Abstract—In thisarticle, the problem of controlling a team of Quadrotors that cooperativelygrasps and transports a common payload is considered. In this regard, a novel decentralizednonlinear robust optimal controller is designed based on Second Order SlidingMode (SOSMC) approach which is capable of facing external disturbances. Themethod combines control vectors, calculated for each Quadrotor usingMoore-Penrose theory, with individual control vectors aiming at cooperativegrasping of an object.
A second order sliding mode controller is used to obtainthe individual control vectors while using a nonlinear observer based onExtended Kalman-Bucy Filter (EKBF) to estimate the unmeasured states. This approachleads to robustness against model uncertainties along with high flexibility indesigning the control parameters to have an optimal solution for the nonlineardynamic of the system. Design of the controller is based on Lyapunov techniqueswhich can provide the stability of the end-effecter during tracking of thedesired trajectory. Finally, simulation results are given to illustrate theeffectiveness of the proposed method.Keywords—Quadrotor, Second Order Sliding Mode Control, ExtendedKalman-Bucy Filter, Cooperative Decentralized Control, External Disturbances1. IntroductionThe cooperative control of multiplevehicle systems despite its wide range of practical applications requirestackling important theoretical and practical challenges which have attractedmany researchers in recent years.
Some research works have focused oncentralized and leader-follower schemes to create interaction mechanismsbetween the object and cooperative Quadrotors 1-6. Nevertheless, these methodssuffer from several disadvantages including high computational complexity ofcentralized methods in large-scale systems and possibility of disappearing thegroup formation in leader-follower strategy due to not receiving the positionof leader by the followers. Inthis regard, there is a plethora of research in cooperative multi-robotcontroller design based on decentralized control methods which can solve asignificant number of problems in cooperative control strategies and benefitalso from the advantages such as decreased number of sensors and fastperformance.
In 7, the problem of cooperation by a team of ground robots isaddressed under quasi-static assumptions based on decentralize controllers consideringa unique solution to robot and object motion. Transporting a payload by aerialmanipulation using cables based on decentralized control is studied in 8,9.In these papers, the focus is on finding robot configurations and ensuringstatic equilibrium of the payload at a desired position while taking intoaccount the constraints on the tension. In addition, cooperative aerial towinghas been studied in 10. 11 employs bilinear matrix inequalities to presentoptimal solutions for decentralized nonlinear multi-agent systems. Despite thenumerous advantages of decentralized methods there are still some criticalissues related to the performance of the control system in presence ofdisturbances and uncertainties. Sliding mode control(SMC) is a well-known control strategy robust against disturbances and dynamicuncertainties 12-16. In this method, the chattering phenomenon can poseproblems; therefore, it should be avoided or at least reduced.
A high ordersliding mode concept can effectively reduce the chattering while keeping theinvariant characteristics in the sliding mode. This control strategy shifts theswitching function into higher order, whereas in standard SMC the switchingfunction is in the first order. This high order (second order) SMC has beenapplied in a quadcopter 17-18. In 19, a robust second order sliding modecontroller was proposed for the attitude stabilization of a four-rotorhelicopter. This controller was able to overcome the chattering phenomena inclassical sliding mode control while preserving the invariance property ofsliding mode. In addition, using an equivalent approach, the performance ofcontrol system in 20 was improved by using a second-order sliding modecontrol (2-SMC) for second-order uncertain plants.
An adaptive second ordersliding mode controller (SOSMC) with a nonlinear sliding surface was introducedin 21 and a linear switching surface was proposed for similar underactuatedsystems in 22. Wind as a disturbance source has also been considered in theflight process of the quadrotor to demonstrate the robustness of the controlalgorithm 23,24. In this paper, a noveldecentralized robust optimal controller is designed for a group of UAVs in aformation which enables manipulation of a common payload in three dimensions.Our cooperative control algorithm is comprised of two parts.
In the first part,fundamental control vectors, which guarantee the optimal performance of thecoupled system, are determined using Moore-Penrose theory. In the second part,individual control vectors are designed for each Quadrotor which can guaranteethe robustness of the coupled system against uncertainties and disturbances.Moreover, a combination of second order sliding mode controller (SOSMC) with an optimal cooperative algorithm is employed to control all of the Quadrotorssuch that a soft touching occurs between the object being carried and thecooperative Quadrotors.
A nonlinear filter equivalent to well-known Extended Kalman-Bucy filter (EKBF) in a continuous-time stochasticsystem has also been developed in our work aiming toimprove the estimation accuracy and robustness of the system. The proposed controller successfully rejects the externaldisturbances (e.g. wind effect) for cooperative Quadrotors. The advantages ofusing this approach are ranging from simple implementation and flexibleparameter learning to design criteria customization. The simulationresults show the effectiveness and superior performance of the proposed controlstrategy for the cooperative Quadrotors grasping and transporting a commonpayload in various maneuvers.The subsequent partsof this paper are as follows: at first developing a model for asingle Quadrotor and modeling of a team of UAV’s rigidly attached to a payloadis discussed in Sec.
2. Sec.3 proposes cooperative Control Algorithm laws defined with respect to thepayload that stabilizes the payload along three dimensional trajectories basedon a SOSMC incorporated with optimal Moore-Penrose theory. Wereview and analyze experimental study with teams of Quadrotors cooperativelygrasping, stabilizing, and transporting payloads in Sec.4.
It includespoint-to-point path planning and trajectory tracking. Finally, Sec.5 concludes the paper with some future improvementsuggestions.2. Mathematical Modeling and Coordinate Reference FrameFig.
1 shows the configuration of our quadrotor system that can fly inall directions and has no limit on maneuver. Two main reference frames areconsidered in order to analyze the dynamic: model frame, , and the body-fixed reference frame, . In this nomenclature, ? and are the position and orientation vectors in reference .The three Euler angels are named roll angle ,pitch angle and yaw angle . is the thrust forcedproduced by four rotors. The full Quadrotordynamic model with the motions asoutcome of a pitch, roll and yaw rotation is 25: (1) The model (1)can be rewritten in a state-space form, , byintroducing which is state vectorof the system: (2) Finally, thecontrol inputs are calculated as: (3) In this equation and represent thethrust and drag coefficient, respectively. The input is related tothe total thrust; while the inputs are related tothe rotations of the Quadrotor and are externaldisturbances, denotes theinertia of the z-axis and is the overallresidual propeller angular speed. Finally,the equation of motion for thecoupled system is obtained as follows 25: (4) where is the net body force and , , are the body moments in the direction.
Also, is the total force from each Quadrotor and , , are moments about each of the Quadrotor’s bodyframe axes.3. The Cooperative ControlAlgorithm In this section, theperfect state cooperative guidance controller of chasers and target UAV presentedin 26 will be developed to achieve a decentralized control strategy with superior results regarding state estimation. The proposed control algorithm includes three parts. In the first part, thecontrol fundamental vectors for n Quadrotors using the Moore-Penrose and EKBF methods aredetermined. Thesecond part is composed of determining individual control vectors by SecondOrder Sliding Mode Controller (SOSMC). Finally, in last part, the cooperativecontrol algorithm can be extracted by combining the individual control vectorsand the control fundamental vectors.3.
1. Control Fundamental Vectors ofCooperative Control StrategyAfter determining the set of all forcevectors of the coupled system in (4), four equations each one with 4n unknown variables canbe described as: (5) where includes 4 control input vectors for each Quadrotor: (6) and is a constant vector containing the states of translational androtational subsystems, and is obtained from EKBF in Theorem 1.Theorem 1: A Nonlinear Observer Based on EKBF 27, 28Consider a nonlinear stochastic system with uncorrelated Gaussian processand measurement noise as follows: (7) where is state vectorand is themeasurements vector. denotes the statenoise and the vector represents themeasurement noise.
Then states of the system can be estimated by an optimal(minimum variance) recursive algorithm designed by determining the filteringand prediction equations. In the filtering step, the mean at a given instantof time is calculated bystate estimation: (8) in which the initial conditions are: (9) and estimation of error covariance matrix is determined as: (10) and Kalman filter gain is computed as: (11) where represents thecovariance matrix of the sensor noise. In prediction step, we have: (12) (13) where, is the Jacobeanmatrix of evaluated at .
Now, consider the following cost function to minimize the controlinputs 26: (14) in which are the weight factors. By using the mathematical theory of Moore-Penrose inverse, an optimal vector to reach thedesired values can be written as: (24) where: , (15) and thecolumns of the matrix are the control fundamental vectors . Thus,the optimal control input vector can be rewritten as follows: (16) Finally,after calculating the four fundamental control vectors, the cooperativecontrol vectors for n collaboration Quadrotors is obtained as follows: (17) in which are the control fundamental vectors includingthe total force, roll, pitch and yow moments generated for the coupled system of n Quadrotors.3.2. Individual Control Vectors ofCooperative Control StrategyNow, to complete the cooperative control algorithm, individual control vectors aredetermined by using nonlinear second order sliding mode controller (SOSMC) strategy in order to generate arobust solution. 3.
2.1 Second Order Sliding Mode Control of Quadrotors In this paper,determining individualcontrol vectors of Cooperative Control Strategy based on second order slidingmode techniques is desired to preform asymptoticposition and attitude tracking of the quadrotors. Using tracking errors ofstate variable(s), a switching sliding surface is considered for the wholesubsystem (rotational or transitional subsystems). The sliding manifolds based on second order sliding mode controlare chosen asfollows 20, 21: (18) where the coefficients . The time derivatives of the sliding manifolds of rotational subsystem areobtained; (19) By making and the corresponding control laws are designed; (20) where the coefficients of the exponential approach laws Now for transitional subsystem, theobjective is to guarantee the state variables converge to the desired values , respectively. The time derivatives of the sliding manifolds of rotationalsubsystem are obtained; (21) By making and the corresponding control laws are designed; (22) where the coefficients of the exponential approach laws .
In addition, the terms are: (23) To synthesize a stabilizing control law by secondorder sliding mode, the sufficient condition for thestability of the system must be verified. Time derivative of satisfying where ? is the positive value . So, a Lyapunov function can be written as , that one can describe as 29: (24) The chosenlaw for the attractive surface is the time derivative of (25) Therefore, according to (25) the time derivative of final Lyapunov function is as: (26) Thus, under the control laws, allthe system state trajectories can reach, and thereafter, stay on thecorresponding sliding surfaces, respectively.
Finally using (26) the input controlvectors extracted from second sliding mode approach can be expressed asfollows: (27) Remark 1. Note thatsignum function which appears in second order sliding mode controllers and filterscan cause chattering phenomena, or high-frequency oscillations of controlvariables. This problem can be avoided by replacing discontinuous signum functionwith appropriate continuous approximation, like; .Finallyindividual control vectors are generated by SOSMC method which has beenexplained in the (28)and they are represented by parameters that are equivalent of . Thus the individual control vectors arederived as follows: (29) Inthese equations is the state vector. Therefore: (30) and: (31) 3.
3. Cooperative Control VectorsNow, from (17) the cooperative control vectors for eachindependently designed Quadrotor can be described as follows: (32)