## 2 Materials and Methods
### Propulsion System's Inverse Dynamic Modeling  
The dynamic modeling of parallel manipulators is not straightforward since these systems are based on closed-loop structures. Motion from one limb should be restricted by the other limbs' motions and the desired pose of the platform. Hence, coupled relationship between system parameters, high nonlinearity in system dynamics and kinematics must be considered [27]. In this article, the reduced inverse dynamic modeling based on the Lagrange formalism was computed. This approach allows one to calculate the actuator forces \(\mathcal{F}_{i}\) while it deals with the passive joints as a set of equations of closed loop chains. With this analysis, forces applied to the line of action on each actuator can be properly computed to create the desired displacement inside the linear actuators. Figure 4 describes the position of the centers of mass from each actuator and the line of action to which force is applied on each limb. Figure 3: Revolute joints attached on the base of the fixed platform. Two rotations are studied from each universal joint. From Figure 4, the parameter \(\delta\) determines the total length from the actuators' hull to which center of mass \(C_{m1 Li}\) is placed at the middle. Moreover, \(\varepsilon\) determines the total distance between the attaching point \(B_{i}\) and the center of mass \(C_{m2 Li}\) from the moving piston inside the linear actuator. \(\mathcal{F}_{i}\) represents the force applied to the actuator and required to generate motion. Hence, the position of each center of mass \((\mathcal{A}_{m1 Li}^{A},P,\mathcal{A}_{m2 Li}^{A}P)\) inside each limb measured from the inertial frame \(\{O_{A}\}\) can be defined as:  
\[\mathcal{A}_{m1 Li}P=\left[ \begin{array}{c}R_{A}c(\Phi_{\text{A}i})+(\delta/2 )s_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})-(\delta/2)c_{1i}s_{2i}\\ (\delta/2)c_{2i}\end{array} \right],\qquad\quad\mathcal{A}_{m2 Li}P=\left[ \begin{array}{c}R_{A}c(\[PERSON]_{\text{A}i})+(d_{i}-\varepsilon)s_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})-(d_{i}-\varepsilon)c_{1i}s_{2i}\\ (d_{i}-\varepsilon)c_{2i}\end{array} \right]\qquad for\;i=1, \tag{11}\]  
\[\mathcal{A}_{m1 Li}P=\left[ \begin{array}{c}R_{A}c(\[PERSON]_{\text{A}i})-(\delta/2 )c_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})-(\delta/2)s_{1i}s_{2i}\\ (\delta/2)c_{2i}\end{array} \right],\qquad\quad\mathcal{A}_{m2 Li}P=\left[ \begin{array}{c}R_{A}c(\[PERSON]_{\text{A}i})-(d_{i}-\varepsilon)c_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})-(d_{i}-\varepsilon)s_{1i}s_{2i}\\ (d_{i}-\varepsilon)c_{2i}\end{array} \right]\qquad for\;i=2, \tag{12}\]  
\[\mathcal{A}_{m1 Li}P=\left[ \begin{array}{c}R_{A}c(\[PERSON]_{\text{A}i})+(\delta/2 )c_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})+(\delta/2)s_{1i}s_{2i}\\ (\delta/2)c_{2i}\end{array} \right],\qquad\quad\mathcal{A}_{m2 Li}P=\left[ \begin{array}{c}R_{A}c(\[PERSON]_{\text{A}i})+(d_{i}-\varepsilon)c_{1i}s_{2i}\\ R_{A}s(\[PERSON]_{\text{A}i})+(d_{i}-\varepsilon)s_{1i}s_{2i}\\ (d_{i}-\varepsilon)c_{2i}\end{array} \right]\qquad for\;i=3. \tag{13}\]  
From Equations (11)-(13), \(c_{1i},\;c_{2i},\;s_{1i},\;s_{2i}\) are short notations for \(\cos(\theta_{1i}),\cos(\theta_{2i}),\)\(\sin(\theta_{1i}),\;\sin(\theta_{2i})\) respectively. Furthermore, the Lagrange formalism is applied from the concepts of work and energy present on the whole mechanism. The Lagrange equation describes the difference between the kinetic and potential energies (\(E_{K},\;E_{P}\), respectively) from all the moving parts of the system. As previously stated, the parallel mechanism consists of three independent symmetric manipulator arms, whose centers of mass are placed on the same position for each actuator. Additionally, for the present study, it is assumed that the friction forces at all joints are negligible.