*7.2. Turning While Walking*

This last subsection shows how to generate turning while walking using the SIP model. The walking trajectory is described in part III of [20]. It is obtained concatenating arcs of circle, whose ray can be changed at the beginning of each step. The control is based on a running local reference frame with respect to which the SIP is constrained to maintain a rectilinear walk. The local running frame, at each instant, rotates along the *z* axis with respect to the world space to have its x axis tangent to the trajectory, and moves its origin to follow an involute of the path curve. Let *s*(*t*) and *s*˙(*t*) be the curvilinear coordinate and its velocity on the path, their values also represent the motion on the abscissa of the local frame, and Θ*z*(*t*) the orientation on the *z* axis of the local frame at each instant *t*.

At time *T*(*k*), immediately before the step *k* starts, let indicate with *AxL*(*k*), *AyL*(*k*), *BxL*(*k*), *ByL*(*k*) the x and y coordinates of the last supporting foot, and of the swing foot at the contact (it is going to be the next support), in the local frame, respectively, and with *θz*(*k*) = *θ<sup>z</sup>* <sup>+</sup> the current orientation angle after the contact.

*δω* is fixed to obtain the desired average *cadence* of the gait, *δα* the average *step length*, and *δsway* the average *distance between f eet*, averaging 1/(*T*(*j*) − *T*(*j* −1)), *BxL*(*j*) − *AxL*(*j*), (*ByL*(*j*) − *AyL*(*j*)) · *u*, respectively, over the period 1 ≤ *j* ≤ *k*. (*ByL*(*k*) + *AyL*(*k*))/2 measures the *o f f set* of the trajectory with respect to the centerline in the local frame and (*θz*(*k* − 1) + *θz*(*k*))/2 the mean *direction* of walk at time *T*(*k*).

From those measures a mild proportional, integral feedback is set to *δy*(*k*) and *δγ*(*k*) to maintain *o f f set* to zero, and *direction* to follow Θ*z*(*k*). The result is the following Figure 12.

#### **8. Conclusions**

Starting from the ideas of [4,5] of using a SIP model that emphasizes energies, the gait and the SFPE were reviewed with a solution based on the exact computation of the kinetic energy, the momentum projected on the z axis and the foot collision. Let note, that with this approach of SFPE the complicate handling of the projection of the angular momentum on the horizontal axis, contained in the original development, is not any more needed and arbitrary central inertia matrices are allowed. Consider that the solution of the impact (Equation (20)) offers the reaction forces at the collision, also. They are not used in this paper, but they will be in future extensions.

Very compact code for the equations have been obtained adopting the Kane's method and using his software environment. The solution of these equations, through a non-linear least square algorithm such as Levenberg-Marquardt, allows to introduce box bounds of the variables, to exploit the Jacobian, and to test for the feasibility of the solution. Moreover, the dependency of these equations on measurable angle positions and velocities and knowledge one step ahead of the ground level consents a real-time implementation to control robots or exoskeletons.

The introduction of the angle *αz*, taking into account the angle *θz*, has extended the approach to non-periodic and omnidirectional gait and to recover the balance from a push in an arbitrary direction.

This paper, originally motivated by the need to help to recover balance, when disturbances are present, in a lower limb exoskeleton [26], is just the starting point. Future exentions, in order of complexity, are: introducing in the pantograph model the width of the pelvis in the distance between the two legs, adding a mass to the legs (sometime balance is recovered by inertia, just extending one leg), running and jumping.

The extension of the real-time control of the gait in the Cartesian space to all five control variables (30) with Model Predictive Control [27] will be also considered. So far we did not succeed, and only two variables have been controlled with classical PI techniques.

The approach was tested on a dynamic simulation of the SIP. Moreover, as the final objective is to offer a flexible Motion Generator for biped robot walking, the results have, also, been compared with the simulation of a fully actuated 12 DOF of a biped robot. In the comparison, the similarity of the COG behaviour in the sagittal plane suggests that actuated and underactuated controls can be intermixed [28], transforming the instantaneous increment of rotational velocities after the flying foot collision into a proper torque of the ankle of the supporting foot during double support and in the phase of single support, when the foot is flat. However, to obtain a nontrivial finite period of double support, compliance has to be added to the actual anelastic collision. As compliance is essential for that [11,29].

The behaviour of the COG in the frontal plane, that is consequence of a pure ballistic trajectory, will require a further investigation, as the resulting sway of the COG in the frontal plane seems excessive.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.
