The paper discusses a geometrical solution to perform the path analysis for an industrial 6R serial robot; the proposed solution represents, for the particular manipulator, an alternative to the general efficient numerical methods. The solution is conceived on the basis of a mechanism geometric decomposition, to calculate an input-output trigonometric equation in the last wrist angular joint co-ordinate. The roots are numerically singled out. By the found roots, representing the last wrist co-ordinates, all the joint sets (up to sixteen), solving the displacement analysis, are analytically worked out. The algorithm has been conceived and written to control an industrial painting robot; it has been tested and applied in production, with a worst-case running time amounting to eight milliseconds.

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