Engineering Notes Use of Cartesian-Coordinate Calibration for Satellite Relative-Motion Control

M ANY of the foundational developments in spacecraft navigation and guidance that were established by Dr. Richard H. Battin and colleagues at theMIT Instrumentation Laboratorywere posed in terms of Cartesian components of the position and velocity vectors. This representation provided advantages in direct relationships with the observation data (for use in navigation algorithms) and with maneuver accelerations and velocity changes (for use in guidance algorithms). Another advantage was the easy inclusion of perturbative accelerations into numerical trajectory solutions, famously using the early development of onboard digital computers. Of courseDr. Battin’swork alsomade prodigious use of orbital elements as another motion representation, taking advantage of their convenient encapsulation of Keplerian motion, such as in the solution of Lambert’s problem [1]. An example of this balanced approach is the combination of the Apollo cross-product steering, which compared the current velocity vector to a required impulsive velocity to determine the velocity-to-be-gained to steer the craft during maneuvers, and the Lambert guidance, which used orbital-elementbased solutions to determine the required impulsive velocity [2,3]. In continuation of this philosophy, this paper presents another approach to combine the use of Cartesian and orbital-element representations in the control of satellite relative motion. The motion of a “deputy” satellite relative to a “chief” satellite can be described using either Cartesian coordinates or orbital-element differences. The evolution of both descriptions obeys nonlinear dynamics, and either set of equations of motion can be linearized for close proximity. The transformation between the two descriptions is a nonlinear mapping, and this mapping can also be linearized for close proximity. These transformations have previously been used to define a calibrated set of Cartesian states. Linearized propagation of these calibrated Cartesian states can contain lower linearization error than linearized propagation of the true Cartesian states [4,5]. Control laws for the motion of the deputy relative to the chief are often designed using the linearized dynamic model for Cartesian coordinates. Implementation of these controllers in the presence of the nonlinear dynamics can result in degraded performance, increased fuel consumption, or even instability. However, the fact that the calibrated Cartesian states can provide a better linear approximation of the impending motion suggests their use in control-law implementation. The calibration process is related to the lower nonlinearity of the orbital-element differences, which was reported by Junkins et al. [6], and additional work in the literature describing the degree of nonlinearity in multiple parameterizations of various systems [7–10]. An implication of that work was that controllers should be designed using parameterizations with low nonlinearity. Here, however, it is shown that even a controller designed using a highly nonlinear parameterization can be improved using the calibration process. This paper demonstrates the improvements in performance, reduction of fuel consumption, and extension of the domain of validity of linearized control laws using calibrated Cartesian coordinates. First, the Cartesian coordinate calibration is reviewed. Next, a linear-quadratic control law designed using linearized dynamics is reviewed, and the implementation of this control law in the presence of nonlinear dynamics is illustrated. Finally, single-impulse rendezvous design using linearized dynamics is reviewed, and the implementation of the maneuver in the presence of nonlinear dynamics is illustrated. In the following, parameters related to the deputy’s orbit about the central body are indicated with a subscript d. For notational compactness, parameters related to the chief’s orbit about the central body are left without subscript.