lecture08 - Earth’s rotation and velocity vectors When...

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Unformatted text preview: Earth’s rotation and velocity vectors When two frames are moving with respect to one another, time derivates in one do not trans- form to time derivatives in the other in the same way that postion vectors do, because the time rate of change of the coordinate systems must be taken into account. See Curtis Sec. 1.5 and 1.6. When one frame is rotating with respect to the other (common origin), there is an additional term, a cross product, so that ˙ r IJK = R z (- θ G )(˙ r ECEF + ω ♁ × r ECEF ) or, to solve for the ECEF velocity, ˙ r ECEF = R z ( θ G )˙ r IJK- ω ♁ × r ECEF . We will encounter this phenomenon again when we do relative motion (Hill’s equations) and at- titude dynamics (Euler’s equations). L. Healy – ENAE404 – Spring 2007 – Lecture 8 (Feb. 20) 1 Transform velocity of previous example Suppose, in the example above, we have a ve- locity vector ˙ r =- 4 . 2357 ˆ I- 3 . 2532 ˆ J + 5 . 5310 ˆ K km / s . We already have the rotation matrix; we need to apply it to this vector R z ( θ G )˙ r IJK = . 98257 0 . 18590 0- . 18590 0 . 98257 0 0 1 - 4 . 2357- 3 . 2532 5 . 5310 = - 4 . 7667- 2 . 4091 5 . 5310 km / s and compute the cross product term, ω ♁ × r ECEF = 7 . 292116 × 10- 5 / sec × - 4657 . 4391 .- 2113 . 8 = - . 32020- . 33960 . 0000 km / s L. Healy – ENAE404 – Spring 2007 – Lecture 8 (Feb. 20) 2 Transformed velocity result The ECEF velocity is the difference ˙ r ECEF = - 4 . 7667- 2 . 4091 5 . 5310 - - . 32020- . 33960 . 0000 = - 4 . 4465- 2 . 0695 5 . 5310 km / s . L. Healy – ENAE404 – Spring 2007 – Lecture 8 (Feb. 20) 3 Subsatellite points A line drawn from a satellite at any point on its orbit to the center of the earth pierces the surface at the subsatellite point . The calculation of this point uses our ability to convert from IJK to ECEF coordinates. • Start with IJK Cartesian coordinates. • Transform satellite position to ECEF Carte- sian coordinates. • Transform from rectangular to spherical po- lar coordinates; read off longitude and lat- itude. L. Healy – ENAE404 – Spring 2007 – Lecture 8 (Feb. 20) 4 Calculating subsatellite point The subsatellite point is computed from the spherical polar form (longitude, latitude, dis- tance) of the satellite’s earth-centered earth- fixed coordinates....
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lecture08 - Earth’s rotation and velocity vectors When...

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