Three-Dimensional Biased Proportional Navigation Guidance Based on Spatial Rotation of Predicted Final Velocity

This article presents the design of 3-D biased proportional navigation guidance laws for arrival at a stationary target along a desired direction based on the spatial rotation of the predicted final velocity vector. The focus is on full constructive derivation using vector-form expressions without introducing the local representation of rotation, such as Euler angles or quaternions. The proposed approach synthesizes the bias command in the form of an angular velocity vector through the realization of the predictive control design philosophy, the direction that has been unexplored in a 3-D setting. The proposed approach avoids heuristic choices and approximations in the design process and, hence, overcomes the limitation of earlier studies. The vector-form design approach provides theoretical and practical advantages, including rigor in derivation, clear geometric understandings about the problem provided by identification of the most effective direction for rotation of final velocity, independence from the selection of a fixed coordinate system, avoidance of singularities in local representations, more direct trajectory shaping, and simple implementation.