1 1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3S-CNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN, Tarek HAMEL, Claude SAMSON INRIA Sophia Antipolis, December 09, 2009
2 2 / 23 Outline 1 Motivations 2 3 4
3 Motivations Underactuated Aerial Robots A PPLICATIONS : Surveillance, Inspection, Searching and rescuing, Cartography (SLAM),... Development challenges B Mechanical and geometrical conception. B Modeling of aerodynamic forces. B State reconstruction. B Control design. 3 / 23
4 How do they fly? 4 / 23 1 Thrust force to monitor the translational motion and to counterbalance the weight. 2 Actuation system to generate torque for orientation control. EXAMPLES: Tail-rotor Helicopters Ducted fan Tailsitters Quadrotor Helicopters
5 How do they fly? 5 / 23 1 Thrust force to monitor the translational motion and to counterbalance the weight. 2 Actuation system to generate torque for orientation control. EXAMPLES: Tail-rotor Helicopters Ducted fan Tailsitters Quadrotor Helicopters
6 How do they fly? 6 / 23 1 Thrust force to monitor the translational motion and to counterbalance the weight. 2 Actuation system to generate torque for orientation control. EXAMPLES: Tail-rotor Helicopters Ducted fan Tailsitters Quadrotor Helicopters
7 Motivation 1 Control design 7 / 23 OBJECTIVE: Unifying control approach robust w.r.t. external perturbations, modeling errors, and measurement/estimation errors. with large domain of stability.
8 Motivation 2 Attitude estimation 8 / 23 Attitude information is required for feedback control. In the 2D-case (ex. Ships), attitude can be directly measured (ex. using a magnetic compass). In the 3D-case (ex. VTOL drones), obtaining attitude is more complicated. OBJECTIVE: Robust and effective attitude estimation algorithms using an Inertial Measurement Unit (IMU) and a GPS.
9 System modeling (1/2) 9 / 23 FORCES APPLIED T : Thrust force control F e : sum of all external forces TORQUES APPLIED (EXPRESSED IN B) Γ R 3 : Torque control input Γ e R 3 : sum of torques created by all external forces
10 System modeling (2/2) 10 / 23 EQUATIONS OF MOTION: ẋ m v Ṙ = J ω v TRe 3 + F e Rω ω Jω + Γ + Γ e, ( ) e CONTROL DIFFICULTIES: Unavailability of a precise model of F e and Γ e valid in a large domain. Fast and unpredictable variation of external forces. Limited actuation power to counter environmental perturbations. Parameter uncertainties (ex. m, J). Measurement/estimation errors of the vehicle s state.
11 Existing control approaches 11 / 23 LINEAR CONTROLLERS Local stability. Difficulty of stabilization of unknown dynamical equilibria. Singularities of attitude minimal parametrization. NONLINEAR CONTROLLERS Feedback exact linearization (Koo et al., CDC 1998). Backstepping-based (Mahony and Hamel, CDC 1999). Hierarchical (Pflimlin et al., ICRA 2006). Nested-saturation (Marconi et al., Automatica 2002).
12 Contributions 12 / 23 Unifying control framework in the sense of taking into account a large class of environmental forces. Assumption: F e depends only on ẋ and t example: a spherical vehicle counter-example: airplanes with lift forces depending on the angle of attack Global stability for simple geometries (spherical body), by exploiting dissipativity of drag forces. Compensation of measurement/estimation error of F e and modeling errors by using a novel integral correction technique. Robustification w.r.t. situations when F e crosses zero, or becomes small.
13 13 / 23 ZOOM ON NOVEL INTEGRAL CORRECTION TECHNIQUE OBJECTIVE: stabilization of the equilibrium x = 0 despite the poor knowledge of the external forces. SOLUTIONS: compensation by means of integral correction. Classical integrator z = x can yield large overshoot. Saturated integrator z = sat ( x ) : no large overshoot but slow desaturation. Proposed integrator: z solution to z + 2k zż + k 2 z (z sat (z)) = f ( x) with k z > 0, f ( ) a bounded monotonic function z, ż, z are bounded and can be desaturated rapidly If z z + 2k zż = f ( x) If z > z + 2k zż + kz 2z = f ( x) + k2 z : left part is a critically damped system
14 Aspects of the proposed control design 14 / 23 Thrust direction control (joystick mode). Velocity control (joystick mode). Horizontal velocity and altitude control (joystick mode). Position control, i.e. tracking trajectory (automatic flight). Non-invertibility of the thrust direction is taken into account. High-gain estimator of F e.
15 Simulation results 15 / 23 Trajectory tracking for a ducted fan tailsitter model in wind gusts
16 Estimation objective 16 / 23 Estimation of the rotation matrix R SO(3) between the body frame and the inertial frame Ṙ = Rω
17 Classical solutions 17 / 23 Integrating the kinematics of rotation using the angular velocity measurement: Estimation error diverges. Precise gyroscopes reduce the divergence rate, but are expensive and heavy. Algebraic solutions requiring at least two vectors (Black, 1964), (Wahba, 1965) Fusion solutions using the angular velocity measurements and two vectors observations, and exploiting complementary characteristics of sensors (ex.: frequency, precision). The choice of sensors and solutions may depend on applications.
18 Classical solution for aerial robots IMU = Gyrometers + Accelerometers + Magnetometers Gyrometers: measure angular velocity ω. Magnetometers: measure the geomagnetic field m B = R m I Accelerometers: measure specific acceleration a B = R ( v g) COMMON SOLUTIONS: Use Accelerometers as Inclinometers Assumption (weak acceleration): v 0 a B R g Classical solutions can be applied. Precision problem in the case of strong accelerations. (Martin, Salaun, IFAC 2007) use the measurement of v (given by GPS) and IMU measurements to estimate v and improve the precision of the estimated attitude. 18 / 23
19 Attitude estimation contributions 19 / 23 Inspired by the solutions of (Martin, Salaun, IFAC 2007) and (Mahony et al., CDC 2005), two novel attitude estimators are proposed 1st solution (high-gain type) ensures: Semi-global convergence and stability. 2nd solution (non high-gain type) ensures: Almost-global convergence, Exponential stability when the acceleration is constant.
20 Simulation results Circle trajectory perfect measures 20 / 23
21 Simulation results Circle trajectory noisy and biased measures 21 / 23
22 Perspectives 22 / 23 CONTROL DESIGN: Extensions to vehicles subjected to important lift forces (ex. airplanes). Extensions for the situation when external forces vanish: The system is Small-Time-Locally-Controllable, but its linearized system is not controllable. Using non-classical techniques: Transverse Function (Morin, Samson, CDC 2005). Pursue works on the on-line estimation of F e. ATTITUDE ESTIMATION: Compensation of measurement biases (gyro., accelero. biases). Robustness w.r.t. magnetic perturbation (Martin, Salaun, IFAC 2007).
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