Page 159 - Autonomous Mobile Robots

142 Autonomous Mobile Robots

Although the linearization of Equation (3.78) is used to formulate the third row
of Equation (3.76), for fixed ψ the equation

(cos(ψ) cos(δψ) + sin(ψ) sin(δψ))δˆa u
− (sin(ψ) cos(δψ) − cos(ψ) sin(δψ))δˆa v = 0

defines a surface of [δˆa u , δˆa v , δψ] values such that δ˙v n = 0. The δv e dynamics
provide a second such null surface. As long as the EKF drives the vector
[δˆa u , δˆa v , δψ] to the intersection of these two surfaces, the net effect of these
errors are balanced. For this 2D example, the intersection of the two surfaces
is defined by

0 cos ψ − sin ψ cos δψ sin δψ δˆa u
= (3.79)
0 sin ψ cos ψ − sin δψ cos δψ δˆa v
In particular, the EKF causes the accelerometer bias estimation errors δˆa u
and δˆa v to converge to zero, but δψ need not converge to zero. This is the
practical result of the fact that, without acceleration, the yaw error is not
observable.
In real 3D applications, the situation is more complex, since without
maneuvering the errors in estimating pitch and roll have similar effects as
accelerometer bias errors. Therefore, the linearized dynamics have unobserv-
able subspaces. As the vehicle maneuvers, the null surfaces change. Over time,
if the null surfaces change sufficiently, then the yaw and bias estimation errors
will converge toward zero (until the maneuvering stops).

Note that if GPS measurements are unavailable, the integration of the IMU
measurements by the INS is not interrupted. Therefore, this approach does
increase the availability of the state estimate (higher frequency and no dropouts
due to missing satellites). The bandwidth of the state estimate is also increased
since it is determined by the bandwidth of the IMU not the bandwidth of the
GPS receiver. The accuracy of the integrated solution will depend on the quality
of the IMU and the GPS receiver. The length of time that the INS can main-
tain a specified level of accuracy after losing reception of GPS signals will
predominantly depend on the quality of the IMU.

3.6 CHAPTER SUMMARY
This chapter has reviewed the use of the KF and EKF as a tool for fusing
information from various sensors that provide information about the state of
a dynamic system. Preconditions necessary for the use of these methods are
analytic models for the state dynamics and the relation between the state and
the measurement. One prominent application of these tools that satisfies these

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 142 — #44