## 5 Method for Designing Manipulators for Man-Portable UGVs
### Load Cases Definition
\begin{bmatrix}-F*(L_{3}\cdot c_{23}+L_{4}\cdot c_{234})\\ F*(L_{3}\cdot s_{23}+L_{4}\cdot s_{234})\end{bmatrix}  \tag{2}\]  
determined in the global coordinate system. To determine the moment in the coordinate system related to joint 3, the joint we are interested in, we need to rotate it by angle beta, which is  
\[\beta=atan2(L_{3}\cdot c_{23},L_{3}\cdot s_{23}) \tag{3}\]  
Representing \(M_{F3}\) in the coordinate system associated with arm 3, marked as \(M_{F33}\), we obtain  
\[M_{F33}= \begin{bmatrix}F*sin(\beta)\sigma_{1}-F*cos(\beta)\sigma_{2}\\ F*sin(\beta)\sigma_{2}+F*cos(\beta)\sigma_{1}\end{bmatrix}  \tag{4}\]  
where  
\[ \begin{split}&\sigma_{1}=L_{3}*s_{23}+L_{4}*s_{234},\\ &\sigma_{2}=L_{3}*c_{23}+L_{4}*c_{234}.\end{split}  \tag{5}\]  
Since, to perform further optimization of the joint body, it is necessary to determine the largest loads that can be achieved in selected directions, the angle parameter \(\alpha\) is introduced to control the angle at which the greatest loads are sought. After projecting the moment vector onto the direction we are interested in, rotated by the angle \(\alpha\), we obtain the radial moment in joint 3 in the \(\alpha\) direction:  
\[M_{F33\alpha}= \begin{bmatrix}\frac{cos(\alpha)*\sigma_{1}}{|cos(\alpha)|^{2} +|sin(\alpha)|^{2}}\\ \frac{sin(\alpha)*\sigma_{1}}{|cos(\alpha)|^{2}+|sin(\alpha)|^{2}}\end{bmatrix}  \tag{6}\]where  
\[ \begin{split}&\sigma_{1}=cos(\alpha)*(F*sin(\beta)*\sigma_{2}-F*cos( \beta)*\sigma_{3})+\\ & sin(\alpha)*(F*sin(\beta)*\sigma_{3}+F*cos(\beta)*\sigma_{2})\\ &\sigma_{2}=L_{3}*s_{23}+L_{4}*s_{234}\\ &\sigma_{3}=L_{3}*c_{23}+L_{4}*c_{234}\end{split}  \tag{7}\]  
To find the largest radial moments in joint 3, an optimization procedure is used in which the searched parameters are the coordinates \(q_{2},q_{3},q_{4}\), for which the value of the objective function, which is \(M_{F33\alpha}\), in the \(alpha\) direction we are interested in is the largest. \[M_{F33\alpha}^{*}=f(q_{2}^{*},q_{3}^{*},q_{4}^{*},\alpha)=maxf(q_{2},q_{3},q_{ 4},\alpha) \tag{8}\]  
Limitations resulting from geometric and force constraints are introduced as unequal constraints: Minimum and maximum values of internal coordinates (limit positions) are introduced as constraints resulting from geometry, and configurations in which the joints and gripper tip collided with the mobile platform are eliminated. As inequality constraints resulting from statics, the maximum force that can be exerted on an obstacle, which results from the specified maximum friction force of the robot platform against the ground, is limited and a limitation of the moment resulting from the action of this force on the arm is introduced, resulting from the adopted configuration of the manipulator. It is assumed that, after a certain moment, the platform would start to tip over or rotate. The constraints can be written as follows:  
\[ \begin{split}& q_{i_{min}}\leq q_{i}\leq q_{i_{max}}\\ & abs(T_{i})\leq T_{i_{max}}\\ & abs(M)\leq M_{max}\\ & abs(F_{1r})\leq F_{max}\end{split}  \tag{9}\]  
where  
\(q_{i_{min}}\), \(q_{i_{max}}\)--position limits at joint \(i\)  
\(T_{i}\)--torque at joint \(i\)  
\(T_{i_{max}}\)--maximum torque at joint \(i\)  
\(M_{1r}\)--radial torque at joint 1  
\(M_{1 max}\)--maximum radial torque at joint 1  
\(F_{1r}\)--radial force at joint 1  
\(F_{1 max}\)--maximum radial force at joint 1  
The calculations are performed using the Fmincon function in MATLAB R2023b. The procedure to find the optimal value of a nonlinear function with inequality constraints uses the interior point algorithm [29]. This is an algorithm that allows finding the local minimum of the function. Therefore, to find the global minimum of the \(M_{F33\alpha}\) function, a modification is introduced in the algorithm that causes the search to be run multiple times from different starting points selected in the \(q_{i}\) coordinates space. As a result of the algorithm, configurations are obtained that generated the highest radial moment loads for the directions defined by angle \(\alpha\).