SECOND ORDER NONLINEAR BOUNDARY VALUE SOLUTION FOR RELATIVE MOTION USING VOLTERRA THEORY

Application of Volterra multi-dimensional convolution theory to the nonlinear unperturbed circular relative motion boundary value problem is considered in this paper. A solution for the initial three-dimensional relative velocity that executes a transfer between specified three-dimensional initial and fmal relative positions in a specified time period is investigated. A previously generated analytic second order solution for the time dependent relative motion deputy positions in terms of linear, quadratic, and bilinear combinations of the initial conditions and the chief orbital elements is used as the basis for the solution. This framework converts the differential boundary value problem to an algebraic one. The resulting nonlinear equations are examined analytically and numerically in various ways to reveal solution characteristics for several cases. Governing relations are analytically structured in a triple variable, double variable, or single variable formulation. Solution multiplicity and visualization using these optional formulations are discussed. The Clohessy-Wiltshire linear boundary value solution is found to be embedded within the broader nonlinear solution. A family of transfer trajectories parameterized by the transfer time are analyzed numerically, and high sensitivity is discovered near the orbital half-period. For the investigated cases, accuracy of the second order solution improves significantly on that for the linear solution.