1. Background:

Kinematics is of fundamental importance for trajectory planning, motion control, etc.. However, multiple solutions 
exist for serial manipulators. The desirable or optimal solution has to be selected and matched from multiple solutions using additional algorithms, such as closest solution, energy-saving. As for the continuous trajectory planning in the Cartesian space, several methods exist, such as the interpolation of quarternion, Frenet-Serret formulas and axis - angles methods. However, they might encounter severe problems for kinematically deficient manipulators. Meanwhile, trajectory planned in joint space makes path between via pints complex and unpredictable.

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2. Proposal:

2.1 Kinematics Analysis and Control: Proposing the reservation method to settle the uniqueness in serial robotics
The desirable and optimal solutions can be directly obtained without the selection and match from multiple solutions.

• 5-DOF Manipulator: KUKA youBot manipulator from KUKA; Pioneer-arm from Mobile Robots

• 6-DOF Manipulator & Humanoid Leg: PUMA560 from Unimation; DARwIn-OP humanoid robot from Robotis

2.2 Continuous Trajectory Planning：Proposing DRO & SRO to achieve continuous Cartesian trajectory. 
The dynamic reference orientation or static reference orientation is proposed to plan the continuously time-varying orientation, which is coupled with position of end-effector. Therefore, the continuous complicated trajectory planning can be directly achieved in the Cartesian space using the above-mentioned inverse kinematics without the interpolation in the joint space. Therefore, it can avoid the unpredictable and complicated shape of end-effector.

3. Summary:  

The above one is the fastest algorithm but the look-up table because of the specified uniqueness. It has been granted
 with Patent of Invention, and also displayed on the official website of KUKA youBot (youBot Newsletter 2015 No. 03).