Singularities of a Manipulator with Offset Wrist

The singularities of manipulators with offset wrists are difficult to enumerate. This article presents a numerical study to illuminate singularity problems when using an offset wrist with an articulated regional arm. Ironically, the regional manipulator singularity problem is worsened when using a singularity-free offset wrist. No wrist singularities exist (they are forced by design to lie outside of joint limits). However, the existing regional arm singularities become skewed from the well-known singular configurations of common manipulators. The wrist offset also skews the wellknown wrist singularities (though they hopefully still lie outside of joint limits, their locations are no longer easily determined). Thus, a zero-offset singularity-free wrist is preferable with regard to overall manipulator singularities. INTRODUCTION Manipulator singularities can be found from the manipulator Jacobian matrix J: J = 0 for nonredundant manipulators and JJ T = 0 for kinematically-redundant manipulators. The linearized rate relationship is expressed { } [ ]{ } v J T ω = Θ , where v are the translational Cartesian velocities,ω are the rotational Cartesian velocities, and { } Θ are the joint rates. For a manipulator with a spherical (zero-offset) wrist, the upper right Jacobian submatrix is the 3x3 zero matrix, which is the velocitydomain manifestation of position/orientation decoupling. Singularities are classified as regional arm 3 singularities (found from JUL = 0) and wrist singularities (found from J LR = 0) (Stanisic and Duta, 1990). [ ] [ ] [ ] [ ] [ ] J J J J UL LL LR =         0 ; J J J UL LR = (1) For a manipulator with an offset wrist the Jacobian matrix [ ] J is fully populated as in Eq. 2. This is because some wrist joints participate in translation of the last wrist frame in addition to orientation. In this case, the singularities can no longer be classified as separate regional arm singularities and wrist singularities (Stanisic and Duta, 1990). [ ] [ ] [ ] [ ] [ ] J J J J J UL UR LL LR =         ; J J J UL LR ≠ (2) Singularity-free, offset double universal joint (DUJ) wrists (Fig. 1) have been designed and built by Trevelyan, et. al. (1986), Milenkovic (1987), and Rosheim (1987). The Jacobian matrix determinant for the DUJ wrist alone is J c c DUJ = 4 5 6 2 ( ci i = cosθ ) so the singularity conditions are θ5 90 = ± (all angles in this article are given in degrees) or θ6 90 = ± . If these are forced to lie outside of joint limits, the wrist is singularity-free (Williams, 1990). In this paper, the 3-axis DUJ wrist is mounted on an articulated 3-axis regional arm (Fig. 2; PUMA with no waist-shoulder offset, i.e. L0=0) to form a 6-dof manipulator. There are eight rows in the DH parameters (Table I, Craig (1989) convention) because the DUJ wrist mechanically couples the two universal joints (which are separated by offset L). This mechanical coupling is usually accomplished via two gear pairs, shown conceptually in Fig. 1. The joint angle offsets in Table I are included to define the zero position as straight up. The regional arm singularities for the first three joints alone are found symbolically from ( ) J L L L s L s s reg = + = 1 2 1 2 2 23 3 0 , where si i = sinθ and ( ) s23 2 3 = + sin θ θ . The non-trivial