## 3 Approximate Inverse Kinematics Solution for Imposed End-Effector State
### Determination of Approximate Robot Spine State Configuration  
The solution of the three inextensible segments for the continuum robot starts with a consideration of initial conditions. First, the definition of continuum robot spine full-length \(l_{\text{r}}\) is necessary. \(l_{\text{r}}\) is the sum of all three segments' lengths \(L_{n}\) (Figure 2(a)). The three segments' lengths. \(L_{1}\), \(L_{2}\), and \(L_{3}\), explicitly define the full robot spine length. Point \(G\) in 3D space represents the target point that the robot end-effector has to reach. Furthermore, \(\overset{\wedge}{\mathbf{o}}_{r}\). defines the desired robot end-effector orientation at the point \(G\). Figure 4 depicts all the necessary initial conditions and state. The robot spine's length is a fixed-length estimate and should match \(l_{\text{r}}\) because the robot segments are inextensible. Knowing this and having information about the preset target point \(G\) and orientation \(\overset{\wedge}{\mathbf{o}}_{r}\) parametric curve equations are necessary. First, the authors propose an empiric tuning parameter \(\lambda_{\text{a}}\) and its expression (1) that will play in final spine state expressions:  
\[\lambda_{\text{a}}=\frac{5}{6}\left(-l_{\text{r}}+\frac{\sqrt{k_{\text{r}} \left(l_{\text{r}}^{2}\left(\sigma_{\text{rx}}^{2}+\sigma_{\text{ry}}^{2}+ \left(1+o_{\text{rz}}\right)^{2}\right)-24l_{\text{r}}\big{(}\,o_{\text{rx}}x _{\text{g}}+o_{\text{ry}}y_{\text{g}}+z_{\text{g}}+o_{\text{rz}}z_{\text{g}} \big{)}+144\left|\mathbf{g}\right|^{2}\right)}}}{k_{\text{r}}}\right). \tag{1}\]Here, \(k_{\rm r}=-31+120v+\alpha_{\rm rx}^{2}+\alpha_{\rm ry}^{2}+\alpha_{\rm rz}(10+ \alpha_{\rm rz})\). The target point is \(G=\left(x_{\rm g},\ y_{\rm g},\ z_{\rm g}\right)\) (in the Cartesian coordinate system). \(\alpha_{\rm r}=\left(\alpha_{\rm rx},\ \alpha_{\rm ry},\ \alpha_{\rm rz}\right)\) is the unit vector, which defines the robot's third segment endpoint's direction. \(v\) is a theoretic speed constant (\(v\ =\ 1\)). The trunk-type robot spine's full length is \(l_{\rm r}\), which equals the sum of all three segments' lengths. The closed-form expressions for coordinates of point \(P_{\rm e3}\) are contingent upon tuning the coefficient \(\lambda_{\rm a}\) expression in Equation (1). Point \(P_{\rm e3}\) expresses the estimated robot spine's endpoint. The latter point is the closest solution to the target point \(G\). Point \(P_{\rm e3}\) coordinates' equations are necessary for creating estimated robot spine curve parametric equations as functions of time:  
\[x_{\rm e3}=\frac{\lambda_{\rm a}\ l_{\rm r}\ o_{\rm rx}\ +\ 10\ l_{\rm r}\ x_{\rm g}}{ 12\ \lambda_{\rm a}\ +\ 10\ l_{\rm r}}, \tag{2}\]  
\[y_{\rm e3}=\frac{\lambda_{\rm a}\ l_{\rm r}\ o_{\rm ry}\ +\ 10\ l_{\rm r}\ y_{\rm g}}{ 12\ \lambda_{\rm a}\ +\ 10\ l_{\rm r}}, \tag{3}\]  
\[z_{\rm e3}=\frac{l_{\rm r}\left(\lambda_{\rm a}\ +\ \lambda_{\rm a}\ o_{\rm rz} \ +\ 10\ z_{\rm g}\right)}{ 2\left(6\lambda_{\rm a}\ +\ 5\ l_{\rm r}\right)}, \tag{4}\]  
\[P_{\rm e3}=(x_{\rm e3},y_{\rm e3},z_{\rm e3}). \tag{5}\]  
Finally, Equations (6)-(8) define robot spine curve parametric equations as a function of time, which will define the spine's curvature:  
\[\hat{x}_{\rm ec}(t)=\frac{t^{2}(x_{\rm e3}(-2t+3l_{\rm r})+(t-l_{\rm r})l_{ \rm r}\ o_{\rm rx})}{l_{\rm r}^{3}}, \tag{6}\]  
\[\hat{y}_{\rm ec}(t)=\frac{t^{2}(y_{\rm e3}(-2t+3l_{\rm r})+(t-l_{\rm r})l_{ \rm r}\ o_{\rm ry})}{l_{\rm r}^{3}}, \tag{7}\]  
\[\hat{z}_{\rm ec}(t)=\frac{t(z_{\rm e3}t(-2t+3l_{\rm r})+l_{\rm r}(-t+l_{\rm r })(l_{\rm r}-t(1+o_{\rm rz})))}{l_{\rm r}^{3}}, \tag{8}\]  
\[\hat{P}_{\rm ec}(t)=(\hat{x}_{\rm ec}(t),\hat{y}_{\rm ec}(t),\hat{z}_{\rm ec }(t)), \tag{9}\]  
Figure 4: Three segments of full-length spine \(l_{\rm r}\) (the robot is in the initial position), goal point \(G\), and orientation vector \(\mathbf{o}_{r}\), defined as initial conditions.