JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 29, No. 2, March–April 2006 Application of the Cayley Form to General Spacecraft Motion Andrew J. Sinclair, ∗ John E. Hurtado, † and John L. Junkins ‡ Texas A&M University, College Station, Texas 77843 The study of N-dimensional rigid-body motion is a well-developed field of mechanics. Some of the key results for describing the kinematics of these bodies come from the Cayley transform and the Cayley-transform kinematic relationship. Additionally, several forms of the equations of motion for these bodies have been developed by various derivations. By using Cayley kinematics, the motion of general mechanical systems can be intimately related to the motion of higher-dimensional rigid bodies. This is done by associating each point in the configuration space with an N-dimensional orientation. An example of this is the representation of general orbital and attitude motion of a spacecraft as pure rotation of a four-dimensional rigid body. Another example is the representation of a multibody satellite system as a four-dimensional rigid body. Introduction W HEREAS THE STUDY of mechanics has been motivated by the desire to explain the three-dimensional, physical universe, the mathematical models that have resulted are in no way limited to three dimensions. Higher-dimensional bodies can be kine- matically defined, and by assuming that principles such as conserva- tion of angular momentum and Hamilton’s principle apply in higher- dimensional spaces, their dynamics can also be developed. In par- ticular, one can consider the mechanics of an N -dimensional rigid body, which can be defined as a system the configuration of which can be completely defined by an N × N proper orthogonal matrix. The descriptions of N -dimensional bodies are not simply mathe- matical curiosities: they can be used to describe real systems. This is done by linking the motion of general systems to the rotation of a higher-dimensional rigid body. Three-dimensional analogs to this approach have been used in the past. For example, Junkins and Turner developed an analogy between spacecraft orbital motion and rigid-body rotations. 1 In that work, a physical reference frame was defined using the spacecraft position and velocity vectors. The or- bital motion could then be studied by describing the evolution of this frame. Because of the osculation constraint implied in the def- inition of this frame, however, its motion does not fully capture the orbital dynamics. The approach required explicit reintroduction of Newton’s second law to describe the behavior of the radial distance. Additionally, whereas the kinematic analogy to a rigid body is clear, dynamically it was found that the gyroscopic equations contained variable inertia attributable to changes in the radial distance. This paper presents a new analogy, called the Cayley form, 2 be- tween the combined attitude and orbital motion of a spacecraft and rotational motion of a four-dimensional, rigid body. In addition to incorporating both the attitude and orbital motion, the new analogy more fully incorporates the dynamics in a general sense (i.e., oscu- lation constraints are not imposed nor are explicit reintroductions required). Incorporating both attitude and orbital motion in a single dynamic representation could be seen as a disadvantage, because Presented as Paper 2004-182 at the AAS/AIAA Spaceflight Mechanics Meeting, Maui, HI, 8–12 February 2004; received 7 April 2004; revision received 17 April 2005; accepted for publication 25 April 2005. Copyright c  2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/06 $10.00 in correspondence with the CCC. ∗ Graduate Student, Department of Aerospace Engineering, TAMU-3141; sinclair@tamu.edu; currently Assistant Professor, Department of Aerospace Engineering, Auburn University, 211 Aerospace Engineering Building, Auburn, AL 36849; sinclair@auburn.edu. Member AIAA. † Assistant Professor, Department of Aerospace Engineering. Senior Member AIAA. ‡ Distinguished Professor and George Eppright Chair, Department of Aerospace Engineering. Fellow AIAA. it does not take advantage of the decoupling that occurs between these two motions for the special case of unforced dynamics. Many other choices for representing translational and rotational motion, however, also exhibit coupling in the motion variables, for example, the body components of translational velocity. 3 The disadvantage is mitigated, however, by the fact that for most spacecraft systems the attitude and orbital motions are in fact coupled by both naturally oc- curring forcing terms, such as the rigid-body gravity potential, and control terms, such as fixed-direction thrusters. A second example is presented to further illustrate the new analogy. It involves the at- titude motion of a satellite containing three momentum wheels that is also related to the rotational motion of a four-dimensional body. The main motivation for the Cayley form, which is still under in- vestigation, is to provide a new set of motion variables to be used in the design of controllers for nonlinear systems. Establishing global results for nonlinear control is in general very difficult. Behavior must typically be established on a system-by-system basis or for each value of initial conditions and system parameters. Perhaps one of the most-studied systems in nonlinear control, however, is space- craft attitude motion. By relating the motion of general nonlinear systems to rotational motion, the Cayley form provides the possibil- ity of leveraging this work in spacecraft attitude control to broader classes of problems. This is perhaps easiest to conceive of in ap- plying conventional attitude controllers to other three degree-of- freedom (DOF) systems. An example of this in Ref. 4 demonstrated stabilization of a three-link manipulator using angular velocity feed- back of the associated rotational motion defined by the Cayley form. It was shown that this controller, designed using the Cayley vari- ables, had superior performance to other controllers designed using traditional motion variables. Of course, the Cayley form relates more complicated systems to higher-dimensional rotations. Applying the proposed approach for controller design to these systems requires generalization of attitude controllers to higher dimensions. 4 The ad- vantage of this approach is to provide a common geometric frame- work for the design of controllers for generic nonlinear systems. This paper presents the necessary first step of relating spacecraft motion to higher-dimensional rotations. In each of the examples covered in this paper, the relations are made by associating each point in the six-dimensional configura- tion space with a particular orientation in four-dimensional space. Similar to the Junkins and Turner analogy, the new analogy does not match all of the dynamics properties associated with rigid bodies. In the following sections of this paper the concepts of N -dimensional kinematics and dynamics are reviewed and their relationship to gen- eral systems is discussed. This is then used to analyze general space- craft motion. First, however, some mathematical preliminaries are covered. The Numerical Relative Tensor χ j ik Index notation is a useful shorthand notation for manipulating matrix operations. Matrices or higher-order tensors are expressed 368