2.2 Closed Loop Kalman Mechanization 
The closed loop Kalman mechanization, shown in Figure 2-3, differs from the previous configuration 
in two aspects. First the INS supplies velocity and attitude information in real time to assist the 
GPS tracking loops for acquisition and tracking purposes. This allows the GPS set to have much higher 
quality (dynamically unbiased) pseudo-range and range rate measurements. Tracking loop bandwidths can 
be made smaller resulting in lower RMS phase error. The higher quality measurements are processed by 
the Kalman filter to result in state estimates X that are free of dynamically induced biases. The 
‘closed loop' occurs when the INS states X resident in the INS computer are corrected by the best 
estimate of their errors X, computed by the Kalman filter. This assures that the signals which aid 
the GPS tracking loops are of the highest quality. 
The advantage of this mechanization is performance. System initialization and navigation can be 
performed to ultimate inherent precisions, as the systems mutually aid each other. Position, velocity 
and attitude exhibit smaller RMS error. Because these errors are smaller, the Kalman filter can be 
simplet as second-order effects become insignificant. (Figure 2-4). 
The disadvantage with this system is that hardware and software integration is a greater task, 
requiring intimate knowledge of the GPS signal processor, and a respect for the physical limitations 
of the INS. The GPS set ideally should have digitally controlled and programmable tracking loops. 
The major design task would be to ensure that velocity aiding is done in a way that assures the 
accurate removal of the dynamically induced phase error transient. The receiver tracking loop would 
only be required to zero dynamic error. As will be explained in Section 3, improper design can lead 
to system instability. 
Hence those applications requiring the utmost in position, velocity or attitude information under 
dynamic conditions will require a closed loop Kalman feedback mechanization. The remainder of this 
paper deals with this mechanization, specifically stabilization aspects and considerations for design 
and field implementation. 
3. STABILITY OF MUTUALLY AIDED HYBRID 
To illustrate the stability paradox of the closed loop Kalman filter mechanization a simple one- 
dimensional example is shown in Figure 3-1. The GPS receiver is represented by a phase-lock loop with 
low pass filter F(S) and a VCOl ^ The INS is a horizontal channel represented by two integrators © - 
The velocity is updated by a Kalman gain K, which processes the innovation found by differencing INS 
position X and GPS range R. 
Careful examination of Figure 3-1 indicates that an instability is due to the slow response time 
(correlation time) of the tracking loop. The fact that the measurement R has a time correlated 
component, that is, the phase error, which is not observable by the Kalman filter, is the responsible 
cause. The filter poorly models system and measurement dynamics, giving rise to the instability. 
Several approaches have been proposed which provide stable response modes of the system. Carroll 
& Michelson suggest a compensation network which yields a well damped overall system response. 
Widnall suggests the use of reduced Kalman gains which is motivated by noting that with zero INS 
feedback gains, the systems decouple. If it were possible to augment the Kalman filter states to 
estimate the code loop dynamics, the correlation would be accounted for. This technique would result | 
in two extra states per channel for code and probably three states per channel for the carrier. That I 
is, 20 extra states would be necessary for a 4 channel receiver. i 
The method proposed herein is unique with respect to the above techniques in that the pseudo range 
measurements are decorrelated. By feeding forward the phase error e (t) and summing with R we recreate | 
the front-end signal (Figure 3-2): 
R=R+e (t) UM 
The result is a measurement which is independent of the INS states and uncorrelated in time. The 
major drawbacks are 
a) The full noise is present in the measurement. 
b) The power spectrum of this noise must be estimated to optimally filter the noise from the 
measurement. 
The decorrelation process allows the Kalman filter to better represent the dynamics of the system 
without further augmentation. The Kalman filter is optimal and the stability margin improved. 
Because signal and noise power (PSD) will generally vary, the filter can be of an adaptive nature, 
where the PSD is estimated real time, or can be of fixed gain where the statistics chosen will assure 
adequate system response over the entire range of the PSD. 
Figure 3-3a illustrates mode behaviour for the conventional system. The steady state gains are 
computed from the discrete Riccati equation and eigenvalues for the continuous system are found. 
As can be seen, within the operational range of the code loop, unstable modes exist, in the form of 
an unbounded oscillation. These correspond to the inertial navigator states. A third mode resides 
near the origin on the real axis. The code loop modes are strongly coupled to the navigator for this 
configuration. The modes lie on the real axis, indicating an overdamped response. 
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