Automatic selection of the Groebner Basis' monomial order employed for the synthesis of the inverse kinematic model of non-redundant open-chain robotic systems

The methods most commonly used to synthesize the Inverse Kinematic Model (IKM) of open-chain robotic systems strongly depend on the robot's geometry, which make them difficult to systematize. In a previous work we presented a systematic procedure that relies on Groebner Bases to synthesize the IKM of non-redundant open-chain robots. This study expands the developed procedure with a methodology for the automatic selection of the basis' monomial order. The procedure's inputs are the robot's Denavit-Hartenberg parameters and the movement range of its actuators, while the output is the synthesized IKM, ready to be used in the robot's control system or in a simulation of its behavior. This procedure can synthesize the IKM of a wide range of open-chain robotic systems, such as Cartesian robots, SCARA, non-redundant multi-legged robots, and all non-redundant manipulators that satisfy the in-line wrist condition. The procedure's performance is assessed through two study cases of open-chain robots: a walking hexapod and a PUMA manipulator. The optimal monomial order is successfully identified for all cases. Also the output errors of the synthesized IKMs are negligible when evaluated in their corresponding workspaces, while their computation times are comparable to those required by the kinematic models calculated by traditional methods.