三维空间旋转公式

Sensors 2011, 11 8546 15. Zhao, L.; Yan, L.; Cheng, J.; Wang, X. The Research of Inertial Navigation System Based on Submarine Space Motion. In Proceedings of the 2008 Pacific-Asia Workshop on Computational Intelligence and Industrial Application PACIIA ’08, Wuhan, China, 19–20 December 2008; pp. 751-755. 16. Grenon, G.; An, P.E.; Smith, S.M.; Healey, A.J. Enhancement of the inertial navigation system for the Morpheus autonomous underwater vehicles. IEEE J. Ocean. Eng. 2001, 26, 548-560. 17. Barshan, B.; Durrant-Whyte, H.F. An Inertial Navigation System for a Mobile Robot. In Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems IROS ’93, Yokohama, Japan, 26–30 July 1993; pp. 2243-2248. 18. Park, S.K.; Suh, Y.S. A zero velocity detection algorithm using inertial sensors for pedestrian navigation systems. Sensors 2010, 10, 9163-9178. 19. Beatty, M.F., Jr. Principles of Engineering Mechanics; Plenum Press: New York, NY, USA, 1986; Volume 1. 20. Tomažič, S.; Stančin, S. Simultaneous orthogonal rotation angle. Electrotech. Rev. 2011, 78, 7-11. 21. Kuipers, J.B. Quaternions and Rotation Sequences; Princeton University Press: Princeton, NJ, USA, 1999. 22. Mathematica Edition: Version 7.0; Wolfram Research, Inc.: Champaign, IL, USA, 2008. Appendix Derivation of Equivalent Single Rotation Axis and Angle Let us consider simultaneous rotations around three orthogonal axes x, y, and z for angles φx, φy, and φz, and let us represent their composite using a rotation matrix R(φ, v), where φ is the rotation angle in the positive direction around the rotation axis defined by the unit vector v. Due to rotation non-commutativity, R(φ, v) cannot be obtained by considering these three simultaneous rotations sequentially. Therefore, we decompose the total rotation into a sequence of n small rotations in which each such small rotation includes rotations for angles φx/n, φy/n, and φz/n. Because all of these small rotations are equal, the axis and angle of their equivalent single rotation are the same. It then holds that: ( , ) ( / , )n nϕ ϕ=R v R v (A.1) If n is sufficiently large that the rotations become nearly commutative, R(φ, v) can be approximated using simultaneous rotations for angles φx/n, φy/n, and φz/n, sequentially and in any preferred order. Thus, we can write: app app( , ) ( ( ), ( )) n n nϕ ϕ= ΔR v R v (A.2) app app( ( ), ( )) ( , ) ( , ) ( , )x x y y z zn n n n nϕ ϕ ϕ ϕΔ = ⋅ ⋅R v R u R u R u (A.3) where vapp(n) and Δφapp(n) denote approximations of v and φ/n, respectively; ux, uy, and uz denote the x y z coordinate system base vector; and R(φx/n, ux), R(φy/n, uy), R(φz/n, uz), represent rotation matrices: Sensors 2011, 11 8547 1 0 0 ( , ) 0 cos ( ) sin ( ) 0 sin( ) cos( ) x x x x x x n n n n n ϕ ϕ ϕ ϕ ϕ ⎡ ⎤⎢ ⎥ = −⎢ ⎥⎢ ⎥⎣ ⎦ R u (A.4) cos( ) 0 sin( ) ( , ) 0 1 0 sin( ) 0 cos( ) y y y y y y n n n n n ϕ ϕ ϕ ϕ ϕ ⎡ ⎤⎢ ⎥ = ⎢ ⎥⎢ ⎥ −⎣ ⎦ R u (A.5) cos( ) sin( ) 0 ( , ) sin( ) cos( ) 0 0 0 1 z z z z z z n n n n n ϕ ϕ ϕ ϕ ϕ −⎡ ⎤⎢ ⎥ = ⎢ ⎥⎢ ⎥⎣ ⎦ R u (A.6) Inserting Equations (A.4)–(A.6) into Equation (A.4) yields ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) app app c c c s s s s c c s c c s s s s c c s c s s s c c s s c c ( ( ), ( )) ( , ) ( , ) ( , ) y y yz z y y yx x x x xz z z z y y yx x x x xz z z z x x y y z z n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅ − ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ Δ = ⋅ ⋅ = ⎡ ⎤⎢⎢⎢⎢⎢⎢⎢⎣ ⎦ R v R u R u R u ⎥⎥⎥⎥⎥⎥⎥ (A.7) where c and s represent the cosine and sine trigonometric functions according to: ( ) ( ) ( ) ( ) c cos s sin α α α α = = (A.8) Rotating a vector vapp(n) coinciding with the rotation axis results in vapp(n) itself. Thus, we can write: app app app app( ( ), ( )) ( ) ( )n n n nϕΔ ⋅ =R v v v (A.9) Based on Equation (A.9), it is obvious that vector vapp(n) is the rotation matrix R(Δφapp(n), vapp(n)) eigenvector for the eigenvalue λ = 1 (every rotation matrix has such an eigenvalue [19]): app app app) 0( ( ( ), ( )) ( )n n nϕ − ⋅ =Δ IR v v (A.10) To obtain the rotation angle Δφapp(n) from R(Δφapp(n), vapp(n)), let us choose an arbitrary vector p orthogonal to v and perform an R(Δφapp(n), vapp(n)) rotation of this vector: app app( ( ), ( ))n nϕ= Δ ⋅q R v p (A.11) where q is the newly obtained vector as illustrated in Figure A.1. Because p and q are both orthogonal to the rotation axis, the angle between them is equal to the rotation angle φ/n. If these vectors are normalised, we can write: appcos( ( ))nϕΔ = ⋅p q (A.12) 1 1 = = p q (A.13) Sensors 2011, 11 8548 Figure A.1. Rotation angle determination. Increasing n beyond all limits, we find that the small rotations become infinitesimally small and thus commutative [20]. The approximations of vapp(n) and φapp(n) then approach the exact values for v and φ/n: applim( ( ))n n→∞=v v (A.14) applim( ( ))n n nϕ ϕ→∞= ⋅Δ (A.15) Given Equation (A.10) and Equation (A.14), we can then write: app app ) 0(lim( ( ( ), ( )))n n nϕ→∞ − ⋅ =Δ IR v v (A.16) Inserting Equation (A.7) into the above eigenvector system of equations and solving it for a normalised vector v using Mathematica [22] yields: 2 2 2 1 x y x y z z ϕ ϕ ϕ ϕ ϕ ϕ ⎡ ⎤⎢ ⎥ = ⎢ ⎥+ + ⎢ ⎥⎣ ⎦ v (A.17) From Equation (A.12) and Equation (A.15), it follows that: 1lim( cos ( )) n nϕ − →∞ = ⋅ ⋅p q (A.18) Using the l’Hospital rule for limits, the above equation can be expressed as: ( )1 2 2 1 ( )(cos ) 1 ( ) lim lim1 1( ) n n dd dndn d dn n n ϕ − →∞ →∞ ⎛ ⎞⎛ ⎞ − ⋅ ⋅⎜ ⎟ ⋅⎜ ⎟ − ⋅⎜ ⎟ = =⎜ ⎟ ⎜ ⎟⎜ ⎟ −⎜ ⎟⎝ ⎠ ⎝ ⎠ p qp q p q (A.19) A vector orthogonal to vapp(n) is also orthogonal to v in limit Equation (A.14). We can then set: Sensors 2011, 11 8549 1 0 ; 0 ⎡ ⎤⎢ ⎥ = ⊗ =⎢ ⎥⎢ ⎥⎣ ⎦ pp v p p (A.20) where v is defined according to Equation (A.17). It should be noted that we are considering general simultaneous rotations around all three coordinate system axes. Where φy = 0 and φz = 0, the above equation must be suitably changed to obtain the required vector orthogonality, for example: 0 1 0 ⎡ ⎤⎢ ⎥ = ⊗ ⎢ ⎥⎢ ⎥⎣ ⎦ p v Inserting normalised vectors p and q given with Equations (A.20), (A.17), and (A.11) into Equation (A.19) yields the following expression: 2 2 lim ( )n A B C D E F G H I ϕ →∞ + + + + = − + − (A.21) where: 3 2 2 2 cos sin cos ( sin ) cos sin ( sin ) cos cos ( sin ) sin cos ( sin ) sin sin cos ( y yx z y y z x z yx z z z x yx z y z x y yx z y x y z yx z z y z x z A n n n n B n n n C n n n D n n n E n n n ϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕ ϕϕ ϕϕ ϕ ϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕ ⎛ ⎞ = ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅⎜ ⎟⎝ ⎠ = ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ − + ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ ⋅ + ⋅ ⋅ + 2 2 sin ) sin ( sin ) cos ( cos sin ) cos ( cos sin ) sin sin ( cos sin ) y yz y z x y z x xz z z y y x x y y z y x xz z y z n n n F G n n n H n n n I n n n n ϕ ϕϕϕ ϕ ϕ ϕ ϕ ϕ ϕϕϕ ϕ ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ ϕ ϕϕϕ ϕ ϕ ⎛ ⎞ ⋅ − ⋅ ⋅ + ⋅⎜ ⎟⎝ ⎠ = + = ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ (A.22) Using Mathematica [22], we obtain the following result: 2 2 2 x y zϕ ϕ ϕ ϕ= + + (A.23) © 2011 by the authors; licensee MDPI, Basel, Switzerland. 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