*1.3. State of the Art of Balance, Stepping and Walking*

In humanoid robotics the generality of the techniques covers basic walking (flat-footed) on flat surfaces in the absence of disturbances. They mostly track for the whole stride a preview signal based on the ZMP of the LIPM [8–10].

At difference, human-like gait, with its mix of fully actuated and underactuated phases (where walking during one of the phases is a "controlled falling") is more complex [11].

Push recovery, walking on rough terrain, and agile footstep control are active research topics [12].

For push recovery in 2-D, along with [2], using the LIPM, see also [13], and, by adding a flywheel [14]. In 3-D the concept of "N step-capturabiliy" has been introduced in [15]. However, the LIPM is used, i.e., the hight of the COG is assumed constant, and no inertia is accounted for.

Ref. [11] discusses theoretically the problems of underactuation and collision in the walk, applied in practice in this paper. The very recent reference [12] contains a comprehensive review of the literature in this field and approaches the walk, as is done here, through foot placement.

#### *1.4. Aims and Organization of the Paper*

The novel approach of processing with a numerical solver the basic energy equations of the SFPE does not involve dynamics and offers several advantages in generating the walk. It does not follow an a priory trajectory, the gait style can be changed at each step, so also cadence and step length, allowing aperiodic walk. It is robust to disturbances. Flat ground has been considered here for simplicity; however, looking one step ahead (watch your step!), also non-flat ground can be accounted for. Maneuvering has been demonstrated in the last section.

This paper has not the intention to offer a solution for the control of a complete robot, but to pave the road, with a motion generation, for a future integration in the gait of three aspects: a realist stride with finite double stance periods, energy efficiency, and a mixture of fully actuated and underactuated phases. The idea to add a small energy at each step to maintain the walk is similar to passive-dynamic walkers [16], and in the line of the frameworks of hybrid zero dynamics [17–19].

The paper is organized as follows: Section 2 discusses the existing results of the SFPE; Section 3 introduces the Kane's method and the symbolic environment Autolev; Section 4 presents the spherical inverted pendulum (SIP) model used in this paper; Section 5 describes the equations needed to estimate the foot placement to reach the balance point; Section 6 applies the approach to find the balance point by stepping in the presence of disturbances in any direction; Section 7 applies the SIP to generate a gait with arbitrary

trajectory and pace, and compares the gait generated from the SIP with the classical gait of an equivalent biped robot based on the ZMP of the LIPM and previews signals [20]; Section 8 concludes the paper, and outlines future works to embed the SIP in the generation of the gait for a complete biped.

#### **2. The Spherical Foot Placement Estimation**

Traditionally, the two degrees of freedom of the SIP (sometime three degrees of freedom are defined, involving, also, the length of the pendulum) were obtained with two rotations on the horizontal axes of the inertial frame [8]. As the final objectives were the two projections of the COG trajectory on the ground, this model originated the celebrated ZMP expressions of the LIPM [9].

In the present case, in order to exploit energies and momenta, rotations along the vertical and one of the horiziontal axes have been chosen. Indicated with *γ* and *ω* the rotation velocities of the SIP, on the vertical and the horizontal axis on the frontal plane of the biped, the approach of [4,5] is based on the projection of the angular momentum on the pivot point of the pendulum on these two directions, and expressing the kinetic energy as function of these two projections. Cleverly, noting that the total energy and the momentum projected on the vertical axis remain constant during the periods before and after the collision of the swing foot with the ground, approaching the collision and the switching of the pivot foot, the problem is solved by writing the equations of the pre-impact, impact and post-impact phases.

#### *Discussion*

To render the problem manageable the two projections of the angular momentum are expressed as a function exclusively of *γ* or *ω* and the two projections of the central inertia are considered constants. i.e., the central inertial matrix is assumed diagonal, and *I*<sup>11</sup> and *I*<sup>33</sup> have identical values (see Appendix A). The impact is solved approximatively. Moreove, no importance is given to the angle of rotation on the vertical axis.
