# Reactive Balance Control for Legged Robots under Visco-Elastic Contacts

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## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Overview

#### 1.2. State of the Art

#### 1.3. Contributions

#### 1.4. Paper Structure

## 2. Background: Dynamics and Contact Model

#### 2.1. Visco-Elastic Contact Model

#### 2.2. Importance of Stiffness vs. Damping

#### 2.3. Centroidal Dynamics

#### 2.4. Whole-Body Dynamics

## 3. State of the Art

#### 3.1. Inverse Dynamics with Rigid Contacts (TSID-Rigid)

#### 3.2. Admittance Control (Adm-Ctrl)

## 4. Inverse Dynamics Admittance Control (TSID-Adm)

## 5. Flexible TSID (TSID-Flex-K)

#### 5.1. Feedback Linearization

#### 5.2. Accounting for Force Variations during the Time Step

#### 5.3. Linear Feedback Regulator

#### 5.4. Friction Force Constraints

#### 5.5. Summary

## 6. Results

- TSID-Rigid: a state-of-the-art approach, see Section 3.1.
- Adm-Ctrl: a state-of-the-art approach, see Section 3.2.
- TSID-Adm: a novel approach, see Section 4.
- TSID-Flex-K: a novel approach, see Section 5.

#### 6.1. Simulation Environment

#### 6.2. Gain Tuning

#### 6.3. Test Description

- realistic encoder quantization errors and white Gaussian noise on force sensing and gyroscope (see Table 3);
- an Extended Kalman Filter (explained in Appendix B) to estimate the robot state, with the covariances specified in Table 4;
- limited torque bandwidth by filtering the desired joint torques with a first-order low-pass filter with a cut frequency of 30 Hz. The best torque-tracking bandwidths that have been reported for high-performance actuators are between 40 Hz and 60 Hz (e.g., 40 Hz for hydraulic actuators [2], 46 Hz for electric motors with harmonic drives [21], 60 Hz for series elastic actuators [22]).
- joint Coulomb friction of about 1% of the maximum joint force/torque (0.4 Nm for hip joints, and 4 N for knee joints, which are prismatic).

#### 6.4. Test A: TSID-Rigid

#### 6.5. Test B: TSID-Flex-K and TSID-Adm

#### 6.6. Test C: Adm-Ctrl

#### 6.7. Test D: All Controllers

#### 6.7.1. Soft Contacts

#### 6.7.2. Medium Contacts

#### 6.7.3. Stiff Contacts

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TSID | Task-Space Inverse Dynamics |

CoM | Center of Mass |

Adm-Ctrl | Admittance Control |

## Appendix A. Gain Tuning

- the robot dynamical model is perfect,
- ${w}_{f}$ and ${w}_{post}$ are sufficiently small not to significantly affect the momentum task,
- the inequality constraints (e.g., friction cones) are not active.

#### Appendix A.1. TSID-Flex-K

#### Appendix A.2. Admittance Control

#### Appendix A.3. TSID-Admittance

- contact damping is negligible: $B\approx 0$;
- ${K}_{d}^{adm}$ and $K{K}_{f}$ are diagonal matrices;
- all entries of ${K}_{d}^{adm}$, and $K{K}_{f}$ corresponding to the same direction (X, Y, Z) have the same value; for instance, the admittance gain ${K}_{d}^{adm}$ in direction Z must be the same for all contact points.

#### Appendix A.4. TSID-Rigid

#### Appendix A.5. Cost Function

- for TSID-Flex-K, $y=(x,\dot{x},\ddot{x},\stackrel{\u20db}{x})$ and $u={x}^{\left(4\right)}$,
- for TSID-Adm, $y=(c,\dot{c},\ddot{c},\stackrel{\u20db}{c})$ and $u={c}^{\left(4\right)}$,
- for TSID-Rigid, $y=(x,\dot{x})$ and $u=\ddot{x}$.

## Appendix B. Estimation

## Appendix C. Order-of-Magnitude Analysis on Importance of Stiffness vs. Damping

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**Figure 6.**Summary of all results. Pareto-optimal tests are depicted in red, while the others in blue.

**Figure 7.**TSID-Adm, soft contacts ($\alpha =0.01$), noiseless simulation and low feedback gains (${w}_{u}={10}^{-6}$). CoM tracking is good despite the significant delay in force tracking.

**Figure 8.**Results for soft contacts ($\alpha =0.01$) and noisy simulation, but without Coulomb friction. Coulomb friction has a detrimental effect to the performance of TSID-Flex-K and TSID-Adm, while it helps stabilizing TSID-Rigid.

**Figure 9.**Comparison between contact forces given by TSID-Rigid and TSID-Adm for medium contacts, noisy simulations.

Symbol | Meaning | Value |
---|---|---|

$\delta {t}_{sim}$ | Simulation time step | 0.0001 s |

$\delta {t}_{ctrl}$ | Control time step | 0.001 s |

$\mu $ | Force friction coefficient | 0.3 |

K | Contact stiffness | $\alpha $ 200,000 |

$\zeta $ | Contact damping ratio | 0.3 |

T | Simulation time | 6 s |

**Table 2.**Controller Parameters. $\mathrm{diag}\left({M}_{0}\right)$ is the diagonal part of the mass matrix evaluated at $q\left(0\right)$.

Symbol | Controller | Meaning | Value |
---|---|---|---|

${w}_{post}$ | TSID-Flex-K | Postural task weight | 0.3 |

${w}_{post}$ | TSID-Adm | Postural task weight | 1 × 10^{−3} |

${w}_{post}$ | Adm-Ctrl | Postural task weight | 1 × 10^{−3} |

${w}_{post}$ | TSID-Rigid | Postural task weight | 1 × 10^{−2} |

${w}_{f}$ | TSID-Rigid | Force regularization weight | 1 × 10^{−4} |

${K}_{p}^{m}$ | Adm-Ctrl | Proportional momentum gain | 30.7 |

${K}_{d}^{m}$ | Adm-Ctrl | Derivative momentum gain | 10.3 |

${K}_{p}^{j}$ | Adm-Ctrl | Proportional joint position gain | ${10}^{4}$$\mathrm{diag}\left({M}_{0}\right)$ |

${K}_{d}^{j}$ | Adm-Ctrl | Derivative joint position gain | 200 $\mathrm{diag}\left({M}_{0}\right)$ |

${K}_{f}$ | Adm-Ctrl | Proportional force gain | 8 × 10^{−3} |

${K}_{p}^{post}$ | All | Proportional posture gain | 10 |

${K}_{d}^{post}$ | All | Derivative posture gain | 6 |

Symbol | Meaning | Value |
---|---|---|

${\sigma}_{fy}$ | Force sensor (Y axis) | 1.3 × 10^{−2} |

${\sigma}_{fz}$ | Force sensor (Z axis) | 1.0 |

${\sigma}_{\omega}$ | Gyroscope | 6.4 × 10^{−3} |

${\delta}_{fy}$ | Force sensor (Y axis) | 1.8 × 10^{−2} |

${\delta}_{fz}$ | Force sensor (Z axis) | 7.3 × 10^{−2} |

${\delta}_{\omega}$ | Gyroscope | 1.0 × 10^{−3} |

${\delta}_{q}$ | Encoders | 8.2 × 10^{−5} |

Symbol | Meaning | Value |
---|---|---|

${\sigma}_{c}$ | CoM position measurement | 1 × 10^{−3} |

${\sigma}_{\dot{c}}$ | CoM velocity measurement | 1 × 10^{−2} |

${\sigma}_{l}$ | Angular momentum measurement | 0.1 |

${\sigma}_{f}$ | Force measurement | 1 |

${\sigma}_{u}$ | Control | 1 × 10^{4} |

${\sigma}_{\ddot{c}}$ | CoM acceleration disturbance | 10 |

PROS | CONS |
---|---|

TSID-Flex-K and TSID-Adm | |

• Easy to tune (unified pos-force feedback). | • Need high frequency for hard contacts. |

• Best for soft/medium contact. | • Unstable for hard contact with noise. |

TSID-Rigid | |

• Ok for hard/medium contacts. | • Unstable for soft contacts. |

• Easy to tune (assuming perfect joint torque tracking). | • Undesired oscillations (no force feedback). |

Adm-Ctrl | |

• Always stable. | • Hard to tune/analyze. |

• Good for soft contacts. | • Never the best. |

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**MDPI and ACS Style**

Flayols, T.; Del Prete, A.; Khadiv, M.; Mansard, N.; Righetti, L.
Reactive Balance Control for Legged Robots under Visco-Elastic Contacts. *Appl. Sci.* **2021**, *11*, 353.
https://doi.org/10.3390/app11010353

**AMA Style**

Flayols T, Del Prete A, Khadiv M, Mansard N, Righetti L.
Reactive Balance Control for Legged Robots under Visco-Elastic Contacts. *Applied Sciences*. 2021; 11(1):353.
https://doi.org/10.3390/app11010353

**Chicago/Turabian Style**

Flayols, Thomas, Andrea Del Prete, Majid Khadiv, Nicolas Mansard, and Ludovic Righetti.
2021. "Reactive Balance Control for Legged Robots under Visco-Elastic Contacts" *Applied Sciences* 11, no. 1: 353.
https://doi.org/10.3390/app11010353