An algorithm for aircraft trajectory prediction is presented that is based on two algebraic representations of the aircraft c.g. trajectory in the Frenet frame: (1) a cylindrical helix, during steady flight segments, and (2) a third-order accurate expansion, during transient maneuvering phases. This technique allows for an efficient evaluation of future c.g. positions in a time interval the extension of which depends on the aircraft maneuver state, without requiring the implementation of the aircraft dynamic model. A Kalman filtering technique with a fixed-lag smoother is used for simultaneously filtering the sensor noise and estimating accelerometer and rate-gyro signal derivatives. The effect of a step variation of commanded load factor and/or roll angle on the aircraft position at following times is displayed to the pilot as a visual aid for representing achievable future positions during the maneuver. The algorithm is demonstrated by computer simulation of reverse turn maneuvers of an F-16 fighter aircraft model.
NomenclatureA = agility vector (time derivative of the acceleration); state matrix of the continuous-time Markov process a = acceleration vector a acc = accelerometer output, (a acc,x , a acc,y , a acc,z ) T a i, j = element of A a n = normal component of the acceleration B = input matrix of the continuous-time Markov process b = binormal unit vector C = output matrix D = Kalman filter dynamic matrix g = gravity acceleration vector h = altitude I = identity matrix K = steady-state Kalman filter gain matrix K ARI = aileron-rudder interconnect gain K W = weight coefficient k = discrete time k 1 , k 2 = intrinsic variables (curvature and torsion) L = Kalman smoother lag L B I , L B F , L F I = rotation matrices m = number of estimated time derivatives of the signal x(t) n = normal unit vector n z = normal load factor Q = covariance matrix of the discrete-time system q i, j = element of Q R = position vector of c.g. in the inertial frame R A = position of the accelerometers in body frame, (x A , y A , z A ) T r = autocorrelation function of the innovation process s = curvilinear abscissa T = sampling time t = time t = tangent unit vector V = velocity modulus V = velocity vector v = observation (Gaussian white) noise W = process (Gaussian white) noise of the discrete-time system w = process (Gaussian white) noise X = discrete-time state vector, x(kT ) x = estimated variable x = continuous-time state vector, (x, dx/dt, d 2 x/dt 2 , . . . , d m x/dt m ) T (x F , y F , z F ) T = position vector in the Frenet frame y = noise corrupted measure of x α = angle of attack δ A , δ R = aileron and rudder deflection λ max = dominant eigenvalue of D ρ = radius of the helix σ 2 v = measurement noise variance of the continuous-time Markov process σ 2 w = process noise variance of the continuous-time Markov process Φ = state matrix of the discrete-time system φ = roll angle ω = angular velocity, ( p, q, r ) T Subscripts A = in the accelerometer position acc = accelerometers B = body frame est = estimated F = Frenet frame...