Analytical solutions to a guidance problem

Analytical solutions to the nonlinear equations of relative motion of a constant-velocity target and an interceptor that uses a form of proportional navigation as a guidance law are obtained for navigation gains of 3 and 4. The solutions, in terms of elliptic functions and integrals, are new closed-form solutions to a much investigated guidance problem and, although relatively complicated, may be readily evaluated. An additional result is a condition for intercept that is based on the initial conditions. Numerical results for an exoatmospheric intercept example are included.