Colaboration:

• TU Delft
• burgeon

This application concerns kinematic optimization of a Stewart platform manipulator for a flight simulator. The Stewart Platform consists of six prismatic actuators and two platforms as shown in Figure 1. Each actuator is connected by a universal joint (two orthogonal intersecting revolute joints) to the lower (base) platform whereas a spheric joint (three orthogonal intersecting revolute joints) is used to connect it to the upper (mobile) platform.

To optimize the manipulator, a procedure based on the maximum workspace criterion proposed by Markine et al. (1998) and Markine (1999) has been used. The goal of optimization is to design a manipulator with maximum workspace whose characteristics are defined a priori according to the manoeuvres of a real aircraft to be simulated. The workspace of the manipulator assessed by the size of the inscribed hyperellipsoid similar to the one containing all required trajectories is then characterized by the corresponding ratio of similitude of the inscribed hyperellipsoid. Thus, for the optimal manipulator the inscribed hyperellipsoid should have maximum ratio of similitude (RS). The design variables comprised the geometry of upper and lower platforms as well as the lengths L_min(i) (i=1, 2, 3) and attachment points of the actuators (Figure 2).

The optimal design has to satisfy a dexterity (manipulability) requirement along with the constraints to avoid the interference between the actuators. The dexterity (DI) can be defined as the ability of the manipulator to arbitrarily change its position and orientation, or apply forces and torque's in arbitrary directions. The position and rotation of the Stewart platform is completely defined by the lengths of its actuators. However, the control of the manipulator might become difficult if a singular configuration has been approached (Markine, 1999). The problem has been solved for the workspace required for the simulation of a large transport aircraft.

A difficulty occurs when a singular configuration of the manipulator is approached. The constraints and objective function cannot be evaluated for such a design of the manipulator. An optimum design of the Stewart Platform has been found in by Markine (1999) using the MARS method (Toropov et al. 1993 and 1996, van Keulen and Toropov 1998). To overcome the difficulty, a reduced initial search subregion has been used while performing a series of 50 optimization runs starting from randomly assigned points. A typical optimization run contained 27 iterations.

The optimum design using the modified plan of experiments presented above has been obtained in 15 iterations. The results of both runs are given in Table 1. It should be noted that for the optimal design the ratio of similitude has been slightly greater than one that means that the obtained manipulator could perform all the required manoeuvres.

References

• van Keulen, F.; Toropov, V.V. (1998). New Developments in Structural Optimization using Adaptive Mesh Refinement and Multi-Point Approximations. Engineering Optimization, 29, 217-234.
• Markine V.L. (1999). Optimization of the Dynamic Behaviour of Mechanical Stystems. PhD Thesis, Delft University of Technology, The Netherlands. Shaker Publishing B.V. ISBN 90-423-0069-8
• Markine, V.L., Meijers, P., Toropov, V.V. (1998). Optimization of Workspace of a Manipulator for a Flight Simulator using Multipoint Approximation Method. Paper AIAA-98-4955, 7th AIAA/USAF/NASA/ISSMO Symp. on Multidisciplinary Analysis and Optimization, St. Louis (USA), September 2-4, part 3, 1922-1932, AIAA.
• Toropov, V.V., Filatov, A.A. and Polynkin, A.A. (1993). Multiparameter Structural Optimization Using FEM and Multipoint Explicit Approximations. Structural Optimization 6, 7-14.
• Toropov, V.V., Keulen, F. van, Markine, V.L., de Boer H. (1996). Refinements in the Multi-Point Approximation Method to Reduce the Effects of Noisy Structural Response. Proceedings of the 6-th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2, 941-951.