quadrotor control

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  A Lyapunov-based approach forTime-Coordinated 3D Path-Following of Multiple Quadrotors Venanzio Cichella, Isaac Kaminer, Enric Xargay, Vladimir Dobrokhodov,Naira Hovakimyan, A. Pedro Aguiar, and Ant´onio M. Pascoal  Abstract —This paper focuses on the problem of developingcontrol laws to solve the  Time-Coordinated 3D Path-Following task for multiple quadrotor UAVs in the presence of time-varying communication networks and spatial and temporalconstraints. The objective is to enable a fleet of quadrotorsto track predefined spatial paths while coordinating to achievesynchronization in both time and heading. One scenario is asymmetric exchange of position by four quadrotors initiallypositioned in four corners of a square room. When the missionstarts, every quadrotor is required to execute collision freemaneuvers and arrive at the opposite corner at the same desiredinstant of time. In this paper, the time-coordination task issolved by adjusting the second derivative of the coordinationvariable along the desired paths. Conditions are derived underwhich the coordination and path-following errors converge toa neighborhood of zero. Flight test results are presented tovalidate the theoretical findings. I. I NTRODUCTION Avoiding harm’s ways requires the employment of in-telligent autonomous vehicles. This, along with recent ad-vances in miniature technology, brings a global spotlighton the development of Unmanned Aerial Vehicles (UAVs).Currently, the use of UAVs plays a crucial role in pre-venting exposure of human beings to uncertain and hostileenvironments, therefore avoiding any danger to the lives of operators. For instance, after being struck by the biggestrecorded earthquake and a devastating tsunami, Japan hasbeen fighting a potential nuclear catastrophe by deployingUAVs in situations where the presence of human operatorswas hazardous.From a design point of view, and with a slight abuse of terminology, UAVs can be classified in two main categories:fixed-wings and rotatory-wings.Compared to the fixed-wings–which cannot freely move in any direction (rotate) orhold a constant position–, rotorcrafts can be deployed ina much wider variety of scenarios. Among rotatory-wingsaircraft, quadrotors play an important role in research areasas prototypes for real-life missions, including monitoring andexploration of small areas.A quadrotor consists of four blades, whose motion controlis achieved by adjusting the angular rate of one or morerotor discs. Control of quadrotors is quite challenging and Research is supported in part by USSOCOM, ONR, AFOSR, ARO, andCO3AUVs of the EU.V. Cichella, E. Xargay, and N. Hovakimyan are with UIUC, Urbana,IL 61801, e-mail:  { cichell2,xargay,nhovakim } I. Kaminerand V. Dobrokhodov are with NPS, Monterey, CA 93943, email: { kaminer,vldobr } P. Aguiar and A. Pascoal are with IST, Lisbon,1049 Portugal, email: pedro, has been addressed in many recent publications. To mentiona few, in [1] and [2] a stabilization and control algorithmis developed using Lyapunov stability theory. In [3] and [4]PD 2 and PID architectures are compared with LQR basedcontrol theory. Backstepping control is proposed in [5],while in [6] and [7] a visual-based feedback control law ispresented using camera measurements for pose estimation.Fuzzy-logic control techniques are proposed in [8]. Intelli-gent control, based on neural networks, is introduced in [9]to achieve vertical take-off and landing. Integral sliding modeand reinforcement learning control are presented in [10]as solutions for accommodating the nonlinear disturbancesfor outdoor altitude control. Finally, in [11] a trajectory-tracking control algorithm is formulated using the SpecialOrthogonal group  SO ( 3 )  for attitude representation, leadingto a simple and singularity-free solution for the trajectorytracking problem.Cooperation between multiple unmanned vehicles has alsoreceived significant attention in the control community inrecent years. Relevant work includes spacecraft formationflying [12], UAV control [13], [14], coordinated controlof land robots [15], and control of multiple autonomousunderwater vehicles [16], [17]. However, much work remainsto be done to overcome numerous critical constraints. Forexample, one of the crucial problems is the presence of time-varying communication networks that arise due to temporaryloss of communication links and switching communicationtopologies [18], [19].Motivated by these challenges, we address the problemof   Time-Coordinated 3D Path-Following  (TCPF), where aset of quadrotor UAVs are requested to  converge to and  follow desired prespecified paths under stringent temporalconstraints . In the solution adopted, the path-following (PF)and time-coordination (TC) problems are almost decoupled.At the PF level, we assume there exists a control law capableof steering a quadrotor along its assigned path. At the TClevel, the synchronization problem is solved by adjustingthe commanded position and velocity of the quadrotorsinvolved in the mission, thus obtaining –indirectly– vehiclecoordination. Figure 1 captures the key concept describedabove.This paper is organized as follows. In Section II, wedefine the PF and TC control problems, and present stability-related properties that the PF closed-loop system must sat-isfy. Then, a formal definition of the TCPF problem isgiven. In Section III we propose a solution for the TC prob-lem. Section IV formulates a PF algorithm that enables an  Report Documentation Page Form Approved OMB No. 0704-0188  Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.   1. REPORT DATE   DEC 2012   2. REPORT TYPE   3. DATES COVERED   00-00-2012 to 00-00-2012 4. TITLE AND SUBTITLE   A Lyapunov-based approach for Time-Coordinated 3D Path-Following of Multiple Quadrotors   5a. CONTRACT NUMBER   5b. GRANT NUMBER   5c. PROGRAM ELEMENT NUMBER   6. AUTHOR(S)   5d. PROJECT NUMBER   5e. TASK NUMBER   5f. WORK UNIT NUMBER   7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)   UIUC,Urbana,IL,61801   8. PERFORMING ORGANIZATIONREPORT NUMBER   9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)   10. SPONSOR/MONITOR’S ACRONYM(S)   11. SPONSOR/MONITOR’S REPORT NUMBER(S)   12. DISTRIBUTION/AVAILABILITY STATEMENT   Approved for public release; distribution unlimited   13. SUPPLEMENTARY NOTES   Preprint Conference on Decision and Control, December 2012,Government or Federal Purpose Rights License.   14. ABSTRACT   15. SUBJECT TERMS   16. SECURITY CLASSIFICATION OF:   17. LIMITATION OF ABSTRACT   Same asReport (SAR)   18. NUMBEROF PAGES   7   19a. NAME OFRESPONSIBLE PERSON   a. REPORT   unclassified   b. ABSTRACT   unclassified   c. THIS PAGE   unclassified   Standard Form 298 (Rev. 8-98)  Prescribed by ANSI Std Z39-18  PathGeneration  PF i TimeCoordinationNetworkExchangeQuadrotorsPFerrors x d,i ( γ  i )  x i u i γ  i γ  i γ  i γ  Fig. 1. TCPF Control Scheme. AR.Drone quadrotor to follow a desired path, and shows thatthe convergence properties of the TCPF system hold for thisparticular vehicle. Section V presents and discusses flighttests results that illustrate the effectiveness of the proposedPF and TC algorithms. Finally, Section VI contains the mainconclusions.II. P ROBLEM  F ORMULATION  A. 3D Path-Following for a single quadrotor  Let  I   denote an inertial reference frame, andlet  x i ( t )  ∈ R 3 be the position of the center of massof the  i th quadrotor in this inertial frame, resolved in  I  .Also, let  B  i  =  {  b 1 ,  b 2 ,  b 3 }  denote the body frame with itssrcin located at the center of mass of the  i th quadrotor;vector  b 3  is normal to the plane defined by the centers of thefour rotors –pointing upwards in non-inverted flight–, whilevectors   b 1  and   b 2  lie in this plane, with   b 1  pointing out thenose and   b 2  completing the right-hand system. Further, let x d,i ( γ  i )  ∈ R 3 be a desired path parameterized by  γ  i , andassume that   ∂x d,i ∂γ  i  ≤  v d max ,i  ,  (1)for some  0  < v d max ,i  < v max , where  v max  is the maximumoperational speed of the quadrotors. The choice of theparameterizing variable  γ  i  will be discussed later.Then, we can define the position error vector as e x,i  =  x d,i ( γ  i ) − x i  ∈ R 3 (2)and the velocity error vector as e v,i  =  ∂x d,i ( γ  i ) ∂γ  i ˙ γ  i  −  ˙ x i  = ˙ x d,i ( γ  i ) −  ˙ x  ∈ R 3 .  (3)Additionally, similar to [11], define the errors e ˜ R,i  = 12  R ⊤ d,i R i  − R ⊤ i  R d,i  ∨ ,  (4) e Ω ,i  = Ω i  − R ⊤ i  R d,i Ω d,i  ,  (5)where  R i  ∈  SO ( 3 )  and  Ω i  ∈ R 3 are, respectively, the ro-tation matrix from the body-fixed frame  B  i  to the inertialframe  I   and the angular velocity of the  i th quadrotor in thebody-fixed frame  B  i ;  R d,i  ∈  SO ( 3 )  represents the desiredattitude of the  i th quadrotor with respect to the inertial frameand is generally expressed as a function of the position andvelocity errors,  e x,i  and  e v,i , as well as the desired head-ing  ψ d,i ;  Ω d,i  satisfies  S  (Ω d ) =  R ⊤ d ˙ R d ; while the operators ( · ) ∨ and  S  ( · )  denote the  vee  and  hat   maps [11].With the above notation, we define the path-followinggeneralized error vector x PF,i  =  e ⊤ x,i , e ⊤ v,i  , e ⊤ ˜ R,i  , e ⊤ Ω ,i  ⊤ ∈ R 12 .  (6)The dynamics of the  i th vehicle’s PF error vector can bemodeled as ˙ x PF,i  =  f  i ( x PF,i ,u i ) ,  (7)where  f  i ( · )  is a general nonlinear vector map and  u i  is thecontrol signal vector. Then, the PF control problem can bedefined as: Problem 1 (Path-Following Problem):  Consider the i th quadrotor UAV and a given trajectory  x d,i ( γ  i ) satisfying (1). We say that a controller  u i ( t )  solves thePF control problem if the generalized PF error vector  x PF,i with the dynamic described in (7) satisfies  x PF,i ( t )  ≤  k  x PF,i (0)  e − λ PF  t , for some parameter  k >  0 , rate of convergence λ PF   >  0 , anddomain of attraction D  =  { x PF,i  ∈ R 12 :   x PF,i  ≤  r } , r >  0 .  Assumption 1:  We assume that there exists a control law u i ( t )  that solves the PF problem defined in Problem 1.  B. Time-Coordination We now address the TC problem of a fleet of   n  quadrotorUAVs. As will become clear, this problem will be solvedby adjusting –for each vehicle– the second derivative of theparameterizing variable  γ  i ( t ) .As described earlier, the desired path assigned to eachvehicle is parameterized by a variable  γ  i ,  i  = 1 ,...,n . Thechoice of the parameter  γ  i  is such that, if   γ  i ( t ) − γ  j ( t ) = 0 , ∀ i,j  ∈ { 1 ,...,n } ,  i   =  j  and  ˙ γ  i ( t ) = 1 ,  ∀ i  ∈ { 1 ,...,n } ,at some time  t , then all the vehicles are synchronized andevolve at the desired speed.To achieve synchronization, the coordination variables  γ  i have to be exchanged among the quadrotors over a sup-porting communications network. Using tools from graphtheory, we can model the information flow as well as theconstraints imposed by the communicationtopology.We startby assuming that the  i th UAV communicates only with aneighboring set of vehicles, denoted by  N  i . We also assumethat the communication between two UAVs is bidirectionalwith no delays. The reader is referred to [20] for key conceptsand details on algebraic graph theory.Following the notation used in [21], we now let L ( t )  ∈ R n × n be the Laplacian of the graph  Γ( t ) .Let  Q  ∈ R ( n − 1) × n be a matrix such that  Q 1 n  = 0 , QQ ⊤ =  I  n − 1 , and define  ¯ L ( t ) =  QL ( t ) Q ⊤ ; it can be shown  that  ¯ L  ∈ R ( n − 1) × ( n − 1) has the same spectrum as the Lapla-cian  L ( t )  without the eigenvalue  λ 1  = 0 . Finally, we let  ¯ L ( t ) satisfy the persistency of excitation (PE) assumption:    t + T t ¯ L ( τ  ) dτ   ≥  µI  n − 1  .  (8)Next, letting  γ  ( t ) = [ γ  1 ( t ) ,...,γ  n ( t )] ⊤ and  ˙ γ  ( t ) = [˙ γ  1 ( t ) ,...,  ˙ γ  n ( t )] ⊤ , we define the coordinationerror vectors ξ  ( t ) =  Qγ  ( t )  ∈ R n − 1 ,  (9) z ( t ) = ˙ γ  ( t ) − 1 n  ∈ R n .  (10)From the definition of   Q  it follows that, if   ξ  ( t ) = 0 n ,then  γ  i  − γ  j  = 0 ,  ∀ i,j  ∈ { 1 ,...,n } . Note that convergenceof   z ( t )  to zero implies that the individual parameterizingvariables  γ  i ( t )  evolve at the desired rate 1.With the above notation, the coordination problem cannow be defined as: Problem 2 (Time-Coordination Problem):  Given a set of  n  3D desired trajectories  x d,i ( γ  i ) , design feedback controllaws for  ¨ γ  i  for all vehicles such that the coordinationerror vectors  ξ   and  z , defined in (9) and (10) respectively, converge exponentially to a neighborhood of zero with rateof convergence  λ TC   >  0 . C. Time-Coordinated 3D Path-Following Considering the PF and TC problems described above, wecan now define the combined TCPF control problem for afleet of quadrotor UAVs. Problem 3 (Time-Coordinated Path-Following Problem): Consider a set of   n  quadrotor UAVs and a set of  n  3D desired trajectories  x d,i ( γ  i ) . Assume the quadrotors cancommunicate over a communications network satisfying (8).Design feedback control laws  u i ( t )  and  ¨ γ  i ( t )  such that1) for each vehicle, the generalized PF error vec-tor  x PF,i ( t )  defined in (6) converges to a neighborhoodof zero;2) the coordination error vectors defined in (9) and (10) converge exponentially to zero.III. T IME -C OORDINATED  3D P ATH -F OLLOWING :M AIN  R ESULT To solve the TCPF problem, we let the evolution of   γ  i ( t ) be given by ¨ γ  i  =  − b (˙ γ  i  − 1) − a  j ∈N  i ( γ  i  − γ  j ) − d  ¯ α i ( x PF,i ) ,γ  i (0) = 0 ,  ˙ γ  i (0) = 1 , where  a ,  b ,  d  are positive coordination control gains, while ¯ α i ( x PF,i )  is defined as ¯ α i ( x PF,i ) =˙ x ⊤ d,i e x,i  ˙ x d,i  +  δ   +˙ x ⊤ d,i e v,i  ˙ x d,i  +  δ  , with  δ   being a positive design parameter. The dynamics of  γ  ( t )  can be written in compact form as ¨ γ   =  − bz − aLγ   − d ¯ α ( x PF  ) , γ  (0) = 0 n ,  ˙ γ  (0) = 1 n  , (11)where x PF   = [ x ⊤ PF, 1 ,...,x ⊤ PF,n ] ⊤ ∈ R 12 n , ¯ α ( x PF  ) = [¯ α 1 ( x PF, 1 ) ,...,  ¯ α n ( x PF,n )] ⊤ ∈ R n . Then, the Lemma below states the main result of thispaper:  Lemma 1:  Consider a set of   n  quadrotor UAVs and a setof   n  3D desired trajectories  x d,i ( γ  i ) . Given  n  PF algorithmssatisfying Assumption 1 and the coordination control lawdescribed in (11), then there exist control gains  a ,  b ,  d , and  δ  that solve the TCPF control problem 3. In particular, it can beshown that the vector  x TCPF   = [ x ⊤ PF  ,ξ  ⊤ ,z ⊤ ] ⊤ convergesexponentially fast to a neighborhood of zero with rate of convergence λ  = min( λ PF   , λ TC  ) ,  (12)where λ TC   < µ 2 T  (1 +  n 2 T  ) 2  ,  (13)and with domain of attraction D c    x TCPF   ∈ R 14 n − 1 :   x PF,i  ≤  r   .  (14) Proof.  An outline of the proof is given in the Appendix.   Remark 1:  The rate of convergence  λ PF   depends on theproperties of the adopted PF control law. If the PF controllaw has a rate of convergence greater than  λ TC  , then the rateof convergence of the TCPF system is equal to the rate of convergence of the TC algorithm.  Remark 2:  Note that the rate of convergence of the TC al-gorithm strictly depends on the quality of the communicationnetwork (parameters  µ  and  T  ).IV. I LLUSTRATIVE EXAMPLE : TCPF  WITH  AR.D RONES To test the performance of the algorithm presented in theprevious section, we adopted the flying robot architecturerealized by Parrot AR.Drone company. To this end, we firstdeveloped a PF algorithm that satisfies the conditions de-scribed in Section II and that uses the control input providedby the AR.Drone autopilot, which accepts control commandsfor linear velocity along the inertial vertical channel  ˙ z , Eulerangles  θ  and  φ  for the horizontal motion, and yaw rate  ˙ ψ .Next, we reformulate the PF problem presented in Section IIfor this particular platform, and derive a PF algorithm basedon simple linear control.  A. PF Error Dynamics For simplicity, we write separately the horizontal andvertical motions: x  = [(Π x ) ⊤ , e ⊤ 3  x ] ⊤ ,v  = [(Π v ) ⊤ , e ⊤ 3  v ] ⊤ ,
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