A general construction framework is introduced to provide three-dimensional balance criteria for multi-segmental legged robots. The approach is based on a reduced-order dynamic model that includes a 3-D mechanism with minimal number of degrees of freedom (DOFs) capable of representing the equivalent center-of-mass (COM) dynamics of a generic legged robot. Systematic mappings from the legged system to the reduced-order model include COM workspace, foot support region, ground reaction forces and moments, center of pressure, joint angle and actuation limits, and angular momentum. A numerical optimization algorithm is established to solve for the minimum and maximum initial COM velocity components of the 3-D mechanism that satisfy nonlinear constraints such as center of pressure boundaries, positive normal reaction, friction cone inequality, and the ability to reach a final static equilibrium. The resulting velocity extrema are the boundaries of the balanced state domain of the given legged robot, and provide the criteria of balanced versus falling state. The balanced state domain, constructed as a viability kernel, is the reachable superset of all possible controller-based domains, and represents the necessary and sufficient condition for balancing without stepping. The approach is validated for 1- and 2-DOF legged systems in sagittal plane, and applications are illustrated for a planar 4-DOF biped system.