Platform stewart

The platform of Stewart , also known under the name of positioner hexapode , is a parallel type of Robot consisted of six actuators. It has six degrees of freedom: three coordinates of translation as well as the angles of pitching, bearing and lace (Angles of Euler). The six legs are actuated to change languor and to vary the orientation of the platform, to a given position of the higher plate corresponds six single lengths of the legs. The platforms of Stewart have applications in the technology of machine tool, the technology of crane, underwater research, the delivery of air-with-sea, the flight simulation, the parabolic aerial placing, the telescopes and the orthopedic surgery

One of the mathematical models of the platform of Stewart:

One combines the ordering of the six actuators in order to be able to impose on the higher plate of the platform the desired position. It is thus a question of determining the lengths of each axis which correspond to the six coordinates of Euler.

The figures hereafter give us the position of the reference marks chosen to carry out calculations.

The angles of Euler model the rotary displacement of the higher platform compared to the fixed base. This basic change is determined by the angles ψ, θ and φ, according to the figures:

One distinguishes two reference marks, a mobile and fixed. The vector which corresponds to axis I is:

Each end of the axis is defined by these coordinates:

The angles and are defined by the following formulas:

The matrices of basic change in rotation respectively around the axes Z, there and X are:

An angular displacement of value (ψ, θ, φ) compared to the position (0,0,0) is modelled by the matrix. Thus the coordinate of a vector are given by:

The length of a leg Li is given by:

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