A classification of a family of 3-revolute (3R) positioning manipulators is established. This classification is based on the topology of their workspace. The workspace is characterized in a half-cross section by the singular curves of the manipulator. The workspace topology is defined by the number of cusps and nodes that appear on these singular curves. The design parameters space is shown to be partitioned into nine subspaces of distinct workspace topologies. Each separating surface is given as an explicit expression in the DH-parameters.
This paper focuses on positioning 3R manipulators with orthogonal joint axes (orthogonal manipulators). Orthogonal manipulators may have different global kinematic properties according to their link lengths and joint offsets. Unlike usual industrial manipulators, orthogonal manipulators may be cuspidal, that is, they can change their posture without meeting a singularity (Parenti andInnocenti, 1988, Burdick, 1988). This property was unknown before 1988 (Borrel and Liégeois, 1988). Several years later, some conditions for a manipulator to be noncuspidal were provided, which include simplifying geometric conditions like parallel and intersecting joint axes (Burdick, 1995) and also nonintuitive conditions (Wenger, 1997). A general necessary and sufficient condition for a 3-DOF manipulator to be cuspidal was established in (El Omri and Wenger, 1995), namely, the existence of at least one point in the workspace where the inverse kinematics admits three equal solutions. The word "cuspidal manipulator" was defined in accordance to this condition because a point with three equal IKS forms a cusp in a cross section of the workspace (Arnold, 1981). The categorization of all generic 3R manipulators was established in (Wenger, 1998) based on the homotopy class of the singular curves in the joint space. Wenger, 1999 proposed a procedure to take into account the cuspidality property in the design process of new manipulators. More recently, Corvez and Rouillier, 2002 applied efficient algebraic tools to the classification of 3R orthogonal manipulators with no offset on their last joint. Five surfaces were found to divide the parameters space into 105 cells with the same number of cusps in the workspace. The equations of these five surfaces were derived as polynomials in the DH-parameters using Groebner Bases. A kinematic interpretation of this theoretical work showed that, in fact, only five different domains exist : two domains of noncuspidal manipulators, one domain where manipulators have two cusps and two domains where they have four cusps (Baili et al, 2003). However, the authors did not provide the equations of the true separating surfaces in the parameters space. On the other hand, they did not take into account the occurrence of nodes, which play an important role for analyzing the number of IKS in the workspace.
The purpose of this work is to classify a family of 3R positioning manipulators according to the topology of their workspace, which is defined by the number of cusps and nodes that appear on the singular curves. The design parameters space is shown to be divided into nine domains of distinct workspace topologies. In each domain, the distribution of the number of IKS is the same. This study is of interest for the design of new manipulators.
The rest of this article is organized as follows. Next section presents the manipulators under study and recalls some preliminary results. The classification is established in section 3. Section 4 synthesizes the results and section 5 concludes this paper.
The manipulators studied in this paper are orthogonal with their last joint offset equal to zero. The remaining lengths parameters are referred to as d2, d3, d4, and r2 while the angle parameters 2 and 3 are set to -90° and 90°, respectively. The three joint variables are referred to as 1, 2 and 3, respectively. They will be assumed unlimited in this study.
Figure 1 shows the kinematic architecture of the manipulators under study in the zero configuration. The position of the end-tip (or wrist center) is defined by the three Cartesian coordinates x, y and z of the operation point P with respect to a reference frame (O, x, y, z) attached to the manipulator base as shown in Fig. 1.
The determinant of the Jacobian matrix of the orthogonal manipulators under study is det(J) = (d3 + c3d4)(s3d2 + c2(s3d3 -c3r2)), where c i =cos( i ) and s i =sin( i ). A singularity occurs when det(J)=0. The contour plot of det(J)=0 forms a set of curves in
, the first factor of det(J) cannot vanish and the singularities form two distinct curves S1 and S2 in the joint space (El Omri, 1996), which divide the joint space into two singularity-free open sets A1 and A2 called aspects (Borrel and Liégeois, 1988). The singularities can also be displayed in the Cartesian space (Kholi andHsu, 1987, Ceccarelli, 1996). Thanks to their symmetry about the first joint axis, a 2-dimensional representation in a half cross-section of the workspace is sufficient. The singularities form two disjoint sets of curves in the workspace. These two sets define the internal boundary WS1 and the external boundary WS2, respectively, with WS1=f(S1) and WS2=f(S2). Figure 2 shows the singularity curves when d2=1, d3=2, d4=1.5 and r2=1. For this manipulator, the internal boundary WS1 has four cusp points. It divides the workspace into one region with two IKS (the outer region) and one region with four IKS (the inner region).
If d3d4, the operation point can m
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