  The nonlinear optimization problem with linear constraints has many
applications in engineering fields such as the visual-inertial navigation and
localization of an unmanned aerial vehicle maintaining the horizontal flight.
In order to solve this practical problem efficiently, this paper constructs a
continuation method with the trusty time-stepping scheme for the linearly
equality-constrained optimization problem at every sampling time. At every
iteration, the new method only solves a system of linear equations other than
the traditional optimization method such as the sequential quadratic
programming (SQP) method, which needs to solve a quadratic programming
subproblem. Consequently, the new method can save much more computational time
than SQP. Numerical results show that the new method works well for this
problem and its consumed time is about one fifth of that of SQP (the built-in
subroutine fmincon.m of the MATLAB2018a environment) or that of the traditional
dynamical method (the built-in subroutine ode15s.m of the MATLAB2018a
environment). Furthermore, we also give the global convergence analysis of the
new method.
