A Unified Approach for Multi-Point Cartesian Compliance Control of Serial Manipulators Phongsaen Pitakwatchara* Department of Mechanical Engineering Chulalongkorn University, Thailand Phongsaen.P@chula.ac.th Abstract— This paper proposes a unified approach for con- trolling the Cartesian compliance of multiple points assigned along the linkage chains of the fully actuated serial manipulator. With this method, a number of points (as much as the total number of moving linkages) and their desired compliances may be specified. Then, their motion to perform a particular operation are designated, of which the compatibility is not required. Rather, it will be accounted for automatically by the controller. The controller attempts to accomplish higher- priority tasks with the available control space while the lower- priority tasks might not be completely achieved due to task confliction or inadequate dimension of the control vector space. Additionally, with the use of generalized inverse, the method is capable of controlling the manipulator at the singularities seamlessly. I. I NTRODUCTION Tasks for today’s manipulators are challenging. Many of them need to work within the uncontrolled environment in contrast to the conventional ones working in the well- designed manufacturing workcells. Some typical examples are the humanoid robots working cooperatively with human [1], [2], the critical task of manipulating the surgical robots among organs inside the patient’s body [3], or the adaptable locomotion and grasping tasks of the hyper-redundant robots [4]. One common aspect of these seemingly different tasks is that the robots entail unplanned and complicated dynamic interaction with the unstructured environment. Typically, a serial manipulator performs operations with the motion of a particular point, usually called the end effector point. However, some advanced operations require the coordinated motion of multiple points along the arm. For example, to successfully perform the laparoscopic surgery, the tele-operated robot needs both the control of the mo- torized forceps end point and the control of the point on the forceps rod interacting with the trocar. As another example, the object-embracing operation of the whole-arm manipulator requires the control of multiple contact points. Task-priority framework [5] is therefore employed to reg- ulate the coordinated motion of multiple points. The method takes into account the confliction among tasks by assigning the priority to each of them. However, only the kinematical level of compatibility among subtasks is considered. [6] introduced the dynamically consistent generalized inverse of * This work is partially supported by the Ratchadapisek New Faculty Initiative Fund of Chulalongkorn University. the Jacobian matrix for the operational space formulation. This particular inverse may replace the pseudo-inverse in the task-priority framework to ensure the dynamical level of compatibility. The extended Jacobian matrix [7] causes the associated extended inertia matrix be a block-diagonal matrix. Conse- quently, control torque in the extended space will not directly affect the acceleration of the task space. Using this fact, [8] designed the multi-point Cartesian compliance controller by projecting the lower prioritized subtasks controller onto the torque space of the extended dynamics. The above mentioned research essentially demonstrated the methods dealing with two different task priorities, i.e. higher and lower ones. In general, each motion may have been associated with different task priority. Thus, the im- proved control system should be capable of controlling each task with different precedence. Recently, [1] presented a natural way to handle the multi-level task priority. Task- consistent Jacobian was introduced and used to derive the dynamical equation of the consistent tasks. The development was applied to the control of the humanoid robot [2] by cat- egorizing the tasks into three different priorities: constraint, task, and posture primitives. In this paper, a task means the process of controlling the motion of a rigid link in the serial chain. Using the pro- posed method, this can be accomplished by the compliance control of a body-fixed point. The operation, e.g. sewing the wound inside the abdomen, is performed by the coordinated motion of multiple points. Therefore an operation involves the control of multiple tasks in a consistent manner. In turn, multiple points on rigid linkages along the arm are being controlled: hence the name of this method ‘Multi-Point Cartesian Compliance Control’. In section II, the consistent task space is derived using the task-priority framework. By the use of the generalized inverse theory [9], the formulation is unified in the sense that it may be applicable to all cases of the under-specified, fully-specified, and over-specified manipulator, no matter it be at the singularities or not. In view of the unscheduled and complex interaction with the unknown environment, the impedance control framework [10] is employed as ex- plained in section III. Basically the task consistent stiffness and damping matrices are determined and implemented. Section IV discusses some practical issues relevant to the 2010 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Montréal, Canada, July 6-9, 2010 978-1-4244-8030-2/10/$26.00 ©2010 IEEE 1087