This paper is dedicated to the relationship between the external force applied on a point of a robot end-effector and its consequent displacement in static conditions. Both the force and the displacement are herein considered in the Euclidean space $E(3)$. This fact represents a significant simplification of the approach, since it avoids some problems related to the absence of a natural positive definite metric on the Special Euclidean Group $SE(3)$. On the other hand, such restriction allows the method to find closed-form solutions to a large class of problems in robot statics. The peculiar goal of this investigation consists of setting up a procedure which guarantees at least one pose at which any force applied (in $E(3)$) to an end-effector point is always parallel to its consequent displacement (also in $E(3)$). This property, which will be referred to as isotropic compliance in $E(3)$, makes the robot tip static behavior uniform with respect to all directions, namely, isotropic, although not homogeneous, since it holds only in some poses. Achieving isotropic compliance in $E(3)$ is a task more general than the classical problem of finding a pose with unit condition number, which does not include the case of different elements in the diagonal joint stiffness matrix. For this reason, the object of the present investigation could not be furtherer simplified to the classical kinetostatic problem in terms of the jacobian matrix alone. The paper reveals how the force–displacement parallelism can be achieved by using a method based on a simple proportional-derivative (PD) controller strategy. The method can be applied when the passive and active stiffness act, on the joints, either in parallel or in series, and the magnitude of the displacement response can be chosen by imposing appropriate values for the overall joints compliance. Results show that for the three analyzed examples, namely, the RR, RRP, and RRR manipulators, with arbitrary lengths of the links, there is, at least, one pose for which the sought property is achieved.

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