*3.3. Simulation Results*

The model of an exoskeleton with a patient used in [5,7] was considered, with parameter *wn* = 3.34 and choosing a speed gain for the velocity loop of *kv* = 250. In this experiment, the linear model of the block diagram of Figure 2 was simulated. In an erect posture, with a preview reference computed as suggested by [2], a step transition on the sagittal plane of *p* of 0.2 m moving the *c* from the heels to the tips of the feet at time *t* = 1.25 s was imposed. Then at *t* = 3.5 s, an external horizontal force disturbance acting in the sagittal plane on the *COG*, tries to move the *CoP* outside the feet support by an additional 0.1 m, causing—if not compensated—a loss of balance.

The experiments are compared using the identical tracking gain *kc* of the *COG* loop and the best (for damping) value of *kp* with Choi's control and feedback from a robust estimator obtained from the extended system of Figure 2. The gain parameters adopted in the case of Choi's feedback were *kc* = 60, *kp* = 5, with a resulting damping ratio of the dominant poles of 0.05; and in the case of observer feedback, *kc* = 60, *kp* = 3, *kcv* = 4, with a resulting damping ratio of the dominant poles of 0.7.

The figures represent reference (dashed) and actual *COG* (blue), *ZMPideal* (red), and *CoP* (cyan), *COG* speed estimate (green), and the estimate of the disturbance effect (violet).

Figure 4 shows the results adopting the Choi control. The low damping of the closed-loop poles is clearly visible. Note that, ignoring the disturbances, the *CoP* does not follow the reference path and exits from the tip of the feet.

**Figure 4.** Choi's control without any disturbance compensation.

Vice versa, when disturbances are also estimated, after a short interval of time depending on the filtering bandwidth of the estimator, the *ZMP* returns to the desired value. Figures 5 and 6 show the feedback from the extended estimator with compensation of disturbances, obtained with the approaches of Sections 3.1 and 3.2, respectively.

The transition of the force disturbance was chosen to be unrealistically steep to evidence that, because of the estimator bandwidth, the compensation of the disturbance can't be perfect, depending on the values assigned to the weighting functions in the extended system.

The delay in the estimation of *δ*, as a consequence of the disturbance, is shown in Figure 7, where *δ* is in blue and its estimate is in green.

**Figure 5.** Extended observer feedback with disturbance compensation.

**Figure 6.** Robust control with disturbance compensation.

**Figure 7.** Effect of disturbance on the *CoP* and its estimation.

#### **4. Control of a 3 DOF Biped**

The approach was tested on the real three-DOF small-scale mock-up of an exoskeleton (Figure 8) and on the simulation of the full-scale exoskeleton. The chains, in both cases, are composed of two joint feet, legs, thighs, and one trunk. The 3 × 3 Jacobian matrix relating *COG* to joints embedding knee motion and trunk attitude is the following:

$$
\begin{bmatrix}
\dot{c} \\
\dot{\theta}\_2 \\
\dot{\theta}\_{trunk}
\end{bmatrix} = \begin{bmatrix}
I\_{\text{org}} \\
0 & 1 & 0 \\
1 & 1 & 1
\end{bmatrix} \cdot \begin{bmatrix}
\dot{\theta}\_1 \\
\dot{\theta}\_2 \\
\dot{\theta}\_3
\end{bmatrix} \tag{6}
$$

where *θ*1, *θ*2, *θ*<sup>3</sup> are the angles of the ankle, knee, and hip, *Jcog* is the Jacobian of *c*, and *θtrunk* is the attitude of the trunk. Joints are controlled by velocity servos, with their references being obtained through the inverse of the Jacobian matrix (6) driven by speed feedback signals. The *COG* speed feedback is similar to the one used for the linearized inverted pendulum (5), where the measures of the *ZMPm* were obtained from the *CoP* and that of the *COMm* from the joint-angle measures *θ*1*m*, *θ*2*m*, *θ*3*m*. The remaining two feedbacks, from the knee angle and trunk attitude measures *θ*2*<sup>m</sup>* , *θtrunkm* , are simply proportional feedbacks, the last measure being obtained from an inertial sensor:

$$
\begin{bmatrix} u\_{\theta\_2} \\ u\_{\theta\_{\text{trank}}} \end{bmatrix} = \begin{bmatrix} k\_{\text{knee}} \cdot (\theta\_{2\_{\text{ref}}} - \theta\_{2\_m}) \\ k\_{\text{trank}} \cdot (\theta\_{\text{trank}\_{\text{ref}}} - \theta\_{\text{trank}\_m}) \end{bmatrix} \tag{7}
$$

where *θ*2*ref* and *θtrunkref* are the references chosen according to the desired postural exercise.

**Figure 8.** A small-scale 3 degrees of freedom (DOF) leg of the exoskeleton.
