Multi-body dynamic modelling and flight control for an asymmetric variable sweep morphing UAV

In this paper, the multi-body dynamic model of an asymmetric variable sweep wing morphing UAV is built based on Kane’s method. This model describes the UAV’s transient behaviour during morphing process and the dynamic characteristic of the variable sweep wings. An integrated design of trajectory tracking control via constrained backstepping method is presented then. The idea of aircraft roll control through asymmetric wing sweep angle changes rather than traditional aileron is explored and used in the flight control design. The control of variable sweep wings is designed as well based on the presented dynamic model. Command filters are used in the backstepping design procedure to accommodate magnitude, rate and bandwidth constraints on virtual states and actuator signals. Stability of the closed-loop system can be proved in the sense of Lyapunov. Simulation of tracking a desired trajectory which contains two manoeuvres demonstrates the feasibility of the proposed protocol and the morphing wing roll controller. The AeronAuTicAl JournAl June 2014 Volume 118 no 1204 683 Paper No. 4020. Manuscript received, 18 June 2013, accepted 4 July2014. Multi-body dynamic modelling and flight control for an asymmetric variable sweep morphing UAV L. Tong [email protected] H. Ji [email protected] Department of Automation University of Science and Technology of China Hefei, China 684 The AeronAuTicAl JournAl June 2014 NOMENCLATURE (p, q, r) angular velocity (u, v, w) body co-ordinate airspeed (x, y, z) co-ordinate of Of in {I} α attack angle β sideslip angle ηl ,ηr time derivatives of Hl and Hr Γ variable sweep UAV system Q matrix relates [p q r]T to [ḟ θ̇ φ̇]T f, θ, Y attitude angles w→1, ... , w →F 8 partial angular velocities of FL V airspeed v→1, ... , v →F 8, partial velocities of Of {I} earth co-ordinate system {B} body co-ordinate system CD drag coefficient CL lift coefficient Fj( j = 1, ..., 8) generalised active forces of Γ Ff j (j = 1, ..., 8) generalised active forces of FL Fl j (j = 1, ..., 8) generalised active forces of Wl Fl * j (j = 1, ..., 8) generalised inertia forces of Wl Fr j (j = 1, ..., 8) generalised active forces of Wr Fr * j( j = 1, ..., 8) generalised inertia forces of Wr FL fuselage kL drag-to-lift ratio Mf mass of FL O CG of Γ O1,O2 connecting points of wings and fuselage Of CG of FL Ol, Or CG of Wl and Wr P, Q, F, G matrix functions t time UAV unmanned air vehicle Wl ,Wr two wings δη differential wing sweep angle morphing δe, δr control-surface deflections γ, χ, μ flight path angles l f central inertia dyadic of FL Hl, Hr wing sweep angles α→f angular acceleration of FL Ω→f angular velocity of FL a→Of acceleration of Of G→f gravity of FL R→*f inertia force of FL R→l, R →* r inertia forces of Wl and Wr T→*f inertia torque of FL Tong And Ji mulTi-body dynAmic modelling And flighT conTrol for An AsymmeTric VAriAble... 685 T→*l,T →* r inertia torques of Wl and Wr V→Of velocity of Of in {I} F*j (j = 1, ..., 8) generalised inertia forces of Γ Ffj *(j = 1, ..., 8) generalised inertia forces of FL JF, JA, JT, EG, KV, KΩ, Kη, N coefficient matrices ω→L1, ..., ω →L 8 partial angular velocities of Wl ω→R1, ..., ω →R 8 partial angular velocities of Wr D→f aircraft zero lift drag d→F, d → Y, d → L, d → D position vectors F→B thrust force of aircraft engines L→t lift force of the horizontal tail M→η roll moment generated by differential wing sweep angle R→f equivalent force of FL R→wl, R → wr equivalent forces of Wl and Wr T→f equivalent torque of FL T→l, T → r control torques of variable sweep wings T→wl, T → wr equivalent torques of Wl and Wr v→L1 , ..., v → L8 partial velocities of Ol v→R1 , ..., v → R8 partial velocities of Or Yt The aerodynamic force of the rudder SM, SR Saturation functions Bz→ The unit vector along {B}’s z axis B I R The rotation matrix from {I} to {B} 1.0 INTRODUCTION With the development of control science, material science and microelectronics, unmanned air vehicles (UAV) have been widely used in both military and civil applications(1,2). Natural advantages, including no human physiological limits needed to be considered, simplicity of design and relatively low price have made UAVs an excellent experimental platform for new technologies. One of the rising topics of interest for researchers is the combination of UAV and aircraft morphing. As has been proved by many studies and aircraft like the F-14 Tomcat, morphing technologies, such as variable sweep, variable span and other kinds of large-scale morphing, can highly improve an aircraft’s performance in different flight scenarios(3-7). The weak sides of morphing that prevent it from been widely used, on the other hand, include the complexity of mechanism, difficulties in modelling and control design and lack of safety guarantee. However, when applied to a UAV, many defects of morphing can be neglected. Nowadays, much researchabout morphing is carried out on a UAV platform. Lockheed Martin flight-tested a UAV with folding-wing morphing, while DARPA used a UAV as a technology demonstrator of morphing in flight(8). M. Abdulrahim and R. Lind et al built a UAV with a biologically-inspired morphing mechanism and their studies showed significant benefit from morphing for small vehicles(8-11). C.H. Hong and M. Cheplak et al designed a multi-body UAV capable of significant geometric morphing to incorporate wing area and sweep angle change(12). While most of the studies consider that the shape change was done in an open-loop manner, some researchers have begun to use structural changes as additional (dynamic) aerodynamic forcing inputs for aircraft control(13). A.C. Hurst et al presented an adaptive aircraft with morphing control input that allows it to land through a perching manoeuvre (14). P. Bourdin et al presented a novel 686 The AeronAuTicAl JournAl June 2014 method for the control of aircraft by using articulated split wing-tips(15). J.J. Henry explored the idea of aircraft roll control through asymmetric variable span morphing(16). The effects of the asymmetric variable sweep mechanism were studied in Ref. 17 and the idea of using it as the roll controller was presented in Ref. 18. The basic idea is that the asymmetric wing sweep morphing can lead to a lift difference between two wings, which yields a roll moment without the help of ailerons. In this paper, the asymmetric wing sweep morphing is used as the roll controller and incorporated with trajectory tracking control of the UAV. For a morphing UAV capable of large scale shape change, it is not suitable to treat it as a single rigid body because of complex interactions between inertial forces and aerodynamic forces. Here the asymmetric variable sweep UAV is considered as a three-body system with its two wings allowed to sweep independently. There are many ways to model a multi-body system, such as the Newton-Euler method or Lagrange’s Equations. In this paper, a classic approach proposed by Thomas R. Kane during the 1970s and 1980s is used to model the UAV system(19). There have been many discussions among Kane’s method, Lagrange’s equations and Gibbs-Appell’s equations. Essentially, they give the same result for a multi-body system but Kane’s method is more suitable for computer programming and analysis(20-22). The presented model describes the transient behavior of the UAV during the wing sweep morphing process. Dynamic equations of the asymmetric sweep wings are also contained, which makes it possible for the morphing control design. The guidance equations used in this paper can be found in Ref. 23. Trajectory tracking is often required for a UAV in various kinds of mission scenarios. To achieve better stability guarantee and improve performance, here the guidance and control laws are designed in an integrated way(24). As a Lyapunov function-based design method, backstepping method is suitable for handling high-order nonlinear systems with lower-triangular form. However, it can not handle physical limitations of control surfaces, especially the wing morphing limitations when applied to the asymmetric variable sweep UAV control design. The constrained backstepping approach presented in Refs 25-28 uses command filters to accommodate magnitude, rate and bandwidth constraints on intermediate states and actuator signals. An additional advantage of this method is that in stead of tedious analytic computations, the virtual control signal’s derivatives can now be acquired by command filters. The paper is outlined as follows. First, dynamic modelling of the UAV via Kane’s method is discussed in Section 2. A review of constrained backstepping method is presented in Section 3. In Section 4 the trajectory control is derived and the control of the morphing wings is presented in Section 5. Simulation examples are shown in Section 7. Finally, conclusions are given in Section 8. Figure 1. Asymmetric variable sweep UAV model Γ. Tong And Ji mulTi-body dynAmic modelling And flighT conTrol for An AsymmeTric VAriAble... 687 2.0 ASYMMETRIC VARIABLE SWEEP UAV MODELLING The purpose of this section is to build a multi-body dynamic model of an asymmetric variable sweep UAV based on Kane’s method. Consider the variable sweep UAV system Γ in Fig. 1, which can be treated as a multi-body system composed of three major parts: the left wing Wl , the right wing Wr and the fuselage FL. Wings are hinged to FL at O1, O2 by electric motors or other kind of actuators that provide torques, allowing wing sweep angles Hl, Hr to change independently. Wing dihedral angle is assumed to be zero and the wind field around the model is assumed to be static. CG (centre of gravity) of Γ, FL, Wl and Wr are O, Of , Ol and Or respectively. 2.1 Preliminary knowledge Since wings and fuselage are hinged together, system Γ is subject to two configuration constraints: 1. Distances between O1(O2) and any particle belonging to Wl (Wr) are constant. 2. The angle between Bz→ which is the unit vector along body co-ordinate system {B}’s z axis, and the velocity vector of any particle belonging to Wl (Wr) is constant. (Since the wing dihedral angle is assumed to be zero, this angle would be 90 degrees.) Choose generalised co-ordinates (x, y, z, f ,θ, ψ, Hl, Hr): (x, y, z) is the co-ordinate of Of in the earth co-ordinate system {I}, f is the roll angle, θ is the pitch angle, ψ is the yaw angle, Hl is the sweep angle of the left wing and Hr is the sweep angle of the right wing. Since the position of system’s CG O varies with different wing sweep angles, here we choose the coordinate of Of as generalised co-ordinates rather than O’s co-ordinate. Accordingly, the origin of body axes co-ordinate system is also chosen as Of . Every assignment of values to these quantities and time t corresponds with a definite admissible configuration of Γ in {I}. Define generalised speeds (u, v, w, p, q, r, ηl ,ηr), where (u, v, w) is the airspeed V → expressed in {B}, (p, q, r) is the angular velocity expressed in {B}, ηl and ηr are time derivatives of Hl and Hr, respectively. We have