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4 Desired Position x d y d x y Position y d z d Fig.. f d q d A. State Space Model y f q Attitude Heading Altitude z U U 3 U 4 W 1 4 Rotors Speed Calculation Control Architecture The state vector is defined as [4] Quadrotor Model X = [ φ φ ] θ θ ψ ψ z ż x ẋ y ẏ (1) The control input vector is defined as where U = [ U U 3 U 4 ] ( = k F Ω 1 + Ω + Ω 3 + Ω 4 ( ) U = k F Ω + Ω 4 ( ) U 3 = k F Ω 1 Ω 3 ( ) U 4 = k M Ω 1 Ω + Ω 3 Ω 4 The state space model is given by where f (X, U) = ) (11) (1) Ẋ = f (X, U) (13) x x 4 x 6 a 1 + x 4 Ω r a + b 1 U x 4 x x 6 a 3 + x Ω r a 4 + b U 3 x 6 x x 4 a 5 + b 3 U 4 x 8 g U1 m cos x 1 cos x 3 x 1 U1 m (sin x 1 sin x 5 + cos x 1 sin x 3 cos x 5 ) x 1 m (sin x 1 cos x 5 cos x 1 sin x 3 sin x 5 ) (14) a 1 = (I yy I zz ) I xx a = J r I xx a 3 = (I zz I xx ) I yy a 4 = J r I yy a 5 = (I xx I yy ) b 1 = l I xx b = l I yy b 3 = 1 B. Outer Control Loop I zz I zz (15) For position control, a PD controller with a saturation function is implemented to generate the reference roll φ d and pitch θ d. The reference angles are tracked by the inner loop controller [19]. Based on the desired waypoint, the position controller calculates the desired accelerations ẍ d and ÿ d ẍ d = k p (x ref x) + k d (ẋ ref ẋ) ÿ d = k p (y ref y) + k d (ẏ ref ẏ) (16) where (x d, y d ) is the desired waypoint, (ẋ d, ẏ d ) is the desired velocity and k p and k d are the proportional and derivative controller gains respectively. The reference roll and pitch angles can be solved for using the desired accelerations m [sin φ d sin ψ + cos φ d sin θ d cos ψ] = ẍ d m [sin φ d cos ψ cos φ d sin θ d sin ψ] = ÿ d (17) Using the small angle assumption around the hover position, the previous equations can be simplified m [φ d sin ψ + θ d cos ψ] = ẍ d (18) m [φ d cos ψ θ d sin ψ] = ÿ d [ ] [ ] sin ψ cos ψ φd = m [ ] ẍd (19) cos ψ sin ψ ÿ d which has a closed form solution for φ d and θ d. A saturation function is needed to ensure that the θ d

5 reference roll and pitch angles are within specified limits φ d = sat(φ d ) () θ d = sat(θ d ) where v v v max sat(v) = (1) sign(v)v max v > v max such that C. Inner Control Loop 1 v < sign(v) = 1 v () The attitude, heading and altitude controllers (see Figure ) are derived using the Backstepping approach. The idea behind Backstepping is to design a controller recursively by considering some of the state variables as virtual controls. Then, intermediate stabilizing control laws are designed for the virtual controls []. At the last step, the actual control input is used to stabilize the whole system. The design procedure is systematic. Therefore, we only present the design procedure for roll control. Pitch, yaw and altitude controllers can be derived similarly. For clarity of presentation, the nonlinear system under investigation is extracted from the state space model derived in (14). ẋ 1 = x ẋ = x 4 x 6 a 1 + x 4 Ω r a + b 1 U (3) which is in the strict feedback form ẋ 1 = x ẋ = f (X) + b 1 U (4) The backstepping procedure transforms the closed loop system to the following coordinates z 1 = x 1 x 1d z = x α 1 (5) where α 1 is the virtual control input and x 1 x 1d is the roll tracking error. We introduce the following Lyapunov function V 1 = 1 z 1 (6) whose derivative along the system trajectories is given by V 1 = z 1 ż 1 = z 1 (z + α 1 ẋ 1d ) The virtual control input can be chosen as such that (7) α 1 = c 1 z 1 + ẋ 1d (8) V 1 = c 1 z 1 + z 1 z (9) We augment the previous Lyapunov function in the following way V = 1 z z (3) The derivative along system trajectories is given by V = z 1 ż 1 + z ż = c 1 z 1 + z 1 z + z (ẋ α 1 ) = c 1 z 1 + z (z 1 + f (X) + b 1 U α 1 ) (31) Now, the actual control input is at our disposal for the overall roll control such that U = 1 b 1 ( c z z 1 f (X) + α 1 ) (3) V = c 1 z 1 c z (33) IV. STABILITY ANALYSIS In this section, the stability and performance of the proposed inner loop controller are analyzed using Lyapunov s stability theorem [7], []. The closed loop system is analyzed in the transformed coordinate system [ ] [ ] [ ] ż1 c1 1 z1 = (34) ż 1 c z The closed loop system is an autonomous system in the error coordinates. Stability can be analyzed using LaSalle invariance theorem. Recall (3), it is clear that the Lyapunov function is positive definite, decresent and radially unbounded. Recall (33), the Lie derivative of the Lyapunov function is negative definite. According to Lyapunov s second method, the closed loop system is Globally Asymptotically Stable (GAS). Also, the system trajectories z 1 and z are bounded. From (8),

6 the boundedness of the virtual input α 1 can be deduced. From (3), the boundedness of the actual control input can be deduced. Moreover, the Lyapunov function has a minimum dissipation rate V (Z) σv (Z) (35) Inner Loop Rotor Speeds Quadrotor Dynamics States Forces Moments Open Dynamics Engine Gazebo API Forces and Moments States Gazebo Visualizer where c 1 1, c 1 and Z = [ z 1 z ]. Therefore, the closed loop system is Globally Exponentially Stable (GES) with bounded control input as long as (4) holds. Next, a bound on the transient tracking error is derived in terms of the controller design parameters. From (33), the dissipation rate of the Lyapunov function satisfies Reference Angles Angular Positions Angular Velocities Outer Loop Fig. 3. Position Velocity Simulator Framework V (Z) c 1 z 1 (36) The L norm of the transient tracking error is given by z 1 = z 1 (τ) dτ 1 c 1 V ((Z (τ)) dτ 1 c 1 [V (Z ()) V (Z ( ))] 1 c 1 V (Z ()) (37) V. SIMULATION FRAMEWORK A simulation framework for testing the quadrotor s stabilization and navigation controllers is developed. It is based on the open-source 3D robotics simulator Gazebo [14] and the ODE library [15]. The quadrotor dynamics model derived earlier is implemented in the Gazebo simulator to calculate the forces and moments acting on the vehicle. Using the Gazebo API interface, the derived forces and moments are input to the ODE. It solves the body s equations of motion and sends the vehicle s states back to Gazebo for visualization. The stabilization and navigation controllers are implemented directly in Gazebo such that they can use the vehicle s states for feedback and send the control commands to the dynamics module. The simulator framework is shown in Figure 3. Gazebo offers plugins for different sensor packages that can be mounted on the quadrotor to provide sensor feedback for higher levels of autonomy. For example, the quadrotor can be equipped with onboard camera and laser sensors to provide feedback for Fig. 4. Simulation Environment visual servoing and simultaneous localization and mapping. Figure 4 shows a screen shot of the simulation environment where the quadrotor is equipped with additional sensors, camera and laser. VI. SIMULATION RESULTS In this section, simulation results for the inner and outer loop controllers are presented. A. Inner Loop Results Figure 5 shows the closed loop response of the roll, pitch, yaw and altitude to a step input. The steady state is reached within approximately seconds using bounded control inputs as shown in Figure 6. The Backstepping controller asymptotically regulates the roll, pitch, yaw and altitude

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