# A Sampling-Based Unfixed Orientation Search Method for Dual Manipulator Cooperative Manufacturing
## 3 Mathematic Model  
Here, the following motion for an arbitrary dual manipulator cooperative system is preset: manipulator A starts from \(\mathbf{P}_{A}^{start}\) heading towards \(\mathbf{P}_{A}^{target}\), in parallel, manipulator arm B is also moving on its initial path from the starting point \(\mathbf{P}_{B}^{start}\) to the target point \(\mathbf{P}_{B}^{target}\). As we consider generating reasonable geometrical paths for all manipulators, the processing of finding the optimal position points around the current positions in their Cartesian space with a distance of \(L_{s}\) will be focused on during each planning period. Figure 3 visualizes the core for building the mathematical model of our proposal with the circumvention strategy of the dual manipulator system in a certain sampling period \(t\sim(t+t_{\rm s})\). Assume the search step length during a sampling time interval \(t_{\rm s}\) in the planning space for both manipulators is \(L_{\rm s}^{t}\) and we make it constant during every sampling period before reaching the target \(L_{\rm s}^{t}=L_{\rm s}\) (\(t\in[1,2,\ldots,N]\)). Figure 2: Schematic of a local search map within a sampling period in SbOSMS and our proposal. The set of all blue dots located on the surface of the green sphere in the right half of the diagram forms the search map for the SbOSMS method during a single sampling period. This paper attempts to extend the local search map within a single sampling period from these “special point sets” to the entire green sphere surface area. Figure 3: Influence of the relative positions of the dual manipulator EEFs on the obstacle avoidance path-planning strategy. The upper part of Figure 4 (picture **a,b**) of the dual manipulator system have crossed initial paths, while the bottom half (**c,d**) is set to initial parallel paths. Meanwhile, the left half of picture a and c corresponds to the interval scenario, (**b,d**) fall under the interact scenario specified in Figure 4. Different initial paths and coexistence scenarios give rise to various choices of obstacle avoidance points \(P_{ie}^{t+1}(i\in\{A,B\})\). For the interval scenario, the relative position vector \((P_{Be}^{t}-P_{Ae}^{t})\) between manipulator B and A is outside the working space of manipulator A (consisting of vectors \((P_{Ae}^{t}-P_{A}^{\text{base}})\), \((P_{A}^{target}-P_{A}^{\text{base}})\) and \((P_{A}^{target}-P_{Ae}^{t})\) together). In this case, it is only necessary to consider how to reach its terminal \(P_{A}^{target}\) as quickly as possible while avoiding the obstacle to ensure that manipulator A has the safest tendency to move towards its base. However, when it comes to interact scenarios, the relative position vector is in the working space of manipulator A, then the first thing that should be considered for manipulator A is to evade immediately along the direction of its base and proceed to its target after it has wholly bypassed the danger area. When the minimum distance between the two manipulators is smaller than a predefined value, the dual manipulators are considered in the danger area in which there is a high probability of collision. The overlap between the two working areas is defined as the danger area. Therefore, when the two manipulators move into the danger area, the EEFs of the dual manipulators should move towards their respective bases to avoid the possible collision. Under this circumstance, the two manipulators are regarded as obstacles to each other. According to Figure 4, the non-linear optimization equations set of dual manipulator path planning will be established:  
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