Variable Structure Control of Pendulum-driven Spherical Mobile Robots

Transcription

1 rd International Conerence on Computer and Electrical Engineering (ICCEE ) IPCSIT vol. 5 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V5.No..6 Variable Structure Control o Pendulum-driven Spherical Mobile Robots Yu Tao +, Sun Hanxu and Jia Qingxuan School o Automation, Beijing University o Posts and Telecommunications Beijing, China, 876 Abstract In this paper, we propose a hierarchical sliding mode control approach or a ixed point regulation o pendulum-driven spherical mobile robots. A simpliied dynamic model is established or their longitudinal motion and the controller is designed to have double layer structure because the system is divided into two subsystems. In the irst layer, the sliding suraces are hierarchically designed or each subsystem, and in the second layer, the whole sliding surace is designed as the linear combination o the two subsystem suraces. The asymptotic stability o the whole system is veriied by Lyapunov analysis, the stability o sliding suraces and convergence o output variables are proved utilizing mathematical techniques. Finally, we carry out a simulation to illustrate the eectiveness o the proposed control scheme. Keywords-spherical robots; longitudinal motion; asymptotic stability; sliding mode control; underactuated systems. Introduction Spherical mobile robots are special cases o exoskeleton robots, which are those robots with an external skeleton. The external skeleton provides eicient rolling security cover or the driving mechanisms and sensory equipments inside. Spherical mobile robots are nity designs, and they have several advantages over their wheeled counterparts: they adapt well to all kinds o terrain and are completely sealed, making them ideal or planetary exploration, and have higher resistances towards object collisions. Spherical mobile robots can be categorized into dierent categories according to their driving mechanisms, which may include rotor type [],[], car type [], [4] and slider type [5]. Recent surveys o spherical mobile robots and ball-shaped robots are presented in [6], [7]. The spherical mobile robots that have these types o driving mechanisms are o the same category in eect, and we call them pendulum-driven spherical mobile robots that utilize mass point relocation as the propulsion mechanism. The structure o a pendulum-driven spherical mobile robot is shown in Figure. Omni-directional rolling o spherical mobile robots can be realized by combinations o their longitudinal and lateral motions. Our research in this paper is conined to the longitudinal motion control problems. We propose a hierarchical sliding mode control approach or the pendulum-driven spherical mobile robot system, which is a single-input multi-output nonlinear underactuated system. In the proposed control system, the double layer structure is adopted to guarantee the stability o the whole system. In addition, the sliding suraces are utilized to drive the output tracking errors to zero. Here, the sliding suraces o the irst layer are designed at each subsystem, while the whole sliding surace is the linear combination o the two subsystem sliding suraces. + Corresponding author. address: yutaolanjie@6.com

2 Fig.. Structure o a pendulum-driven spherical mobile robot. The rest o this paper is organized as ollows: In Section II, Euler-Lagrange ormulation is utilized to derive the dynamic model o the pendulum-driven spherical mobile robots. In Section III, the proposed hierarchical sliding mode controller is described and the asymptotic stability o the whole system is simply proved. In Section IV, the stability o sliding suraces or each subsystem and convergence o output variables are analyzed. The proposed controller is utilized to control a spherical mobile robot and the simulation results are presented in Section V. Finally, the conclusion is given in Section VI.. Dynamic Analysis Let us start with a simpliied model, only considering no slip longitudinal motion on perectly lat suraces. Figure illustrates the simpliied model with a side view o the spherical mobile robot. It represents the spherical shell with its center o mass Ce and a pendulum (composed o a massless link and a point mass at its end) with center o mass Cp and the axis attached at the center o the sphere. The deinitions o model parameters are listed in Table I. TABLE I. θ β M m I R l g DEFINITIONS OF MODEL PARAMETERS Rolling angle o the spherical shell Sway angle o the pendulum Mass o the spherical shell Mass o the pendulum Moment o inertia o the spherical shell Radius o the spherical shell Length o the pendulum link Gracitational acceleration τ τ Rolling riction toque applied to the spherical shell Input toque applied to the pendulum by driving motor τ Ce Y θ β Cp O X P Fig.. Simpliied model or longitudinal motion.

3 We choose θ and β as the two generalized coordinates or this system. Utilizing the Euler-Lagrange ormulation, the dynamic equations or the longitudinal motion can be expressed as M( q) q + N( q, q ) = τ () where MR + mr + I mrl cos β M( q) = mrl cos β ml mrl sin ββ τ Nqq (, ) = τ = mgl sin β τ Although we cannot precisely model the riction torque τ due to its high nonlinearity, we just express it as τ = μrnsgn( θ) + d ( t) () where μ r is the rolling riction coeicient and N is the normal orce with the ground, d denotes the sum o unmodeled riction torques and disturbances. And d is upper-bounded, i.e., d () t D. Here D is a positive real value. Here i z > sgn( z) = i z = i z < Analyzing the orces in relation to Y reerence rame, the normal orce with the ground N can be expressed as N = ( m+ M) g+ m( βlsinβ + β lcos β) Deine the state vector x = ( θ θ β β) T, output vector y = ( θ β ) T and the control input u = τ. Substituting () into (), () can be transormed into the ollowing state space orm x = x x = ( x) + b( x) u+ d( t) x = x4 () x 4 = ( x) + b( x) u+ d( t) T y = ( x x) where mrl sin xx4 + mrg cos xsin x μrn sgn( x) ( x) = MR + mr sin x + I mlr cos xsin xx4 + ( MR + mr + I) g sin x μrnr cos xsgn( x) ( x) = + lmr ( + mrsin x + I) lmr ( + mrsin x + I) Rcos x b ( x) = lmr ( + mr sin x+ I) MR + mr + I ( x) = b ml ( MR + mr sin x + I ) d() t = d () t MR + mr sin x + I Rcos x d () t = d () t lmr ( + mrsin x + I) N = ( m+ M) g+ m( x 4lsinx + x lcos x) From (), we can obtain d() t D d() t D where D and D are positive real values given as R D = D D = D MR + I lmr ( + I). Control Design

4 Considering the system in (), we can transorm the system into two subsystems with state variable groups ( θ θ) T and ( β β) T, or whose we construct the ollowing linear unctions as sliding suraces, which we call the irst layer sliding mode and are given as s = λe + e (4) s = λe + e4 (5) where λ and λ are real positive design parameters. And the errors are deined as e = x θd e = x θd e = x βd e4 = x4 βd Here, θ d, θd, β d and βd are desired values or θ, θ, β and β respectively. Dierentiating (4), (5) and equalizing to zero, we can obtain the equivalent control as ( x) + λe θd u = b ( x) (6) ( x) + λe βd u = b ( x) (7) Here, θd, βd are desired values or θ, β respectively. For the single-input multi-output nonlinear underactuated system in (), it is usually diicult to control outputs [9]. Hence, we design the whole sliding surace o the second layer as the linear combination o the two subsystem suraces S = αs+ αs (8) where α and α are sliding mode parameters. Theorem : Suppose that the given system in () is controlled by the ollowing control input αbu + αbu u = + us αb+ αb (9) ( η+ αd+ αd)sgn( S) ks us = αb+ αb () Here u s is called the switching control. Then the system in () is asymptotically stable. Proo: We consider the ollowing Lyapunov unction candidate V = S () Utilizing (4), (5) and (6), (7), we can deine the dierential orm o the sliding surace o the second layer as ollows S = αs + αs = α( e + λe ) + α( e 4 + λe ) () = ( αb+ αb) u s + αd+ αd Taking the time derivative o the Lyapunov unction in (), and substituting () into (), V can be represented as V = SS = α( ds D S) + α( ds Dsgn S) η S ks () α( d S D S ) + α( d S Dsgn S ) η S ks < Consequently, according to the Lyapunov stability criterion, asymptotic stability o the system is guaranteed.. Stability and Convergence Analysis.. Stability o s and s From () and (), it is obvious Vt () V() < (4) And rom () and (), we can obtain I s s, then it is true = ( + ) Sdξ α s α s dξ = ( α s + αα s s + α s ) dξ < (5)

5 sgn( αs) = sgn( αs) (6) I (6) is suicient, rom (5) we can easily obtain i s dξ < Then si L (7) Integrating both sides o (), we can obtain S L (8) I (6) is suicient, we may also have si L (9) From () and () we have S L () Dierentiating (4), (8), we can obtain s = λe + e () S = αs + αs () Then e and e can be regarded as the velocity and acceleration respectively, which are controlled by the orce s L () From () and () we can obtain S L (4) Then rom () and (4), we can obtain s L (5) Then rom () and (5), we can obtain s i L (6) From (7), (9) and (6), according to Barbalat s lemma, we can obtain lim s = (7) lim s = (8).. Convergence o x and x From (4), (5) and (7), (8), we have 4 lim λ e + e = (9) lim λ e + e = () where λ >, λ >. From (9) and (), we can ind the ollowing two cases: ) I sgn(e ) = sgn(e ) lim λe+ e= lim e= lim e= () Similarly when sgn(e ) = sgn(e 4 ), we can obtain lim e = lim e = () ) I sgn(e ) sgn(e ) Then And or e = e, we have 4 lim λe + e = lim e = lim λe () λ( t t) lim e ( t) e ( t ) (4) λ( t t) lim e ( t) λ e ( t ) (5) This proves that e, e are locally exponentially stable with λ as the convergence rate, so lim e = lim e = (6) Similarly when sgn(e ) sgn(e 4 ), we can prove that e, e 4 are locally exponentially stable with λ as the convergence rate, so lim e = lim e = (7) 4 We can conclude that output variables converge to their desired values. 4. Simulation

6 In this section, we apply the proposed control scheme to a pendulum-driven spherical mobile robot, to demonstrate the eectiveness o the proposed control system. Here d is selected as a sinusoidal signal with an amplitude o. in this simulation. Parameters o the spherical mobile robot are selected as listed in Table II. Here, we properly choose the initial values and desired values which are expressed as θ(), β() and θ d, β d. In addition, the dierentiated initial values θ (), β () and the desired values θd, βd are chosen as listed in Table III. TABLE II. PARAMETERS OF THE SPHERICAL ROBOT Mass o the spherical shell M Mass o the pendulum m 7.9kg 6.kg Moment o inertia o the spherical shell I.455kgm Radius o the spherical shell R.5m Length o the pendulum link l.568m Gracitational acceleration g 9.8m/s TABLE III. CONTROL PARAMETERS AND CONDITIONS α. α. λ λ. k η. D.8 D.54 θ() θ d β() β d θ () β () θd βd θ(rad) θ(rad/s)

7 .6.4. β(rad) β(rad/s) Fig.. Tracking results or the spherical robot. The tracking results o the rolling angle and the sway angle regulation perormance are shown in Fig. We can ind that the rolling angle and the sway angle are asymptotically converged to their desired values, the tracking errors o them are asymptotically converged to zero. We can also ind that all sliding suraces are asymptotically converged to zero in Fig 4. And it is clear that the controller is able to overcome the eect o the torque disturbances during rolling s s

8 S Fig.4. Time evolution o the sliding suraces. 5. Conclusion In this paper, we proposed a hierarchical sliding mode control approach or the longitudinal motion o pendulum-driven spherical mobile robots. We used the double layer structure to strictly guarantee the stability o the whole system. The control development was based on the construction o cascaded sliding mode controller made o two layer sliding suraces. Based on Lyapunov analysis, this paper proved that the last sliding layer and consequently the tracking errors o output variables converge asymptotically to zero. Simulation results conducted on a spherical mobile robot veriied the eectiveness o the proposed sliding mode controller. Future work aims to develop an adaptive sliding mode control strategy or spherical mobile robots with parametric variations. 6. Acknowledgment The authors wish to acknowledge the support provided by National Natural Science Foundation o China (5775) and the Cultivation Fund o the Key Scientiic and Technical Innovation Project, Ministry o Education o China (78). 7. Reerences [] Y. Ming and Z. Q. Deng, Introducing HIT Spherical Robot: Dynamic Modeling and Analysis Based on Decoupled Subsystem, IEEE International Conerence on ROBIO '6, Kunming, China, pp. 8 86, Dec 6. [] H. X. Sun, A. P. Xiao, Q. X. Jia and L. Q. Wang, Omnidirectional kinematic analysis on a bi-driver spherical robot, Journal o Beijing University o Aeronautics and Astronautics, vol., pp , July 5. Language: Chinese. [] A. Bicchi, A. Balluchi, D. Prattichizzo and A. Grelli, Introducing the Sphericle: An Experimental Testbed or Research and Teaching in Nonholonomy, Proc. IEEE Int. Con. on Robotics and Automation, Albuquerque, New Mexico, pp. 6 65, 997. [4] J. Alves and J. Dias, Design and Control o a Spherical Mobile Robot, Proc. o the IMechE Part I: Journal o System & Control Engineering, Vol. 7, No. 6, pp ,. [5] A. H. Javadi A. and P. Mojabi, Introducing glory: A Novel Strategy or an Omnidirectional Spherical Rolling Robot, Trans. o the ASME. Journal o Dynamic Systems, Measurement and Control, Vol. 6, No., pp , 4. [6] K. Husay, Instrumentation o a Spherical Mobile Robot, Master s Thesis, Department o Engineering Cybernetics, Trondheim [7] J. Suomela and T. Ylikorpi, Ball-shaped robots: A historical overview and recent development at TKK, Field and Service Robots, vol. 5, No. 6, pp. 4 54, 6.

9 [8] Laplante, J.-F., Masson, P. and Michaud, F., Analytical longitudinal and lateral models o a spherical rolling robot, Technical Report, Department o Electrical Engineering and Computer Engineering. 7. [9] W. Wang, J. Vi, D. Zhao and D. Liu, Design o a Stable Sliding mode controller or a Class o Second-order Underactuated Systems, Control Theory and Applications, IEE Proceedings, vol. 5, no. 6, pp , 4.