Singularity Loci of Spherical Parallel Mechanisms Ilian A. Bonev Department of Automated Manufacturing Engineering ´ Ecole de technologie sup´ erieure 1100 Notre-Dame Street West, Montreal, QC, Canada H3C 1K3 ilian.bonev@etsmtl.ca Cl´ ement M. Gosselin Department of Mechanical Engineering Universit´ e Laval Quebec City, QC, Canada G1K 7P4 gosselin@gmc.ulaval.ca Abstract— This paper presents the computation and repre- sentation of the Type 2 singularity loci of symmetric spherical parallel mechanisms based on a not-well-known intuitive orientation representation. The latter, previously introduced under the name of the Tilt-and-Torsion angles, is briefly described. Then, to illustrate the approach, the two most basic spherical parallel mechanisms are considered and their Type 2 singularities are fully analyzed for various designs. Index Terms— spherical parallel mechanisms, wrists, sin- gularities, orientation representation, Euler angles. I. I NTRODUCTION Recently, there has been a certain breakthrough in the synthesis of spherical parallel mechanisms (SPMs), accom- plished mainly by a couple of researchers in the field [1]– [3]. This completes a series of older works on more basic 3-DOF SPMs [4]–[6]. Still, there are most certainly new SPMs to be discovered, yet the design and analysis of even the most basic SPMs are far from complete. One of the least studied problems related to SPMs is their singularity analysis or, more specifically, finding their singularity loci. For example, even though the prototype of the Agile Eye (a 3-DOF 3-R RR 1 SPM with orthogonal joint axes) was built in 1993 [7], it was not until nearly a decade later that its singularities were fully studied [8]. Yet, how can one adopt a specific design without exact knowledge of its singularity-free workspace. The only other works that focus on the Type 2 singularity loci 2 of SPMs (of 3-UP S/S type) are presented in [10] and [11]. In these works, analytic expressions for the singularity loci are obtained in the ZYZ and XYZ Euler angles but are very hard to interpret. The works illustrate clearly why the singularity loci of SPMs are so unattractive to study—it is not intuitive to represent them. By offering a geometric tool for computing and representing the properties of SPMs, we overcome this difficulty. In the following section, we describe this geometric tool that should become a standard for the orientation representation of symmetric SPMs. Then, in Section III, we apply this tool for the computation and representation of the singularity loci of 3-UP S/S symmetric SPMs. Similarly, in Section IV, we analyze 3-R RR symmetric SPMs. Finally, a conclusion is given in Section V. 1 It is common to denote parallel mechanisms by using the symbols U, R, S, and P, which stand respectively for universal, revolute, spherical, and prismatic joint. When a joint is actuated, its symbol is underlined. 2 It is common to classify the singularities of parallel mechanisms into two types [9]: Type 1 (or serial) and Type 2 (or parallel) singularities. II. ORIENTATION REPRESENTATION A novel three-angle orientation representation, later called the Tilt-and-Torsion (T&T ) angles, was proposed in [12] in 1999, in conjunction with a new method for computing the orientation workspace of symmetric spatial parallel mechanisms. It was shown that the T&T angles take full advantage of a mechanism’s symmetry. These angles were also independently introduced in [13] and [14] in 1999. Later, it was found out that the angles had been proposed in [15] in 1984 under the name halfplane- deviation-twist angles. The author of that reference pro- posed the set due to its indisputable advantages in modeling the limits of human body joints. Yet, again in 1999, these angles were proposed independently in [16] as a new standard in modeling angular joint motion, and particularly that of the spinal column’s vertebra. These angles are also used for computer animation of articulated bodies, known as the swing-and-twist representation. Finally, in [17], the advantages of the T&T angles in the study of spatial parallel mechanisms were further demonstrated. It was shown that there is a class of 3-DOF mechanisms that have always a zero torsion. The T&T angles are defined in two stages—a tilt and a torsion. This does not, however, mean that only two angles define the T&T angles but simply that the axis of tilt is variable and is defined by another angle. In the first stage,+ illustrated in Fig. 1(a), the body frame is tilted about a horizontal axis, a, at an angle θ, referred to as the tilt. The axis a is defined by an angle φ, called the azimuth, which is the angle between the projection of the body z ′ axis onto the fixed xy plane and the fixed x axis. In the second stage, illustrated in Fig. 1(b), the body frame is rotated about the body z ′ axis at an angle σ, called the torsion. x y z φ x ∗ y ∗ z ∗ θ a (a) x y z φ x ∗ y ∗ z ′ ≡ z ∗ x ′ y ′ σ σ θ a (b) Fig. 1. The successive rotations of the T&T angles: (a) tilt, (b) torsion. Proceedings of the 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April 2005 0-7803-8914-X/05/$20.00 ©2005 IEEE. 2968