# Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation

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## Abstract

**:**

## 1. Introduction

## 2. Dual-Objective Convergence Equation for the Biped Robot

#### 2.1. Model Configuration

#### 2.2. Dual-Objective Convergence Equation

#### 2.3. Convergence Equation for Sliding

#### 2.4. Convergence Equation for Standing

#### 2.5. Torso Recovery Strategy

- First, reduce the difference (i.e., error) between U and ${U}^{ref}$ so that E begins to converge in the desired way.
- Then, wait for the amplitude of E to gradually decrease to within a permissible range, which indicates that the robot enters a stable state.
- Finally, add an equilibrium offset $\Delta U$ to the desired equilibrium target ${U}^{ref}$ according to the feedback states of the torso. Therefore, the angle of the torso $\theta $ gradually recovers to the reference value.

## 3. QP Controller with Multiple Constraints

#### 3.1. Qp Process

#### 3.2. PD Controller

#### 3.3. Dynamic Constraints

#### 3.4. Contact Constraints

#### 3.4.1. Contact Constraint in Sliding

#### 3.4.2. Contact Constraint in Standing

## 4. Results

#### 4.1. Balance in Sliding on Uneven Terrain

#### 4.2. Balance Recovery on Terrain with a Variable Coefficient of Friction

#### 4.3. Balance Recovery in Standing

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Block diagram of the balance control framework. ${\tau}^{*}$ is the optimized joint torque.

**Figure 2.**Coordinates and structural configuration. ${X}_{b}$ and ${Y}_{b}$ are the x and y floating-base coordinates. $\theta $ is the angle between the y-axis of the world frame and the torso. ${q}_{1}$ and ${q}_{2}$ are the joint coordinates of the hip and knee. m, I and l are the mass, inertia and length of a rigid body.

**Figure 5.**Snapshots of sliding on uneven terrain: (

**a**) sliding from the bottom to the uphill, (

**b**) sliding from the uphill to the flat-top, (

**c**) sliding from the flat-top to the downhill, (

**d**) sliding from the downhill to the bottom, (

**e**) sliding having stopped.

**Figure 12.**Trajectory of the estimated friction coefficient when the robot slides on terrain with a variable friction coefficient.

**Figure 13.**Trajectory of the equilibrium variable when the robot slides on terrain with a variable friction coefficient.

**Figure 14.**Trajectory of the torso angle when the robot slides on terrain with a variable friction coefficient.

**Figure 15.**Snapshots of the posture recovery when the robot stands on ground with a low-friction coefficient.

**Figure 18.**Horizontal contact force when the robot is sliding on terrain with a variable friction coefficient.

Description | Symbol | Value | Unit |
---|---|---|---|

torso mass | ${m}_{0}$ | 15.6 | Kg |

thigh mass | ${m}_{1}$ | 6.8 | Kg |

calf mass | ${m}_{2}$ | 4.5 | Kg |

torso inertia | ${I}_{0}$ | 0.49 | Kg · m${}^{2}$ |

thigh inertia | ${I}_{1}$ | 0.11 | Kg · m${}^{2}$ |

calf inertia | ${I}_{2}$ | 0.09 | Kg · m${}^{2}$ |

torso length | ${l}_{0}$ | 0.40 | m |

thigh length | ${l}_{1}$ | 0.33 | m |

calf length | ${l}_{2}$ | 0.33 | m |

Description | Symbol | Value |
---|---|---|

dual-objective proportion | $\epsilon $ | 0.025 |

torso control parameter | ${K}_{\theta}$ | [0.50, 0.25, 0.04, 0.02, 10$\pi $/180, 500$\pi $/180, 0.0125, 0.0156] |

equilibrium variable PD parameter | ${K}_{{U}_{sl}}$ | [2.5, 0.02, 3.0, 0.5, 50.0, 1.0] |

knee PD parameter | ${K}_{{q}_{2}}$ | [$\pi $, 10$\pi $/180, 1.5$\pi $, 0.5, 20$\pi $, 200$\pi $/180] |

equilibrium variable relaxations weight in QP | ${\rho}_{{U}_{sl}}$ | 1000 |

knee relaxations weight in QP | ${\rho}_{{q}_{2}}$ | 1 |

max Vertical contact force proportion | ${\alpha}_{max}$ | 2.5 |

min Vertical contact force proportion | ${\alpha}_{min}$ | 0.5 |

Description | Symbol | Value |
---|---|---|

dual-objective proportion | $\epsilon $ | 0.01 |

torso control parameter | ${K}_{\theta}$ | [1.0, 0.5, 0.02, 0.01, 10$\pi $/180, 1000$\pi $/180, 0.002, 0.003] |

equilibrium variable PD parameter | ${K}_{{U}_{st}}$ | [1.0, 0.02, 1.5, 1.5, 9.8, 0.2] |

knee PD parameter | ${K}_{{q}_{2}}$ | [$\pi $, 10$\pi $/180, 1.5$\pi $, 0.5, 20$\pi $, 200$\pi $/180] |

equilibrium variable relaxations weight in QP | ${\rho}_{{U}_{st}}$ | 1 |

knee relaxations weight in QP | ${\rho}_{{q}_{2}}$ | 1 |

max Vertical contact force proportion | ${\alpha}_{max}$ | 2.5 |

min Vertical contact force proportion | ${\alpha}_{min}$ | 0.5 |

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**MDPI and ACS Style**

Lu, Y.; Gao, J.; Shi, X.; Tian, D.; Liu, Y.
Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation. *Appl. Sci.* **2021**, *11*, 4016.
https://doi.org/10.3390/app11094016

**AMA Style**

Lu Y, Gao J, Shi X, Tian D, Liu Y.
Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation. *Applied Sciences*. 2021; 11(9):4016.
https://doi.org/10.3390/app11094016

**Chicago/Turabian Style**

Lu, Yizhou, Junyao Gao, Xuanyang Shi, Dingkui Tian, and Yi Liu.
2021. "Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation" *Applied Sciences* 11, no. 9: 4016.
https://doi.org/10.3390/app11094016