# Research on an Optimal Path Planning Method Based on A* Algorithm for Multi-View Recognition
## 3 Optimal Path Planning for Multi-View Recognition Based on the A\({}^{*}\) Algorithm
### Objective Functions and Constraints
Some of these image pairs are randomly selected as training samples, and other image pairs are used as test samples to evaluate the recognition performance of the constructed BM2 DKPCANet + Softmax classifier. Based on the above approach, the final recognition results of the multi-view classifier after integrated learning can be expressed by the following equation:  
\[\widehat{R}(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{k})=\prod(\sum_{i=1}^{k-1} \mathbf{p}_{i}(\mathbf{y}|\mathbf{x}_{i},\mathbf{x}_{i+1})) \tag{16}\]  
where \(\left\{\mathbf{x}_{1},\mathbf{x}_{2},\ \ldots,\ \mathbf{x}_{k}\right\}\) is a sequence of input multi-view images. \(\prod(\cdot)\) is the index of the position of the maximum element of the input vector. \(\mathbf{p}_{i}(\mathbf{y}\ |\ \mathbf{x}_{i},\mathbf{x}_{i+1})\) is the C-dimensional posterior probability vector derived from the image classification by the base classifier, expressed as  
Figure 4: Two-view classifier. \[\mathbf{p}_{i}(\mathbf{y}|\mathbf{x}_{i},\mathbf{x}_{i+1})=\left[ \begin{array}{l}p_{i}(\mathbf{y}_{1 }|\mathbf{x}_{i},\mathbf{x}_{i+1})\\ p_{i}(\mathbf{y}_{2}|\mathbf{x}_{i},\mathbf{x}_{i+1})\\ \vdots\\ p_{i}(y_{C-1}|\mathbf{x}_{i},\mathbf{x}_{i+1})\\ p_{i}(y_{C}|\mathbf{x}_{i},\mathbf{x}_{i+1})\end{array} \right] \tag{17}\]  
where \(p_{i}(y_{h}|\mathbf{x}_{i},\mathbf{x}_{i+1})\) (h = 1, 2,...,C) is the category posterior probability recognized of the input image by the basic classifier. It can be expressed as:  
\[p_{i}(y_{h}|\mathbf{x}_{i},\mathbf{x}_{i+1})=p_{i}(y_{h}|\mathbf{z})=\frac{\exp(z_{h})}{ \sum\limits_{c=1}^{C}\exp(z_{c})} \tag{18}\]  
where \(\mathbf{z}\) is the input of the classifier of the \(i\)-th identification network. The constraint of viewpoint seeking path planning for robotic arms requires that the planned path points are in the target observation space, and also belong to the reachable workspace of the robotic arm. The difference in viewpoints between adjacent path points cannot exceed the number of base classifiers. The arm joint angles are within the joint limit angles. The speed, acceleration and pulse speed of each joint cannot exceed the corresponding maximum value. In summary, the mathematical model for transforming the viewpoint seeking path planning problem into a multi-objective optimization problem can be expressed as:  
\[\left\{ \begin{array}{l}\max\widetilde{R}(\mathbf{X})=\max(\widetilde{R}(\mathrm{ P}_{1}^{\mathrm{p}},\mathrm{P}_{2}^{\mathrm{p}},\ldots,\mathrm{P}_{k}^{\mathrm{p}}))= \max(\sum R^{\hat{d}}(\Phi_{ij},\Phi_{(i+d_{\theta})(j+d_{\lambda})}))\\ \min\widetilde{D}(\mathrm{P}_{c},\mathrm{P}_{1})\\ \vdots\widetilde{\theta}^{s}(t)|\leq J_{\max}^{s}\end{array} \right. \tag{19}\]  
where \(\Omega\) is the viewpoint position search space, W is the accessible workspace of the manipulator, \(\Phi_{ij}\) is the range of view angle, \(d_{\theta}\) is the view angle difference of latitude, \(d_{\lambda}\) is the view angle difference of longitude, \(k\) is the number of view angles in path planning, \(\mathrm{C}_{l}\) is the number of basic classifiers, \(\theta_{s}\) is the joint angle of the \(\mathrm{sth}\) joint, \(\theta_{s}^{d}\) and \(\theta_{s}^{u}\) are the upper and lower boundary of the angle movement for the \(\mathrm{sth}\) joint, respectively. \(\widetilde{\theta}^{s}(t),\widetilde{\theta}^{s}(t)\) and \(\widetilde{\theta}^{s}(t)\) are the actual velocity, acceleration and pulse of the \(\mathrm{sth}\) joint of the manipulator at \(t\) time, respectively. \(V_{\max}^{s}\), \(A_{\max}^{s}\) and \(J_{\max}^{s}\) are respectively the maximum velocity, maximum acceleration and maximum pulse of the \(\mathrm{sth}\) joint of the manipulator.