A Simulation Study for Practical Control of a Quadrotor

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1 A Siulation Study for Practical Control of a Quadrotor Jeongho Noh* and Yongkyu Song** *Graduate student, Ph.D. progra, ** Ph.D., Professor Departent of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang City, South Korea. Orcid ID: **Corresponding author Abstract In this paper a thorough siulation study is perfored and analyzed for practical control of a quadrotor. After a systeatic construction of a control syste, linear and nonlinear techniques are copared to see their perforances on position control of a quadrotor. In the siulation headinghold and heading-varying coands are tried and the feedback effects of Euler angular rates and body rates are also exained. inally an effective technique for long distance guidance is proposed. INTRODUCCTION Recently ore and ore quadrotors (or ulticopters) are adopted to various applications. To perfor safe and satisfactory issions a good flight controller should be designed. irst a brief survey on quadrotor control is described and then quadrotor dynaics and control structure are discussed. After the controller is set various situations are siulated and analyzed. In the literature several techniques such as linear, nonlinear, learning-based, hybrid schees are used to control quadrotors. As linear techniques PID or PD controller is ost used and Linear Quadratic or H schees often used. Those linear techniques, in fact, are good enough for ost of quadrotor control [1]. Bouabdallahet al. showed that both PID and Linear Quadratic techniques work well on the control of indoor quadrotor under soe disturbances[1,2]. Li et al. used H and Model Predictive Control to stabilize attitude and position control under weak wind[3]. Since quadrotor dynaics are inherently nonlinear, several nonlinear techniques are tried. Bouabdallah and Siegwart showed that backstepping and sliding-ode techniques can be used to control a quadrotor indoor under soe disturbances[4]. ang et al. cobined backstepping with an adaptive controller to deal with odel uncertainty and disturbance, reducing outputs overshooting, delay, and steady-state errors[5]. eedback linearization techniques first linearize the nonlinear parts of the quadrotor dynaics and adopt linear control for the linearized syste. Kendoul et al. showed that feedback linearization is able to control a quadrotor in several flight tests[6]. As a learning-based technique, Efe used neural networks to siplify PID controller, thus reducing coputation tie[7]. As seen in the above a flight controller based-on a singletechnique such as PID does not show a good perforance other than a given flight condition. In this paper thus a thorough siulation study is perfored and analyzed for practical control of a quadrotor. After a systeatic construction of a control syste, linear and nonlinear techniques are copared to see their perforances on position control of a quadrotor. In the siulations heading-hold and heading-varying coands are tried and the feedback effects of Euler angular rates and body rates are also exained. inally an effective technique for long distance guidance is proposed. QUADROTOR DYNAMICS In this section siple quadrotor dynaics are introduced using conventional Euler angles [8]. irst we consider 4 fraes shown in ig. 1. By rotating the Earth-fixed frae(x f, Y f, Z f, unit vectors i f, j f,z f ) through yaw angle ψ about Z f -axis, one gets 1-frae ( X 1, Y 1, Z 1, unit vectors i 1, j 1,z 1 ), and by rotating 1-frae through pitch angle θ about Y 1 -axis, one gets 2-frae ( X 2, Y 2, Z 2, unit vectors i 2, j 2,z 2 ), and finally by rotating 2-frae through roll angle φ about X 2 -axis, one gets 3-frae ( X 3, Y 3, Z 3, unit vectors i 3, j 3,z 3 ), which is equal to the body-fixed frae (X b, Y b, Z b, unit vectors i b, j b, z b ). igure 1: Relationships between fraes by Euler angles 11598

To derive quadrotor dynaics let s denote the forces and torques by each otor as i, τ i as shown in ig. 2. Then the forces driven by otors and gravity are gives as Thrust Gravity = ( )k b (2.

2 To derive quadrotor dynaics let s denote the forces and torques by each otor as i, τ i as shown in ig. 2. Then the forces driven by otors and gravity are gives as Thrust Gravity = ( )k b (2.1) g = gk f (2.2) where = f + r + b + l. Therefore the translational equations of otion are given as r = ( )k b + gk f (2.3) where r = x f i f + y f i f + z f k f is the position vector fro the origin of the Earth-fixed frae to the center of ass of the quadrotor. These equations can be used in giving heading coands. If the body-fixed frae for a quadrotor is a principal axes syste then the rotational equations of otion are given as Where p = (I y I z ) qr + 1 τ I x I φ x q = (I z I x ) I y rp + 1 I y τ θ (2.6) r = (I x I y ) pr + 1 τ I z I ψ z Rolling torque τ φ = l( l r ) (2.7) Pitching torque τ θ = l( f b ) (2.8) Yawing torque τ ψ = τ r + τ l τ f τ b (2.9) And the relationship between the angular velocities in the body frae (p, q, r) and the Euler rates are given as φ 1 sin φ tan θ cos φ tan θ p [ θ 0 cos φ sin φ ] [ q] (2.10) ψ 0 sin φ sec θ cos φ sec θ r NONLINEAR CONTROL O A QUADROTOR igure 2: orces and torques by otors Then the equations are resolved as x f = ( )(sin ψ sin φ + cos ψ sin θ cos φ) y f = ( )( cos ψ sin φ + sin ψ sin θ cos φ) (2.4) z f = g + ( ) cos θ cos φ These equations can be expressed as using the rotation between the Earth-fixed frae and 1-frae. x f 0 sin θ cos φ [ y f sin ψ cos ψ 0] [ sin φ ] = z f g + ( ) cos θ cos φ 0 x 1 [ sin ψ cos ψ 0] [ y 1 ] (2.5) z 1 or the control of a quadrotor we first consider the attitude control and the position control is then designed based on it. or the attitude control let s assue that the Euler angles and their rates are sall then one can assue that the following holds. φ p [ θ ] [ q] (2.11) ψ r Then the rotational equations of otion can be expressed as φ = (I y I z ) θ ψ + 1 τ I x I φ x θ = (I z I x ) I y ψ φ + 1 I y τ θ (2.12) ψ = (I x I y ) φ θ + 1 τ I z I ψ z or the attitude control one can use a stable PD controller as φ = (I y I z ) I x θ ψ + 1 I x τ φ = k d1 φ k p1 (φ φ d ) θ = (I z I x ) I y ψ φ + 1 I y τ θ = k d2 θ k p2 (θ θ d ) (2.13) ψ = (I x I y ) I z φ θ + 1 I z τ ψ = k d3 ψ k p3 (ψ ψ d ) 11599

3 Since (p, q, r) are available fro the gyros one can still use the to eliinate the nonlinear ters as τ φ I x { k d1 φ k p1 (φ φ d )} (I y I z )qr τ θ I y { k d2 θ k p2 (θ θ d )} (I z I x )rp (2.14) τ ψ I z { k d2 ψ k p3 (ψ ψ d )} (I x I y )pq Or one can easily feedback (p, q, r) if the aneuver is not so big as τ φ I x { k d1 p k p1 (φ φ d )} (I y I z )qr τ θ I y { k d2 q k p2 (θ θ d )} (I z I x )rp (2.15) τ ψ I z { k d2 r k p3 (ψ ψ d )} (I x I y )pq or position control we first assue that the heading is held in the North direction, i.e., ψ = 0. Then the equation (2.5) is given as x f = ( )(sin θ cos φ) igure 3: Structure for ulticopter control In case the heading coand is not held zero but varys it can be handled as. After rearranging the equation (2.4) [ x f y f sin ψ cos ψ ] [ use PD control (2.17) as sin θ cos φ sin φ ] (2.22) y f = ( ) sin φ (2.16) z f = g cos θ cos φ Now the position control is perfored by PD controller as x f = ( )(sinθ cos φ) k d4x f k p4 (x f x d ) u x y f = ( ) sin φ k d5 y f k p5 (y f y d ) u y (2.17) Or [ x f y f sin ψ cos ψ ] [ [ sin θ cos φ cos ψ sin φ sin ψ sin θ cos φ u x sin φ u ].(2.23) y sin ψ cos ψ ] [u x ] y = [ u x cos ψ + u y sin ψ u x sin ψ + u y cos ψ ] [u x u y ] (2.24) z f = g cos θ cos φ k d6z f k p6 (z f z d ) u z That is, feedback linearization is accoplished by = g u z cos φ cos θ (2.18) u y = g u z tan φ (2.19) cos θ u x = (u z g) tan θ (2.20) Thus the pitch and roll coands for position control can be given by φ d = tan 1 ( u y cos θ ), θ g u d = tan 1 ( u x ) (2.21) z u z g Then the systeatic controller block diagra is given as Thus the pitch and roll coands can be coputed as φ d = tan 1 ( u y cos θ ), θ g u d = tan 1 ( u x ). (2.25) z u z g This technique can be used for arbitrary heading coands. SIMULATION O ATTITUDE CONTROL In order to see the perforance of attitude controller we adopt PD controller of the equation (2.14) ipleenting MATLAB Siulink as (ig. 4.). or the siulation the paraeters in [9] are used. Paraeter Values [kg] I x [kg 2 ] I y [kg 2 ] I z [kg 2 ] 11600

4 kp=16, kd=4.8 kp=100, kd=16 1 phi, theta (rad) igure 4: Attitude control of ulticopter Tie (sec) In the siulation two sets of control gains were tried as. Gains Siulation1 Siulation 2 k p1, k p2, k p k d1, k d2, k d psi, (rad) kp=16, kd=4.8 kp=100, kd=16 Pitch and roll responses in ig. 5. are good enough and yaw response is also good though a little overshoot. With a set of big gains the response ties are within 0.5 seconds, which see fast enough for the translational aneuver of a quadrotor Tie (sec) igure 5: Siulation results of attitude control POSITION CONTROL WITH NONLINEAR SCHEME or the position control we need to ipleent the equations (2.17) and (2.21), which was done using MATLAT Siulink as in the following figure. igure 6. Position control block-diagra for ulticopter 11601

5 In the siulation (ig. 7.) the altitude and heading coands were set to zero and x d = 1, y d = 2. The position coands were accoplished in 1 second and the heading is kept zero. The next siulation was done with the heading coand of first view to the target, i.e., ψ co = atan(2,1). In this case the equations (2.18), (2.24), and (2.25) are ipleented because the heading is not held zero. The siulation (ig.8.) shows that the heading catches up the coand quickly and the pitching and rolling works together with the heading. LINEAR CONTROL In this section we try linear control techniques. irst we linearize the quadrotor dynaics (Eqn. (2.12)) assuing that the heading is kept zero and the tri is held for φ 0 = 0, θ 0 = 0. Then the rotational dynaics are linearized as igure 8: Position control with nonlinear schee (ψ co = atan(2,1), Euler rates feedback) ϕ = 1 I x τ φ θ = 1 I y τ θ (2.26) ψ = 1 I z τ ψ Then if we use PD control as in eqn. (2.17) x f = g θ k d4 x f k p4 ( x f x d ) u x y f = g φ k d5 y f k p5 ( y f y d ) u y (2.28) z f = k d6 z f k p6 ( z f z d ) u z the attitude angle and thrust coands are given as θ d = u x g φ d = u y g (2.29) d = u z In case the heading is not kept zero one ay use the equation (2.22) assuing that the variables are sall as igure 7: Position control with nonlinear schee (ψ co = 0, Euler rates feedback) And the translational equations of otion are linearized as x f = g θ y f = g φ (2.27) z f = 0 [ x f y f sin ψ cos ψ ] [ θ ] 0 φ = [ sin ψ cos ψ ] [ g θ g φ ] [ u x u ] (2.30) y That is, [ g θ g φ cos ψ sin ψ sin ψ cos ψ ] [ u x u ] (2.31) y Therefore the attitude coands are given as θ d = u x cos ψ+ u y sin ψ g φ d = u x sin ψ+ u y cos ψ g (2.32) The siulation results with linear control are siilar to those 11602

6 with nonlinear one as seen in ig. 9. the target sees unstable due to the sensitivity of the heading coputation near the target. igure 8: Position control with linear schee (ψ co = atan(2,1), Euler rates feedback) igure 101: Position control with linear schee (ψ co = atan(y, x), Euler rates feedback) The next siulation (ig. 10) shows the effects of p,q,r feedback rather than Euler rates φ, θ, ψ and soe overshoots are shown because in this case the exact 2 nd feedback syste is not ade. POSITION CONTROL VIA SPEED CONTROL If we try to send the quadrotor farther away (for exaple, x d = 20, y d = 30 ), big overshoots results in with previous gains (ig. 12.) because the closed-loop syste works as a lightly daped syste due to the liits of the attitude coands (our case φ d 20, θ d 20 ) If the gains are tuned again as in ig. 13., a good tracking can be ade. However, if the target is changed into a far distant position (for exaple, x d = 100, y d = 200 ), then big overshoots coe again even with the tuned gains. Thus another gain tuning should be tried. And on can see that it is not easy to tune a proper set of gains for each target position. Therefore it is necessary to find a consistent way to be applied to any target distance. or this purpose a position control via speed control is proposed. That is, a speed control is used until the quadrotor reaches the target within a prespecified radius and then the previously used PD control is applied to precisely send the quadrotor to the target. That is, using the equation (2.27), one can let the feedback keep the speed 10 /sec, for exaple, as in case ψ = 0. igure 90. Position control with linear schee (ψ co = atan(2,1), Angular rates feedback) Another siulation is tried to see the effect of varying heading coand which is given to direct the target all tie (ψ co = atan (y, x)). With this heading coand the final approach to 11603

7 [ x f y f sin ψ cos ψ ] [ g θ g φ ] We can use the following feedback = [ sin ψ cos ψ ] [ x 1 ] (2.35) y 1 [ x 1 g θ d k( x 1 10) ]. (2.36) y 1 g φ d 0 Thus once the flying gets started the quadrotor turns the heading to the target line, i.e., ψ c = tan 1 ( x d x f y d y f ). igure 112: Position control with big overshoots ((ψ co = 0, Euler rates feedback) and then the speed control (2.36) is operated until the quadrotor reaches near the target and then PD controller used. ig. 14 shows this siulation results. igure 14: Position control via speed control (ψ co = atan (y, x)) igure 123: Position control with gains tuned. ((ψ co = 0, Euler rates feedback) [ x f g θ d k( x f 10) ] (2.33) y f g φ d 0 In case the heading is not kept zero one ay use the coordinate transforation between the Earth frae and the 1- frae, that is, [ x f sin θcosφ y f sin ψ cos ψ ] [ ] sin φ = [ sin ψ cos ψ ] [x 1 ] (2.34) y 1 Or using the following linearized one One can still use different speed coand schedule according to the reaining distance. CONCLUSION In this work various situations were siulated for quadrotor control. irst the siulations show that nonlinear and linear control schees are siilar in perforance. As far as the heading coand is concerned constant heading coands, whether it is zero or not, are not a proble but real-tie varying heading coand to the target sees to bring the instability near the target due to the coputational sensitivity. or the attitude control it sees better to feedback the Euler rates rather than the body rates. inally in order to send the quadrotor to a target with arbitrary distance the proposed control strategy (position control via speed control) sees working well whereas the gains should be changed in the gain tuning ethod

8 ACKNOWLEDGEMENT This work was supported by Korea Institute of Planning and Evaluation for Technology in ood, Agriculture and orestry (IPET) through Advanced Production Technology Developent Progra, funded by Ministry of Agriculture, ood and Rural Affairs(MARA)(grant nuber : ) REERNNCES [1] Paul Edward Ian Pounds Design, construction and control of a large quadrotor icro air vehicle, Phd. thesis, Australian National University, [2] S. Bouabdallah, A. Noth, and R. Siegwart, PID vs LQ control techniques applied to an indoor icro quadrotor, In Intelligent Robots and Systes, (IROS 2004). Proceedings IEEE/RSJ International Conference on, volue 3, pages vol.3, doi: /IROS [3] Jun Li and Yuntang Li, Dynaic analysis and PID control for a quadrotor, In Mechatronics and Autoation (ICMA), 2011 International Conference on, pages , doi: /ICMA [4] S. Bouabdallah and R. Siegwart. Backstepping and sliding-ode techniques applied to an indoor icro quadrotor, In Robotics and Autoation, ICRA Proceedings of the 2005 IEEE International Conference on, pages , doi: /ROBOT [5] Zheng ang and Weinan Gao, Adaptive backstepping control of an indoor icroquadrotor, Research Journal of Applied Sciences, 4, [6] arid Kendoul, Zhenyu Yu, and Kenzo Nonai, Guidance and nonlinear control syste for autonoous flight of inirotorcraft unanned aerial vehicle, Journal of ield Robotics, 27(3): , [7] M.O. Efe, Neural network assisted coputationally siple pid control of a quadrotor uav, Industrial Inforatics, IEEE Transactions on, 7(2): , ISSN doi: /TII [8] R. C. Nelson, light Stability and Autoatic Control, 2nd Ed., McGraw-Hill, [9] A. Das, K. Subbarao, and. Lewis. Dynaic inversion with zero-dynaics stabilisation for quadrotor control. Control Theory & Applications, IET, 3(3): , [10] C. Balas, Modeling and Linear Control of a Quadrotor, MSc Thesis, Cranfield University,

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