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By Marina Cohen

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40 the geometrica} parameters in a general nR manipulator are assumed the link lengths ai ~ O (i=1, ... ,n), the link offsets di (i = 2, ... ,n), and the twist angles ai~ k x/2 (i=1, ... ,n-1), (k=O, ... ,4). d1 rnay not be considered since it shifts the hyper-ring up and down, only. A reference point H can be selected on the extreme link of a manipulator chain for workspace determination. A ring is generated by revolving a torus about an axis. RiRJ(H), [1], (2) Thus, the boundary 8W3R(H) of a ring can be thought as the envelope of toroidal surfaces generated by revolution of the generating torus or, alternatively, it can be obtained by an envelope of a torus family traced from the parallel circles of the generating torus, yet, as (3) where "env" expresses an envelope operator.

J with ' 2 ' +Gi+lEi+I(Ci+lzi+l +Gi+ 1 )/Ei+l +Ei+ 1 -2zi+lzi+l sj+l = 2(zj+l +dj+l )z~+l (Il) The symbol' is the derivative operator with respect to the torus family parameter 8n. Eqs. (11) can be computed through the equation coefficients Ej+ 1, Fj+ 1, Gj+ 1, Oj+ 1 and their derivatives Ej+ 1, F]+ 1, Gj+ 1, Q]+ 1, whose expressions and vălues can be âeduced from rj+ 1 and Zj+ 1 formulation. f11) ana by using, in addition, the second derivatives of the ring equation coefficients. The radial rj+ 1 and axial zj+ 1 reaches, the ring coefficients and derivatives of the (n-j)R workspace ooundaty may be expressed iteratively, by means of the envelope development, from the (n-j-1)R hyper-ring boundaty and so on up to the extreme ring workspace due to the last three revolute joints in the chain.

J. Lenareic and B. B. ), Advances in Robot Kinematics and Computationed Geometry, 49-58. © 1994 Kluwer Academic Publishers. 50 Acknowledgemen ts: We wish to acknowledge the considerable help we received from Dr. R. J. Morris and Dr. N. P. Kirk in producing computer renderings for the unfoldings of the corank 2 germs. II. An Engineering Example Figure 1 represents the double four-bar, a planar linkage with two degrees of freedom. Associated to the coupler point C is a coupler region in the plane together with a curve called the critical image, which includes the boundary of the coupler region.

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