**Giuseppe Menga**

Department of Control and Computer Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; menga@polito.it; Tel.: +39-011-090-7272

**Abstract:** The gait and the Foot Placement Estimation (FPE) has recently been extended to 3-D spaces by adopting a specific form of a spherical inverted pendulum (SIP). The approach is very attractive, as it does not involve dynamics, but it is based solely on energies and momenta, however the authors (DeHart et al.) introduced several questionable approximations, in order to reach a manageable solution. The scope of the present paper is to revisit this spherical inverted pendulum applied to biped walking, offering an exact solution to the gait and the FPE by using symbolic computation. This is facilitated by exploiting the Kane's approach to dynamical modelling, and his software environment for symbolic manipulation, called Autolev. It generates explicit formulas describing the energies and angular momenta before/after the impact, along with the mechanics of the impact. As the resulting equations, function of (measurable) angular positions and velocities, are very compact, embedded in a numerical nonlinear solver, are suitable to be implemented in real time and used in practice to control biped robots or lower limb exoskeletons. The two main contributions of the paper are: the recovery of the balance by stepping, in the presence of a push in an arbitrary direction and omnidirectional walking. In this last respect, this specific form of SIP emphasizes the expenditure of energy in the walk. For the first time, at our knowledge, the walk of the SIP, based on energy, has been compared to the simulation of a 12 degrees of freedom biped robot tracking preview signals using the Zero Moment Point (ZMP) of the Linear Inverted Pendulum (LIPM). This quantitatively shows the inefficiency, in terms of energy, of the ZMP-based walk, and the gain due to the recovery of the collision of the flying foot. Similarity in the sagittal plane and differences in the frontal plane of the center of mass trajectories of the two approaches are shown, to open the road to an integration of fully actuated and underactuated controls, for an efficient full-dimensional robot gait to be developed in a future paper.

**Keywords:** humanoid and bipedal locomotion; legged robots; passive walking; foot placement estimation
