Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Chapter 4 / Target Tracking
_ _
Although the least-squares method provides smooth estimates they are
biased in this dynamic environment. An α−β−γ filter was used to initialise
the IMM filters with a bandwidth of 0.4 Hz and a 3 s settling time. The
IMM filter position covariances are determined from the Polar to Cartesian
transformation,
E
A
A 2 ⎛ P ⎞
t
2
2
2
( P ) : ⎜
∂
= ⎟ ⋅ diag ( 4 ⋅ σ , 4 ⋅ σ , 4 ⋅ σ )
∆ t
R
Θ
Ψ
⎜
⎝
Z ~
∂
⎟
⎠
4-8
Z ~
⎛
⎜
∂ P
⋅
⎜
⎝ ∂
A
t
⎞
⎟
⎠
T
Equation 4.3-5
The measurement uncertainty is doubled to accommodate any initialisation
transients.
2 2 2
2
2
3
( , σ , σ ) : = ( 10 , 0.
003 , 0.
003 )
σR Θ Ψ
A 2
2
( ∆Pt ) : = 4 ⋅ 50 I 3
&
E ⋅
Equation 4.3-6
Equation 4.3-7
Normally the acceleration uncertainty reflects the sustained acceleration
capability of the target, particularly if the radar can discriminate between
target types. However, as a result of the initialisation pre-filtering used in
this example,
A 2
2
( ∆Pt ) : = 4 ⋅ 50 I 3
&
E ⋅
Equation 4.3-8
A better approach to filter initialisation might be to isolate the IMM
acceleration filter by setting its mode probability to 1 and transitional
probabilities to 0. This method utilises the range-rate measurement thereby
accelerating convergence and improving accuracy. The initial velocity and
acceleration would be zero, and their uncertainties set commensurate with
target capabilities; 300 m/s and 7 g for aircraft, increasing to 20 g for
missiles.
4.4 Stochastic Filtering
Kalman [K.3] (1964) developed the most widely used minimum variance
solution to a set of linear differential equations assuming Gaussian
initialisation, process and measurement noise. When used for non-linear
tracking the process and measurement models are linearised in a form
known as an EKF, a form often reported to be sub-optimal when updated
using measurements corrupted by non-Gaussian errors. Much research has
been undertaken to mitigate the effect of non-linear measurements on