A Unified Method for Computing Position and Orientation Workspaces of General Stewart Platforms

The workspace of a Stewart platform is a complex sixdimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, such workspace is difficult to compute and represent, so that comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constant-orientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms involving mechanical limits on the active and passive joints. Additional advantages over previous methods include the ability to determine all connected components of the workspace, and any motion barriers present in its interior. INTRODUCTION Due to their advantages in terms of dynamic properties, load-carrying capacity, high accuracy, and stiffness, Stewart platforms are widely used as flight simulators, high-precision positioning devices, mining machines, or surgical robots [1–7]. The assembly constraints imposed by their kinematic design, however, substantially reduce the set of poses that such platforms can attain, leading to highly-constrained workspaces in most of the cases. The availability of proper tools to accurately compute and represent such workspaces is thus of utmost importance, not only to assist the robot designer during the conception of the platform, but also to be able to implement trajectory planners more efficiently [8], once an adequate design has been chosen for a particular application. The workspace of the Stewart platform is hard to compute and visualize. Its large dimension and complex shape, which may encompass several connected components, difficult any attempt of computing it exhaustively, due to the curse of dimensionality. In many situations, fortunately, the platform either operates with a fixed orientation or rotates about a fixed point, so that it can be assumed that three of the six pose parameters are held constant, leading to three-dimensional workspaces that are easier to obtain and represent. The constant-orientation workspace, in particular, is clearly understood, and fast geometric algorithms exist for computing its boundary [9], even in the presence of joint limits in the passive joints, or potential link-link interferences [10]. Interval analysis methods have been given too, to compute the interior of such workspace [11]. The constant-position workspace, also known as the orientation workspace, has also been studied, but its computation and visualization turn out to be more problematic, due to the complexity of the intervening equations, and to the difficulty of representing orientations in an intuitive way. Previous methods either assume one of the orientation angles held fixed [12, 13], thus producing two-dimensional sections of the workspace only, or let the three angles vary [14, 15], but all methods rely on some sort of discretization, which leads to incomplete or less accurate output in some situations. Recently, a fast method providing appealing visualizations of the orientation workspace has been given [16], but mechanical limits in the passive joints are neglected, so that the computed workspace is, actually, an overestimation of the real workspace. Some of the methods, finally, are only devised to compute the connected component of the workspace that is achievable from a pre-defined configuration [16–18], which limits the ability to characterize the movement of the platform under alternative assembly modes. Despite the literature on the topic is rich, three important requirements are not fully met by previous approaches. First of all,