Formation-Flying Using Potential Functions

A group of small spacecraft able to change their relative position and attitude through the use of the potential function method is discussed. The spacecraft shapes, sizes and manoeuvring capabilities are not identical, although all are assumed to manoeuvre using continuous thrusters. A hyperbolic form of the attractive potential function is used to reduce actuator effort by using natural orbital motion to approaching the goal configuration. A superquadric repulsive potential with 3D a rigid object representation is then used to provide an accurate representation of the shape of spacecraft in the potential function. As the spacecraft start away from their goal, a parabolic attractive potential is inefficient as the control force increases with distance from the goal. Using a hyperbolic attractive potential, the control force is independent of the distance to goal, ensuring smooth manoeuvring towards the goal with a bound actuator effort. INTRODUCTION The concept of artificial potential fields was described initially for robot motion planning and manipulators by Khatib. The original global potential field was composed of two independent functions, attractive and repulsive. As robotics technology advances, manoeuvring requirements become more demanding. Since the original work of Khatib modifications to the original potential function have been introduced in areas such as dynamic environments, real object shapes and object rotations. The attractive potential generates a goal configuration with a final specified position and orientation. Typically, the translational attractive potential has not been coupled to the rotational attractive potential. This coupling has been generated through the repulsive potential which can incorporate relative position and orientation for effective collision avoidance. The repulsive potential should be defined whenever more than one manoeuvring object exists in the work space. Many functions are suitable for obstacle representation such as FIRAS, superquadric deformable functions, Gaussian functions, and Laplace functions. In order to enhance the autonomy of manoeuvring spacecraft, enabling them to have a better choice of control action when in proximity to an obstacle or other manoeuvring objects a continuous control approach is discussed in this paper using a hyperbolic attractive potential along with superquadric obstacle fields. The hyperbolic attractive potential provides global stability, and maintains a bounded control force, which vanishes whenever the natural orbital dynamics are sufficient to drive the spacecraft to the goal configuratiuon. The superquadric model then forms repulsive potential fields according to the actual spacecraft shape and dimensions in 3D. The superquadric potential couples the rotational and translation motion of the spacecraft, allowing spacecraft to avoid collisions via translations and/or rotations. HYPERBOLIC ATTRACTIVE POTENTIAL Minimum control activity, bounded actuator effort and global stability are key requirements for effective autonomous spacecraft control. These demands are satisfied using the hyperbolic form of the attractive potential function. The hyperbolic function is smooth in the neighbourhood of its vertex, the specified goal point, ensuring stability at goal. In addition, the hyperbolic surface asymptotes to a cone away from the vertex enabling bounded actuator effort as will be seen later, Fig. 1. The hyperbolic attractive potential function suitable for continuous control is defined as: