This paper addresses the dimensional synthesis of an adaptive mechanism of contact points ie a leg mechanism of a piping inspection robot operating in an irradiated area as a nuclear power plant. This studied mechanism is the leading part of the robot sub-system responsible of the locomotion. Firstly, three architectures are chosen from the literature and their properties are described. Then, a method using a multi-objective optimization is proposed to determine the best architecture and the optimal geometric parameters of a leg taking into account environmental and design constraints. In this context, the objective functions are the minimization of the mechanism size and the maximization of the transmission force factor. Representations of the Pareto front versus the objective functions and the design parameters are given. Finally, the CAD model of several solutions located on the Pareto front are presented and discussed.