## 4 Robot Bending Simulation Results
The light blue curve is an estimated robot spine curve, and calculated robot segments arc curves are in blue, red, and green: (**a**) robot spine view when \(G=(50,\;-33,100)\) and \(\overset{\wedge}{\mathbf{o}}_{r}=(0,\;0,\;1)\); (**b**) robot spine view when \(G=(94,\;-54,20)\) and \(\overset{\wedge}{\mathbf{o}}_{r}=(0.6509,\;-0.6509,\;-0.3906)\); (**c**) robot spine view when \(G=(55,\;-55,80)\) and \(\overset{\wedge}{\mathbf{o}}_{r}=(-0.6155,\;0.6155,\;0.4924)\); (**d**) robot spine view when \(G=(40,\;40,\;40)\) and \(\overset{\wedge}{\mathbf{o}}_{r}=(0,\;0,\;-1)\). where the first vector is \(\mathbf{a}\) and the second vector \(\mathbf{b}\) is the angle \(\theta\) between them. Redesigning the dot product via Equation (59) produces the orientation error expression:  
\[e_{\mathrm{o}}=\ \cos^{-1}\frac{\overset{\overset{\mathbf{A}}{\mathbf{S}_{\text{orienN3 }}}}{||\overset{\mathbf{A}}{\mathbf{S}_{\text{orienN3}}}||}\overset{\overset{ \mathbf{A}}{\mathbf{S}_{\text{orienN3}}}}{||\overset{\mathbf{A}}{\mathbf{S}_{ \text{orienN3}}}||}}. \tag{60}\]  
Table 2 exposes orientation errors between vectors \(\overset{\mathbf{A}}{\mathbf{o}_{\text{r}}}\) and \(\mathbf{s}_{\text{orienN3}}\) and position errors between goal point \(G\) and bent robot third segment endpoint \(P_{3}\). From Table 2, it is clear that the algorithm imposes both constraint conditions: soft for segment goal point and hard for robot end orientation. However, substantial position errors, defined as the distance between desired and achieved robot endpoints, can be seen in some situations, Table 2. For example, the fifth simulation results contain a 27.0281 cm position error. Three main concerns affect the bent robot position accuracy. First, some goal points that have to reach the robot end effector's target position exceeded the robot's configuration space. Second, the position error is noticeable because the robot position is soft-constrained, while the primary condition is orientation. Figures 18-21 show that reaching the desired locations is possible in most simulations but not with the desired robot end orientations. This is the main reason for the substantial position errors in some cases. metal wires (tendons) by using Equations (48)-(57) for all three robot segments. Such open-loop (prediction/planning) motion profiles will automatically impose final rigid body motion that practically solves the inverse dynamic problem. That is why this section further elaborates on simulation results dedicated to motor speed control profiles. First, this study presents angular speed profiles for any generic electric motor installation with or without a gearbox. The only customization parameter value is the radius of a motor rotor or a gear. The parameter helps translate rotational mechanical motion to longitudinal motion, which directly controls a tendon. Accordingly, the speed motion profiles and the kit radii provide an opportunity to plan torque profiles for electric motors and their gearboxes, if any, in the future. Next, an expression for tendon lengths' rate of change is necessary to arrive at the angular speed contours. Based on Equations (48)-(57), the approximation of the first-order derivative of cable lengths, as a function of time (Figure 22), is as follows:  
\[v_{clumi}=\frac{\Delta l}{\Delta t}=\frac{l_{nm,i}-l_{nm,i-1}}{\Delta t}, \tag{61}\]  
where a cable length \(l_{nm}\) belongs to the robot's \(n\)-th segment and its \(m\)-th cable, the observation's discrete-time index is \(i\), and \(\Delta t\) is the sampling interval in seconds. Calculating the robot cable length longitudinal speed motor angular speed is then done as follows:  
\[\omega_{nm}=\frac{v_{clumi}}{r_{\text{ms}}}, \tag{62}\]  
where the motor rotor or gear radius is \(r_{\text{ms}}\). Motor shaft angular speed also involves the representation of rotational speed:  
\[\omega_{nm}=n_{nm}\left(\frac{2\pi}{60}\right), \tag{63}\]  
where motor or gear rotational speed is \(n_{nm}\), which belongs to the \(n\)-th segment and its \(m\)-th cable.