*4.1. Joint Angles–COGm Map Fitting and Control*

In order to apply the results of the previous section to a biped device, it is necessary to have a reliable evaluation of the *COG* from the joint angles, consistent with the *CoP*. Its value depends on the position of the center of masses and weights of each link of the chain. Those data are scarcely known in advance but can be identified with a series of a priori experiments. This approach is called statically equivalent serial chain (SESC) modeling [9] (see also [16,17] for applications to rehabilitation).

Espiau et al., in [9], showed from experiments measuring the projection of the *COG* on a force table that the physical parameters of a kinematic chain cannot be identified uniquely. What can be identified is only a set of expressions of them, representing classes of equivalent (with respect to the *COG*) chains. These expressions appear in linear form in the SESC model. Considering a three-joint kinematic model composed of feet, leg, thigh, and HAT (head, arms, trunk) for motions in the sagittal plane, the parameters of the SESC model can be identified using least squares with two equations and two sets of experiments: collecting and recording a series of joint-angle positions with the corresponding measures of the *CoP* in the steady state and a set of samples of joint angles and *CoP* trajectories in motion spanning the operating area at random. A slight modification of the model presented in [16] is proposed here, where the first equation, expressing explicitly *COGx*, refers to steady-state experiments, while the second equation, expressing *COGz* indirectly (Equations (1) and (2) can be rewritten as *COGx* <sup>−</sup> *ZMPx* <sup>=</sup> *COG*¨ *<sup>x</sup>*/9.81 · *COGz* ), refers to dynamical ones.

The equations are:

$$
\begin{bmatrix}
\text{COG}\_{\text{x}} \\
\text{COG}\_{\text{x}} - \text{ZMP}\_{\text{x}}
\end{bmatrix} = \\
$$

$$
\begin{bmatrix}
1 & 0 \\
0 & \text{C}\dot{\text{OG}}\_{\text{x}}\text{/} 9.81 \\
0 & 1 \\
\end{bmatrix}.
$$

$$
\begin{bmatrix}
1 & 0 & \sin(\theta\_{1}) & \sin(\theta\_{1} + \theta\_{2}) & \sin(\theta\_{1} + \theta\_{2} + \theta\_{3}) \\
0 & 1 & \cos(\theta\_{1}) \cdot b & \cos(\theta\_{1} + \theta\_{2} + \theta\_{3}) \cdot b \\
\end{bmatrix}.
$$

$$
\begin{bmatrix}
r\_{1x} \\ r\_{1z} \\ r\_{2} \\ r\_{3} \\ r\_{4}
\end{bmatrix},
\tag{8}
$$

with

$$\begin{aligned} r\_{1x} &= \left(m\_0 \cdot x\_0 + \left(m\_1 + m\_2 + m\_3\right) \cdot x\_1\right) / m\_{tot}, \\ r\_{1z} &= \left(m\_1 + m\_2 + m\_3\right) \cdot z\_1 / \left(m\_1 + m\_2 + m\_3\right), \\ r\_2 &= \left(m\_1 \dot{l}\_{10} + \left(m\_2 + m\_3\right) \cdot l\_1\right) / m\_{tot}, \\ r\_3 &= \left(m\_2 \cdot l\_{20} + m\_3 \cdot l\_2\right) / m\_{tot}, \\ r\_4 &= m\_3 \cdot l\_{30} / m\_{tot}, \\ b &= m\_{tot} / \left(m\_1 + m\_2 + m\_3\right), \end{aligned} \tag{9}$$

where *l*1, *l*<sup>2</sup> are the length of legs and thighs, *m*0, *m*1, *m*2, *m*<sup>3</sup> are the masses of feet, legs, thighs, and trunk (HAT), (*mtot* = *m*<sup>1</sup> + *m*<sup>2</sup> + *m*<sup>3</sup> + *m*4), *x*<sup>0</sup> is the center of mass of the feet, *x*1, *z*<sup>1</sup> are the coordinates of the ankle, and *l*10, *l*20, *l*<sup>30</sup> are the distances from the center of mass to the distal joints for the leg, thigh, and proximal joint for HAT. Coefficient *b* accounts for the difference (the feet do not move during the dynamical experiments) in sensing the *COG* statically and dynamically. From this model, the six parameters of (9) are identified recursively with a non-linear least squares technique such as Levenberg–Marquardt, where *COG*¨ is obtained approximately from numerical differentiation of *COG*. The actual small-scale leg was first identified, with results (statical and dynamical) contained in Figures 9 and 10.

**Figure 9.** Results of estimating the statically equivalent serial chain (SESC) model in static experiments.

Then a control exercise was carried out, maintaining the *CoP* position fixed and the posture erect while performing an up-and-down motion (such as sit-to-stand) of the body. The results (*CoPx* and *COGx*), based on the identified model and the proposed control scheme, when a disturbing force is applied in the sagittal plane are shown in Figure 11. The action of the feedback on the *COG* to compensate the disturbance is clear.

**Figure 10.** Results of estimating the SESC in dynamic experiments.

**Figure 11.** Maintaining the *CoP* position during a stand-to-sit like exercise in the presence of a disturbance force.

#### *4.2. Simulation of a Stand-to-Sit Exercise*

In order to validate the results on the linearized inverted pendulum of Section 3.3, the proposed control (with the same estimator and feedback parameters) was applied to the non-linear simulation of a multi-chain with 3 DOF, having as average the same *COGz*. It represents a biped in the sagittal plane emulating the first phase of a stand-to-sit exercise to test the balance control of a future full-scale exoskeleton with a patient. While the pelvis is lowered from a standing posture to reach the chair and the trunk attitude assumes a natural bending forward, the *COGx* is shifted from heels to tips and a disturbance force is applied, as in the previous experiment of Figures 5 and 6. The animation of the exercise can be seen in Figure 12. The resulting response of the *COG* − *ZMP* in Figure 13 is very similar to that of the linearized inverted pendulum. Particularly, in the final phase of the exercise, the reaction of the exoskeleton to preserve equilibrium against the push forward of the disturbing force is particularly visible.

**Figure 12.** Animation of the 3 DOF kinematics during a stand-to-sit exercise.

**Figure 13.** COG–ZMP during stand-to-sit and shift of *CoP* in a nonlinear 3 DOF kinematics-robust control.

Vice versa, Choi's original feedback with identical *COG* gain, applied to the nonlinear simulator, has not been able to guarantee stability.
