Guidance Model

The UAV Toolbox™ uses the North-East-Down (NED) coordinate system convention, which is also sometimes called the local tangent plane (LTP). The UAV position vector consists of three numbers for position along the northern-axis, eastern-axis, and vertical position. The down element complies with the right-hand rule and results in negative values for altitude gain.

The ground plane, or earth frame (NE plane, D = 0), is assumed to be an inertial plane that is flat based on the operation region for small UAV control. The earth frame coordinates are [x_e,y_e,z_e]. The body frame of the UAV is attached to the center of mass with coordinates [x_b,y_b,z_b]. x_b is the preferred forward direction of the UAV, and z_b is perpendicular to the plane that points downwards when the UAV travels during perfect horizontal flight.

The orientation of the UAV (body frame) is specified in ZYX Euler angles. To convert from the earth frame to the body frame, we first rotate about the z_e-axis by the yaw angle, ψ. Then, rotate about the intermediate y-axis by the pitch angle, ϴ. Then, rotate about the intermediate x-axis by the roll angle, ϕ.

The angular velocity of the UAV is represented by [p,q,r] with respect to the body axes, [x_b,y_b,z_b].

For fixed-wing UAVs, the following equations are used to define the guidance model of the UAV. Use the derivative function to calculate the time-derivative of the UAV state using these governing equations. Specify the inputs using the state, control, and environment functions.

The UAV position in the earth frame is [x_e, y_e, h] with orientation as heading angle, flight path angle, and roll angle, [χ, γ, ϕ] in radians.

The model assumes that the UAV is flying under a coordinated-turn condition, with zero side-slip. The autopilot controls airspeed, altitude, and roll angle. The corresponding equations of motion are:

V_a and V_g denote the UAV air and ground speeds.

The wind speed is specified as [V_wn,V_we,V_wd] for the north, east, and down directions. To generate the structure for these inputs, use the environment function.

k_* are controller gains. To specify these gains, use the Configuration property of the fixedwing object.

From these governing equations, the model gives the following variables:

These variables match the output of the state function.

For multirotors, the following equations are used to define the guidance model of the UAV. To calculate the time-derivative of the UAV state using these governing equations, use the derivative function. Specify the inputs using state, control, and environment.

The UAV position in the earth frame is [x_e, y_e, z_e] with orientation as ZYX Euler angles, [ψ, ϴ, ϕ] in radians. Angular velocities are [p, q, r] in radians per second.

The UAV body frame uses coordinates as [x_b, y_b, z_b].

The rotation matrix that rotates vector from body frame to world frame is:

The cos(x) and sin(x) are abbreviated as c_x and s_x.

The acceleration of the UAV center of mass in earth coordinates is governed by:

m is the UAV mass, g is gravity, and F_thrust is the total force created by the propellers applied to the multirotor along the –z_b axis (points upwards in a horizontal pose).

The closed-loop roll-pitch attitude controller is approximated by the behavior of 2 independent PD controllers for the two rotation angles, and 2 independent P controllers for the yaw rate and thrust. The angular velocity, angular acceleration, and thrust are governed by:

This model assumes the autopilot takes in commanded roll, pitch, yaw rate, and a commanded total thrust force, F^c_thrust. The structure to specify these inputs is generated from control.

The P and D gains for the control inputs are specified as KP_α and KD_α, where α is either the rotation angle or thrust. These gains along with the UAV mass, m, are specified in the Configuration property of the multirotor object.

From these governing equations, the model gives the following variables:

These variables match the output of the state function.