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Let p represent the vector from the i-frame origin to the point P The aframe is another arbitrary reference frame The relative rate of rotation of frame a with respect to frame i represented in frame a is a The notation ia a = [ a ] is de ned in eqn (B15) It should be reviewed now as it will ia ia be used frequently throughout the chapter From eqn (261) in Section 262, the second derivative of p in the i and a frames are related by pi = d2 ra a Ri 2 a va + a a ra + ia ra + ia ia ia a dt2 (1123)

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where r represents the vector from the a-frame origin to the point P The i term from eqn (261) has been dropped in eqn (1123) In eqn (261) represented the vector from the i-frame origin to the a-frame origin In the instances in which this equation is used in this chapter, the origins of the a and i frames will be coincident, so that = 0 The symbol will be reserved for another use later in the chapter From eqn (115) we have that pi = f i + Gi Solving eqn (1123) for the second derivative of ra yields d2 ra dt2 = a f a + Ga 2 a va a a + ia ra ia ia ia a ia ra (1124) (1125)

FIGURE 2-11

These two equation allow a brief comparison between mechanized and strapdown inertial navigation systems In a mechanized INS, the accelerometers and gyros are attached to a platform that is designed to maintain its alignment with the a-frame as the vehicle maneuvers Assuming that the platform is initially aligned with the a-frame, a main idea of the design is to apply torques to the platform so that the outputs of the gyros remain at zero (ie, platform to a-frame alignment is maintained) This approach requires that the gyro biases be accurately calibrated With the mechanized approach, the accelerometers measure the speci c force in the a-frame Given initial conditions va (0) and pa (0) and alignment of the platform frame with the a-frame, eqn (1124) can be integrated to compute va (t) and pa (t) The vehicle attitude with respect to the a-frame could be determined by measuring the angles between the platform and the vehicle In a strapdown INS the accelerometers and gyros are mounted on a platform that is rigidly attached to the vehicle In this approach, the gyros experience the full rotational rate of the vehicle as it maneuvers; hence gyro scale factor accuracy (as well as bias accuracy) is important In the strapdown implementation, the accelerometers measure the body frame (bframe) speci c force f b which can be transformed into the a-frame using f a = Ra f b The strapdown INS approach requires initial conditions va (0) b and pa (0) and Ra (0) The gyro outputs (and navigation state) are used b to compute b (t) so that the INS can maintain Ra (t) computationally by ab b integrating Ra = Ra b b b ab (1126)

The strapdown INS integrates eqn (1125) to compute va (t) and pa (t) The vehicle attitude is represented by Ra b This chapter will focus on strapdown inertial navigation Throughout the presentation, the direction cosine representation of attitude will be assumed Other representations exist (eg, Euler angles, quaternions) Readers are encouraged to consider the quaternion representation (see Appendix D) It may have a steeper learning curve, but provides a computationally e cient and singularity-free attitude representation approach The error modeling approach presented in this chapter is applicable to any of the attitude representations Equations for speci c choices of the a-frame will be considered in the following subsections For each frame, a di erential equation will be derived for each of the position, velocity, and attitude

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