## 1 Introduction
A control scheme based on a feedback linearization method is then used to generate smooth reference link positions that excite the mode corresponding to the desired periodic motion that is performed employing a small amount of energy. Linearization and modal analysis techniques are used also in [25; 26] to minimize the consumed mechanical energy of actively controlled multibody systems moving between two given points. In particular, the optimal spring design, the optimal trajectory, and the operating time are determined using the optimal control theory. A concurrent optimization of the springs and the trajectory is performed also in [27; 28] where both the trajectory and the nonlinear spring profile are parameterized by means of polynomial functions whose coefficients are determined to minimize actuation torques. All methods provide efficient ways to tune the elastic elements added to the system for minimizing the energy consumption, but they also impose constraints on the kind of path that the robot must follow, i.e., they do not allow to use an arbitrary trajectory for the operation. This can be a drawback  
Figure 1: Serial (**a**) and parallel (**b**) spring configuration as shown in [PERSON] et al. [19]. in several scenarios, e.g., when an obstacle avoidance planning is necessary, if the robot controller allows only for the implementation of standard motion primitives, if an optimal motion planning with a different objective function is to be used, such as minimum execution time [29] or smoothness of the motion profile [30]. Few works in the literature deal with the optimal design of springs when the system moves following an arbitrary, given trajectory. Additionally, all these works have been developed for a specific type of system. In [31], a constrained optimization problem is formulated to reduce the peak torque requirement of a one-dof rotating link by finding the stiffness value and placement of a linear spring to be added in parallel with the joint actuator. In [32], springs in parallel with the actuators are used to minimize the mechanical energy required by the motors of a four-dof parallel robot performing predefined pick-and-place trajectories. In [33], an analytical method has been developed with reference to a one-dof rotating beam provided with a regenerative drive. The method allows to find the optimal design of the spring balancing the system that minimizes the electric energy of rest-to-rest motions. The authors of the present work in [34] achieved energy consumption minimization retrofitting a Delta-2 robot with both regenerative drives recovering braking energy and springs, properly tuned for a chosen trajectory by means of numerical optimization, storing elastic energy. As promising results have been obtained (e.g., energy reduction up to 70%), this paper further develops and generalizes such an approach for robotic systems performing cyclic tasks. In particular, the electromechanical model of the system provided with regenerative drives and linear springs mounted in parallel with the actuators is described. It is demonstrated analytically that the addition of the springs is always beneficial for reducing the energy consumption of the system if they are properly designed in terms of stiffness and preload. The optimal values for the spring parameters are analytically determined through a bounded optimization problem whose objective function is the reduction of the electrical energy consumption. No constraints are set on the system trajectory, so the energy efficiency can be improved for any arbitrary trajectory (e.g., the current one or an energy-optimal motion profile to further boost the energy savings). Numerical results of the two test cases, a five-bar mechanism and a SCARA robot, show that the consumed energy can be significantly reduced. The paper is organized as follows. Section 2 reports the electromechanical model of robots provided with regenerative drives.