Journal for Geometry and Graphics 10 (2006), No. 2, 173--182
Copyright Heldermann Verlag 2006
Kinematic Analysis of a Pentapod Robot
Gert F. Bär
Institute for Geometry, Technical University, 01062 Dresden, Germany
gert.baer@tu-dresden.de
Gunter Weiß
Institute for Geometry, Technical University, 01062 Dresden, Germany
gunter.weiss@tu-dresden.de
The investigated milling robot with an axial spindle as platform is a parallel manipulator. Five legs carry and control the spindle. An algebraic solution of the direct kinematic problem is given by the help of vector calculus. The solutions are determined by the roots of 5 polynomials of degree 4. Therefore, together with a quadratic normalizing condition the number of solutions is not greater than 2048. Compared to our first result this number is strongly reduced but still large. However, numerical solutions of the polynomial equation system show a stable and fast convergence using Newton methods.
Then, the inverse kinematic problem is solved. Four of the five leg lengths are determined by solutions of two quadratic equations. Some geometrical considerations and additionally technical restrictions allow to prove that a unique solution exists.
Furthermore, the velocity and shakyness is studied. Using Ball's screw we show how for a given rate of change of the leg lengths the velocities are determined. The special design of the spindle causes that the Pentapod robot is architecturally shaky with respect to a revolution about the spindle axis. This fact is no technological lack because the spindle axis is identical with the actuated milling axis. Finally, all singular positions are characterized.
Keywords: Robotics, spatial mechanisms, constraint parallel manipulators.
MSC: 53A17
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