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Table C4: Calculated Satellite Position in ECEF Coordinates
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APPENDIX C
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CALCULATION OF GPS SV POS & VEL
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Reference Frame Consistency
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Section C2 presented the standard equations use to compute the position pi (ti ) of the i-th satellite vehicle at the time of transmission ti as represented in the ECEF frame at time ti A subtlety in the equations presented in Table C2 is the fact that the computed position for each satellite is in a distinct frame-of-reference and are therefore not compatible The ECEF is not an inertial frame Due to the rotation of the frame, the ECEF frame a time t1 is distinct from the ECEF frame at time t2 Let [Pi ]e(t) denote the coordinates of the point Pi with respect to the ECEF frame at time t Even in the case that P1 is xed in inertial space, in general [P1 ]e(t1 ) = [P1 ]e(t2 ) for t1 = t2 Use of time-of-transit to measure range is premised on the fact that the speed-of-light through a vacuum is constant relative to inertial reference frames Therefore, if the coordinates of two points were known in an ineri i tial frame [P1 ] and [P2 ] the time for light to transit between the points through a vacuum would be [P1 ] [P2 ] c
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When the coordinates of the two points are known in distinct reference a b frames a and b, [P1 ] and [P2 ] , neither of which is an inertial frame, then they could be transformed to an inertial frame: Ri [P1 ] Ri Ra [P2 ] a a b c
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Because the norm of a vector is invariant under rotational transformation, this is equivalent to a b [P1 ] Ra [P2 ] b = c There are two important points First, to calculate the distance between these two points P1 and P2 , they must rst be represented in the same reference frame Second, there is not a unique best frame The following discussion will use the ECEF frame at an arbitrary time denoted as ta Often the selected time is tr the time at which the receiver measured the pseudoranges e(t ) e(t ) Given [P1 ] 1 and [P2 ] 2 the coordinates of P1 and P2 with respect to the ECEF frame at time ta are [P1 ] [P2 ]
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e(ta ) e(ta )
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= Re(ta) [P1 ] 1 = Re(ta) [P2 ] 2
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e(t )
e(t )
e(t1 ) e(t2 )
C4 SATELLITE VELOCITY
The coordinate transformation matrix e e cos ( ie (ta ti )) sin ( ie (ta ti )) 0 e(ta ) e e e Re(ti ) = [ ie (ta ti )]3 = sin ( ie (ta ti )) cos ( ie (ta ti )) 0 0 0 1 is a plane rotation that accounts for the the rotation of the Earth over the time span (ta ti ) at angular rate ie around the ECEF z-axis In the case where ta = tr , then for the i-th satellite, its position is roe e tated by the angle ( ie ti ) = ie (tr ti ) where ti and ti are the quantities p sv p sv de ned in eqn (C1) referred to the i-th satellite Neglecting to transform all satellite positions to the same frame-of-reference can result in range errors of up to 40m
Satellite Velocity
Use of the GPS Doppler measurements requires knowledge of the velocity of the satellite That issue is addressed in this section
C41
Equations from Ephemeris
Equations for the satellite position as a function of time are shown in Table C2 The satellite velocity can be computed as the derivative of the satellite position as summarized in Table C5
Ek k uk rk k X Yk
dik dt
= = = = = = = = = = =