Adaptive Command Filtered Backstepping Trajectory . - Atlantis Press
Advances in Intelligent Systems Research (AISR), volume 141International Conference on Applied Mathematics, Modelling and Statistics Application (AMMSA 2017)Adaptive Command Filtered Backstepping TrajectoryTracking Control of Hexarotor UAVChengshun Yang1,* and Zhong Yang21School of Electric Power Engineering, Nanjing Institute of Technology, Nanjing, ChinaCollege of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China2Abstract—In this paper, a hexarotor unmanned aerial vehicle(UAV) is concerned to solve such problems as smaller payloadcapacity, lack of both hardware redundancy and anti-crosswindcapability for quad-rotor. Considering the under-actuated andstrong coupling nonlinear system with external disturbance andparameter uncertainty properties of the hexarotor UAV, a nesteddouble-loops trajectory tracking control strategy is proposed. Aposition error PID controller is designed as the outer-loopcontroller, of which the task is to compare the desired trajectorywith real position of the hexarotor UAV and export the desiredattitude angles to the inner-loop. And, an adaptive commandfiltered backstepping controller is designed as the inner-loopcontroller which makes use of parameter update laws to estimatethe disturbances of the hexarotor UAV. The simulation resultsshow that the proposed control strategy enhances the trajectorytracking performance by controlling the external disturbanceand parameter uncertainty.Keywords- Hexarotor UAV, trajectory tracking control,backstepping, adaptive, command filterI.INTRODUCTIONRotary wing type UAVs are classified into multi-rotor type(such as quad-rotor and hexarotor), co-axial helicopter, andtraditional helicopter, etc. The design of the vehicle is simplerthan for normal helicopters in that the quad-rotor does not usemechanical linkages to vary the rotor blade pitch angle as theyspin and this reduces maintenance time and cost. Further, itsrelatively low-cost feature make it attractive candidates forswarm operations, a field of ongoing research in the UAVcommunity[1-4].The purpose of this paper is to design a trajectory trackingcontrol system forcing the hexarotor UAV (The structure asshown in Fig.1) to track the desired trajectory accurately. Torealize this purpose, the highly stable and nonlinear controllersare required. In consideration of the under-actuated and strongcoupling characteristics of the hexarotor UAV, the nesteddouble-loops trajectory tracking control strategy is introducedin this paper. Accordingly, the outer-loop refers to the positionloop while the inner-loop refers to the attitude control loop. Inthe outer-loop, a position error proportional-integral-derivative(PID) controller is developed, of which the task is to comparethe desired trajectory with real position of the hexarotor UAVand construct the command signals to the inner-loop.In the approach of attitude controller design, dynamicinversion[5], feed linearization and sliding mode control[6],model reference adaptive[7] have been widely used.Backstepping control design[2-4], due to its simple, hasbecome an effective approach for controller design. Therotational dynamics of the hexarotor UAV satisfies the strictfeedback form, so it can be used backstepping to design theattitude controller. However, the traditional backsteppingcontrol also has several of inadequacies which limit thebackstepping technique in practical applications. First, theanalytic derivative expressions of virtual control variables areusually overly complicated or unknown especially for systemswith uncertain or noise. Second, considering the practicalapplication, the states of the UAV especially for the attitudeangles and angular rate are usually needed to limit. For instance,in the inspection of the transmission lines using the hexarotorUAV, in order to facilitate the camera carried by the UAVfocusing on the components of the transmission lines, theattitude angles and angular rate of the hexarotor UAV cannotchange too much. Third, the problem of the control saturationis not considered [8,9]. Especially the last defect, it may lead toserious problems in the actual control systems. If the generatedcontrol command is not fully implemented by actuators, theaccumulation of errors may lead to the system unstable.However, these problems are not considered in the reference[2-4]. To solve these problems, Farrell, etc.[10,11] introduced aconstrained command filter into backstepping control systems.The command filter is used to eliminate the impact ofderivative of the virtual control signals and control saturation.Motivated by the above analysis, an adaptive commandfiltered backstepping attitude controller for the hexarotor UAVis designed to overcome the problems of input and stateconstraints, avoid calculating the virtual control signalderivative analytically and increase the robustness of thedisturbances. The adaptive backstepping method proposed inthis paper is a recursive, Lyapunov-based, nonlinear controllerdesign approach which makes use of parameter update laws toestimate the disturbances of the rotational dynamics. With thecommand filter, it is possible to control the limit of the attitudeand angular rate, at the same time under the actuatorconstraints[10]. And, the derivatives of the pseudo controlsignal are numerically calculated by the command filter insteadof calculating it analytically. At the same time, an auxiliaryfilter is also introduced to compensate for the command filtererror simultaneously satisfying the overall stability requirement.II.MATHEMATICAL MODELINGTo simplify the modeling of the hexarotor UAV and makethe controller design easier, several reasonable assumptions areCopyright 2017, the Authors. Published by Atlantis Press.This is an open access article under the CC BY-NC license 03
Advances in Intelligent Systems Research (AISR), volume 141made: 1) hexarotor UAV is a rigid. Then the nonlineardynamics can be derived by using Newton-Euler formulas. 2)The structure of the hexarotor UAV is symmetrical withrespect to the axes ox , oy and oz . 3) The height between therotors and the plane of the hexarotor UAV is ignored.z p v T v -gze m Rze WΩ 1 D1 1 1 1 b J ( Ω JΩ ) J Ga J 2 D2 zgxgoogygyxIn (1), p ( x, y.z )T and v (vx , v y , vz )T are the positionand velocity of the hexarotor UAV in the E-frame, respectively.g is the gravitational acceleration, T is the resulting force inthe B-frame (excluding the gravity force) acting on theairframe and ze (0, 0,1)T is a unit vector expressed in the Eframe. R Î SO(3) is the orthogonal rotation matrix to orientthe hexarotor UAV and defined as followsFIGURE I. THE STRUCTURE OF HEXAROTOR UAV AND THEASSOCIATED FRAMESADMl2vl3l1B c c R c s s s c c s s s s c c s c c where ξ (f ,q ,y )T denotes the vector of three Euler anglesand s. and c. are abbreviations for sin( ) and cos( ).xyEOAnd, in (2) Ω ( p, q, r )T is the angular rate, W is an Eulermatrix [4] and given by 1 sin tan W 0cos 0 sin sec NCs c s s c s s s c c c s FFIGURE II. THE SIMPLE STRUCTURE OF HEXAROTOR UAVFirstly, two frames have to be defined: a body-fixed frame(B-frame) and an earth-fixed frame (E-frame). Let B {oxyz}denote the body-fixed frame whose origin o is at the center ofmass of the hexarotor and E {og xg yg z g } denote the earthfixed frame, as shown in Fig. 1. Therefore, under theassumption 3), the structure of the hexarotor UAV can besimplified as shown in Fig. 2.In Fig.2, l1 denotes the length of OB and OE, l2 denotesthe length of AM, DM, CN and FN, l3 denotes the length ofOM and ON, a represents the included angle between AM andOM.Then, the translational dynamics and the rotationaldynamics [12] of the hexarotor UAV can be expressed ascos tan sin cos sec J Î R 3 3 is the total inertial matrix of the hexarotor UAV.Under the assumption 2), it is a diagonal positive definiteconstant matrix expressed in the B-frame. Di , (i 1, 2) arecomposite disturbances, including aerodynamic moments,external disturbances and parameter uncertainties. b i , (i 1, 2)are the input disturbance matrices. And the vector Gaexpresses the gyroscopic torque given by: Ga J r (Ω ze )( i 3,4,5i ) i 1,2,6 iwhere J r and i are, respectively, the rotor inertia and therotor speed.From (1) and (2), It is obvious that the hexarotor UAV is anunder-actuated mechanical system with six degree of freedom(DOF) and four main control inputs. It is characterized by onemain control force T generated by the six propellers in the free404
Advances in Intelligent Systems Research (AISR), volume 141air and three main control torques τ b ( , , )T . So, thecontrol force T can be expressed as66i 1i 1T f i b i2 command filter error simultaneously satisfying the overallstability requirement.To give a clear idea of the overall design procedure, a flowchart is depicted as Fig.3 PcPeAt the same time, the reactive torque caused by air drag isalso generated by each propeller and given by Qi d i2 . Thus,the total reactive torque by the six propellers is given as follow: Q i 3,4,5Qi Qi 1,2,6 d(i i 3,4,5 i2 2ii 1,2,6) ξc cΩc D̂1τbΩe2 III. 22 23 24 25 2 T6 z g cos cos ΩξPD2Tm x (cos sin cos sin sin )Tm y (cos sin sin sin cos )Tm The altitude subsystem (6) containing vertical force input Twhich can be linearized by selecting T asTRAJECTORY TRACKING CONTROL STRATEGYConsidering the under-actuated and strong couplingcharacteristics of the hexarotor UAV, the nested double-loopstrajectory tracking control strategy is introduced. And, therelationship between the outer-loop and inner-loop isestablished based on the position error PID controller. Then anadaptive command-filtered backstepping controller for thehexarotor UAV is designed to track the desired attitude anglesgenerated by the outer-loop. The controller makes use ofparameter update laws to estimate the composite disturbancesof the rotational dynamics of the hexarotor UAV. And, anauxiliary filter is also introduced to compensate for theωBased on (1), we have 21T-1A. Position controller designIn this section, a position controller based on PID in theouter-loop will be developed, of which the task is to comparethe desired trajectory with real position of the hexarotor UAVand construct the desired attitude angles to the inner loop. Thetranslational dynamics (1) will be treated as the plant to thecontroller which calculates the desired attitude angles andcontrol force based on the desired trajectory. bbbb bl2 sin bl2 sin bl1bl2 sin b(l2 cos l3 ) b(l2 cos l3 ) 0 b(l2 cos l3 ) dddd MD̂2 Ω D1wherebb bl sin bl12 M b(l2 cos l3 )0 d d ξcFIGURE III. THE FLOW CHART OF THE DESIGN PROCEDUREIn order to facilitate the computation of the real controlinputs, that is, i , (i 1, 2, , 6) , (4) and b are put together:(T , , , )T M Tξce1where i is the rotor speed, b and d are thrust and dragfactors[3], respectively.222222 bl2 sin ( 4 6 1 3 ) bl1 ( 5 2 ) b b(l2 cos l3 )( 12 42 32 62 ) d ( 32 42 52 12 22 62 ) ddtΩd*e1 ξ Therefore, according to Fig. 2 and (5), it is easy to obtainthe airframe torques generated by the six propellers given by:P T mgu cos cos cos cos The necessary condition for (9) is cos cos 0 , where u ,a PD controller, is given by u K d z K p ( z zd ) 405
Advances in Intelligent Systems Research (AISR), volume 141where Kp and Kd are the proportional and the derivativepositive gain and zd the desired altitude.The derivatives of e1 and e2 are given byPosition subsystem is given by (7) and (8). Let x d and y dbe the desired speed in x and y direction, respectively. Then,the errors at desired and actual speed are separately given by e 1 W β1 D1 ξ c ex x d x e 2 J 1 (Ω JΩ ) J 1Ga J 1 τ b β2 D2 Ω c To give a clear idea of the controller design procedure, thefollowing steps are given.ey y d y The desired roll and pitch angles in terms of errors betweenactual and desired speed are, thus, separately given by c arcsin(ue sin ue cos )x c arcsin(Step 1: In this step, the task is to stabilize (17) with respectto the Layapunov function 1 T1e1 e1 D 1TΦ1 D 122 yuexcos cos sin sin )cos cos where D 1 D1 Dˆ1 , Φ1 is a positive definite design parametermatrix.The time derivative of V1 with respect to time is given bywhere, uex and uey areuex V1 K y ey mK x ex m, ue y TT eT e D TΦ D V11 11 1 1 e1T (WΩ β1 D1 ξ c ) D 1TΦ1 Dˆ1 where Kx and Ky are the positive constants and T is the desiredvertical force input by the altitude control.Then, the virtual controller and parameter update law forD1 can be designed asB. Attitude controller designFirst, a reasonable assumption and a definition are given asfollows. Ωd* W 1 (c1e1 ξ c β1 Dˆ1 ) Assumption 1: The derivatives of the disturbancesD i 0, i 1, 2 . (This means that the disturbances changeslowly.) D̂1 Φ1 1 β1T e1 Definition 1: D i Di Dˆ i , i 1, 2 are the estimation errorsof the disturbances.Consider the rotational dynamics given by (2), and define,respectively, the attitude and angular rate tracking errors asfollows:where Ωd* is the desired angular rate and c1 is a positivedefinite matrix to be designed.Substituting (21) and (22) into (20) yields eT ( c e β D ) D T β T eV111 11 11 1 1 e1T c1e1 0 e1 c e2 Ω - Ωc where ξ c ( c , c , c )T are the desired attitude anglescomputed by (12)and (13). Ωc is the filtered-command of Ωd*and it will be defined later. And we can get the conclusion that (21) and (22) canguarantee the stability of (17).In order to overcome the problems of input and stateconstraints and avoid calculating the virtual control signalderivative analytically, a constrained command filter isintroduced into the procedure of the adaptive backsteppingcontrol design. Ωd* and Ωc are the input and output of thecommand filter as described in Fig.4. In addition, the derivateof Ωc , namely Ω c can also be generated by the command filter.406
Advances in Intelligent Systems Research (AISR), volume 141*dΩωn2ζ1s2ζωn1sΩ cΩcSelect the Lyapunov function aswhere and n are damping ratio and natural frequency ofthe command filter, respectively.Thus, the state space of the command filter can berepresented as[11] Ωc x Mx M1 T111e1 e1 D 1TΦ1 D 1 e2T e2 D 2TΦ2 D 22222 and the time derivation of V2 is given by e T e D TΦ D eT e D TΦ D V21 11 1 12 22 2 2 Then we can design the control torques τ b and parameterupdate law for D2 as Ωc 0 * 2 Ωd 2 n Ωc n D̂2 Φ2 1 β2T e2 Based on (21), (24) and (25), we havee 1 W β1 D1 ξ c c1ε W (Ωc Ωd* ) WΩ* W (Ω Ω ) β D ξ c ε At the same time, in consideration of the influence causedby the command filter, an auxiliary filter is also introduced tocompensate for the command filter error simultaneouslysatisfying the overall stability requirement.ε c1ε W (Ωc Ωd* ) Thus, the attitude tracking error and the parameter updatelaw for D1 are redefined as followse1 ξ ξ c ε where c2 is a positive definite matrix to be designed.1It is obviously that the system (23) is a linear stable system.Accordingly, if Ωd* is bounded, Ωc and Ω c are bounded andcontinuous. where D 2 D2 Dˆ 2 , Φ2 is a positive definite designparameter matrix.d x MIn the interval of linear change of the SR ( ) and SM ( ) , thestate space of the command filter can be described as V2 τ b J [c2 e2 J 1 (Ω JΩ ) J 1Ga Ω c W T e1 β2 Dˆ 2 ] And, the definition of SR ( ) is similar to SM ( ) . Ω c 0 2 Ωc n where SM ( ) and SR ( ) denote magnitude and rate limitedfunction, and SM ( ) is defined as M , SM ( x ) x, M , Step 2: In this step, the task is to design the control torquesand achieve Ω tracking Ωc asymptotically.FIGURE IV. COMMAND FILTER BLOCK DIAGRAM Ω c Ω c n2 * 2Ω n SR c 2 SM (Ωd ) Ωcn D̂1 Φ1 1 β1T e1 c c1e1 β1 D 1 We2 11c1 Using the expression (29) and substituting it into (18) yieldse 2 c2 e2 β2 D 2 W T e1 Substituting (26), (30)-(32) into (28), we obtain e T c e eT c e 0V21 1 12 2 2 Thus, the attitude and angular rate tracking errors e ande2 converge to zero exponentially.IV.SIMULATION RESULTS AND DISCUSSIONIn this section, in order to verify the validity and efficiencyof the control law and update laws for the composite407
Advances in Intelligent Systems Research (AISR), volume 141disturbances Di , (i 1, 2) , simulations of a typical trajectorytracking task is performed on Matlab/Simulink.In the simulation, the initial positions and attitude anglesare p0 (0, 0, 0)T , ξ 0 (0, 0, 0)T , respectively, and so arelinear and angular velocities, respectively. The compositedisturbances inputs are chosen as D1 (t ) [0.05, 0.1, 0.15]T ,2z/mThe hexarotor UAV model parameters are taken asm 1.7Kg, g 9.81m/s2, l1 281.5mm, l2 287.6mm,l3 40.08mm, 45 , J diag{0.0019,0.0019,0.031}Kg.m2,J r 3.357 10-5 Kg.m2, b 2.98 10-6N s2/rad2, d 1.14 107Nm s2/rad2.40-232y/mpcParameterpqr nMag.limitRate.limit2 10-3 5rad/s 10rad/s22 10-3 5rad/s 10rad/s22 10-3 5rad/s he position controller and attitude controller parameters insimulations are fixed at Kp 0.98, Kd 0.265, Kx 1.0,Ky 0.97, c1 diag{2, 2, 2} , c2 diag{3,3,3} . The update lawparametersarechosenasΦ1 diag{0.2, 0.2, 0.2} , Φ2 diag{0.05, 0.05, 0.05} and thecommand filter parameters are shown in Table 1.The comparison of the simulation results is demonstrated inFig.5 and Fig.7 where dashed-dot-dot lines represent thereference trajectory and the desired attitude angles, the solidlines correspond to the responses with update laws, and thedashed lines represent the responses without update laws. Thedetailed time responses for the trajectories of x, y, z directionare presented in Fig. 6.p without update lawsFIGURE V. THE 3D FLIGHT TRAJECTORY AND DESIREDTRAJECTORYZ/mTABLE I. COMMAND FILTER PARAMETERS0p with update laws: Desired trajectory: Actual flight trajectory with update laws: Actual flight trajectory without update lawsThe desired horizontal rectangle trajectory is given by:where fsg is an interval function and expressed assign( x a ) sign( x b).fsg( x, a, b) 22 x/m0D2 (t ) [0.1cos(0.5t), 0.01sin(0.4t) 0.1,0.05cos(0.3t) 0.15]T .And the disturbance matrices b i , (i 1, 2) are all unit matrix. xc 0.8(t 5)fsg(t ,5,10) 4fsg(t ,10,15) 0.8(20 t )fsg(t ,15, 20) yc 0.6(t 10)fsg(t ,10,15) 3fsg(t ,15, 20) 0.6(25 t )fsg(t , 20, 25) zc 0.6tfsg(t , 0,5) 3fsg(t ,5,30) c 0.2rad4110pp20t/s: Desired trajectory: Actual flight trajectory with update laws: Actual flight trajectory without update lawsFIGURE VI. ACTUAL TRAJECTORY AND DESIRED TRAJECTORY OFX, Y, Z DIRECTIONFrom Fig. 5, the trajectory tracking without the update lawsseems to work fine; however, the desired performance is notobtained. In Fig. 6, the trajectories of x, y, z direction and theresponses of the attitude angles exhibit significant deviations.Contrarily, the actual flight trajectory with the update lawsclosely follows the desired trajectory even under the compositedisturbances. So, it is obviously that the control law performsbetter with the update laws under the composite disturbancesinputs. At the same time, the simulation results verify that theproposed control strategy has the same efficiency to trackdifferent trajectory types.408
Advances in Intelligent Systems Research (AISR), volume 141[6] /rad)0.10 /rad-0.10.20.10-0.1 /rad0.40102030400102030400.20 c01020t/s 30 40: Desired attitude angles produced by Eq.(21): Responsesof attitude angles with update laws: Responses of attitude angleswithout update lawsD. Lee, H. J. Kim, S. Sastry. Feedback linearization vs. adaptive slidingmode control for a quadrotor helicopter. International Journal of Control,Automation and System, 2009, 7(3): 419-428.[7] B. T. Whitehead, S. R. Bieniawski. Model reference adaptive control ofa quadrotor UAV. AIAA Guidance, Navigation and Control Conference,Toronto, Ontario Canada, August 2-5, 2010.[8] D. Swaroop, J. K. Hedrick, P. P. Yip, J. C. Gerdes. Dynamic surfacecontrol for a class of nonlinear systems, IEEE Transactions onAutomatic Control, 2000, 45(10): 1893- 1899.[9] D. Wang, J. Huang. Neural network-based adaptive dynamic surfacecontrol for a class of uncertain nonlinear systems in strict feedback form,IEEE Transactions on Neural Networks, 2005, 16(1): 195-202.[10] J. A. Farrell, M. Polycarpou, M. Sharma, W. Dong. Command filteredbackstepping, IEEE Transactions on Automatic Control, 2009, 54(6):1391-1395.[11] J. A. Farrell, M. Polycarpou, M. Sharma, Longitudinal flight path controlusing on-line function approximation, AIAA Journal of Guidance,Control and Dynamics, 2003, 26(6): 885-897.[12] C. S. Yang, Z. Yang, D. Z. Xu, L. Ge. Trajectory tracking control for anovel six rotor aircraft. Systems Engineering and Electronics, 2012,34(10): 2098-2105. (in Chinese)FIGURE VII. RESPONSES OF THE ATTITUDE ANGLES FORTRACKING THE DESIRED TRAJECTORYV.CONCLUSIONSAn adaptive command filtered backstepping controlstrategy is applied for the trajectory tracking problem of anunder-actuated and strong coupling nonlinear hexarotor modelwith verification by the experiments. The hexarotor rotationaldynamics is developed to include the composite disturbances.Update laws are established for the composite disturbanceswhich include aerodynamic moments, external disturbance andparameter uncertainties. Simulation results demonstrate that theproposed control strategy provides desirable trajectory trackingperformance levels even under the composite disturbancesinputs.ACKNOWLEDGMENTThis work was supported by the Open Research Fund ofJiangsu Collaborative Innovation Center for Smart DistributionNetwork(XTCX201714), Nanjing Institute of Technology andHigh-level Scientific Research Foundation for the Introductionof Talent of Nanjing Institute of Technology (YKJ201412).REFERENCES[1][2][3][4][5]E. Altug, J. P. Ostrowski, C. J. Taylor. Control of helicopter using dualcamera visual feedback. The International Journal of Robotics Research,2005, 24(5): 329-341.H. Bouadi, M. Bouchoucha, M. Tadhine. Sliding mode control based onbackstepping approach for an UAV type-quadrotor. International Journalof Applied Mathematics & Computer Sciences, 2008, 4(1): 12-17.A. A. Mian, D. B. Wang, Modeling and backstepping based nonlinearcontrol strategy for a 6 DOF quadrotor helicopter. Chinese Journal ofAeronautics, 2008, 21(3): 261-268. (in Chinese)Z. Zuo. Trajectory tracking control design with command-filteredcompensation for a quadrotor. IET Control Theory and Applications,2010, 4(11): 2343-2355.A. Das, K. Subbarao, F. Lewis. Dynamic inversion with zero-dynamicstabilization for quadrotor control. IET Control Theory and Application,2009, 3(3): 303-314.409
In the approach of attitude controller design, dynamic inversion[5], feed linearization and sliding mode control[6], model reference adaptive[7] have been widely used. Backstepping control design[2-4], due to its simple, has become an effective approach for controller design. The rotational dynamics of the hexarotor UAV satisfies the strict
nonlinear control methods that have been applied to linear induction motor is the backstepping design [8, 9, 10]. Backstepping is a systematic and recursive design methodology for nonlinear feedback control. This approach is based upon a systematic procedure for the design of feedback control strategies suitable for the
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(like hiking and dining) or different transportation modes, such as walking and driving. We show examples of trajectory classification in Section 7. Trajectory Outlier Detection: Different from trajectory patterns that frequently occur in trajectory data, trajectory ou
Review Autocorrelation Spectrum White Bandwidth Bandstop Shape Summary Outline 1 Review: Power Spectrum and Autocorrelation 2 Autocorrelation of Filtered Noise 3 Power Spectrum of Filtered Noise 4 Auditory-Filtered White Noise 5 What is the Bandwidth of the Auditory Filters? 6 Auditory-Filtered Other Noises 7 What is the Shape of the Auditory Filters? 8 Summary
focused on backstepping control and sliding mode control. In [25], a backstepping technique was proposed to control the un-deractuated USV under constant environmental disturbances. In [3], a siding mode control was proposed to address the underactuated USV control problem, and experiments were carried out to verify the effectiveness.
where a neural network adaptive controller is used for the transition from horizontal ight to hover, and [16], where a nonlinear dynamic inversion approach is used for formation ight. Within this context, an autopilot con guration for longitudinal and latero-directional aircraft control based on nonlinear backstepping is pre-
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