# Estimation and Closed-Loop Control of COG/ZMP in Biped Devices Blending CoP Measures and Kinematic Information

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## Abstract

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## 1. Introduction

#### 1.1. Paper Contributions

## 2. Choi’s Approach

## 3. An Extended System for COG–ZMP Robust Estimation and Control

#### 3.1. Separate Estimator and Feedback

#### 3.2. ${\mathcal{H}}_{\infty}$ Robust Control

#### 3.3. Simulation Results

## 4. Control of a 3 DOF Biped

#### 4.1. Joint Angles–$CO{G}_{m}$ Map Fitting and Control

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

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Choi’s Feedback Limitations

- with high values of the ${k}_{v}$ gain, the dominant closed-loop poles depend on the pair ${k}_{c},{k}_{p}$ only, as they are highly insensitive to ${k}_{v}$;
- the steady-state $COG$ loop gain (A2) is proportional to the rate ${k}_{c}/{k}_{p}$ (independent from ${k}_{v}$);
- the steady-state gain of both sensitivities related to $COG$ are inversely proportional to ${k}_{c}-{k}_{p}$;
- however, if the gain ${k}_{c}$, or more precisely, the difference ${k}_{c}-{k}_{p}$, increases, then the damping ratio of the dominant closed-loop poles decreases.

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

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

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

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**MDPI and ACS Style**

Menga, G.; Ghirardi, M.
Estimation and Closed-Loop Control of *COG*/*ZMP* in Biped Devices Blending *CoP* Measures and Kinematic Information. *Robotics* **2019**, *8*, 89.
https://doi.org/10.3390/robotics8040089

**AMA Style**

Menga G, Ghirardi M.
Estimation and Closed-Loop Control of *COG*/*ZMP* in Biped Devices Blending *CoP* Measures and Kinematic Information. *Robotics*. 2019; 8(4):89.
https://doi.org/10.3390/robotics8040089

**Chicago/Turabian Style**

Menga, Giuseppe, and Marco Ghirardi.
2019. "Estimation and Closed-Loop Control of *COG*/*ZMP* in Biped Devices Blending *CoP* Measures and Kinematic Information" *Robotics* 8, no. 4: 89.
https://doi.org/10.3390/robotics8040089