Strapdown Navigation Using Geometric Algebra: Screw Blade Algorithm

Affiliation(s)

Institute of Automation, National University of Defense Technology, Changsha, China..

Institute of Automation, National University of Defense Technology, Changsha, China..

ABSTRACT

The rigid body motion can be represented by a motor in geometric algebra, and the motor can be rewritten as a trinometric function of the screw blade. In this paper, a screw blade strapdown inertial navigation system (SDINS) algorithm is developed. The trigonometric function form of the motor is derived and utilized to deduce the Bortz equation of the screw blade. The screw blade SDINS algorithm is proposed by using the procedure similar to that of the conventional rotation vector attitude updating algorithm. The superiority of the screw blade algorithm over the conventional ones in precision is analyzed. Simulation results reveal that the screw blade algorithm is more suitable for the high-pre- cision SDINS than the conventional ones.

The rigid body motion can be represented by a motor in geometric algebra, and the motor can be rewritten as a trinometric function of the screw blade. In this paper, a screw blade strapdown inertial navigation system (SDINS) algorithm is developed. The trigonometric function form of the motor is derived and utilized to deduce the Bortz equation of the screw blade. The screw blade SDINS algorithm is proposed by using the procedure similar to that of the conventional rotation vector attitude updating algorithm. The superiority of the screw blade algorithm over the conventional ones in precision is analyzed. Simulation results reveal that the screw blade algorithm is more suitable for the high-pre- cision SDINS than the conventional ones.

Cite this paper

D. Wu and Z. Wang, "Strapdown Navigation Using Geometric Algebra: Screw Blade Algorithm,"*Positioning*, Vol. 3 No. 2, 2012, pp. 13-20. doi: 10.4236/pos.2012.32003.

D. Wu and Z. Wang, "Strapdown Navigation Using Geometric Algebra: Screw Blade Algorithm,"

References

[1] J. E. Bortz, “A New Mathematical Formulation for Strap- down Inertial Navigation,” IEEE Transactions on Aero- space and Electronic Systems, Vol. AES-7, No. 1, 1971, pp. 61-66. doi:10.1109/TAES.1971.310252

[2] M. B. Ignagni, “Optimal Strapdown Attitude Integration Algorithms,” Journal of Guidance, Vol. 13, No. 2, 1990, pp. 363-369. doi:10.2514/3.20558

[3] M. B. Ignagni, “Efficient Class of Optimized Coning Com- pensation Algorithm,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 2, 1996, pp. 424-429. doi:10.2514/3.21635

[4] J. W. Jordan, “An Accurate Strapdown Direction Cosine Algorithm,” NASA TN-D-5384, 1969.

[5] J. G. Lee, Y. J. Yoon, J. G. Mark and D. A. Tazartes, “Ex- tension of Strapdown Attitude Algorithm for High-Fre- quency Base Motion,” Journal of Guidance, Vol. 13, No. 4, 1990, pp. 738-743. doi:10.2514/3.25393

[6] R. B. Miller, “A New Strapdown Attitude Algorithm,” Journal of Guidance, Vol. 6, No. 4, 1983, pp. 287-291. doi:10.2514/3.19831

[7] P. G. Savage, “Strapdown Inertial Navigation Integration Algorithm Design Part 1: Attitude Algorithms,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 1, 1998, pp. 19-28. doi:10.2514/2.4228

[8] P. G. Savage, “A Unified Mathematical Framework for Strapdown Algorithm Design,” Journal of Guidance, Con- trol, and Dynamics, Vol. 29, No. 2, 2006, pp. 237-249. doi:10.2514/1.17112

[9] Y. Wu, X. Hu, D. Hu, T. Li and J. Lian, “Strapdown Iner- tial Navigation System Algorithms Based on Dual Qua- ternion,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 41, No. 1, 2005, pp. 110-132. doi:10.1109/TAES.2005.1413751

[10] D. Wu and Z. Wang, “Strapdown Inertial Navigation Sys- tem Algorithm Based on Geometric Algebra,” Advances in Applied Clifford Algebras, Vol. 22, No. 1, 2012, pp. 1- 17.

[11] L. Dorst, D. Fontijne and S. Mann, “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry,” 2nd Edition, Morgan Kaufmann Publishers, San Francisco, 2007.

[12] E. Bayro-Corrochano, “Geometric Computing for Wave- let Transforms, Robot Vision, Learning, Control and Ac- tion,” Springer Verlag, London, 2010.

[13] D. Hestenes, “New Foundations for Classical Mechan- ics,” 2nd Edition, Kluwer Academic Publishers, Dordrecht, 1999.

[14] M. B. Ignagni, “Duality of Optimal Strapdown Sculling and Coning Compensation Algorithms,” Journal of the In- stitute of Navigation, Vol. 45, No. 2, 1998, pp. 85-95.

[15] P. G. Savage, “Strapdown Inertial Navigation Integration Algorithm Design Part 2: Velocity and Position Algo- rithms,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 208-221. doi:10.2514/2.4242

[16] Y. Wu, P. Wang and X. Hu, “Algorithm of Earth-Cen- tered Earth-Fixed Coordinates to Geodetic Coordinates,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 39, No. 4, 2003, pp. 1457-1461. doi:10.1109/TAES.2003.1261144

[17] R. A. McKern, “A Study of Transformation Algorithms for Use in a Digital Computer,” M.S. Thesis, MIT, Cam- bridge, 1968.

[18] Y. Qin, “Inertial Navigation (in Chinese),” Science pub- lisher, Beijing, 2006.

[1] J. E. Bortz, “A New Mathematical Formulation for Strap- down Inertial Navigation,” IEEE Transactions on Aero- space and Electronic Systems, Vol. AES-7, No. 1, 1971, pp. 61-66. doi:10.1109/TAES.1971.310252

[2] M. B. Ignagni, “Optimal Strapdown Attitude Integration Algorithms,” Journal of Guidance, Vol. 13, No. 2, 1990, pp. 363-369. doi:10.2514/3.20558

[3] M. B. Ignagni, “Efficient Class of Optimized Coning Com- pensation Algorithm,” Journal of Guidance, Control, and Dynamics, Vol. 19, No. 2, 1996, pp. 424-429. doi:10.2514/3.21635

[4] J. W. Jordan, “An Accurate Strapdown Direction Cosine Algorithm,” NASA TN-D-5384, 1969.

[5] J. G. Lee, Y. J. Yoon, J. G. Mark and D. A. Tazartes, “Ex- tension of Strapdown Attitude Algorithm for High-Fre- quency Base Motion,” Journal of Guidance, Vol. 13, No. 4, 1990, pp. 738-743. doi:10.2514/3.25393

[6] R. B. Miller, “A New Strapdown Attitude Algorithm,” Journal of Guidance, Vol. 6, No. 4, 1983, pp. 287-291. doi:10.2514/3.19831

[7] P. G. Savage, “Strapdown Inertial Navigation Integration Algorithm Design Part 1: Attitude Algorithms,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 1, 1998, pp. 19-28. doi:10.2514/2.4228

[8] P. G. Savage, “A Unified Mathematical Framework for Strapdown Algorithm Design,” Journal of Guidance, Con- trol, and Dynamics, Vol. 29, No. 2, 2006, pp. 237-249. doi:10.2514/1.17112

[9] Y. Wu, X. Hu, D. Hu, T. Li and J. Lian, “Strapdown Iner- tial Navigation System Algorithms Based on Dual Qua- ternion,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 41, No. 1, 2005, pp. 110-132. doi:10.1109/TAES.2005.1413751

[10] D. Wu and Z. Wang, “Strapdown Inertial Navigation Sys- tem Algorithm Based on Geometric Algebra,” Advances in Applied Clifford Algebras, Vol. 22, No. 1, 2012, pp. 1- 17.

[11] L. Dorst, D. Fontijne and S. Mann, “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry,” 2nd Edition, Morgan Kaufmann Publishers, San Francisco, 2007.

[12] E. Bayro-Corrochano, “Geometric Computing for Wave- let Transforms, Robot Vision, Learning, Control and Ac- tion,” Springer Verlag, London, 2010.

[13] D. Hestenes, “New Foundations for Classical Mechan- ics,” 2nd Edition, Kluwer Academic Publishers, Dordrecht, 1999.

[14] M. B. Ignagni, “Duality of Optimal Strapdown Sculling and Coning Compensation Algorithms,” Journal of the In- stitute of Navigation, Vol. 45, No. 2, 1998, pp. 85-95.

[15] P. G. Savage, “Strapdown Inertial Navigation Integration Algorithm Design Part 2: Velocity and Position Algo- rithms,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 208-221. doi:10.2514/2.4242

[16] Y. Wu, P. Wang and X. Hu, “Algorithm of Earth-Cen- tered Earth-Fixed Coordinates to Geodetic Coordinates,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 39, No. 4, 2003, pp. 1457-1461. doi:10.1109/TAES.2003.1261144

[17] R. A. McKern, “A Study of Transformation Algorithms for Use in a Digital Computer,” M.S. Thesis, MIT, Cam- bridge, 1968.

[18] Y. Qin, “Inertial Navigation (in Chinese),” Science pub- lisher, Beijing, 2006.