# Tolerance Synthesis of Delta-like Parallel Robots Using a Nonlinear Optimisation Method
## 2 Materials and Methods
### Modelling the Delta Parallel Robot
#### 2.1.2 Kinematic Equations  
To parametrise the Delta parallel robot (using the Khalil-Kleinfiger parameterisation [27] for the robot with complicated chains), it is sufficient to open the kinematic closed loops at the mobile platform while taking into account the limitations imposed by the other chains. Illustrated in Figure 3, the Khalil-Kleinfiger parameterisation rule is used to formulate the kinematic equation of each subchain, and the geometric parameters are listed in Table 1.  
The forward kinematic equation of the \(i\)th chain, in terms of its Khalil-Kleinfiger parameters, is given by [27]:  
\[ \begin{array}{l}P_{i}(q_{a},q_{p},v)=^{0}_{A}T_{i}\ \ \begin{array}{l}A\\ 1\end{array}T_{i}\ \begin{array}{l}A\\ 1\end{array}T_{i}\ \begin{array}{l}2\\ 2\end{array}T_{i}\ \begin{array}{l}2\\ 3\end{array}T_{i}\ \begin{array}{l}3\\ 4\end{array}T_{i}\ \begin{array}{l}4\\ 5\end{array}T_{i}\ \begin{array}{l}5\\ 5\end{array}T_{i}\times[0\ 0\ 0\ 1]^{T},\\ (i=1,2,3)\end{array}  \tag{1}\]  
which can also be rewritten as:  
\[ \begin{array}{l}P_{i}(q_{a},q_{p},v)=^{0}_{A}T_{i}\ \begin{array}{l}A\\ 1\end{array}T_{i}\ \begin{array}{l}A\\ 2\end{array}T_{i}\ \begin{array}{l}2\\ 6\end{array}T_{i}\ \begin{array}{l}6\\ 7\end{array}T_{i}\ \begin{array}{l}7\\ 5\end{array}T_{i}\ \begin{array}{l}5\\ 5\end{array}T_{i}\times[0\ 0\ 0\ 1]^{T},\\ (i=1,2,3)\end{array}  \tag{2}\]  
\[ \begin{array}{l}q_{a}=q_{1}i,q_{p}=\{q_{2i},q_{3i},q_{4i},q_{5i},q_{6i},q_{7 i}\},\\ q_{v}=\{La_{i},\ L_{b},\ R_{a},\ R_{b},\ d_{i},\ \theta_{i},\ \alpha_{i},\ \alpha_{ 3i},\ \alpha_{5i},\ \alpha_{6i},\ \alpha_{E}\},\\ q_{1i}=q_{i},q_{3i}=\gamma_{i}\end{array} \]  
where \(P_{i}\) is the mobile platform position and \(\overset{j-1}{j}T_{i}\), \((j=1,\ldots,5)\ (i=1,\ 2,\ 3)\) is the homogeneous transformation matrix (HTM) between the \(j\)th and \(j+1\)th coordinate frame in the \(i\)th subchain. \(\overset{5}{T}_{i}\) is the HTM between the end-effector and the 5 th coordinate frame. The variables \(q_{a}\), \(q_{p}\), and \(q_{v}\) represent the robot's active joint variables, passive joint variables, and parametric variables, respectively. Each chain from the base frame to the mobile platform can be considered as a serial robot. The elementary matrices to describe a kinematic chain \(i\) are as follows:  
\[ \begin{array}{l}\overset{0}{{}_{A}}T_{i}=\left[\begin{array}{cccc}cos( \theta_{i})&-sin(\theta_{i})&0&0\\ sin(\theta_{i})&cos(\theta_{i})&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]_{A}T_{i}=\left[\begin{array}{cccc}cos(q_{1i})&- sin(q_{1i})&0&R_{a}\\ 0&0&-1&0\\ sin(q_{1i})&cos(q_{1i})&0&0\\ 0&0&0&1\end{array}\right],\\ \overset{1}{2}T_{i}=\left[\begin{array}{cccc}cos(q_{2i})&-sin(q_{2i})&0&0\\ sin(q_{2i})&cos(q_{2i})&0&0\\ 0&0&1&d\\ 0&0&0&1\end{array}\right]_{A}\overset{2}{3}T_{i}=\left[\begin{array}{cccc} cos(q_{3i})&-sin(q_{3i})&0&0\\ 0&0&1&0\\ -sin(q_{3i})&-cos(q_{3i})&0&0\\ 0&0&0&1\end{array}\right],\\ \overset{3}{4}T_{i}=\left[\begin{array}{cccc}cos(q_{4i})&-sin(q_{4i})&0&Lb\\ sin(q_{4i})&cos(q_{4i})&0&0\\ 0&0&1&d\\ 0&0&0&1\end{array}\right]_{A}\overset{4}{5}T_{i}=\left[\begin{array}{cccc} cos(q_{5i})&-sin(q_{5i})&0&0\\ 0&0&-1&d\\ sin(q_{5i})&cos(q_{5i})&0&0\\ 0&0&0&1\end{array}\right],\\ \overset{2}{6}T_{i}=\left[\begin{array}{cccc}cos(q_{6i})&-sin(q_{6i})&0&0\\ 0&0&1&0\\ -sin(q_{6i})&-cos(q_{6i})&0&0\\ 0&0&0&1\end{array}\right],\\ \overset{5}{E}T_{i}=\left[\begin{array}{cccc}1&0&0&Rb\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&1\end{array}\right]\end{array} ,\\ \overset{6}{5}T_{i}=\left[ \begin{array}{cccc}1&0&0&Rb\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&1\end{array} \right]\end{array},\\ \overset{6}{7}T_{i}=\left[\begin{array}{cccc}cos(q_{7i})&-sin(q_{7i})&0&Lb\\ sin(q_{7i})&cos(q  
Indeed, by considering a restriction on the ends of the three serial subchains, the forward kinematics of the parallel mechanism can be determined. The end points of the three subchains, \(P_{i}\) (\(Px_{i},\ Py_{i},\ Pz_{i}\)), for \(i=1,\,2,\,3\), are the same and are represented by the expression:  
\[\left[ \begin{array}{c}x-Px_{i}\\ y-Py_{i}\\ z-Pz_{i}\end{array} \right]=\left[