Workspace of Tendon-driven Stewart Platforms: Basics, Classification, Details on the Planar 2-dof Class

Conditions for the workspace of tendon-driven Stewart platform manipulators are given. For a general 2dof planar manipulator, these criteria are calculated analytically and illustrated graphically. The methods are developed such that they may be generalized, for mechanisms with more dofs. INTRODUCTION State of the Art Theory and applications of Stewart platforms (Fig. 1) have been widely examined since [15]. Main advantages are very high rigidity, low structure weight and high payload. They have a very good positioning accuracy and a low energy consumption. Today there are applications in manufacturing, telescopes, and other elds [10, 4]. Many research e orts have been devoted to forward kinematics [7, 14], in particular for special cases [6]. FIGURE 1: Stewart Platform FIGURE 2: Tendon-driven serial manipulator Tendon-driven mechanisms (Fig. 2) reduce the weight of the moving structure by another principle: they have actuators mounted on the base, so they are not moved. This also reduces the size of the moving parts [9]. But serial tendon-driven mechanisms present some di culties when guiding tendons around joints [3]. An important aspect of their theory is the distribution of force among the tendons [13]. FIGURE 3: tendon-driven Stewart Platform The synthesis of both concepts into tendon-driven Stewart Platforms (Fig. 3) has been developed quite recently. [1] points out that this concept can be used for applications like shipbuilding, where robot arms with heavy rigid links are not appropriate, and simple one-tendon-cranes are di cult to control because of their very low sti ness and accuracy. Another eld of application are extremely fast moving micromanipulators, as described in [8], permitting accelerations higher than 40 g. In the past, most research about tendon-driven Stewart platforms has been directed towards particular applications, as in the papers cited. A rst approach of a general classi cation has been given in [11]. As pointed out there, the fact that tendons represent unilateral constraints implies that for n dofs, at least n+1 tendons are necessary to build a \Completely Restrained Positioning Mechanism" (CRPM) { the same result for serial manipulators is stated in [13] { unless external forces like gravity are involved to get additional (non-kinematic) constraints; in this case, we have an \Incompletely Restrained Positioning Mechanism" (IRPM). From now on, we also distinguish between \CRPMs" in a strict sense, with exactly m = n+ 1 tendons, and \Redundantly Restrained Positioning Mechanisms" (RRPMs) with more than n+ 1 tendons. The Department of Mechatronics of the University of Duisburg is going to start a systematic research on tendon-driven Stewart platforms. The objective is to give the basics for developing tendondriven Stewart platforms for a wide range of applications; in addition to those cited above, a eld of increasing interest is theater technology, where heavy components like walls must be moved fast, precisely and silently. In a later stage, this topic is likely to be covered in collaboration with industrial partners. Main Issue: Workspace One of the major problems of tendon-driven Stewart platforms is the rather small workspace. It is not just the area where the platform can be positioned without exceeding joint limits, but it implies at least the following conditions: It must be possible to exert force and torque. Tendon forces must be positive. Tendon forces should lie between pretension and maximum tension. The structure must be su ciently sti . The mechanism must avoid singularities. Tendons must not intersect with each other. Joint limits do not appear here because they depend on the realization of a mechanism, rather than theoretical limits. The same is true for collisions of tendons with the support structure, the platform, or obstacles. The authors intend to develop, in the future, optimal designs for general-purpose manipulators, where the tasks are described only by the number and type of dofs. Thus, it is important to ful ll a minimum standard for the above conditions in a wide area of space, rather then to have excellent standards in a small area needed for a particular task. Knowledge about such optimal designs is useful because many of the applications mentioned before are general-purpose, and also because it is the basis for the development of task-speci c manipulators. The following focuses on CRPMs, like in [12], because from a mathematical point of view, these are the simpler ones. MATHEMATICAL MODEL De nitions All geometric quantities are shown in Fig. 4: KB;KP are coordinate frames attached to base and platform, respectively; b1; : : : ; bm point from KB to the base . . .