# Natural Motion for Energy Saving in Robotic and Mechatronic Systems
## 4 Design Optimization with Natural Motion  
In the previous sections, we have analyzed the mechanical design of both robotic and mechatronic systems for natural motion and of desired natural dynamics to ensure energy efficiency. It should be noted that, in the majority of the works, the exploitation of natural motion is strictly related to the definition of an optimal design problem. Indeed, spring parameters (with variable or non-variable compliance and linear or non-linear behavior) as well as trajectory parameters are designed using an optimization strategy. In this section, we provide an overview on the optimization problems adopted for the design of trajectory and spring parameters. Table 2 reports a summary on the optimal design problems, showing design variables, objective and constraint functions and algorithms adopted in the reviewed works. Furthermore, in the last column of the table the results obtained in terms of energy (or torque) efficiency are reported. Before analyzing the optimization problems in detail, it is necessary to make a distinction between works in which the design formulation includes spring parameters, works in which it considers trajectory parameters or both. In problem formulations in which the spring parameters are to be determined, both stiffness and equilibrium position are typically design variables. The second design formulation finds the optimal values for the trajectory and task time, frequency or coefficients of the motion profile are design variables. Another formulation is the concurrent design optimization of spring parameters and trajectory properties. This is the most complex optimization formulation to be implemented, since the properties of elastic elements usually influence the trajectory and vice versa. Despite the complexity, this formulation allows for a high degree of design flexibility and, therefore, potential to further reduce energy consumption. Several methods are used to solve the optimal design problem with natural motion. Due to their efficiency with convex problems, the majority of the works about natural motion utilize gradient-based optimization algorithms. Due to their use of gradients and also use of Hessians, these algorithms are referred to as first and second order algorithms. [PERSON] et al. [81] adopt the gradient-based trust-region-dogleg algorithm on an unconstrained problem, whereas [PERSON] et al. [73], [PERSON] et al. [89] and [PERSON] et al. [80] use MATLAB's gradient-based fmincon function. [PERSON] et al. [40], as well as [PERSON] et al. [65] and [PERSON] et al. [83] adopt a gradient descent algorithm. The unconstrained gradient-based shooting method was used by [PERSON] et al. in Reference [91]. [PERSON] et al. [61, 102] adopts the sequential quadratic programming (SQP) algorithm. A gradient-free method is implemented in Reference [103], where a downhill simplex algorithm is considered. In some cases, an analytical solution is found to the optimal design problem, avoiding the cost of setting up and solving numerical optimization problems. These papers include References [79, 86, 96, 98, 99]. Figure 9: The 1-DOF mechanism with variable compliance presented by [PERSON] et al. [105]. The linear spring acts as a parallel rotational spring at the joint and its stiffness can be tuned by controlling the ball screw mechanism.