Dynamic optimisation problem with characteristic times, widely existing in many areas, is one of the frontiers and hotspots of dynamic optimisation researches. This paper considers a class of dynamic optimisation problems with constraints that depend on the interior points either fixed or variable, where a novel direct pseudospectral method using Legendre-Gauss (LG) collocation points for solving these problems is presented. The formula for the state at the terminal time of each subdomain is derived, which results in a linear combination of the state at the LG points in the subdomains so as to avoid the complex nonlinear integral. The sensitivities of the state at the collocation points with respect to the variable characteristic times are derived to improve the efficiency of the method. Three well-known characteristic time dynamic optimisation problems are solved and compared in detail among the reported literature methods. The research results show the effectiveness of the proposed method.