## 1 Introduction  
One important problem in the operation of many industrial manipulators is the transmission of vibrations to the fixed frame during their high-speed motion. These vibrations are produced by the unbalanced inertia forces that increase the shaking force and shaking moment. Thus, a traditional and still very important research topic in robot design is finding new methods to remove or decrease these alternating dynamic loads transmitted to the base And, in this way, to reduce wear, allowing to increase the operating velocities. The shaking force balancing, i.e., the reduction or, in the best case, elimination of the variation in the force transmitted to the base, is a problem that has been solved in different ways. One of the most popular and effective is mass redistribution. This method proposed by different authors was successfully introduced in Reference [1] using the \"linear independent vectors\" method. In general terms, the objective is to find the appropriate location of the center of mass of the links of the mechanisms in order to make the general Center of Mass (CoM) stationary for any motion of the manipulator. Practical ways to move the CoM of the system and to make it stationary have been to add a set of counterweights [2; 3; 4; 5], to add auxiliary structures [6; 7; 8; 9], or to modify the form of the links from the early design phase [10; 11; 12]. A comprehensive review of the methods with illustrative examples can be seen in Reference [13]. Specific applications to robotics can be seen in Reference [14; 15; 16; 17; 18; 19; 20], and a review of these kinds of applications is in Reference [21]. However, all these methods imply a considerable increment in the mass of the system, causing the shaking moment and, hence, the driving torque to grow; see Reference [22]. An additional drawback of these methods is that they also imply modification of the original system, making its application somehow hard or complicated, leading to relatively complex mechanical systems increasing their size. Recently, a new approach has been proposed. It is based on the optimal control of the manipulator global CoM [23; 24]. The method is based on planning the displacements of the total CoM of the moving links. Its trajectory is defined as a straight line with \"bang-bang\" motion profile, ensuring solely the initial and final positions of the end-effector. This approach allows the reduction of the maximum value of the acceleration of the CoM, leading to a reduction in the shaking force. In this work, this approach is resumed and presented using fully Cartesian coordinates, also known as natural coordinates [25]. The application of natural coordinates to the dynamic balancing of mechanisms in the plane has been explained in Reference [26]. An application to obtain the balancing conditions of planar mechanisms is described in Reference [27]. A specific application to the dynamic balancing of the four-bar mechanism, explaining the way to obtain the dynamic reactions to be used in an optimization procedure, can be found in Reference [28]. Initial efforts to use this approach to obtain the balancing conditions in spatial mechanisms can be seen in Reference [29]. The methods presented in References [23; 24] use reference-point coordinates to calculate the position of the global CoM. Then, this position has to be expressed in terms of the degrees of freedom and some convenient relative coordinates. In this process, the kinematic constraints (vector closed-loop equations) that relate the reference-point coordinates and the selected relative coordinates are used. The process becomes somehow cumbersome when dealing with closed-loop mechanisms, especially in three-dimensional space. One advantage of using fully Cartesian coordinates is that, in complex multibody systems, points and vectors can be chosen at the joints usually shared by contiguous bodies, contributing to a definition of joints with few or no constraint equations while keeping moderate the number of variables.