A new method for performing digital control system attitude computations using quaternions

Space vehicle attitude control system performance is often limited by the computational speed of airborne digital computer hardware. The application of quaternions to digital atti- tude control problems involving coordinate system transformations can reduce computation time by more than 40% over the equivalent direction cosine matrix solution due to a new method of treating quaternions which allows the order of multiplication to be interchanged so as to isolate the most rapidly varying parameters. This paper presents a formulation for control equations in which the control error is expressed as a and all coordinate system transformations are performed using quaternions. The new principle of quaternion algebra which allows interchanging the order of multiplicatio n is developed and used to simplify the control equations. The control equations are applied to a three- gimbal IMU example and the results are compared to an equivalent direction cosine matrix solution. T HE problem of describing the relationship between two coordinate systems is one of the most basic concepts en- countered in the field of navigation and guidance. Until re- cently Euler angles formed the most widely used method of describing the rotation between two coordinate systems. The use of three Euler angles to fix the attitude of a body with re- spect to an inertial or reference coordinate system has the advantage of being well-defined geometrically and fairly simple to visualize. The 3X3 direction cosine matrix, how- ever, was more readily adaptable to high-speed digital com- putation and has replaced the Euler angle method in the solution of all but the most simple navigation problems. The direction cosine matrix approach is particularly useful to de- scribe several successive rotations of a body with respect to a fixed reference system. A third, but infrequently applied, approach to establishing body orientation utilizes the qua- ternion, first devised by Hamilton. The ap- proach makes use of Euler's Theorem which states that any real rotation of one coordinate system with respect to another may be described by a rotation through some angle about a single fixed axis. The is a compact form for rep- resenting the single fixed axis and angle referred to by Euler's Theorem. The may be handled much the same as the direction cosine matrix, in that successive rotations result in successive multiplication. The ad- vantage of the lies in its ability to define the ro- tational relationship between two coordinate systems using only four numbers as opposed to the nine elements of a direc- tion cosine matrix. This results in a similar simplification when the effect of several successive rotations is being com- puted. The principal impediment to use of quaternions has been that the direction cosine matrix, rather than the qua- ternion, is the desired end product of computation. Usually, the computation saved by using quaterion multiplication to perform successive coordinate system rotations is lost when the resultant must be put in the form of a direction cosine matrix. When the object of the computation is to obtain the control error it is not necessary to define the direction cosine matrix explicitly. In fact, optimum control, in the least angle sense, is performed by defining the control error in terms of the single axis and angle of Euler's Theorem. This may be done by formulating the problem completely in terms of quater- nions. The formulation is shown to be particu- larly advantageous due to a new method for treating quater- nions which allows the order of multiplicatio n to be inter- changed.