The article describes the device and the mathematical model of a walking mobile robot of a fundamentally new construction. The difference between this technical solution is the ability to move the robot on surfaces of arbitrary orientation in different coordinate systems: rectangular Cartesian, cylindrical and spherical. In the proposed design, the walking robot mechanism is made in the form of flexible pedipulators and allows the mobile to perform transitions on surfaces that have an arbitrary orientation at different angles to the horizon. The technique for calculating the construction parameters of the walking robot mechanism and the trajectories of the robot legs bending is given. This walking robot is designed to perform technological operations in extreme conditions.