Study of the Antagonistic Stiffness of Parallel Manipulators with Actuation Redundancy

In this paper parallel manipulators, consisting of a basic antropomorph kinematic chain and parallel chains with redundant in number linear actuators are investigated. Antagonistic stiffness, as a result of the actuation redundancy is also studied. A kinematic model of parallel manipulators and a stiffness model are created. Conditions for specification of the antagonistic stiffness are considered and the possibilities for generation of a desired stiffness matrix are investigated. Approaches for specification of a desired compliance along a given direction in the operation space and for specification of the biggest compliance in the operation space are proposed. A scheme for stiffness control of manipulators with actuation redundancy is developed on the basis of the above cited approaches. Computer experiments are performed and a graphic interpretation of the results is presented. Keywordsrobot, manipulator, parallel, actuation, redundancy, antagonistic, stiffness, control, compliance. 1.INTRODUCTION. To execute high-value added operations, such as fine motion control and force control in assembly, deburring, and many other manufacture tasks, one needs to design control strategies which would provide accurate and stable motion. One of the natural solutions of these tasks is to incorporate into the robot manipulator inputs whose number is larger than the needed minimum. If a manipulator possesses degrees of mobility that outnumber the dimensions of the task space, the redundancy thus obtained is called “kinematic redundancy”. On the other hand, if the number of the actuators is larger than that of the degrees of mobility, the resulting redundancy is called “actuation redundancy” [1]. Additional degrees of mobility allow the realization of gross motion, dexterity, obstacle avoidance, etc. Additional actuators significantly enhance the load handling capacity of the manipulator. However, actuation redundancy is used to minimize joint torques or to satisfy actuator constrains [2],[3],[4]. At the same time, it enables the system to modulate the end-effector stiffness by realizing an internal load distribution [5], [6],[7]. Parallel manipulators [8],[9] are designed with actuation redundancy, while kinematic redundancy is a characteristic for serial manipulators. Parallel structures are created also by serial manipulators contacting the environment, walking machines, multi-fingered end-effectors or coordinated multiple manipulators [10]. The hybrid (serial-parallel) manipulators combine the properties of both types and admit kinematic, as well as, actuation redundancy. The hybrid manipulators with an anthropomorphic serial kinematic chain and with actuation chains, fixed parallel to the serial one, copy to a certain extent the structure and actuation of the human limb. It is known that the human limb is characterized by both kinematic and actuation redundancy. Thus, the human body admits gross and dexterity motion, and at the same time undergoes large loads and has variable stiffness. Purposeful motion under external mechanical disturbances and in the absence of any offered feedback is executed in human and in animal natural limb[11]. This motion is possible because a given level of coactivation of antagonistic muscles defines an equilibrium condition of the joint. Furthermore deviation of the limb from this equilibrium state results in the generation of a restoring torque which is a function of the mechanical properties of the muscle and is independent of other feedback. The shift to the next equilibrium position is linked with the specification of a new appropriate set of muscle activities, required to define the new equilibrium position. Moreover, it enables the inherent mechanical properties of the muscle-skeletal systems to take care of the details of motion. Similarly, the provision of an appropriate stiffness or compliance in an industrial manipulator considerably simplifies tasks, involving manipulation of objects against kinematic constraints such as, for instance, the insertion of a shaft into a bearing . Manipulation against kinematic constraints will be considerably simplified if it is possible to specify motion of the end effector in response to arbitrary disturbance forces. Control strategies, most generally known as stiffness control by redundant actuation [12], are developed for this purpose. The strategies are part of impedance control schemes [13] where one can directly follow the ratio “force-displacement”, and not each quantity separately. These considerations imply the necessity to describe the desired dynamic relationship between the end-effector force and position. Stiffness control schemes realized by employing redundant actuation are basically divided into: passive stiffness control, feedback stiffness control, and stiffness control realized by means of antagonistic actuation. First, the passive stiffness control is not exactly “control” but rather an analysis of the end-effector characteristics, taking into account the stiffness of the system elements. The end-effector stiffness is a result of the transformation of the joint stiffness and of the additional compliant elements introduced into the manipulator structure [14], [15]. For example, such a scheme is used in the design of remote center compliance (RCC) devices for assembly [16]. Second, feedback stiffness control is a scheme where the characteristics of the end-effector stiffness are controlled by choosing proportional coefficients in the positioning joint controllers that would correspond to the desired characteristics. An example of the development of such a scheme is the so-called “direct compliance control (DCC)”, [17], [18]. One of the important features of such a control law is that the desired arm dynamics is based on the stiffness parameters defined in the operation space. One can obtain the latter by individually adjusting the stiffness parameters of each joint. One can also formulate the DCC law by analyzing the relationship between the arm stiffness parameters defined in the operational space and those of the actuation joint. One can obtain more compliance adjustable joints by using actuation redundancy. A disadvantage of such a scheme, however, is the occurrence of constraints of the feedback control, on the one hand, and the existence of stabilization problems, on the other hand. Third, antagonistic actuation stiffness control uses a specific characteristic consisting in of the fact that the actuation redundancy yields antagonistic forces in the mechanism. The internal forces balance each other in a closed mechanism and do not perform any effective work but generate end-effector stiffness [5]. The following solutions of the present scheme are known: open-loop stiffness control[12]; stiffness control which utilizes the internal forces to modulate the stiffness at variable contact locations[19], and lower bound stiffness control[20]. Open-loop stiffness control scheme introduces a feed-forward stiffness controller to alleviate the feedback burden [12]. The open-loop stiffness control involves off-line planning of antagonistic actuator loads, such that one can obtain the characteristics of the desired object stiffness, as well as the desired net effective load acting on the object. Open-loop stiffness planning produce restoring forces against position disturbances or offsets. Small perturbations or non-modeled dynamics remain to be compensated by the feedback controller. A further continuation of [12] is the next scheme which utilizes the internal forces to modulate the stiffness at variable contact locations on the object held by dual arm robots [19]. Except for a positioning feedback controller, the scheme includes a stiffness controller with feedback with respect to position. The required internal forces, that keep the contact stiffness close to the desired stiffness, can be determined by minimizing an objective function with respect to the magnitude of the internal force. Another control scheme [20] employs antagonistic actuation of three actuated legs and guarantees a lower bound of the stiffness in all directions within the two dimensional task space. The control of the smallest singular value of the end-effector stiffness guarantees the stiffness lower bound. Here, similarly to the previous scheme, the stiffness model improves when applying feedback with respect to position. The objective of the up-to-date work of the author is the investigation of parallel manipulators with actuation redundancy. A procedure for structural and geometric synthesis of parallel manipulators with linear drive modules is developed [21]. Investigations of the passive compliance of these manipulators are performed [22]. The influence of the joint stiffness is shown considering the manipulator structure and position over the end-effector stiffness variation. The objective of the present work is to investigate the antagonistic stiffness produced by actuation redundancy and to create an approach for specification of a desired stiffness in the operation space by means of a magnitude variation of the antagonistic drive forces, suitable for a redundant manipulators control. This paper is structured in the following way. Chapter 2 presents a kinematic model of the parallel manipulators with a basic chain and a redundant number of parallel drive chains. Chapter 3 develops a model of the end-effector stiffness, considering the stiffness and forces in the drive joints. Chapter 4 reveals the conditions for specification of desired end effector stiffness. Chapter 5 presents some approaches for specification of the stiffness and manipulator control. Conclusions are made in chapter 6. 2. A KINEMATIC MODEL OF PARALLEL MANIPULATORS. The studied manipulators[21] presented in Fig.1. have a basic kinematic chain including the immovable base 0 and two movable links connected by means of two rotational joints. The class of the kinematic joints is not shown in Fig.1. The choice of rotational joints of class 3, 4 or 5 determines the number of degrees of mobility of the basic chain. The manipulator end-effector is situated in the last link 2 of this chain, which moves in a ν-dimensional operation space. Parallel kinematic chains m in number are attached to the immovable basis and to the links of the basic chain. These chains consist of similar type drive modules with one single drive joint for a linear motion P and two passive rotational joints R1 and R2. The number of the degrees of freedom of each parallel chain is equal to three in the planar case and to six in the spatial one. In this way the number of the manipulator degrees of mobility is defined by the number of the degrees of mobility of the basic chain h. The number of degrees of mobility of the studied manipulators with actuation redundancy h should be less than the number of the drive modules m ( h<m ). The parameters of the relative motion in the drive linear joints of the parallel chains are sellected as generalized parameters for the kinematic system definition: λ = [λ1, ....., λm] (1) and the parameters of the relative motion in the joints of the basic chain q = [q1, ...., qh] . (2) Every closed loop in the parallel manipulator structure produces links between the generalized parameters (1) and (2). These links can be defined by means of the vector function (3) ( ) q Φ λ = which allows to calculate the parameters (1) from parameters (2). Part of the parameters (1) with number h [ T h 1 ,...,λ λ = u λ ]