Use of a reduced IMU to aid a GPS receiver with adaptive tracking loops for land vehicle navigation

Abstract

A reduced inertial measurement unit (IMU) consisting of only one vertical gyro and two horizontal accelerometers or three orthogonal accelerometers can be used in land vehicle navigation systems to reduce volume and cost. In this paper, a reduced IMU is integrated with a Global Positioning System (GPS) receiver whose phase lock loops (PLLs) are aided with the Doppler shift from the integrated system. This approach is called tight integration with loop aiding (TLA). With Doppler aiding, the noise bandwidth of the PLL loop filters can be narrowed more than in the GPS-only case, which results in improved noise suppression within the receiver. In this paper, first the formulae to calculate the PLL noise bandwidth in a TLA GPS/reduced IMU are derived and used to design an adaptive PLL loop filter. Using a series of vehicle tests, results show that the noise bandwidth calculation formulae are valid and the adaptive loop filter can improve the performance of the TLA GPS/reduced IMU in both navigation performance and PLL tracking ability.

Appendix

In the following, the formulae of Doppler aiding error-induced phase errors are derived. In the formula derivation, ξ = 0.707 is the damping ratio of the second-order system, ω _n is undamped natural frequency, T is Kalman filter measurement update period.

Random ramp-induced phase error

The corresponding steady state error, in response to the step of frequency derivative, can be obtained from Fig. 1, and given by:

$$ e_{ss} = \mathop {\lim }\limits_{t \to \infty } e(t) = \mathop {\lim }\limits_{s \to 0} sE(s) = \mathop {\lim }\limits_{s \to 0} {\frac{{ - s^{2} }}{{s^{2} + 2\xi \omega_{n} s + \omega_{n}^{2} }}}\,\Updelta F_{d} (s) = {\frac{{ - d(\Updelta f_{d} (t))/dt}}{{\omega_{n}^{2} }}} $$

(16)

From (16), the phase error can be obtained.

First-order Gaussian Markov-induced phase jitter

In the following, the phase jitter induced by first-order GM Doppler aiding error is derived. From (1), the power spectral density (PSD) of phase jitter can be expressed as

$$ P_{\delta \varphi } (\omega ) = \left| {H(j\omega )} \right|^{2} P_{\Updelta fd} (\omega ) $$

(17)

where $H(s) = {\frac{s}{{s^{2} + 2\xi \omega_{n} s + \omega_{n}^{2} }}}$, $P_{\Updelta fd} (\omega )$ is the PSD of first-order GM Doppler error.

In sampling system, Doppler error $\Updelta f_{d} (t) = u_{1} , 0 \le t < T,\Updelta f_{d} (t) = u_{2} ,T \le t < 2T , \ldots , \Updelta f_{d} (t) = u_{i} ,(i - 1)T \le t < iT, \ldots ,$ where $u_{i}$ is a constant in $(i - 1)T \le t < iT,\;i = 1, \, 2, \, 3, \ldots ,N$, So the Fourier transform of $\Updelta f_{d} (t)$ can be expressed as

$$ \Updelta F_{d} (j\omega ) = \sum\limits_{i = 1}^{N} {\int\limits_{(i - 1)T}^{iT} {u_{i} e^{ - j\omega \tau } } d\tau } = {\frac{{1 - e^{ - j\omega T} }}{j\omega }}\sum\limits_{i = 1}^{N} {u_{i} e^{ - j\omega (i - 1)T} } = {\frac{{1 - e^{ - j\omega T} }}{j\omega }}U(e^{j\omega T} ) $$

(18)

From z-transform definition, when $u_{i} \left( {i = 1, \, 2, \, 3,\ldots} \right)$ is a first-order GM sequence, $U(e^{j\omega T} )$ can be expressed as (Girod et al. 2001)

$$ U(e^{j\omega T} ) = \sigma_{w} {\frac{{e^{j\omega T} }}{{e^{j\omega T} - e^{ - \alpha T} }}} $$

(19)

where $\alpha$ is the inverse of correlation time of the GM sequence, $\sigma_{w}$ is driving noise sequence standard derivation. Since $u_{i}$ is sampling random signal, its variance is $\sigma_{u}^{2} \delta (t)$ (Mandal and Asif 2007). It can be approximated as $\sigma_{u}^{2} /T$ for its equivalent discrete signal since$\int_{0}^{T} {\delta (t)\,dt = } \int_{0}^{T} {{\frac{1}{T}}\,dt = 1}$. Therefore (Gelb 1974),

$$ \sigma_{w} = \sigma_{u} \sqrt {(1 - e^{ - 2\alpha T} )/T} $$

(20)

Where $\sigma_{u}$ is standard derivation of $u(t)$, where $u(t)$ is the GM process.

So the PSD of $\Updelta f_{d} (t)$ can be expressed as

$$ \begin{gathered} P_{\Updelta fd} (\omega ) = \left| {\Updelta F_{d} (j\omega )} \right|^{2} = \sigma_{w}^{2} {\frac{1}{{(1 + e^{ - 2\alpha T} ) - 2e^{ - \alpha T} \cos (\omega T)}}}{\frac{2 - 2\cos (\omega T)}{{\omega^{2} }}} \hfill \\ \, \approx {\frac{{e^{{\alpha T_{k} }} \cdot \sigma_{w}^{2} }}{{\omega^{2} + \gamma^{2} }}} \, \hfill \\ \end{gathered} $$

(21)

where $\gamma = \sqrt {e^{\alpha T} } \cdot \alpha.$ Therefore, the standard derivation of phase jitter is

$$ \sigma_{\delta \varphi }^{2} = {\frac{1}{2\pi }}\int\limits_{ - \infty }^{ + \infty } {P_{\delta \varphi } (\omega )d\omega } \approx {\frac{{e^{\alpha T} \cdot \sigma_{w}^{2} }}{{\omega_{n}^{4} + \gamma^{4} }}}\left( {{\frac{{\gamma^{2} }}{{2\sqrt 2 \omega_{n} }}} + {\frac{{\omega_{n} }}{2\sqrt 2 }} - {\frac{\gamma }{2}}} \right) $$

(22)

In the above derivation, the following formula is used (Chiou 2005):

$$ \int\limits_{0}^{ + \infty } {{\frac{{x^{m - 1} }}{{1 + x^{n} }}}\,dx = {\frac{\pi }{{n\sin \left( {m\pi /n} \right)}}}} ,\;0 < m < n $$

(23)