Coordinate Systems and Their Transformations

Coordinate systems are pervasive in computer simulations of aerospace vehicles. In contrast to frames with their base point and base vectors, coordinate systems have no physical substance. They are just mathematical schemes of relabeling the coordinates of tensors. However, we have seen, if base vectors are expressed in pre­ferred coordinate systems, they take on a particularly simple form. This relationship invites us to display geometrically the direction and positive sense of the coordi­nate axes. Remember, however, that the coordinate axes do not have to emanate from the base point, although it will be convenient for us to do so most of the time.

We are already acquainted with the triads of the heliocentric, inertial, Earth, and body frames. Over these triads we superimpose the preferred coordinate systems with the same names. The axes are labeled 1,2,3, rather than x, y, z, withacapital superscript indicating the associated frame. We let the coordinate axes pierce the unit sphere and connect the piercing points to create surface triangles like orange peels. After some practice it will not be necessary to draw the axes any longer. The sketches of the orange peels will suffice to help you visualize the coordinate systems. At least that’s what happened to me. Professor Stuemke, University of Stuttgart and formerly Peenemuende, was an orange lover and demanded from his students to think of coordinate systems as orange peels. Having dealt over the past 40 years with many coordinate systems, I am thankful that he did not waver in his devotion.

Rather than introducing individual coordinate systems, I pair them up and show mutual relationships that lead to coordinate transformations. We begin with the two most important reference systems.