# Zero Moment Line—Universal Stability Parameter for Multi-Contact Systems in Three Dimensions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Zero Moment Point

#### 2.1.1. The Standard Definition

#### 2.1.2. The Line Definition

#### 2.1.3. The Angular Definition

#### 2.2. Stability and the Support Polygon

## 3. Results—Application of the Line Definition

#### 3.1. Standing Up with Help

#### 3.2. Two Approaches for Treating a Balanced System

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ZMP | Zero moment point |

ZML | Zero moment line |

COM | Center of mass |

SP | Support polygon |

## Appendix A

## References

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**Figure 1.**A humanoid, which can be either a human or a humanoid robot, supported by his feet on the ground and by his hands at a higher location. The SP, shown with the red curve, extends from the heel to the hands of the humanoid. The ZMP line passes through the humanoid and its SP, which indicates that the humanoid is stable.

**Figure 2.**A humanoid with some exemplar external forces and torques. The orientation of the fixed coordinate system defining the space is shown in the lower left corner. See the text for the explanation of the symbols.

**Figure 3.**The linear inverted pendulum model with the accelerations of the COM and the accelerations produced by the ground reaction force acting at ${p}_{x}^{g}$.

**Figure 4.**A schematic representation of two COPs on two horizontal planes at heights ${z}_{1}$ and ${z}_{2}$ and the COP line. ${\overrightarrow{F}}_{1}$ and ${\overrightarrow{F}}_{2}$ are two external forces, ${\overrightarrow{M}}_{1}$ and ${\overrightarrow{M}}_{2}$ are two external torques, the vectors ${\overrightarrow{r}}_{1}\left({z}_{1}\right)$, ${\overrightarrow{r}}_{2}\left({z}_{1}\right)$, ${\overrightarrow{r}}_{1}\left({z}_{2}\right)$, and ${\overrightarrow{r}}_{2}\left({z}_{2}\right)$ point from the centers of the two planes to the locations where the external forces and torques act, ${\overrightarrow{F}}^{\mathrm{tot}}$ is the sum of ${\overrightarrow{F}}_{1}$ and ${\overrightarrow{F}}_{2}$, while ${\overrightarrow{r}}^{\mathrm{cop}}\left({z}_{1}\right)$ and ${\overrightarrow{r}}^{\mathrm{cop}}\left({z}_{2}\right)$ are the vectors pointing from the centers of the two horizontal planes to their corresponding COP locations.

**Figure 5.**The ZML passing through the COM, with three different locations of the ZMP, such that ${z}_{\mathrm{zmp}}>{z}_{\mathrm{com}}$, $0<{z}_{\mathrm{zmp}}<{z}_{\mathrm{com}}$, and ${z}_{\mathrm{zmp}}<0$, respectively. ${\phi}_{\mathrm{zmp}}$ is defined as the angle between the ZML and the vertical line, which is for a clearer representation passing through the COM.

**Figure 6.**The ZMP angle and the coordinates, ${x}_{\mathrm{heel}}$ and ${x}_{\mathrm{toes}}$, and the angles, ${\phi}_{\mathrm{heel}}$ and ${\phi}_{\mathrm{toes}}$, of the edges of the SP, being the heel and the toes of the foot of the humanoid, with respect to its COM.

**Figure 7.**Support polygons (red curves) around a humanoid in its sagittal plane for different scenarios when the humanoid is standing up and is (

**a**) supported only by its feet, (

**b**) supported by its feet and pulled by its hands, (

**c**) supported by its feet and pushed by its hands, (

**d**) supported by its feet and its buttocks, (

**e**) supported by its feet and by its buttocks while being pulled by its hands, and (

**f**) supported by its feet and by its buttocks while being pushed by its hands. The black arrows represent the external forces acting on the humanoid.

**Figure 8.**Outtakes of the recorded motion of a subject sitting down and standing up. The first row shows the photos of the measurement, while the second and the third rows show the sagittal and lateral planes, respectively. The photo and the figures from the same column correspond to the same time frame, noted at the top of the column. The bright dots positioned on the recorded subject, on the force plates, and the force sensors are the reflective markers of the Optitrack system. The black dots in the outtakes from the second and the third row represent the reconstructed joints of the recorded subject, the black rectangles represent the force plates positioned on the ground and the bench, respectively, the blue arrows emerging from the force plates represent the forces measured with the force plates, the blue arrows emerging from the hands represent the forces measured with the force sensors mounted on the double handles, and the red dotted line is the ZML, calculated as described in Section 2.1.2.

**Figure 9.**Two subjects stand on separate force plates and hold on to the double handles while leaning backward. The blue and green arrows represent the external forces acting on the subject to the left and right, respectively. The red dotted line represents the ZML of both subjects treated as one system, while the blue and green dotted lines represent the ZMLs of the subject to the left and right, respectively, when treated as separate systems.

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**MDPI and ACS Style**

Brecelj, T.; Petrič, T.
Zero Moment Line—Universal Stability Parameter for Multi-Contact Systems in Three Dimensions. *Sensors* **2022**, *22*, 5656.
https://doi.org/10.3390/s22155656

**AMA Style**

Brecelj T, Petrič T.
Zero Moment Line—Universal Stability Parameter for Multi-Contact Systems in Three Dimensions. *Sensors*. 2022; 22(15):5656.
https://doi.org/10.3390/s22155656

**Chicago/Turabian Style**

Brecelj, Tilen, and Tadej Petrič.
2022. "Zero Moment Line—Universal Stability Parameter for Multi-Contact Systems in Three Dimensions" *Sensors* 22, no. 15: 5656.
https://doi.org/10.3390/s22155656