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It is advantageous to validate the theories of dynamic movement using measurements of mechanical pendulum movements instead of human movements. Physical pendulums can be manufactured to have uniform density and known properties, while it is not possible to specify the physical properties of the human body. The mass and moment of inertia of a mechanical (e.g. metal) segment can also be accurately measured and determined, while the mass of a body segment (in vivo) cannot be directly measured, and the derivations of the centre of mass and moment of inertia required certain assumptions. Using physical pendulums allows for measurements to give minimal error in model parameter values.

4.1.6.1 Pendulum models

In this thesis, three physical pendulum segments were produced and it was possible to analyse the dynamics of a three segments pendulum. However, only the modelling of the single segment pendulum (section 4.1.3) and a two segments pendulum (section 4.1.4) were described. The assumption was that if the model of a two segments pendulum does not produce realistic results, then models and simulations of systems consisting of more segments would not produce realistic results. Therefore this work focused on improving the model of the two segments pendulum.

In section 4.1.3, the method for simulating the movement of the single pendulum can be simplified by computing the moment of segment A in Eqn 4.5 around A0 instead of A*, then the force of N acting on A in Eqn 4.4 would not need to be considered, and Eqn 4.4 is unnecessary. However the presented method described in section 4.1.3 is scalable to a multiple-segment system, where the equations can be applied to other segments in a multiple segment pendulum, such as the two segments pendulum described in section 4.1.4.

4.1.6.2 Simulated results

The results in Figure 4.6 and Figure 4.8 have shown that the single pendulum model produced results that agree with measured data, while the two segments model did not produce results that agree with measured data.

The measured and simulated movement of the single pendulum in Figure 4.6 show that the measured amplitude at 10s had decreased by about 10% from the original amplitude, due to friction. This was not included in the model, and therefore the measured data after 10s from the start of the experiment was not used for comparison with simulated movement.

A good agreement can be seen between the measured and simulated movement of the single pendulum in Figure 4.6. The model parameter values were measured or calculated and did not have to be obtained through model fitting or parameter estimation. This showed that the Newton-Euler method to model the movement of a single pendulum can provide good agreement with measured data.

Figure 4.7 showed the effect of the simulated movement of the two segments pendulum with and without the term _A( )t in the calculation of _B( )t in Eqn.4.12. It can be seen that the initial measured movement of segment B swings towards 3

4 rad

 . The simulation with the _A( )t term has shown better agreement with this movement characteristic than the simulation without the _A( )t term. Furthermore the measured movement of segment A did not resemble a sine wave, however the simulated movement of segment A without the _A( )t resembled a sine wave with very small distortion. This suggests the movement dynamics of A is significantly affected by the movement of segment B, which can be seen in the simulated movement of segment A with the _A( )t term in the calculation of _B( )t .

Six seconds of simulated and measured data of the two segments pendulum (with the _A( )t term in the calculation of _B( )t ) are shown in Figure 4.8. The shape of the trajectories of the segments are similar between the measured and simulated result, however there is a noticeable difference in the oscillation frequencies between the measured and simulated data. The oscillation frequency of the simulated data was about twice of the oscillation frequencies of the measured data. From the theory of second order system oscillation (Cartwright, 2001), a second order system of the form as shown in Eqn 4.14, where x t( ) is the system variable and u t( ) is the system input, has an undamped

natural frequency of _n for a step input.

2 2 2 1 ( ) 2 ( ) ( ) ( ) n n dx t dx t x t u t dt dt      (Eqn 4.14)

If Eqn 4.14 is rearranged into

2 2 2 ( ) 2 ( ) ( ) ( ) n n dx t dx t x t u t dt dt       _   _   (Eqn 4.15)

then this can be compared with Eqn 4.12 where x t( ) is equivalent to ( )t and

2

n

 is equivalent to 1/I_A and 1/I_B . This suggests that the mathematical derivations of the moment of inertias for the two segments pendulum may be incorrect and caused the error in oscillation frequency. This suggests that the method to calculate the moment of inertias for the segments should be an area of investigation if the Newton-Euler method is to produce simulated dynamics that agree with measurements for two or more segments pendulums and multi- linked rigid bodies.

In this work, the single pendulum forward simulation has produced results that agreed with measured data, while it has not been the case for the double and triple pendulum. This suggests that accumulated numerical error in the ODE solver is not the cause for the error, but instead additional theory is required to

support the Newton-Euler’s equation of dynamics motion to produce realistic results. A suggestion for this is that the calculation of the moment of inertia for segment B and any other distal segments in a multi-segment system should be reassessed.

Several new methods had been used in this work in attempting to identify and rectify the causes of the disagreement in dynamics, including the derivations of linear accelerations using centripetal accelerations in Eqn 4.4 and Eqn 4.11, and the inclusion of the angular acceleration of segment A in the calculation of the angular acceleration of segment B in Eqn 4.12. However the end results were still unsatisfactory.

4.1.6.3 Alternative methods and work by other researchers

The Lagrange-Euler method has not been used in this work. However, the Lagrange-Euler and the Newton-Euler methods have been shown to be equivalent though symbolic analysis (Silver, 1982), therefore if the Newton-Euler method cannot give good agreement between the measured and simulated data, then this implies the Lagrange-Euler method cannot produce good agreement either.

Other researchers in biomechanics (John et al., 2012) have also used the Newton-Euler method in multi-segment musculo-skeletal modelling, for inverse analysis to compute joint forces, and forward simulation to predict movement. John et al has encountered the problem that using the Newton-Euler’s equations of motion in inverse dynamics analysis has resulted in residual forces and acceleration in the system. They have used a least square estimation technique to minimise the error, however the error reduction method was not a deterministic and robust approach, and this does not fully solve the problem. External to biomechanics, the Newton-Euler method is widely used in robotics (Niku, 2001), however hardware feedback implantations have been the norm to correct angular, positional and velocity errors. Subsequently there has not been

a need in the field of automation and robotics to rectify the problem in the Newton-Euler method identified in this work and (John et al., 2012).

Clearly if one wishes to successfully predict multi-segment movement and apply this in biomechanics, further investigations into the multi-segment modelling theory are required.