# Finite-Horizon Kinetic Energy Optimization of a Redundant Space Manipulator
## 1 Introduction
[14], who introduced the dynamic models of hydraulic and electrical actuators and showed that a major advantage of energy minimization compared to joint velocities minimization is that it tends to move joints with lower inertia more than those with higher one, thus reducing energy consumption. As pointed out by [PERSON] [2] in his review, these early studies made it very clear that a trade-off between local optimization performance and global stability is generally unavoidable: pointwise minimization must always be relaxed in order not to get stuck in singularities. Several works have been published on stabilization of joint torques minimization methods: most of them are aimed at minimizing joint torques without considering any robot payload (worthy of notice are [15] by [PERSON] and [16] by [PERSON] et al. ); nevertheless, the minimization of torques in the presence of external loads has also been addressed ([PERSON] et al. [7]). However, the solutions they propose are only marginally useful for the methods that minimize kinetic energy. Other recent developments of pseudoinverse-based methods are aimed at addressing specific shortcomings of existing formulations; for example [PERSON] et al. [6] presented a solution to minimize computational times by setting a variable priority value for each end-effector task. Alongside pseudoinverse-based methods, other inverse kinematics methods based on Quadratic Programming have been proposed. These are considered promising since they have been proven capable of solving inverse kinematics problems with equality and inequality constraints [17]. These methods, however, have been proposed for joint torques minimization [18] rather than kinetic energy minimization. Alongside local inverse kinematics methods, offline methods based on optimal control have been pursued. These techniques are slower, but allow for better optima of the cost function, although at the price of much longer computation times. Their cost function is usually a sum of the end-effector position error and a control cost (usually kinetic energy or actuators torque) computed on a specific set of points. The set may correspond to the whole trajectory or to a finite number of points after the last completed integration step. In [19], [PERSON] et al. presented an optimal control method for energy optimization along a trajectory subject to holonomic constraints based on Euler-Lagrange equations. However, this method has only proven efficient for non-redundant manipulators. A different optimal control approach, based on Pontryagin maximum principle, is proposed in [20] by [PERSON] et al. for electromechanical linear actuators. In this work, a mathematical model of the actuators to be included in the cost function is presented, but the adaptability of this approach to manipulators with revolute joints is doubtful. Several other works address the issue of minimizing the energy required to perform the whole robot trajectory using offline methods. Among those, an early attempt was presented by [PERSON] et al. [21], featuring an optimal control based minimization of the squared joint velocities along the trajectory, as a means to reduce the kinetic energy in the operations of a non-redundant manipulator. In [22], [PERSON] et al. investigated an offline path planning method for non-redundant manipulators using a multi-objective cost function involving both trajectory time and mechanical energy. In [23], [PERSON] et al. presented a solution of the energy minimization problem based on dynamic programming. Their intuition does not serve the purpose of finding the global optimum, since each one of the iterations only explores a subspace of the search space; however, it manages to avoid mediocre local minima, and keeps the execution time reasonably low avoiding the curse of dimensionality usually involved with dynamic programming. [PERSON] et al. [24] recently presented an energy optimal solution for hydraulically powered redundant manipulators based on dynamic programming.