Parallel Robots

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Parallel Robots Frank C. Park Robotics Laboratory, Seoul National University, Seoul, Korea

Abstract Parallel robots are closed chains consisting of a fixed and moving platform that are connected by a set of serial chain legs. Parallel robots typically possess both actuated and passive joints and may even be redundantly actuated. Although more structurally complex and possessing a smaller workspace, parallel robots are usually designed to exploit one or more of the natural advantages they possess over their serial counterparts, e.g., higher stiffness, increased positioning accuracy, and higher speeds and accelerations. In this chapter we provide an overview of the kinematic and dynamic modeling of parallel robots, a description of their singularity behavior, and basic methods developed for their control.

Keywords Closed kinematic chain; Closed loop mechanism; Parallel manipulator

Introduction A parallel robot refers to a kinematic chain in which a fixed platform and moving platform are connected to each other by several serial chains, or legs. The legs, which typically have the same kinematic structure, are connected to the fixed and moving platforms at points that are distributed in a geometrically symmetric fashion. The Stewart-Gough platform (Fig. 1) is a wellknown example of a parallel robot: each of the six legs is a UPS structure (i.e., consisting of rigid links serially connected by a universal, prismatic, and spherical joint), with the prismatic joint actuated. Other examples of parallel robots include the 6  RUS platform of Fig. 2, the haptic interface device of Fig. 3, and the eclipse mechanism of Fig. 4. Parallel robots can be regarded as a special class of closed chain mechanisms (i.e., chains that contain one or more closed loops) and are purposely designed to exploit the specific advantages afforded by the closed chain structure, e.g., for improved stiffness, greater positioning accuracy, or higher speed. Parallel robots should be distinguished from two or more cooperating serial robots that may form closed loops during execution of a task (e.g., a robotic hand grasping an object). Some of the fastest velocities and accelerations recorded by industrial robots have been achieved by parallel robots, primarily by

J. Baillieul, T. Samad (eds.), Encyclopedia of Systems and Control, DOI 10.1007/978-1-4471-5058-9, © Springer-Verlag London 2015

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Parallel Robots, Fig. 1 Stewart-Gough platform

placing the actuators on the fixed platform and thereby minimizing the mass of the moving parts. Many of the model-based techniques developed for the control of traditional serial chain robots are also applicable to a large class of parallel robots. On the other hand, kinematic and dynamic models for parallel robots are inherently more complex. Parallel robots also possess features not found in serial robots, e.g., passive joints, the possibility of redundant actuation, and a diverse range of singularity behavior, that need to be considered when designing a control law. We therefore begin with a brief overview of the kinematic and dynamic modeling of parallel robots before discussing their control.

Modeling Kinematics Whereas the kinematic degrees of freedom, or mobility, of a serial chain robot can be obtained as the sum of the degrees of freedom of each of the joints, the situation is somewhat more complex for parallel robots and closed chains in general, since only a subset of the joints can be independently actuated. The mobility of a parallel robot corresponds to the total degrees of freedom of the joints that can be independently actuated. In some cases the number of actuated joint degrees

Parallel Robots

of freedom may exceed the kinematic degrees of freedom, in which case we say that the robot is redundantly actuated. A parallel robot with a designated end-effector frame also has a notion of forward and inverse kinematics. While for serial chains the forward kinematics is a well-defined mapping and the inverse kinematics can typically have multiple solutions, for parallel robots the situation is less straightforward. For the Stewart-Gough platform of Fig. 1, in which the leg lengths can be adjusted by actuating the prismatic joints, the inverse kinematics is unique and straightforward to obtain, whereas the forward kinematics will have multiple solutions. For other types of parallel robots in which the legs themselves contain one or more closed loops, both the forward and inverse kinematics can have multiple solutions. The notion of kinematic singularities for parallel robots is also much more involved than the case for serial robots. Whereas kinematic singularities for serial chain robots are characterized by configurations at which the forward kinematics Jacobian (i.e., the linear mapping relating joint velocities to end-effector frame velocities) becomes singular, for parallel robots and closed chains in general, there exist other notions of singularities not found in serial chains. For example, given a parallel robot with kinematic mobility m – if the parallel robot consists only of one degree-of-freedom joints, this implies that exactly m joints can be actuated – there may exist configurations in which these m joints cannot be independently actuated. Conversely, even if the m actuated joints are each fixed to some value, the parallel robot may fail to be a structure, i.e., some of the links may be able to move. In the above scenario, choosing a different set of m actuated joints may remedy this situation, in which case such singularities are referred to as actuator singularities. Configurations at which singularity behavior occurs regardless of which joints are actuated are denoted configuration singularities. The final class of singularities are endeffector singularities, which correspond to the usual serial chain notion of kinematic singularity, in which the end-effector loses one or more degrees of freedom of available motion.

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Revolute joint

Universal joint Ball joint

Parallel Robots, Fig. 2 6  RUS platform

Prismatic Joint

Universal Joint

Universal Joint

P Parallel Robots, Fig. 3 A 3  P U U haptic interface

Dynamics In the case of a parallel robot whose actuated degrees of freedom coincides with its kinematic mobility m, it is possible to choose an independent set of generalized coordinates of dimension m, denoted q 2 Rm and typically identified with the actuated joints, and to express the dynamics in the standard form M.q/qR C C.q; q/ P qP C G.q/ D ;

(1)

where  2 Rm denotes the vector of input joint torques, M.q/ denotes the n  n mass matrix, the matrix-vector product C.q; q/ P qP denotes the vector of Coriolis terms, and G.q/ 2 Rm

Parallel Robots, Fig. 4 The 3  PPRS eclipse parallel mechanism

denotes the vector of gravitational forces. The structure of the dynamic equations is identical to that for serial chain robots. Also like the case for serial chain robots, the Coriolis matrix term C.q; q/ P 2 Rmm is not unique, so that one should ensure that the correct C.q; q/ P is used in, e.g., any control law whose stability depends on the matrix MP .q/  2C.q; q/ P being skewsymmetric.

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It is also important to keep in mind that the q must satisfy the kinematic constraint equations imposed by the loop closure constraints. That is, if  2 Rn denotes the vector of all joints (both actuated and passive), then q 2 Rm , m  n, will be a subset of  whose values can only be obtained by solution of the kinematic constraint equations; depending on the nature of the kinematic constraints, one may have to resort to iterative numerical methods. If the parallel robot is redundantly actuated, then the dynamics are subject to a further set of constraints on the input torques. Letting qe denote the set of independent generalized coordinates and qa be the vector of all actuated joints, the vector of actuated joint torques a must then further satisfy S T a D W T , where  denotes the vector of joint torques for an equivalent tree structure system that moves identically to the redundantly actuated parallel robot and W and S are defined, respectively, by SD

@ ; @qe

W D

@qa : @qe

(2)

Compared to the dynamics for serial chain robots, the dynamics for parallel robots is, in general, considerably more complex and computationally involved. The recursive algorithms that are available for computing the inverse and forward dynamics of serial chain robots can also be used to develop similar recursive algorithms for parallel robot dynamics; however, the computations will be considerably more involved and require multiple iterations.

Motion Control Exactly Actuated Parallel Robots For parallel robots whose actuated degrees of freedom match the kinematic mobility (this excludes the set of all redundantly actuated parallel robots), most control laws developed for serial chain robots are also applicable. This is not altogether surprising in light of the similarity in the structure of the kinematic and dynamic equations between serial and parallel robots. Control

laws for serial robots are also covered in this handbook, and we refer the reader to  Linear Matrix Inequality Techniques in Optimal Control for the essential details. Here we summarize the most basic control laws and point out any additional computational or other requirements that are needed when applying these laws to parallel robots. Note that other control laws and techniques developed for serial chain robots, e.g., robust, sliding mode, can also be applied with the same additional considerations and requirements outlined below: 1. Computed torque control: Computed torque control for parallel robots has the same control law structure as for serial robots, i.e.,    D M.q/ Kp e  Kv eP C ff ;

(3)

where e denotes the tracking error, Kp and Kv are the proportional and derivative feedback gain matrices, and ff denotes the feedforward term required to cancel the nonlinear dynamics. Robust versions of computed torque control are also applicable to parallel robots under the same set of conditions, e.g., establishing appropriate bounds on the mass matrix eigenvalues and on the norm of the Coriolis matrix. 2. Augmented PD control: The augmented PD control law for serial robots is also applicable to parallel robots, i.e., P qP C G.q/;  D Kp e  Kv eP C M.q/qRd C C.q; q/ (4) where qd is the reference trajectory to be tracked and Kp and Kv are the proportional and derivative feedback gains. Asymptotic stability is also established under the same conditions. 3. Adaptive control: Because the dynamic equations for parallel robots are also linear in the link mass and inertial parameters, i.e., M.q/qR C C.q; q/ P qP C G.q/ D ˆ.q; q; P q/p; R (5)

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where p denotes the vector of link mass and inertial parameters, adaptive control laws developed for serial robots can also be used. 4. Other control methods: There exist numerous control methods developed for serial robots, e.g., task space or operational space control, sliding mode control, and various nonlinear control techniques; with few exceptions most of these algorithms can also be applied to exactly actuated parallel robots with minimal modification.

Redundantly Actuated Parallel Robots As described earlier, parallel robots exhibit a much more diverse range of singularity behavior than their serial counterparts, many of which depend on the choice of actuated joints (actuator singularities). One way to eliminate actuator singularities is via redundant actuation, i.e., the total degrees of freedom of the actuated joints exceeds the kinematic mobility of the mechanism. Redundant actuation offers some protection in the event of failed actuators and, when combined with an appropriate control law, offers an effective means of reducing joint backlash, increasing speed and payload and stiffness, controlling compliance through the generation of internal forces, and even improving power efficiency (as an analogy, the human musculoskeletal system is redundantly actuated by antagonistic muscles). Of course, redundant actuation introduces a new set of control challenges, since the control inputs must be designed so as not to conflict with the kinematic constraints inherent in the parallel robot; loosely speaking, the actuated joints can no longer be independently controlled, since the consequences of unintended antagonistic actuation may be catastrophic. The control of cooperating manipulators (see  Optimal Control and Mechanics) has a long history in robotics, and many of the control techniques developed for such multi-arm systems can also be applied to redundantly actuated parallel robots. One can also apply the control strategies developed for exactly actuated parallel robots to the redundantly actuated case, but modifications

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are necessary to account for the different structure of the dynamic equations. Like all model-based control algorithms, the above control laws are subject to model uncertainties. Whereas in serial chains the effects of model uncertainty simply lead to errors in tracking, for redundantly actuated parallel robots, the consequences can lead to internal forces in addition to end-effector tracking errors. Perhaps the most significant effect of any modeling errors is that, unlike the serial chain case, the kinematic errors can potentially alter the shape of the configuration space (recall that the configuration space will in general be a curved space for closed chains) and also interfere with any PD feedback introduced into the control. The development of control laws that are robust to such modeling errors and disturbances remains an open and ongoing area of research in parallel robot control.

Force Control Both hybrid force-position control and impedance control are well-known and widely applied concepts in serial robots and can be extended in a straightforward manner to exactly actuated parallel robots. Recall that the basic feature of hybrid force-position control is that the task space is decomposed into force- and positioncontrolled directions, whereas in impedance control, the goal is have the robot maintain a certain desired spatial stiffness in the task space. Controllers that combine aspects of forceposition and impedance control have also been proposed and developed for both serial robots and exactly actuated parallel robots. Modeling errors will cause deviations in both the forceand position-controlled directions – leading to motions in force-controlled directions and forces in position-controlled directions – which can be addressed by, e.g., a switching control strategy. The problem of force control for redundantly actuated parallel robots, which encompasses both force-position and impedance control, has also received some attention in the literature.

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The main difference with the exactly actuated case is that internal forces can now be generated, which requires a more detailed and coordinateinvariant examination of stiffness. Control methods that combine elements of force-position and impedance control for redundantly actuated parallel robots have received only limited attention in the literature.

Cross-References  Linear Matrix Inequality Techniques in Opti-

mal Control  Optimal Control and Mechanics  Optimal Control via Factorization and Model

Matching  Optimal Sampled-Data Control

Recommended Reading The monograph (Merlet 2006) offers a detailed and comprehensive treatment of all aspects of parallel robots, with a particularly thorough treatment of the kinematics and singularity analysis. Mueller (2008) provides an excellent survey of the dynamics and control of redundantly actuated parallel robots and is based on the preceding work (Mueller 2005). Cheng et al. (2003) examines in detail the dynamic model for redundantly actuated parallel robots and the basic control strategies; Nakamura and Ghodoussi (1989) also examines dynamic models for redundantly actuated parallel robots. Stiffness analysis and control of redundantly actuated parallel robots are addressed in Yi and Freeman (1993), Chakarov (2004), and Fasse and Gosselin (1998). Analysis of specific parallel robots engaged in various control tasks includes Caccavale et al. (2003), Honegger et al. (1997), Kim et al. (2001), and Satya et al. (1995). The basic references on robot control are Murray et al. (1994), Spong et al. (2006), Anderson and Spong (1988), and Ghorbel (1995) focuses on PD control for closed chains.

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Bibliography Anderson RJ, Spong MW (1988) Hybrid impedance control of robotic manipulators. IEEE J Robot Autom 4:549–556 Caccavale F, Siciliano B, Villani L (2003) The Tricept robot: dynamics and impedance control. IEEE/ASME Trans Mechatron 8:263–268 Chakarov D (2004) Study of the antagonistic stiffness of parallel manipulators with actuation redundancy. Mech Mach Theory 39:583–601 Cheng H, Yiu Y-K, Li Z (2003) Dynamics and control of redundantly actuated parallel manipulators. IEEE Trans Mechatron 8(4):483–491 Fasse ED, Gosselin CM (1998) On the spatial impedance control of Gough-Stewart platforms. In: Proceedings of the IEEE international conference on robotics and automation, Leuven, pp 1749–1754 Ghorbel F (1995) Modeling and PD control of closedchain mechanical systems. In: Proceedings of the IEEE conference on decision and control, New Orleans Honegger M, Codourey A, Burdet E (1997) Adaptive control of the Hexaglide, a 6-DOF parallel manipulator. In: Proceedings of the IEEE international conference on robotics and automation, Washington, DC, pp 543–548 Kim J, Park FC, Ryu SJ, Kim J, Hwang JC, Park C, Iurascu CC (2001) Design and analysis of a redundantly actuated parallel mechanism for rapid machining. IEEE Trans Robot Autom 17(4):423–434 Merlet JP (1988) Force feedback control of parallel manipulators. In: Proceedings of the IEEE international conference on robotics automation, Philadelphia, pp 1484–1489 Merlet JP (2006) Parallel robots. Springer, Heidelberg Mueller A (2005) Internal prestress control of redundantly actuated parallel manipulators and its application to backlash avoiding control. IEEE Trans Robot 21(4):668–677 Mueller A (2008) Redundant actuation of parallel manipulators. In: Wu H (ed) Parallel manipulators: towards new applications. I-Tech Publishing, Vienna Murray R, Li ZX, Sastry S (1994) A mathematical introduction to robotic manipulation. CRC Press, Boca Raton Nakamura Y, Ghodoussi M (1989) Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators. IEEE Trans Robot Autom 5(3):294–302 Satya SM, Ferreira PM, Spong MW (1995) Hybrid control of a planar 3-DOF parallel manipulator for machining operations. Trans N Am Manuf Res Inst/SME 23:273–280 Spong MW, Hutchinson S, Vidyasagar M (2006) Robot modeling and control. Wiley, New York Yi B-J, Freeman RA (1993) Geometric analysis of antagonistic stiffness in redundantly actuated parallel mechanisms. J Robot Syst 10:581–603