# Finite-Horizon Kinetic Energy Optimization of a Redundant Space Manipulator
## 3 Results and Discussion  
### Simulation Setup  
A 3-DOF planar manipulator model (with three revolute joints) is considered, which corresponds to a real robot prototype for space applications that has been used in previous works [3; 4; 5; 25]. The two end-effector Cartesian coordinates are controlled, \(x\) and \(y\), thus leaving one degree of kinematic redundancy. Geometrical and inertial characteristics of the manipulator, which is depicted in Figure 1, are presented in Table 3.  
A simulator of the manipulator inverse kinematics has been developed using the Matlab programming language, except for the code used for steps 4-5 of the PMKE method described in Section 2.1, which corresponds to the minimization of the kinetic energy integral: the routine for this optimization has been first written in Matlab exploiting the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\Delta t\)**[s]** & \(\Delta T\)**[s]** & \(h\) & \(I\)**[s]** \\ \hline  & 0.01 & 1 & 2 & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main parameters of the PKME method.  
Figure 1: Space robot prototype considered in this work.  
_fminon_ function, and then translated into C, compiled, and interfaced to the inverse kinematics simulator.  
In order to show the effectiveness of the presented method, four trajectories have been simulated with both the PMKE and a traditional method, based on the Moore-Penrose pseudoinverse and performing the minimization of the joint velocity vectors through local least-squares optimization. This method is called here Least Squares Velocities (LSV) minimization method. Moreover, a further comparison is made with the results computed with a global optimal method [25] recently published by the authors.  
Simulations 1-2 feature two short end-effector trajectories aimed at demonstrating the PMKE method general performance. Simulation 3 features a trajectory in which the LSV method encounters a singularity. Finally, simulation 4 features a trajectory whose length causes a significant degradation of the performance of the LSV method. All of the proposed trajectories start from the same end-effector position \(x_{init}=[0.468;\ 0]\) m and robot initial configuration \(q_{init}=[0;\ 0.327;-0.754]\) rad, and run for a total time T = 10 s. The considered trajectories and their characteristic dimensions are presented in Table 4.  
The end-effector velocity profiles have been chosen in order to have sufficiently smooth accelerations (continuous and with continuous derivative), as shown in Figure 2. The equation of the derivative of the curvilinear abscissa of the end-effector used is:  
\[\frac{ds}{dt}=\left\{\begin{array}{ll}2*\left(\frac{\frac{1}{ 2}}{\left(\frac{1}{2}\right)^{2}}\right)*\left(\frac{\frac{2}{2}}{\tau}+\ \left(2*\beta\right)^{-2}\ *\cos((2*\beta)*\mathrm{t})-\ \left(2* \beta\right)^{-2}\right),&t\leq\frac{T}{2}\\ \\ \mathrm{L}\ -\left(2*\left(\frac{\frac{1}{2}}{\left(\frac{1}{2}\right)^{2}} \right)*\left(\frac{\left(\mathrm{T}-\mathrm{t}\right)^{2}}{2}+\ \left(2*\beta\right)^{-2}\ *\cos \left(\left(2*\beta\right)*\left(\frac{T}{2}-\mathrm{t}\right)\right)-\ \left(2* \beta\right)^{-2}\right)\right),&t>\frac{T}{2}\end{array}\right. \tag{5}\]  
where T is the total motion time, L the overall length of the considered trajectory, and \(\beta=\frac{2\pi}{T}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Simulation nr.** & **Shape** & **Characteristic Dimension** & **Length [m]** \\ \hline
1 & Rectilinear & Length & 0.10 \\
2 & Circular & Radius & 0.05 \\
3 & Rectilinear & Length & 0.40 \\
4 & Rectilinear & Length & 0.70 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulated trajectories.  
Figure 2: End-effector curvilinear abscissa and its derivatives.