## 2 Materials and Methods
### Propulsion System's Inverse Dynamic Modeling
The total kinetic energy of the whole system is defined as:  
\[E_{K}=E_{Kp}+\sum_{i=1}^{3}E_{K_{Li}}, \tag{14}\]  
where \(E_{Kp}\) is the kinetic energy from the platform and \(E_{K_{Li}}\) is the kinetic energy from limb \(i.\) Since the platform only presents rotational motion, the equation of kinetic energy \(E_{Kp}\) is defined as:  
\[E_{Kp}=\frac{1}{2}I_{P}\bigg{(}\dot{\alpha}^{2}+\dot{\beta}^{2}+\dot{\gamma}^{ 2}\bigg{)}, \tag{15}\]  
Figure 4: View of an actuated limb and the fourth restrictive limb inside the parallel mechanism; position of centers of mass and force applied to the actuator’s line of action are displayed. where \(I_{P}\) is the inertial moment computed for the solid disk that compounds the moving platform and \(\omega=\left(\dot{a},\dot{\beta},\dot{\gamma}\right)^{T}\) is the angular velocity of the platform. The center of mass on the moving platform is placed at the origin of the reference frame \(\{O_{M}\}\). On the other hand, the equation of kinetic energy on each limb \(i\) will consequently be defined as:  
\[ \begin{array}{rl}E_{K_{Li}}&=\frac{1}{2}\left(\dot{\theta}_{1,i}\right)^{2} \left[m_{1}(\delta/2)^{2}(s_{2i})^{2}+I_{L}+m_{2}(d_{i}-\varepsilon)^{2}(s_{2 i})^{2}+I_{U}\right]\\ &+\frac{1}{2}\left(\dot{\theta}_{2,i}\right)^{2}\left[m_{1}(\delta/2)^{2}+I_{ L}+m_{2}(d_{i}-\varepsilon)^{2}+I_{U}\right]+\frac{1}{2}\left(\dot{d}_{i} \right)^{2}m_{2}.\end{array}  \tag{16}\]  
In Equation (16), \(I_{L}\) represents the inertial moment from the first part of the limb (actuator's hull) while \(I_{U}\) represents the inertial moment of the upper part of the actuator. Furthermore, the potential energy for the whole system will be defined as:  
\[E_{P}=E_{P_{P}}+\sum_{i=1}^{3}E_{I_{Li}}. \tag{17}\]  
Since the parallel mechanism is designed to only follow rotational motions, the potential energy at the moving platform \(E_{P_{P}}\) at its center of mass will not vary over time. It will be defined as:  
\[E_{P_{P}}=M_{P}g_{O_{M}}^{A}P; \tag{18}\]  
where \(M_{P}\) is the mass of the moving platform, \(g\) is the gravitational acceleration and \(\overset{A}{O_{M}}P\) is the height presented between platforms. The potential energy at the centers of mass from each limb \(i\) is defined as:  
\[E_{P_{Li}}=m_{1}g[(\delta/2)c_{2i}]+m_{2}g[(d_{i}-\varepsilon)c_{2i}]. \tag{19}\]  
Finally, using Equations (14)-(19), the Lagrange equation \(\mathcal{L}\) for the whole system is defined as:  
\[\mathcal{L}=E_{K}-E_{P}. \tag{20}\]  
Derivation of Limb's Force Equations  
Actuating forces \(\mathcal{F}_{1},\ \mathcal{F}_{2},\ \mathcal{F}_{3}\) applied to the limbs are to be found based on the derivation of the Lagrange equation from the previous analysis. For the present design, the normalized limb lengths \(d_{i}\) are the independent generalized coordinates and the revolute \(\theta_{1,i},\ \theta_{2,i}\) inside the universal passive joints are the dependent generalized coordinates. Then, the Lagrange equations of motion are defined by:  
\[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial q_{j}}\right)-\left( \frac{\partial\mathcal{L}}{\partial q_{j}}\right)+\sum_{k=1}^{6}\lambda_{k} \,\frac{\partial f_{k}}{\partial q_{j}}=\mathbb{Q}_{j};\ \ \ \ j=1,\ldots,9, \tag{21}\]  
where:  
\[q_{j}= \begin{cases}d_{i}&j=i=1,2,3\\ \theta_{1,i}&j=4,5,6;i=1,2,3\\ \theta_{2,i}&j=7,8,9;i=1,2,3\end{cases} \ \ \ \ \text{and}\ \ \ \ \ \mathbb{Q}_{j}= \begin{cases}\mathcal{F}_{i}&j=i=1,2,3\\ 0&j=4,\ldots,9\end{cases} .\]  
In real-time applications when controlling these types of mechanisms, the platform's orientation, velocity, and acceleration are known or predetermined. It is based on such predispositions where the interest of determining the forces required to actuate the limbs is of paramount importance. By properly defining the force necessary to actuate the links, the platform will follow a desired trajectory. Equation (21) defines a system of 9 equations to which the first three serve the purpose of computing the values of the forces applied on each actuator (when \(j=1,2,3\)).