Coordinated Turn
The Coordinated Turn Motion Model in Radar Tracking
The phrase has two related meanings. To a pilot, a coordinated turn is one flown with the right balance of bank, rudder, and elevator so the aircraft has zero side-slip and the apparent gravity stays in the plane of symmetry. To a radar tracking engineer, the coordinated turn (CT) model is the mathematical idealization of that maneuver: speed is constant, the heading rotates at a fixed angular rate ω, and the resulting ground track is a circular arc. This idealization matters because a target that turns invalidates the constant-velocity assumption built into a basic tracker, causing the predicted position to fall outside the next measurement gate and the track to break.
In a CT tracker the state vector is typically [x, vx, y, vy, ω], position and velocity in a level Cartesian frame plus the turn rate. Over a sample interval T the velocity vector rotates by ωT while its magnitude is preserved, so the speed V = sqrt(vx2 + vy2) is conserved and the path curves with radius R = V/ω. The transition is nonlinear because ω appears inside trigonometric functions and multiplies the velocity states, which is why CT models are paired with an extended Kalman filter (analytic Jacobian) or an unscented Kalman filter (sigma points).
Real targets do not turn at one fixed rate forever, so a single CT model is rarely used alone. Instead, an interacting-multiple-model (IMM) estimator runs a constant-velocity model alongside several CT models tuned to different turn rates, computes the likelihood of each from the innovation, and blends them. During straight flight the CV model dominates; the instant the target banks, probability shifts to the CT models, the blended estimate curves, and the track survives the maneuver with only a small transient.
Constant Turn-Rate State Transition
R = V / ω ω = g × √(n2 − 1) / V
CT velocity update (rotation by ωT):
vx(k+1) = vxcos(ωT) − vysin(ωT)
vy(k+1) = vxsin(ωT) + vycos(ωT)
CT position update:
x(k+1) = x + (sinωT / ω)vx − ((1 − cosωT) / ω)vy
y(k+1) = y + ((1 − cosωT) / ω)vx + (sinωT / ω)vy
Where V = target speed, ω = turn rate (rad/s), n = load factor (g), g ≈ 9.81 m/s2, T = sample interval. As ω → 0 the equations reduce to the constant-velocity model (sinωT/ω → T). Example: V = 300 m/s at n = 9 g gives ω ≈ 0.29 rad/s (≈ 17°/s) and R ≈ 1.03 km.
Motion Models for Radar Trackers Compared
| Model | State vector | Linearity | Filter | Best for | Weakness |
|---|---|---|---|---|---|
| Constant Velocity (CV) | x, vx, y, vy | Linear | Linear KF | Cruise / non-maneuvering | Lags during turns |
| Constant Acceleration (CA) | + ax, ay | Linear | Linear KF | Straight accel/decel | Poor on sustained turns |
| Coordinated Turn (CT), fixed ω | x, vx, y, vy | Linear in state | KF per ω | Known turn rate | Needs model bank |
| Coordinated Turn (CT), unknown ω | + ω | Nonlinear | EKF / UKF | Adaptive turn tracking | Init sensitivity |
| IMM (CV + CT bank) | Mode-mixed | Hybrid | IMM | Mixed maneuver profiles | Higher compute load |
Frequently Asked Questions
How does the coordinated turn model differ from a constant velocity model?
A constant-velocity (CV) model assumes a straight path with zero acceleration, giving a linear transition that tracks cruising targets well but lags during turns. The CT model adds a turn rate ω and rotates the velocity vector at that rate while holding speed constant, so the prediction curves. The CT transition is nonlinear because it contains sin(ωT) and cos(ωT) terms, so it needs an extended or unscented Kalman filter rather than a plain linear one.
How is the turn rate omega estimated in a coordinated turn tracker?
Two approaches dominate. In the augmented-state CT model, ω is added as a state and estimated jointly with position and velocity by an EKF or UKF that adapts to the observed curvature. In the multiple-model approach, several CT filters with fixed turn rates (for example −6, −3, 0, +3, +6 °/s) run in parallel and an IMM algorithm blends them by posterior probability. The augmented method uses fewer filters but is more init-sensitive; the model bank is more robust but heavier.
What turn rates and accelerations does a coordinated turn model need to cover?
Turn rate relates to load factor by ω = g×√(n2−1)/V. An airliner in a standard-rate turn changes heading at 3 °/s, which at a typical terminal-area speed near 95 m/s is about 0.5 g of lateral acceleration; the same 3 °/s at cruise demands a steeper bank and more g. A fighter sustaining 9 g at 300 m/s turns near 17 to 18 °/s with a radius around 1 km, and air-to-air missiles can briefly exceed 30 to 40 g. A practical aircraft tracker spans roughly −20 to +20 °/s, with process noise sized to the expected jerk.