## 2 Materials and Methods
### Propulsion System's Inverse Dynamic Modeling
Additionally, to correctly solve the system, six constraint equations \(f_{k}\) that are functions of \(d_{i},\ \theta_{1,i},\ \theta_{2,i}\), and their time derivatives, should be properly defined. Moreover, the six Lagrange multipliers \(\lambda_{k}\) should be solved by solving iteratively six simultaneous equations, i.e., Equation (21) when \(j=4,\ldots 9\). Based on the schematic of the kinematics chains from Figure 2, two vectorial closed loop constraint equations may be defined as follows:  
\[\{O_{A}^{\rightarrow}\}A_{1}+A_{1}^{\rightarrow}B_{1}+B_{1}\{\overset{ \rightarrow}{O}_{M}\}+\{O_{M}\}B_{2}+B_{2}^{\rightarrow}A_{2}+A_{2}\{ \overset{\rightarrow}{O}_{B}\}=0, \tag{22}\]  
\[\{O_{A}^{\rightarrow}\}A_{2}+A_{2}^{\rightarrow}B_{2}+B_{2}\{\overset{ \rightarrow}{O}_{M}\}+\{O_{M}^{\rightarrow}\}B_{3}+B_{3}^{\rightarrow}A_{3}+A_ {3}\{\overset{\rightarrow}{O}_{B}\}=0. \tag{23}\]  
Vectorial Equations (22) and (23) are rewritten as scalar equations to generate the six constraint equations \(f_{k}\) where \((k=1,\ldots,6)\):  
\[R_{A}c(\[PERSON]_{\text{A}1})+d_{1}s_{11}s_{21}+d_{2}c_{12}s_{22}-R_{A}c(\[PERSON]_{\text {A}2})-{}_{M}^{A}R_{B_{1}}^{M}P_{x}+{}_{M}^{A}R_{B_{2}}^{M}P_{x}=0; \tag{24}\]  
\[R_{A}s(\[PERSON]_{\text{A}1})-d_{1}c_{11}s_{21}-{}_{M}^{A}R_{B_{1}}^{M}P_{y}+{}_{M} ^{A}R_{B_{2}}^{M}P_{y}+d_{2}s_{12}s_{22}-R_{A}s(\[PERSON]_{\text{A}2})=0; \tag{25}\]  
\[d_{1}c_{21}-d_{2}c_{22}-{}_{M}^{A}R_{B_{1}}^{M}P_{z}+{}_{M}^{A}R_{B_{2}}^{M}P_{ z}=0; \tag{26}\]  
\[R_{A}c(\[PERSON]_{\text{A}2})-d_{2}c_{12}s_{22}-{}_{M}^{A}R_{B_{2}}^{M}P_{x}+{}_{M} ^{A}R_{B_{2}}^{M}P_{x}-d_{3}c_{13}s_{23}-R_{A}c(\[PERSON]_{\text{A}3})=0; \tag{27}\]  
\[R_{A}s(\Phi_{\text{A}2})-d_{2}s_{12}s_{22}-{}_{M}^{A}R_{B_{2}}^{M}P_{y}+{}_{M} ^{A}R_{B_{3}}^{M}P_{y}-d_{3}s_{13}s_{23}-R_{A}s(\[PERSON]_{\text{A}3})=0; \tag{28}\]  
\[d_{2}c_{22}-d_{3}c_{23}-{}_{M}^{A}R_{B_{2}}^{M}P_{z}+{}_{M}^{A}R_{B_{3}}^{M}P_{ z}=0. \tag{29}\]  
Finally, the actuating force along the \(i\)th limb is computed by directly solving Equation (21), using the computation of \(\mathcal{L}\) and \(f_{k}\) and its derivatives with respect to \(d_{i}\). The three equations of force required per limb on the designed system are then defined as:  
\[\mathcal{F}_{1}=\ddot{d_{1}}m_{2}-\left(\dot{\theta}_{1,1}\right)^{2}m_{2}(d_ {1}-\varepsilon)(s_{21})^{2}-\left(\dot{\theta}_{2,1}\right)^{2}m_{2}(d_{1}- \varepsilon)+m_{2}gc_{21}-\lambda_{1}s_{11}s_{21}+\lambda_{2}c_{11}s_{21}- \lambda_{3}c_{21}; \tag{30}\]  
\[\mathcal{F}_{2}=\ddot{d_{2}}m_{2}-\left(\dot{\theta}_{1,2}\right)^{2}m_{2}(d_ {2}-\varepsilon)(s_{22})^{2}-\left(\dot{\theta}_{2,2}\right)^{2}m_{2}(d_{2}- \varepsilon)+m_{2}gc_{22} \tag{31}\]  
\[\{O_{A}^{\rightarrow}\}A_{2}+A_{2}^{\rightarrow}B_{2}+B_{2}\{\overset{ \rightarrow}{O}_{M}\}+\{O_{M}^{\rightarrow}\}B_{3}+B_{3}^{\rightarrow}A_{3}+A_ {3}\{\overset{\rightarrow}{O}_{B}\}=0. \tag{23}\]0  
\[\{O_{A}^{\rightarrow}\}A_{2}+A_{2}^{\rightarrow}B_{2}+B_{2}\{\overset{ \rightarrow}{O}_{M}\}+\{O_{M}^{\rightarrow}\}B_{3}+B_{3}^{\rightarrow}A_{3}+A_ {3}\{\overset{\rightarrow}{O}_{B}\}=0. \tag{23}\]1  
Equations (30)-(32) might be rearranged so that the robotic system can be defined by a dynamic equation in matrix form. Hence, the forces required to actuate each limb will be defined as:  
\[\{O_{A}^{\rightarrow}\}A_{2}+A_{2}^{\rightarrow}B_{2}+B_{2}\{\overset{ \rightarrow}{O}_{M}\}+\{O_{M}^{\rightarrow}\}B_{3}+B_{3}^{\rightarrow}A_{3}+A_ {3}\{\overset{\rightarrow}{O}_{B}\}=0. \tag{23}\]2  
where \(F_{i}\) is the (3 \(\times\) 1) matrix containing the required forces \((\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3})^{T}\), \(D\) is a (3 \(\times\) 3) diagonal mass matrix, \(C\) is a (3 \(\times\) 6) Coriolis and centrifugal forces matrix, \(G\) is the (3 \(\times\) 1) gravitational forces matrix, and \(\lambda_{j}\) is the (3 \(\times\) 1) Lagrange multipliers and constraints matrix.