the process of selecting pairs of individuals which will be mated together during reproduction. Selection is based probabilistically on a gene's fitness value; the higher the fitness of a gene, the more likely it can reproduce. After selecting two parents, crossover is performed according to a crossover probability. If crossover is to be performed, offspring are constructed by copying portions of

*i*

*<sup>f</sup> <sup>N</sup>* 

*N*

min

<sup>2</sup>

*yt yt*

() () ˆ

**2.5. Estimation and optimization of passive properties using GA** 

 and min 1 

 and min <sup>1</sup> 

from the model. The output prediction error is defined as:-

physical output (passive torque, *pas*

algorithms might get stuck in (Lee, 1993).

minimizing the mean-squared error (MSE);

a. S1 is between max 1

b. S2 is between max 1

c. S3 is between 1 and 50.

follows:-


**Table 3.** Rule base of the fuzzy model NB=Negative big NS=Negative small Z=Zero PS=Positive small PB=Positive big

**Figure 5.** A fuzzy expert system model

The configuration of the fuzzy expert system model is shown in Figure 5. In the fuzzification step, crisp inputs are fuzzified into linguistic values to be associated to the input linguistic variables. After fuzzification, the inference engine refers to the fuzzy rule base containing fuzzy IF-THEN rules to derive the linguistic values for the intermediate and output linguistic variables. Once the output linguistic values are available, the defuzzifier produces the final crisp values from the output linguistic values. The defuzzification method was based on calculating the centre of gravity of the fuzzy output. Scaling factors is applied to ensure that the domain of discourse covers the whole range (Reznik, 1997). Therefore, two input scaling factors are used to transform the crisp inputs into the normalised inputs so as to keep their value within -1 and +1. The scaling factors are S1 for knee angle, and S2 for knee angular velocity, . An output scaling factor S3 provides a transformation of the defuzzified crisp output from the normalised universe of the model output into an actual physical output (passive torque, *pas* ). These scaling factors influence the performance of the model and optimise simultaneously using GA. The ranges of the scaling factors are set as follows:-


328 Viscoelasticity – From Theory to Biological Applications

**Knee angle** 

**Table 3.** Rule base of the fuzzy model NB=Negative big NS=Negative small Z=Zero PS=Positive small PB=Positive big

> S1

> S2

knee angular velocity,

**Figure 5.** A fuzzy expert system model

Knee angle

Knee angular velocity

30 parameters that determine the centre and width of the 15 Gaussian membership functions, 25 weights associated with the fuzzy rules, and 3 scaling factors for the normalization and denormalization of 2 inputs and 1 output of fuzzy model respectively. The weights of the 25 fuzzy rules have been optimised between 0 and 1. The rules are identified based on expert knowledge that refers to relationship between knee angle and velocity and passive torque. Table 3 summarizes the rule base for the fuzzy model

> **Knee angular velocity** NB NS ZO PS PB

> > Inference

The configuration of the fuzzy expert system model is shown in Figure 5. In the fuzzification step, crisp inputs are fuzzified into linguistic values to be associated to the input linguistic variables. After fuzzification, the inference engine refers to the fuzzy rule base containing fuzzy IF-THEN rules to derive the linguistic values for the intermediate and output linguistic variables. Once the output linguistic values are available, the defuzzifier produces the final crisp values from the output linguistic values. The defuzzification method was based on calculating the centre of gravity of the fuzzy output. Scaling factors is applied to ensure that the domain of discourse covers the whole range (Reznik, 1997). Therefore, two input scaling factors are used to transform the crisp inputs into the normalised inputs so as

to keep their value within -1 and +1. The scaling factors are S1 for knee angle,

Fuzzy Rule Base

Engine Defuzzification

. An output scaling factor S3 provides a transformation of the

S3

Viscoelasticity

and S2 for

NB PB PB PB PS ZO NS PB PB PS ZO NS ZO PB PS ZO NS NB PS PS ZO NS NB NB PB ZO NS NB NB NB

encompassing possible AND combinations of the input fuzzy values.

Fuzzification

#### **2.5. Estimation and optimization of passive properties using GA**

The GA is based on natural selection and population genetics theory (Goldberg, 1989). This evolutionary algorithm is chosen to estimate passive knee joint properties of paraplegic because the search space is large and complex. The advantage of the GA approach is robust, searches many points simultaneously, and is able to avoid local optima that the traditional algorithms might get stuck in (Lee, 1993).

GA is used to estimate a dynamic characterization of passive viscoelasticity of the knee joint using fuzzy model and estimation of the anthropometric inertia parameters such as foot mass, shank mass, moment of inertia about COM and positions of COM along the segmentals length of the lower limb as shown in Figure 6. The goal of GA optimization process is to minimize the error between the knee angle obtained experimentally and and from the model. The output prediction error is defined as:-

$$e(t) = y(t) - \hat{y}(t) \tag{4}$$

where *y*( )*t* is the experimental data and *y*ˆ( )*t* is the estimated current output of knee angle. The 'goodness of fit' of the identified model is determined using the objective function by minimizing the mean-squared error (MSE);

$$f = \min \left\{ \frac{\sum\_{i=1}^{N} \left( y(t) - \hat{y}(t) \right)^2}{N} \right\} \tag{5}$$

The GA optimization procedure is shown in Figure 7 and implemented in MATLAB with GA Toolbox. First, an initial population of individuals is generated. Each individual corresponds to a chromosome, which is a set of specific genes from the biological point of view. Then, the performance of each member of the population is assessed through an objective function imposed by the problem. This fires the process of selecting pairs of individuals which will be mated together during reproduction. Selection is based probabilistically on a gene's fitness value; the higher the fitness of a gene, the more likely it can reproduce. After selecting two parents, crossover is performed according to a crossover probability. If crossover is to be performed, offspring are constructed by copying portions of

An Approach for Dynamic Characterisation of Passive Viscoelasticity and Estimation of Anthropometric Inertia Parameters of Paraplegic's Knee Joint 331

parent genes designated by random crossover points. Otherwise, an offspring copies its entire gene from one of the parents. As each bit is copied from parent to offspring, the bit has the probability of mutating. Mutation is believed to help in reinjection of any

In this section, the results of the new method to model the passive viscoleasticity and estimate the anthropometric inertia parameters by GA optimization process are presented. Population size of GA was set to 50 and crossover and mutation probabilities were 0.8 and 0.001 respectively. The automatic GA optimization process was generated up to 200 generations of solutions. The best solution was kept and the rest were discarded until there is no significant change in the mean square error (MSE) was observed after the 165th generation. The minimum MSE achieved was 1.87. The response of the model was tested and the result is shown in Figure 8. The results showed that the model parameters were estimated well and that the fit between the model and the experimentally data was

information that may have been lost in previous generations (Goldberg, 1989).

**Figure 8.** Responses of the pendulum test and the model response

**3. Results** 

good.

**Figure 6.** Optimization of passive properties

**Figure 7.** The GA optimization procedure

parent genes designated by random crossover points. Otherwise, an offspring copies its entire gene from one of the parents. As each bit is copied from parent to offspring, the bit has the probability of mutating. Mutation is believed to help in reinjection of any information that may have been lost in previous generations (Goldberg, 1989).

### **3. Results**

330 Viscoelasticity – From Theory to Biological Applications

*t*)(

*t*)(

**Figure 6.** Optimization of passive properties

**Figure 7.** The GA optimization procedure

*te* )(

*f*

*MM ds MM gi*

ˆ *ty* )(

*ty* )(

In this section, the results of the new method to model the passive viscoleasticity and estimate the anthropometric inertia parameters by GA optimization process are presented. Population size of GA was set to 50 and crossover and mutation probabilities were 0.8 and 0.001 respectively. The automatic GA optimization process was generated up to 200 generations of solutions. The best solution was kept and the rest were discarded until there is no significant change in the mean square error (MSE) was observed after the 165th generation. The minimum MSE achieved was 1.87. The response of the model was tested and the result is shown in Figure 8. The results showed that the model parameters were estimated well and that the fit between the model and the experimentally data was good.

**Figure 8.** Responses of the pendulum test and the model response
