Workspaces of Planar Parallel Manipulators
This paper presents geometrical algorithms for the determination of various workspaces of planar parallel manipulators. Workspaces are defined as regions which can be reached by a reference point C located on the mobile platform. First, the maximal worksp
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1 Introduction Parallel manipulators have been proposed as mechanical architectures which can overcome the limitations of serial robots [5]. Parallel manipulators lead to complex kinematic equations and the determination of their workspace is a challenging problem. Some researchers have addressed the problem of the determination of the workspace of parallel manipulators ([2]; [8]; [11]; [12]), especially for computing the workspace of the robot when its orientation is fixed. In this paper, the problem of the determination of the workspaces of planar 3 d.o.f parallel manipulators is addressed. Algorithms are proposed for the determination of the maximal workspace, a problem which has been elusive to previous analyses. Planar 3 d.o.f. parallel manipulators are composed of three kinematic chains connecting a mobile platform to a fixed base. The manipulator of particular interest in this study is referred to as the 3 - RP R manipulator. In this manipulator, the mobile platform is connected to the base via three identical chains consisting of a revolute joint attached to the ground followed by an actuated prismatic joint which is connected to the platform by a revolute joint (figure 1). Henceforth, the center of the joint connecting the ith chain to the ground will be denoted A; and the center of the joint connecting the ith chain to the platform will be referred to as B;. Others types of planar parallel manipulators are: the 3- RRR robot ([1];[9];[5]) in which the joints attached to the ground are the only actuated joints and the 3 - P RR robot in which the prismatic joints are actuated. A fixed reference frame is defined on the base and a moving reference frame is attached to the platform with its origin at a reference point C. The position of the moving platform is defined by the coordinates of point C in the fixed reference frame and its orientation is given by the angle () between one axis of the fixed reference frame and the corresponding axis of the moving frame. For the 3 - RP R manipulator, the workspace limitations are due to the limitations of the prismatic actuators. The ~aximum and minimum lengths of the prismatic actuator of the jth chain are denoted pim=, flmin· These values will be referred to as extreme values of the joint coordinates. A. Morecki et al. (eds.), ROMANSY 11 © Springer-Verlag Wien 1997
J-P. Merlet, C.M Gosselin, N. Mouly
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Figure I: The 3 - RP R parallel manipulator. Furthermore, an annular region £ is defined as the region which lies between two concentric circles with different radii. The circle £e with the largest radius will be referred to as the external circle and the smaller circle£' (if it exists) will be referred to as the internal circle. The dimensions of the manipulators which are used in the examples are given in the appendix. In what follows, the presentation of the various workspaces will focus on the 3 - RP R manipulator.
2 Maximal workspace The maximal workspace is defined as the region which the reference point C can reach with at least one orientation. It shall be
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