Trajectory Control of a Quadrotor Using a Control Allocation Approach

Transcription

1 Trajectory Control of a Quadrotor Using a Control Allocation Approach Haad Zaki 1, Mustafa Unel 1,2 and Yildiray Yildiz 3 Abstract A quadrotor is an underactuated unanned aerial vehicle with four inputs to control the dynaics. Trajectory control of a quadrotor is a challenging task and usually tackled in a hierarchical fraework where desired/reference attitude angles are analytically deterined fro the desired coand signals, i.e. virtual controls, that control the positional dynaics of the quadrotor and the desired yaw angle is set to soe constant value. Although this ethod is relatively straightforward, it ay produce large and nonsooth reference angles which ust be saturated and low-pass filtered. In this work, we show that the deterination of desired attitude angles fro virtual controls can be viewed as a control allocation proble and it can be solved nuerically using nonlinear optiization where certain agnitude and rate constraints can be iposed on the desired attitude angles and the yaw angle need not be constant. Siulation results for both analytical and nuerical ethods have been presented and copared. Results for constrained optiization show that the flight perforance is quite satisfactory. I. INTRODUCTION Nowadays Unanned Aerial Vehicles (UAV) are utilized in any civilian and ilitary applications. Quadrotor is one of the ost used kinds of UAV because of its ability to coplete difficult tasks due to its vertical take off and landing (VTOL) capability. Quadrotor is a highly nonlinear syste, so trajectory tracking is a challenging task. In recent years, there have been a nuber of papers dealing with various probles inherent to the exploitation of quadrotors dynaics. Trajectory control of quadrotors has gained great interest in the unanned air vehicle counity. Dynaic odeling issues were addressed in [1] and [2]. Bouabdallah [1] used the linear odel and copared the results for PID controller with LQ controller. The results showed that hovering control was satisfactory for PID controller but for stability in the presence of disturbances, results were not so efficient. Different nonlinear control techniques such as sliding ode control, backstepping control and feedback linearization were used [3] [6]. The sliding ode approach provided average results to stabilize the attitude while the structural changes affected the control quality because of the high-frequency disturbances. Coparison of feedback linearization and adaptive sliding ode to estiate uncertainty such as ground effects and noisy conditions was presented in [7]. Madani [8] used the technique of dividing 1 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla Istanbul, Turkey 2 Integrated Manufacturing Technologies Research and Application Center, Sabanci University, Istanbul, Turkey {haadzaki,unel}@sabanciuniv.edu 3 Departent of Mechanical Engineering, Bilkent University, Cankaya, Ankara, Turkey yyildiz@bilkent.edu.tr quadrotor dynaics into any linearly connected subsystes and proposed a full-state backstepping and sliding ode control technique based on the Lyapunov stability theory for quadrotor to track the desired trajectories. Adaptive fuzzy backstepping technique was used [9], in which a control law for the odel was generated by fuzzy syste using backstepping approach. Robust trajectory tracking was presented which used cobination of integral backstepping and PID controller to stabilize the dynaics [1]. Global trajectory tracking control [11] was proposed without linear velocity easureents but efficiency in ters of position errors was not so good. Flatness-based control of a quadrotor via feedback linearization was perfored in [12]. H and odel predictive control were used to solve the proble of trajectory control [13]. Design and developent of a tilt-wing quadrotor and its robust control using a hierarchical structure were considered in [14] [16]. It consists of two parts: upper level control for positional dynaics which generates virtual controls and lower level control which provides tracking of desired attitude angles coputed fro virtual controls. Desired or reference angles were obtained through analytical forulas. More recently adaptive nonlinear hierarchical control of a tilt-wing quadrotor was developed in [17] and [18]. Chan also applied hierarchical control where otor speed control and attitude control were utilized [19]. Optiized trajectory planning algoriths had been applied using nonlinear optiization and results were shown for linear path where desired angles were obtained through analytical ethod [2]. In this work, a control allocation type technique is forulated on the nonlinear dynaics of the quadrotor to obtain optiu values for the desired attitude angles fro coand signals (virtual controls) developed for positional dynaics. Different fro an existing study [21] where control allocation approach is applied to distribute the total control effort aong real actuators optially, here the coand signals are utilized to get the values for desired attitude angles. With the help of hierarchical control, controller design is divided into two parts: position control and attitude control. Positional dynaics of the quadrotor is considered as the underdeterined part as it contains ore unknown variables than the equations. In control allocation, nonlinear constraint optiization is used to obtain required actuator inputs according to coand signals by solving an underdeterined syste [22]. In our approach, high level controller is designed to obtain the desired coand signals fro the positional dynaics. Nonlinear constrained optiization is used to get desired attitude angles (φ r, θ r, ψ r ) according to the coand signals. Sequential quadratic prograing (SQP ) algorith is used in nonlinear constraint optiiza-

2 tion. Low level controllers are ipleented to ensure that the attitude angles are adjusted according to desired trajectory. Analytical ethod for the calculation of reference angles is also applied and results are shown for the sae trajectories. Siulation results for both analytical and nuerical ethods have been presented for coparison to show the efficiency of the proposed technique. The organization of the paper is as follows: Section II outlines the dynaics of the quadrotor. Section III presents the control allocation and nonlinear optiization. Section IV presents the controller design. Section V and VI present the siulation results and the suary of the paper respectively. II. QUADROTOR DYNAMICS The quadrotor is a type of aerial robot which consists of four rotors separated in a cross structure. The crossed configuration shows robustness even the echanically linked otors are heavier than the frae [23]. Reduction gears are used to connect propellers to the otors. The otion of the quadrotor depends upon the direction of the rotation of the propellers of rotors. Front and rear propellers rotate counter clockwise, while the left and the right ones turn clockwise. There is no need of the tail rotor because of opposite pairs directions configuration unlike the standard helicopter structure. Fig. 1 shows the odel in a hovering state, where all the propellers have the sae speed. The otion of a Fig. 1: Quadrotor Dynaics (Adapted fro [24]) quadrotor is described by two frae of references, one of which is fixed and called inertial frae and the other one which is oving, called body frae. The vertical force w.r.t body-fixed frae is provided by thrust coand (U 1 ) which is used to raise (or lower) the quadrotor by increasing and decreasing the speed of the propellers equally respectively. Siilarly, increasing (or decreasing) the left propeller speed and decreasing (or increasing) the right one results into roll coand (U 2 ), which akes the quadrotor to turn due to the torque with respect to the x axis. Pitch coand (U 3 ) is very siilar to the roll but in this case, increase (or decrease) in the rear propeller speed and decrease (or increase) in the front one leads to a torque with respect to the y axis which akes the quadrotor to turn. In order to enable the quadrotor to turn along z axis, torque is provided by the yaw coand (U 4 ) which is provided by increasing (or decreasing) the front-rear propellers speed and by decreasing (or increasing) the speed of the left-right couple propellers. Detailed description of the quadrotor dynaics can be found in [23]. The quadrotor positional dynaics expressed in world frae and attitude dynaics expressed in body frae are given as Ẍ = (sin ψ sin φ + cos ψ sin θ cos φ) U 1 Ÿ = ( cos ψ sin φ + sin ψ sin θ cos φ) U 1 Z = g + (cos θ cos φ) U 1 ṗ = I yy I zz qr J prop + U 2 I xx I xx I xx q = I zz I xx pr + J prop + U 3 I yy I yy I yy ṙ = I xx I yy I zz pq + U 4 I zz where U 1, U 2, U 3 and U 4 are defined as U 1 = b(ω ω ω ω 2 4) U 2 = lb( ω ω 2 4) U 3 = lb( ω ω 2 3) U 4 = d( ω ω 2 2 ω ω 2 4) where ω 1, ω 2, ω 3 and ω 4 are the speed of front, right, rear and left propellers; l, b and d are the length of the rotor ar, thrust factor and drag factor, respectively. First three equations represent the positional dynaics and last three equations describe the orientation of the quadrotor in (1). The relationship between angular velocities of quadrotor in body frae (η) and the Euler rates in world frae (Ω) is given by η = E(Θ)Ω (3) where E(Θ) is the velocity transforation atrix defined as 1 sinφtanθ cosφtanθ E(Θ) = cosφ sinφ sinφ cosφ cosθ cosθ where Θ = [φ, θ, ψ] T, Ω = [ φ, θ, ψ] T and η = [p, q, r] T. III. CONTROL ALLOCATION AND NONLINEAR A. Control Allocation OPTIMIZATION Motion control is used to control the otion of the echanical systes. All echanical systes require control forces which are produced by the end effectors. Actuators are the devices which are used to control these forces. Soeties there are ore actuators than required. In that case syste becoes underdeterined due to ore nuber of unknown variables than the nuber of equations. Control allocation approach is used for such underdeterined systes. The purpose of the control allocation is to distribute the desired (1) (2)

3 control effort to all redundant actuators. In order to provide optial solution in the presence of coupling aong the actuators due to underdeterined nature, optiization techniques are used for this purpose. Control allocation is a hierarchical type algorith which consists of the following three parts [25]. High level controller is used to produce virtual coand inputs. Optiization is used to distribute the total virtual coand aong the actuators through linear and nonlinear optiization depending upon the cost function to be iniized and constraints. Low level controller is used to produce required force at each end effector with the help of actuators. Control allocation algorith is advantageous due to the following reasons. 1) It is used to solve probles like actuator position and rate saturation. 2) It is used to produce fault tolerant syste in case of failure of any actuators. 3) It is used to produce size and cost effective echanical design by choosing the optial set of the actuators rather than the sall nuber of the actuators. In this work positional dynaics of the quadrotor is exploited to apply the control allocation approach. If we look at the positional dynaics of the quadrotor in (1), it consists of three equations and four unknown variables (φ, θ, ψ, U 1 ), so it can be considered as underdeterined syste. In [22] control allocation approach had been used to solve underdeterined syste where nonlinear optiization proble had been forulated. As underdeterined part of the quadrotor consists of nonlinear equations, nonlinear optiization is required to get the optial solution. The purpose of the control allocation is to generate coand inputs that ust be produced jointly by all actuators which in our case are φ, θ, ψ and U 1. Optiization proble along with the nonlinear and linear constraints is forulated as J(ζ) = 1 2 (st s) (4) where J(ζ) is the cost function to be iniized and s is a slack variable. In our proble ζ is given by: ζ = [φ, θ, ψ, U 1 ] (5) Cost function is iniized subject to the following nonlinear and linear constraints s = τ B(ζ) (6) ζ in ζ ζ ax (7) ζ C (8) where τ is the desired coand inputs that is provided by the high level controller of the hierarchical control. τ and B(ζ) are given as τ = [Ẍ Ÿ Z] T (9) (sin ψ sin φ + cos ψ sin θ cos φ) U1 B(ζ) = ( cos ψ sin φ + sin ψ sin θ cos φ) U1 g + (cos θ cos φ) U1 (1) ζ in and the ζ ax are the constrained range for ζ. Actuator rate constraint ζ is included in the forulation by liiting the change in the control inputs ζ fro the last sapling instant to soe constant C. B. Nonlinear Optiization Optiization deals with finding the feasible solution for n variables to iniize or axiize any function. Nonlinear prograing is a atheatical tool, used to iniize the cost function subject to linear and nonlinear constraints. Feasible regions show the set of optiized variables which lie in the range of constraints. In unconstrained optiization feasible solutions converge to soe finite values. In constrained optiization, nonlinear proble is converted into any subspaces and then different algoriths are applied to obtain optiu solution according to constraints. Sequential quadratic prograing (SQP ) is a nonlinear optiization tool which is one of the ost effective iterative ethods. Advantage of using SQP is to find good initial point for feasible solution. Active set ethod is used in SQP. It converts nonlinear proble into the subspace based on Lagrange approxiation. Lagrange function is given by: L(ζ, λ) = J(ζ) + λ i.g i (ζ) (11) i=1 where g i (ζ) is a gradient and λ i is a Lagrange ultiplier. Detailed description of SQP is given in [26], briefly it consists of following three parts: Hessian Matrix part uses positive definite quasi Newton optiization ethod for Lagrange function using Broyede- Flecher-Golfarb-Shanon(BF GS) [26] ethod at every instant k. where H k+1 = H k + q kq T k q T k l k i=1 H kl k l T k H k l T k H kl k (12) l k = ζ k+1 ζ k (13) q k = J(ζ k+1 ) + λ i.g i (ζ) ( J(ζ k ) + λ i.g i (ζ)) i=1 This equation is used to neutralize the gradients between cost function and active constraints, so for balancing, the agnitudes of the Lagrange ultipliers are necessary. Positive definiteness of the Hessian atrix is ensured providing q T k l k is positive at each iteration. When q T k l k is not positive, q k is odified on eleent by eleent basis so that q k is positive [26]. Quadratic Prograing Solution part consists of two steps: first step gives the feasible point and the second step produces iterative process of feasible points to converge

4 within the constraints. During this part of nonlinear prograing, proble is converted to subprobles and then QP is used to solve, which is forulated as in 1 2 dt H k d + J(ζ k ) T d g i (ζ k ) T d + g i (ζ k ) =, i = 1, 2,..., e g i (ζ k ) T d + g i (ζ k ), i = e + 1,..., The following QP for is used at every iteration. (14) in q(d) = 1 2 dt Hd + c T d (15) subject to following equality and inequality constraints. A i d = b i, i = 1, 2,..., e A i d b i, i = e + 1,..., (16) The solutions of quadratic prograing will give d k which is feasible region search direction. Active constraints are updated at every iteration to for a basis for new search direction d k. Optial Line Search part of the prograing is used to produce new iteration using vector d k. ζ k+1 = ζ k + αd k (17) where α is the step length paraeter, which ensures that ζ k+1 reains in the feasible region. IV. CONTROLLER DESIGN Hierarchical control is used as control approach for controller design. PID controllers are used as high level controller to provide virtual coand inputs which are used to get reference angles (φ r, θ r, ψ r ) with the help of nonlinear optiization. Acquired reference angles are used to produce desired trajectory with the help of the low level controller as shown in closed loop control structure in Fig. 2. A. Position Control Fro the positional dynaics of quadrotor in (1) µ X = Ẍ = (sin ψ sin φ + cos ψ sin θ cos φ)u 1 µ Y = Ÿ = ( cos ψ sin φ + sin ψ sin θ cos φ)u 1 µ Z = Z = g + (cos θ cos φ) U 1 Errors are defined as e X = X r X e Y = Y r Y e Z = Z r Z Error dynaics can be forulated as ė X = Ẋr Ẋ ë X = Ẍr Ẍ = Ẍr µ X (18) (19) ė Y = Ẏr Ẏ ë Y = Ÿr Ÿ = Ÿr µ Y (2) ė Z = Żr Ż ë Z = Z r Z = Z r µ Z µ X, µ Y and µ Z consist of feed forward and feedback ters, which are defined as µ X = Ẍr + K p,x e X + K d,x ė X + K i,x e X dt µ Y = Ÿr + K p,y e Y + K d,y ė Y + K i,y e Y dt (21) µ Z = Z r + K p,z e Z + K d,z ė Z + K i,z e Z dt In [15] analytical ethod was used to calculate the desired attitude angles of the aerial vehicle fro desired acceleration vector (τ) by assuing yaw angle (ψ) to be soe fixed value (ψ ). Desired angles are calculate as U 1 = µ 2 X + µ2 Y + (µ Z + g) 2 (22) B. Attitude Control θ r = arcsin( µ X U 1 ) (23) µ Y φ r = arcsin( ) (24) U 1 cos θ r In order to develop controllers for attitude control, fro (1) attitude dynaics can be linearized around hover conditions i.e. φ, θ, ψ and φ, θ, ψ. Angular accelerations in body and world fraes will be approxiately equal after linearization i.e. ṗ φ, q θ, ṙ ψ [15]. Resulting attitude dynaics is given by: Errors are defined as φ = U 2 I xx, θ = U 3 I yy, ψ = U 4 I zz (25) e φ = φ r φ, e θ = θ r θ, e ψ = ψ r ψ (26) Siilarly error dynaics are defined as ė φ = φ r φ ë φ = φ r φ ė θ = θ r θ ë θ = θ r θ ė ψ = ψ r ψ ë ψ = ψ r ψ Attitude controllers are designed as U 2 = I xx ( φ r + K p,φ e φ + K d,φ ė φ + K i,φ U 3 = I yy ( θ r + K p,θ e θ + K d,θ ė θ + K i,θ U 4 = I zz ( ψ r + K p,ψ e ψ + K d,ψ ė ψ + K i,ψ V. SIMULATION RESULTS ) e φ dt ) (27) e θ dt (28) ) e ψ dt A control allocation type approach has been ipleented where reference angles are generated through nuerical ethod using nonlinear optiization. Analytical ethod for reference angles calculation is also considered for coparison. For siulation, the following two scenarios are taken into account. 1) During the first scenario yaw angle (ψ) is fixed to a constant value (ψ ) for both nuerical and analytical ethods.

5 ψ ψ ψ X r,y r,z r - Σ High level Controller τ Non-linear Optiization φ,,ψ - Σ Low level Controller Quadrotor φ,,ψ X,Y,Z Fig. 2: Closed Loop Control Syste 2) During the second scenario, nuerical ethod has been taken into account and yaw angle (ψ) is not fixed and constraints have been put on all reference angles. For nonlinear optiization, stopping criterion specifications are suarized in the TABLE I. TABLE I: Nonlinear Optiization Specifications Specifications Range Max Iterations 4 F unction T olerance 1e-6 Constraint T olerance 1e-6 Max F unction Evaluations 1 F irst Order Optiality Measure 1e-6 For trajectory it is assued that quadrotor takesoff vertically fro the spot along the z axis, then hovers at desired altitude, takes a straight flight along the x axis, oves in a circular loop and lands at the desired spot after taking soe ore straight flight. Trajectory has been shown in Figure 3. x() y() z() Fig. 4: Cartesian positions of the quadrotor vs Tie (Desired=green, Analytical=blue, Nuerical=red.) z() e y () e x () y() x() Fig. 3: Trajectory Tracking (Desired=red, Actual=blue ) During the first scenario of the siulation yaw angle (ψ) is taken to be fixed at 3.5 for the calculation of reference angles through analytical and nuerical ethod. Cartesian positions of the quadrotor during the trajectory for both ethods have been shown in Figure 4 which shows that quadrotor successfully tracked the desired trajectory for both ethods. Position errors in Figure 5 show the efficiency of tracking for the nuerical ethod. It has been observed that cartesian e z () Fig. 5: Position Errors positions of the quadrotor deonstrate siilar behavior during the circular path but during the transition fro hovering

6 TABLE II: Euler Angles and Total Thrust (U 1 ) Bound Values Euler Angles Range Rate φ -14 to 14 (deg) 8.5 (deg/saple) θ -14 to 14 (deg) 8.5 (deg/saple) ψ -7 to 7 (deg) 5.5 (deg/saple) U 1 to 2 1 state to straight flight, nuerical ethod showed soother results than analytical ethod of finding reference Euler angles which can be seen during 1 to 2 seconds and after 85 seconds in Figure 6. x() y() z() Fig. 7: Cartesian positions of the quadrotor vs Tie for bounded angles optiization (Desired=solid, Actual=dashed) Fig. 6: Euler Angles (Analytical=solid, Nuerical=dashed) In the second scenario, yaw angle (ψ) is not taken to be fixed; instead constraints have been put on all reference angles which are given in TABLE II. Siulation results for trajectory curves have been shown for constrained optiization in Figure 7, which shows that quadrotor successfully tracked the trajectory. In Figure 8 coparison of Euler angles for constrained optiization in the first scenario when the yaw angle (ψ) is fixed and for constraint optiization for range of values for all reference angles including yaw angle (ψ), shows that Euler angles not only change in a sooth fashion but also reain within the bounds for the sae trajectory. In Figure 9 control efforts for nuerical ethod when yaw angle (ψ) is not fixed are plotted and it is clear that each one of the stays withing phsyical bounds. VI. CONCLUSION AND FUTURE WORK In this paper, a control allocation like ethod using nonlinear constraint optiization is ipleented on the nonlinear dynaics of the quadrotor, which consists of high level control and low level control. High level controller is used to get necessary desired generalized coand signals. Nonlinear constraint optiization is used at every instant to get the Fig. 8: Euler Angles (For fixed psi (ψ )=dashed, for bounded values of psi (ψ)=solid)) reference angles for the desired trajectory tracking according to generalized coands. Low level controllers are used for attitude control. Hierarchical control allows us to design separately the controllers for position and attitude for highly coupled nonlinear dynaics of quadrotor. For trajectory, it has been considered that vehicle oves in a circular loop after hovering at the desired altitude. For trajectory tracking, two scenarios have been taken into account based on yaw

7 U 2 U 3 U U 12 Fig. 9: Control Efforts angle (ψ). Siulation results for both scenarios have been shown for both analytical and nuerical ethods of finding reference angles for coparison. Fro results it has been inferred that both ethods provided successful trajectory tracking but nuerical ethod showed soother changes in Euler angles than the analytical ethod. Siulation results have also been presented for the second scenario when the yaw angle is not taken as fixed; instead bounded constraints have been considered for optiization. Results showed that quadrotor successfully tracked the trajectory with the desired attitude angles reaining within the bounds for the whole tie period. As future work, we will work on aking the proposed technique real-tie and then test it on a physical quadrotor. REFERENCES [1] S. Bouabdallah, P. Murrieri, and R. 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