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Kinematics Model of Nonholonomic Wheeled Mobile Robots for Mobile

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Kinematics Model of Nonholonomic Wheeled Mobile Robots for Mobile

    Kinematics Model of Nonholonomic Wheeled Mobile Robots for Mobile

    Manipulation Tasks

    Dimitar Chakarov

    Institute of Mechanics- BAS, 1113 Sofia, “Acad. G.Bonchev” Str., Block 4

Abstract. In the work presented in this paper a class of robots is revealed scheduled for mutual interaction with the human

    being. These are robots that can join together mobility and manipulation. Mobile manipulator systems hold promise in

    many industrials and service applications including manipulations, assembly, inspection and work in hazardous

    environments. Virtual variants of wheeled mobile robots WMR are derived in the present investigation. A generalised

    kinematics model of nonholonomic WMR is presented in this work. The direct and reverse problem of the kinematics of

    nonholonomic WMR is also shown. The usage of the kinematics model for control of WMR in mobile tasks or in mobile

    manipulation tasks is suggested.

    1. Introduction 2. Holonomic and nonholonomic WMR.

    The robotized technologies are very quickly The wheeled mobile robots (WMR) are divided into spreading for domestic, service and entertainment needs. A two basic types - holonomic and nonholonomic. big number of scientific investigations and scientific Theoretically, the holonomic mechanical systems comprise activities form a new scientific field during the recent years links, that impose restrictions on the limb velocities, and devoted to mutual interaction among robots and the human after integration these restrictions can be reduced to being. This is an interdisciplinary scientific field that covers restrictions only on the limb locations. When WMR do not robotics, computer sciences, and the science of knowledge, impose restrictions on the motion velocities in the 2D physiology and sociology. (planar) solution they are called holonomic. The holonomic

    Robots are going very soon to assist the human being WMR possess maximal number of degrees of freedom in on a wide range of problems, which are not attractive, they the 2D (planar) solution h=3. In the field of the mobile

    are dangerous, not well paid or boring to humans. Robots robots the term holonomic is used as an abstract term for assistants are going to work in the future as patient sitters, WMR with three degrees of freedom. Thus, every WMR as security guards, as rescuers and fire rescuers, in surgery with three degrees of freedom in plane is called a and rehabilitation, in domestics and in offices, in mining, in holonomic one. Various mechanisms are used as universal building as well as in stores and museums. or omni wheels, orthogonal or ball wheels in order to

    In order to work together with and to assist and to achieve a holonomic motion. [5]. Тhe holonomic WMR

    interact with people the new robot generation must posses a allow easier motion planning in plane. A holonomic WMR mechanical structure that is suitable for this partnership in is shown on Fig.1. the human not organised and unknown surrounding The nonholonomic mechanical systems comprise links environment. The robot’s successful invasion in this restricting the system velocities; thus these restrictions environment depends on the development of competent and cannot be integrated. In this way the nonholonomic WMR practical systems that can be very well trusted, which are impose restrictions on the velocities of the motions in plane. reliable and simple for usage [1,2,3,4]. Wheeled mobile robots (WMR) are long ago invented in areas, where they

    interact with humans as service robots for remote book

    reading in a library or for serving tea and meals, human

    following and guiding robots, entertainment and cleaning

    robots.

    The compatible with people robots must integrate

    mobility and manipulation. Mobile manipulator systems

    hold promise in many industrials and service applications

    including manipulations, assembly, inspection and work in

    hazardous environments. The integration of a manipulator

    and a mobile robot base places special demands on the

    vehicle's mechanical system.

    The objective of the present paper is to evaluate the possible solutions of mobile robot bases and to build a kinematics model of WMR suitable for mobile Fig.1. Holonomic WMR manipulation tasks.

    Due to this reason the nonholonomic WMR possess The distance between the driving wheels along the is 2b, and the angular velocities of the left and the less than three degrees of freedom in plane h < 3. They are 1axis Y??simpler in construction and thus cheaper, with less right wheel are respectively ,. θθlrcontrollable axes and ensure the necessary mobility in plane. When one wheel is rolling on a straight line without Due to this reason a kinematics model of nonholonomic ?slipping with angular velocity θ, its centre is moving with WMR with two degrees of freedom h=2 are derived in the velocity V. The velocity of the oscillate point T with the cpresent work. A nonholonomic WMR is shown on Fig.2. plane L is 0 and thus the following equation is fulfilled:

    ? (1) V?V?θr?0Tc

    The upper equation can be integrated and can be presented

    as a link among the angular and the linear position of the

    wheel: l - θr = 0.

     When the wheel is rolling along a curved line the

    T??linear velocity of its centre V?[X;Y] in the base co-ccc

    ordinate system OXY depends on the wheel orientation in

    the plane defined by the angle ф

     ??X?θrcosφ?0c (2) ??Y?θrsinφ?0Fig.2. Nonholonomic WMR c

     In Fig. 3 a generalised model of a nonholonomic WMR The upper equations can not be integrated in order to define with h=2 is presented. It includes two symmetrically relations only between the wheel positions. In the plane allocated driving wheels with radii r. The nonholonomic motion on the wheel velocities are imposed restrictions, WMR include a various number universal wheels for thus the mobile devices from the type shown in Fig. 3, are keeping up the balance in plane. The investigated WMR called nonholonomic WMR. possess a single universal wheel. This wheel is not a driving

    one and it is not included in the kinematics model.3. Direct and reverse kinematics task of nonhoonomic . In the

    WMR. robot centre P is connected a local co-ordinate system

     PXY, where Xis along the axis of symmetry, and Y is 11 1 1The nonholonomic mobile devices include two co-axial along the axis of the driving wheels. The angle between the r ldriving wheels, the velocities of their centres Vи V are axis Xand the axis Xof the immovable co-ordinate system cc1 co-linear with the axis X of the local co-ordinate system OXY is denoted with ф. 1PXY. The velocity V of the centre P of the mobile 11p platform is also co-linear with the axis X. The plan of X11 1 Zvelocities of WMR in the plane OXY is presented in Fig.1. c The following equations can be derived: L V?rl V?12(V?V) (3) pcc rl? 2bφ?V?VC (4) ccr T If we derive the upper equations along the axes of

    the base co-ordinate system OXY, where the velocity of the

    0T??centre VV?[X;Y] is presented by the co-ordinates , ppppY1 Y r land the velocities Vи V of the right and the left wheel is ccl V?cdefined with the help of (2), thus equations are derived l rdefining the kinematics of WMR in the base co-ordinate X1 Vsystem [6] P b???rrX?cosφθ?cosφθr prlV22P c ?ф ???rr (5) Y?sinφθ?sinφθprl 22b??rr?φ?θ?θrl2b2b

     ?If the velocity V of the centre of the mobile platform and rprX O ?its velocity of rotation we combine in the vector:

    T ????X?[X;Y;φ] (6) ppFig.3. A general model of non-holonomic WMR. ??and the velocities of the driving wheels θθ, we combine lr

    in the vector

     As it can be clearly seen by the developed study on T??? (7) θ?[θ;θ]rlthe three independent parameters (6) that define uniquely the positioning of WMR in the plane n=3, that is imposed

    then the direct task of the kinematics of WMR is presented one nonholonomic restriction (11) m=1, from where the

    degrees of freedom of the nonholonomic WMR in plane are by the vector equation

    defined as two h=n-m=2.

    ?? (8) X?Sθ4. Conclusion where rr??cosφ;cosφThe developed kinematics model of the nonholonomic 22??rr ?? (9) S?sinφ;sinφWMR can be used for control creation in mobile tasks or in 22??mobile manipulation tasks. Control can be based on the rr;???2b2b??direct kinematics task by means of using equation (5). In this case the equilibrium of the given velocities of the ?? The direct task allows for each position of WMR , guarantee motion along a straight line, and lrwheels definition of (3x1) the vector of velocities (6) of the centre the difference between them defines robot rotation. This P from (2x1) the vector of velocities (7) of the driving control mode belongs to a lower level and is more wheels. convenient for derivation of on-line tasks of motion. It is necessary to solve the reverse task of the Control can be build up on the inverse kinematics task kinematics in order to plan the robot motion and the robot by using equation (10). In this case control models are control. In order to find a solution for the reverse task of the derived including tracing a path considering the non-+kinematics of WMR, the pseudo inverse matrix S of (9) holonomic restrictions (11). These control schemes are can be used, because matrix (9) is not a quadratic one. more sophisticated, they can consider also the dynamics of ??? (10) WMR, they can include adaptiveness and robustness. θ?SX

     It is necessary to use an additional restriction on parameters

     (6) defined by the non-holonomic links of WMR [7],

    References: because in this case the number of the input parameters (6)

     is bigger than the number of the output parameters (7).

    ?1. Khatib, O., K.Yokoi, O.Brock, K. Chang, A. Casal. (11) AX?0+Robots in Human Enviroments: Basic Autonomous Matrixes S and A can be easily defined by using Capabilities. Int.Journ. of Rob. Res., Vol.18, No.7, the equalities (3) and (4) presented in the form below: pp.175-183, 1999. r?V?bφ?Vpc2. Takanishi, A., Hirano, S., and Sato, K. 1998. (12) l?V?bφ?VDevelopment of an anthropomorphic head-eye system pc

    foå humanoid robotrealization of human-like head-eye when we define the upper equations along the axes of the motion using eyelids adjusting to brightness. In local co-ordinate system PXY. The velocity of the centre 11Proceedings of the International Conference on T??V?[X;Y] is defined along the axes X and Yby 11 pppRobotics and Automation, Vol. 2, Leuven, Belgium, pp.

    means of consideration of their angle of rotation ф. In this 13081314. way equations (12) along axis X1present: 3. Nishiwaki, K., Sugihara, T., Kagami, S., Kanehiro,

     F., Inaba, M., and Inoue, H. Design and development

    ?of research platform for perception-action integration in ???Xcosφ?Ysinφ?bφ?rθppr (13) humanoid robot: H6. In Proceedings of the ????Xcosφ?Ysinφ?bφ?rθpplInternational Conference on Intelligent Robots and

     Systems, Akamatsu, Japan, Vol. 3, pp. 15591564, 2000.

    and along axis Y present: 4. Khatib O., O.Brock, K.Chang, D.Ruspini, L.Sentis, 1

     S.Viji. Human-Centered Robotics and Interactive

    ??Haptic Simulation, The Int.Journ.of Robotics Res., ?Xsinφ?Ycosφ?0 (14) ppVol.23, No.2, pp.167-178, 2004. 5. Holmberg R., O. Khatib. Development of a Equation (13) express the reverse task in kinematics (10), Holonomic Mobile Robot for Mobile Manipulation +from where for the matrix S we can derive: Tasks. Pros. of the Int. Conf.on Field and Service Robotics - FSR'99, Pattsburg, PA, Aug.1999. cosφ;sinφ;b??1?6. Katsura S, K.Ohnishi. Human Cooperative Wheelchair (15) S????cosφ;sinφ;brfor Haptic Interaction based on Dual Compliance ??

    Control. IEEE Trans. on Industr. Electr., vol.51, No.1,

    Febr., 2004. Equality (14) expresses the equation of the links (11)

    7. Coelho P. , U. Nunes. Path-Following Control of between the velocities of the nonholohomic WMR

    Mobile Robots in Presence of Uncertainties. IEEE Trans. including the matrix:

    on Robotics, vol.21, No.2, April., 2005.

     (16) ??A??sinφ;cosφ;0

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