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for t [0, 8)s for t [8, 10)s for t [0, 8)s for t [8, 10)s
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These conditions result in the vehicle driving 8m north, performing a radian rotation without changing position, driving 8m south, and performing another radian rotation without changing position The vehicle is now back at its original location and orientation The process repeats every 10 seconds In this case, for t [8, 10)s the unobservable space is two dimensional with basis vectors de ned by [000, 000, 100, 000, 000]
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97 GENERAL 3-D PROBLEM and [000, 000, 000, 071, 071]
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The predicted navigation accuracy for the same parameter settings is essentially the same as that indicated in Figure 94
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The main body of this chapter has presented a GPS aided encoder-based dead-reckoning approach built on the assumption of a planar operating surface The approach is easily generalized to nonplanar surfaces when roll angle and the pitch angle can be measured The main changes to the approach would occur in the kinematic model derived in Section 93 The kinematic model of eqn (93) would be replaced by cos( ) cos( ) p = sin( ) cos( ) u (944) sin( ) = cos( ) cos( ) (945)
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Eqn (944) was derived using vt = Rt vb with the assumption that vb = b [u, 0, 0] Eqn (945) is derived using eqn (274) and the assumption that sin( ) the product cos( ) q is negligibly small where q represents the angular rate about the body frame v-axis In addition to accounting for the cos( ) terms in eqns (99) and (918), an additional equation was required to propagate the vertical position component h Note that attitude sensors provide the roll and pitch relative to the geodetic frame When the region of operation is small, the di erence between the tangent frame and the geodetic frame would be inconsequential for this approach When the area of operation is large enough for the di erence to be important, the designer can either compute the tangent frame relative attitude from the geodetic frame relative attitude or modify the approach of this chapter to work in the geodetic frame
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10
AHRS
An attitude and heading reference system (AHRS) is a combination of instruments capable of maintaining an accurate estimate of the vehicle roll , pitch , and yaw as the vehicle maneuvers Various AHRS design approaches are reviewed in 9 of [78] This chapter considers a single approach wherein gyro outputs are integrated through the attitude kinematic equations and aided with accelerometer (ie, gravity direction in body frame) and magnetometer (ie, Earth magnetic eld in body frame) measurements The vehicle attitude describes the relative orientation of the axes of the body (b-frame) and navigation (n-frame) frames-of-reference This book describes three equivalent attitude representations Each attitude representation could be used to implement the method of this chapter: integration of eqn (254) yields the direction cosine matrix from which the Euler angles could be computed by eqns (245 247); integration of eqn (274) yields the Euler angles directly; or, integration of eqn (D28) yields the quaternion representation of the rotational transformation from which the Euler angles could be computed using eqns (D16 D18) All three approaches are theoretically equivalent In this chapter we choose to use the quaternion approach due to its computational e ciency and lack of singularities Various useful quaternion related results are presented in Appendix D A block diagram illustration of the approach described in this chapter is shown in Figure 101 A triad of gyros will measure the angular rate vector b of the b-frame relative to the i-frame represented in the b-frame Inteib gration of the gyro measurements through the attitude kinematics yields the rotation matrix Rn from which the Euler angles can be computed Body b frame accelerometer and magnetometer measurements are transformed to the navigation frame to predict the known Earth gravitational acceleration and magnetic eld vectors A Kalman lter uses the residual measurements to estimate the attitude error and sensor calibration factors Due to the 353
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