739. Design of adaptive sliding mode control for spherical robot based on MR fluid actuator

Transcription

1 739. Design of aaptive sliing moe control for spherical robot base on MR flui actuator M. Yue,, B. Y. Liu School of Automotive Engineering, Dalian University of echnology 604, Dalian, Liaoning province, China State Key Laboratory of Robotics an Systems, Harbin Institute of echnology 5000, Harbin, Heilongjiang Province, China (Receive 8 November 0; accepte 4 February0 Abstract. Inner suspension platform of a spherical robot unergoes ynamic oscillation process when the robot is rolling ahea, which significantly reuces stability an precision of the motion. o suppress these vibrations, this stuy consiers an aaptive sliing moe control metho for a spherical robot base on application of magnetorheological (MR flui actuator. After analyzing the mechanical structure of the spherical robot, a ynamic moel escribing its rolling motion is erive. Consiering the uncertainty of the isturbances introuce by moeling error an outsie perturbation, an aaptive scheme is propose to estimate the ynamic isturbances. Sliing moe technology is applie to construct system controller characterize by robustness an parameter insensibility. Simulations are performe in orer to verify the effectiveness of the propose control methos. Keywors: spherical robot, MR flui actuator, sliing moe control, aaptive. Introuction Spherical robots recently attracte a lot of research attention owing to its special configuration an movement performances [-3]. Most spherical robots aopt inner suspension platform to install sensors an other evices. In this case the suspension is in an uner-actuate state by the riving motor s reaction torque [4-6]. his uner-actuate state causes violent vibration of inner suspension. Such vibration may cause great amages to the robotic imaging system an may even lea the failure of etection mission, especially, when the spherical robot is operating within unstructure environment. A lot of research work has been performe to solve the aforementione problems. In [4], a mechanical structure was esigne in spherical robot to form a stabilize platform that can always keep a stable attitue automatically. In [5], spherical robot was equippe with two cameras which can hol a state attitue when the robot moves. However, because spherical robot internal space is very narrow, the aitional mechanical structure will increases the complexity of the robotic system. In [7], a state feeback controller was esigne base on sigmoial function, which ensures globally asymptotic convergence of the system state to a esire small neighborhoo. In [8], base on the linear ynamic moel, a state feeback controller was esigne, an this controller can realize arbitrarily pole assignment. Such treatments for holing the balance of the inner suspension platform are complex an are ifficult to realize in practice. Motivate by [9, 0], we chose to apply a MR flui actuator to suppress the vibration of inner suspension platform of a spherical robot. MR flui, as a type of smart material, has a property that the rheological properties can be continuously an reversibly change within several millisecons by applying or removing external magnetic fiels. Owing to this superior characteristic, MR flui has been wiely investigate an applie in practice recently [9-]. herefore, we attempte to implement the MR effect into a spherical robot because it can achieve active vibration suppression within a much smaller space. argete at a typical spherical robot, a mechanical structure is evelope to realize the MR flui actuator, followe by the evelopment of the ynamic moel escribing the rolling motion. 96 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

2 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. Another ifficulty to implement MR flui actuator is the control problem. Enhancement of system aaptability an robustness is a challenging task, especially, in the complex an ynamic unstructure environment. herefore, a esign metho for control system of the spherical robot with MR flui actuator is also iscusse in this stuy. Furthermore, the sliing moe control technology is applie to erive the controller owing to their better robustness an parameter insensibility [-4]. In aition, the isturbances are unavoiable an it is unrealistic to achieve their practical value ahea in unstructure exploration environment. Consiere these facts, we evelope a isturbance aaptive control scheme to estimate the isturbance on line. he aaptive control schemes are erive via Lyapunov stability theorem, an it can significantly enhance the aaptability an robustness of the robot system. Moreover, the inherent chattering behavior on sliing moe surface is suppresse because continuously ifferentiable hyperbolic tangent function is applie in sliing moe controller. his paper is organize as follows: in section, a MR flui actuator applie in spherical robot system is presente. Dynamic moel of the spherical robot is erive in section 3. Section 4 aresses the sliing moe control esign process via Lyapunov stability theorem. In section 5, the simulation stuy is conucte to emonstrate the effectiveness an feasibility of control metho. Finally, some conclusions are provie. Spherical robot with MR flui actuator hough mechanical structures of various spherical robots are ifferent from each other, the robots with inner suspension platform are much more similar. In this case, we iscuss a typical spherical robot with suspension platform, which is escribe in more etail in []. Generally, the movement of the spherical robot can be separate as rolling ahea an turning motions, among which the rolling ahea movement is a more important motion moe (a moe where the oscillations occur. herefore, we mainly aress this motion moe to improve the accuracy an performance of the robot. he mechanical structure of the MR flui actuator in spherical robot propose in this stuy is illustrate in Fig.. Fig.. Structural iagram of the spherical robot he motor housing is esigne as an input component of the MR flui actuator, which consists of the MR flui, the electromagnetic coil an the yoke. When the external magnetic fiel is applie to the electromagnetic coil, the corresponing magneto-motive force is generate. hen, the shear force is generate in the opposite irection of the rotary motor housing motion, i.e., in the presence of the magnetic fiel, aitional shear force occurs ue to VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

3 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. the MR effect. As a result, the torque is transmitte from the input component to output one. Since the yiel stress of the MR flui changes accoring to the strength of the magnetic fiel, the output torque can be continuously an reversibly controlle by ajusting external magnetic fiel. herefore, in Fig., the torque generate by the motor can be transmitte from the motor to the inner suspension continually. hen, the inner suspension coul be riven by the output torque, so the uner-actuate characteristics are change to the active ones. Dynamic moel of the spherical robot A typical spherical robot in rolling ahea motion can be escribe in Fig., which has one motor to rive the shell an a MR flui actuator to rive the inner suspension. In Fig., the inner suspension is connecte with the MR flui actuator housing rigily. he clearance between motor housing an MR actuator housing is full of MR flui, which is represente by shae parts shown in Fig.. he equation of ynamic behavior of the robot base on MR actuator is governe by: Μ ( q + C ( q, + G( q + W( t + = τ ɺ ( where M( q R is a symmetric an positive efinite inertia matrix, C( q, R is a amping matrix, G( q R is a stiffness matrix, an W ( t R is a isturbance matrix. τ R is a control input matrix an state vector is q = [ ϕ θ ]. Specific parameters are as follows. a bcosθ 0 bθ M =, = ɺ sinθ C, G = [ 0 sinθ ], W = [ w w ], τ = [ τ τ ], b cosθ c 0 0 where a = (5m+ 3m r, 3 b= m rl, c= m l, an = m gl. ϕ represent the angle that the shell rolls from the start point, an θ is the angle that the inner suspension sways away from the vertical irection; m is the mass of shell an m is the mass of inner suspension; r represents the shell raius an l escribes the istance between shell center an inner suspension centroi; g is acceleration of gravity. τ an τ are the control input of motor an the output of the MR flui actuator respectively, an τ, which can be controlle by external magnetic fiels, is restricte by τ. Fig.. Rolling ahea movement of a spherical robot with the MR flui actuator Design of controller for the spherical robot Sliing moe controller. he goal of the spherical robot control system is to track the esire spee an restrain the irregular vibration of the inner suspension. Meanwhile, sliing moe 98 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

4 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. control is applie to erive the system ynamic controller owing to its better robustness an parameter insensibility characteristics. hen, we apply sliing moe control technology to erive the control algorithm of the spherical robot [5-7]. Suppose that e = [ e e ] = q q, where q an q represent the esire state value an the actual state value respectively. o achieve the performance mentione above, the sliing surface can be efine by s = eɺ + Γe ( λ where Γ = is a iagonal positive matrix. It is obvious that the tracking error e 0 if the λ positive matrix is selecte properly. Suppose that the isturbance is known an it can be irectly applie in the controller esign. he control input can be efine as: τ= M( + Γeɺ + C( + Γe + G+ W+δ tanh( κs+ δ s (3 where δ, δ an κ are positive constants. Note that the continuously ifferentiable hyperbolic tangent function tanh(κs is use to replace the sign function here, which coul largely suppress the chattering behavior of the control system on sliing moe surface. Accoring to the above analysis, we can give a theorem as follows: heorem. he errors q will asymptotically converge to zero, if the controller (3 is e applie to the system (. Proof: A Lyapunov function is efine as V = s Ms (4 Clearly, V > 0. Differentiating V with respect to time yiels ɺ s Ms ɺ s Msɺ s Mɺ V = + = ( C s+ s Cs+ s Msɺ (5 Let matrix A= M ɺ C, then the matrix M ɺ an A can be erive as follows: 0 θɺ ɺ b sinθ 0 bθɺ sinθ M =, A = bθɺ sinθ 0 bθɺ sinθ 0 Consiering that matrix A has the characteristic of A = A, then we can raw the conclusion that matrix A is a skew symmetric matrix. herefore, we can obtain s As = 0, i. e., s ( Mɺ C s= 0 (6 Substituting (6 into (5 yiels Vɺ = s ( Cs+ Msɺ = s [ Cs+ M( ɺɺ e+ Γeɺ ] = s = s ( Ceɺ + CΓe+ Mɺ = s [ C( eɺ + Γe + M( ɺ = s [ C( + Γe + M( ɺ + C + G+ W τ+ MΓeɺ + Γeɺ + C + G+ W τ] + Γeɺ + G+ W τ] [ Cs+ M( ɺ ɺ + MΓeɺ ] VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN (7 99

5 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. Substituting (3 into (7 achieves that V ɺ = δ s tanh( κs δ s s (8 Clearly, V ɺ 0. By LaSalle s theorem [8], the errors q e converge to zero globally, an then the theorem is prove. A isturbance aaptive scheme. Although the theorem 4. is certifie correct above, the isturbance is impossible to measure correctly because of its uncertainty ue to the complex unstructure environment. Consiering the above limitations, the controller (3 is impractical for control of the spherical robot system (. hough the isturbance is uncertain, the change rate of isturbance is much slower in practical exploration. hen, we attempt to use an aaptive control scheme to estimate the isturbance value consiering this case. Let Ŵ represents the estimate vector of ~ isturbance W, the estimate error coul be efine by W= W Wˆ. hen, the isturbance an ~ the estimate error can be governe by W ɺ = W ɺ ˆ. herefore, the control law can be rewritten as τ= M( + Γeɺ + C( + Γe + G+ Wˆ + δ tanh( κs + δ s (9 Aaptive law can be given as W ɺˆ = λs (0 where δ, δ > 0 an κ, λ> 0. he parameter λ etermines the convergence time, an the bigger value of the parameter is set, the shorter time of converging to the isturbance is obtaine, but the bigger overshoot of estimation of isturbances is generate. Accoring to the above analyses, we can formulate a theorem as follows: heorem. he errors q will asymptotically converge to zero, if the control law (9 an e aaptive law (0 are applie to the system (. Proof: A new Lyapunov function coul be efine as follows: ~ ~ V = s Ms+ W W ( λ Clearly, V > 0. Similarly, ifferentiating V with respect to time yiels ɺ ɺ s C Γe M ɺ Γeɺ G Wˆ ~ ~ V= [ ( + + ( W τ] W Wˆ λ ɺ s C Γe M ɺ Γeɺ G Wˆ ~ ~ = [ ( + + ( τ] + s W W Wˆ λ ~ ~ ɺ s [ C( Γe M( ɺ Γeɺ G Wˆ = τ] + W s W Wˆ λ ( Substituting (9 an (0 into ( provies: V ɺ = δ s tanh( κs δ s s 00 Clearly, V ɺ 0. By LaSalle s theorem [8], the errors q converge to zero globally, an then e the theorem is prove. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

6 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. Simulation stuy he parameters of the spherical robot an the control coefficients are liste in able. Suppose that initial state vector of spherical robot is [ ϕ ϕɺ θ θɺ ] = [ ] an, consiering that the rolling resistance is existent in unstructure environment, θ is set to 0. ra to eal with the rolling resistance. herefore, the esire value of spherical robot is set as [ ϕ ϕɺ θ θɺ ] = [5t ]. Meanwhile, the isturbances to the system are hypothesize as a step change form, which represents a worst situation to verify the effectiveness of the control algorithms. Furthermore, white noise is introuce to the system to simulate the real application environment as much as possible. Fig. 3 shows curves of states of spherical robot with aaptive sliing moe control. he simulation results emonstrate that the erive controller with isturbances estimation can control states to their esire states. he control effect is closely relate to the control parameters matrix Γ which etermine overshoot an response time of the control system. he overshoot in Fig. 3 b an c is.4% an.85% respectively, an the response time is no more than 0. secon. With the propose metho, the inner suspension can hol a relatively constant angle, which can ensure a goo imaging effect of spherical robot in unstructure environment. he enlarge rawing shown in Fig. 3 b further emonstrates that the actual value can converge to the esire value quickly even though the white noise is introuce, which emonstrate the robustness of the propose control scheme. able. Spherical robot parameters Spherical robot Value Mechanical parameters m = 3, m = 5, r = 0.75, l = 0.094, g = 9.8. Control coefficients λ = λ = 0, λ = 90, δ = 0., δ = 50, κ = 50. Fig. 3. States variables of the controlle spherical robot VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

7 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. Fig. 4 provies error curves of the shell angular velocity an the inner suspension swing angle. he errors can converge to zero globally even when the isturbances change violently in 4 secons an 8 secons. he enlarge rawings, which are shown in Fig. 4 a an b, emonstrate that the propose control scheme can eal with white noise effectively emonstrating strong robustness of the control methos. Fig. 4. racking errors of ϕɺ anθ Fig. 5 escribes the estimation curves of isturbances. Here, the isturbances with step form are suppose to affect both the shell system an the inner suspension system thereby representing an extremely unfavorable conition. he isturbance w acting on shell subsystem occurs between N an 4 N at 4 secons an 8 secons. he isturbance w acting on inner suspension subsystem occurs between 0.5 N an.5 N at 4 secons an 8 secons. It implies that the converging time an accuracy can be improve by increasing λ, while the overshoots woul increase abruptly. Slight fluctuation occurs after the isturbances are estimate well owing to the white noise. However, the estimation can quickly converge to the estimate value. Fig. 5. Disturbance estimation process for an arbitrary step isturbance Fig. 6 shows control torque of the shell subsystem an inner suspension subsystem. Overshoot occurs at the beginning ue to the inertia of the system. he results inicate that τ, which rives inner suspension before introucing MR flui actuator, is larger than τ. herefore, τ is transmitte from τ an coul be controlle by external magnetic fiels, which are 0 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

8 739. DESIGN OF ADAPIVE SLIDING MODE CONROL FOR SPHERICAL ROBO BASED ON MR FLUID ACUAOR. mentione in section. Conclusion Fig. 6. System control input A MR flui actuator was introuce an an aaptive sliing moe control was propose in this stuy. Firstly, we investigate a esign metho of the mechanical structure using MR flui actuator to change the uner-actuate suspension platform to the active one. his metho can save more space an was prove to be more reasonable. Seconly, the ynamic moel, which can escribe the rolling ahea motion of the spherical robot, was erive. he moels accounts for the sphere rolling an penulum angles. hirly, an aaptive control scheme was propose, which can estimate the uncertain isturbances in an aaptive manner. his aaptive control metho may consierably enhance the robustness an aaptability of the spherical robot system. Such works will affor basic theoretical founation for application of the MR flui actuator, an further work will be concerne with experimental stuy in orer to verify the propose metho by using a real-life spherical robot. Acknowlegment his financial supports for this research project by the National Natural Science Founation of China (Grant No an No. 6750, an State Key Laboratory of Robotics an System (SKLRS-00-MS03 are all gratefully acknowlege. References [] Halme A., Schonberg., Wang Y. Motion control of a spherical mobile robot. In Proc. of 4th Int. Workshop on Avance Motion Control, (su, Jpn., 996, p [] Yue M., Deng Z. Q., Yu X., Yu W. Introucing hit spherical robot: ynamic moeling an analysis base on ecouple subsystem. In IEEE International Conference on Robotics an Biomimetics, (Kunming, China, Dec. 006, p [3] Joshi V. A., Banavar R. N., Hippalgaonkar R. Design an analysis of a spherical mobile robot. Mechanism an Machine heory, Vol. 45, Issue, 00, p [4] Li. J., Liu W. Q. Design an analysis of a win-riven spherical robot with multiple shapes for environment exploration. Journal of Aerospace Engineering, Vol. 4, Issue, 0, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 0. VOLUME 4, ISSUE. ISSN

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