Design of planar 3-DOF 3-RRR reactionless parallel manipulators
Introduction
In high-speed mechanical systems, mass balancing of the moving links brings about a reduction of the variable dynamic loads on the frame and, as a result, a reduction of vibrations. Different approaches and solutions have been developed and documented [1], [2], [3] but, despite its long history, mechanism balancing theory continues to develop and new approaches and solutions are constantly being reported. A new field for their application is the design of fast parallel manipulators, which are very efficient for advanced robotic applications. Previous work on the problem of balancing of parallel manipulators may be arranged in the following groups:
- (a)
Shaking force balancing by counterweights mounted on the movable links of the parallel manipulator [4], [5], [6], [7], [8]. The aim of these balancing methods is the redistribution of movable masses by adding counterweights to the links, which allows the fixation of the common centre of mass of the moving links of the manipulator. After such a redistribution of the masses, the gravitational and inertia forces are cancelled.
- (b)
Gravitational force balancing by springs mounted on the movable links of the parallel manipulator [7], [8], [9], [10]. Such a balancing can be defined as when the weights of the links do not produce any force on the actuators for any configuration of the manipulator, i.e. potential energy of the parallel manipulator is constant for all possible configurations. It should be noted that many results in the field of balancing of robotic arms and linkages [11], [12], [13], [14], [15], [16], [17] can be successfully applied to the balancing problems of parallel manipulators.
- (c)
Gravitational force balancing by secondary mechanisms coupled with the parallel manipulator [18], [19], [20], [21], [22]. In this case the balancing element, which can be a spring [18], a counterweight [19] or an actuating power cylinder [20], [21], [22], is mounted on the links of the secondary mechanism. In these studies the added system is a pantograph linkage which allows the gravitational forces to be balanced.
These approaches have been developed for inertia or gravitational force balancing of parallel manipulators. In the case of shaking force balancing the mentioned methods allow the cancellation of the resultant of all reaction forces at the frame. However, the unbalanced angular moments create a moment on the frame, which can also be significant.
Among several works on this subject, studies devoted to the design of reactionless parallel manipulators [23], [24] should be highlighted. These manipulators are of interest because the inertia forces are cancelled together with the total angular momentum of the manipulator. Such a design enables the cancellation of the reaction forces and torques at the frame of the parallel manipulator.
In this paper, the design of reactionless 3-DOF 3-RRR planar parallel manipulators is considered and design equations and techniques are developed.
Section snippets
3-DOF 3-RRR planar parallel manipulator and dynamic model with concentrated masses (a special shape of the moving platform)
The moving platform of a planar 3-DOF 3-RRR parallel manipulator is connected to its legs by three revolute joints Pi (i = 1, 2, 3) (Fig. 1) [25]. Each leg comprises two links connected by revolute joints Ai (i = 1, 2, 3) and they are mounted on the frame by revolute joints Oi (i = 1, 2, 3). The input parameters of such a manipulator are defined by the joint angles θi (i = 1, 2, 3) of each leg and the output parameters by the pose of the moving platform, i.e. its orientation ϕ and position of one point of the
Balancing by counter-rotations
The suggested balancing method is based on balancing of the inertia forces by means of counterweights mounted on the links and balancing of the total angular momentum by means of counterweights with planetary gear trains to generate the counter-rotations.
Balancing by inertia flywheel
In this section we consider the shaking moment cancellation of the fully force balanced 3-DOF 3-RRR parallel manipulator by an inertia flywheel with prescribed rotation. It is evident that this solution is constructively more efficient.
Fig. 9 shows the fully force balanced 3-DOF 3-RRR parallel manipulator and balancing inertia flywheel, which is mounted on the base of the manipulator. The conditions for balancing the shaking moment of the manipulator are determined from the following
Conclusions
A new field for shaking force and shaking moment balancing is the design of fast parallel manipulators, which are very efficient for advanced robotic applications. In this paper, the shaking force and shaking moment balancing approach is developed for planar 3-DOF 3-RRR parallel manipulators. It is based on the dynamic substitution of the mass of the platform by three concentrated masses situated at the axes of the revolute joints of the legs. By application of this approach the dynamic model
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