Stability Region Estimation of Statically Unstable Two ...

Proceedings of the 2011 IEEE International Conference on Robotics and Biomimetics December 7-11, 2011, Phuket, Thailand

Stability Region Estimation of Statically Unstable Two Wheeled Mobile Robots Zareena Kausar, Student member, IEEE, Karl Stol, and Nitish Patel

Abstract— Most of the research so far on two wheeled mobile robots is on the control of the robot body. However when a robot is implemented in real environment the range of initial conditions of different states of the robot need to be known for safe operation. This paper presents the estimation of safe region of initial conditions of states for a statically unstable two wheeled mobile robot using a standard Lyapunov direct method. The method uses a known Lyapunov Function. Variation of stability region with the Lyapunov functions requires the selection of a good Lyapunov function for a system. Twotypes of Lyapunov functions, Quadratic and Energy based were formulated. The simulation results for the stability regions were evaluated using these Lyapunov functions and were compared with true region, determined using ode solver. A Lyapunov function that showed less conservative stability region was selected to show the theoretical stability region. The experiments verified the results. I.

T

INTRODUCTION

inherent instability, nonlinear and complex dynamics of two wheeled mobile robots (TWMRs) provide an interest for the researchers. A TWMR consists of two drive wheels, joined by an axle, and a robot body called intermediate body (IB). The IB rotates freely with respect to vertical plane when the TWMR either stands or traverses along a path. Both the configurations of TWMR, statically stable [1] and statically unstable [2], need the position control of IB. The stability control of IB is another important problem of statically unstable TWMRs. Concerning these problems, many control algorithms have been proposed in literature. In particular, state space, optimal and intelligent controllers have been implemented [3]-[7]. In Grasser et al. [3] dynamic equations were linearized around an operating point to design a controller made up of two decoupled state-space controllers. A pole placement controller was proposed by Nawawi et al. [4], but pole placement controllers worked out non-robust with respect to un-modeled dynamic uncertainties. To avoid this problem a linear quadratic regulator (LQR) has been designed [5]. Since the presence of nonlinearities in the system leads to limit cycles or even instability, nonlinear controllers such as Fuzzy logic [6] and Gain scheduling [7] have been designed HE

Manuscript received August 9, 2011. This work was supported by the Higher Education Commission (HEC), Pakistan. Zareena Kausar and Karl Stol are with the Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand. (e-mail: [email protected]; [email protected]). Nitish Patel is with the Department of Electrical Engineering and Computer Science, University of Auckland, Auckland, New Zealand. (e-mail: [email protected]).

‹,(((

to improve the stability control of TWMRs. However, the region of ‘safe’ operation of the TWMR has yet to be described for the physical implementation. This region is known as the stability region, a set of initial conditions for which the corresponding state trajectories converge to a locally stable equilibrium point [8]. TWMR has applications in human interaction and surveillance areas due to small footprint and zero-radius turning capability. These autonomous applications demand the robot to know the region of its stability. This paper presents the estimation of stability region for TWMRs using different methods. The methods to determine the stability region of a stable equilibrium point for nonlinear systems are generally categorized as Lyapunov function[9] and non Lyapunov function[10] methods. The Lyapunov stability methods are prominent in literature [11]-[12]. The direct method of Lyapunov has been considered as clean and short method but for some systems it is not straight forward to find a Lyapunov function. Various methods proposed in literature for the construction of Lyapunov function were based on conventional [13] and numerical [14] approaches. The numerical methods have been applied in many applications [15] to avoid computational problems of conventional method. The challenge of selecting a ‘good’ Lyapunov function is related to the maximization of the stability region where the negativeness of the derivative of Lyapunov function is ensured. The purpose of this work was twofold: one to select a ‘good’ Lyapunov function and second to estimate the region of initial states of closed loop dynamics of a TWMR. Section II of this paper presents the dynamics model of the TWMR. Section III provides an insight to the controller designed to stabilize upright position of a TWMR. A couple of feasible Lyapunov functions have been computed in section IV to select a Lyapunov function which imparts stable region closer to the true region of attraction.The stability regions simulated using numerical method and their experimental verification have been presented in section V. The contribution of this note is to design the procedure for the selection of a Lyapunov function and determination of stability region for statically unstable wheeled mobile robots. II.

MODELING

A. Platform Model In order to model the dynamics of a TWMR, the motion of the robot was considered in x-y plane. Schematic diagram of a statically unstable TWMR has been shown in Figure 1. The

1379

system system was modelled as an inverted pendulum on twoon two was modelled as an inverted pendulum wheels.wheels. Both the IB and wheels were assumed to be Both the IB and wheels were assumedrigid to be rigid bodies. bodies. The mass eachofwheel, m, wasm,assumed to be to be The ofmass each wheel, was assumed located located at the centre the wheel a radiusa radius . Ther. The at the of centre of the having wheel having mass ofmass IB ( of) IB was(M) believed to be located at a distance was believed to be located at a distance l from the axlethejoining two wheels. The mass of from axle joining two wheels. The moment mass moment of inertia inertia for IB for and IB wheels has been as and and wheels hasdenoted been denoted as I p , and I, respectively. respectively. It has been assumed that thethat wheels rotate atrotate certain It has been assumed the wheels at angle certain angle without slippingslipping and remain in contact with thewith ground at a 0without and remain in contact the ground at a single point. motorA has been to applytoa apply torquea torque singleApoint. motor hasconsidered been considered ( ) on wheel navigate the robot distance alongxthe x- the x(T) on to wheel to navigate thea robot a distance along plane. As a reaction to the applied torque the IB rotates by an plane. As a reaction to the applied torque the IB rotates by an Fig. 1. A schematics of statically unstable two Fig. 1. A schematics of statically wheeled mobile robot. unstable two angle .angleB. This angle been pitchasinpitch the remaining Thishas angle hascalled been as called in the remaining wheeled mobile robot. part of part the paper. of the TWMR in x- in xof the The paper.linear The motion linear motion of the TWMR direction and rotation of the IB produce accelerations. In III. CONTROLLER direction and rotation of the IB produce accelerations. In III. CONTROLLER order toorder determine these accelerations the dynamics of the to determine these accelerations the dynamics of the The objective of controller design was to stabilize the TWMRTWMR was modelled using Newton-Euler method.method. The The The position objectiveofof was speed to stabilize the was modelled using Newton-Euler upright IB controller of TWMRdesign with robot tracking. resulting equations of motion using kinematic constraint of upright position of IB of TWMR with robot speed tracking. resulting equations of motion using kinematic constraint A of Full State Feed Back (FSFB) controller based on linear no-slip no-slip (1) have(1)been standard form ofform writing a A aFull State Feed Back (FSFB) controller based on linear havegiven been ingiven in standard of writing quadratic regulator (LQR) was selected for this study. This is nonlinear model as (2). as (2). quadratic regulator (LQR) was selected for this study. This is nonlinear the similar controller used by [7]. The variation is that the x=0rmodel (1) the similar controller by [7]. Thethat variation is that the (1) gains were tuned for aused system model includes motor M(q)ii + C(q, q) + G(q) = T (2) gains were tuned for a system model that includes motor (2) dynamics. The dynamic model (3) was linearized about dynamics. The dynamic model (3) was linearized about upright position of TWMR, a zero pitch angle. As the robot The matrices of (2) have been given (3) in (3) The matrices of(2) have beeningiven upright position of TWMR, zero pitch angle. the As the position control was not oura main objective, staterobot of  position control was not our main objective, the state of robot position has been omitted from the set of possible Mircos(B) M(q) = + robot position has been omitted from the set of possible states. A state vector, used for gain evaluation instead -Micos(B) -(Mi 2 + I p ) states. A vector, used foras gain evaluation instead included an state integral of speed error following: included an integral of speed error as following:

1

[r(m+M :2)



C(q, q) =

G(q) =

[-Mlrs~n (8)8

2

]

(3)          ( ሻ

x=

vref )dt :i (J

r

e

This additional state has been used to track the speed of This additional state has been used to track the speed of the robot and compensate the mass imbalance. The resulting the compensate the mass imbalance. The resulting state robot space and model has been given in (5).

[MlgS~n (8)]

state space model has been given in (5).



o1 o c2g o0 - ab-c 2 o0 o g(cMor+MlI) o0 ab-c 2 o

 = [T]

T



B. Motor Model B. Motor Model The motors to drive the wheels were assumed to be The motors to drivemotors, the wheels were through assumed motor to be brushless DC (BLDC) controlled brushless DC (BLDC) motors, controlled through motor drivers. The motor drivers match the motor coil current to drivers. The motor match corresponding the motor coildirectly currenttoto the controller output drivers thatis current the controller output thatis current corresponding directly motor torque. The motor behaviour has been modelled as (4)to motor torque. The motor behaviour has been modelled as (4) [16]: [16]: (4) (4)

In (4) Toutis the motor output torque, Ktis the torque In (4) is the motor output torque, is the torque constant of the motor, uis the current demanded by the constant of the motor, is the current demanded by the controller, Kfis the friction constant and {jJ is the motor controller, is the friction constant and is the motor angular velocity. The sensors in the system were assumed to angular velocity.The sensors in the system were assumed be perfect and the effect of noise and delays in sensors wasto be perfect and the effect of noise and delays in sensors was ignored for simplicity.

ignored for simplicity.

[J (x -

c(l+r)+lpr ab-c 2

o

[T]

c+b - ab-c 2

(5) (5)

= Mi 2 + I p ; b = M or 2 + I; M o = M + m ; ; vref = desired speed c = Mlr : ;

In (5), a

In (5),

A block diagram of the control system implemented using A block diagram of the control system implemented using control law u = -KX has been shown in Figure 2. The control has been Figure 2. The controllerlaw gains have been tunedshown using inlinear quadratic controller gains have been tuned using linear quadratic regulator (LQR). The state and input weighting matrices for regulator (LQR). The state and input weighting matrices for LQR were selected such that they fulfil the control objective. LQRgains were selected such that they fulfil the objective. The were tuned implementing thecontrol controller on The gains were tuned implementing the controller on nonlinear dynamics of TWMR.

nonlinear dynamics of TWMR.

1380 1380

Ref. velocity

Referencing Tracking Controller

++

T

indirect method. method. According According to to Lyapunov Lyapunovindirect indirect method methodthe the indirect set of of equations equations (7) (7) must must be be satisfied satisfied for for aa system system to to be be set stable. stable.

Two Wheeled Robot Model

= xtvx » 0 = xTpx + xrt»: = -xrox < 0 -Q = PI + ITp QT = Q

VeX) VeX)

States FSFB Controller

Measured velocity

Sensor

Velocity Sensor

Fig. 2.2.Block Blockdiagram diagramrepresentation representation of ofcontrol controlsystem. system. Fig.

The resulting resulting optimized optimized gains gains showed showed minimum minimum steady steady state state The error and overshoot of pitch. The desired speed for controller error and overshoot of pitch. The desired speed for controller tuning was was assumed assumedzero. zero. The The state state matrix matrixof ofthe the closed closedloop loop tuning system with the designed controller was computed as system with the designed controller was computed as I = A - B * K.. The The closed closed loop loop system system matrix, matrix, JJ was was ofthe the transfer transfer function function of ofsystem system Hurwitz and and all all the the poles poles of Hurwitz have negative real parts. Hence, a necessary and sufficient have negative real parts. Hence, a necessary and sufficient condition for for aa closed closed loop loop system system to to be be stable stable has has been been condition verified. verified. IV. SSTABILITY REGION IV. TABILITY REGION The above above analysis analysis of of linearized linearized closed closed loop loop system system The suggested that the system was stable for infmite region of suggested that the system was stable for infinite region of attraction. But the response of nonlinear system at different attraction. But the response of nonlinear system at different states initial initial conditions conditions suggested suggested aa limitation limitation on on stability stability states region. This disparity was the motivation to study the region. This disparity was the motivation to study the stability region of closed loop nonlinear system to assess an stability region of closed loop nonlinear system to assess an operating envelope of statically unstable two wheeled mobile operating envelope of statically unstable two wheeled mobile robots for for the the proposed proposed controller. controller. Stability Stability region region tells tells how how robots far away from the equilibrium point a controller would be far away from the equilibrium point a controller would be able to stabilize the system at different initial conditions of able to stabilize the system at different initial conditions of the system states. the system states. The Lyapunov Lyapunov direct direct method method has has been been used used for for the the The evaluation of stability region in this study. According to the evaluation of stability region in this study. According to the Lyapunov direct direct method, method, for for aa known known Lyapunov Lyapunov function function Lyapunov that isis scalar, scalar, positive positive definite definite and and differentiable differentiable the the that nonlinear system system would would have have aa globally globally asymptotically asymptotically nonlinear stable equilibrium, equilibrium, ififfollowing following exists: exists: stable (i) Lyapunov Lyapunovfunction function VeX) > 0 (i) ofLyapunov function VeX) < 0 (ii) Derivative Derivative of (ii) Lyapunov function To devise devise aa Lyapunov Lyapunov function function for for TWMRs TWMRs two two To Lyapunov function function candidates candidates have have been been considered considered in in this this Lyapunov study to to select select one one “good” "good" Lyapunov Lyapunov function. function. A A “good” "good" study Lyapunov function function would would be be one one that that gives gives less less conservative conservative Lyapunov stabilityregion. region. stability

A. Lyapunov Lyapunov Function Function Candidates Candidates A. 1) Quadratic Quadratic function function 1) The quadratic quadratic Lyapunov Lyapunovfunction function isisexpressed expressedas as (6) (6) The VeX) = xrex (6) (6) In (6) X is state vector and P is symmetrically scalar matrix. In (6) X is state vector and P is symmetrically scalar matrix. The formulation formulation of of quadratic quadratic function function uses uses the the Lyapunov Lyapunov The

(7) (7)

Apositive definite definite matrix matrix QQ has has been been selected selected as as an an Apositive identity matrix matrix and and then then solved solved the the Lyapunov Lyapunov equation equation identity PI + ITp = -I,, for for P. The quadratic quadratic Lyapunov Lyapunov function function . The established using using PP would would be be aa positive positive definite definite scalar scalar established differentiable function function as as the the evaluated evaluated Pwas positive differentiable was aa positive definite matrix. matrix. Using Using the the quadratic quadratic formula formula given given in in (6) (6) the the definite Lyapunovfunction function generated generatedwas was (8) (8) Lyapunov ~(q,q)

= 19.85xf + 0.70x~ +

0.976x~

+

7.735x~

+

0.686xlX3 + 0.534xlX4 + 0.534x 3x2 - 1.39 X4X2 - 0.5x 3x4 ) (8) (8) X = [Xl' X 2' X 3 ' x 4]Tiissthe the state state vector vector of ofthe the system. system. The The displacement error, error, x 2 isis linear linear speed speed of ofrobot, robot, X 3 state Xl isis displacement state pitch, and and X 4 isis pitch pitch rate. rate. The The derivatives derivatives of of Lyapunov Lyapunov isis pitch, function for for linearized linearized closed closed loop loop dynamics dynamics of ofTWMR TWMR isis as as function given in in (9). (9). given ~ (q, q) = 39.7 Xl Xl + 1.4 x 2 x2 + 1.95 x 3 x3 + 15.47 X4 X4 - Xl X2 - X2 Xl - 1.37 Xl X3 - 1.37 X3 Xl + 1.07 Xl X4 + 1.07 X 4 Xl - 1.07 x 3 x2 - 1.07 x 2 x3 2.78 x 4 x2 - 2.78 x 2 x4 - x 3 x4 - x 4 x3 (9) (9) The Lyapunov function derivative for the nonlinear The Lyapunov function derivative for the nonlinear system was was determined determined by by replacing replacing x2 and and x4 in in (8) (8) with with x system and iJ,, the the nonlinear nonlinear expressions expressions of of which which have have been been and and at at zero zero reference reference produced from from (3). (3). Moreover, Moreover, x3 = X 4 and produced speed Xl = x 2. • The The resulting resulting derivative derivative of of Lyapunov Lyapunov speed function isis(10) (10) function ~(q, q) = - (Xl - 1.4x2 + 1.07x3 + 2.78x4 )x -

2 (-0.5X 1X2

-

(2.78x 2 + x 3 - 15.47x4 - 1.07xl )iJ - X2 2 - X4 2 + 39.7 X1X2 + 1.95 x 3 x 4 - 1.37 Xl X4 - 1.37 x 2 x 3 (10) (10) ofcontrol control input input and and All the the states states in in (8) (8) & & (10) (10) are are function function of All . xand and iJare function of control input and states X & X are function of control input and states 3& 4.

2) Energy Energy function function 2) The Energy Lyapunov function function was was developed developed using using The Energy Lyapunov energy method. energy method.

=GX22(m + :2)) + Gxl(Ml2 + I

Energyof ofTWMR Energy TWMR =

p)

+

12Mx22+Mlcosx3x2x4 ͳʹ ʹʹ൅ …‘• ͵ ʹ Ͷ The state state vector vector X = [x 2 , X 3 ' X 4 ] T represents represents linear linear The of robot. The values of speed, pitch angle, and pitch rate speed, pitch angle, and pitch rate of robot. The values of m,M,l,r,Iandlphave been given in table 1. The energy have been given in table 1. The energy function used used as as the the Lyapunov Lyapunov function function candidate candidate has has been been function therefore expressed expressedas as shown shown in in (11). (11). therefore ~(q,q) = 26.5893x 2 2 + 0.2924x4 2 + 1.032 X2X4 cos(x 3 ) (11) (11) ofEnergy EnergyLyapunov LyapunovFunction Functionisis (12) (12) The derivative derivative of The

1381 1381

~(q, q)

= 53.1786x2X2 + 0.5848 X4 X4 + 1.032 [X 2 X4 cos(x3 ) +

Table Table1: 1:Physical Physicalparameters parametersof ofaaTWMR TWMR

x4x2cosx3 - x2x4x3sinx3 (12)  Ͷ ʹ…‘• ͵െ ʹ Ͷ ͵•‹ ͵ (12) In (12) x = x x and x The expressions , = , and 4=iJ. = . The expressions for for x In (12) 3 4 , 2=x, and represent nonlinear nonlinear system system (3). (3). Hence, Hence, the the derivative derivative and iJ represent of ofthe theLyapunov Lyapunov function function appeared appearedas as (13). (13). ~ (q, q) = [53.18x 2 + 1.03x4 cos(x 3 )]x + [O.58x 4 + 1.03x2cosx3&(13) ͳǤͲ͵ ʹ…‘• ͵ െx2x42sinx3 ʹ Ͷʹ•‹ ͵ (13) All the states in (11) and (13) have been taken as All the states in (11) and (13) have been taken as aa function function of and iJ have have been been function function of of of control control input input whereas whereas x and control control input input and and states states X 3 and andx 4 ..

B. B. Stability StabilityRegion Region As and ~(q,q) were were scalar, scalar, positive positive definite definite and and As ~(q,q)and differentiable functions, the stability has been concluded differentiable functions, the stability has been concluded without withoutknowing knowingthe the solutions solutions of ofthe the equations equationsgoverning governingthe the system but looking at the values of Lyapunov functions system but looking at the values of Lyapunov functions and and their their derivative derivative atat different different initial initial conditions. conditions. ItIt isis worth worth mentioning that~ (q, q) or ~ (q, q) represents mentioning that or represents energy energy of of the the TWMR TWMR and and itit corresponds corresponds to to the the wheel wheel and and IB IB rotation rotation that that need need to to be be stabilized. stabilized. This This shows shows that that the the successful successful stabilization stabilization of ofthe theset set S = {(q, q): V(q, q) = O} leads leads to to the the solution solution of ofthe the problem. problem. To To assure assure the the stabilization stabilization of of S, , the the function function ~ (q, q) or or ~ (q, q) decrease decrease along along the the closed closed loop system solution while satisfying the controller loop system solution while satisfying the controller performance. or ~ (q, q) has has to to be be performance. In In this this case case ~ (q, q) or negative negativealong alongthe theclosed closed loop loop system system solution. solution. Let be any any point point in in the the state state space. space. Consider Consider the the Let [qo,qo] be solution solution [q(t), q(t)] of of the the closed closed loop loop system system with with (q(O),q(O)) = (qo,qo). . Then Then the the derivative derivative of of VV along along the the solution took the the form form V(q(t), q(t)) that that showed showed that that solution q(t) took the the solution solution q(t) had had aa non-empty non-empty limit limit set set and and that that V(q(t), q(t)) tend tend toto some some constant constant value value as as t ~ 00. . This This was was because because V isisnon-increasing non-increasing along along the the solution solution q(t). . As As the the function function (q(t), q(t)) was was differentiable, differentiable, the the value value of of derivative of the function tend to zero as t ~ 00. This derivative of the function tend to zero as . This implied impliedthat that state state derivatives derivatives approach approachto tozero zeroas ast ~ 00. . This Thiswas was concluded concludedhere herethat that for for aa TWMR TWMRwith with aa FSFB FSFB& & speed tracking controller, the limit set for any solution speed tracking controller, the limit set for any solution of ofthe the closed closed loop loop system system consist consist of ofthe the set set (14) (14)

_ {Xo: [v(q(t), q(t)) > 0 at t = 0 &V(q(t), q(t)) = 0 at t = 00];}

S-

[V(q(t), q(t))

< 0 at t = 0 &V(q(t), q(t)) = 0 at t = 00] (14) (14)

In In (14) (14) Xo represents represents the the initial initial values values of of all all the the states. states. This This stability region is not unique with any controller stability region is not unique with any controller and and parameters. parameters. But But for for one onecontroller controller and and one one fixed fixed set set of ofvalues values of of parameters, parameters, the the stability stability regions regions are are comparable. comparable. AA settling time with the designed controller was settling time with the designed controller was expected expected to to be be 44 seconds. We therefore required comparing the energy seconds. We therefore required comparing the energy required required to to accomplish accomplish the the same same work work and and fmd find the the boundaries of initial conditions of states, which accomplish boundaries of initial conditions of states, which accomplish the the conditions conditions of of stability stability within within 44 seconds. seconds. This This has has been been presented in next sections. presented in next sections.

v. V. NUMERICAL NUMERICALSIMULATIONS SIMULATIONSAND ANDEXPERIMENTS EXPERIMENTS The The functionst/, functions (q, q)and and ~ (q, q) have havebeen been used used as asthe the

Name Name Intermediate IntermediateBody: Body: Mass MassofIB of IB Mass Mass moment moment of of inertia inertia of of IB IB Distance of center center of of mass mass Distance of ofIB of IBfrom fromwheel wheelaxle axle Wheels: Wheels: Mass Massof ofeach eachwheel wheel Mass Mass moment moment of of inertia inertia of of each eachwheel wheel f each Radius Radius0of eachwheel wheel Motors Motorsfor forwheels: wheels: Torque Torqueconstant constantofmotor of motor Friction Frictionconstant constantofmotor of motor Current Currentdemand demandlimits limits

Symbol Symbol

Value Value

t,Ip

MM

43.0 43.0 0.56 0.56

Kg Kg Kg.m Kg.m22

Il

0.024 0.024

M M

MM II

2.77 2.77 0.0891 0.0891

Kg Kg Kg.m Kg.m22

rr

0.196 0.196

M M

x,Kt

0.1911 0.1911 0.0292 0.0292 ±5 ±5

Nm/A Nm/A Nm/rad.s Nm/rad.s AA

KKf f --

Unit Unit

candidate candidate Lyapunov Lyapunov functions functions to to find find stability stability of of aa TWMR. TWMR. The The resulting resulting stability stability regions regions have have been been compared compared toto those those obtained obtained from from solutions solutions of of the the equations equations governing governing the the systems. systems. In In order order to to determine determine the the stability stability region region and and compare compare the the results results for for different different Lyapunov Lyapunov functions functions simulations simulationswere wererun runusing usingMATLAB MATLAB and andSimulink. Simulink. Since Sincethe the problem problem isis four four dimensional dimensional and and itit was was difficult difficult toto plot plot stability stability region region for for four four states states on on aa single single plot. plot. The The initial initial conditions for two of the states, therefore, were considered conditions for two of the states, therefore, were considered atat aa time time assuming assuming remaining remaining two two states states to to have have zero zero initial initial condition. The simulations for each of the given set condition. The simulations for each of the given set of of initial initial conditions conditions were were run run for for 55 seconds. seconds. For For simulation simulation of of the the derived non-linear equations of motion the MATLAB ode45 derived non-linear equations of motion the MATLAB ode45 solver solver has has been been used. used. This This solver solver integrates integrates ODE's ODE’s with with specified initial conditions and variable step size. specified initial conditions and variable step size. As As the the equations equations represent represent the the dynamics dynamics of of aa statically statically unstable unstable TWMR TWMR system, system, so sothe the stability stabilityregion region obtained obtained from from solving solving these equations has been considered the true these equations has been considered the true region region of of attraction. attraction. The The stability stability criterion criterion was was that that the the IB IB of of TWMR TWMR isis stabilized stabilizedwithin within 44 seconds secondswith with zero zerosteady steadystate stateerror error and and with minimum possible overshoot. The proposition with minimum possible overshoot. The proposition of of Lyapunov Lyapunov direct direct method method was was considered considered to to determine determine the the stability stability region region using using Lyapunov Lyapunov function. function. The The simulation simulation results of stability regions projected to xl '"'-'x2 results of stability regions projected to x1~x2 phase phase plane plane =x2=O, x2'"'-'x3 with with x3=x4=O x3=x4=0 ,, x3'"'-'x4 x3~x4 phase phase plane plane with with xl x1=x2=0, x2~x3 phase =x4=O and phase plane plane with with xl x1=x4=0 and x2'"'-'x4 x2~x4 phase phase plane plane with with xl =x3=Ohave been plotted as shown in Figure 4. x1=x3=0 have been plotted as shown in Figure 4. In In order order to to verify verify the the results results based based on on numerical numerical simulations, simulations, the the stability stability regions regions were were determined determined for for aa real real system system experimentally. experimentally. We We chose chose aa TWRP TWRP (Two (Two wheeled wheeled robotic robotic platform) platform) developed developed for for research research in in the the Dynamics Dynamics & & Control Control Research Research Lab. Lab. of of the the Mechanical Mechanical Engineering Engineering department department atat UOA UOA [16]. [16]. The The two two independently independently driven driven wheels wheels of of the the system system are are coupled coupled directly directlytoto the the output output shaft shaft of of the the gearbox gearbox of of each each motor. motor. The The motors, motors, motor motor drivers, drivers, circuit circuit boards boards and and sensors sensors are are placed placed on on the the intermediate intermediate body. body. By By design, design, the the centre centre of of massof massof the the intermediate intermediate body body isis above above the the line line joining joining two two wheel wheel centres, centres, hence hence called called statically staticallyunstable. unstable. The The two two wheels wheels are are actuated actuated toto allow allowthe the robot robot to to move move and and balance balance the the intermediate intermediate body body applying applying aa

1382 1382

VI. RESULTS AND DISCUSSION The plots in Figure 3 represent the variation of Lyapunov functions and derivative of Lyapunov functions of TWMR at certain specific set of initial conditions consisted of Pitch rate ( )=0; Linear displacement ( )=0; Linear velocity ( )=0 and Pitch ( )=30 degrees. Energy Lyapunov function resulted with smaller initial value as compared to the initial value of quadratic Lyapunov function. Lyapunov function represents physically the energy thus this result means that the same task has been realized with smaller energy consumption for Energy Lyapunov function as compare to very large initial energy consumption for Quadratic Lyapunov function. The function also tends to zero faster and with slightly smaller oscillations for Energy Lyapunov Functions in comparison with Quadratic. Figure 4(a) shows the estimates of stability region for pitch and speed using the Lyapunov functions and ode solver. Figure 4(b) shows three estimates of stability regions for pitch and pitch rate. Figure 4(c) shows three estimates of stability regions for pitch rate and speed.

70

0 Energy based

50 40

Derivative of energy

60 Total energy (J)

torque supplied by both the motors of driving wheels. The system has same parameters as used for simulation study. The controller was implemented in real time using Simulink and xPC target toolboxes of MATLAB. The xPC target has hardware in-the-loop property providing advantages such as ease of use, reliability, and flexibility. The sensors used on the platform include IMU and laser scanner. IMU measures the pitch and pitch rate but laser scanner measures the displacement. This displacement is filtered to minimize the noise and differentiated to find the speed of the TWRP. The experiments were run as hardwarein-loop architecture that imposed some limitations to the parameter variation. The position of platform has been limited to 1.5m due to the length of the cables used to supply power to the platform and communicate data between host and target PCs. Moreover, to provide safety from fall of IB the safety bumpers have been added to the platform. The bumpers limit the pitch of the platform to 30 degrees. The experiments were run by operating the TWRP at different sets of initial conditions of the states. The first test was run with zero initial condition of states i.e. the platform was standing upright at origin. Then the platform operated at different initial pitch angles, 0 to 30o, and observed whether the platform comes back to its initial position and stands upright, the stable state. The second test was started same as first but when the platform started moving at certain speed a step disturbance in pitch was given ensuring the platform was at origin with IB not rotating. At different speeds of platform disturbance up to 30o was imparted to IB to test the stability of platform. The tests were repeated for variable initial pitch rates too. Next set of experiments was completed with platform moving at different speeds initially and having IB rotating at different rates for each speed. The regions of stability have been plotted along the simulated stability regions, shown in Figure 4 as shaded areas.

Quadratic LF

30 20

0.2

-200 Quadratic

-300

Energy 10 LF

0

-100

0.4 0.6 Time (s)

0.8

-400

0.2 0.4 0.6 0.8 Time (s)

Fig. 3. Comparison of Lyapunov functions and derivatives of Lyapunov functions in the context of energy and its derivative.

Figure 4(d) shows three estimates of stability regions for speed and displacement. The boundaries of the regions, in all the cases, depict that the largest stability region is the one obtained solving the equations of motion formulated using dynamic model, the true region. The second largest region shows the boundary of stability region resulted from the Energy Lyapunov function. The smallest stability region is obtained from the Quadratic Lyapunov function. It turns out that the energy Lyapunov function yields better results. The Energy Lyapunov function resulted to a ‘good’ Lyapunov function for TWMRs that leads to a less conservative and closer to the true stability region. The experimental results show that the region of stability is wider than the region of stability obtained through the use of Quadratic Lyapunov function but is within the boundary of stability region determined using energy Lyapunov function. Although the experimental results do not give full region of stability due to practical constraints on robot parameter variations and ignorance of noise and delay in sensors but they support the conclusion obtained from simulation results. The improvements in hardware, to get full stability region by running the platform standalone, will be considered in future work. VII. CONCLUSIONS In real applications it is important to know the safe region of operation of a robot. In relation to this the determination of stability region of a TWMR was addressed in the present study. Direct Lyapunov method was used to determine this region. To implement Lyapunov direct method a Lyapunov function is required. Energy function and Quadratic function were formulated as candidate Lyapunov functions for use in the method of finding stability region.The selection of the Lyapunov function for TWMRs based on stability region is devised that shows that the Quadratic Lyapunov function realized the same task with more energy consumption and has conservative stability region as compare to Energy based Lyapunov function. It has been concluded, therefore, that

1383

an Energy Lyapunov function may be a “good” Lyapunov Function for statically unstable two wheeled mobile robots. The stability region obtained using the proposed Lyapunov function might be useful in comparing the performance of different controllers proposed for therobot and to visualize its safe region of operation. The proposed Lyapunov function may also provide the infrastructure for design of Lyapunov-based nonlinear robust controllers. The future work would extend the application of recommended method for estimation of stability region to the application of TWMR on inclined and rough terrain.

100 Ode solv er 50 Initial pitch, deg

Quadratic 0 Experimental Energy

-50

-100 -3

-2

-1

0 1 Initial speed, m/s

2

3

REFERENCES

(a)

[1]

100 Energy

[2]

Initial pitch, deg

50

Quadratic

0

[3]

Experimental

[4]

-50 Ode solv er -100 -150

-100

-50 0 50 Initial Pitch rate, deg/s

100

[5]

150

(b)

[6]

100 Experimental

Initial pitch rate, deg/s

Ode solv er 50

[7]

0 Quadratic

[8] [9]

-50 Energy

-100 -3

[10] -2

-1

0 1 Initial speed, m/s

2

3

[11]

(c) 6

Initial speed, m/s

[12]

Ode solv er

4

[13]

2 Quadratic 0

[14]

Experimental

[15]

-2 Energy

[16]

-4 -6 -3

-2

-1

0 1 Initial position, m

2

3

(d) Fig. 4. Stability regions of a TWMR estimated using ode solver, quadratic and Energy based Lyapunov functions and measured experimentally. 1384

A. Salerno and J. Angeles, "A new family of two-wheeled mobile robots: Modeling and controllability," IEEE Transactions on Robotics, vol. 23, pp. 169-173, 2007. S. W. Nawawi,M. N. Ahmad andJ. H. S. Osman., "A PI sliding mode tracking controller with application to a 3 DOF direct-drive robot manipulator," in TENCON 2004. 2004 IEEE Region 10 Conference, 2004, pp. 455-458 Vol. 4. F. Grasser, A. D'Arrigo, S. Colombi and A. C. Rufer, "JOE: A mobile, inverted pendulum," IEEE Transactions on Industrial Electronics, vol. 49, pp. 107-114, 2002. S. W. Nawawi, M. N. Ahmad andJ. H. S. Osman., "Real-time control system for a Two-Wheeled Inverted Pendulum Mobile Robot," inAdvanced Knowledge Application in Practice, Igor Fuerstner (Ed.) , INTECH, Nov 2010, pp. 299-312. Y. Kim, S.H. Kim and Y.K. Kwak, "Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot," Journal of Intelligent and Robotic Systems: Theory and Applications, vol. 44, pp. 25-46, 2005. R. Xiaogang and C. Jianxian, "Fuzzy Backstepping Controllers for Two-Wheeled Self-Balancing Robot," in Informatics in Control, Automation and Robotics, 2009. CAR '09. International Asia Conference on, 2009, pp. 166-169. Z. Kausar, K. Stol, and N. Patel, "Performance enhancement of a statically unstable two wheeled mobile robot traversing on an uneven surface," in 2010 IEEE Conference on Robotics, Automation and Mechatronics, RAM 2010, 2010, pp. 156-162. H. K. Khalil, Nonlinear Systems 3ed.: Prentice Hall 2001. H. D. Chiang and J. S. Thorp, "Stability regions of nonlinear dynamical systems: a constructive methodology," Automatic Control, IEEE Transactions on, vol. 34, pp. 1229-1241, 1989. G. Chesi, "Estimating the domain of attraction for uncertain polynomial systems," Automatica, vol. 40, pp. 1981-1986, 2004. G. Chesi.,A. Garulli, A. Tesi and A. Vicino, "LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems," International Journal of Robust and Nonlinear Control, vol. 15, pp. 35-49, 2005. J. J. E. Slotine and W. Li, Applied Nonlinear Control System. Englewood Cliffs: Prentice Hall Inc., 1991. G. Golub, S. Nash, and C. Van Loan,"A Hessenberg-Schur method for the problem AX + XB= C," Automatic Control, IEEE Transactions on, vol. 24, pp. 909-913, 1979. Z. Tian and C. Gu, "A numerical algorithm for Lyapunov equations," Applied Mathematics and Computation, vol. 202, pp. 44-53, 2008. M. A. Hassan and C. Storey, "Numerical determination of domains of attraction for electrical power systems using the method of Zubov," International Journal of Control, vol. 34, pp. 371-381, 1981. D. R. Jones and K. A. Stol, "Modelling and Stability Control of TwoWheeled Robots in Low-Traction Environments," presented at the Australasian Conference on Robotics and Automation, Brisbane, Australia, 2010.