# A Sampling-Based Unfixed Orientation Search Method for Dual Manipulator Cooperative Manufacturing
## 3 Mathematic Model
\begin{array}{rl}min&\sum\limits_{i\in(A,B)}\mu_{i}\bigg{[}\Big{(}a_{i} \textbf{V}_{ie}^{t}+\beta_{i}\Big{(}\textbf{P}_{i}^{base}-\textbf{P}_{ie}^{t} \Big{)}\Big{)}\cdot\Big{(}\textbf{P}_{ie}^{t}-\textbf{P}_{ie}^{t+t_{s}}\Big{)} ^{T}+\parallel\textbf{P}_{i}^{target}-\textbf{P}_{ie}^{t+t_{s}}\parallel^{2} \bigg{]}\\ s.t.&\textbf{P}_{Ae}^{t}-\textbf{P}_{e}^{t+t_{s}}=steplength;\\ \textbf{P}_{Be}^{t}-\textbf{P}_{Be}^{t+t_{s}}=steplength;\\ \textbf{d}_{t+t_{s}}^{min}\geq L_{min}\end{array}  \tag{1}\]  
where \(\textbf{V}_{Ae}^{t}\), \(\textbf{V}_{Be}^{t}\) are the linear velocities of EEFs of manipulator \(A\) and \(B\) at time \(t\);  
\(\textbf{P}_{Ae}^{t}\), \(\textbf{P}_{Be}^{t}\) are the Cartesian coordinates of EEFs of manipulator \(A\) and \(B\) at time \(t\);  
\(\textbf{P}_{Ae}^{t+t_{s}}\), \(\textbf{P}_{Be}^{t+t_{s}}\) are the Cartesian coordinates of EEFs of manipulator \(A\) and \(B\) at time \((t+t_{s})\);  
\(\textbf{P}_{A}^{target}\), \(\textbf{P}_{B}^{target}\) are the target Cartesian coordinates of manipulator \(A\) and \(B\);  
\(steplength\) indicates the length of the search step of the manipulators' EEF within one sampling cycle \(t_{s}\);  
\(d_{t+t_{s}}^{min}\) (\(i\in int(1\sim n)\)) is the minimum distance between the two manipulators during sampling period \(t\sim(t+t_{s})\);  
\(L_{min}\) is the threshold of the minimum distance value;  
\(\mu_{A}\), \(\mu_{B}\) are the positive circumventing weighting factors that meet \(\mu_{A}+\mu_{B}=1\). \(\mu_{A}=0\) means manipulator A will always follows its initial path without modification and vice versa. \(\alpha_{A}\), \(\alpha_{B}\), \(\beta_{A}\) and \(\beta_{B}\) are positive deviation direction weighting coefficients for manipulators A and B, which take values in the set of {0,1}. We simplify the Formula (1):  
Figure 4: Three dual-manipulator coexistence scenarios. The blue dotted line represents the initial path of manipulator A, and the red dotted line represents the initial path of manipulator B. The blue area is the operation space of manipulator A, and the red area is the operation space of manipulator B. In the scenario of “Overlap”, the initial path of manipulator A is in the operation space of manipulator A and the initial path of manipulator B is in the operation space of manipulator B. In scenario “Interval”, the two initial paths overlap and the EEF points of the two manipulators should be driven back towards their points of departure. In scenario “Interact paths”, the avoidance strategy pushes the EEFs of the two manipulators into the working space of another manipulator, undoubtedly increasing the possibility of collision. If \(\mathbf{\omega}<0\mid\mid\varphi<0\), (\(\mathbf{P}_{A}^{target}-\mathbf{P}_{Ac}^{t}\)) is out of the space region defined by vectors \(\left(\mathbf{P}_{A}^{base}-\mathbf{P}_{Ac}^{t}\right)\) and \(\left(\mathbf{P}_{Be}^{t}-\mathbf{P}_{Ac}^{t}\right)\). In this case, the mere maintenance of the motion trend towards the termination points do not guarantee manipulator A circumventing vector \(\left(\mathbf{P}_{Be}^{t}-\mathbf{P}_{Ac}^{t}\right)\). Therefore, forgoing the motion option of heading towards the terminals and temporarily seeking avoidance towards the base of the manipulator A is a better motion strategy, making \(\alpha_{A}=0\), \(\beta_{A}=1\). Such derivations and conclusions also apply to manipulator \(B\)