Robotica (1989) volume 7, pp 323-325 Isoelastic behavior of parallel robots F. Artigue, M. Y. Amirat and J. Pontnau LIMRO; 9, Avenue de la division Leclerc, 94230 Cachan {France) (Received in Final Form: October 28, 1988) SUMMARY This paper presents a particular architecture of a parallel robot which is characterized by a diagonal stiffness matrix. This result suggests the use of a parallel robot in the final phase of insertion as a passive compliance device. The stiffness rate of this device is controlled by the gain of the feedback loop. As the correction of small angular misalignments due to contact forces do not generate lateral errors and vice versa, we have the equivalent of an isoelastic swivel. KEYWORDS: Parallel robot; Isoelastic behaviour; Diagonal stiffness; Compliance device. 1. INTRODUCTION In the assembly tasks framework parallel robots, which possess concurrent mobilities, offer numerous advantages compared with sequential robots (additive mobilities). 1 Among these advantages it may be quoted a high positioning accuracy and a force feedback control easy to implement. The purpose of this paper is to show how a particular mechanical architecture of a parallel robot may exhibit a diagonal stiffness matrix. In this case, an external force induces a displacement in the same direction as the force applied. In the same way a pure force induces a translation and a pure torque a rotation. This result is important because it suggests the use of parallel manipulators as passive compliance devices. 2 The stiffness rate of the device is controlled by the gain of the feedback position loop, so the correction of small misalignments due to contact forces does not generate lateral errors and vice versa. Next, this property is proved to belong to a class of parallel manipulators. 2. PARALLEL MANIPULATOR DESIGN AND KINEMATIC MODELING Parallel manipulators include a static and a mobile part linked by linear actuators. Their mechanical architecture differs mainly by the disposition of linear actuators and the type of joint. Two configurations are generally used. The first one (Figure 1) includes a swivel at each end of the actuator. This mechanical configuration which includes two swivels rigidly linked allows five degrees of freedom like this: 2x2/? (rotations) +17? (common rotation to the swivels along the rigid link). In the second one (Figure 2), the linear actuators are rigidly tied on the base, while the mobile part is linked to the actuator through a C 5 link, i.e. a swivel and two perpendicular sliding plates (3/? + 2 T). Afterwards a general modeling of this kind of manipulator is presented and then applied to the parallel manipulator with C 5 links. A displacement is described by a tensor operator included in an Euclidean frame, center (O), a translation vector f and a rotation vector (/?). 3 The displacement of a point Mi tied to the mobile part is defined by a Di vector: RR{OMI) is obtained from OMi by rotation R. For small amplitude displacements, a first order approximation allows us to write: RR(OM) = OMi + R A OMi (2.2) For each linear actuator, the following notations are used: Di is the displacement vector common to the actuator and the mobile part. P is the required lengthening of the actuator. n is the unit vector along the initial direction of the linear actuator. swivel linear actuator swivel static part Fig. 1. Parallel robot with C 3 links. sliding plates swivel linear actuator static part Fig. 2. Parallel robot with C 5 links.