Quadrature discretization method in tethered satellite control

A pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented. Traditional pseudospectral methods expand the state and control trajectories using global Lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the Jacobi family, such as Legendre or Chebyshev polynomials, which are orthogonal with respect to a specific weight function over a fixed interval. Although these methods have many advantages, the location of the collocation points are more or less fixed. The method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval. The trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method.