Dynamics of parallel manipulators by means of screw theory
Introduction
Over the last two decades parallel manipulators have received increasing attention by kinematicians as demonstrated by the relevant amount of published works on this subject, see for instance [1], [2], [3], [4], [5], [6], [7], [8], [9] among many others.
As it is well known a general parallel manipulator is a mechanism composed of a mobile platform connected to the ground by several independent kinematic chains, called serial connector chains. Each serial connector chain can be regarded as a serial manipulator with both actuated and passive joints, the former providing the actuation to the mobile platform. Despite of a reduced workspace and a more complex solution of the direct kinematic problem than serial manipulators, the higher stiffness, accuracy and payload/weight ratio which can be achieved by parallel manipulators make them attractive systems for applications ranging from motion simulators to positioning robotic systems. The first suggestion of this class of mechanisms is historically attributed to Gough, who constructed a prototype of a parallel mechanism for measuring the tire wear and tear under different conditions [2]. Afterwards a similar mechanism was proposed by Stewart, in 1965, as a flight simulator; and nowadays mechanisms that employ the same architecture of the Gough mechanism are without doubt the most studied type of parallel manipulators, universally known in literature as Gough–Stewart platforms.
The dynamic analysis of parallel manipulators has been traditionally carried out through several different methods, i.e. the Newton–Euler method, the Lagrange formulation and the principle of virtual work, which nevertheless present some drawbacks. The Newton–Euler method usually requires large computation time, since it needs the exact calculation of all the internal reactions of constraint of the system, even if they are not employed in the control law of the manipulator. On the other hand, both the Lagrangian and the principle of virtual work formulations are based on the computation of the energy of the whole system with the adoption of a generalized coordinate framework, whereby the system dynamics equations are expressed. Such an energy approach to the analysis of parallel manipulator can be further simplified and standardized by means of the theory of screws.
In this work the analysis of parallel manipulators is developed through a novel methodology based on the theory of screws. The kinematics is approached by extending results previously obtained by the authors [10], [12], [13], in the analysis of open serial and closed chains to the kinematics of parallel manipulators. Then the dynamics is approached by an harmonious combination of screw theory with the principle of virtual work. Finally, in order to show the effectiveness of the method two applications are given, the first one dealing with a 2-DOF spherical parallel mechanism, and the second one dealing with a Gough–Stewart platform.
Section snippets
Preliminary concepts: kinematics of open serial and closed chains
This section summarizes a few results, obtained via screw theory, dealing with the kinematics of open serial and closed chains.
Consider two rigid bodies i and i+1, that are connected to each other by means of a helicoidal joint described by the normalized screw . The screw can be expressed in terms of Plücker coordinates as:where is a unit vector defined on the instantaneous rotation axis and is usually given as a function of the screw pitch
Kinematics of parallel manipulators
The results obtained in Section 2 for open serial and closed chains can be applied to solve kinematics of parallel manipulators.
Consider a parallel manipulator composed of k serial connector chains, as shown in Fig. 3, and suppose that the mobile platform has an instantaneous motion with respect to an inertial reference frame OXYZ, fixed to the base link, body 0, given byAfter expressing all the screws with respect to the point O, the inverse velocity analysis can be
Dynamics of parallel mechanisms
Recently the inverse dynamics of a Gough–Stewart platform has been significantly simplified by means of the principle of virtual work [7], [9]. In this section it is shown how this approach can be further simplified by applying the Klein form, a bilinear symmetric form of the Lie algebra.
Assume that and are two elements of the Lie algebra, e(3). The Klein form KL($1,$2), a non-degenerate symmetric bilinear form, is defined as
Example 1, 2-DOF spherical mechanism
This section shows a first application of the method to a 2-DOF spatial parallel mechanism used as a haptic interface to simulate a virtual car gearshift. The calculation of the system dynamics will be approached by solving the direct kinematic equation of the system. The kinematic representation of the spherical mechanism is shown in Fig. 4 together with the ground coordinate system OXYZ and the joint axes.
The differential kinematic of the mechanism is easily derived using screw algebra. Since
Example 2, Gough–Stewart manipulator
This section presents an application of the presented method to the analysis of a Gough–Stewart manipulator Fig. 5.
The mobile platform is connected to the base link by means of six serial chains. Each serial connector chain is composed of a lower spherical joint, a prismatic actuated joint and an upper universal joint. The prismatic actuated joints provide six degrees of freedom to the mobile platform, so that it can assume an arbitrary pose. The actuators lengths are defined by 12 points,
Conclusions
This paper shows how the screw theory and the principle of virtual work can be used to systematically solve the dynamics of parallel manipulators. Unlike other procedures reported in the literature, the proposed method does not require the computation of internal forces of constraint, which is an unnecessary task when the goal of the analysis is neither the dimensioning of the elements nor the computation of the kinetic energy of the whole system. Two examples were analyzed, and the general
Acknowledgements
This work was supported by CONCyTEG, México, and by MURST, Italy.
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