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This paper presents the 3D foot and center of gravity motion planning for the humanoid robot called the "local axis gait" (LAG) algorithm. It permits walking on different kinds of surfaces, such as planes, ramps or stairs. Furthermore, continuous change of the step length and orientation in real time will be possible, due to the real-time linear dynamics model of the walking pattern of the humanoid. The robot model is based on the cart table formulation for planning the center of gravity (COG) and zero moment point (ZMP) motion. The proposed algorithm takes into account physical robot constraints such as joint angles, angular velocity and torques. Torques are computed by the Lagrange method under screws and Lie groups. The LAG is divided into several stages: computation of the footprints; the decision of the ZMP limits around the footprints; the dynamic humanoid COG motion generation based on the cart table model; and joining the footprints of the swing foot by splines. In this way it is possible to generate each step online, using the desired footprints as input. In order to compute the joint torque limits, the Lagrangian method is used under the Lie groups and screw theory. The paper presents and discusses some successful results on the LAG in the full-size humanoid robot Rh-1 developed in the Roboticslab of University Carlos III of Madrid.