# Solution to the Inverse Problem of the Angular Manipulator Kinematics with Six Degrees of Freedom
## 2 Methods and Algorithms
### Kinematic Diagram of Manipulator  
Figure 1 illustrates an example of a typical welding manipulator, designed according to a common scheme proposed by the developers of the PUMA 560 manipulator [1].  
Such a manipulator has six angular DOF, which are provided by six kinematic rotational pairs of the fifth class (see Figure 1): the rotation of the column (\(C\)) occurs around the vertical axis at an angle \(\varphi_{1}\), the rotation of the arm (\(S\)) and elbow (\(L\))--around the horizontal axes at angles \(\varphi_{2}\) and \(\varphi_{3}\), the rotation of the forearm (\(F\))--around its longitudinal axis at an angle \(\varphi_{4}\), and the rotation of the wrist (\(W\))--around the axis transverse to the forearm at an angle \(\varphi_{5}\). Finally, the rotation of the gripper (\(B\)) is performed around its longitudinal axis at an angle of \(\varphi_{6}\). The working tool (\(T\)) is fixed in the gripper.  
Let us denote the point of intersection of the axis of rotation of the gripper at an angle of \(\varphi_{6}\) and the axis of rotation of the wrist at an angle of \(\varphi_{5}\) with the letter \(K\), as shown in Figure 1. Let us assume that this point also lies on the axis of symmetry of the initial section of the working tool. In some designs of manipulators, the axis of rotation of the forearm through the angle \(\varphi_{4}\) also passes through this point. In this case, a closed-form analytical solution to the inverse problem of the kinematics of the manipulator can be derived, which will be presented below. Taking into account the absence of intersection of the axis of rotation of the forearm with point \(K\) leads to the fact that to solve such a problem analytically it is necessary to solve the transcendental equation regarding the angle of rotation \(\varphi_{4}\).  
Let us define two further points of the working tool: the break point \(E\) and end point \(D\) (see Figure 1). We associate three coordinate systems (CSs) of the working tool with the triangle \(\Delta KED\). In all three, the abscissa and applicate axes lie in the plane of this triangle; in two of them, the beginning is located at point \(D\), and the third at point \(K\) (this is _Kxyz_--shown in Figure 1). We will call the last CS the output coordinate system of the working tool since it is obtained by sequential transformation of the original absolute CS (\(OXY\) in Figure 1) as a result of its displacements and rotations in accordance with the dimensions and sequence of rotations of the links from the stand to the gripper. The first  
Figure 1: Six DOF universal welding manipulator.  
and second CSs are shown in Figure 2, and have their origin at point \(D\), which coincides with the end point of the working tool.  
The first CS will be termed the main coordinate system of the working tool, where the \(x\)-axis is a continuation of the section \(DE\). The second CS will be designated as the backup coordinate system of the working tool, where the \(x\)-axis is a continuation of the straight line \(DK\). For both axes, the applicate is directed in the plane of the triangle \(\Delta KED\) towards the orientation of tool bend at point \(E\). The meaning of introducing these two CS will be clear in the following considerations.