Adaptive Robust Control (ARC) For An Altitude Control Of A .
2011 11th International Conference on Control, Automation and SystemsOct. 26-29, 2011 in KINTEX, Gyeonggi-do, KoreaAdaptive Robust Control (ARC) for an Altitude Control of a Quadrotor TypeUAV Carrying an Unknown PayloadsByung-Cheol Min, Ji-Hyeon Hong, and Eric T. MatsonMachine-to-Machine(M2M) LabDepartment of Computer and Information Technology, Purdue University, West Lafayette, IN 47907, USA(Tel : 1-765-586-5944; E-mail: minb@purdue.edu)Abstract: This research deals with an altitude controller of a quadrotor type UAV with an unknown total mass of thestructure. We assume that the uncertainty results from the flight mission in which the UAV carries unknown payloads.Since the quadrotor type UAV involves both translational and rotational motions due to its inherent dynamics, it is ofimportance to know accurate information on the vehicles the moment of inertia and the total mass in order to guaranteethe UAVs attitude and position controls. An Adaptive Robust Control (ARC) is utilized to compensate for the parametricuncertainty. Then, Lyapunov based stability analysis shows that the proposed control design guarantees asymptotic tracking error for the UAVs altitude control. Numerical simulation results which are time-based are presented to illustrate thegood tracking performance of the designed control law.Keywords: Adaptive Robust Control, ARC, UAV, Quadrotor, Altitude control1. INTRODUCTIONtrol [7-10]. Bouadi et al. presented stabilizing controllaws synthesis by sliding mode based on the backstepping approach [3]. S. Bouabdallah et al. used the integral backstepping technique for attitude, altitude andposition control [6]. In this paper, the results of thistechnique showed a powerful and flexible control structure. Furthermore, he showed that OS4 quadrotor wasable to perform autonomous hovering with altitude control and autonomous take-off and landing E. Altug et al.implemented a proportional derivative (PD) controller tocontrol attitude and altitude using dual cameras via feedback linearization [4]. There is another approach combining a PID controller, the backstepping and the slidingmode controller together [4,7,9]. The authors divided thequadrotor model into several subsystems. U. Ozguner etal., for instance, divided the model into two subsystems:a fully-actuated subsystem and an under-actuated subsystem [9]. Then, he controlled them with a PID controllerand a sliding mode controller, respectively. As a result,this combined controller stabilized the overall quadrotorsystem with the model errors, parametric uncertaintiesand other disturbances.As the quad-rotor helicopter has both translational androtational motions and operates, in three dimensional environments with the rotors, it is of importance for us toknow accurate information on the vehicle’s moment ofinertia and the total mass. We may obtain them by experiments if they can be measured in advance and notchanged. If the mission of the UAV is to carry an unknown load or object, it is hard to know the exact valueof the its weight in advance. If we fail to obtain the exactvalue, the controller for UAV’s attitude and position willbe collapsed with inaccuracy parameters of the UAV because they are closely associated with the rotational motion and the translational motion. Thus, in this paper, weapply the ARC control law to the UAV in case of whichwe do not know the exact value of its mass. Due to theparameter estimation we will obtain the exact value ofNumerous research efforts have been conducted onUnmanned Aerial Vehicles (UAVs) due to their wide usein many practical applications, e.g., research, surveillance and reconnaissance in specific regions. This family of UAVs can be classified into two categories by theirshapes and flying methods: fixed-wing type aircraft andvertical take-off and landing (VTOL) type aircraft. Fixedwing type aircrafts are generally deployed where extensive areas are to be covered in a short time because theycan fly with considerable speeds. On the other hand,VTOL type aircrafts (single rotor helicopter, coaxial typehelicopter and quad-rotor) are suitable for a limited areaor application that requires a stationary or a slow flightbecause they have specific characteristics like hoveringand vertical landing.Focusing on the VTOL category of UAVs, a quad-rotorhelicopter is representative of a VTOL type helicopter.By using four rotors which can rotate with individuallycontrollable speeds, the mechanical structure of the quadrotor aircraft becomes simpler than that of the single rotoraircraft or the coaxial type helicopter [1]. Such advantages result in making the quad-rotor type UAV a simpleand light rotor actuator system. Furthermore, because ofits high payload and relatively easily balanced center ofmass, the quad-rotor helicopter is often used in missionswhich carry objects [2]. To achieve such a mission, however, a highly sophisticated attitude and position controlis required. The position control of the quad-rotor helicopter is highly linked with its attitude. If rolling andpitching moments are produced, the helicopter may flytowards x or y direction, respectively. If yawing momentis produced, the head of the helicopter may turn at thesame place.Over the years, various researches on quad-rotor helicopter control have been conducted, including a backstepping and sliding mode control [3-7,9] and a PID con-978-89-93215-03-8 98560/11/ 15 ICROS1147
the UAV’s mass and will use the value in an altitude controller.The rest of this paper is arranged as follows. The dynamics of the quad-rotor type UAV will be introduced inSection 2 so that the adaptive robust control for the altitude control of the helicopter can be designed in Section3. Then, Section 3 will discuss an adaptive robust controllaw, and Lyapunov based stability analysis shows that thedesigned control law guarantees a zero tracking error forthe UAVs altitude control. In Section 4, to verify the performance of the control, the results of numerical simulations will be given by describing the operations of thedesigned controller. Finally, the conclusions and futurescope of this research will be summarized in Section 5.F4{B}"x! : Pitch: RollF2mJ{E} zyxFig. 1 The coordinate system with an earth frame {E}and a body frame {B}.where Ωr (w1 w2 w3 w4 ), and Ui (i 1, 2, 3, 4)are control inputs of the model as follows: U1 F1 F2 F3 F4 U F F231(3) U3 F4 F2 U4 F1 F3 F2 F4 . Thebody is rigid and symmetrical.rotors are rigid, i.e. no blade flapping occurs. The difference of gravity by the spin of the earth isminor. The center of mass and body fixed frame origin coincide. The3. ADAPTIVE ROBUST CONTROL (ARC)The ARC control strategy is implemented for an altitude control of the quadrotor UAV, with the total masswhich is unknown. This idea of using ARC control lawwas first proposed in [11], and applied in [12]. The controller for the altitude system is illustrated in Fig. 2.In Fig. 2, ua can be considered as the correct modelcompensation that is needed for the UAV system to tracka time-varying trajectory along z-axis. With this approximate model compensation ua , however, a perfect tracking may not be achieved. Thus, certain feedback actionis needed to keep the output tracking error e z zd .zd is the desired altitude to be tracked by z. Since theUAVs dynamics along z-axis dynamics (2) is of secondorder, we can introduce a Proportional Derivative (PD)feedback control design p shown in Fig. 2 [13]. p isgiven by p ė k1 e ż żeq , żeq żd k1 e wherek1 is a positive feedback gain. Since the transfer functionfrom p to e, Gp (s) e(s)/p(s) 1/(s k1 ) is stable,if we are able to regulate p well, it will guarantee that theoutput tracking error e will go to zero. Then, control input and the model uncertainty with p in the motion of theUAV along z axis (13) are as follows:These assumptions can be formed because of slowerspeed and lower altitude of the quad-rotor aircraft whosebody having 6 DOF (Degree of Freedom) is rigid as compared to a regular aircraft. Under these assumptions, it ispossible to describe the fuselage dynamics, and the coordinate system can be divided into an earth frame {E} anda body frame {B} as shown in Fig. 1. Using the formalism of Newton-Euler, the dynamic equations are writtenin the following form [3]:vFf Fd FgREB S(Ω) Ω JΩ Γf Γa Γg .: YawF3Deriving mathematical modeling or differential equations is necessary for the control of the quad-rotor helicopter position and altitude. However, it is hard for thecomplicated structure of the quad-rotor type to expressits motion with simple modeling. In addition, since thequad-rotor type aircraft includes highly nonlinear factors,we need to consider several assumptions in order to get adesired model [3]. zy2. MODELINGξ mξ ṘEBJΩF1(1)As a consequence, the complete dynamic model whichgoverns the quad-rotor helicopter is as follows: (Cφ Sθ Cψ Sφ Sψ )U1 ẍ m (Cφ Sθ Sψ Sφ Cψ )U1 ÿ m (CC)U φθ1 g z̈ m(2)θ̇ψ̇(Iy Iz ) dU2 Kax φ̇2 Jr θ̇Ωr φ̈ Ix 2 φ̇ψ̇(I I) dU zx3 Kay θ̇ Jr φ̇Ωr θ̈ Iy 2 ψ̈ φ̇θ̇(Ix Iy ) U4 Kaz ψ̇Izmṗ m(z̈ g) m(z̈eq g) U m(z̈eq g)(4)where U is control input of the model, represents aninput disturbance that results from the tilted body of theUAV, c.f, Cφ Cθ , m is the total mass of the UAV that is themodel uncertainty, and z̈eq z̈d k1 ė . Let the unknownparameter and the known basis function be θ m and1148
!#. Then, it is bounded above byεVs exp( λt)Vs (0) [1 exp( λt)]λ!" # %&' ()*3URMHFWLRQ W\SH GDSWDWLRQ /DZ9 -where λ 2k2 /θmax . The time derivative of Vs is then,9 -/ ?@41RQOLQHDU 5REXVW )HHGEDFN9 89 0;. 01; 2 39:0RGHO &RPSHQVDWLRQV̇s k2 p2 p{us2 ϕθ̃ }. ,-V̇s k2 p2 ε λVs εTherefore, by Barbalats lemma [13], p 0 as t ,which implies the output tracking error asymptoticallygoes to zero. (5)4. SIMULATIONwhere θmin and θmax are known as lower and upperbound of θ, and d(t) is an time-varying disturbance.Then, the ARC law that is shown in Fig. 2 can beobtained as follows:Uus ua us , ua ϕθ̂, ϕ (z̈eq g) us1 us2 , us1 k2 p(6)in which us1 represents a nominal stabilizing feedback.us2is the additional feedback that is needed to achieve a guaranteed robust performance when model uncertainty exists. Thus, it should satisfy the following robust performance conditionspus2 0p{us2 ϕd θ̃ } εOutput10.5Output Tracking Error0(8)where θ̂ is the estimate of θ, and Γ 0 is a diagonal matrix. Let θ̃ denote the estimation error (i.e., θ̃ θ̂ θ). Ifthe desired trajectory satisfies the persistent exciting (PE)condition that is introduced in [13], the parameter estimates asymptotically converge to their true values (i.e.,θ̃ 0 as t ).Let Lyapunov theorem be considered and a positivedefinite (p.d.) function define, in order to show that thedesigned control law guarantees asymptotic tracking error for the UAV’s altitude control1Vs p2 .2Actual Output zDesired Putput zd1.5(7)where ε is a design parameter that can be arbitrarilysmall. Then, a projection type adaptation law is givenby [13] θ̂ Pr ojθ̂ (Γτ ), τ ϕpIn this section, the results of a numerical simulationin terms of time history are presented in order to demonstrate the performance of the ARC control law utilizedin this paper. The desired altitude is first set as zd (t) 1m with a disturbance (t) (1 cos φ cos θ)(sin(t2 ))where φ and θ are assumed to be less than 5 degreesrespectively, which comes from roll and pitch angleswhen the body of the UAV is fluctuated. To test thetracking performance of the controller with different unknown loads, then, the desired altitude is set as zd (t) 0.5(1 cos(π(t)) with the same disturbance. In this test,Altitude (m)i.ii. pṗ θeT Γ 1 θ̂(14) ( ϕT θe k2 p)p θeT Γ 1 Γϕp k2 p2 .V̇aAssumption. The unknown parameter lies in the knownbounded region Ωθ and unknown nonlinear function isbounded by the known function δ(t) Ω { : (t) δ(t)d(t)}(12)which leads to (10) and proves that all signals arebounded. Let another positive definite (p.d.) functiondefine as1(13)Va Vs θ̃T Γ 1 θ̃.2Then, the derivative of Va satisfiesϕ z̈eq g, respectively. In [14], (t) is assumed to bebounded a known function. The assumption is as follws: Ωθ {θ : θmin θ θmax }(11)With ii of (7) and λ min{2k2 /θmax }, we can have,/LQHDU 6WDELOL]LQJ )HHGEDFN5 AB AC*60./01 2 34 # 567" 2 ,8 70;3' )HHGEDFN3ODQWFig. 2 The structure of the ARC control for the altitudecontrol.θ(10)0123456Time (sec)789107891078910Output Tracking Error10.500123456Time (Sec)Control InputControl Input402000123456Time (Sec)Prameter Estimates MEstimated Para m hatActual Para mMass (kg)10.50123456Time (sec)78910Fig. 3 ARC controller with a constant desired outputand true parameter m 0.550.(9)1149
Output10.5012356Time (sec)789x 100200123456Time (Sec)7890.510Output Tracking Error-3441Output Tracking ErrorOutput Tracking Error01001234Control Input23456Time (Sec)7Prameter Estimates M910123456Time (sec)789x 1000123789x 10Altitude (m)0.500123456Time (Sec)78910Control Input123456Time (Sec)7Prameter Estimates M9123456Time (sec)0123456Time (Sec)78910Estimated Para m hatActual Para m123456Time (sec)78910Actual Output zDesired Putput zd10.501235456Time (sec)789107891078910Output Tracking Error-4x 100-50123456Time (Sec)78910500123456Time (Sec)Prameter Estimates MEstimated Para m hatActual Para m10.50100.510Estimated Para m hatActual Para m1Mass (kg)8Mass (kg)Control Input1009Control InputControl Input200010Output Tracking ErrorAltitude (m)Output Tracking Error56Time (sec)81.5Actual Output zDesired Putput zdOutput Tracking Error-3147100356Time (Sec)Fig. 6 PID controller with a sinusoidal desired output,true parameter m 0.550, Γ 5000.0.5242001110OutputOutput1.509210Fig. 4 PID controller with a sinusoidal desired output,true parameter m 0.720, Γ 5000.0810.507Prameter Estimates MEstimated Para m hatActual Para m1Mass (kg)8Mass (kg)Control Input10156Time (sec)Control Input2004Output Tracking Error-3Control Input0Actual Output zDesired Putput zd1.5Altitude (m)Altitude (m)OutputActual Output zDesired Putput zd1.50.5010123456Time (sec)78910Fig. 5 ARC controller with a sinusoidal desired output,true parameter m 0.720, Γ 5000.Fig. 7 ARC controller with a sinusoidal desired output,true parameter m 0.550, Γ 5000.we first employ a classical PID controller and then, utilize the ARC control law for comparison. The gains ofthe PID controller are set as P 0.04, I 0.006, and D 0.009, respectively. The bounds of the parameter variations are set as θmin 0.500 and θmax 0.800. The initialparameter estimates are set as 0.5000, for all cases. Theadaptive rate, Γ, is 5000.As a result, the responses and tracking errors of thosesimulations are shown in Fig. 3-7. As shown in the Fig.3, when a desired output zd was a constant, the UAVcould track the desired output with almost zero trackingerrors. With a constant zd , however, the parameter estimate did not converge to its true value because the PEcondition [13] was not satisfied with a constant desiredoutput. Next, a desired output zd is sinusoidal. Whenthe PID controller was employed for an altitude control,the outputs could converge to the desired value with errors about 2 10 3 m, shown in Fig.4 and Fig. 6. Whenthe ARC control law was employed, the outputs couldconverge to the desired value with the smaller trackingerrors than those of the PID controller, shown in Fig.5and Fig. 7. As the parameter estimates could nearly approach to their true values, which are 0.720 and 0.550, respectively, the tracking errors with the ARC control lawcould become almost zero. Since the desired output wasof sinusoidal, the PE condition could be satisfied.5. CONCLUSIONThe final goal of our research is the success of theflight mission in that a quad-rotor helicopter carries un-1150
known payloads. However, since the quad-rotor helicopter involves both translational and rotational motions,knowing of the accurate information on the vehicle, suchas the moment of inertia and the total mass, is very significant, in order to guarantee the vehicles attitude andposition controls.Generally, if parameter uncertainty exists in a vehicles system in terms of a total mass of the structure, thenthe uncertainty is compensated by introducing an integral (I) controller in the PID controller. If the I gain hasto increase highly to compensate the model uncertainty,however, it might make the stability of the system worse,making the system unstable overall.Therefore, an adaptive robust controller was employedin case there is the parametric uncertainty in the quadrotor type UAV system. As shown in numerical simulationresults, by utilizing the ARC law, the UAV could trackthe desired altitude in spite of the parameter uncertaintyof the model. By comparing results of the PID controllerwith those of the ARC controller, tracking errors of theARC controller could become smaller than those of thePID controller as parameter estimates nearly approachedto their true values.The parameter estimates could successfully convergeto their true values when the desired outputs were differentiable. However, when the desired output was constant, the parameter estimate could not approach to itstrue value. In such a case, the quadrotor UAV systemagain involved the parametric uncertainty, even thoughthe UAV could track the desired output.Our future project will focus on the design of an attitude controller, by extending the ARC control strategythat is introduced in this paper.Systems, 2006 IEEE/RSJ International Conferenceon, pp. 3255-3260, 2006.[6] S. Bouabdallah and R. Siegwart, “Backsteppingand sliding-mode techniques applied to an indoormicro quadrotor,” in Proc. (IEEE) InternationalConference on Robotics and Automation (ICRA05),Barcelona, Spain, 2005.[7] A.A. Mian, M.I. Ahmad, D. Wang, “Backstepping based PID Control Strategy for an Underactuated Aerial Robot,” Proceedings of the 17th WorldCongress, The International Federation of Automatic Control, Seoul, Korea, July 6-11, 2008.[8] B. Erginer, E. Altug, “Modelling and PD Controlof a Quadrotor VTOL Vehicle,” Proceedings of the2007 IEEE Intelligent Vehicles Symposium, Istanbul,Turkey, June 13-15, 2007.[9] R. Xu and U. Ozguner, “Sliding mode control of aquadrotor helicopter,” Proc. of the 45th IEEE Conference on Decision and Control, pp. 4957-4962,2006.[10] B. C. Min, C. H. Cho, K. M. Choi, and D. H.Kim, “Development of a Micro Quad-Rotor UAV forMonitoring an Indoor Environment,” Lecture Notesin Computer Science, Vol. 5744/2009, pp. 262-271,2009.[11] Bin Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automatica, Vol 33, Issue 5, pp. 893900, 1997.[12] Li Xu and Bin Yao, “Adaptive robust precision motion control of linear motors with negligible electrical dynamics: theory and experiments,” IEEE/ASMETransactions on Mechatronics, Vo. 6, Issue: 4, pp.444-452, 2001.[13] Bin Yao, “Advanced Motion Control: From Classical PID to Nonlinear Adaptive Robust Control (Plenary Paper),” The 11th IEEE Internaltional Workshop on Advanced Motion Control, Nagaoka, Japan,2010.[14] Bin Yao and M. Tomizuka, “Adaptive Robust Control of a Class of Multivariable Nonlinear Systems,”In IFAC World Congress, Vol. F, pp. 335-340, 1996.ACKNOWLEDGEMENTSome materials featured in this paper were presentedin the final project report in the ME 68900/Spring 2011instructed by Professor Bin Yao at Purdue University.REFERENCES[1] Kimon P.Valavanis, “Advances in Unmanned AerialVehicles,” 1st Ed., Springer Ltd., 2007.[2] D. Mellinger, M. Shomin, N. Michael, and V. Kumar, “Cooperative Grasping and Transport usingMultiple Quadrotors,” in Distributed AutonomousRobotic Systems, Lausanne, Switzerland, 2010.[3] H. Bouadi, M. Bouchoucha and M. Tadjine, “SlidingMode Control based on Backstepping Approach foran UAV Type-Quadrotor,” Intern
a fully-actuated subsystem and an under-actuated subsys-tem [9]. Then, he controlled them with a PID controller and a sliding mode controller, respectively. As a result, . Section 2 so that the adaptive robust control for the alti-tude control of the helicopter can be designed in Section 3. Then, Section 3 will discuss an adaptive robust .
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