Figure 3 Kinematics of a three-legged parallel robot
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end of the leg. Fig. 2 shows two such possible robot configurations constructed by using standard joint mod- ules, custom-designed links and mobile platforms. The Fig. 2: Three-legged parallel robot configurations Fig. 4: A forward singularity configuration: passive joint axes pass through leg-end points Example 1: Fig. 4 shows a particular three-legged parallel robot in which the three passive joint axes 813, $23, and $33 always pass through their corre- sponding leg-end points A;, Ay, and A3. As a re- sult, wig = 23 = ugg = 0 and A, = 0. In fact, this forward singularity is caused by a particular robot architecture and exists for all configurations inside the entire robot workspace. This is, therefore, such an architecture singularity and can be avoided by design. Fig. 6: A forward singularity configuration: two passive joint axes are parallel to Az A3 simultaneously parallel with the vector defined by points A; and A;, p;; (or p;;). It follows that both uz and u,;3 will be perpendicular to vector p;; (or pj). At least one of the row vectors in Jp (Eq. (25)) becomes a null vector and hence, A,=0. Example 3: Fig. 6 shows a specific configuration of a three-legged parallel robot, in which two passive joint axes S23 and 833 are simultaneously parallel with vector A» Az, Le., wes \\wss3 \\pes- Asa result, u23 L po3 and ug3 1 po3, and the determinant. of J, will vanish. A forward singularity occurs in this specific configuration. Fig. 8: An inverse singularity configuration: three join axes in a leg are parallel and coplanner of a three-legged parallel robot, in which the three revolute joint axes in leg 1, i.e., $11, S12, and 813, are parallel and coplanar. Therefore, uj1,uj2, and u;3 are linearly dependent such that A; = 0. A inverse singularity occurs in this particular configuration. Fig. 7: An inverse singularity configuration: one joint axis passes through its corresponding leg-end point
