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A Gauss Pseudospectral Collocation for Rapid Trajectory Prediction and Guidance

Bradley T. Burchett

2017
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AIAA Atmospheric Flight Mechanics Conference
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unpublished

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
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... tion. 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. Nomenclature I xx , I yy roll and pitch inertia expressed in the projectile reference frame (sl-ft 2 ) A, B Linear State Space Matrices C N A normal force aerodynamic coefficient C X0 axial force aerodynamic coefficient C LP roll rate damping moment aerodynamic coefficient C LDD fin rolling moment aerodynamic coefficient C M A pitch moment due to AOA aerodynamic coefficient C M Q pitch rate damping moment aerodynamic coefficient D projectile characteristic length (ft) f vector of non-linear algebraic constraints g gravitational constant = 32.2 (ft/s 2 ) I identity matrix L N N th order Legendre polynomial m projectile mass (sl) N number of collocation points p, q, r angular velocity vector components expressed in the fixed plane reference frame (rad/s) S = πD 2 /4, projectile reference area (ft 2 ) SL cg stationline of the projectile c.g. location (ft) SL cp stationline of the projectile c.p. location (ft) s downrange distance (calibers) u, v, w translation velocity components of the projectile center of mass resolved in the fixed plane reference frame (ft/s) u 1 , u 2 =C N α,can δ canz , C N α,can δ cany canard commands in no roll frame without effectiveness scaling V = √ u 2 + v 2 + w 2

doi:10.2514/6.2017-0246
fatcat:kgofebzizzcelikioenwuprpbi