# Finite-Horizon Kinetic Energy Optimization of a Redundant Space Manipulator
## 1 Introduction  
In space robotics missions, both for space servicing or debris capture, the minimization of the robot energy consumption and the minimization of the reactions transferred to the base spacecraft are of crucial importance [1]. Both of these targets are related to a longer system lifetime in orbit: while energy minimization allows longer operations by definition, reactions minimization allows to keep the base spacecraft orientation unchanged, which in turn means fuel savings due to the fact that attitude corrections are not necessary. In the case of trajectory tracking of redundant manipulators, inverse kinematics methods are usually adopted, which however only provide local minima along the end-effector trajectory. Thus, as trajectories increase in length, they fail to follow the lowest energy path in the robot joints space, and tend to be singularity prone. Differential kinematic inversion has been extensively used for the solution of the real-time motion of redundant manipulators, while minimizing a certain cost function [2, 3, 4, 5, 6, 7]. It has been used in plenty of variations, as it has several interesting properties: it has fast execution times, can be used to prioritize specific tasks over others, and it can be used to optimize a variety of different cost functions while driving the robot end-effector through a predefined path. However, this comes at a price: it only provides local optima, which can be quite far from the global one [2], and it is subject to the problem of singularities. In fieldapplications [8], like space or underwater robotics, avoiding singularities and using as little energy and power as possible is important as robots usually run on limited energy sources, such as batteries or solar panels. In such applications, local inverse kinematics frameworks are not ideal, and, on the other hand, global optimization methods are too computationally demanding. Minimization of energy consumption of robotic systems has been subject of intense research, since it helps reducing operational costs, and allows us to use robots even when resources are limited. Focusing on robot manipulators, many attempts to minimize kinetic energy through inverse kinematics have been focused on minimizing the joint velocities norms. To the best of the authors' knowledge, the first systematic review of this topic has been proposed by [PERSON] and [PERSON] [9], which is a milestone in the research on the use of the pseudoinverse of the Jacobian matrix for robot control. The paper outlines how the pseudoinverse generally keeps the required instantaneous power low, since it minimizes joint velocities, which has been considered as equivalent to approximately minimize the kinetic energy. It also states that the pseudoinverse method, since it minimizes joint velocities, keeps the manipulator away from singularities, which was however proven incorrect by [PERSON] et al. [10]. Many of the subsequent works developed their algorithms starting from the concept of weighted pseudoinverse, as presented by [PERSON] [11]. In particular, several researchers have followed the direction of pointwise minimizing the joint torques as a means to minimize the actuator energy consumption, associating such an algorithm with positive results regarding the minimization of the kinetic energy integral along the whole trajectory. Extensive analysis on this approach is presented by [PERSON] and [PERSON] [12], who however found out that a dynamics-based inverse kinematics resolution method can lead to instabilities on longer trajectories. A more strictly energy-focused approach was followed by [PERSON] et al. [13], who proposed a method for joint torques minimization that holds some global properties. However, it does not allow to minimize kinetic energy when initial conditions on joint displacements are imposed and, although less instable than the previous work, still presents stability issues. The first research providing actuator models for energy consumption minimization was presented by [PERSON] et al.