# Scheme of Operation for Multi-Robot Systems with Decision-Making Based on Markov Chains for Manipulation by Caged Objects

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## Abstract

**:**

## 1. Introduction

## 2. Manipulation Scheme

## 3. Dynamic Model

## 4. Multi-Robot Tasks

#### 4.1. Equality Tasks

#### 4.1.1. Geometric Shape Formation

#### 4.1.2. Regulation Task

#### Individual

#### Cooperative

#### 4.2. Inequality Tasks

#### 4.3. Obstacle Avoidance

#### 4.4. Collision Avoidance

#### 4.5. Geometric Permissible Region

## 5. Hierarchical Quadratic Programming

Algorithm 1 Task phase |

**Require:**- State of the group ($\xi ,\dot{\xi}$), Task phase $\mathbb{S}$, $\delta t$
**Ensure:**- Control input $\tau $
Get ${\mathcal{R}}_{p}^{o}$ with (2) $\eta ={\mathcal{R}}_{p}^{o}\dot{\xi}$ Get $M,\phantom{\rule{0.222222em}{0ex}}h$ with (6) Indicate the order of the tasks: $\mathbb{S}=[{e}_{oa},{e}_{ci},\cdots ]$ Get task Jacobians: ${J}_{s}=[{J}_{oa},{J}_{ci},\cdots ]$ Get time derivative task Jacobians: ${\dot{J}}_{s}=[{\dot{J}}_{oa},{\dot{J}}_{ci},\cdots ]$ Get $\tau $ with (30) Get $\dot{\eta}$ with (6) |

## 6. Action Phases

## 7. Discrete-Time Markov Decision Agent

**Definition 1**

**Definition 2**

**Definition 3**

**Definition 4**

**Theorem 1.**

**Sketch of the Proof.**

## 8. Simulation Results

#### 8.1. Phase 1: Formation Phase

#### 8.2. Phase 2: Grip Phase

#### 8.3. Phase 3: Manipulation Phase

#### 8.4. Phase 4: Waiting Phase

#### 8.5. Variance Value and Phase Measure

**Figure 14.**Close up of the Measure from Figure 15.

#### 8.6. Comparison of DTMC and FSM in the Presence of Unexpected Failures

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DTMC | Discrete-Time Markov Chain |

HQP | Hierarchical Quadratic Programming |

FSM | Finite-State Machine |

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**Figure 1.**The Manipulation scheme involves six interconnected blocks: 1. Robots, representing the omnidirectional robots; 2. Environment, the operational space for the robots; 3. Tasks, the specific goals to be achieved by the robots; 4. Phases, the stages involved in the manipulation scheme; 5. DTMC, which utilizes a discrete-time Markov chain; and 6. HQP, which computes the control of the robots to carry out simultaneous tasks.

O | Inertial Frame |

P | Referential in the robot C.o.M. |

n | number of robots |

$\xi $ | robot extended state |

${q}_{i}$ | i-robot state |

${x}_{i}$ | i-robot position along x-axis |

${y}_{i}$ | i-robot position along y-axis |

${\theta}_{i}$ | i-robot orientation along z-axis |

${R}_{O}^{P}\left(\theta \right)$ | Rotation Matrix from Frame O to P |

${\mathcal{R}}_{O}^{P}\left(\theta \right)$ | Extended Rotation Matrix |

${M}_{i}$ | i-robot inertia Matrix |

${M}_{\varphi i}$ | i-robot wheels inertia Matrix |

${h}_{i}$ | non-linear Vector from i-robot |

${\tau}_{i}$ | i-robot control input |

${\tau}_{\varphi i}$ | i-robot control input for each wheel |

M | Extended inertia Matrix |

h | non-linear extended vector from i-robot |

${e}_{Q}$ | Q-task error |

${J}_{Q}$ | Q-task Jacobian |

${\dot{J}}_{Q}$ | Time derivative from ${J}_{Q}$ |

${u}_{Q}$ | Q-task auxiliary control |

$\mathbb{S}$ | Behaviour state space |

${\mathbb{S}}_{j}$ | j-state of behaviour |

${\alpha}_{j}$ | j-state distribution |

Phase | Task | Error | Hierarchy |
---|---|---|---|

Phase 1 Formation | Obstacle avoidance | ${e}_{oa}$ | 1 |

Collision avoidance | ${e}_{om}$ | 2 | |

Geometric formation | ${e}_{ci}$ | 3 | |

Phase 2 Grip | Obstacle avoidance | ${e}_{oa}$ | 1 |

Collision avoidance (equidistant) | ${e}_{om}$ | 2 | |

Geometric formation | ${e}_{ci}$ | 3 | |

Phase 3 Manipulation | Obstacle avoidance | ${e}_{oa}$ | 1 |

Collision avoidance | ${e}_{om}$ | 2 | |

Permissible space | ${e}_{pi}$ | 3 | |

Cooperative regulation | ${e}_{r}$ | 4 | |

Phase 4 Waiting | Obstacle avoidance | ${e}_{oa}$ | 1 |

Collision avoidance | ${e}_{om}$ | 2 | |

Individual regulation | ${e}_{ri}$ | 3 |

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**MDPI and ACS Style**

Arreguín-Jasso, D.; Sanchez-Orta, A.; Alazki, H. Scheme of Operation for Multi-Robot Systems with Decision-Making Based on Markov Chains for Manipulation by Caged Objects. *Machines* **2023**, *11*, 442.
https://doi.org/10.3390/machines11040442

**AMA Style**

Arreguín-Jasso D, Sanchez-Orta A, Alazki H. Scheme of Operation for Multi-Robot Systems with Decision-Making Based on Markov Chains for Manipulation by Caged Objects. *Machines*. 2023; 11(4):442.
https://doi.org/10.3390/machines11040442

**Chicago/Turabian Style**

Arreguín-Jasso, Daniel, Anand Sanchez-Orta, and Hussain Alazki. 2023. "Scheme of Operation for Multi-Robot Systems with Decision-Making Based on Markov Chains for Manipulation by Caged Objects" *Machines* 11, no. 4: 442.
https://doi.org/10.3390/machines11040442