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DOMAINS OF OPERATION AND INTERFERENCE FOR BODIES IN MECHANISMS AND MANIPULATORS E.J. HAUG, F.A. ADKINS, AND C.M. LUH Center for Computer-Aided Design and Department of Mechanical Engineering The University of Iowa Iowa City, Iowa 52242

Abstract. Domains associated with one or more working bodies of a mech-

anism or manipulator are de ned that characterize the range of operation of the bodies and interference that may occur between the bodies. The equations de ning domains are derived in terms of equations de ning the kinematics of the mechanism and the geometry of the working bodies. Analytical criteria de ning the boundaries of such domains are derived and numerical methods for computing families of generators on boundaries are outlined. The criteria and numerical methods presented are applied to planar mechanism and manipulator examples.

1. Introduction Criteria for boundaries of workspaces of mechanisms and manipulators, using conditions associated with singularity of constraint Jacobian or velocity transformation matrices, have been developed by a number of authors in the recent past. Litvin [6] used the implicit function theorem to de ne singular con gurations of mechanisms as criteria for boundaries of workspaces. Analytical conditions associated with special features of the geometry of speci c manipulators have been used by a number of authors to obtain explicit criteria for boundaries of workspaces (see [2], [7], [8], and [10]). Singularity of the velocity transformation between input and output coordinates has likewise been used to characterize singular surfaces of manipulators (see [5] and [9]). Numerical methods for mapping boundaries of workspaces of

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E.J. HAUG, F.A. ADKINS, AND C.M. LUH

mechanisms and manipulators have recently been presented (see [3], [4], and [11]). These methods are summarized here, with accompanying numerical methods.

2. Analytical Conditions For Working Domains And Boundaries In order to de ne working domains in bodies that move with an underlying mechanism, whose con guration is de ned by a generalized coordinate q, the shape of the domain of the working bodies is parameterized by a vector h iT T T . De ning the nz-vector z = q ;  , the nu-vector u to a working point on a working body is given analytically in the form u = g (z), where the vector function g(z) is twice continuously dierentiable. The kinematic constraint equations for the mechanism and equations involving parameterization of the shapes of working bodies, which may account for inequalities with slack variables [4] to de ne domains, are written in the form  (z) = 0, where the m-vector of functions  (z) is twice continuously dierentiable. The system of nu + m equations that de ne points in a domain speci ed by the associated criteria is   u ? g ( z ) (u; z)   (z) = 0 (1) Thus, the domain of interest is the set D = fu 2