__Mathematical Model of Fixed-Wing Aircraft__

The first step in designing any control system is to mathematically model the process in question. Thus, this preliminary section concerns the kinematics, dynamics, and forces and moments acting upon a rigid fixed-wing aircraft that allow for the derivation of such an aircraft’s six-degrees-of-freedom (6-DoF) equations of motion. Subsequently, the equations of motion will be used to form a state space model of the aircraft composed of 12 state variables which will be utilized throughout the majority of this report.

Aircraft dynamics are composed of translational and rotational motions. Translational motion is comprised of three position states and three velocity states for a total of 6 state variables (and 3-DoF). Rotational motion is comprised of three angular positions and three angular rates for an additional 6 state variables (and 3-DoF). Thus, the resulting state space model is composed of 12 state variables and corresponding equations that describe an aircraft’s 6-DoF motion.

An aircraft’s translational position is comprised of inertial north position (*p _{n}*), inertial east position (

*p*), and inertial down position

_{e}*(p*) – note that the inertial down position will often be referred to as the aircraft’s altitude or height (

_{d}*h*) above the Earth’s surface. Translational velocity is comprised of forward velocity (

*u*), lateral velocity (

*v*), and upward velocity (

*w*). Together, these 6 state variables describe the aircraft’s translational motion. An aircraft’s rotational position is comprised of roll angle (), pitch angle (), and yaw angle (). Rotational velocity is comprised of roll rate (

*p*), pitch rate (

*q*), and yaw rate (

*r*). Together, these 6 state variables describe the aircraft’s rotational motion. Figure 1 below depicts the 12 state variables of an aircraft’s translational and rotational motion shown schematically on a MATLAB animation of a Zagi HP Electric Flying Wing aircraft designed specifically for this project.

Figure 1: State Variables Shown Schematically on Flying-Wing Model

External forces and moments must also be accounted for in a UAV’s equations of motion, and thus, applied to the 12-state equations resulting from the aircraft’s translational and rotational motions. External forces are a combination of gravitational, aerodynamic and propulsion effects, whereas external moments are a simply a combination of aerodynamic and propulsion effects. Figure 2 below shows the control surfaces on a flying wing aircraft – known as elevons. Actuating the left and right elevons together has the same effect as elevators on traditional fixed-wing aircraft – allowing the aircraft to change pitch and therefore altitude. This type of control actuation effects the aircraft’s longitudinal dynamics which include the up-down and pitching motions.

Figure 2: Control Surfaces on Zagi HP Electric Flying Wing – Elevator Action

Figure 3 below depicts driving the elevons of a flying wing aircraft differentially which has the same effect as ailerons on traditional fixed-wing aircraft – changing the aircraft’s roll angle and therefore course heading. This control actuation effects the lateral-directional dynamics of an aircraft which includes side-to-side or turning motions, and the rolling and yawing motions.

Figure 3: Control Surfaces on Zagi HP Electric Flying Wing – Aileron Action

Effecting the control surfaces creates aerodynamic forces and moments that allow the aircraft to change direction and altitude. Thus, the resultant forces and moments of control surface actuation are a major consideration that must be accounted for when mathematically modeling any aircraft.

The other major contributor (apart from gravitational effects) to the forces and moments acting on an aircraft is the propulsion effect. Figure 4 below shows the motor housing and propeller used to create thrust for the Zagi HP Electric Flying Wing aircraft. A thorough derivation of the mathematical models for lateral and longitudinal control surface actuation, and propeller thrust can be found in [2] or any textbook on aircraft dynamics as well as Appendix C of this report. The coefficients used in defining the forces and moments acting upon the Zagi HP Electric Flying Wing were provided by [2] and can be additionally found in Appendix A of this report.

Figure 4: Motor Housing and Propeller on Zagi HP Electric Flying Wing

The equations given in Figure 5 below represent the 12 state equations used to describe the translational and rotational motion of a fixed-wing aircraft including external forces and moments. These equations represent the complete mathematical model of a fixed-wing aircraft that will be utilized in simulation and subject to control algorithms and designs for this project.

Figure 5: State Space Equations with Forces and Moments Acting on Aircraft from [2]

A thorough derivation of the UAV equations of motion used throughout this project can be found in [2] and additionally in Appendix B of the attached report.

__Model Linearization__

In control system design it is desirable to have simplified, linear mathematical models in order to apply classical control techniques. The state space representation resultant of section 2.1 is a highly nonlinear model which would be impractical for classical control design techniques due to its complexity. Although nonlinear control methods exist and are often useful, classical control techniques (i.e. successive loop closure) are most effective for linear time invariant (LTI) systems – thus, the focus of this section is to linearize the aircraft state space model about trim conditions. Trim conditions refer to the reconfiguration of the aircraft’s equations of motion about operating points that are in a specified equilibrium with the aerodynamic, propulsive, and gravitational forces and moments constantly acting on the rigid aircraft. Additionally, in this section, the state space model derived in section 2.1 is decoupled into lateral and longitudinal equations of motion – a common practice in flight control system design that allows for more suitable design models. The lateral-directional state space model includes roll and course-heading angles (and their derivatives) while the longitudinal state space model includes pitch angle, altitude and airspeed. The resulting models will subsequently be used in the design of various autopilots that are paramount to the objectives of this project.

The UAV is considered to be in equilibrium, or at a suitable trim condition, when it is moving with constant-airspeed and climbing at a constant flight path angle (or at a constant altitude). Figure 6 below shows equilibrium conditions suitable for computing trim states and control inputs schematically. Calculating trim involves using Jacobian operators on lateral and longitudinal transfer functions resulting from the decomposition of state space equations into lateral and longitudinal equations. Taking partial derivatives and evaluating at these specified trim conditions results in linear state space models for lateral and longitudinal dynamics of the aircraft. The linearized state space models are easily converted to the Laplace domain in order to create lateral and longitudinal transfer function models which are subsequently used in the successive loop closure control design method. Note that conversion to transfer function models is not required for the linear quadratic regulator control design method as only the linearized state space models are used.

Figure 6: Linearization via Trim Conditions of Flying Wing Type Aircraft from [2]

A thorough derivation of trim conditions, linearized state space models, and transfer function models used throughout this project can be found [2] and in Appendix C of the attached report.