# Energy Optimization of Functionally Redundant Robots through Motion Design
## 1 Introduction  
The optimization of robotic operation is a topic that has drawn a considerable attention and has been the central topic of countless works, also in the light that several metrics can be defined to measure the performance of a robot. Most often the only tool available for optimizing a robotic operation in the trajectory planning, as most industrial robots are programmed by defining their motion as a sequence of via-points or by composing the motion using a pre-defined set of motion primitives. Within this framework, traditionally trajectory optimization has been used to minimize the execution time of a task [1, 2], or the smoothness of the motion profile [3, 4] for minimum motion-induced vibrations. Later on, the minimization of the actuator effort, usually defined as the mechanical energy required to drive the robot joint or a quadratic torque norm, has been investigated in several works, such as [5, 6, 7]. Recently, the attention has shifted to the investigation and optimization of the energy consumption of a robotic system [8]. Such works are motivated by the energy saving policies supported by the European Union, as well as by the clear economic advantage associated with increased efficiency. The review work [8] lists several methods to reduce energy consumption, among which are the use of regenerative motor drives and energy sharing on a bus [9, 10], the use of mechanical energy storage devices [11, 12, 13, 14, 15]. Such methods are listed in the work [8] as hardware solutions, meaning that the energy efficiency enhancement is obtained by introducing some physical modifications to the system. On the other hand, all methods that do not require any physical alteration are listed as software solutions, being based just on the modification of the software that handles the robotic operation. Among them, the most common solution is to define an energy-optimal motion profile: within this approach the estimated energy saving can be very significant, since improvements up to 33% are testified by [PERSON] in the work [16]. The energy optimization can be used also together with the exploitation of redundancy, that can be either intrinsic or functional. An intrinsically redundant robot is one for which the dimension of the joint space is larger than the dimension of the operational space, as in the case of a seven degrees of freedom robot. A manipulator is then said to be functionally redundant when the dimension of the operational space is larger than the dimension of the task space: in this case the number of degrees of freedom required by the task is larger than the number of degrees of freedom of the end-effector of the robot [17]. The key concept behind the use of redundancy as a tool to optimize a robot performance metric is that among the infinite solutions of the inverse kinematic problem an optimal one can be defined. One example is provided in the work [18], where the use of a kinematically redundant SCARA robot is proposed and it is shown that redundancy can be exploited to maximize the capability of the robot to produce high-speed motion. The topic of energy saving in a kinematically redundant robotic cell has been investigated by two of the authors of this work in [19], in which it is shown that by adding an additional degree of freedom to a SCARA robot through a moving platform, its energetic performance when executing a pick and place task can be significantly improved. According to the classification proposed above, such work can be classified as a mixed approach, since it combines both hardware modifications by the added degree of freedom, and software modifications through the trajectory modification. This work explores a similar topic by focusing on functional redundancy. Several operations that are commonly performed by robots in industry are actually characterized by functional redundancy, considering that a radial symmetry of the end-effector tool is sufficient to define a functional redundancy.