Task-space/Joint-space Damping Transformations for Passive Redundant Manipulators

We consider here passive mechanical wrists, capable of imparting a desired dumping matrix to a grasped workpiece. Previous work 112. 131 has shown how to select a dumping matrix such that an assembly operatwn can be modc force-guided Thc passive mechanical wrist is to be progmnvnable it can adopt a widc range of damping matrices by virtue of a number of t u d e dampers which inrcrconneet the joints. We have been studying the range of dmnping matrices that such a wrist can adopt. purely by tuning its dampers. We find that a redundant wrist has a broader range of realizable damping matrices than a non-rewrist. A kinematic Jacobian relates the task-space damping matrix to a similar matrix in the hydraulic space of the tunable dampers (jointspace). For redudmt wrists the transformation of damping matrices between task-space and joint-space is not straightfonvard. In this paper we identify the causal directions along which the transformations are linear. We show that the joint-space matrices which are obtained as linear tranrfonnations of desired task-space matrices are all singular. Many realizable joint-space matrices (corresponding to a &sired task-space damping matrix) are shown to exist which are not discovered by linear transformations. 1.0 BACKGROUND AND MOTIVATION The motion with which a robot responds to forces encountered during assembly may bring the workpiece closer to or farther from correct assembly. We have been studying methods of designing the accommodation (inverse damping) properties of a grasped workpiece, such that the forces arising during assembly always cause motions which move the workpiece closer to correct assembly. The design of an accommodation matrix for force-guided assembly was described in [13]. Implementing a suitable accommodation behavior is a form of force control 171. Force control schemes in which the robot mimics a passive physical system are known to enjoy inherent advantages in interactive robotic tasks such as automated assembly. Colgate and Hogan showed that onZy a passive system remains stable at all frequencies when coupled to an a rb i t rq passive environment [3]. Robot controllers may emulate a passive system in order to take advantage of this fact [I, 11.141. Unfortunately, the speed of a softwarecontrolled system is limited by the control system bandwidth [17]. This motivates the use of mechanical elements, such as springs, dampers etc., in order to implement force control. One of the best known mechanical devices used for a class of assembly tasks is the remote center of complimce (RCC) device 1161. Work on the analysis and design of devices with desirable compliance, accommodation, or inertia properties suitable for different classes of interactive tasks are found in [2. 6,9,12,13,18]. A suitable force control law is task-specific. An accommodation matrix that works for a particular task is not necessarily useful (in fact it may be detrimental) for another. 1050-4729~93 $3.00 Q 1993 IEEE Therefore, robots must be able to adopt a broad range of accommodation matrices in order to perform a variety of tasks. A disadvantage of mechanically implemented force control is the loss of simple software programmability. This motivates the need for mechanical elements with programmable parameters, e.g. spring stiffness, damping coefficient etc. For example, Cutkosky and Wright developed a programmable RCC wrist for introducing variable compliance in a robot [4]. We have been studying the range of accommodation matrices attainable by coupling the joints of a robot (or more practically of a wrist) via a passive network of programmable dampers (see Fig. 1). The network directly determines the wrist's joint-space accommodation matrix. This matrix describes the force-velocity relationship of the individual joints and does not involve their geometry or interconnections. Figure I . A simple parallel 2 MIF passive mechanism. The ports of the hydraulic cylinders are interconnected through constrictions with tunable damping. The accommodation matrix of the workpiece as viewed by the environment is called the task-space accommodation matrix. This matrix relates the forces on and the velocities of a firmly grasped rigid workpiece. The task-space matrix is related to the joint-space matrix by the manipulator's Jacobian. Our objective is to achieve a wide range of task-space accommodation matrices by programming the network of dampers that couple the joints. However we find that passive networks may adopt only a particular class of accommodation matrices [5 ] . We have proposed kinematic redundancy as a means of increasing the range of force control laws that may be implemented by a passive device. 2.0 OBJECTIVE AND SUMMARY In this paper, we study the relationship between accommodation (or damping) matrices in joint-space and taskspace for passive redundant manipulators. Our analysis can be immediately applied to networks of springs (imparting a compliance) or of masses (imparting an inertia matrix). Just as we use a manipulator's Jacobian matrix to transfonn forces and velocities between its joint-space and task-space, we can imagine similar transformations between the spaces for its accommodation or damping matrices. By analogy to the term "forward kinematics." the computation of the task-space accommodation matrix from a given joint-space accommodation matrix will be called the forward transformation problem. The problem of determining the joint-space accommodation matrix from a desired task-space matrix will be called the inverse tramformation problem. The inverse transformation problem is relevant when. as described above, a desired accommodation matrix is specified in task-space in order to make an assembly operation forceguided. The desired matrix is transformed to the robot's joint space. To implement the resulting joint-space accommodation matrix one still has to program the network appropriately as described in 151. The tasks-space accommodation (or damping) matrices of a manipulator are related to their joint-space counterparts through a congruence transformation. This is a linear transformation involving the manipulator's Jacobian and is sensitive to the manipulator's pose. For non-redundant manipulators in non-singular poses, the forward and inverse transformations are simple one-to-one mappings between jointFor redundant manipulators, however, the transformations are not always straightforward. Kinematic redundancy imposes constraints on joint-space velocities (in parallel manipulators) or forces (in serial manipulators). These constraints give rise to preferred causal directions along which linear transformations of accommodation and damping matrices may take place. The causal directions depend on the structure of the manipulator (serial or parallel) as well as on the type of matrix being transformed (accommodation or damping). For example, in a parallel manipulator, an accommodation matrix maps linearly from task-space to joint-space but not in the reverse direction. A damping matrix, on the other hand, maps linearly from joint-space to task-space. Dual results exist for serial manipulators. For redundant manipulators some of the transformations are many-to-one. For instance, for a serial redundant manipulator, many joint-space accommodation matrices map to a single task-space matrix. To implement a desired task-space matrix, one has a choice of many joint-space matrices, and hopefully some of them are realizable by a passive network of dampers. Unfortunately the causal linear transformations do not directly identify all of the corresponding matrices in the case of many-to-one transformations. As an example, the inverse transformation (which is a linear congruence transformation) of a desired task-space accommodation matrix for a parallel manipulator yields only one matrix. However, infinitely many joint-space matrices exist that also correspond to the given task-space matrix. In order to take full advantage of redundancy, one must therefore look beyond the linear transformation. In the next section we give a simple physical example to point out some of the important characteristics space and task-space. exhibited by redundant passive mechanisms. Section 4.0 discusses the nature of force and velocity transformation between joint-space and task-space of redundant manipulators. An understanding of force and velocity transformation is important for identifying the causal directions in which accommodations and damping matrices transform. We discuss the latter in Section 5.0. Finally, in Section 6.0 we apply our results to passive force control.