[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】球形移动机器人的设计与分析-外文文献

wheeledeffectivelyhas beencycle kinematics and is driven with the help of two stepper motors. All the robots described up till now work on the principleof change of center of mass with the help of IDU for rolling the robot. Rollmob, designed and developed by Ferrire et al. 6is a spherical ball driven by an universal wheel equipped with rollers. The rotation of the roller wheel drives the spherearound the direction parallel to the wheel axis, while the sphere rotates freely around the direction perpendicular to this0094-114X/$ - see front matter C211 2009 Elsevier Ltd. All rights reserved.* Corresponding author. Tel.: +91 22 25767888.E-mail addresses: vrundasc.iitb.ac.in (V.A. Joshi), banavarsc.iitb.ac.in (R.N. Banavar).Mechanism and Machine Theory 45 (2010) 130136Contents lists available at ScienceDirectMechanism and Machine Theorydoi:10.1016/j.mechmachtheory.2009.04.003dian Institute of Technology Bombay, India.A typical construction of the spherical mobile robot has a spherical shell with some internal driving unit (IDU) mountedinside the spherical shell. The robot in 1 is composed of a spherical shell and an arch shaped body. The driving unit consistsof a pendulum and a controlling arch. The arched body and pendulum can control the pitch angle and the controlling archbelow the pendulum controls the roll angle simultaneously with the pitch angle. In 2,3, the IDU consists of a wheel rollinginside the spherical shell which is in turn driven by a motor. The robot moves due to the disturbance of the system equilib-rium due to unbalance of the inside construction. The Rollo, a spherical mobile robot designed and developed by Harmoet al. 4, the IDU is hanging from the rim. The rim can be rotated around the two axes of the IDU. The spherical shell rollsaccording to the movement of the rim. A small car like structure has been used as IDU in Sphericle 5. The car has the uni-1. IntroductionMobile robotics is one of the importantwalking, hopping, sliding are used. Rollingis less, the set of configurations is reachableis conservative. As compared to singleto the spherical shape, the robot recoversspherical shell and the robot can beautonomous spherical mobile robotbranches of robotics. For mobility of the robots, different motions like rolling,motion has certain advantages over other motions. The problem of wear and tearby lesser number of inputs as the system is nonholonomic and the role of frictionrobots like gyroscopes 1, the spherical structure is statically stable and duefrom the collisions with unknown obstacles. Any sensor can be mounted inside theused. Due to these reasons and for the purpose of constructing a testbed, andesigned and developed by our group at Systems and Control Engineering, In-Design and analysis of a spherical mobile robotVrunda A. Joshia, Ravi N. Banavara,*, Rohit HippalgaonkarbaSystems and Control Engineering, Indian Institute of Technology Bombay, Mumbai, IndiabMechanical Engineering, Indian Institute of Technology Bombay, Mumbai, Indiaarticle infoArticle history:Received 4 July 2008Received in revised form 2 April 2009Accepted 2 April 2009Available online 13 May 2009Keywords:Spherical robotNonholonomic systemsEuler parametersabstractA spherical mobile robot, rolling on a plane with the help of two internal rotors and work-ing on the principle of conservation of angular momentum has recently been fabricated inour group. The robot is a classic nonholonomic system. Path planning algorithms exist inthe literature for certain classes of nonholonomic systems like chained form systems, nil-potent systems and differentially flat systems. The model of this spherical mobile robothowever, does not fall into any of these classes and hence these existing algorithms arerendered inapplicable to this system. The final objective is to make this robot as a testbedfor feasible path planning and feedback control algorithms.C211 2009 Elsevier Ltd. All rights reserved.journal homepage: /locate/mechmtaxis. The construction of the robot designed by Bhattacharya et al. 7 is driven by a set of two mutually orthogonal rotors,attached to the spherical shell of the robot from inside. Along the Z axis there is a single rotor and along X axis there are tworotors which are rotated in tandem as a single rigid connected body. When rotors are rotated, the spherical robot rolls in theopposite direction due to the conservation of angular momentum. GroundBot, a spherical robot developed by Rotundus isdesigned for extraterrestrial exploration. The center of gravity of this robot is kept close to the ground with the help of acontrolled pendulum. When it is raised, the ball rolls forward. When the pendulum is moved sideways, the ball turns. Sphe-robot, a spherical mobile robot by Mukherji et al. 8 has radial spokes along which masses are placed. Radial movement ofthese masses create a moment about the center causing the motion of the robot. Cyclops 9 has two degrees of freedom inits locomotion system. It can pivot in place along its vertical axis and roll forward and backward along a fixed horizontal axisvia a small motor and gear-head fixed inside. The review papers discussing constructional details of available spherical mo-bile robots are 10,11 and a concise review on the existing path planning algorithms and feedback algorithms for the systemcan be obtained from 12,13. In this paper, we present a systematic study of a spherical mobile robot, designed, fabricatedand analyzed in our laboratory. The organization of the paper is as follows. Section 2 describes the construction and designdetails of the robot. Section 3 describes the mathematical modeling of the system using quaternion. In Section 4, the prop-erties of the robot are discussed in the quaternion space. The experimental setup along with the discussion on the experi-mental results is reported in Section 5. Section 6 provides concluding remarks.2. DesignV.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 130136 131The spherical mobile robot designed in our laboratory works on the principle of conservation of angular momentum. Therobot has two internal rotors similar to a few constructionsmentioned above. The robots spherical shell is made up of acrylicmaterial having 4 mm thickness. The inner radius of the robot is 30 cm. A crucial aspect of the design is to place the internalcomponents such that the center of mass of the robot is exactly at the geometric center of the sphere. This is very importantso that the robot will not tip over on its own. The easiest way to achieve this is to place all the parts symmetrically. There aretwo internal rotors along two mutually orthogonal axes of the sphere. These rotors are driven by two D.C. motors MAXON EC/32, Brushless, 80 W. Two batteries of the type Polyquest PQ08003 TWENTY LIPO PACK of capacity 800 mAh each, are usedfor supplying power to one motor. So in all there are four batteries. The two speed control units which control the speed ofthe motors receive control signals from the external controller. For symmetrically placing the components, the motor alongwith rotor and one speed control unit is placed on one side and the battery along with dead weight is placed on the diamet-rically opposite direction as shown in Fig. 2. Similarly another pair of motor assembly and battery + dead weight are placedin diametrically opposite directions. The robot is fabricated in two hemispheres as shown in Fig. 1. Each hemisphere consistsof one motor assembly and one battery + dead weight assembly. The total weight of the robot is 3.4 kg. Fig. 3 shows the sche-matic of the robot developed. As discussed in Section 1, there are different driving mechanisms available in the literature.Some spherical robots work on the principle of change in the center of gravity while some work on the principle of conser-vation of angular momentum. As already stated, our spherical robot works on the principle of angular momentum. Otherspherical robots working on a similar principle are reported in 7,14. In these designs two rotors are mounted in oppositedirections for mass balancing and motors moving these rotors must run in tandem which may create problems at the time ofpractical implementation. As against this, we have a single rotor in the X as well as the Z direction for overcoming this prob-lem. We provide dead weights on the opposite sides to adjust the imbalance. In addition, the transparent acrylic sphericalshell enables students to understand the internal mechanism while in motion. It is absolutely critical that there be no rel-ative motion between the two hemispheres while in motion. This can be achieved by an arrangement for screwing a con-necting rod along the axis of the sphere as shown in Fig. 3.Fig. 1. Placing of components in a half shell.132 V.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 1301363. Mathematical modelingThis section describes the development of an analytical model of the spherical rolling robot using quaternion. Consider aspherical robot rolling on a horizontal plane as shown in Fig. 4. An inertial coordinate frame is attached to the surface anddenoted as xIyIzIwith its origin at a point O. The body coordinate axes xbybzbare attached to the sphere and have their originat the center of the sphere G. The set of generalized coordinates describing the sphere consists of 15Fig. 2. Construction of the spherical robot.Fig. 3. Construction of the spherical robot.Fig. 4. Sphere rolling on a surface.C15 CoordinatesC15 AnyEulerwhereformingLet ib;w.r.t.We assumeWe assumeEqs.6 7It canV.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 130136 133the robot) are located along X and Z axis of the body frame. The robot is symmetric in construction and we therefore considerxby 0, reducing the system equations from (3) to_p X1pxbx X2pxbz; 4wherep xye0e1e2e3C138T; 5abe shown that the matrix Q is an orthogonal matrix and hence is invertible. The internal rotors (which are actuators for00C02e2C02e32e02e100C02e32e2C02e12e04 5Q 00C02e12e02e3C02e2666777:01C02e12e0C02e32e200 2e02e12e22e366677710 2e2C02e3C02e02e12637_e30 0 1whereQ_y_e0_e1_e26666666647777777750010666666664777777775xbx0001666666664777777775xby0000666666664777777775xbz; 3_x2 302 302 302 3_y C0 2e1_e0 2e0_e1C0 2e3_e2 2e2_e3 0: 2b(1) and (3) describe the kinematics of the sphere fully giving a set of state equations asreduce to_x 2e2_e0C0 2e3_e1C0 2e0_e2 2e1_e3 0; 2a_y C0 2re1_e0 2re0_e1C0 2re3_e2 2re2_e3 0:the sphere to have unit radius without any loss of generality. For a unit sphere, the no-slip constraint equations_x 2re2_e0C0 2re3_e1C0 2re0_e2 2re1_e3 0:time using the relationship given in 17,200xbxxbyxbz2666437775 2e0e1e2e3C0e1e0e3C0e2C0e2C0e3e0e1C0e3e2C0e1e02666437775_e0_e1_e2_e32666437775: 1the sphere to roll without slipping and no-slip constraints for a sphere of radius r are given byThe projection of the angular velocity vector on the body axes can be related to the rate of change of the Euler parametersx; y are the coordinates of the contact point I and e0; e1; e2and e3are the Eulers parameters describing orientationunit quaternion such thate20 e21 e22 e23 1:jb; kbbe the unit vectors of the body frame and x be the angular velocity of the sphere given byx xbxibxbyjbxbzkb:eters form a unit quaternion and can be manipulated using quaternion algebra 1619. Using a set of Euler parameters fordefining orientation, we get the set of generalized coordinates asp x; y; e0; e1; e2; e3T;parameters have the advantage of being a nonsingular two to one mapping with the rotation. In addition, Euler param-We use Euler parameters (instead of Euler angles) which is a set of 4 parameters to describe the orientation of the sphere.of the contact point I on the plane.set of variables describing the orientation of the sphere.4. Properties4.1. ControllabilityBeforenects21. InConsiderwhere6e0e3 e2e1 4C0e21 e20C0 e23 e226 7It canbutiondescribewhichusinget motions134 V.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 130136X53e23 e22C0 e20C0 e21C012e112e012e3C012e2666666664777777775; X68C0e1e2 e0e3C012e2C012e312e012e1666666664777777775:be observed that all higher order brackets can be expressed in terms of X1; X2; X3; X4; X5giving the rank of the distri-as 5. As the Euler parameters are used, the number of the state variables is 6. We are using a system of 4 parameters tothe orientation. But all possible orientations lie on a hyper surface defined by a level sete20 e21 e22 e23 1; 7is a 3 sphere. This gives the dimension of the configuration space as 5 which is equal to the rank of the distribution. SoChows Theorem the system is controllable and can be taken from any position to any arbitrary position using Lie Brack-described by the vector fields (6).X32e21C0 2e20 2e23C0 2e224e2e1C0 e0e312e212e3C012e0C012e12666666666437777777775; X46e0e1C0 e2e36e1e3 e0e212e3C012e212e1C012e026666666643777777775;2 32 3D X1; X2; X3; X4; X5; X6C138;A distribution is formed using the above generated Lie Bracket vector fields given byX3X1; X2C138; 6aX4X1; X3C138; 6bX5X2; X3C138; 6cX6X1; X4C138: 6dwhere p; X1p and X2p are given by (5). We compute the following Lie Brackets using a Philip Hall convention 21,221x2y_p X pxb X pxb;X2pQC0100001666666664777777775C02e1e3 e0e2C012e312e2C012e112e0666666664777777775: 5cof the modelwe proceed to path planning of the spherical robot, it is essential to check whether there exists a path that con-any arbitrary configurations of the sphere. The question can be answered using a result known as the Chows Theoremthis section, we use the algorithm given in 22 to answer the question.the system described by Eq. (4).004 512e3C012e24 502 32e2e3C0 e0e12 3X1pQ166677712e0666777; 5bC010002666663777772e0e3 e2e1e23 e22C0 e20C0 e21C012e12666663777774.2. Conversion into a chained formFor determining the degree of nonholonomy, we construct a distribution associated with the control system (4) asD spanfX1; X2g:We then construct filtrations associated with the distribution D asE0 D; F0 D;E1 E0E0; E0C138; F1 F0F0; F0C138;E2 E1E1; E1C138; F2 F1F1; F0C138;E3 E2E2; E2C138; F3 F2F2; F0C138;E4 E3E3; E3C138; F2 F3F3; F0C138:According to 23, a feedback transformation which puts a driftless two input nonholonomic system into a chained form ex-ists if and only ifdimEi dimFi i 2; i 0; .; n C0 2: 8In this particular casecertainC15 PivotingThispositionspeedgradualV.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 130136 135communicationchannelFig. 5. Experimental setup at the laboratory.Blue toothPCPIC16F877DACDACSpeedControlSpeedControlOptical encoderMOTOR1MOTOR2Hall sensormaneuvers, for which the trials were carried out on the experimental setup. These maneuvers are,maneuver:When one of the rotors is in the vertical position, the robot is expected to spin about the vertical axis.particular maneuver is known as pivoting maneuver. The experiments were carried out with the rotor in the verticalfor this particular maneuver. It has been observed that the robot responds to the sudden change in the angularby spinning about the vertical axis as expected. However, the result particularly is not satisfactory for slow andchanges in the magnitude of the angular velocity.For i 0; dimF02;For i 1; dimF13;For i 2; dimF25i 2;For i 3; dimF35;For i 4; dimF45i 2:It can be observed that the condition (8) is not satisfied and it is not possible to convert the model into a chained form.5. Experimental setup and discussionRigorous experimentation has been carried out on the experimental setup shown in Fig. 5. The main controller in the sys-tem is a bluetooth enabled PC, which generates control signals according to the algorithm programmed. The control signalsare transmitted to the microcontroller PIC16F877 through a bluetooth modem BlueSMiRF Gold. According to the signal re-ceived from the PC, the microcontroller controls the speed of the D.C. motors using a Digital to Analog Converter (DAC). Forenabling feedback, there are Hall sensors that provide information about the speed of the motors and optical encoders thatgive position of the rotors. During the experimentation, it was possible to control speed of the motors. We have worked onthe path planning problem of the robot and the related work is reported in 24. In this work we are mainly interested inC15 Equatorial maneuver:When one of the rotors is in the horizontal position, the robot is expected to roll along a straight lineon the plane and the contact point should follow the great circle on the sphere surface, in the plane perpendicular to theaxis of the rotor. According to the experiments carried out, the results of this maneuver do not match with the expectedresults. The probable reasons are imperfections of the sphere surface and some design related critical issues such as rotorparameters.136 V.A. Joshi et al./Mechanism and Machine Theory 45 (2010) 1301366. ConclusionDesign and constructional features of a spherical mobile robot rolling on a plane are presented in this paper. The kine-matic model of the system is developed using quaternions for the description of the orientation of the robot. It can be ob-served that the model is nonsingular and valid everywhere. It is shown that the model is fully controllable and can be takenfrom any arbitrary configuration to any arbitrary configuration within the unit 3-sphere in the quaternionspace. Further, it isnot nilpotent and also can not be converted into a chained form. According to the experimentation carried out on the robot,we observe that it works for some maneuvers while for some maneuvers, the results do not match with the expected results.This is due to some imperfections in the fabrication of the robot. We plan to work on these issues. We also w