## 1 Introduction  
The ever-growing awareness of the negative consequences on the environment of the energy needs of manufacturing systems has led to the study and development of methodologies to reduce energy consumption. Industrial robots make up a significant part on the total electrical energy consumed in the manufacturing sector [1]. Studies to reduce the energy consumption of these systems have been recently reviewed by [PERSON] et al. [2]. Two main categories have been identified: software and hardware solutions. The software approaches rely mainly on the optimization of system motion planning and, therefore, require a modification of just the software that handles the robotic operations. Improvements in energy efficiency can be achieved by optimizing the location of the task with respect to the base of the manipulator [3] or by exploiting the system intrinsic or functional kinematic redundancy [4; 5; 6]. Other approaches are based on the trajectory optimization toward minimizing the energy consumption of point-to-point [7], multipoint [8] trajectories, and maximizing the energy sharing in multi-axis robots provided with a common Direct Current (DC) bus [9; 10]. The energy efficiency enhancement by means of hardware solutions is obtained by introducing physical modifications to the system. An example of hardware solution is the lightweight design of the system and its components [11] by means of, e.g., the use of novel materials [12]. The energy efficiency of an industrial robot can be improved by replacing its traditional drives with regenerative drives [13; 14; 15; 16]. These devices convert the braking energy of the system into regenerative electric energy that is returned to the system when required, instead of dissipating it as heat. Another way to store energy for transferring it to the system consists in providing the system with energy-storing devices such as flywheels [17] or compliant elements as, e.g., springs. The adoption of springs is by far most used in robotic systems with respect to flywheels. They are typically installed at robot joints in a serial or parallel connection [18], see Figure 1. The serial arrangement generally requires to substitute stiff actuators with compliant ones. Such a configuration is typically adopted to reduce force peaks due to impacts as for example in human-robot interaction. Conversely, the parallel configuration does not require the substitution of the original actuators, resulting in an easier and cheaper installation. The addition of springs to robotic systems for reducing the actuator effort was first adopted for gravity balancing [20; 21]. Gravity balanced robots are more energy efficient than non-equilibrated ones as they do not need actuator effort to stay in equilibrium at any configuration. Methods that use springs to improve the efficiency of robotic systems in repetitive operations are based on the concept of the natural motion [19]. The basic idea of these methods is to shape by means of a proper tuning of the spring stiffness the free vibration response of the robot so that it matches with the repetitive tasks that the robot has to perform. An iterative procedure to exploit the system free response for performing pick-and-place tasks between two given points in a prescribed amount of time is presented in [22; 23]. A multibody simulator is used to simulate the free response of the system for the given initial position and for guess values of the springs mounted in parallel with the actuators. Simulations are stopped when the optimal spring values are found, i.e., when after the desired time the position of the system is close enough to the final given position. In [24], adjustable compliant actuators mounted in series with the joints are used to exploit the natural dynamics of serial robots. The actuator stiffnesses are tuned by imposing the desired natural frequencies in the decoupled modal model of the system obtained through linearization of the dynamic model.