Quaternion Based Robust Tracking Control of Kinematically Redundant Manipulators Subject to Multiple Self-Motion Criteria † H¨ usn¨ u T¨ urker S ¸ahin, Ufuk ¨ Ozbay and Erkan Zergero ˘ glu Abstract— In this study, a model based robust control scheme is developed for kinematically redundant robot manipulators that also enables the use of self motion of the manipulator to perform multiple sub-tasks in order to increase the manipulabil- ity and/or performance of the system. The proposed controller ensures uniformly ultimately bounded end-effector and sub-task tracking despite the parametric uncertainty associated with the dynamic model. A Lyapunov based approach has been utilized in the controller design and extension to a non minimum set of parameters for orientation representation has been presented to illustrate the flexibility of the approach. The capabilities and performance of the resulting controller is demonstrated by simulation results. Index Terms— Robot control, kinematically redundant ma- nipulator, self-motion control, robust/variable structure control. I. INTRODUCTION Robotic manipulators are highly nonlinear multi-input multi-output systems subjected to uncertainties associated with their dynamics. Moreover, external disturbances are inevitable under practical operation conditions. For this reason an efficient tracking controller for robot manipula- tors should achieve sufficient robustness versus parametric uncertainties in the dynamics, as well as external addi- tive disturbances. Kinematically redundant manipulators are complicated robotic systems with more degrees of freedom (DOF) than required to perform a task in the operation space. Owing to their extra DOF they can achieve much better performance in more dextrous operations, and/or have the increased flexibility for the execution of sophisticated tasks. Since the dimension ( i.e., n) of their link position variables is greater than the dimension (i.e., m) of the operational space variables, the null space of their Jacobian matrix has a minimum dimension of n − m. Any link velocity in the null space of the manipulator Jacobian will not effect the operational space velocity and hence is referred to as self- motion. As stated in [1], [2], and [3], there are generally an infinite number of solutions for the inverse kinematics of a redundant manipulator. Thus, given a desired end effector trajectory, it is difficult to select a reasonable joint space trajectory that satisfies both control constraints (i.e., stability and boundedness of all signals) and mechanical constraints (i.e., singularities and joint limit avoidance). Therefore an H¨ usn¨ u T¨ urker S ¸ ahin, Ufuk ¨ Ozbay and Erkan Zergero ˘ glu are with the Department of Computer Engineering, Gebze Institute of Technology, PK. 141, 41400 Gebze/Kocaeli-Turkey. E-mail: {ezerger}@bilmuh.gyte.edu.tr † This work is mainly supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Project No: 104 E061. efficient controller for kinematically redundant robot ma- nipulators should be robust to parametric uncertainties and external disturbances, and achieve accurate end effector tracking, while reserving the self motion of the manipulator available for performance enhancement. Many robotic manipulators are non-planar in geometry. For accurate operation in 3D task space, singularity free representation of the position and orientation (pose) infor- mation is also important. Frequently adapted orientation representation techniques such as Euler angles and Ro- driguez parameters, have singularities for certain parts of operational space, which degrade controller operation. These singularities can be avoided by the utilization of 4 parameter based unit quaternion representations, however this slightly complicates the overall control design. Due to the challenging nature of the fore-mentioned con- trol design problem, many researches have proposed alter- native solutions. To name a few; in [4], Khatib proposed a control scheme based on the dynamic model of a manipulator in Cartesian space and extended this result for redundant manipulators by using the pseudo-inverse of the Jacobian matrix. In [5], Seraji proposed the configuration control approach in which the end-effector motion in task-space is augmented by any n − m dimensional additional tasks, such as optimization of kinematic and dynamic objectives or pos- ture control. In [6], Hsu et.al. proposed a dynamic feedback linearizing control law that guarantees end-effector tracking and also provides control of redundant link velocities. In [7], Oh et.al. proposed a disturbance observer based robust controller that controls both the motion of the end effector and the null space motion of the redundant manipulator. In [8], Zergeroglu et.al. presented a model based controller that achieves exponential end-effector and sub-task tracking. The extensions for adaptive and model based output feedback type of controllers were also presented. However these controllers either required the exact knowledge of the dy- namics, or assumed linearly parameterizable uncertain robot dynamics and the absence of external disturbances. From an examination of these previous research, we can group the control strategies for kinematically redundant manipulators in two categories. The first approach is based on extending the dimension of the operational space by incorporating additional constraints, so that the overall system becomes non-redundant. This approach usually introduces additional algorithmic singularities in the extended Jacobian matrix, and hence can cause the control input to become unbounded even away from manipulator singularities. The second approach is the generalized/pseudo-inverse based control formulations