**3. The Kane's Method and Autolev**

In this work, the so-called Kane's method [6] was adopted to model the spherical inverted pendulum. This method is particularly interesting in this case because it is equally applicable to either holonomic and non-holonomic systems and, for non-holonomic systems, without the need to introduce Lagrangian multipliers. Briefly, the main contribution of the Kane's method is that, through the concepts of motion variables (later called generalized speeds), the vectors of partial velocities and partial angular velocities, generalized active forces and generalized inertia forces, the dynamical equations are automatically determined, enabling forces and torques with no influence on the dynamics to be eliminated early in the analysis. Early elimination of these noncontributing forces and torques greatly simplifies the mathematics and enables problems with greater complexity to be handled.

#### *3.1. Generalized Coordinates and Speeds*

A multi-body system, which possesses *n* degrees of freedom, is represented by a state with a *n*-dimensional vector **q** of configuration variables (*generalized coordinates*) and an identical dimension vector **u** of *generalized speeds* called also *motion variables*, that could be any nonsingular combination of the time derivatives of the generalized coordinates that describe the configuration of a system. These are the kinematical differential equations:

$$\mu\_r = \sum\_{i=1,\cdots,n} \mathbf{Y}\_{ri} \mathbf{q}\_{i\prime} r = 1, \cdots, n \tag{1}$$

*Yri* may be in general nonlinear in the configuration variables so that the equations of motion can take on a particularly compact (and thus computationally efficient) form with the effective use of generalized speeds.

#### *3.2. Partial Velocities and Angular Velocities*

Partial velocities of each point (partial angular velocity of each body) are the *n* threedimensional vectors expressing the velocities of that point (angular velocity of that body) as a linear combination of the generalized speeds. Let be **v***<sup>B</sup>* the translational velocity of a point *B* and *ω<sup>P</sup>* the rotational velocity of a body *P* with respect to the inertial reference frame, then

$$\begin{array}{l} \mathbf{v}^{B} = \sum\_{r=1\ldots n} \mathbf{v}\_{r}^{B} \boldsymbol{u}\_{r} \\ \boldsymbol{\omega}^{P} = \sum\_{r=1\ldots n} \boldsymbol{\omega}\_{r}^{P} \boldsymbol{u}\_{r} \end{array} \tag{2}$$

where **v***<sup>B</sup> <sup>r</sup>* and *ω<sup>P</sup> <sup>r</sup>* are the *r*th partial velocity and partial angular velocity of *B* and *P*, respectively.

#### *3.3. Generalized Active and Inertia Forces*

The *n* generalized forces acting on a system are constructed by the scalar product (projection) of all contributing forces and torques on the partial velocities and partial angular velocities of the points and bodies they are applied to.

Let us consider a system composed by *N* bodies *Pi*, where the torque **T***Pi* , and force **R***Bi* applied to a point *Bi* of *Pi* are the equivalent resultant ("*replacement*" [6]) of all active forces and torques applied to *Pi*. Then

$$F\_r^{P\_i} = \omega\_r^{P\_i} \cdot \mathbf{T}\_i^{P\_i} + \mathbf{v}\_r^{B\_i} \cdot \mathbf{R}\_r^{B\_i} \tag{3}$$

is the *r*th generalized active force acting on *Pi* and

$$F\_r = \sum\_{i=1,\cdots,N} F\_r^{P\_i} \tag{4}$$

the *r*th generalized active force acting on the whole system. Identically for the inertia forces, indicated as *F*∗ *r* .

The dynamical equations for an *n* degree of freedom system are formed out from generalized active and inertial forces *F*∗ *r*

$$F\_r + F\_r^\* = 0,\\ r = 1, \dots, n. \tag{5}$$

These are known as Kane's dynamical equations.

They result in a *n*-dimensional system of second order differential equations (2*n* order state variable representation) on generalized coordinates and speeds

$$
\bar{\mathbf{M}}(\mathbf{q})\dot{\mathbf{u}} + \bar{\mathbf{C}}(\mathbf{q}, \mathbf{u})\mathbf{u} + \bar{\mathbf{G}}(\mathbf{q}) - \bar{\mathbf{I}}(\mathbf{q}, \mathbf{u}, \tau) = \mathbf{0},\tag{6}
$$

where the parameter definitions are similar but not identical of the classical Lagrangian form and more efficient computationally [21].

#### *3.4. Non-Holonomic Constraints*

When *m* constraints on the motion variables are added to the model, only *n* − *m* generalized speeds are independents. The system is, then, called a non-holonomic system. The *non-holonomic* constraints are expressed as a set of *m* linear relationships between dependent and independent generalized speeds of the type

$$u\_r = \sum\_{i=1,\cdots,p} A\_{ri} u\_{i\cdot}, \\ r = p+1, \cdots, n,\tag{7}$$

with *p* = *n* − *m*. In this case, selected the independent speeds, the Kane's method immediately offers the minimal 2*p* order state variable representation from

$$F\_r + F\_r^\* = 0,\\ r = 1, \dots, p,\tag{8}$$

where Kane calls *F<sup>r</sup>* and *F*<sup>∗</sup> *<sup>r</sup>* non-holonomic generalized active and inertial forces, while the remaining *m* original redundant equations resolve themselves in the expressions of the *m* reaction forces/torques returned by the constraints. Because the Kane's method is fundamentally based on the projection of forces on a tangent space on which the system dynamics are constrained to evolve, spanned by the partial velocities, reaction forces/torques result from the projection on its null-space.

Moreover, it is always possible to handle an holonomic (configuration) constraint as if it is non-holonomic, that is, to treat it as a motion constraint. This is particularly advantageous to represents the spherical inverted pendulum with a pantograph during a step, where in the first phase non-holonomic constrains allow pivoting on the supporting leg, and in the second phase, releasing the non-holonomic constraints the impact of the swing leg with the ground can be represented.

#### *3.5. Unilateral Constraints and Collision*

As a consequence of switching between different non-holonomic models during gait, unilateral constraints and collisions cannot be ignored.

Clearly, adopting non-holonomic dynamics assuming points of the feet fixed to the ground is valid for bilateral constraints (ignoring eventual detachment from the ground and slipping). In the approaches known as hybrid complementarity dynamical systems based on forward dynamics [22] the necessary conditions for satisfying unilateral constraints are directly embedded into the model. Vice versa, a minimalistic view is adopted here, noting that in a physiological gait, normally, bilateral constraints on the feet are not assumed to be violated. Hence, we design a priori walking strategies and we test through the simulator that this effectively occurs, by monitoring, a posteriori, reaction forces for the conditions:

$$F\_{z\_{fout\_i}} > 0, i = 1,2\tag{9}$$

and

$$\|F\_{\dot{\jmath}\_{foot\_{\dot{i}}}}\| < \mu F\_{z\_{foot\_{\dot{i}}}, \dot{\jmath}} = \ge \mu, \dot{\jmath} = 1, 2. \tag{10}$$

Obviously, the control we propose cannot adapt itself to pathological conditions, such as a slipping surface.

For the second point, mechanics of the collision of the swing foot to the ground has to be considered, when switching to the next step causes the transfer of final conditions of the generalized speeds of one phase to the initial conditions of the successive. With reasonable assumptions of non-slipping and anelastic restitution the reaction impulsive force **F***<sup>B</sup>* at the impact point *B* and the initial conditions of the generalized speeds for the new phase **u**(*t* +) can be computed. Also for this aspect, Autolev offers all needed mechanical expressions.

The following analysis is based on two concepts: *generalized impulse* and *generalized momentum* [6,23]. Indicate, as usual, with **v***<sup>B</sup> <sup>r</sup>* the r-th component of the partial velocity vectors of the point *B* (the swing foot), the *generalized impulse* at the point *B* at the contact with the ground at instant *t* − is defined as the scalar product of the integral of the reaction impulsive force **<sup>F</sup>***Bδ*(*<sup>t</sup>* − *<sup>τ</sup>*) in the time interval *<sup>t</sup>* <sup>−</sup> ÷ *t* <sup>+</sup> with the corresponding partial velocities

$$I\_r \approx \mathbf{v}\_r^B (t^-)^T \cdot \mathbf{F}^B,\\ r = 1, \dots, n,\tag{11}$$

the *generalized momentum* is defined as the partial derivative of the kinetic energy *K* with respect to the r-th generalized speed

$$p\_r(t) = \partial K / \partial \mu\_r, r = 1, \dots, n,\tag{12}$$

then, Kane proves that

$$I\_r \approx p\_r(t^+) - p\_r(t^-). \tag{13}$$

Indicate the matrices

$$\mathbf{V}^{B} = (\mathbf{v}\_1^{B}(t^{-}) \cdot \cdots \cdot \mathbf{v}\_n^{B}(t^{-})) \tag{14}$$

$$\mathbf{P} = \{\partial p\_i(t^-) / \partial u\_j\}, i, j = 1, \cdots, n \tag{15}$$

of vectors of partial velocities, and of partial derivatives of *pr*(*t*) with respect to the generalized speeds, and the vectors

$$\mathbf{I} = \begin{bmatrix} I\_1 \cdots \cdot I\_n \end{bmatrix}^T = \mathbf{V}^{B^T} \cdot \mathbf{F}^B \tag{16}$$

$$\mathbf{u}(t) = [\boldsymbol{u}\_1(t) \cdot \cdots \cdot \boldsymbol{u}\_n(t)]^T \tag{17}$$

$$\mathbf{v}^{B}(t) = \mathbf{V}^{B} \cdot \mathbf{u}(t) \tag{18}$$

$$\begin{bmatrix} p\_1(t), \cdots, p\_n(t) \end{bmatrix}^T = \mathbf{P} \cdot \mathbf{u}(t) \tag{19}$$

of *generalized impulses*, of *generalized speeds*, of the velocity of point *B* and of *generalized momenta*, respectively.

Then, taking into account from (16) to (19), considering that **v***B*(*t* −) is known and **v***B*(*t* +) is zero, assuming non-slipping condition and inelastic collision, the following system of equations is solved to derive the unknown **F***<sup>B</sup>* and **u**(*t* +):

$$
\begin{bmatrix}
0
\end{bmatrix} = \begin{bmatrix}
\mathbf{V}^{B}(t)^{T} & -\mathbf{P} \\
0 & \mathbf{V}^{B}(t)
\end{bmatrix} \cdot \begin{bmatrix}
\mathbf{F}^{B} \\
\mathbf{u}(t^{+})
\end{bmatrix} \tag{20}
$$

An essentially similar equation was discussed in [11]. At the solution, along with the velocity **u**(*t* +) after the impact, it must be verified that the impulsive force **F***<sup>B</sup>* satisfies the conditions of unilateral constraint (9) and (10).

#### **4. The Spherical Inverted Pendulum Model**

In describing the spherical inverted pendulum the same notation used in [4,5] was adopted. In addition, explicitly, *θ<sup>z</sup>* indicates the angle of rotation with respect to the vertical axis, and two degrees of freedom of the swing leg relative to the pendulum were introduced, with the angles *α<sup>z</sup>* and *α*, as shown in Figure 1, with the kinematics of the joints in Figure 2a. The configuration Figure 2b will be used in special situations, only to perform side shuffle.

The angle position and velocity of the pendulum on the z and y axes of the inertial frame, and the two rotations of the swing leg with respect to the local axis z, and y of the supporting leg are *θz*, *θ*, *γ*, *ω*, *αz*, *α*, respectively. The configuration variables of the model are *θz*, *θ*, *x*, *y*, *z*, as the swing leg is considered without mass and inertia, where *x*, *y*, *z* are the coordinates of the pivot foot. The motion variables are *γ*, *ω*, *u*1, *u*2, *u*3, where imposing a non-holonomic constraint to the pivot foot, the velocities , *u*<sup>1</sup> = *x*˙, *u*<sup>2</sup> = *y*˙, *u*<sup>3</sup> = *z*˙, are zero during the swing phase, but are released at the impact of the swing foot with the ground.

In the next sections the SFPE and the gait based on this model are described.

**Figure 1.** The spherical inverted pendulum.

**Figure 2.** The kinematics of the joints—(**a**) forward/bacward motion of the flying leg, (**b**) side shuffle.

#### **5. The Estimation of the Balance Point**

Before the impact the motion variables have value *γ*−, *ω*−, 0, 0, 0 and after *γ*+, *ω*+, *u*<sup>+</sup> <sup>1</sup> , *<sup>u</sup>*<sup>+</sup> <sup>2</sup> , *<sup>u</sup>*<sup>+</sup> <sup>3</sup> . The total energy and the projection on the vertical axis of the angular momentum, *k<sup>γ</sup>* (the notation of [4] is maintained, even if it is shown in Appendix A that in general both speeds are present in this projection, and here an explicit dependency on *γ* is no more needed), are constant before and after the impact, however, they have a reduction during the impact.

To simplify the computations and to avoid spurious solutions, the next procedure is started after the pendulum reaches the vertical position and *θ* > 0. Let us say that at time *t*<sup>0</sup> the state variables assume the values *θz*0, *γ*0, *θ*0, *ω*0, the total energy *T*0, and the moment on the vertical axis *k<sup>γ</sup>* <sup>0</sup> (these last two values are the same, also, at the unknown instant of the impact *t* −). This gives the first equation, linking all state variables at the pre-impact.

$$T\_0 = T(\theta\_z^{-}, \gamma^{-}, \theta^{-}, \omega^{-}) \tag{21}$$

At the impact the swing foot, indicated with the point B, touches the ground. The vertical coordinate of *B* offers the second equation, linking the pre-impact angle *θ*− to *α* and *α<sup>z</sup>*

$$B\_z(\theta^-, \mathfrak{a}, \mathfrak{a}\_z) = 0 \tag{22}$$

The constant momentum *k<sup>γ</sup>* offers the third equation, linking *γ*<sup>−</sup> to the other preimpact motion variable

$$k^{\gamma}{}\_0 = k^{\gamma}(\gamma^-, \theta^-, \omega^-) \tag{23}$$

Switching the pivot foot after the impact, the relationship between the angles *θ<sup>z</sup>* <sup>+</sup>, *θ*<sup>+</sup> of the new pivot leg from *θ<sup>z</sup>* <sup>−</sup>, *θ*−, *α*, *α<sup>z</sup>* is obtained equating the three projections, with respect to the inertial axes, of swing and support legs (i.e., the swing leg becomes the new support leg)

$$\mathcal{S}SI(\theta\_{\overline{z}}^{+}, \theta^{+}) = \mathcal{S}\mathcal{W}(\theta\_{\overline{z}}^{-}, \theta^{-}, \alpha, \alpha\_{z}) \tag{24}$$

The solution of the impact equation (performed symbolically) (20) gives the motion variables after the impact, hence the total energy and the angular momentum *kγ*. The total energy after the impact can be evaluated before the switching of the pivot foot so it does not require the value of *θ*<sup>+</sup> and *θ*<sup>+</sup> *<sup>z</sup>* after switching. The angular momentum is computed on the new pivot point, so it requires the new value of *θ*<sup>+</sup> and *θ*<sup>+</sup> *<sup>z</sup>* after the switching. This gives the third and fourth equations.

$$\begin{array}{c} TE^{+} = TE(\theta\_z^{-}, \theta^{-}, \gamma^{+}, \omega^{+}, \mu\_1 + , \mu\_2 + , \mu\_3 +) \\\ k^{\gamma+} = k^{\gamma}(\theta\_z^{+}, \gamma^{+}, \theta^{+}, \omega^{+}) \end{array} \tag{25}$$

Moreover, by imposing velocity zero of the swing foot, after the impact, angles before the impact can be related to motion variables after, with a further relationship

$$\left[\left[\mathcal{B}\_{\mathbf{x}}, \mathcal{B}\_{y}, \mathcal{B}\_{z}\right]^{T} = 0 = F(\theta\_{z}^{-}, \theta^{-}, \mathfrak{a}, \mathfrak{a}\_{z}, \gamma^{+}, \omega^{+}, \mathfrak{u}\_{1}^{+}, \mathfrak{u}\_{2}^{+}, \mathfrak{u}\_{3}^{+})\right.\tag{26}$$

To estimate the foot placement to reach the balance in an erect posture after the impact, with *ω* = 0, *γ* = 0, and *θ* = 0, noting that *kγ*<sup>+</sup> is zero by the last condition (28), it is imposed that the total energy after the impact is equal to the maximal potential energy

$$TE^{+} = m \cdot \lg \cdot L \tag{27}$$

Finally, to impose that *γ* be zero at the balance point (but not necessarily after the impact, as it will be seen in the next Figure 4), from the impact the last equation is set

$$k^{\gamma^+} (\theta\_z^+, \gamma^+, \theta^+, \omega^+) = 0 \tag{28}$$

From the previous relationships, the unknown variables *θ<sup>z</sup>* <sup>−</sup>, *θ<sup>z</sup>* <sup>+</sup>, *γ*−, *θ*−, *θ*+, *ω*−, *α*, *α<sup>z</sup>* are determined, using non-linear least squares, with some numerical solver such as the Levenberg-Marquardt algorithm [24,25].

#### **6. Recovering Balance**

The first application of the SFPE is to recover balance in the presence of an impulsive disturbance. It is assumed that the biped is in quiet standing balance, conventionally oriented in the direction of the x axis. An impulsive horizontal force in an arbitrary direction generates a velocity of the COG. The biped, in reorienting himself, will react in two different ways, with a minimal rotation around the z axis, according to the relative direction of the pulse: if the direction is closer to his sagittal plane he will mostly move the free leg forward (or backward), if it closer to the frontal plane the motion will be mostly laterally with a side shuffle. To emulate these two distinct situations if the direction is less than 45◦ to the sagittal plane the model adopted is the standard one of Figure 2a, otherwise Figure 2b. When the angle *θ* falls below a safety value and the direction is detected from the falling velocity, let say [*vx*, *vy*, *vz*] *<sup>T</sup>*, the model (a) or (b) is selected according to *atan*2(*vy*, *vx*), and initial velocities are assigned to *ω* and *γ* from the inverse of the first two rows of the matrix of partial velocities (29)

$$\mathbf{V}^{\rm COG} = [\mathbf{v}\_{\omega}^{\rm COG} \mathbf{v}\_{\gamma}^{\rm COG}].\tag{29}$$

Then, the SFPE algorithm is run. This defines, among other variables, the swing foot angles *α*, *α<sup>z</sup>* for the impact that allows recovering the balance in one step. The numerical algorithm does not converge in two cases: when the total energy after the impact is lower than the maximum potential energy, or when one step is not enough to recover the balance. In the first case the condition (27) cannot be satisfied. In the second case the bound on the maximum allowable *α* is not satisfied.

Two examples are presented using the two models (Figure 3). The situation of a push perfectly aligned with either the sagittal or the frontal plane is not considered, as it can be simply solved with the classical 2-D approach. In both cases the initial velocity is of 0.5 m/s and it is detected, at instant 0.5 s, when the falling angle reaches 0.1 rad. The first example, using model (a), presents the response to a push at an angle of 20◦ from the x axis, in the second, with model (b), the angle is of 110◦.

**Figure 3.** Recovering an impulsive disturbance-COG behaviours.

Apart from a difference in the sign of the angles, and the velocities the two behaviours are very similar, so the case of model (b) is detailed, only.

The original central inertia matrix of the example, the same of the biped of [20], given in the Appendix A, had the elements I12 and I23 equal zero, the correct responses of angles (speeds and positions) in Figure 4a,b are represented with solid lines. To see the differences, fictitious values different from zero have been assigned to I12 and I23, and the example rerun. The responses are indicated in the legends with a X, and plotted with dashed lines.

From the SFPE, the flying foot final position can be forecasted and the SIP COG trajectory used as a reference, knowing the kinematics, to control the joints of a real biped robot or exoskeleton to recover balance.

**Figure 4.** Angle velocity and position behaviours-with the effect of off-diagonal elements different from zero in the inertial matrix.

### **7. The Gait**

At difference of other works, only some of the expressions of Section 5 are exploited for generating the gait, the SFPE is only used to impose a halt at the end of the walk.

*α* is chosen to achieve the desired step length, *α<sup>z</sup>* to achieve the COG sway from −*θzMax* to *θzMax* and to control the offset with respect to the baseline of walk. The gait is initiated giving an initial condition to *θz*, *γ*, *θ*, *ω*, or, simply, from a standing up balance by leaving the pendulum to fall forward.

Each step is concluded when the swing foot touches the ground (the vertical coordinate of point B becomes zero, Equation (22)). From the impact Equation (20) the new motion variables *γ*+, *ω*+, *u*<sup>+</sup> <sup>1</sup> , *<sup>u</sup>*<sup>+</sup> <sup>2</sup> , *<sup>u</sup>*<sup>+</sup> <sup>3</sup> are determined, and from Equation (24) the starting values of *θ<sup>z</sup>* <sup>+</sup>, *θ*<sup>+</sup> for a new step are computed. The gait is maintained by increasing at each step, after impact, the resulting *ω*<sup>+</sup> to compensate for the reduction of kinetic energy due to the impact, controlling, also, the gait cadence, and perturbing *γ*<sup>+</sup> to correct the angle of direction of the walk.

In the present model, the two legs have no mass and inertia. Therefore, the motions of the angles *α* and *α<sup>z</sup>* are instantaneous and energy free. The only energy contribution to maintain the walk is given by proper impulsive forces and torques just after the impact to modify the velocities *ω*<sup>+</sup> and *γ*<sup>+</sup> resulting from the impact. This emulates, in a real walk, the contribution given by the biped in the brief double support phase and in the period of single support when the foot is flat and able to transfer torques.

Five control variables are identified to control the five objectives of the walk: *cadence*, *step length*, *distance between f eet*, *y o f f set* with respect to the baseline of walk, and *direction o f walk* ( even if interacting each other, each of the five variables predominantly controls one of the five objectives). After each impact, at the start of step *k* they are

$$\begin{array}{c} \delta\_{\omega}(k) \Rightarrow \omega(k) = \omega^{+} + \delta\_{\omega}(k)(cadence) \\ \delta\_{\mathfrak{a}}(k) \Rightarrow \mathfrak{a}(k) = \mathfrak{a}0 + \delta\_{\mathfrak{a}}(k)(step\ length) \\ \delta\_{\mathfrak{s}\text{may}}(k) \cdot \mathfrak{u} + \delta\_{\mathfrak{y}}(k) \Rightarrow \mathfrak{a}\_{\mathfrak{z}}(k) = (\mathfrak{a}\_{\mathfrak{z}0} + \delta\_{\mathfrak{s}\text{may}}(k)) \cdot \mathfrak{u} + \delta\_{\mathfrak{y}}(k) \\ \qquad (spacing\ between\ feet\ and\ y\ offset) \\ \delta\_{\mathfrak{y}}(k) \Rightarrow \gamma(k) = \gamma^{+} + \delta\_{\gamma}(k)(direction) \end{array} \tag{30}$$

where *u* assume the values +1, −1 according to the right or left foot support.

It must be noted that no periodic reference is tracked. The whole gait style (cadence, length of the step, offset with respect to the baseline of walk-through a side shuffle, spacing between the two feet and direction) can be changed at each step. The energy consumption of the gait is measured by the difference, after the foot collision, of the kinetic energies before and after the application of the control (in particular *δω*(*k*) and *δγ*(*k*)).

The next Figures 5 and 6 show a sample of a typical rectilinear walk, terminating with a halt. In particular, the Figure 7 show the energy needed at each step to maintain the gait, provided by *δω*(*k*) and how the gait collapses after two steps if no maintenance is performed.

(**a**) COG along the x and y axes (**b**) Details of the projection of the COG on the ground and pivot foot posizion

**Figure 5.** The COG behaviour.

**Figure 6.** Angle position and velocity behaviors.

(**a**) The energies when the gait is maintained. (**b**) The decay of the energies in absence of maintainment.

#### *7.1. Comparison between SIP and ZMP Based Gaits*

An interesting question is how the SIP gait compares with the classical one based on the ZMP of a linear inverted pendulum. In previous papers, a 10/12 degrees of freedom biped model was simulated in rectilinear and curved trajectories [20]. The technique adopted was classical, by controlling the model to follow a preview trajectory based on the ZMP.

The total mass, the central inertia and the COG height of the biped were used to model the SIP. The parameters that control the gait of the SIP were adjusted to synchronize the two walks. To compare the center of pressure (COP) on the shoes in the two cases, the SIP is mounted on the ankle of the same foot of the biped (see Appendix A), no torque is transferred from the joints of the SIP, but the force on the ankle, returned from the non-holonomic constraint, are balanced by an identical force in the ZMP on the sole. The comparison is shown in the next figures.

In Figures 8a,b and 9 the two trajectories are superimposed. In particular, the right and left supports of the SIP and the periods of double support of the preview based simulation are also indicated.

The COG behaviours along the x a z axes are very similar, but not along the y axis. In fact, the details of the two COGs projected on the ground, and the relationships between the COP and the foot support placement, compared in the Figure 10a,b, show marked differences.

**Figure 9.** COGz, comparisons.

To achieve a similar sway of the *COG* on the *y* direction in the two cases, the SIP keeps the feet closer than in the preview case. Let consider that the SIP, passed the erect position is in free fall and the COP jumps suddenly on the new foot. Vice versa, in the preview-based gait the biped is always controlled to maintain the COP close to the supporting foot, and to transfer it, almost continuously from single to double support.

Finally, the energies were compared in Figure 11. As expected, the preview based on the ZMP demands a greater expenditure of energy.

**Figure 11.** Comparison of energies.
