The subject of this paper is the application of the mechanics of static stability to determine the potential upset of computer-controlled quadruped walking vehicles which are under development for use in very uneven terrain by the European construction and forestry industries. The margin of static stability is measured using normalized values for virtual power, each value corresponding to the motion about an axis of rotation which is associated with one mode of potential overturning. Static stability is treated for spatial positioning of the feet and for both three and four feet in contact with the ground. The inertial loads from a heavy tool, such as an excavating arm, are included in a quasi-static manner. The values for normalized virtual power are then related to a geometrical description of stability of the vehicle as if it were operating on level ground. Recommendations are made for visually displaying the geometrical description to the operator of the machine. The required measurements of force and position are listed.