Provision of Controlled Motion Accuracy of Industrial Robots and Multiaxis Machines by the Method of Integrated Deviations Correction

There is a developed method of correction of the integrated motion deviations of industrial robots and multiaxis machines, which are caused by the primary geometrical deviations of their segments. This method can be used to develop a control system providing the motion correction for industrial robots and multiaxis machines.


Introduction
It is impossible to avoid deviations in the geometrical size and form, known as primary deviations, when producing and assembling parts and units of segments entering into the composition of such complex mechanical systems as industrial robots. These geometrical deviations result in a difference of the motion parameters of a real industrial robot from the motion parameters of its nominal model made in accordance with the design documents (3-D model). To improve the accuracy of controlled motions it is necessary to make corrections of the nominal model of an industrial robot taking into account measuring primary geometrical deviations of its segments [1].  Figure 2).   Matrix Ai,i* that shows a transformation of homogeneous coordinates from coordinate system Si* into coordinate system Si, considering primary linear and angular deviations of the segment, will represent a mathematical model of primary deviations.

A Mathematical Model of Primary Deviations
(1) Character ~ in expression (1) denotes that the matrix contains primary geometrical deviations.
When drawing up the matrix it is unnecessarily to measure all nine angular deviations. It suffices to measure only three of them. The remaining six angular deviations can be calculated. For this it is necessary to solve six nonlinear equations. These equations represent orthonormal vectors constituting rows and columns of the matrix.
Besides, positioning deviations influence the accuracy of multilink mechanisms. Positioning deviations are associated with deviations of generalized coordinates ( Figure 3).

Figure 3. Positioning deviations
The matrix taking into account the positioning deviations has the following view: where q igeneralized coordinate, Δq ipositioning deviation for the i-th generalized coordinate, β i - A mathematical model represents a matrix multiplication:

An Alternative Model
Instead of the mathematical model described by matrix (1), an alternative model can be offered. Let us consider the model in Figure 4.
Angles φ(i) are set by the initial configuration of the multilink mechanism, and angles γ(i) reflect the primary deviation of the links form and distortions in the joints. Linear deviations ρ(i), used in model (3) As in case of (1) when drawing up matrix (3) is necessary to measure three angular deviations.

A Method of Correction of Integrated Deviations
Mathematical models considering primary deviations of segment geometrical parameters in complex mechanical systems with a daisy-chain structure can be used to determine the integrated deviations of segment positions arising because of the error accumulation in the positions of each previous segment of a kinematic chain. An integrated deviation of a complex mechanical system should be defined as the guidepath deviations of segment characteristic points and the segment orientation deviations from the specified orientation caused by primary geometrical deviations ( Figure 5). Applying the mathematical support to realize this method of correction in the automatic control systems of industrial robots or multiaxis machines allows improving an enduring accuracy.

Conclusion
Mathematical models (1), (2) and (3) may be used in the parametrization methods of geometric models of multilink mechanisms. Applying mathematical support to realize the suggested method of correction in the systems of automatic control of industrial robots and multiaxis machines allows one to improve enduring accuracy. A detailed realization of the method of integrated deviations correction is given in the papers [2][3][4][5][6][7].