## 2 Motion Planning of XXY Stage Using the ANN
### Motion Planning Using ANN  
An artificial neural network (ANN) is invented based on the operation of biological neural networks, and it is defined as a collection of parallel processors connected in the form of a network such that the structure provides the nonlinear mapping between the input and output [16]. Recently, neural network techniques are applied to a complex control problem in the automotive industry and robotic motion planning. A sensor based navigation scheme which makes use of a global representation of the environment by means of a self-organizing network is presented in [3]. The aim of the application of the mobile robot for a motion planning problem is to obtain a collision-free path among moving obstacles with dynamic constraints to limit robot motions. Based on the idea of the ANN based motion planning in [16], this study investigates the use of the ANN to achieve nonlinear mapping between the image coordinates and the global coordinates for an image-servo problem. For training the ANN models, there are three types of algorithms, such as supervised learning algorithm, unsupervised learning algorithm, and associative memory learning algorithm. This study is suitable for using the supervised learning ANN, and the back-propagation neural network (BPNN) is investigated to establish the relationship between the motor command's and the actual position in the global frame. The main objective is to find a suitable mapping between the desired movement \([ \begin{array}{cc}\delta_{x}&\delta_{y}&\delta_{0}\end{array} ]^{T}\) and the necessary motor command \([dx1~{}dx2~{}dy]^{T}\). The used ANN is the multilayer perceptron and the structure of the neural network, which contains four layers consisting of input, output and two hidden layers as shown in Figure 4. The layers are connected by synaptic weights and the learning operation is realized by a backpropagation (BP) algorithm based on the error-correction principle. The input-output relationship is described as follows. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Command (pulse)**} & \multicolumn{3}{c}{**Displacement in the Global Space**} \\ \hline
**X1 Motor** & **X2 Motor** & **Y Motor** & \(\delta\)**0 (degree)** & \(\delta\)**x (\(\mu\)m)** & \(\delta\)**y (\(\mu\)m)** \\ \hline
0 & 200 & 0 & 0.07 & 102.7 & \(-\)75.3 \\
0 & 400 & 0 & 0.14 & 199.7 & \(-\)150.5 \\
0 & 600 & 0 & 0.22 & 296.4 & \(-\)220.6 \\
0 & 800 & 0 & 0.30 & 404.5 & \(-\)296.5 \\
0 & 1000 & 0 & 0.39 & 506.5 & \(-\)373.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rotational and translation movement for only \(X1\)-axis actuation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Command (pulse)**} & \multicolumn{3}{c}{**Displacement in the Vision Space (pixel)**} \\ \hline
**X1 Motor** & **X2 Motor** & **Y Motor** & **X1** & **Y1** & **X2** & **Y2** \\ \hline
0 & 200 & 0 & 30 & 2 & 19 & \(-\)23 \\
0 & 400 & 0 & 59 & 4 & 37 & \(-\)46 \\
0 & 600 & 0 & 89 & 6 & 55 & \(-\)69 \\
0 & 800 & 0 & 121 & 8 & 75 & \(-\)86 \\
0 & 1000 & 0 & 153 & 9 & 94 & \(-\)120 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Movement experiment for only \(X1\)-axis actuation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Command (pulse)**} & \multicolumn{3}{c}{**Displacement in the Global Space**} \\ \hline
**X1 Motor** & **X2 Motor** & **Y Motor** & \(\delta\)**0 (degree)** & \(\delta\)**x (\(\mu\)m)** & \(\delta\)**y (\(\mu\)m)** \\ \hline
0 & 200 & 0 & 0.07 & 102.7 & \(-\)75.3 \\
0 & 400 & 0 & 0.14 & 199.7 & \(-\)150.5 \\
0 & 600 & 0 & 0.22 & 296.4 & \(-\)220.6 \\
0 & 800 & 0 & 0.30 & 404.5 & \(-\)296.5 \\
0 & 1000 & 0 & 0.39 & 506.5 & \(-\)373.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Relationship between displacement and the X1-motor’s actuation.