IROS2019国际学术会议论文集 1074

An Approximation Free Simple Control Scheme for Uncertain Quadrotor Systems Theory and Validations Gang Wang Weixin Yang Na Zhao Peng Li Yantao Shen and Chaoli Wang Abstract In this paper a simple tracking control scheme is proposed for quadrotor systems with uncertain dynamics It precludes the necessity for prohibitive analytic computation of the derivatives of the desired virtual attitude that is typically employed in controlling quadrotor systems Moreover this control scheme is approximation free in the sense that it does not incorporate any adaptive laws observers or command fi lters to compensate for unknown parameters in the dynamics and the absence of the analytic differentiation thus exhibiting remarkably low complexity levels and making its implementa tion straightforward The thrust saturation is approached in the position control design which also enables the singularity in desired attitude extraction to be avoided entirely It is demonstrated that based on the proposed scheme the tracking errors can be made arbitrarily small by appropriately selecting design parameters Extensive simulations and experiments are performed to verify the effectiveness of our scheme I INTRODUCTION The advent of ultra fast microprocessors together with advanced sensing devices has stimulated the evolution of autonomous quadrotors due to the numerous applications that can be addressed by such systems like surveillance trans portation or mapping 1 2 3 As for the autonomous fl ight problem a fundamental task is to embed appropriate and stable control schemes Nevertheless from a control perspective designing control schemes for the quadrotor system constitutes a quite challenging issue The diffi culty arises from the fact that these systems possess particular characteristics such as the strong coupling nonlinearity higher order dynamics underactuation and multi variable nature 4 Attempts to approach the stabilization and trajectory tracking control of the quadrotor with uncertain dynam ics have been presented by applying linear and nonlinear control methods Classic linear control algorithms such as proportional derivative control were initially employed for quadrotors to actualize the position control 5 however as remarked in 6 7 these algorithms require a linearization of the quadrotor system at specifi c operation points and can barely perform tracking missions nearby these points G Wang is with the Institute of Machine Intelligence University of Shanghai for Science and Technology Shanghai 200093 China He is also with the Department of Electrical Biomedical Engineering University of Nevada Reno NV 89557 USA2010wanggang W Yang N Zhao and Y Shen are with the Department of Electrical Biomedical Engineering University of Nevada Reno NV 89557 USA ytshen unr edu P Li is with HIT Shenzhen China and his work was partially supported by JCYJ20180507183456108 C Wang is with the Department of Control Science and Engineering University of Shanghai for Science and Technology Shanghai 200093 Chinaclwang Motivated by these facts some effective nonlinear control approaches have been proposed to carry out full envelope tracking assignments for the quadrotors With the help of model free control and sliding mode control a tracking con trol strategy is presented to control the attitude and position of a quadrotor in 8 Nevertheless this eminent strategy is restricted to locally approximated simple quadrotor model Regarding the complicated original nonlinear model of the quadrotor a robust adaptive control scheme is provided in the signifi cant work 9 However the analytic calculation of the second derivatives of the desired virtual attitude is required by this scheme which is prohibitively diffi cult to obtain as emphasized in 10 Instead of using these analytic imple mentations command fi lters are introduced to approximate these derivatives online and an adaptive control algorithm capable of stabilizing attitude and tracking trajectory is then designed for quadrotor system in 11 Unfortunately this algorithm suffers from the singular problems while comput ing the desired attitude for the control torque design By the robust compensator technique a trajectory tracking controller without requiring desired attitude extractions is designed for quadrotors and a perfect tracking performance is achieved and verifi ed through experiments in 12 Under the presence of time varying aerodynamic effect and bounded external disturbance a novel augmented L1adaptive strategy is presented in order to achieve precise trajectory control of quadrotors in 4 But the thrust saturation was not taken into account in 4 12 which may destabilize the closed loop system 13 Dealing with the singular problems introduced by desired attitude extractions while considering the thrust constraint for the control of the quadrotor via a hierarchical control strategy is recently reported in 7 10 Taking account of the quite limited onboard computation power a new simple control scheme is designed to realize trajectory tracking control for the quadrotors with uncertain dynamics in this paper It precludes the necessity for in volved analytic computation of the derivatives of the desired virtual attitude that is typically required in controlling quadrotors Moreover this control scheme is approximation free in the sense that it does not include any adaptive laws observers or command fi lters to compensate for unknown parameters in the dynamics and the absence of the analytic differentiation Owing to the aforementioned attributes it needs quite a few and manageable calculations to generate the control input thus rendering its implementation straight forward Additionally the thrust saturation is taken into account in the position control design by using differentiable saturated functions which also enables the singularity in 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE3078 desired attitude extraction to be completely avoided II PROBLEMSTATEMENT Let B BxByBz represent the body fi xed frame and its origin coincide with the mass center of the quadrotor Let p x y z T R3denote the position of the mass center of the quadrotor expressed in the fi xed inertial frame E ExEyEz where x R y R and z R are respectively the longitudinal latitudinal and vertical positions T R3is the attitude of the quadrotor where R is the roll angle around the Bxaxis R is the pitch angle around the Byaxis and R is the yaw angle around the Bzaxis The dynamic model of the quadrotor can be expressed as follows 6 p 1 m RT ge3 1 f g 2 where e3 0 0 1 Twith Tbeing the transpose operator T R is the thrust on the body T R3 is the total control input for rotational motion m denotes the mass of the quadrotor g is the gravity acceleration Ix Iy Iz R are the moment of inertia of quadrotor Jr R is the moment of inertia of rotor l R denotes the distance from the motor to the center of gravity R represents the unbalanced rotor spinning speed due to imperfect motor installation and yaw maneuvering g diag l Ix l Iy 1 Iz and R cos sin cos sin sin cos sin sin sin cos cos cos f f f f Iy Iz Ix Jr Ix Iz Ix Iy Jr Iy Ix Iy Iz As demonstrated in 6 it is considerably time consuming to measure the moment of inertia Ix Iy and Iz In addition Jr and are also diffi cult to identify accurately due to the unavoidable measurement noises as well as the irreg ular shape of the quadrotor Hence we aim at solving the tracking control problem of the quadrotor system without a priori knowledge of these dynamic parameters Moreover on account of the thrust saturation and the limited onboard computation power the control objective of this paper is to design a predesignated bound of thrust T and a simple control input for the quadrotor modeled by 1 and 2 with unknown parameters Ix Iy Iz Jr and in the dynamics such that it can track the smooth desired trajectory pd xd yd zd T R3and yaw d R Assumption 1 The desired trajectory pd t is available and satisfi es pd t pd t pd t are bounded and zd t M L is defi ned as i L L ii M N and M N and iii L N and N L where M 4 16 M L 3 3 M 2 8 M L M 2 13M 3L 16 It can be checked that is bounded by M and has a bounded fi rst derivative Because the virtual position tracking controller phas been designed in 3 on account of k Rd t k 1 we select the translational thrust T as T t mk p t k 5 Thus given the reference acceleration pd the resulting thrust T is always predesignated bounded and satisfi es 0 T t T t 0 where T m g sup t 0 k p d t k 3s 2M 6 Consider the developed virtual position tracking controller pin 3 According to 4 if z t 6 0 we can extract the virtual roll dand virtual pitch das d t arcsin xsin d ycos d 2 x 2y 2z d t arctan xcos d ysin d z 7 Notice that the design parameters s2and M have been chosen to satisfy s2M 0 which implies that dand dcan always be extracted via 7 3079 without singularity and that k p t k 0 and 0 0 such that e t 5T for all t t1with satisfying 0 0 as t we say that the ultimate bounds of t and t are O 5T t t1 we have that 4x t t t1 Thus applying Lemma 6 in 13 to 11 we can conclude that exand exare ultimately bounded by O Similarly we can deduce that ey s2 k2 s2 ey s1 k1 s1ey 4y t ez s2 k2 s2 ez s1 k1 s1ez 4z t 14 with4y t cos sin sin sin cos cos dsin dsin d sin dcos d Tand4z t cos cos cos dcos d T Usingthemimicking argument as 12 it is straightforward to show that 4y t and 4z t t t1 Hence applying Lemma 6 in 13 to 14 gives that ey ey ez and ezare ultimately bounded by O which completes the proof Step 2 In this step we intend to design the control input such that the attitude can track dwith the prescribed adjustable steady state error bound Besides for the purpose of straightforward implementation of our scheme we do not resort to the complicated analytic computation of the deriva tives of the desired attitude d adaptive laws observers or command fi lters In this direction we utilize the prescribed performance design technique and introduce an auxiliary function 1 1 R with ln 1 1 15 16 Design the virtual angular velocity controller as 1 e 1 t 15 where 1is a positive constant 1 t 1 0 1 e 1t 1 with 1 0 0 1 0 and 1 0 Here 1 is the required maximum steady state error and 1 0and 1 are selected to satisfy 1 0 max e 0 1 and 1 max d 0 2 and 2and 2 being positive constants Lemma 2 Given any desired yaw angle dobeying As sumption 2 if the fi rst derivatives of dand din 7 are bounded then the developed simple control input 16 can ensure that the attitude errors e are bounded and satisfy 1 t e t 0 and xand yare always bounded implies that there exists a positive constant c 3such that 1 t c 3 N we can obtain from 4 that i t i x y z are bounded Assumption 2 dictates that d t is bounded Therefore we conclude from 17 that d t and d t are bounded As a result the conditions in Lemma 2 can always be satisfi ed and we have 1 t e t 1 t Owing to 1 t 1 0 1 e 1t 1 and 1 5 1 T there exists t1 0 such that e t 5T for all t t1 Finally the conditions in Lemma 1 are satisfi ed and the rest conclusions of this theorem can be drawn immediately by applying Lemma 1 V SIMULATIONS AND EXPERIMENTS A Simulation Study To verify the effectiveness of the proposed control scheme we carry out a simulation study in Coppelia Robotics V REP We consider the quadrotor with m 0 32 kg l 0 2 m Ix Iy 0 0069 kg m2 Iz 0 0126 kg m2 Jr 0 0003 kg m2 and 0 rad s The gravity acceleration is set as g 9 81 m s2 The quadrotor is required to track the desired trajectory pd t and yaw d t which are described as pd t sin t cos t 2 sin t 2 Tm and d t 0 rad The control parameters in 4 are set as s1 0 4 s2 2 4 k1 0 2 and k2 1 2 The design parameters of the saturation function in 4 are selected as L 1 M 2 and N 3 It can be checked that the control parameters s2and M satisfy s2M e 0 and 2 0 d 0 As a result 1 0 1 and 2 0 2 1 0 0 t 0 tf and 1 1r 1 we get L 1 1 1 t 1r 2 1 c 1 r 1 t 0 tf Apparently it holds that L 1 c 1 1 As a consequence we obtain that r 1 r 1 max r 1 0 c 1 1 t 0 tf It then follows from 21 that 1 1 e r 1 1 e r 1 1 t er 1 1 er 1 1 1 23 for all t 0 tf Taking the time derivative of gives 1 1 2 t 2 1 d 1 t 1 1 t By 20 and 23 it is straightforward to deduce the boundedness of on 0 tf Step 2 Considering L 2 r2 2 2 we conclude L 2 2 2g r2 2 c 2 r 2 2 t t 0 tf where c 2 is an unknown positive constant satisfying f t d d 1h 1 1 2 t 2 t 2 c 2 for all t 0 tf Thus r 2 r 2 max r 2 0 c 2 2g t 0 tf Correspondingly we can obtain from 21 that 1 1 e r 2 1 e r 2 2 t er 2 1 er 2 1 1 24 for all t 0 tf As an immediate result the control input is bounded on 0 tf By virtue of 23 and 24 we conclude that there is a compact subset so that t t 0 tf Hence by 17 Proposition C 3 6 the solution exists for t 0 i e tf Finally from 23 and 1 e 1 we have the following inequality 1 t e t 1 t for all t 0 REFERENCES 1 V Kumar and N Michael Opportunities and challenges with au tonomous micro aerial vehicles Int J Robot Res 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