## 3 After Failure Strategy Approach
### Connecting Path
#### 3.1.1 Linear Programming Problem
It is indeed possible to evaluate the actuator speed \(\dot{\theta}_{i}\) in terms of the end-effector speed \(v\) from (6) as follows  
\[\dot{\theta}_{i}=\frac{l_{i}}{\rho}=\frac{-\mathbf{u}_{i}^{T}\mathbf{v}}{\rho} \tag{12}\]  
Analogously, the actuator acceleration \(\ddot{\theta}_{i}\) can be expressed as:  
\[\ddot{\theta}_{i}=\frac{\ddot{l}_{i}}{\rho}=\frac{-\mathbf{u}_{i}^{T}\bar{ \mathbf{r}}-\dot{\mathbf{u}}_{i}^{T}\mathbf{v}}{\rho} \tag{13}\]  
Substituting (12) and (13) in (5) the actuator dynamics can be expressed as:  
\[c_{i}+\tau_{i}\rho+b\frac{\mathbf{u}_{i}^{T}\mathbf{v}}{\rho}=-j_{m}\frac{ \mathbf{u}_{i}^{T}\bar{\mathbf{r}}}{\rho}-j_{m}\frac{\dot{\mathbf{u}}_{i}^{T} \mathbf{v}}{\rho} \tag{14}\]  
and the torque \(c_{i}\) of each actuator can be related to \(\mathbf{S}\) by obtaining the acceleration of the end-effector \(\bar{r}\) from its dynamics:  
\[\mathbf{M}\bar{\mathbf{r}}=\mathbf{S}\boldsymbol{\tau}+\mathbf{G} \tag{15}\]  
where \(\mathbf{M}\) is the mass matrix of the end-effector and \(\mathbf{G}\) the weight of the platform, thus obtaining:  
\[c_{i}=-\frac{b}{\rho}\mathbf{u}_{i}^{T}\mathbf{v}-\frac{j_{m}}{\rho}\dot{ \mathbf{u}}_{i}\mathbf{v}-\tau_{i}\rho-\frac{j_{m}}{m\rho}\mathbf{u}_{i}^{T} \mathbf{S}\boldsymbol{\tau}-\frac{j_{m}}{m\rho}\mathbf{u}_{i}^{T}\mathbf{G} \tag{16}\]  
Lastly, the equations of the torques \(c_{i}\) of the actuators can be grouped in matrix form  
\[\mathbf{c}=-\frac{b}{\rho}\mathbf{S}^{T}\mathbf{v}-\frac{j_{m}}{\rho}\dot{ \mathbf{S}}^{T}\mathbf{v}+\left(-\mathbf{I}\rho-\frac{j_{m}}{m\rho}\mathbf{S}^ {T}\mathbf{S}\right)\boldsymbol{\tau}-\frac{j_{m}}{m\rho}\mathbf{S}^{T} \mathbf{G} \tag{17}\]  
The maximum breaking effect can be found by solving the linear programming problem defined as follows  
\[maximize:(-sign(v_{y}^{\prime})f^{\prime}y)=(-sign(v^{\prime}y)\mathbf{w}_{ \mathbf{R}_{y}}^{T}\mathbf{\tau}_{w})\]  
\[\begin{cases}\mathbf{w}_{\mathbf{R}\mathbf{x}}^{T}\mathbf{\tau}_{w}>0\\ \mathbf{w}_{\mathbf{R}\mathbf{z}}^{T}\mathbf{\tau}_{w}=0\\ 0\leq\boldsymbol{\tau}_{\text{{min}}}\leq\boldsymbol{\tau}\leq\boldsymbol{\tau} _{\text{{max}}}\\ \boldsymbol{\tau}_{f}=0\\ \mathbf{c}_{\text{{max}}}+\frac{b}{\rho}\mathbf{S}^{T}\mathbf{v}+\frac{j_{m}}{ \rho}\dot{\mathbf{S}}^{T}\mathbf{v}+\frac{j_{m}}{m\rho}\mathbf{S}^{T}\mathbf{ G}>\left(-\mathbf{I}\rho-\frac{j_{m}}{m\rho}  
evaluate the maximum wrench that can be exerted along a given direction \(d\) while keeping null all the other wrench components. The adopted geometric approach is based on the definition of two n-dimensional polytopes: \(\mathbf{T}\), which represents the acceptable cable tension configurations, and \(\mathbf{\Omega}\), which represents the corresponding forces exerted on the moving platform [15]. The latter can be obtained from \(\mathbf{S}^{T}\) and \(\mathbf{T}\) and provides a graphical evaluation of the maximum and minimum force applicable along a given direction. However, this approach requires the definition of the direction to analyse, which cannot be predetermined when defining the Connecting Path. Hence, this work applies the following iterative procedure to define the appropriate direction:  
* In the first iteration, the algorithm rotates the mobile reference frame of 90\({}^{\circ}\) along the \(z\)-axis, in order to align the \(x\)-axis with the direction opposing \(v^{\prime}_{y^{\prime}}\). * The WEC index geometric formulation is used to evaluate the maximum and minimum force along the new \(x\)-axis;
* The evaluated force limits are verified in terms of cable tensions and actuator torques, i.e., the constraints in (18) must be satisfied;
* If the constraints are not satisfied, a new iteration is carried out, rotating the initial mobile frame along the \(z\)-axis of an angle lower than the previous one, i.e., \(<\)90\({}^{\circ}\), until the constraints are fulfilled. Despite its iterative nature, the proposed approach has showed to be an order of magnitude faster than the linear programming approach in several tests.