Aircraft turn displacement from non-linear turn rate (instantaneous roll rate)

Question

I've been trying to figure out an aircraft kinematics problem to estimate the x and y offset relative to current position after completing a turn. The turn is a specific change in heading, finishing in level flight but not necessarily starting in level flight.

Assumptions:

• Instantaneous maximum roll rate
• Constant velocity
• No wind
• End turn in straight level flight
• Begin turn at any roll angle (not necessarily level)

Provided inputs/constants:

• Current velocity
• Max roll angle
• Max roll rate
• Gravity
• Desired change in heading
• Initial roll angle

What I tried:

• Split problem into 3 parts: (a) transition from current roll angle to max roll angle (b) sustained max roll angle (c) transition from max roll angle to level.
• For smaller maneuvers, max bank angle won't be reached and phase b will be skipped. It should (I hope) be fairly straightforward to derive the max roll angle reached, substitute for max roll angle, and solve.
• I found the turn angle by integrating turn rate as a function of time w=g tan(b)/v.
• My hope was to use the arc equation (x,y)=(R cos(T),R sin(T) and integrate wrt time but it was pointed out that this is only true for a constant radius arc and is not applicable in this scenario (plus the integral was insane - Wolfram Alpha couldn't solve it).

I believe this problem would be classified as non-uniform angular acceleration in a spiral with constant radial velocity but might have that wrong?

Answer 1

As suggested by Mike, using the turn rate of the function of time rather than arc equation worked for the spiral components. I ended up using the arc equation for the constant radius portion. The small angles tan approximation and a couple of Taylor series were required to get a formal solution. MATLAB simplified analysis of the Taylor series (Using parameters specific to my use case so not necessarily broadly applicable hence not providing a specific formula here).

One key consideration is that the offsets from each phase of the turn have to be rotated to the angle of turn completed in the previous turn phases before summation of the phase-wise offsets.

Answer 2

Without taking the trouble to figure it out, it sounds like a simple ODE problem. Integrated variables would be roll angle, heading, x, and y. I would take the horizontal acceleration as g*tan(roll angle), which assumes constant altitude. I don't see why you need the arc equation.

P.S. This is an old problem in highway/railway design, where turns consist of a circular arc in the center with transition spirals at the ends. The transition spirals have linearly changing curvature. (I programmed those in pre-history.)