DYNAMICS AND CONTROL OF A SUCTION-TYPE WALL-CLIMBING ROBOT

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DYNAMICS AND CONTROL OF A SUCTION-TYPE WALL-CLIMBING ROBOT DYNAMICS AND CONTROL OF A SUCTION-TYPE WALL-CLIMBING ROBOT DYNAMICS OF WALL-CLIMBING DYNAMICS AND AND CONTROL CONTROL OF A A SUCTION-TYPE SUCTION-TYPE WALL-CLIMBING ROBOT ROBOT Yunafi’atul Aniroh*, Andika P. Yudha*, Hung-Chyun Yunafi’atul Aniroh*, Andika P. Yudha*, Hung-Chyun DYNAMICS AND CONTROL OF A SUCTION-TYPE WALL-CLIMBING ROBOT Yunafi’atul Aniroh*, Andika P. Yudha*, Hung-Chyun

Chou* ,Chung-Hsien Kuo*, Felix L. Chernousko**, V.G. Yunafi’atul Aniroh*, Andika P. Yudha*, Hung-Chyun Chou* ,Chung-Hsien Kuo*, Felix Chernousko**, V.G. Yunafi’atul Aniroh*, Andika P. L. Yudha*, Hung-Chyun Chou* ,Chung-Hsien Kuo*, Felix L. Chernousko**, V.G. Gradetsky** and Nikolay Bolotnik** Chou* ,Chung-Hsien Kuo*, Felix L. Chernousko**, V.G. Yunafi’atul Aniroh*, Andika P. Yudha*, Hung-Chyun Gradetsky** and Nikolay Bolotnik** Chou* ,Chung-Hsien Kuo*, Felix L. Chernousko**, V.G. Gradetsky** and Nikolay Bolotnik** Gradetsky**Kuo*, and Nikolay Nikolay Bolotnik** Chou* ,Chung-Hsien Felix L.Bolotnik** Chernousko**, V.G. Gradetsky** and *Department of Engineering, National *Department of Electrical Electrical Engineering, National Taiwan Taiwan Gradetsky** and Nikolay Bolotnik** *Department ofand Electrical Engineering, NationalROC Taiwan University of Science Technology, Taipei, Taiwan, (e-mail: *Department of Electrical Engineering, NationalROC Taiwan University of Science and Technology, Taipei, Taiwan, (e-mail: *Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC (e-mail: {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) University of Science Scienceofand and Technology, Taipei, Taiwan, ROC (e-mail: *Department Electrical Engineering, NationalROC Taiwan {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) University of Technology, Taipei, Taiwan, (e-mail: {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) **The Institute for Problems in Mechanics RAS, Moscow, Russia {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) University of Science and Technology, Taipei, Taiwan, ROC (e-mail: **The Institute for Problems in Mechanics RAS, Moscow, Russia {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) **The Institute for{chern, Problems in Mechanics RAS, Moscow, Russia (e-mail: gradet, bolotnik}@ipmnet.ru) **The Institute for Problems in Mechanics RAS, Moscow, Russia {m10007808,m10207813, d9907308, chkuo}@mail.ntust.edu.tw) (e-mail: {chern, gradet, bolotnik}@ipmnet.ru) **The Institute Problems in Mechanics RAS, Moscow, Russia (e-mail:for{chern, gradet, bolotnik}@ipmnet.ru) (e-mail:for{chern, {chern, gradet, bolotnik}@ipmnet.ru) **The Institute Problems in Mechanics RAS, Moscow, Russia (e-mail: gradet, bolotnik}@ipmnet.ru) (e-mail: {chern, gradet, for bolotnik}@ipmnet.ru) Abstract: This paper presents the trajectory tracking approach a wall-climbing robot by using adaptive control schemes. Abstract: Abstract: This paper presents presents the the trajectory trajectory tracking tracking approach approach for for aa wall-climbing wall-climbing robot robot by by using using adaptive adaptive control control schemes. This paper schemes.

The most important important consideration for controlling the wall-climbing wall-climbing robot is to to makerobot sure by thatusing the wheels wheels cancontrol be always always well Abstract: This presents trajectory tracking approach wall-climbing adaptive schemes. The most controlling the is sure that the can be well Abstract: This paper paperconsideration presents the the for trajectory tracking approach for for aarobot wall-climbing adaptive schemes. The most important consideration for controlling the wall-climbing robot is to make makerobot sure by thatusing the wheels cancontrol be always well Abstract: This presents theof trajectory tracking approach for asacrificing wall-climbing robot by using adaptive control schemes. contacted to thepaper wall regardless of the slope conditions without sacrificing robot’s mobility. To consider different slope The most important consideration for controlling the wall-climbing robot is to make sure that the wheels can be always well contacted to the wall regardless the slope conditions without robot’s mobility. To consider different slope The most important consideration for controlling the wall-climbing robot is to make sure that the wheels can be always well contacted to the wall regardless of the slope conditions without sacrificing robot’s mobility. To consider different slope The most important consideration forthe controlling thecontrol wall-climbing robot is to make sure that so theTo wheels can be always well conditions of the wall, this paper proposes proposes adaptive control schemessacrificing to alter alter therobot’s vacuum force so that different gravity effects contacted to wall regardless of slope conditions without mobility. consider different slope conditions of wall, this paper adaptive schemes to the vacuum force that different gravity effects contacted to the wall regardless of the slope conditions without sacrificing robot’s mobility. To consider different slope conditions of the the wall, this paper proposes adaptive control without schemessacrificing to alter therobot’s vacuummobility. force so To thatconsider different different gravity effects contacted to wall with. regardless of thepressure slope conditions slope can be be properly properly dealt with. Practically, pressure andcontrol IMU sensors sensors are used to provide provide the vacuum force and spatial spatial posture conditions of wall, this paper proposes adaptive schemes to alter the vacuum force so that different gravity effects can dealt Practically, and IMU are used to the vacuum force and posture conditions of the wall, this paper proposes adaptive control schemes to alter the vacuum force so that different gravity effects can be properly dealt Practically, pressure andcontrol IMU sensors are to provide the vacuum force and spatial posture conditions offor therealizing wall, with. thisadaptive paper proposes adaptive schemes to used alter the vacuum force so that different gravity effects can be dealt with. Practically, pressure and IMU are used to vacuum force and posture Finally, MATLAB simulations andthe real tests for dealing with different different information for realizing adaptive control schemes can be properly properly dealt with. Practically, pressure and IMU sensors sensors are used to provide provide the vacuum force and spatial spatial posture .. Finally, MATLAB simulations and real tests for dealing with information control schemes Finally, MATLAB simulations and real tests for dealing with different information for realizing adaptive control schemes can be properly dealt with. Practically, pressure and IMU sensors are used to provide the vacuum force and spatial posture surface slope slopefor conditions were performed with the triangle triangle trajectories. .. Finally, MATLAB information realizing adaptive control schemes surface conditions performed with the trajectories. Finally, MATLAB simulations simulations and and real real tests tests for for dealing dealing with with different different information realizingwere adaptive control schemes surface slopefor conditions were performed with the triangle trajectories. . Finally, MATLAB simulations and real tests for dealing with different information for realizing adaptive control schemes Keywords: Wall climbing robot, adaptive control, trajectory tracking, inertial measurement unit surface slope conditions were performed with the triangle trajectories. Keywords: Wall climbing robot, adaptive control, trajectory tracking, inertial measurement unit surface slope conditions were performed with the triangle trajectories. © 2015, slope IFAC (International Federation of Automatic Control) Hosting Elseviermeasurement Ltd. All rightsunit reserved. Keywords: Wall climbingwere robot, adaptive control, trajectory tracking,byinertial surface conditions performed the triangle trajectories. Keywords: Wall climbing adaptive control, trajectory tracking, Keywords: Wall climbing robot, robot, adaptivewith control, trajectory tracking, inertial inertial measurement measurement unit unit Keywords: Wall climbing robot, adaptive control, trajectory tracking, inertial measurement unit

NTRODUCTION 11 IINTRODUCTION NTRODUCTION 11 IINTRODUCTION NTRODUCTION The wall climbing robots are are able to to perform INTRODUCTION The robots The wall wall 1climbing climbing robots are able able to perform perform dangerous operation, such asrobots the inspection The wall climbing are able of to high-rise perform

dangerous operation, such the The wall climbing are able of to high-rise perform dangerous operation, such as asrobots the inspection inspection of high-rise wall climbing robots are ableof togas perform building,The spray painting and sand blasting ofof gas tanks, dangerous operation, such as the inspection high-rise building, spray painting and sand blasting tanks, dangerous operation, such as the inspection of high-rise building, spray painting sand blasting ofofgas tanks, dangerous operation, suchand asfacilities, the inspection high-rise maintenance of nuclear surveillance and building, spray painting and sand blasting of gas tanks, maintenance of nuclear facilities, surveillance and building, spray painting and sand blasting of gas tanks, maintenance of nuclear facilities, surveillance and building, spray painting and sand blasting of gas tanks, reconnaissance,…etc. The robotics team of the City maintenance of of nuclear nuclear facilities, team surveillance and reconnaissance,…etc. The robotics of the City maintenance facilities, surveillance and reconnaissance,…etc. The robotics team of the City maintenance of York nuclear facilities, surveillance and College of New (CCNY) has developed the wall reconnaissance,…etc. The robotics team of the City College of New York (CCNY) has developed the wall reconnaissance,…etc. The robotics team of the City College of New YorkThe (CCNY) has developed the wall reconnaissance,…etc. robotics team of the City climbing robot with the function of transit from two College of New York (CCNY) has developed the wall climbing function transit two College ofrobot New with Yorkthe (CCNY) hasof the wall climbing robot the function ofdeveloped transit from from two College of New with York (CCNY) has developed the wall different surfaces, such as grounds, walls, and ceilings. climbing robot with the function of transit from two different surfaces, such as grounds, walls, and ceilings. climbing robot with the function of transit from two different surfaces, such as grounds, walls, and ceilings. climbing robot with the function of transit from two This paper proposes an adaptive control schemes to different surfaces, such as grounds, walls, and ceilings. This paper proposes an adaptive control schemes to different surfaces, such as an grounds, walls, and ceilings. This paper proposes adaptive control schemes to different surfaces, such as grounds, walls, andeffects ceilings. alterThis the vacuum vacuum force, and different gravity effects can be be This paper proposes an adaptive control schemes to alter the force, and different gravity can paper proposes an adaptive control schemes to alterThis the vacuum force, and different gravity effects can be paper proposes andifferent adaptive controleffects schemes to properly dealt with. with. Inand order to implement implement the adaptive alter the vacuum vacuum force, and different gravity effects can be be properly dealt In order to the adaptive alter the force, gravity can properly dealt with. Inand order to implement the adaptive alter the vacuum force, different gravity canslip, be controller, the dynamic dynamic model of the robot as aseffects friction, slip, properly dealt with. In order to implement the adaptive controller, the model of the robot friction, properly dealt with. In order to implement the adaptive controller, the dynamic model robot as friction, slip, properly dealt with. Indetermined. order of to the implement the adaptive and weight needs to be controller, the dynamic model of the robot as as friction, friction, slip, and weight needs to be determined. controller, the dynamic model of the robot slip, and weight needs to be determined. controller, the dynamic model of the robot as friction, slip, adaptive control is not sufficiently mature to and weight needs to be determined. Traditional adaptive control is not sufficiently mature andTraditional weight needs to be determined. Traditional adaptive control is not sufficiently mature to to and weight needs to be determined. solve control problems with challenges which the strict Traditional adaptive control is not sufficiently mature to solve control problems with challenges which the strict Traditional adaptive control is not sufficiently mature to solve control problems with challenges which the strict Traditional adaptive control is not sufficiently mature to performance and the guarantees of robustness are required, solve control problems with challenges which the strict performance and the guarantees of robustness are required, solve control problems with challenges which the strict performance and the guarantees of robustness are the required, solve control problems with challenges which strict because set of unknown constants need to be estimated. performance and the guarantees of robustness are required, because set of unknown constants need to be estimated. performance andunknown the guarantees of robustness areestimated. required, because set of constants need to be performance and the guarantees of robustness areestimated. required, The parameter parameter estimation needs an approach that explicitly because set of of unknown constants need to to that be estimated. The estimation needs an approach explicitly because set unknown constants need be The parameter estimation needs an approach that explicitly because set of estimation unknown constants need to that be estimated. accounts for robust robust performance andapproach stability specifications. The parameter estimation needs an approach that explicitly accounts for performance and stability specifications. The parameter needs an explicitly accounts for robust performance andapproach stability that specifications. The estimation needsthis an explicitlya To parameter achieve thisperformance goal, this paper specifications. proposes accounts for and To achieve this goal, paper proposes accounts for robust robust performance and stability stability specifications. To achieve this goal, this paper proposes aa accounts for robust performance and stability specifications. robust adaptive control approach, sliding controller. To achieve achieve this goal, goal, thissliding papercontroller. proposes The a robust adaptive approach, The To this this paper proposes robust adaptive control control approach, sliding controller. Theaa To achieve this goal, this paper proposes sliding controller is expected to avoid the slips of wheels, robust adaptive control approach, sliding controller. The sliding controller is expected to avoid the slips of wheels, robust adaptive control approach, sliding controller. The sliding controller is expected to avoid thecontroller. slips of wheels, robust adaptive control approach, sliding The as well as to reduce the power of the wheel sliding controller is to avoid the of wheels, as well as to reduce the power consumptions of the wheel sliding controller is expected expected toconsumptions avoid the slips slips as well as to reduce the power consumptions of of thewheels, wheel sliding controller is expected to avoid the slips of wheels, motors and the suction motor. The sliding controller design as well as to reduce the power consumptions of the wheel motors suction sliding as well and as tothe reduce themotor. powerThe consumptions of the design wheel motors suction sliding controller controller as well and asaatothe reduce themotor. powerThe consumptions the design wheel provides systematic approach tosliding solve controller theofproblem problem of motors and the suction motor. The design provides systematic approach to solve the of motors and the suction motor. The sliding controller design provides a systematic approach to solve the problem of motors and suction and motor. Theto controller design maintaining stability and consistent performance while provides a the systematic approach tosliding solve the problem problem of maintaining stability consistent performance while provides a systematic approach solve the of maintaining stability and consistent performance while provides a systematic approach to solve the problem of facing to to the the modeling imprecision. maintaining stability imprecision. and consistent performance while facing modeling maintaining stability and consistent performance while facing to the modeling maintaining stability imprecision. and consistent performance while facing imprecision. facing to to the the modeling modeling imprecision. facing to the2modeling imprecision. YNAMIC ODELLING DYNAMIC MODELLING

2 DYNAMIC MODELLING 22 D ODELLING YNAMIC M ODELLING DYNAMIC Mhas The wall 2climbing climbing robotMhas parameter variations and D YNAMIC ODELLING The wall robot parameter The wall climbing robot has parameter variations variations and and uncertainties which are caused by friction, slip, and weight.

Fig. Fig. 1. 1. The The cases cases of of different different surfaces, surfaces, (a) (a) on on the the floor. floor. (b) (b) Fig. 1. The cases of different surfaces, (a) on the floor. (b) on the slope. (c) on the wall. (d). on the ceiling Fig. 1. The cases of different surfaces, (a) on the floor. (b) on the slope. (c) on the wall. (d). on the ceiling (a) on the floor. (b) Fig. 1. The cases of different surfaces, on slope. (c) onofthe wall. (d). on the ceiling Fig.the The cases different surfaces, (a) on the floor. (b) on the (c) wall. on on the1.slope. slope. (c) on on the the wall. (d). (d). on the the ceiling ceiling on theThe slope. (c) on theequations wall. (d).is on the ceiling differential expressed as The differential equations is expressed as The differential equations is expressed as The differential equations is expressed as & & & x x ⎡⎡ m ⎤ ⎡ ⎤ The differential equations is expressed as ⎤ ⎡⎡ xx&& ⎤⎤ m&&xx&&The ⎡⎢ m equations m&&&yxx&&&⎤⎤⎥⎥⎤⎥ = αdifferential ⎡⎢⎡⎢m ⎡⎢⎢⎡ xx&& ⎤⎥⎥⎤ is expressed as − θ β m P U t 2 ( ) ( ) = − α θ β y P U t 2 ⎢⎢⎡⎢m&yx&⎥⎥⎤⎥ = αP (θ )U (t ) − 2 β ⎢⎢⎢⎡⎢ xyyy&& ⎥⎥⎥⎤⎥ ⎢⎢⎢m m &y&⎥ = αP θ U t − 2 β ⎢d yy&&222θ&& ⎥⎥⎥ θ JJθθ&&y&&⎥⎦⎥⎥ = αP (θ )U (t ) − 2 β ⎢⎣⎢⎢d ⎣⎢⎢⎣m (1) (1) y&2θ&&& ⎦⎥⎥⎦⎦⎥ ⎣⎢⎢ JJθθ&&y&&⎥⎦⎦⎥⎥ = αP (θ )U (t ) − 2 β ⎢⎣⎣⎢⎢d (1) θ ⎦⎥⎦ d 222θ ⎣⎣ Jθ&&⎦⎦ ⎣⎣d (1) (1) &⎥ ⎢ ⎥ ⎢ θ J θ d ⎣ ⎦ ⎣ ⎦ (1) With With With With With ⎡⎡ ⎤⎤⎤ With ⎡⎢⎢⎡⎡− sin θ sin θ ⎥⎥⎤⎤⎥ θ sin ⎢⎢− sin θ sin θ θθ sin ⎥ 2 sin θ sin θ sin 2 ⎢⎡⎢⎢− G sin θ ⎥⎤⎥⎥ sin cos 2 θ G θθ −sin θθθ − cos θ − sin θθ φφφ cos sin − sin sin sin − ⎢ P θ θ cos cos ( ) = G θ sin cos ⎢ cos θ − Pθ = cos θ − cos θ − G sin φ cos 222 θ ⎥⎥⎥ P(θ ) = ⎢⎢⎢−cos cosθθ − cos θ − sin θ 22φ ffcos sinθθ −sin G sin θ ⎥⎥ 1 1 P θ θ θ cos cos ( ) = − − 2 2φ fcos P(θ ) = ⎢⎢ cos θ − cos θ − cos θ− cos θ − G sinM 1 ⎥⎥ f 2 M r ⎢ 2 fr111 θ ⎥⎥⎥ P(θ ) = ⎢⎢ cos θ − cos θ − cos− θ sgn − (θ )M 1 θ 1 1 sgn − r ⎥⎥ ⎢⎢ 1 2df 1 1 − sgn (θ ) 2 dffr1 2M M 1 − ⎥⎦⎥⎦⎦ ⎢⎣⎢⎣⎣ 1 θ )) 2Mdfrr111 1 1 sgn ((θ − sgn dfr11 2df ⎥⎦⎦⎥ ⎢⎣⎣⎢ 1 (2) 1 − sgn (θ ) 2 (2) 1 (2) 2df1 ⎦⎥ ⎣⎢ (2) (2) (2) ⎤⎤ ⎡⎡ ⎤ ( t ) ⎥ ⎢⎡⎡⎢⎡U U 11 ((tt )) ⎤⎥⎤ ⎢U U 11 ((tt )) ⎥⎥⎥⎥⎤ U ((tt )) = U = ⎢⎢⎢⎡⎢U U U (t ) = ⎢⎢⎢U 111222 ((tt ))⎥⎥⎥⎥ 222 ff((11tt ))⎥⎥ U ((tt )) = U2 = ⎢⎢U U ⎥ 2f 1 ⎥ U (t ) = ⎢⎢⎢⎢⎢⎣U2 2α 2f f(111t )⎥⎥⎥⎥⎦⎦ (3) ⎢⎣⎣⎢ 2 (3) (3) f ⎦⎥ ⎢⎣⎣⎢ 2α α (3) α 1 ⎥⎦⎦⎥ (3) ⎣⎢ α dd ⎦⎥is (3) Where is the the distance distance of of the the tracked tracked wheel wheel from from Where Where d is the distance of the tracked wheel from CM; G G is the thed weight of the the robot; robot; mtracked is the the mass mass offrom the Where d weight is the the distance distance of the them tracked wheelof from CM; is of is the Where is of wheel CM; G is thed weight of the robot; is the mass offrom the Where is the distance of them wheel rolling friction between tracked wheel and robot;G CM; G is the the weight weight of the the robot; robot; mtracked is the mass mass of and the fii is rolling friction between tracked wheel robot; CM; of m is the of the f is the rolling friction between tracked wheel and robot; f CM; Gffii is the weight of the moment robot; mproduced is the mass of and the rolling friction between tracked wheel robot; is the rolling friction between tracked wheel and robot; is the resistance by friction surface; M rr is the resistance moment produced by friction surface; M is the resistance moment produced by friction surface;fiii is M the rolling friction between tracked wheel and robot; r surface; is the the resistance resistance moment moment produced produced by by friction friction surface; M M rr is surface; M rr is the resistance moment produced by friction

The has variations and uncertainties which are arerobot caused byparameter friction, slip, slip, and weight. weight. The wall wall climbing climbing robot hasby parameter variations and uncertainties which caused friction, and Thesuitable wall climbing robot has parameter variations and The controller of the robot is determined according uncertainties which are caused by friction, slip, and weight. The suitable controller of the robot is determined according uncertainties which are caused by friction, slip, and weight. The suitable controller the robot is determined according uncertainties which are of caused byproject, friction, slip, and weight. to the dynamics model. In the the wall climbing The suitable controller of the robot is determined according to the dynamics model. In the project, the wall climbing The suitable controller of the robot is determined according to the dynamics model.of In the project, the wallaccording climbing The suitable controller theon robot issurface determined robot is capable to the different to the model. In the project, the wall robot isdynamics capable to move move on the surface with different to the dynamics model. In the project, the with wall climbing climbing robot is capable to move on the surface with different to the dynamics model. In the project, the wall climbing slopes. robot is capable to move on the surface with different slopes. robot is capable to move on the surface with different slopes. robot to move on the surface with different slopes. slopes.is ©capable 2405-8963 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. slopes. Copyright © 2015, IFAC 902 Peer review©under of International Federation of Automatic Copyright 2015,responsibility IFAC 902Control. Copyright © 2015, IFAC 902 10.1016/j.ifacol.2015.05.187 Copyright © 2015, 2015, IFAC IFAC 902 Copyright © 902 Copyright © 2015, IFAC 902

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Yunafi’atul Aniroh et al. / IFAC-PapersOnLine 48-1 (2015) 902–903

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4 EXPERIMENTS RESULT on surface. α (N/V) can be determined as α =

kt

,

JRa rg

Fig. 2 shows experimental and simulation result of climbing robot triangle trajectory. The parameters in the simulations are selected from datasheet as followings: α = 131.5363 N/v; β = 71.1087 Kg/s; Umax = 24.0 V; μ = 0.4; m = 1.7 Kg; G = 16.6 N; l = 58 mm; d = 11.5 cm; ρr = 333.8 N/m; J ≈ 1.08.10-6 Kgm2; Φ = 5000; λ = 5000; η = diag{0.1, 0.1, 0.1}, and suppose the actual values for uncertain parameters are f 21 ∈ [0,88] , fˆ21 = 44 Δfˆ = α = 44 ,

where Ra is the armature resistance of the motors; rg is the gear ratio; kt represents the torque coefficient; J denotes the moment of inertia; Ui is the voltage which applies to the ith motor, and vi is the velocity of the i-th wheel.

3 SLIDING CONTROLLER

21

This part describes the control of nonlinear systems of the form which is derived in chapter II where the models T are imprecise. By defining the vector Z&& = &x& &y& θ&& , the dynamic model (1) is rewritten as the following state space equations: MZ&& = bU − f1 − f 2 ⋅ g 1 − f 3 ⋅ g 2 (4)

]

Where : ⎡ − sin θ b = α ⎢ cos θ ⎢ d ⎣

sin θ ⎤ − cos θ ⎥ ; f1 = BZ& ; f3 = sinφ ; ⎥ d ⎦

⎡2β ⎢ B=⎢ 0 ⎢⎣ 0

0 ⎤ ⎡ sin θ ⎤ 0 ⎡ ⎤ ⎥ 0 ⎥ ; g 1 = ⎢ − cos θ ⎥ ; g = ⎢− G cos 2 θ ⎥ ; ⎢− sgn(θ& )⎥ 2 ⎢ ⎥ 2 β d 2 ⎥⎦ 0 ⎣ ⎦ ⎣ ⎦

2β 0

7

6

6 5 y (m)

4

2

1

1

20

25

-1 -5

0

5

15

20

25

0.2

0

y (m)

-0.2

-0.1

-0.4

-0.6

-0.2

-0.8

-0.3

-1

-0.4

-0.2

0

0.2

0.4

0.6

x (m)

-0.4 -0.5 -0.2

10 x (m)

(b)

0

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(d)

x (m)

(c) Fig. 2. Triangle tracking performance Result of the robot on the different slope (a) Simulation Result on the wall, (b) on ceiling, (c) Experimental result with λ = 24000, (d) Experimental result with PID Controller.

5 CONCLUSION This paper presents a wheeled wall climbing robot robust adaptive control method of trajectory tracking. The second-order nonlinear dynamic model with two unknown parameters and bounded uncertainties are constructed via the Newtonian second motion law. Based on this dynamic model, the adaptive robust controller is synthesized via adaptive gain smooth sliding controller and achieves both trajectory tracking and stabilization. A smooth sliding controller with adaptively update the gains for the switching functions is proposed in order to control the nonlinear systems. Simulation and experiment confirm the theoretical results and conform to the expected robot trajectory planning.

λ is a strict constant. To eliminate the chatter, term

( Φ).

sgn(s) of U, is replaced by sat s

] ( Φ )}

U = − Mb −1 − M −1 f 1 − M −1 fˆ2 g 1 − M −1 fˆ3 g 2 − Z&&d + λe& + η1 s

REFERENCES

(6)

A. Filipescu, L. Dugard, J-M. Dion, “Adaptive gain sliding observer based sliding controller for uncertain parameters nonlinear systems. Application to flexible joint robots. Decision and Control”, IEEE Conference, Volume 4, Issue, 9-12 Dec. 2003 Page(s): 3537–3542. R. Yue, J. Xiao, S. Wang, S. L. Joseph, “Modeling and Path Planning of the City_climber Robot Part I : Dynamic Modeling”, Proc. Of the 2009 IEEE Int. Conf. on Robotics and Biomemetics, Gulin, China, December 19-23, 2009.

If s < Φ

}

U = − Mb −1 − M −1 f1 − M −1 fˆ2 g 1 − M −1 fˆ3 g 2 − Z&&d + λ e& + η1 ⋅ sgn( s )

let’s pick : η1 = α1 + α 2 + η ; η>0

15

0.1

(5)

]

10 x (m)

0.2

]

{[

5

0.3

s& = − M −1 f 1 − M −1 fˆ2 g 1 − M −1 fˆ3 g 2 − Z&&d + λe& − M −1 Δfˆ2 g 1 − M −1 Δfˆ3 g 2 + M −1bU

{[

0

(a)

0 ⎤ k k 0 ⎥ ; and β = e t ⎥ JRa Mr ⎦

s >Φ

0

-1 -5

controller design is translated in terms of finding a control law for the vector U that verifies the individual sliding conditions of the differentiable form of vector s:

[

4 3

2

emf coefficient; attitude and the desired differentiable trajectory is T described by Z d = [xd yd θ d ] . λ is a strict constant. The

2.

8

7

3

ke denotes the backθ is x y plane attitude; φ is slop

If

9

8

0

⎡m 0 0 ⎤ ⎡2 f 0 ⎢ ⎥ M = 0 m 0 ; f2 = ⎢ 0 2 f ⎢0 0 J⎥ ⎢0 0 ⎣ ⎦ ⎣ Where e is equal to Z − Z d ;

1.

9

5

y (m)

0

Δfˆ23 = α 3 = 44 , f 3 ∈ [− 1,1] , fˆ3 = 0 , Δfˆ3 = δ = 1 .

y (m)

[

1

f 22 ∈ [0,88] , fˆ22 = 44 , Δfˆ22 = α 2 = 44 , f 23 ∈ [0,88] , fˆ2 = 44 , 3

(7) (8)

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