Modelling of Ball and Plate System Based on First Principle Model and Optimal Control

Modelling of Ball and Plate System Based on First Principle Model and Optimal Control František Dušek Daniel Honc Rahul Sharma K Department of Process Control Faculty of Electrical Engineering and Informatics University of Pardubice nám Čs legií 565 532 10 Pardubice Czech Republic frantisekdusekupcecz danielhoncupcecz rahulsharmastudentupcecz AbstractThis paper presents modelling of ball and plate systems based on first principles by considering balance of forces and torques A nonlinear model is derived considering the dynamics of motors gears ball and plate The nonlinear model is linearized near the operating region to obtain a standard state space model This linear model is used for discrete optimal control of the ball and plate system the trajectory of the ball is controlled by control voltages to the motor KeywordsBall and plate system first principle model optimal control LQ control I INTRODUCTION Laboratory experiments are an integral part of control education There are lots of educational platforms available eg inverted pendulum magnetic levitation ball and beam system etc Ball and plate system is an upgraded version of ball and beam system where the position of the ball can be manipulated in two directions 1 The educational model ball and plate consists of a plate pivoted at its center such that the slope of the plate can be manipulated in two perpendicular directions The ball and plate is a nonlinear multivariable and openloop unstable system There are basically two control problems point stabilization and trajectory tracking In point stabilization the aim is to carry the ball to a specific position and hold it there In trajectory tracking control the goal is to make the ball follows a predefined trajectory linear square circle and Lissajous curve 25 The first step is finding out a mathematical model which describes the system There are basically two modelling approaches for the ball and plate system in literature the Lagrangian method and the NewtonEuler method The derivation of dynamical equations of ball and plate system by Lagrangian method can be seen in 6 The modelling based on NewtonEuler method is quite rare in the literature Even though NewtonEuler method is quite cumbersome the variables and equations have physical meaning which is suitable for control educational purpose The balance of forces and torques are considered in the NewtonEuler method to derive the mathematical model In this paper a nonlinear mathematical model of ball and plate system is derived by considering the dynamics of the ball and plate system DC motors and gear system based on balance of forces and torques The model is linearized around equilibrium points to arrive at a linear state space model Infinite horizon optimal linear quadratic LQ control is applied to the trajectory tracking problem by penalizing the state and control effort Simulation results of model verification and trajectory tracking control are also provided II MATHEMATICAL MODELLING Mathematical model of nonlinear dynamics takes account of the position of the ball on the plate depending on the voltage of the motors that control the tilt of the plate in two perpendicular axes see Fig 1 A ball of mass mb moment of inertia Jb and radius rb is located on a square surface plane tilting in the two perpendicular axes x and y The origin of the axis is located at the intersection of coordinate axes The moment of inertia of the plate is Jp relative to each axis of rotation On each axis the torques Mx and My are operated Moments are created by two DC geared motors via two cable systems with the same gear ratio of G but with different moments of inertia JGx and JGy The mathematical model is based on the balance of forces and torques acting on the ball and the dynamic model of DC series motor The real behaviour is taken into account by including an approximation of linear mechanical losses depending on the speed of the rotational motion In the case of moving balls the mechanical losses are proportional to the square of opposition translational speed of movement The model is built on the following assumptions a there is no loss of contact area with the ball b the ball is hollow ball pingpong ball c an infinitely large area not considering the rebound d connection of the motor to tilt the axis is perfectly rigid Fg Mx x y β α mb My Fy Fx Fig 1 Ball and plate system 2017 21st International Conference on Process Control PC June 69 2017 Štrbské Pleso Slovakia 9781538640111173100 c2017 IEEE 216 A Balance of forces ball Since the ball is placed on a surface rotating in two axes we 2 need to consider in addition to the inertia of the translational movement the influence of the apparent forces Euler TTT ange centrifugal and Coriolis These forces are caused by the 8 Feeling comer rotational movement ie all the forces in curvilinear motion hoe x Fomx Feyermyx8 consisting of the movement of translation and rotation A Mann x r ful or Cor j general representation of the resultant forces acting upon the MrsXFyra FeraxMogsin8 curvilinear motion on the mass point in vector form V is the translational velocity vector and is the angular velocity F era10gCOSB WW FgraMug FcenMux8 vector is then described 7 Fig 3 Ball and plate movement in the xaxis xz plane and rotation Foxe m mxr Imax torque acting in yaxis at dt dt Inerita force Face Euler force Feu Coriolis force Foor mxxr 1 Face Frot Fios Few Foor Feen ForaaB kgms entrifugal force Fen 2 centrifugal force Fe Translational force Face Mp All the apparent forces acting in a plane are perpendicular 2 apparent in a plane are perp Rotational force Froe ts 2 SE to the axis of rotation Fig 2 shows a situation where all the rh dt rp dt forces are acting in the plane of motion The mass point is at a Rolling resistance Fios kpy perpendicular distance r from the rotational axis and the vector Environmental resistance F c208 ax dx p ax line connecting the point and the axis of rotation forms an los x3Pairb Ge ae 3 at lat angle a from the selected x axis The Euler force F also Centrifugal force Keen Mpwpgx Mpx apparent inertia force acts only when there isa change in the External force gravity Fyrax mpgsin8 speed of rotation w and its direction is perpendicular to the vector Centrifugal force Fre acts at nonzero rotation speed where and its direction is in the direction of the vector Coriolis force Mp Mass of the ball F cor acts when the velocity of motion is not perpendicular to the Tp Radius of the ball vector ie there is a change in the size of the vector Its Sb Total area S nr direction is perpendicular to the direction of translational speed A Thickness of the ball In the arrangement rotational axis in the coordinate axes Jo Torque of ball J 2m27 AA the forces are decomposed in the directions of axes x and y Wy Wy Angular velocity xaxis TW directions The translational velocity v v and rotation speed oe ky Coefficient approximation rolling losses a and qp are either parallel or perpendicular to each other a i il Decomposition of the axes describes the movement in one 605 Coefficient o aerodynamic resistance ba direction only ie the apparent Euler and Coriolis force is Pair12 Density of dry air at 20 C and a pressure of applied only in the balance of moments The situation of 1014 kPa moving in the xaxis is shown in Fig 3 Since it is not a fixed ke cxpairttrs2 Coefficient of resistance of the environment point we need to consider more ball spin moment of inertia g981 Gravitational constant with speeds of rotation and loss due to rolling resistance and environment resistance Rolling resistance is proportional By considering balance of forces on x axis we get to the rotational speed of the ball and resistance is proportional m vk 4 fo Ux kydx pdx 4 mex to the square of environment translational velocity The overall Pat rp dt ry at at lat b Nat balance of forces can be expressed as Mpgsinf Jy dx y dx ky 4 Kex dx By a gi 1 2 dt dt Mp FI x Sr gsinB 2 Foxt Similarly by considering balance of forces on y axis we get ay Jo Py kody 4p 2 dy lay day f Feen mwxwxt Mp G2 t rz dt rp dt keri dt S mMpy cr mpgsina Jo ay 5 dy Kn 4 Kex dy aay a gi Feor 2mwxdrdt 1 2 dat at mp I y G sina 3 f FE Fes mrxcct B Balance of moments plate with ball drdt ace mdvdt a WwW dadt Overall the balance of moments can be expressed as P 2 62 Fig 2 Forces acting on ball and plate Macey Mace Mcor Mios Mmot Mgra kgms 217 da Torque of plate Maccp Up Jex oa I R tL a UUoUy ws Torque of ball Mace Y Feu myy Xf Coriolis moment Mcorp YFcor 2mpy 2 e dt dat Uo 5 I 1 da i Torque losses Mios Kpx a VJ UjUouy Gravitational moment Mg mpyg cos TR where Fig 4 Equivalent circuit of DC motor ae By considering the balance of voltages Kirchhoffs law of 1 2 Jp Plate moment of inertia MI J mpa the motor connected to x axis My Mass of the plate di a i ote oP x i a Pivot length passes through the center axis of R te Lae t ky PGE t Relix ty te Uo JoxnJey Pivot length passes through the center axis L o k OG R Rix Rziy Ux Up 7 Kpx kpy Coefficient of approximation of rotational Similarly by considering balance of voltages on motor losses connected to y axis XY Current position of the ball di do Mex Mey Actual moments of the drives Rely hae thu Pet Reix ty Uy Up u ap a B Current the angle of the platform according to L ky G Rz Riy Rzix Uy Uo 8 the respective axes where By considering balance of forces on x axis we get R Resistance of motor winding Pa a ayaa a R Internal source resistance a Up ex Gz Moy Gs 2mpy oS kox G L Inductance of motor winding Mex Mpyg cosa 1 Magnetic flux constant k Speed constant voltage of the motor IptJax da dy Kpxda Mgx u ee y at 22 me F 7 ygcosa A iyiy Current of the motor Similarly by considering balance of torques on y axis we get By considering the balance of moments moment of inertia ap 5 a2p ax ap ap M rotational resistance proportional to rotational speed Up Jey Ga Myx Te 2myx TT t kpy Fe mechanical losses M and load torque M caused by magnetic Mgy mpx g cosB field which is proportional to current Ipth d7B dx kpydB M d 0x doy Sat 20 Te eg 0088 5 Im Gea ogee Ma ken Pix Substituting 6 to the above equation gives C Gear system 2 2 2 Mex Gkm ImG2da kG da The gear box reduces the angular velocities of motors to a ny mp dt m dt 9 output angular velocities with respect to the gear ratio Similar dering bal Ft t ted Similarly the torques of motors are increased to output torques t IM AMY COSICCTINE DARANCES OF FOTAUCS ON MOLOF CONES 0 y axis i i a d a Ox B oy Im Gat Keo EY My Kym Piy Mcx GM Mcy GM 6 wus where ox oy 6 Substituting 6 gives to the above equation gives G Gear transmission ratio of the drive Mey GKm JmGdB koG a6 10 xy Angle of rotation of the rotor mp mp Y my dt mp at a B Angle of rotation of the plate with respect to where the relevant axis Tin Moment of inertia of rotating parts of the D Balance of energy and moment motor motor A val rcuit ofan ideal Km Torque constant of motor n sauva ent circu of an idea pc series motor is shown ko Coefficient of rotational loss of motor in Fig 4 It consists 0 Tesistance R inductance L and magnetic MM Current load torque of motors field M Each motor is independently controlled by its own x Py Current angle of rotors supply voltage U U taken from a common voltage source Ug through control signal uuy The rotor generated back Combined model electromotive force EMF is in reverse polarity and is The dynamical equations of ball and plate gears and DC proportional to the rotor angular velocity The torque of the motors by substituting 910 to 45 are given by motor is proportional to the current i 218 ty ax a cx 8 Dy 2 24 2 4b byi yg cosa 16 145 at at ory My atl ae gsinB 11 bax Ge Var 2x ae alxygcosa 16 tp vy Kex 4 i b x2 8 2x b x bai x g cos 17 124 ai at mrp my atl tae gsina ty t Vet at 2y at 3ty gcosB 17 di da Lote y da zy 4 Rex KG de at as a t dalx dily doux 18 mp dt dat Mp mp at di d nti yg cosa ds daiy driy douy 19 b 12 tJoytImG 2 k 2 ett x Es 2 Sy ey OE OX II LINEAR STATE SPACE MODEL Mp dt dt Mp mp at Gkm iy xg cosB The nonlinear dynamic equations 1419 can be linearized di da around operating points X9yo by assuming the following Loe t ky OG R Rix Rziy Uz Up 13 approximation diy ap oo Loe t kuPG a Ry Rly Reix Uy Uo 1 At small angles of plate inclination sin 6 6cos 1 By substituting the following parameters 2 At small rate of change and at initial conditions v a k k Wy i Vy B ay 1t 0 a1 mare a F a3 The linearized model can be represented in standard state Int Jax ImG2 Kpx kpG Gk space model in the form of by PS OD H S Ob Mp Mp Mp Mp qx AxB JptJaytImG Kpy koG Uo dt x 20 byy Beet i by OE gy a mp Mp Mp L y x Re RtRe ku ay da L dg 06 wherexXX Vy wy iy y vy B wy iy The dynamics governing the ball and plate system becomes 5 pp oo 3 we OOF Or dx dx dx dp a 4 a a F xS gsins 14 0000 00000 a4 ay dy ay aa2 c 7 OP OT ay 2 a a3 2 yS gsina 15 000001000 0 TABLE PARAMETERS OF BALL AND PLATE SYSTEM Symbol a I Tre O8 Ma low al Mo dive can etna Z Mo ve ax sma A On i é SWO 219 0 10 0 0 0 00 0 O IV OPTIMAL CONTROL OF BALL AND PLATE SYSTEM a o ua 9 027 9 Discretizing the state space model 20 with a sampling time 0 0 0 1 0 0 0 0 O 0 0 00 bex b3 9b1x6 0 0 0 0 Ts we get 2 2 22 Dixt5 Dixt V6 bixy3 xk 1 Axk Buk 21 0 00 d d 0 00 0 d k xk A 09 00 0 0 60 10 0 0 wk xk 0 O 2 0 0 0 2 0 0 0 With the linear state space model an optimal LQ controller can 0 00 0 0 0 00 1 0 be designed for the ball and plate system The aim of the alby9 g QQ 0 0 0 9 controller is generating optimal control voltages by minimizing bsy9 Pry tx9 Pay to the following criteria 0 00 0 d 0 0 0 d d The model parameters used in the simulation is listed in Joo izrlx Ck OQxCk 1 uk DR uk d Table 1 The linearized model 20 is discretized with a The cost function consists of penalization weighting matrix Q sampling time of T 01s and compared with the continuous of state variables and control effort weighting matrix R If the time dynamic model 1519 by applying a series of step control state variables are able to be estimated the optimal control voltages Fig 5 shows the control voltages and ball positions in actions can be calculated by x and y axes of linear and nonlinear model Plate angles and motor currents are shown in Fig 6 Since the system is open loop uk Kxk xwk 22 unstable and has integrating character the quality of Wherex is the desired state variable for reference point at time linearization has to be finally checked by closed loop and K is feedback gain matrix obtained by the following experiments equation Eo K BPB R 1BPA Z 005 The matrix P is the solution of discrete Riccati equation which 04 is given by 0 02 04 06 08 1 12 14 16 18 2 P APA Q APBBPB 4 RBPA 1 ewe In MATLAB the feedback gain can be obtained by 005 K dlqr ABQR x Simulation experiments were conducted on two different 0 OR 0406 0B NR TA TB 8 trajectories square shaped and Lissajous curve shaped 2 trajectory The model parameters used in the simulation are as S listed in Table I with a sampling period of T 01s The 37 0 Td rd weighting matrices are chosen as follows 5 Qeye1010 Reye210 0 02 04 06 08 1 12 14 16 18 2 Q11100 Q66100 Time s Q22100 Q7710 Fig 5 Nonlinear vs Linear model outputs and inputs in open loop verification Trajectory Velocities oo 04 50 Real 0 rc 01 Reference 2 z 01 Ky ye Eo 0 02 0 02 04 06 08 1 12 14 16 18 2 04 0 5 10 15 2 OF KK Plate slopes o 100 oa 7 3 S 50 10 02 01 0 01 02 0 02 04 06 08 1 12 14 16 18 2 m 0 LS 05 2 Control voltages 50 KR Zo 10 0 5 10 15 2 Motor currents 05 15 0 02 04 06 08 1 12 14 16 18 2 5 1 05 i 05 pty r a P P 05 cart 02 04 06 08 1 12 14 16 18 2 0 5 10 15 20 0 5 10 15 20 Time s Time s Time s Fig 6 Plate angles and motor currents in open loop verification Fig 7 Square trajectory control voltages velocities plate slopes and currents 220 Fig 7 shows the simulation results of LQ control trajectory tracking with a trajectory in the shape of square Control voltages ball velocities plate slopes and motor currents are also shown The initial location of the ball was at origin and was different from initial reference point The controller was able to track the ball to the reference trajectory points Fig 8 shows the simulation results of the Lissajous curve shaped trajectory The simulation experiments with both the trajectories show the quality of linearized model which is derived from the nonlinear model is good for control purposes V CONCLUSION The mathematical model of ball and plate system is derived by NewtonEuler method considering balance of forces and torques of ball and plate motors and gears The nonlinear model is linearized around operating points following some approximation Simulation of open loop model verification is performed The linearized model is used to discrete optimal LQ control of the trajectory tracking problem of ball and plate system The simulation result proves the quality of linearization of nonlinear model ACKNOWLEDGMENT This research was supported by project SGS modern methods for simulation control and optimization at FEI University of Pardubice This support is very gratefully acknowledged REFERENCES 1 Humusoft 1996 2014 CE151 Ball Plate Apparatus Users manual 2 Nokhbeh Mohammad and Daniel Khashabi Modelling and Control of BallPlate system Diss PhD thesis Amirkabir University of Technology 2011 3 Jadlovská A Š Jajčišin and R Lonščák Modelling and PID control design of nonlinear educational model Ball Plate Proceedings of the 17th International Conference on Process Control Vol 9 2009 4 Oravec Matej and Anna Jadlovská Model Predictive Control of a Ball and Plate laboratory model Applied Machine Intelligence and Informatics SAMI 2015 IEEE 13th International Symposium on IEEE 2015 5 Liu Dejun Yantao Tian and Huida Duan Ball and plate control system based on sliding mode control with uncertain items observe compensation Intelligent Computing and Intelligent Systems 2009 ICIS 2009 IEEE International Conference on Vol 2 IEEE 2009 6 Hauser John Shankar Sastry and Petar Kokotovic Nonlinear control via approximate inputoutput linearization The ball and beam example IEEE transactions on automatic control vol 373 pp 392398 1992 7 J Šrejtr Technická mechanika II Kinematika 1částSNTL Praha pp 256 1954 Fig 8 Lissajous curve trajectory control voltages velocities plate slopes and currents 221