# Natural Motion for Energy Saving in Robotic and Mechatronic Systems
## 4 Design Optimization with Natural Motion

Table 2: Optimal design formulation with natural motion. An alternative approach is to tackle the problem with an optimal control theory approach. This is used for the optimal design problems by [PERSON] et al. [56; 100], by [PERSON] et al. [82], who along with [PERSON] et al. [101] implement the [PERSON]'s minimum principle (see Section 3.2.1). Most of the reviewed works adopt objective functions based on actuators torque, on consumed energy or power. A few works use objective functions based on task time and positions [103], position and velocity of the end-effector [81] or position and velocity errors [91]. Results shown in the last column of Table 2 indicate that all the reviewed approaches for the optimal design problems in natural motion consistently reduce energy consumption. The vast majority of papers utilize gradient-based optimization algorithms. Despite the great efficiency of these algorithms, the validation is typically carried out on simple test cases (as reported in Table 3) with low number of design variables and a convex objective functions. This review did not result in any research using genetic algorithms or evolutionary strategies employed in the field of natural motion. Although, these algorithms are less efficient, they can often handle non-convex problems, discrete design variables and noisy system functions. As it is the goal to consider complex mechanical systems and expanded design problems, the optimization formulation will require a higher number design variables including the parameters \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Mechanical System** & **Gravity** & **Experimental Tests** & **Ref.** & **1 st Author** & **Year** \\ \hline
1-DOF revolute joint & & & [7] & [PERSON] & 2017 \\  & & & ✓ & [40] & [PERSON] & 2018 \\  & & ✓ & [105] & [PERSON] & 2017 \\  & ✓ & ✓ & [58] & [PERSON] & 2017 \\  & ✓ & & [56] & [PERSON] & 2011 \\  & ✓ & & [77] & [PERSON] & 2007 \\  & ✓ & & [98] & [PERSON] & 2008 \\  & ✓ & & [86] & [PERSON] & 2013 \\  & ✓ & ✓ & [79] & [PERSON] & 2015 \\ \hline
2-DOF planar robot (RR) & ✓ & & [40] & [PERSON] & 2018 \\  & ✓ & & [56] & [PERSON] & 2011 \\  & ✓ & & [100] & [PERSON] & 2016 \\  & ✓ & & [104] & [PERSON] & 2017 \\  & ✓ & ✓ & [94] & [PERSON] & 2016 \\  & ✓ & & [93] & [PERSON] & 2016 \\  & ✓ & & [58] & [PERSON] & 2017 \\  & & & [64] & [PERSON] & 2005 \\  & & & [95] & [PERSON] & 2005 \\  & & & [101] & [PERSON] & 2009 \\  & ✓ & & [96] & [PERSON] & 2017 \\  & ✓ & & [77] & [PERSON] & 2007 \\  & ✓ & & [97] & [PERSON] & 2008 \\  & ✓ & ✓ & [99] & [PERSON] & 2009 \\  & & & [78] & [PERSON] & 2014 \\  & & ✓ & [86] & Velasco & 2013 \\  & & & [79] & Velasco & 2015 \\ \hline
2-DOF planar robot (PP) & ✓ & & [96] & [PERSON] & 2017 \\ \hline
3-DOF SCARA & & ✓ & [16] & [PERSON] & 2012 \\ \hline
3-DOF planar robot (RRR) & ✓ & & [73] & [PERSON] & 2017 \\ \hline
3-DOF spatial robot (RRR) & ✓ & & [102] & [PERSON] & 2012 \\  & ✓ & & [61] & [PERSON] & 2013 \\ \hline
3-DOF spherical robot (RRP) & ✓ & & [96] & [PERSON] & 2017 \\ \hline
4-DOF spatial robot (RRRR) & ✓ & & [73] & [PERSON] & 2017 \\ \hline Slider-crank mechanism & & & [95] & [PERSON] & 2005 \\ \hline Five-bar linkages & & & [103] & [PERSON] & 2017 \\  & ✓ & & [80] & [PERSON] & 2018 \\  & ✓ & & [91] & [PERSON] & 2019 \\  & & & [82] & [PERSON] & 2018 \\ \hline Delta-2 robot & ✓ & & [90] & [PERSON] & 2019 \\ \hline Delta robot & ✓ & & [81] & [PERSON] & 2018 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical systems employed for the numerical and experimental validations, grouped together with respect to their kinematic structure. The check-marks on the second and third columns indicate whether gravity is considered and whether experimental tests are performed, respectively. of the motion law. This may lead to non-convex optimization problems for which gradient-based algorithms are not suitable. Thus, the proper optimization formulation, the system equations and corresponding choice of optimization algorithm will play a critical role moving forward with the optimal design with natural motion.