# Energy Optimization of Functionally Redundant Robots through Motion Design
## 3 UR5 Robot Parameters  
The robot chosen as the test-case here is a Universal Robot UR5 manipulator. Most of its mechanical and electrical parameters are available in the literature including the robot manufacturer datasheet [31], or the components manufacturers' one. The kinematic properties of the manipulator, according to the manufacturers' data, are collected in Table 1.  
The mass properties of the robot are collected in Table 2 by listing the center of mass and the total mass of each link, as declared by the manufacturer.  
As for the other parameters required by Equations (1)-(6), they have been obtained by the commercial catalogs of the components used in the UR5.  
The motors are produced by Kollmorgen and belong to the KBM 'frameless' series [32]: three'size 1' motors, which can exert a peak torque of 28 Nm at the joint, and three'size 3' motors, which can deliver up to 150 Nm at the joint. The actual motors can actually deliver higher torque values, as the torque is limited by the robot control unit to enforce the the maximum safety force limits [31]. To cope with the lack of more specific data, according to the authors' knowledge, the electrical and mechanic parameters of the motors have been estimated through similar motors. Such data are collected in Table 3, including the continuous service torque \(T_{cs}\), the back-emf constant \(k_{b}\), the torque constant \(k_{t}\), winding resistance \(\Omega\), motor shaft inertia \(J_{m}\), static friction torque \(T_{C}\) and viscous friction constant \(f_{\text{p}}\).  
The reducers used in the UR5 are harmonic drive speed reducers [33] and belong to the HFUS-2 SH family with 100:1 reduction ratio [34]. Again, the data used to set-up the dynamic model refer to the reducers that, according to the datasheet, better fit the specifications provided by the robot manufacturer. The main parameters that describe the reducers are reported in Table 4, including the static friction \(T_{C}\), the reducer moment of inertia \(J_{r}\) and the average efficiency \(\eta\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Joint \#** & **\(\theta\) [rad]** & **a [m]** & **d [m]** & **\(\alpha\) [rad]** \\ \hline
1 & 0 & 0 & 0.089159 & \(\pi/2\) \\
2 & 0 & \(-0.425\) & 0 & 0 \\
3 & 0 & \(-0.39225\) & 0 & 0 \\
4 & 0 & 0 & 0.10915 & \(\pi/2\) \\
5 & 0 & 0 & 0.09465 & \(-\pi/2\) \\
6 & 0 & 0 & 0.0823 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Denavit-Hartenberg parameters of the UR5 robot.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Link \#** & **Mass [kg]** & **Center of Mass Position [m]** \\ \hline
1 & 3.7 & \([0,-0.02561,0.00193]\) \\
2 & 8.393 & \([0.2125,0,0.11336]\) \\
3 & 2.33 & \([0.15,0.0,0.0265]\) \\
4 & 1.219 & \([0,-0.0018,0.01634]\) \\
5 & 1.219 & \([0,0.0018,0.01634]\) \\
6 & 0.1879 & \([0,0,-0.001159]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mass properties of the UR5 robot.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Joint \#** & \(T_{cs}\) [Nm] & \(k_{b}\) [V/rad/s] & \(k_{t}\) [Nm/a] & \(R\) [\(\Omega\)] & \(J_{m}\) [kg m\({}^{2}\)] & \(T_{C}\) [Nm] & \(f_{\text{p}}\) [Nm \&/rad] \\ \hline
1 & 2.87 & 0.61 & 1.05 & 9.0 & \(8.8\times 10^{-5}\) & \(7.4\times 10^{-2}\) & \(6.6\times 10^{-5}\) \\
2 & 2.87 & 0.61 & 1.05 & 9.0 & \(8.8\times 10^{-5}\) & \(7.4\times 10^{-2}\) & \(6.6\times 10^{-5}\) \\
3 & 2.87 & 0.61 & 1.05