Chapter 9.5.4: INERTIAL SYSTEMS TECHNOLOGIES: Examples

9.5.4 Examples

9.5.4.1 Damping Vertical Channel Errors The propagation of altitude error δh over time is governed by Eq. 9.80:

where the Schuler period T_Schuler ≈ 5046 s at sea level. The INS initialization procedure can make the initial value δh(t₀) very small (e.g., in the order of a meter). It can also make the vertical component of the initial gravity model error very small (e.g., by making it match the sensed vertical acceleration during alignment). Thereafter, the vertical channel navigation error due to zero-mean white accelerometer noise ω(t) will propagate according to the model

where is vertical velocity error.

The equivalent state transition matrix in discrete time with timestep Δt is

and the corresponding Riccati equation for propagation of the covariance matrix P_vert.chan of vertical channel navigation errors has the form

where q_{accelerometer} is the incremental variance of velocity uncertainty per timestep Δt due to vertical altimeter noise. For example, for velocity random walk errors specified as having VRW meter per second per root hour, we obtain

Figure 9.32 is a plot of altitude uncertainty versus time after INS initialization for a range of accelerometer white noise levels, from 10^-2 to 10² m/s/√hr. All the solid-line plots will increase over time without bound.

Barometric Altimeter for Vertical Channel Damping. If a barometric altimeter is to be used for vertical channel stabilization, then the altimeter error δh_altimeter will not be zero-mean white noise, but something more like a zero-mean exponentially correlated error. This sort of error has a discrete-time model of the form

where τ_altimeter is the correlation time and the white-noise sequence {ω_k} has variance

for steady-state altitude variance .

In this case, the augmented vertical channel state vector

and the resulting Riccati equations for state uncertainty will be

where R_altimeter is mean-squared altimeter noise, exclusive of the correlated component δh_altimeter.

The dotted lines in Fig. 9.32 are plots of altitude uncertainty with vertical channel damping, using a barometric altimeter. The assumed atmospheric and altimeter model parameters are written on the plot. These show much better performance than does the undamped case, over the same range of accelerometer noise levels. In all cases, the damped results do not continue to grow without bound.

Figure 9.32 was generated by the m-file VertChanErr.m on the CD-ROM.

9.5.4.2 Carouseling Accelerometer bias errors δb_acc couple into horizontal navigation errors, as modeled in Eqs. 9.86-9.87 and 9.96:

where δb_acc is the vector of accelerometer biases and can have any of the values given in Eqs. 9.97-9.100.

Figure 9.33 is a plot of fourteen hours of simulated INS position errors resulting from 10-μg north accelerometer bias on a gimbaled INS, with and without carouseling. The simulated carousel rotation period is 5 min, and the resulting navigation errors are reduced by more than an order of magnitude. This shows why carouseling (and indexing) is a popular implementation scheme.

The plot in Figure 9.33 was generated by the MATLAB m-file AccBias-Carousel.m on the CD-ROM. Note that it exhibits the same coriolis-coupled

Schuler oscillations as in Fig. 9.30, which was the result of an initial north velocity error. In this case, it is a north acceleration error, the result of which is that the Schuler oscillation center is offset from the starting point.

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Chapter 9.5.4: INERTIAL SYSTEMS TECHNOLOGIES: Examples

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