The flight of symmetric projectiles is modeled by linear and quasi—linear ODEs known respectively as projectile linear theory and modified projectile linear theory. A Gauss pseudo-spectral collocation may be used to discretize both linear and non-linear ODE models, converting the problem into a set of coupled algebraic equations. Since the approximation is exact at the collocation points, accurate trajectory predictions may be rendered using a small number of points, resulting in very rapid solution. The method allows for solution of high launch elevation trajectories and can account for varying aerodynamic coefficients as well. Results which are compared to a full 6DOF simulation are shown for standard linear, modified linear, and modified linear with varying aero coefficients. By also discretizing the cost function for optimal control, the problem of optimal trajectory design is rendered as an algebraic cost function with algebraic equality constraints. Such a problem is solved by appending equality constraints to the cost function integrand with Lagrange multipliers. The resulting large set of non-linear algebraic equations is then numerically solved. Feasibility of the optimal trajectories was demonstrated by commanding forward canards by a gain scheduled LQR inner loop. The projectile tracked desired trajectories with very little error resulting in a large reduction in dispersion at the target.
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