A Simulation Study for Practical Control of a Quadrotor

In this paper a thorough simulation study is performed and analyzed for practical control of a quadrotor. After a systematic construction of a control system, linear and nonlinear techniques are compared to see their performances on position control of a quadrotor. In the simulation headinghold and heading-varying commands are tried and the feedback effects of Euler angular rates and body rates are also examined. Finally an effective technique for long distance guidance is proposed. INTRODUCCTION Recently more and more quadrotors (or multicopters) are adopted to various applications. To perform safe and satisfactory missions a good flight controller should be designed. First a brief survey on quadrotor control is described and then quadrotor dynamics and control structure are discussed. After the controller is set various situations are simulated and analyzed. In the literature several techniques such as linear, nonlinear, learning-based, hybrid schemes are used to control quadrotors. As linear techniques PID or PD controller is most used and Linear Quadratic or H∞ schemes often used. Those linear techniques, in fact, are good enough for most of quadrotor control [1]. Bouabdallahet al. showed that both PID and Linear Quadratic techniques work well on the control of indoor quadrotor under some disturbances[1,2]. Li et al. used H∞ and Model Predictive Control to stabilize attitude and position control under weak wind[3]. Since quadrotor dynamics are inherently nonlinear, several nonlinear techniques are tried. Bouabdallah and Siegwart showed that backstepping and sliding-mode techniques can be used to control a quadrotor indoor under some disturbances[4]. Fang et al. combined backstepping with an adaptive controller to deal with model uncertainty and disturbance, reducing outputs’ overshooting, delay, and steady-state errors[5]. Feedback linearization techniques first linearize the nonlinear parts of the quadrotor dynamics and adopt linear control for the linearized system. 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 simplify PID controller, thus reducing computation time[7]. As seen in the above a flight controller based-on a singletechnique such as PID does not show a good performance other than a given flight condition. In this paper thus a thorough simulation study is performed and analyzed for practical control of a quadrotor. After a systematic construction of a control system, linear and nonlinear techniques are compared to see their performances on position control of a quadrotor. In the simulations heading-hold and heading-varying commands are tried and the feedback effects of Euler angular rates and body rates are also examined. Finally an effective technique for long distance guidance is proposed. QUADROTOR DYNAMICS In this section simple quadrotor dynamics are introduced using conventional Euler angles [8]. First we consider 4 frames shown in Fig. 1. By rotating the Earth-fixed frame(Xf , Yf , Zf, unit vectors if , ⃗⃗ ⃗ jf ⃗⃗⃗ ,zf ⃗⃗ ⃗) through yaw angle ψ about Zf -axis, one gets 1-frame (X1, Y1, Z1 , unit vectors i1, ⃗⃗ ⃗ j1 ⃗⃗ ,z1 ⃗⃗ ⃗ ), and by rotating 1-frame through pitch angle θ about Y1 -axis, one gets 2-frame (X2, Y2, Z2 , unit vectors i2, ⃗⃗⃗⃗ j2 ⃗⃗ ,z2 ⃗⃗ ⃗), and finally by rotating 2-frame through roll angle φ about X2 -axis, one gets 3-frame (X3, Y3, Z3 , unit vectors i3, ⃗⃗⃗⃗ j3 ⃗⃗ ,z3 ⃗⃗ ⃗), which is equal to the body-fixed frame (Xb , Yb , Zb, unit vectors ib , ⃗⃗⃗⃗ jb ⃗⃗⃗ , zb ⃗⃗ ⃗). Figure 1: Relationships between frames by Euler angles International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 21 (2017) pp. 11598-11605 © Research India Publications. http://www.ripublication.com 11599 To derive quadrotor dynamics let’s denote the forces and torques by each motor as Fi, τi as shown in Fig. 2. Then the forces driven by motors and gravity are gives as Thrust F = (−F)kb ⃗⃗⃗⃗ (2.1) Gravity mg = mgkf ⃗⃗⃗⃗ (2.2) where F = Ff + Fr + Fb + Fl. Therefore the translational equations of motion are given as mr ̈ = (−F)kb ⃗⃗⃗⃗ + mgkf ⃗⃗⃗⃗ (2.3) where r = xfif ⃗⃗ + yfif ⃗⃗ + zfkf ⃗⃗⃗⃗ is the position vector from the origin of the Earth-fixed frame to the center of mass of the quadrotor. Figure 2: Forces and torques by motors Then the equations are resolved as follows mx?̈? = (−F)(sinψ sinφ + cosψ sin θ cosφ) my?̈? = (−F)(− cosψ sinφ + sinψ sin θ cosφ) (2.4) mz?̈? = mg + (−F) cos θ cosφ These equations can be expressed as follows using the rotation between the Earth-fixed frame and 1-frame. [ mx?̈? my?̈? mz?̈? ] = [ cosψ − sinψ 0 sinψ cosψ 0 0 0 1 ] [ −F sin θ cosφ F sin φ mg + (−F) cos θ cosφ ] = [ cosψ − sin ψ 0 sinψ cosψ 0 0 0 1 ] [ mx1̈ my1̈ mz1̈ ] (2.5) These equations can be used in giving heading commands. If the body-fixed frame for a quadrotor is a principal axes system then the rotational equations of motion are given as follows ṗ = (Iy−Iz) Ix qr + 1 Ix τφ ?̇? = (Iz−Ix) Iy rp + 1 Iy τθ (2.6) ?̇? = (Ix−Iy) Iz pr + 1 Iz τψ Where Rolling torque τφ = l(Fl − Fr) (2.7) Pitching torque τθ = l(Ff − Fb) (2.8) Yawing torque τψ = τr + τl − τf − τb (2.9) And the relationship between the angular velocities in the body frame (p, q, r) and the Euler rates are given as [ ?̇? ?̇? ?̇? ] = [ 1 sin φ tan θ cosφ tan θ 0 cosφ − sinφ 0 sin φ sec θ cosφ sec θ ] [ p q r ] (2.10) NONLINEAR CONTROL OF A QUADROTOR For the control of a quadrotor we first consider the attitude control and the position control is then designed based on it. For the attitude control let’s assume that the Euler angles and their rates are small then one can assume that the following holds. [ ?̇? ?̇? ?̇? ] ≈ [ p q r ] (2.11) Then the rotational equations of motion can be expressed as ?̈? = (Iy−Iz) Ix ?̇??̇? + 1 Ix τφ ?̈? = (Iz−Ix) Iy ?̇??̇? + 1 Iy τθ (2.12) ?̈? = (Ix−Iy) Iz ?̇??̇? + 1 Iz τψ For the attitude control one can use a stable PD controller as follows ?̈? = (Iy−Iz) Ix ?̇??̇? + 1 Ix τφ = −kd1?̇? − kp1(φ − φd) ?̈? = (Iz−Ix) Iy ?̇??̇? + 1 Iy τθ = −kd2?̇? − kp2(θ − θd) (2.13) ?̈? = (Ix − Iy) Iz ?̇??̇? + 1 Iz τψ = −kd3?̇? − kp3(ψ − ψd) International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 21 (2017) pp. 11598-11605 © Research India Publications. http://www.ripublication.com 11600 Since (p, q, r) are available from the gyros one can still use them to eliminate the nonlinear terms as follows τφ ≡ Ix{−kd1?̇? − kp1(φ − φd)} − (Iy − Iz)qr τθ ≡ Iy{−kd2?̇? − kp2(θ − θd)} − (Iz − Ix)rp (2.14) τψ ≡ Iz{−kd2?̇? − kp3(ψ − ψd)} − (Ix − Iy)pq Or one can easily feedback (p, q, r) if the maneuver is not so big as follows τφ ≡ Ix{−kd1p − kp1(φ − φd)} − (Iy − Iz)qr τθ ≡ Iy{−kd2q − kp2(θ − θd)} − (Iz − Ix)rp (2.15) τψ ≡ Iz{−kd2r − kp3(ψ − ψd)} − (Ix − Iy)pq For position control we first assume that the heading is held in the North direction, i.e., ψ = 0. Then the equation (2.5) is given as follows x?̈? = (− F m )(sin θ cosφ) y?̈? = ( F m ) sinφ (2.16) z?̈? = g − F m cos θ cosφ Now the position control is performed by PD controller as follows x?̈? = (− F m )(sinθ cosφ) ≡ −kd4x?̇? − kp4(xf − xd) ≡ ux y?̈? = ( F m ) sinφ ≡ −kd5 y?̇? − kp5(yf − yd) ≡ uy (2.17) z?̈? = g − F m cos θ cosφ ≡ −kd6z?̇? − kp6(zf − zd) ≡ uz That is, feedback linearization is accomplished by F m = g−uz cosφ cosθ (2.18) uy = g−uz cos θ tanφ (2.19) ux = (uz − g) tan θ (2.20) Thus the pitch and roll commands for position control can be given by φd = tan −1 ( uy cosθ g−uz ) , θd = tan ( ux uz−g ) (2.21) Then the systematic controller block diagram is given as follows Figure 3: Structure for multicopter control In case the heading command is not held zero but varys it can be handled as follows. After rearranging the equation (2.4) [ x?̈? y?̈? ] = [ cosψ − sinψ sin ψ cosψ ] [ − F m sin θ cos φ F m sinφ ] (2.22) use PD control (2.17) as follows [ x?̈? y?̈? ] = [ cosψ − sinψ sin ψ cosψ ] [ − F m sin θ cos φ F m sinφ ] = [ ux uy ].(2.23)