Mobile manipulators with implicit aboard kinematics Krzysztof Tcho´ n, Janusz Jakubiak and Robert Muszy´ nski Institute of Engineering Cybernetics Wroclaw University of Technology ul. Janiszewskiego 11/17, 50–372 Wroclaw, Poland email: tchon|jjakubia|mucha@ict.pwr.wroc.pl Abstract We study the kinematics of a mobile manipulator composed of a nonholonomic mobile platform and one or more sta- tionary aboard manipulators whose kinematics are defined implicitly. Combined with the platform kinematics this leads to implicitly defined instantaneous kinematics of the mobile manipulator. By extending the endogenous configuration space approach we have defined forward and inverse ana- lytic Jacobians, and designed Jacobian pseudoinverse al- gorithms of forward and inverse mobile manipulator kine- matics. Computer simulations of a kinematic car type plat- form equipped with a five bar mechanism illustrate the the- ory. Keywords: mobile manipulator, implicit kinematics, for- ward kinematics, inverse kinematics, Jacobian algorithm, parallel manipulator 1 Introduction A mobile manipulator is a robotic device consisting of a mobile platform and a stationary manipulator mounted on the platform, that distinguishes itself by remarkable mo- bility and manipulation capabilities. Endowing the mobile platform with a number of cooperating serial manipulators or with a parallel manipulator will further enhance manip- ulation capabilities and reliability of the mobile manipula- tor. Since these kinematic structures form closed kinematic chains, they define implicit aboard kinematics [1, 2]. Con- sequently, the kinematics of the whole mobile manipulator also become defined implicitly. While mobile manipulators with open chain (explicit) aboard kinematics have been studied in the literature for quite a long time, cf. [3, 4, 5, 6, 7] and references therein, we are not aware of any work dealing systematically with mobile manipulators with implicit kinematics. A pair of cooperating manipulators mounted on a platform consid- ered in [6] is a rare and rather specific case study in the context of performance evaluation. This paper is intended as a step towards filling in this gap by using the endoge- nous configuration space approach [7, 8, 9]. Specifically, we shall define the implicit instantaneous kinematics, for- ward and inverse analytic Jacobians, and finally introduce Jacobian forward and inverse kinematic algorithms based on suitable Jacobian pseudoinverses. The algorithms are applied to exemplary inverse kinematic problems involv- ing a mobile manipulator equipped with a five bar mech- anism. Two such inverse kinematic problems have been solved, and in the course of computation of the end effector trajectory the forward kinematics algorithm has been in- voked multiply. It has been demonstrated that the endoge- nous configuration space approach provides useful tools for dealing with the kinematics of mobile manipulators with implicitly defined aboard kinematics. This paper is composed in the following way. In section 2 we define basic kinematic concepts, and state forward and inverse kinematic problems. Section 3 is devoted to Ja- cobian algorithms of forward and inverse kinematics. The performance of these algorithms is illustrated in section 4 with an example of the kinematic car type mobile platform with a five bar mechanism aboard. The paper is concluded with section 5. 2 Basic concepts A mobile manipulator with implicit aboard kinematics will be defined as a robotic system consisting of a nonholo- nomic mobile platform, and of a single or of a collection of stationary manipulators mounted on the platform, whose kinematics are defined implicitly. Given a mobile manip- ulator, we let variables q R n , x R p , y R r denote a posture of the platform, joint positions of the manipulator, and a position and an orientation of the end effector. Un- der assumption that the platform velocities are subject to l n independent constraints, and that the aboard kinemat- ics are described by s r independent constraints imposed 1