# Solution to the Inverse Problem of the Angular Manipulator Kinematics with Six Degrees of Freedom
## 2 Methods and Algorithms
### Determination of Initial Values of Angles Corresponding to Coordinates of Point \(K\)  
The first three angles in the joints of the manipulator \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{3}\) determine the distance of point \(K\) from the manipulator rack; therefore, they are called transition angles. In contrast, the angles \(\varphi_{4}\), \(\varphi_{5}\), \(\varphi_{6}\) are called orientation ones. Therefore, we consider separately the problem of determining the transition and orientation angles. Determination of Transition Angles  
Taking into account that point \(K\) is located in the vertical plane, in the same place as the beginning of the absolute CS (point O in Figure 1) or close to it, we will consider the kinematic diagram of the manipulator in three options as follows:  
1. Point \(K\) is located in a vertical plane coinciding with the axis of rotation of the manipulator column at an angle \(\varphi_{1}\) (Figure 4a, green dash-dotted line)
2. Point \(K\) is located in a vertical plane parallel to the axis of rotation of the column at an angle \(\varphi_{1}\), and is separated from this plane at a distance \(d\) (Figure 4b, green dash-dotted line)
3. Point \(K\) is located at a distance \(e\) from the axis of rotation of the forearm \(B_{2}B_{3}\) at an angle \(\varphi_{4}\), and the axis \(B_{2}B_{3}\) itself is at a distance \(d\) from the vertical plane containing the axis of rotation of the column of the manipulator \(OA_{1}\) at an angle \(\varphi_{1}\) (see Figure 4c, green and red dotted line). That is, point \(K\) is located on a circle of radius \(e\) when link \(B_{2}B_{3}\) is rotated around its axis through an angle \(\varphi_{4}\)  
Figure 3: Determination of the angle \(\psi_{E}\). Figure 5 shows the projections of the manipulator: both onto absolute coordinate planes--vertical \(XOZ\), Figure 5a,c and horizontal \(XOY\), Figure 5b. Moreover, projections of two possible configurations of the manipulator are presented on the vertical plane, ensuring that point \(K\) gets into the given position. One configuration corresponds to the mechanism approaching point \(K\) from above, when the elbow (point \(B\)) is located at the top, see Figure 5a, and the other, which corresponds to the mechanism approaching point \(K\) from below, when the elbow (point \(B\)) is located below, see Figure 5c. Therefore, we will derive the values of the angles at the joints for these two possible configurations of the mechanism. In this case, triangles formed by projections of the corresponding characteristic points of the mechanism onto the indicated vertical and horizontal planes are considered; the names of the projections of the points are preserved by their variable identifiers. We assume that the dimensions of the manipulator links \(h_{0}\), \(h_{1}\), \(h_{2}\), \(h_{3}\), \(h_{4}\), \(h_{5}\), \(h_{6}\), (see Figure 5a), \(d_{1}\), \(d_{2}\), \(d_{3}\) (Figure 5b) and coordinate values of point \(K\) in the absolute reference system \(X_{K}\), \(Y_{K}\), \(Z_{K}\) are known. The dimensions \(\Delta z\) and \(AK\) are  
\[\Delta z=Z_{K}-h_{0}-h_{1},\quad AK=\sqrt{X_{K}^{2}+Y_{K}^{2}+\Delta z^{2}}\]  
Let us use the first assumption about the position of point K. Then, we find the angle \(\delta\) between straight line \(A_{1}K\) and the horizontal plane, see Figure 5a, and the angle \(\alpha\) between the projection of straight line \(OK\) onto the horizontal plane and the \(OX\) axis, see Figure 5b, as follows:  
\[\delta=\arcsin\frac{\Delta h}{AK},\quad\alpha=\arctan\left(\frac{Y_{K}}{X_{K} }\right) \tag{5}\]  
Figure 4: Three options of the model.