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In both cases, the matrix E involved in the computation of the Euler angle derivatives of eqn (274) on page 57 becomes singular For the 2 matrix 1 is large (potentially in nite) which can greatly magnify sensor E errors in the computation of the Euler angle derivatives The following example considers the motion similar to that of the previous example in the sense that the vehicle is maneuvering near = rad, 2 but using a quaternion implementation Example D2 For the initial condition ( , , ) = (0, 0, 0), the initial quaternion is b(0) = [1, 0, 0, 0] Figure D1 shows the body frame angular rates [p, q, r] The quaternion that results from the integration of either eqn (D28) or (D30) is plotted
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D7 REFERENCES AND FURTHER READING
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Figure D1: Angular rates for Example D2
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in Figure D2 The Euler angles computed from the quaternion are plotted in Figure D3 Throughout the simulation, the gain de ned by the matrix b in eqn (D29) is small For example, due to b being a unit quaternion each row has magnitude less than 10
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For an additional example of the use of quaternions, especially for attitudes with near 90 degrees, see the discussion related to Figure 105
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The main sources for the material in this appendix were [23, 45, 50, 103, 115, 118, 137] Attitude representations are compared in [120]
Exercises
Exercise D1 Show that eqn (D7) yields the same result as eqn (D3) Exercise D2 Use the de nitions of eqns (D6) and (D7) with the de nition of quaternion products in eqn (D7) to verify that eqn (D5) Exercise D3 Verify that eqn (D13) (D14) yields the same result as eqn
APPENDIX D QUATERNIONS
12 1 08 06 Quaternion 04 02 0 02 04 06 08 0 05 1 15 2 Time, t, sec 25 3 35 4
Figure D2: Quaternion b(t) for Example D2: b1 is solid, b2 is dash-dotted, b3 is dashed, and b4 is dotted
Roll, , deg
200 0 200 100 0 100 200 0 200
Pitch, , deg
yaw, , deg
2 Time, t, sec
Figure D3: Computed attitude by eqns (D16-D18) for Example D2
D8 EXERCISES
Exercise D4 Let b and b represent a quaternion and its computed value = 1 q b and b = 1 q b with q = [0, ] being the quaternion where b 2 2 representation of the angular rate vector and q = [0, ] being the quaternion representation of the measured angular rate vector = + xg + g The quaternion error can be de ne as b = b b 1 where b [1, ] 1 Use eqn (D14) to show that the small angle rotation corresponding to b is R( b) = (I + 2[ ]) (D37) to rst order 2 Given the product b = b b 1 and eqn (D37), show that R(b) = (I + 2[ ]) R(b) 3 Starting from the equation b = b b 1 + b b 1 use the fact that b 1 = b 1 b b 1 to show that b = 1 (q b b q ) 2 (D38)
4 Show that the vector portion of eqn (D38) reduces to = 1 1 [ ] xg g 2 2 (D39)
after all higher order terms have been dropped
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Bibliography
[1] N Ackroyd and R Lorimer Global navigation: a GPS user s guide Lloyd s of London Press, London; NewYork, 1990 [2] D W Allan Statistics of atomic frequency standards Proceedings of the IEEE, 54(2):221 230, 1966 [3] W G Anderson and E H Fritze Instrument approach system steering computer Proceedings of Institute of Radio Engineers, 41(2):219 228, February 1953 [4] Anonymous Phase I NAVSTAR/GPS major eld test objective report thermostatic correction Technical report, Navstar/GPS Joint Program O ce, Space & Missile Systems Organization, Los Angeles Air Force Station, Los Angeles, California, May 4 1979 [5] Anonymous Recommended practice for atmospheric and space ight vehicle coordinate systems Technical Report AIAA/ANSI R-0041992, American Institute of Aeronautics and Astronautics, Reston, VA, 1992 [6] Anonymous NAVSTAR GPS space segment/navigation user interfaces Technical Report ICD-GPS-200, ARINC Research Corporation, April 1993 [7] Anonymous RTCM recommended standards for di erential navstar GPS service Technical Report 104, RTCM Special Committee, January 1994 [8] Anonymous The global positioning system: A shared national asset Technical report, National Research Council, Washington, DC, 1995 [9] Anonymous IEEE standard speci cation format guide and test procedure for single-axis laser gyros Technical Report IEEE Std 6471995, IEEE, New York, NY, 1995 515
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