The nonlinear equations of motion given by eqs. (4.1) to (4.7), (4.11) to (4.13) and (4.22) to (4.24) describe the dynamics of the aircraft, using the aerodynamic, engine and actuator characteristics stated above.
The state and control variables of the aircraft model are given in eqs. (4.48) and (4.49).
x= [u v w q0q1q2q3p q r x y z]0 (4.48)
u= [δeδaδrδt]
0 (4.49)
The equations of motion are numerically solved for the state variables x using the explicit Euler integration scheme given in eq. (4.50). In eq. (4.50) the term t defines the current simulation time, and ∆t is the simulation time step size. The step size is fixed and taken as 0.005 s for the simulations of both the lower-scale and higher-scale aircraft. The derivative term ˙x(t) is obtained by evaluating the nonlinear equations of motion with the state variables of the respective time instant, i.e. x(t). The integration is initialized with desired initial values of the state variables shown in eq. (4.48).
A preliminary comparison of explicit Euler and fourth order Runge-Kutta inte- gration methods on lower-scale aircraft simulations with aforementioned time step size indicated that, the Runge-Kutta method provides an accuracy gain of less than 1.5 millimeters on all three axes of the follower’s relative position with respect to the leader while causing substantial increase in simulation time. Hence explicit Eu- ler method was chosen as it provides good balance between solution accuracy and computational overhead for the simulation set up with the selected time step size.
x(t + ∆t) = x(t) + ∆t ˙x(t) (4.50)
For initialization of the quaternion state variables, desired initial Euler angles are used. These initial Euler angles are then converted to their quaternion counterparts using eq. (4.51) [119]. q0 q1 q2 q3 =
cos (φ/2) cos (θ/2) cos (ψ/2) + sin (φ/2) sin (θ/2) sin (ψ/2) sin (φ/2) cos (θ/2) cos (ψ/2) − cos (φ/2) sin (θ/2) sin (ψ/2) cos (φ/2) sin (θ/2) cos (ψ/2) + sin (φ/2) cos (θ/2) sin (ψ/2) cos (φ/2) cos (θ/2) sin (ψ/2) − sin (φ/2) sin (θ/2) cos (ψ/2)
(4.51)
Chapter 5
Automatic Control of the Aircraft
This chapter details the automatic control aspects of the leader and follower aircraft. Descriptions of the autopilots of the follower and the leader aircraft, the leader’s path- following algorithm, and the follower aircraft’s formation-hold controller are given. Since the leader and the follower are identical aircraft, their autopilots are identical. The setpoints of the leader aircraft’s autopilot are generated by its path-following algorithm. Follower aircraft’s autopilot is driven by its formation hold controller.As stated previously, two sets of formation flight simulations are performed in this work, each with different scales of aircraft. Both scales of aircraft share the same control systems, which are detailed in this chapter. The only difference between the control systems of the lower-scale aircraft and their higher-scale counterparts are the numerical values of the control system gains and tuning parameters. The numerical values of these entities are presented for each scale of aircraft in Appendix A.
As part of the numerical flight simulation scheme employed in this work, the automatic pilots and the guidance algorithms of the aircraft operate at a frequency of 100 Hz.
5.1
Autopilots
The aircraft autopilots are of velocity-hold autopilot type. The autopilot inputs are the commanded values of the ground speed V on the longitudinal channel, the climb angle γ on the vertical channel, and the course angle χ on the lateral channel. This type of autopilot is selected not specifically for the automated formation flight application, but also for obtaining an autopilot compatible with other common UAV missions such as path following and automated landing.
Successive loop closure approach [79] is used in order to keep the overall structure of the autopilot simple. The method suggests the design of successive simple feedback controllers, from fastest to slowest plant dynamics, instead of a more complex single control system. The resulting controller in cascaded form enables a controlled re-
sponse and better disturbance rejection on faster inner-loop state variables. Integral control actions are used on each channel in order to achieve zero steady-state error on each velocity component. Combined with the relative position controller presented in Section 5.3, the follower aircraft autopilot yields a structure comparable to the feedback-control-based formation control approaches outlined in Section 1.4.1.
For the speed control, a single feedback loop is used. This control loop generates throttle command based on the difference between commanded and actual velocity, as given in eq. (5.1). A proportional and integral control action is used.
δtc = KPV(Vc− V ) + KIV
Z
(Vc− V ) dt (5.1)
The flight path angle control consists of an inner feedback loop for the pitch angle
θ, which generates elevator commands using proportional control action on the pitch
angle error θc− θ. The setpoint of the pitch angle controller is generated by the outer flight path angle control loop. The flight path angle controller uses proportional and integral control action in order to drive the flight path angle error toward zero. Equations (5.2) and (5.3) give the control actions on the vertical channel.
δec = KPθ(θc− θ) (5.2)
θc = KPγ(γc− γ) + KIγ
Z
(γc− γ) dt (5.3)
On the lateral channel of the autopilot, the outermost course angle control loop generates roll angle command using proportional control action. The inner feedback loop applies another proportional control action on roll angle error in order to generate a roll rate command. The innermost control loop on roll rate uses a proportional and integral control action in order to generate the aileron command. Here a stability augmentation loop is included as adding the fed-back roll rate with proportional action on the aileron command, in order to enhance roll rate damping characteristics. The controller is described in eqs. (5.4) to (5.6).
δac = KPp(pc− p) + KIp
Z
(pc− p) dt + Kdampp (5.4)
pc= KPφ(φc− φ) (5.5)
φc= KPχ(χc− χ) (5.6)
The roll rate control loop has an additional importance, considering that the au- topilot described above is used in a formation flight application. As described in Chapter 1 and Chapter 6, depending on the relative position of the follower aircraft with respect to the leader, the follower aircraft may experience significant roll distur-
bance, due to the wake flow field of the leader aircraft. Using a roll-rate control loop helps reject this disturbance, before it causes significant relative position errors.
In the present setting of the autopilot, the use of rudder is not required, which is normally the case also in formation flight applications, such as aerial refueling [52].
In eqs. (5.1) to (5.6), the K-terms are the constant gains for the respective control actions. The gain values that are used for lower and higher-scale aircraft are given in Appendix A.