This paper deals with the in-depth kinematic analysis of a special spherical parallel wrist, called the Agile Eye. The Agile Eye is a three-legged spherical parallel robot with revolute joints in which all pairs of adjacent joint axes are orthogonal. Its most peculiar feature, demonstrated in this paper for the first time, is that its (orientation) workspace is unlimited and flawed only by six singularity curves (rather than surfaces). Furthermore, these curves correspond to self-motions of the mobile platform. This paper also demonstrates that, unlike for any other such complex spatial robots, the four solutions to the direct kinematics of the Agile Eye (assembly modes) have a simple geometric relationship with the eight solutions to the inverse kinematics (working modes).
Most of the active research work carried out in the field of parallel robots has been focused on a particularly challenging problem, namely, solving the direct kinematics, that is to say, finding each possible position and orientation of the mobile platform (assembly mode) as a function of the active-joint variables. A second popular problem has been the evaluation and optimization of the workspace of parallel robots [1], [2]. Unfortunately, the direct kinematic problem and the workspace analysis have most often been treated independently, although they are closely related to each other.
It is well known that most parallel robots have singularities in their workspace where stiffness is lost [3]. These singularities (called parallel singularities in [4] and Type 2 in [3]) coincide with the set of configurations in the workspace where the finite number of different direct kinematic solutions (assembly modes) changes. For parallel manipulators with several solutions to the inverse kinematic problem (working modes), another type of singularity exist and defines what may be generalized as the workspace boundary. These singularities (called serial singularities in [4] and Type 1 in [3]) coincide with the set of configurations in the workspace where the finite number of different inverse kinematic solutions (working modes) changes. While ensuring that a parallel robot stays with the same working mode along a discrete trajectory is straightforward, the notion of “same assembly mode” is not even clear in the general case. Indeed, the direct kinematic solutions are most often obtained by solving a univariate polynomial of degree n > 3, which means that there is no way to designate each solution to a particular assembly mode.
Then, how does one choose a direct kinematic solution for a parallel robot for each different set of active-joint variables?
A natural sorting criterion is that the direct kinematic solution could be reached through continuous motion from the initial assembly mode, i.e., the reference configuration of the robot when it was first assembled, without crossing a singularity. Before [5], it was commonly thought that such a criterion was sufficient to determine a unique solution. Unfortunately, as for serial robots [5], [6], a non-singular configuration changing trajectory may exist between two assembly modes for robots that are called cuspidal. This result later gave rise to a theoretical work on the concepts of characteristic surfaces and uniqueness domains in the workspace [7]. However, it still remains unknown what design parameters make a given parallel robot cuspidal, and it is still unclear how to make a given parallel robot work in the same assembly mode. It will be shown in this paper, that there is a spherical parallel robot (quite possibly the only one), for which there are clear answers to these complex questions.
From the family of parallel wrists [8], the Agile Eye provides high stiffness and is quite probably the only one to provide a theoretically unlimited and undivided orientation workspace. The Agile Eye is a 3-RRR spherical parallel mechanism in which the axes of all pairs of adjacent joints are orthogonal. Based on the Agile Eye design, a camera-orienting device was constructed at Laval University a decade ago [9], an orientable machine worktable was manufactured at Tianjin University, and a wrist for a 6-DOF robot was built at McGill University.
The Agile Eye has been extensively analyzed in literature, but surprisingly some of its most interesting features have not been noticed. One of the key references in this paper is [10], where the simple solution to the direct kinematics of the Agile Eye is presented. Namely, it is shown that the Agile Eye has always four trivial solutions (at which all three legs are at singularity and can freely rotate) and four nontrivial solutions obtained in cascade. A second key reference is [11], where the singularities of the Agile Eye are studied. Unfortunately, in [11], it was mistakenly assumed that the only singular configurations are the four trivial solutions (orientations) to the direct kinematics. Basically, the fact that at each of these four orientations, a special arrangement of the legs can let the mobile platform freely rotate about an axis was overlooked. Indeed, as this paper shows, the singularities of the Agile Eye are six curves in the orientation workspace corresponding to self motions of the mobile platform or lockup configurations.
The Agile Eye at its reference configuration.
When it was realized that the unlimited workspace of the Agile Eye is not divided by singularity surfaces (as is the case for all other spherical parallel robots), yet accommodating four unique assembly modes, it first seemed that the Agile Eye is a cuspidal robot. In this paper, it is shows not only that the Agile Eye is not cuspidal, but that it is straightforward to identify its four assembly modes via a simple relationship to its working modes.
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