**5. Conclusions**

In this paper, Choi's feedback for postural control of a biped robot, based on a linearized inverted pendulum model, has been revised. In practical situations, this feedback generates very undamped closed-loop dynamics. Then, the design problem was reformulated using state estimation and state feedback control. In fact, closing the loop with a state observer of the *COG* and exploiting velocity along with position and acceleration estimates guarantees a greater damping of closed-loop poles, with identical steady-state gain. However, to be effective in fusing *CoP* and *COM* measures, this observer needs to be extended to also estimate external disturbance forces, and the kinematic model of the *COM* needs to be tuned to the actual mass distribution. The former problem was tackled by a robust estimator based on an extended system embedding into the model unknown force disturbances, the latter by identifying a priori the SESC model of the mapping between joint angles and the *COM*. This a priori identification can also be repeated to maintain the mapping up to date in cases of changes in the weight distribution of the biped.

Two approaches to design the feedback were pursued: one is numerical, computing the state feedback for a given observer with a Levenberg–Marquardt algorithm. The second exploits integrally estimator and feedback obtained from a robust control regulator and adapts it to the tracking of

the preview signals. The results show similar performances, with good disturbance compensation. It must be emphasized that the adoption of an extended system with its weighting functions offers a formal technique to set the observer and feedback characteristics, guaranteeing the desired loop gain and bandwidth.

Robustness was shown by applying the control designed for a linearized inverted pendulum to two non-linear systems: a three-DOF kinematic chain of an actual mechanical small-scale leg and the simulation of an exoskeleton constraining a patient to perform a joint-legged stand-to-sit exercise in the sagittal plane. The proposed control correctly integrates the *COG* information, which is poorly reconstructed from joint measures and kinematics of the chain, with actual *CoP* measures, accommodating uncertainties in the model and unknown external force disturbances. Moreover, an identification procedure of the SESC model was proposed and tested.

COG–ZMP and linearized inverted pendulum models continue to be at the basis of balance control of bipeds. However, extended systems and SESC models have not yet been proposed in order to offer robustness to the approach.

The proposed *COG* − *ZMP* control was successfully used by the authors for the balance of turning during walking of biped robots and for a more detailed and complete sit-to-stand exercise described in the companion paper [8]. In particular, future developments will consider haptic exoskeletons, where the action of the patient on some joints, through electromiographical signals, controls the motion of part of the degrees of freedom, while the automatic control discussed here guarantees balance acting on the ankles or on the hips.

Computing robust estimation and robust control, as well as the block diagrams present in the paper, were made with the design environment G++ developed by the authors described in [18] and that can be downloaded from [19]. However, the used technique is fairly standard in the robust control field and can be found in classical textbooks such as [14].

**Author Contributions:** Methodology, G.M. and M.G.; writing—original draft, G.M.; writing—review & editing, M.G.

**Funding:** This research has been partially supported by MIUR, the Italian Ministry of Instruction, University and Research through project ESOPO, and the Piedmont Region through project ESROB.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Choi's Feedback Limitations**

This appendix is devoted to showing that, in spite of stability, Choi's original feedback has a poor damping of the closed-loop dynamics in practical operating conditions. Applying the control strategy of Equation (4) to the model of Figure 1, we obtain a closed-loop system with three design gains *kv*, *kc*, *kp* and one coefficient *wn*, as depicted in Figure A1. In classical linear control theory, it is customary to introduce disturbances in the input and output of the system and to study the closed-loop performance, analyzing the open-loop transfer function (t.f.), and the related closed-loop sensitivity functions linking the output to the input and output noises. The block diagram presents two partial feedback loops, on *COG* and on *ZMP*, that can be analyzed separately by opening (indicated with an *X* in the block diagram) the two feedbacks one at a time and considering the other part of the system.

**Figure A1.** The COG–ZMP model of the linearized inverted pendulum with Choi's feedback control.

The functions related to the *ZMP* loop are not particularly interesting: the feedback of the *ZMP*, although important for stability reasons, does not play any role in disturbance rejection and tracking error because its steady-state loop gain is always lower than 1. In fact, it is given by

$$G\_{a\_{znp}}(s) = \frac{k\_v k\_p (1 - s^2 / w\_n^2)}{s^3 + k\_v s + k\_v k\_c} \, ^\prime \tag{A1}$$

where *kp* < *kc* always holds.

Vice versa, the following t.f.s of the *COG* loop are noteworthy: the open-loop transfer function *Ga*(*s*), the output tracking error sensitivity *S*(*s*), and the output sensitivity to input disturbances *Geq*−*COG* (*s*).

$$G\_{a}(s) = \frac{k\_{v}k\_{c}}{s^{3} + k\_{v}k\_{p}/w\_{n}^{2}s^{2} + k\_{v}s - k\_{v}k\_{p}}\tag{A2}$$

$$S(s) = \frac{s^3 + k\_v k\_p / w\_n^2 s^2 + k\_v s - k\_v k\_p}{s^3 + k\_v k\_p / w\_n^2 s^2 + k\_v s + k\_v (k\_c - k\_p)}\tag{A3}$$

$$G\_{\text{eq}\_{\text{c-COG}}}(\mathbf{s}) = \frac{k\_{\text{v}}}{\mathbf{s}^3 + k\_{\text{v}}k\_p/w\_{\text{n}}^2\mathbf{s}^2 + k\_{\text{v}}\mathbf{s} + k\_{\text{v}}(k\_{\text{c}} - k\_p)}\tag{A4}$$

In order to guarantee stability (negative real part of the roots of the third-order, closed-loop, characteristic polynomial appearing as denominator in Equations (A3) and (A4)), the following condition on the parameter *kp* must be satisfied:

$$\frac{w\_n^2}{w\_n^2 + k\_v} k\_{\mathcal{L}} < k\_p < k\_{\mathcal{L}} \,\,\,\,\tag{A5}$$

Note that condition (A5) is slightly different from Choi's result.

In order to have more insights about the closed-loop poles of Equations (A3) and (A4), consider the root locus, function of *kv*, of the following open-loop transfer function:

$$G\_{\mathfrak{a}\_{\mathbb{B}\_p}}(\mathbf{s}) = \frac{k\_{\mathbb{B}}(k\_p/w\_n^2\mathbf{s}^2 + \mathbf{s} + k\_{\mathbb{C}} - k\_p)}{\mathbf{s}^3}. \tag{A6}$$

Note that the numerator of 1 + *Gakv* (*s*) is exactly the characteric polynomial of (A3) and (A4). When *kv* → ∞, one real closed-loop pole → −∞, but the dominant closed-loop poles are complex conjugate and approach asymptotically the zeroes of the t.f. (A6), i.e., the root of the polynomial

$$(s^2 + w\_n^2 / k\_p s + w\_n^2 (k\_c - k\_p) / k\_{p\prime} \tag{A7}$$

having the damping ratio

$$\zeta = \frac{w\_n}{2} \sqrt{\frac{1}{k\_p (k\_c - k\_p)}}.\tag{A8}$$

Moreover, the root locus shows that for any value of *kv* < ∞, the damping ratio of the pair of dominant complex conjugate poles is always lower than that of these zeroes. From previous results, the following observations can be made:


In conclusion, if a sufficiently high loop gain in the *COG* − *ZMP* control system is imposed, with the feedback proposed by Choi, even if the closed loop remains stable, its behavior becomes highly undamped. However, a high loop gain, and hence a high value of *kc* is needed when a robust control has to be used in exoskeletons to improve postural equilibrium for ill or elderly people, in order to cope with uncertainties.

#### **References**


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