The problem of correct position of the organization is the basic problem of institutional science. It is also the basis of mechanism analysis and synthesis, speed and acceleration solution, and dynamic analysis and error analysis. For parallel robots , the positive position solution is a difficult problem to solve. Given the input position parameters of the parallel robot, the pose parameter of the solver is the positive solution of the position of the parallel robot.
The positive solution of position mainly has numerical solution and analytical method. The analytic method mainly eliminates the unknowns in the mechanism constraint equation by the elimination method, so that the input-output equation of the mechanism becomes a high-order equation with only one unknown [1~4]. The advantage of this method is that it can solve all possible solutions of the mechanism, but the above-mentioned elimination process is generally very cumbersome, and the calculation accuracy is very high when solving the one-dimensional higher-order equation [1]. Generally, the six-degree-of-freedom parallel mechanism has 40 solutions [2], so the final equation is a 40-order equation. The advantage of the numerical method is that it can be applied to any type of parallel mechanism, but the general numerical method uses the optimized search principle and requires a lot of calculations. Time, and can only achieve limited accuracy [5 ~ 8].
The proposed successive approximation method is a numerical calculation method. Each approaching direction of the method is the instantaneous velocity direction of the position, which can approximate the desired pose with arbitrary precision. Because the method adopts successive approximation, it avoids the shortcomings of the optimized search calculation time used in the general numerical method. Therefore, this method has important theoretical significance and practical application value in the positive solution analysis of parallel robot position.
1 positive position analysis of parallel robot
1.1 position inverse solution
Figure 1 shows a 6-SPS parallel mechanism. The dynamic coordinate system p-xyz is built on the upper platform, and the coordinate system o-xyz is fixed on the lower platform. The vector R of any point in the moving coordinate system can be transformed to the vector R of the point in the fixed coordinate system by the coordinate transformation method.
R=°TpR(1)
In the formula:
Where: p is the position vector of the origin of the moving coordinate system in a fixed coordinate system, and T is the cosine matrix of the direction of the moving coordinate system with respect to the fixed coordinate.
When the size of the upper and lower platform structures is given for the mechanism, the coordinate values ​​of the hinge points (pi, si, i = 1, 2, ..., 6) of the upper and lower platforms in the respective coordinate systems are determined. According to the relative pose of the upper and lower platforms, according to the formula (1), the coordinate values ​​of the hinge points of the upper platform in the fixed coordinate system can be obtained. At this time, the six driver rod length vectors Ii (i=1, 2, ..., 6) can be expressed as a fixed coordinate system
(2)
Where: pi and si are the odd-order vectors of the hinge points of the upper and lower platforms in the moving coordinate system and the fixed coordinate system, respectively.
Obtaining the positional inverse equation of the mechanism
(3)
Where: lix, liy, liz, and li are the component and vector lengths of the i-th rod length vector on the x-axis, y-axis, and z-axis.
1.2 speed analysis
Set the speed of the 6 drives to lvi, then
(4)
Figure 1 6-SPS type parallel mechanism
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