KINEMATIC CONTROL OF CONSTRAINED ROBOTIC SYSTEMS Gustavo M. Freitas∗ Antonio C. Leite∗ [email protected] [email protected] Fernando Lizarralde∗ [email protected] ∗Department of Electrical Engineering - COPPE Federal University of Rio de Janeiro Rio de Janeiro, RJ, Brazil RESUMO for parallel robots and cooperating redundant robots is discussed based on the following concepts: forward kine- Controle de Sistemas Robóticos com Restrições matics, differential kinematics, singularities and kine- Cinemáticas matic control. Simulations results, obtained with a Este artigo considera o problema de controle de pos- Four-Bar linkage mechanism, a planar Gough-Stewart tura para sistemas robóticos com restrições cinemáti- platform and two cooperating robots, illustrate the ap- cas. A ideia principal é considerar as restrições cine- plicability and versatility of the proposed methodology. máticas dos mecanismos a partir de suas equações es- truturais, ao invés de usar explicitamente a equação de KEYWORDS: Parallel robots, redundant robots, multi- restrição. Um estudo de caso para robôs paralelos e robot coordination, kinematic singularities. robôs cooperativos é discutido baseado nos conceitos de cinemática direta, cinemática diferencial, singularidades 1 INTRODUCTION e controle cinemático. Resultados de simulação, obti- dos a partir de um mecanismo Four-Bar linkage, uma In advanced robotic systems, accuracy, repeatability plataforma de Gough-Stewart planar e dois robôs coo- and load capacity are fundamental skills for carrying perativos, ilustram a aplicabilidade e versatilidade da out several practical tasks, where the robot end-effector metodologia proposta. has to perform some operation on a constraint surface or to manipulate an object of the environment (Siciliano PALAVRAS-CHAVE: robôs paralelos, robôs redundantes, et al., 2008). The structure of a robot manipulator con- coordenação multi-rôbos, singularidades cinemáticas . sists of a set of rigid bodies or links connected by means of revolute or prismatic joints, integrating a kinematic ABSTRACT chain. In general, one end of the chain is fixed to a base, whereas an end-effector is connected to the other end. This paper addresses the posture control problem for From a topological point of view, a kinematic chain can robotic systems subject to kinematic constraints. The be classified in (i) open or serial, when there is only one key idea is to consider the kinematic constraints of the sequence of links and joints connecting the two ends of mechanisms from their structure equations, instead of the chain, and (ii) closed or parallel, when a sequence explicitly using the constraint equations. A case study of links and joints are arranged such that at least one loop exists. Usually, serial chain robots can present lim- Artigo submetido em 10/03/2011 (Id.: 01293) itations in their reachable workspace, kinematic singu- Revisado em 27/05/2011 Aceito sob recomendação do Editor Associado Prof. Carlos Roberto Minussi larities, reduced accuracy and rigidity, or be sensitive to Revista Controle & Automação/Vol.22 no.6/Novembro e Dezembro 2011 559 scaling. These drawbacks could be overcome by the use and collision with obstacles, as well as to optimize the of parallel robots to increase rigidity, redundant robots robot motion with respect to a cost function (e.g., joint or mobile manipulators (manipulators mounted on a torque energy). Furthermore, considering the pres- mobile robot, e.g., remotely operated vehicles, wheeled ence of mechanical joint limits, redundant manipula- robots) to augment the system workspace and/or avoid tors can also be used to increase the robot workspace singularities in order to accomplish the task of interest (Siciliano, 1990; Chiaverini et al., 2008). In general, (Murray et al., 1994). multiple cooperating robot arms and mobile manipu- lators, for instance, belong to the class of redundant Parallel robots provide a rigid connection between the robots (Caccavale and Uchiyama, 2008). base structure and the payload to be handled by the end-effector, with positioning accuracy superior to the The coordination of multiple robots is an essential activ- one obtained by serial chain manipulators (Merlet, 1993; ity in several industrial applications, such as assembly Merlet and Gosselin, 2008). The main disadvantages of and manufacturing tasks, where multiple robotic arms using parallel robots are: workspace limitation, more are often grasping an object in contact with the envi- complex forward kinematics maps and more involved ronment. Typical examples include: deburring, con- singularity analysis (Wen and O’Brien, 2003; O’Brien tour following, grinding, machining, painting, polish- et al., 2006; Simas et al., 2009). For instance, in contrast ing and object aligning (Namvar and Aghili, 2005) or to serial chain manipulators, the singularities in parallel even robot dexterous hands (Caurin and Pedro, 2009). mechanisms may have different characteristics. In this framework, a study of the differential kinemat- ics and manipulability indexes for multiple cooperating In this context, the singularities can be classified into robot arms with unactuated joints is presented in (Wen three basic types (Gosselin and Angeles, 1990; Liu et al., and Wilfinger, 1999; Bicchi and Prattichizzo, 2000). Ad- 2003): vanced motion and force control of cooperative robotics manipulators with passive joint are considered in (Tinos (i) configuration space singularities: when the rank of et al., 2006; Pazelli et al., 2011). A screw-based system- the structure equations drops and, thus, the end- atic method to derive the relative Jacobian for two co- effector loses the ability to move instantaneously in operating robots is developed in the recently published some directions; work (Ribeiro et al., 2008; Simas et al., 2009). (ii) end-effector singularities: when the end-effector Motivated by several applications to parallel robots and loses degrees of freedom (DOF), that is, the mo- redundant manipulators, this paper provides a control tion of the active joints can result in no motion of methodology for robotic systems under kinematic con- the end-effector; straints based on a novel proposed method (Wen and O’Brien, 2003). The key idea is to consider the kine- (iii) actuator singularities: when the actuator of the matic constraints of these mechanisms from their struc- manipulator or the active joints cannot produce ture equations, rather than explicitly using the con- end-effector forces and torques in some directions. straint equations. The major advantage of this method- ology is its applicability and versatility when different The kinematic and dynamic control problems for paral- robotic systems are considered. Simulation results are lel robots are considered in (Kövecses et al., 2003; Cheng obtained from the kinematic models of a Four-Bar link- et al., 2003; Rosario et al., 2007) and an autonomous age mechanism, a planar Gough-Stewart platform and control approach to reach the dynamic limits of parallel two cooperating robots. mechanisms is presented in (Pietsch et al., 2005). Redundant manipulators have more degrees of freedom1 Terminology and Notation that those strictly necessary to perform a given task. Redundancy is therefore a relative concept for a robot In this work, the following notation, definition and as- manipulator, depending on the particular type of task sumptions will be adopted: to be executed. For example, six DOF are necessary for positioning and orienting a robot end-effector, thus a 6- E¯ = [ ⃗x ⃗y ⃗z ] denotes the orthonormal frame DOF manipulator is considered non-redundant. How- a a a a a and ⃗x , ⃗y , ⃗z denote the unit vectors of the frame ever, if only the positioning task is of concern, the a a a axes. same arm becomes redundant. The extra degrees of freedom provide more dexterity to the robot structure, 1For some authors, this term has the same meaning as the and it can be used to avoid kinematic singularities degrees of mobility. 560 Revista Controle & Automação/Vol.22 no.6/Novembro e Dezembro 2011 For a given vector x∈Rn, its elements are denoted constraints can be locally represented as algebraic con- T by xi for i=1 ··· n, that is, x=[ x1 x2 ··· xn ] . straints in the configuration space Q by ne : number of effective degrees of freedom of the c(θ) = 0 , (2) mechanism2. n r where c(·):R 7→R . Then, the mechanism has ne =n−r nt : number of degrees of freedom required to per- effective degrees of freedom and the following assump- form a task. tion is considered: Definition 1 A robotic system can be classified as: (i) Assumption 3 The number of active joints is equal to non-redundant: ne ≤nt; (ii) redundant: ne >nt. the effective degrees of freedom, that is, na =ne. Assumption 1 For an open-chain mechanism consti- Note that, the constraint described in (2) is an example tuted by i + 1 links connected by n joints, where the of holonomic constraint. In general, a constraint is said link 0 is fixed, each joint provides a single degree of mo- to be holonomic if it restricts the motion of the system bility to the mechanism structure, and n=i. to a smooth hypersurface in the configuration space Q (Murray et al., 1994). Assumption 2 For a closed-chain mechanism consti- tuted by i + 1 links, the number of joints n must be Now, considering that the constraint can be written in greater than i. In particular, the number of closed loops terms of the joints velocity vectors, we have is equal to n−i. ˙ Jc(θ) θ = 0 , (3) ∂c(θ) ∈ Rr×n 2 CONSTRAINED ROBOTIC SYSTEMS where Jc = ∂θ is named the constraint Jaco- bian. This section considers the kinematics of the closed-chain robotic systems subject to kinematic constraints on ve- On the other hand, the end-effector velocity v, com- locity. The general methodology to derive the forward posed by the linear velocity p˙ and the angular velocity kinematics and the differential kinematics equations for ω (defined as R˙ = ω × R (Murray et al., 1994)), can be constraint-based robotics systems is to open the loop related with the velocity θ˙ by of the mechanism, propagate the kinematics along the [ ] p˙ branches and add the kinematic constraints.