## 3 Results
### Workflow Using Commercial FEA Software
#### 3.2.2 Model Reduction  
The fifty modes that were calculated are not all equally relevant for the input/output relations of the system. For example, mode 2 in Figure 10 is important, and mode 3 in the same figure is not. Mode shapes that are large at both the input(s) and the output(s) are relevant. Therefore, one way to proceed with the model reduction is to select the most important modes by just looking at the mode shapes. This works quite well in the SISO case, but becomes tedious if the system has multiple inputs and/or outputs. Automation or at least guidance of the mode selection is possible by using balanced truncation. Balanced truncation is a model order reduction technique in itself, developed in the field of systems and control [27]. It specifically aims to reduce the model while keeping the accuracy of the input/output relations as high as possible. This contrasts with the mode displacement method that is not aware of inputs and outputs. A limitation of balanced truncation is that it can only be applied to models up to 1000 degrees of freedom (DOFs); FEM models typically have many more. To overcome this limitation, the reduction strategy consists of two steps:  
1. Initial reduction using the mode displacement method and all modes up to some frequency that is considered to be high enough. 2. Secondary reduction by balanced truncation. This turns balanced truncation into a mode selection method. The balanced truncation method usually selects linear combinations of the degrees of freedom of the model that it is applied to. Figure 11 shows an example: 11 Balanced truncation DOFs are chosen (or actually 22 first order DOFs), resulting in 11 modes selected from the modal displacement model. Figure 11: Mode selection by balanced truncation. Hankel singular values (**top** figure) show the expected increase in accuracy for each added DOF. The **bottom** plot shows the selected modes: orange for observability and blue for controllability. (The rigid body mode was removed temporarily.) The resulting transfer functions of the reduced model are shown in Figure 12. The results are accurate up to 1 kHz. Peaks above 1 kHz are neglected in the reduced model, but this is acceptable. Figure 13 compares the reduced order model, with added modal damping, to measurements. The amount of damping could be tuned further, but the current match is acceptable. Overall, the match is quite good, except for the peak near 150 Hz. The mode shape that corresponds to this peak suggests that shaft and/or disk coupling is too flexible in reality, or too stiff in the model. It can be seen that the resonances around 1 kHz frequency are not captured well by the model. Further investigation of this mismatch revealed that the real plant exhibits additional bending modes of the manipulator base frame that was not included in the geometry of the FEM model. However, this high-frequency dynamics lies far beyond achievable target bandwidth and thus was decided to be left unmodelled. The derived model achieves a very good match for frequencies up to 500 Hz, which is more than satisfactory for the purpose of control design in our case. Figure 12: Transfer functions of the reduced model: tip displacement/motor torque (**top**), motor angular velocity/motor torque (**bottom**). Figure 14 compares both derived 1 st principle models with the experimental data. It can be seen that the analytical model based on the Euler-Bernoulli beam theory is able to capture first-smode dynamics very well. There is a mismatch of about 25% in the location of the second resonance frequency. The higher bending modes are not modelled well for several reasons:  
* Overly simplistic point two-mass model of the actuator and arm connection that is a source of the oscillatory dynamics around 500 Hz;
* Neglected machine frame dynamics (as in the case of the FEM model). The achieved result shows typical weakness of the analytical modelling approach. Although it may work for simple mechanisms, its applicability seems limited for complex machine geometries.