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hw02 - c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 = q 2...

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EEL 6935 Electronic Navigation Systems Homework #2 Problem 1 An exponentially correlated random process can be generated using ˙ x ( t ) = - βx ( t ) + w ( t ), where E [ w ( t )] = 0 and E [ w ( t ) w ( t + τ )] = 2 βσ 2 δ ( τ ). Then the statistics of x ( t ) are E [ x ( t )] = 0 and E [ x ( t ) x ( t + τ )] = σ 2 e - β | τ | . Use the matrix superposition integral to show that the discretized version of the above system is x k +1 = e - β Δ t x k + w k , where E [ w k ] = 0 and E [ w 2 k ] = σ 2 (1 - e - 2 β Δ t ), and Δ t = t k +1 - t k ; Problem 2 A quaternion can be written as q = q q 4 , where q = q 1 q 2 q 3 . Note that q 4 is called the scalar part of the quaternion, and q is called the vector part of the quaternion. Show that if q is the quaternion product of a and b , q = a · b , then the vector part of q is given by q = a 4 b + b 4 a + a × b , and and the scalar part of q is given by q 4 = a 4 b 4 - a · b . 1

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Problem 3 Recall that a rotation matrix can be calculated from a quaternion as follows:
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Unformatted text preview: c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 = q 2 1-q 2 2-q 2 3 + q 2 4 2( q 1 q 2-q 3 q 4 ) 2( q 1 q 3 + q 2 q 4 ) 2( q 1 q 2 + q 3 q 4 )-q 2 1 + q 2 2-q 2 3 + q 2 4 2(-q 1 q 4 + q 2 q 3 ) 2( q 1 q 3-q 2 q 4 ) 2( q 1 q 4 + q 2 q 3 )-q 2 1-q 2 2 + q 2 3 + q 2 4 Show that 1 + c 11 + c 22 + c 33 = 4 q 2 4 . Show that c 21-c 12 = 4 q 3 q 4 . Find an algorithm to calculate a quaternion from a rotation matrix. In other words, solve for all four q i ’s as a function of the nine c ij ’s. 2...
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hw02 - c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 = q 2...

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