This paper presents a new method to solve the geometric design problem of spatial two degrees of freedom, open loop robot manipulators with revolute joints that perform tasks, which require the positioning of the end-effector in three spatial locations. Tsai and Roth  solved this problem first using screw parameters to describe the kinematic topology of the R-R manipulator and screw displacements to obtain the design equations. The new method, which is developed in this paper, uses Denavit and Hartenberg parameters and 4×4 homogeneous matrices to formulate and obtain the kinematic equations. The loop-closure geometric equations provide eighteen design equations in eighteen unknowns. Polynomial Elimination techniques are used to solve these equations and obtain the manipulator Denavit and Hartenberg parameters and the manipulator base and end-effector geometric parameters. A sixth order polynomial is obtained in one of the design parameters. Only two of the six roots of the polynomial are real and they correspond to two different robot manipulators that can reach the desired end-effector poses.
A New Polynomial Solution to the Geometric Design Problem of Spatial R-R Robot Manipulators Using the Denavit and Hartenberg Parameters
Contributed by the Mechanisms Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 1999. Associate Editor: C. M. Gosselin.
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Mavroidis, C., Lee, E., and Alam, M. (October 1, 1999). "A New Polynomial Solution to the Geometric Design Problem of Spatial R-R Robot Manipulators Using the Denavit and Hartenberg Parameters ." ASME. J. Mech. Des. March 2001; 123(1): 58–67. https://doi.org/10.1115/1.1335484
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