# Stiffness-Based Cell Setup Optimization for Robotic Deburring with a Rotary Table
## 2 Robotic Deburring Cell
### Kinematic Model of Robot ACMA XR701  
The kinematic model of the quasi-serial ACMA XR701 robot is constructed using DH-notation and direct homogeneous transformations. First, a simplified model is constructed in DH-notation by considering only the serial kinematic chain and excluding the kinematic parallelogram. Then, the kinematic parallelogram is added to the simplified model using direct homogeneous transformation matrices (HTM) to construct a detailed kinematic model of the robot (Figure 2).  
The simplified model is sufficient to describe the transformation from the robot base frame to the end-effector frame required for inverse kinematics. Moreover, solving the inverse kinematics using the simplified model with fewer components is computationally more efficient than using the detailed kinematic model of the robot. Therefore, the simplified model is used for computationally intensive optimization, while the detailed model is used for result validation and presentation.  
The DH-parameters for the simplified robot model are shown in Table 1. The simplified kinematic modeling approach considers \(\theta_{3}\) as the actuated joint, which is also intuitively the resulting angle between link two and three. The homogeneous transformation matrix \(\prescript{0}{}T\) between frames zero and six is expressed as follows:  
\[\prescript{0}{}T = \prescript{0}{}T\prescript{1}{}T\prescript{2}{}T\prescript{3}{}T \prescript{4}{}T\prescript{5}{}T\prescript{6}{}T \tag{1}\]  
The detailed kinematic modeling approach with the kinematic parallelogram considers \(\theta_{A}\) as the actuated joint, which is consistent with the kinematics of the real robot. The parallelogram structure is difficult to account for using the DH-approach. A much more convenient approach is to use HTMs to account for the parallelogram. In this way, frames \(A\), \(B\) and \(C\) are added as an open kinematic chain to frame two of the simplified kinematic model to represent the kinematic parallelogram. The HTM-kinematic parameters for the detailed kinematic model are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Link \(i\)** & \(a_{i-1}\) **(m)** & \(a_{i-1}\) **(rad)** & \(a_{i}\) **(m)** & \(\theta_{i}\) **(rad)** \\ \hline
1 & 0.2500 & \(-\pi/2\) & 0.9500 & 0 \\
2 & 1.0000 & 0 & 0 & \(-\pi/2\) \\
3 & 0.2300 & \(-\pi/2\) & 0 & 0 \\
4 & 0 & \(\pi/2\) & 1.3294 & 0 \\
5 & 0 & \(-\pi/2\) & 0 & 0 \\
6 & 0 & 0 & 0.2500 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: DH-kinematic parameters.  
Figure 2: Detailed kinematic model of ACMA XR701.  
Each individual homogeneous transformation matrix \(\overset{i-1}{{}_{i}}T\) is obtained by multiplying the homogeneous transformation matrix \(\overset{i-1}{{}_{i}}T_{t}\) represented by the translation vector \(t_{i}~{}=~{}[x_{i},~{}y_{i},~{}z_{i}]\) and the homogeneous transformation matrix \(\overset{i-1}{{}_{i}}T_{r}\) represented by the Euler angles \(r_{i}~{}=~{}[\alpha_{i},\beta_{i},~{}\gamma_{i}]\), as expressed by the following:  
\[\overset{i-1}{{}_{i}}T~{}=~{}\overset{i-1}{{}_{i}}T\overset{i-1}{{}_{t_{i}}}T _{r} \tag{2}\]  
The homogeneous transformation matrix \(\overset{0}{{}_{C}}T\) between frames 0 and \(C\) can now be expressed as follows:  
\[\overset{0}{{}_{C}}T~{}=~{}\overset{0}{{}_{1}}T\overset{1}{{}_{2}}T\overset{2 }{{}_{A}}T\overset{A}{{}_{B}}T\overset{B}{{}_{C}}T \tag{3}\]  
where the functional dependence of \(\theta_{3}\), \(\theta_{B}\), \(\theta_{C}\) on \(\theta_{2}\) and \(\theta_{A}\) is expressed as follows:  
\[\left\{\begin{array}{ll}\theta_{3}~{}=~{}-(\theta_{2}+\pi/2)+\theta_{A}\\ \theta_{B}~{}=~{}(\theta_{2}+\pi/2)-\theta_{A}\\ \theta_{C}~{}=~{}\theta_{A}\end{array}\right. \tag{4}\]