## 3 Approximate Inverse Kinematics Solution for Imposed End-Effector State
### Robot Cables' (Tendons') Length Calculation According to the Robot Spine Curvature  
The key to controlling the shape of a trunk-type robot is cabling (tendons). One trunk-type robot segment can have three or four (or more) tendons dedicated to control. This section will discuss the robot segments' options with only three cables for the control of each segment. Each segment will have equations for three robot cable lengths \(l_{ni}\) (\(n\)--segment number, \(i\)--cable number). Some of the cable length formulas originate from [33]. The authors adjusted these equations and used them in the robot control approach. The main reason for choosing these equations was that [33] used robot segment spine curvature and orientation information about robot segment construction. In [33], the equations were based on assumptions that their cables are in straight lines between the section connecting disks and not in the arc-like segment spine. However, the authors of [33] assessed the number of cable guide connecting disks in the section (Figure 16), which is essential for accurate robot cable length calculation. In [33], the equations for the first segment cable length's calculation are  
\[l_{11}\overset{\frac{1}{n}=r_{1}}{\rightarrow}q_{1}(r_{1}-D_{1}\sin\varphi_{ 1}), \tag{48}\]  
\[l_{12}\overset{\frac{1}{n}=r_{1}}{\rightarrow}q_{1}\Big{(}r_{1}+D_{1}\sin \Bigl{(}\frac{\pi}{3}+\varphi_{1}\Bigr{)}\Big{)}, \tag{49}\]  
\[l_{13}\overset{\frac{1}{n}=r_{1}}{\rightarrow}q_{1}\Big{(}r_{1}-D_{1}\cos \Bigl{(}\frac{\pi}{6}+\varphi_{1}\Bigr{)}\Big{)}, \tag{50}\]  
where the robot first segment cable lengths are \(l_{11}\), \(l_{12}\), \(l_{13}\); the first segment spine curvature is \(k_{1}\); the first segment circle arc radius is \(r_{1}\); the distance from the robot spine center point to the robot cable center point on the robot connecting disk plane of the first segment is \(D_{1}\); \(\varphi_{1}\) is the angle between a vector \(\overset{\wedge}{\mathbf{s}}_{\text{orien1}}\) and the first segment coordinate system \(x\)-axis on the \(x\)-\(y\) plane (the angle around the \(z\)-axis). The angle \(\varphi_{1}\) range is \([0,2\pi]\) from the \(x\)-axis to vector \(\overset{\wedge}{\mathbf{s}}_{\text{orien1}}\) on the \(x\)-\(y\) plane in a counter-clockwise direction. \(q_{1}\) comes from this equation:  
\[q_{n}=2m_{n}\sin\biggl{(}\frac{k_{n}L_{n}}{2m_{n}}\biggr{)}\overset{\frac{1}{ n}=r_{n}}{\rightarrow}2m_{n}\sin\biggl{(}\frac{L_{n}}{2m_{n}r_{n}}\biggr{)}, \tag{51}\]  
where the \(n\)-th segment length is \(L_{n}\) and the number of spaces between the connecting disks of the \(n\)-th segment is \(m_{n}\). The first robot cable holes' positions on the connecting disks are necessary to determine the cable lengths for the second and third segments of the robot. Cables conclude three groups: for the first robot segment control \(l_{11}\), \(l_{12}\), \(l_{13}\); for the  
Figure 16: A trunk-type robot with three segments, each having three tendons. second robot segment \(l_{21}\), \(l_{22}\), \(l_{23}\), and for the third robot segment \(l_{31}\), \(l_{32}\), \(l_{33}\). According to this, nine cable holes have to be on the robot's first segment connecting disks, six cable holes have to be on the robot second segment connecting disks, and three cable holes have to be on the robot third segment disks. Punches used for robot cables from the same cable group have to be spaced \(120^{\circ}\) along all connecting disks. Cable hole positions were first defined in the first segment connecting disks and later on the second and third segment cable guide disks. The starting robot segment cable hole on the first segment guide disk is on the \(y\)-axis for first segment control cable \(l_{11}\) (Figure 17a). The holes for cables \(l_{12}\), \(l_{13}\) are placed correspondingly. Disk holes for cables \(l_{21}\), \(l_{22}\), \(l_{23}\) on the first segment are simply the cable \(l_{11}\), \(l_{12}\), \(l_{13}\) holes rotated \(40^{\circ}\) clockwise around point \(O\).