# Appendix 18.A Three-Dimensional Constant Turn Rate Kinematics

In this section, we develop a method to generate a trajectory that can include constant turn-rate turns in both the horizontal and the vertical. This will allow us to create moving object trajectories that contain horizontal turns, vertical dives, and spirals.

Assume that an object at time t is moving at a horizontal heading and at a vertical attitude of . Consider the Cartesian East–North–Up coordinate system shown in Figure 18.25, where the East–North plane is the horizontal plane.

## 18.A.1 General Velocity Components for Constant Turn Rate Motion

The Cartesian components of velocity for constant speed in an East–North–Up Cartesian system can be expressed as

where ϑ and γ represent the heading (−180^{ o} ≤ ϑ < 180^{ o}) and attitude (−90^{ o} ≤ γ ≤ 90^{ o}), respectively.

For a constant rate turn, referring to Figure 18.26, we can write

where T t − t_{0} and Here, Ω and Ψ represent the constant turn rates in ...

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