# Finite-Horizon Kinetic Energy Optimization of a Redundant Space Manipulator
## 3 Results and Discussion
### Simulation Results
This behavior is explained by considering that, when a joint is rotated, the same rotation is also transferred to the successive joints; thus, the moment of inertia of a link is constituted by the sum of its own moment of inertia and an additional term for each of the successive joints. While the LSV method simply minimizes the joints velocity norm, the PMKE method, similarly to other methods that minimize kinetic energy, takes into account the fact that the motion of the joints closer to the base are more energetically expensive. Stroboscopic and joint velocities plots in Figures 4 and 5 confirm that the LSV method features a joint motion more uniform than that of the PMKE method. In fact, for trajectory 1, the first joint features a maximum velocity of 0.063 rad/s in the LSV solution and 0.017 rad/s in the PMKE solution, while the third joint reaches up to 0.197 rad/s in the LSV solution and 0.203 rad/s in the PMKE solution. While the maximum velocity of the third joint is around three times the velocity of the first one in the LSV solution, the third joint is more than 10 times faster than the first one in the PMKE solution. Similarly, for trajectory 2, the first joint has a peak velocity of 0.196 rad/s in the LSV solution and a peak velocity of 0.095 rad/s in the PMKE solution, while the third joint velocity reaches 0.529 rad/s in the LSV solution and 0.613 rad/s in the PMKE solution. This behavior is typical of local kinetic energy minimization algorithms. However, PMKE is able to depart from it when required to obtain improved optimization results. A very clear example of this case is provided by trajectory 3, for which the simulation results are presented in Figure 6. This trajectory is singular using the LSV method, while, on the other hand, the PMKE method is able to avoid the singularity. This remarkable behavior is allowed by the fact that the PMKE method, although generally striving to keep the first joint velocity low, moves it (and the other ones) away from singularities when required, if they fall within its prediction window. The stroboscopic plots in Figure 6 show that, while LSV (right) hits a singularity at \(t\) = 6.5 s, PMKE (left) features a different motion altogether, with first and second joints reaching higher velocities (excluding the singularity in LSV): first joint reaches 0.250 rad/s in the PMKE solution and 0.186 rad/s in the LSV solution, while second joint reaches 0.261 rad/s in the PMKE solution and 0.023 rad/s in the LSV solution. Figure 6: Trajectory 3, comparison between PMKE (**left**) and LSV (**right**). First joint: blue lines; second joint: red lines; third joint: yellow lines. Joint velocities at the singularity not shown in LSV. Finally, the PKME-LSV comparison for trajectory 4 is presented in Figure 7. This is the longest trajectory that has been simulated, and it would be impossible to cover its whole length while also limiting the motion of the first joint. In this case PMKE and LSV feature similar joint velocity profiles, and PMKE features a consistently higher first joint maximum velocity (sitting at 0.780 rad/s, against 0.512 rad/s in the LSV solution). In this case, the PMKE method provides a solution with a lower kinetic energy integral by making the robot to assume configurations that feature lower coefficients of the inertia matrix, compared to those obtained with the LSV method. In order to verify that the coefficients of the inertia matrix are lower for the PMKE solution, a possible indicator is the maximum eigenvalue of the manipulator inertia matrix, which is known as its spectral radius. In fact, the spectral radius of a square matrix A is defined as:  
\[\rho(A)=\text{ max }\{|\lambda_{1}|,\ \ldots,\ |\lambda_{n}|\} \tag{6}\]  
where \(\lambda_{1},\ \ldots,\lambda_{n}\) are the eigenvalues of the matrix \(A\). Figure 8 shows the evolution of the spectral radius of the inertia matrix of the robot manipulator along trajectory 4 for both the LSV (dotted line) and PMKE (dashed line) solutions.