An enlarged polygon method without binary variables for obstacle avoidance trajectory optimization

In this article, an Enlarged Polygon/Polyhedron (ELP) method without binary variables is proposed to represent the Convex Polygonal/Polyhedral Obstacle Avoidance (CPOA) constraints in trajectory optimization. First, the equivalent condition of a point outside the convex set is given and proved rigorously. Then, the ELP condition describing the CPOA constraints equivalently is given without introducing binary variables, and its geometric meaning is explained. Finally, the ELP method is used to transform the CPOA trajectory optimization problem into an optimal control problem without binary variables. The effectiveness and validity of ELP method are demonstrated through simulations with both simple linear dynamic model (unmanned aerial vehicle) and complex nonlinear dynamic model (hypersonic glide vehicle). Comparison indicates the computational time of ELP method is only 1%-20% of that of the traditional Mixed-Integer Programming (MIP) method.