# Stability Analysis and Navigational Techniques of Wheeled Mobile Robot: A Review

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## Abstract

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## 1. Introduction

#### 1.1. Stability Analysis of Wheeled Robot

- Kinematic and dynamic models: Stability analysis approaches use mathematical models to describe the robot’s dynamics and kinematics. The kinematic model is solely based on the robot’s motion, disregarding any stresses or torques that may affect the robot’s mobility. Contrarily, the dynamic model considers the pressures and torques that affect the robot’s motion.
- The stability analysis criteria are used to evaluate the stability of the robot. The most common stability analysis criteria used in WMRs are Lyapunov stability, passivity-based stability, and control Lyapunov functions (CLFs). Lyapunov stability analysis uses Lyapunov functions to determine the stability of the robot’s motion. Passivity-based stability analysis is based on the concept of passivity and uses energy functions to evaluate the stability of the robot. CLFs are used to design control laws that ensure the stability of the robot.
- Trajectory tracking: To assess the robot’s ability to follow a desired trajectory, stability analysis techniques are also applied. The discrepancy between the robot’s actual trajectory and its anticipated trajectory is known as the trajectory tracking error. The robot’s stability is measured by its capacity to achieve a trajectory tracking error of zero.
- Control design: Stability analysis methods are used to design control laws that ensure the stability of the robot. The control laws are designed based on the dynamics of the robot and the stability analysis criteria. The control laws are then implemented in the robot’s control system to ensure stable motion.

- Simultaneous localization and mapping (SLAM): Popular mapping and localization methods include SLAM. It entails mapping the surroundings and determining the robot’s position inside that map at the same time. When an existing map of the environment is not accessible, SLAM is very helpful.
- Monte Carlo localization (MCL): For determining the robot’s location inside a known map, MCL is a particular kind of particle filter. It functions by converting the location of the robot into a probability distribution and updating it in response to sensor readings.
- Reactive navigation: Reactive navigation techniques focus on the robot’s immediate environment and are designed to quickly and efficiently respond to changing situations. Examples of reactive techniques include potential fields and behavior-based control.
- Model predictive control (MPC): In order to forecast the robot’s future behavior and optimize control inputs, the MPC control approach employs a mathematical model of the robot’s dynamics. Planning a course and avoiding obstacles are both possible with MPC.
- Reinforcement learning (RL): RL is a machine learning technique that enables the robot to learn how to navigate through trial and error. The robot receives rewards or punishments based on its actions and adjusts its behavior to maximize the rewards.

#### 1.2. Major Objective

- Provide an extensive review that focus on navigational complexities in the context of wheeled mobile robots (WMRs).
- Trace the historical and contemporary developments in WMR research, including the establishment of kinetic stability and the construction of intelligent WMR controllers.
- Examine the importance of stability and intelligent capabilities in WMR controllers and their impact on WMR performance.
- Present a comprehensive overview of stability analysis techniques tailored for WMRs, including discussions on Lyapunov stability analysis and passivation-based control.
- Explore various navigation techniques for WMRs, covering aspects such as path planning, obstacle avoidance, localization and mapping, and trajectory tracking in both indoor and outdoor settings.

#### 1.3. Paper Organization

## 2. Literature Review

#### 2.1. Kinematic Analysis of WMR

#### 2.2. Dynamic Analysis of Mobile Robot

- $B\left({\xi}_{I}\right)$ denotes a matrix or function that is dependent on a $vector{\xi}_{I}.$ You would need to specify B’s precise nature and how it depends on ${\xi}_{I}$ in your particular circumstance. $B\left({\xi}_{I}\right)$ may stand for a transformation matrix or a function associated with a specific component of the system under study, for example.
- τ: This vector represents the force or torque that is being applied. When employed in the context of dynamics, τ usually denotes outside forces or control inputs operating on a mechanical system.
- $JT\left({\xi}_{I}\right):$ In this case, J is a Jacobian matrix.

#### 2.3. Eliminate Modeling Uncertainties and Environmental Disturbances

## 3. Path and Motion Planning

## 4. Navigation of Mobile Robot

#### 4.1. Indoor Navigation

#### 4.1.1. Map-Based Navigation

#### 4.1.2. Map-Building-Based Navigation

#### 4.1.3. Mapless Navigation

#### 4.2. Outdoor Navigation

#### 4.2.1. Outdoor Navigation in Structured Environments

#### 4.2.2. Outdoor Navigation in Unstructured Environments

## 5. Various Navigational Approach of WMR

#### 5.1. Artificial Neural Network (NN) Technique

#### 5.2. Genetic Algorithm Technique

#### 5.3. Fuzzy Logic

#### 5.4. Probabilistic Roadmap (PRM) Planning

- Sample configurations: The PRM algorithm starts by randomly sampling a set of configurations from the robot’s configuration space. Let Q be the configuration space, and q_sample be a randomly generated sample from Q, where q_sample ∈ Q.
- Validate configurations: Each sample is then checked to see if it is a valid configuration that the robot can reach without colliding with obstacles. This is achieved by performing collision detection using the robot’s sensor data or environment maps. Let C(q_sample) be the collision detection function that checks if q_sample is in the collision, where C(q_sample) = True if q_sample is in a collision and False otherwise.
- Connect valid configurations: Valid samples are then connected with edges in the roadmap to create a graph. Two configurations ${q}_{1}\mathrm{a}\mathrm{n}\mathrm{d}{q}_{2}$ are connected if there exists a valid path between them. The edge weight between ${q}_{1}\mathrm{a}\mathrm{n}\mathrm{d}{q}_{2}$ is typically set to the Euclidean distance between the two configurations. Let E be the set of edges between connected configurations, where e = $({q}_{1},{q}_{2})\in EifC\left({q}_{1}\right)$ = False, $C\left({q}_{2}\right)$ = False, and there exists a valid path between ${q}_{1}\mathrm{a}\mathrm{n}\mathrm{d}{q}_{2}.$
- Find a path: The next step is to utilize a search technique, such as Dijkstra’s algorithm, to identify a path connecting the start and goal configurations in the graph. Assume that the graph created in step 3 is G = (V, E), with V being the collection of configurations. Let the start and target configurations be denoted by s and g, respectively. The Dijkstra algorithm determines the shortest route in G between s and g.
- Smooth the path: The final path can be smoothed to remove unnecessary turns and improve efficiency. This is typically achieved using path smoothing algorithms, such as the spline-based or polynomial-based methods.
- The PRM algorithm is probabilistic because it relies on random sampling to generate the roadmap and does not guarantee the optimal path. However, it is often effective and scalable, particularly in high-dimensional configuration spaces where other algorithms may struggle.

- Create a graph: Create a graph where each node represents a location in the 2D environment. The edges between nodes represent possible movements from one location to another.
- Initialize start and goal nodes: Set the start location as the starting node and the goal location as the goal node.
- Calculate heuristic function: Calculate the heuristic function h(n) for each node n. This can be achieved using a distance metric, such as Euclidean distance, to estimate the distance between the node and the goal location.
- Initialize costs: Set the cost of the start node g(start) to 0, and the cost of all other nodes to infinity.
- Add start node to open set: Add the start node to the open set, which is the set of nodes to be evaluated.

- Select the node with the lowest f(n) value from the open set.
- Stop the search and retrace the path if the chosen node is the goal node.
- Otherwise, for each neighbor of the selected node:
- Calculate the tentative cost of the path from the start node to the neighbor node, $g\prime \left(neighbor\right)=g\left(selected\right)+cost(selected,neighbor)$.
- If g’(neighbor) is less than the current cost of the neighbor node, update the cost of the neighbor node to g’(neighbor) and set its parent to the selected node.
- If the neighbor node is not already in the open set, include it and determine the f(n) value. If the goal node is not reached, return failure. Once the A* search algorithm has found a path, it can be represented as a sequence of nodes that correspond to locations in the 2D environment. The robot can then move along this path by following the sequence of locations. If obstacles are detected during movement, the algorithm can be rerun to find a new path around the obstacle.

#### 5.5. Rapidly-Exploring Random Tree (RRT) Planning

Algorithm 1: Rapidly-exploring random tree (RRT) planning. |

Step 1: Initialize the tree with the start configuration as the root node. |

Step 2: Repeat until the goal configuration is reached or a maximum number of nodes are added: |

a. Sample a random configuration in the space. |

b. Find the nearest node in the tree to the sampled configuration. |

c. Generate a new node by extending the nearest node towards the sampled configuration, while ensuring that the robot does not collide with any obstacles. |

d. Add the new node to the tree and connect it to its nearest neighbor. |

Step 3: If the goal configuration is reached, return the path from the start to the goal configuration by backtracking through the tree from the goal node to the root node. |

Step 4: The RRT algorithm is simple and efficient and can handle high-dimensional and complex spaces. It can also deal with moving obstacles by updating the tree in realtime. |

Algorithm 2: Pseudocode for navigation planning. |

S: a set of all valid configurations in the space. |

q_start: the start configuration of the robot. |

q_goal: the goal configuration of the robot. |

K: maximum number of nodes to add to the tree. |

T: tree of configurations, initially containing only q_start. |

r: maximum distance to extend the tree towards a new configuration. |

T ← {${q}_{start}$} |

for k = 1 to K do |

a. ${q}_{rand}$←RandomConfiguration() |

b. ${q}_{near}$←NearestNeighbor(${q}_{rand}$, T) |

c. ${q}_{new}$←Extend(${q}_{near},{q}_{rand},r$) |

d. if CollisionFree(${q}_{near},{q}_{new}$) then |

T ← T ∪ {${q}_{new}$, (${q}_{near},{q}_{new}$)} |

e. if Distance(${q}_{new},{q}_{goal}$) ≤ ε then |

return Path($T,{q}_{start},{q}_{new}$) |

return Failure |

- RandomConfiguration() returns a random configuration in S.
- NearestNeighbor(${q}_{rand},T$) returns the configuration in T that is closest to q_rand.
- Extend(${q}_{near},{q}_{rand},r)$ returns a new configuration that is a distance r away from ${q}_{near}towards{q}_{rand}.$
- CollisionFree$({q}_{near},{q}_{new})$ returns true if the path from q_near to q_new is collision-free.
- Distance$({q}_{new},{q}_{goal})$ returns the distance between ${q}_{new}and{q}_{goal}.$
- ε is a small value that determines the proximity of ${q}_{new}to{q}_{goal}.$
- Path$(T,{q}_{start},{q}_{new})$ returns the path from ${q}_{start}to{q}_{new}$ by backtracking through the tree T.

#### Model Predictive Control (MPC)

- a.
**Assumptions:**

- b.
**Algorithm:**

- The control input ${u}_{k}$ that minimizes the cost function subject to the system constraints is computed using an optimization solver.
- The first element of the optimal control input sequence ${u}_{\left\{k:k+N-1\right\}}$ is applied to the system as the control input, and the process is repeated at the next time step.

- c.
**Assumptions:**

#### 5.6. SLAM (Simultaneous Localization and Mapping)

#### 5.7. Comparative Analysis of Existing Work

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Comparative analysis of methods used in navigation and stability analysis. (

**a**) Comparative analysis of success rate of existing work. (

**b**) Comparative analysis of the control method. (

**c**) Comparative analysis of methods.

Method | Kinematic and Dynamic Models | Stability Analysis Criteria | Trajectory Tracking | Control Design | Computational Time | Modeling Uncertainties | Environmental Disturbances | Success Rate (%) |
---|---|---|---|---|---|---|---|---|

Artificial neural networks | Yes | Lyapunov stability, CLFs | High | Design based on neural network training | Moderate to high | Can handle some uncertainties, depending on training | Can adapt to some disturbances if in training data | 82% |

Genetic algorithms | Yes | Lyapunov stability, passivity-based stability | Moderate | Design based on genetic algorithm optimization | High | Can address uncertainties through the optimization process | May not inherently handle disturbances | 73% |

Fuzzy logic control | Yes | Lyapunov stability, CLFs | High | Design based on fuzzy rules and membership functions | Low to moderate | Designed to handle uncertainties with linguistic variables and rules | Can adapt to some disturbances with appropriate rule sets | 89% |

Firefly algorithm | Yes | Lyapunov stability, CLFs | High | Design based on algorithm optimization | Varies (moderately high) | Not inherently designed to handle uncertainties | Effectiveness may vary depending on the problem and adaptation mechanisms | 76% |

Hybrid approaches | Yes | Lyapunov stability, CLFs | High | Design based on a combination of AI techniques | Varies (Depends on constituent methods) | Depends on the individual methods in the hybrid approach | Depends on the individual methods in the hybrid approach | 92% |

Model predictive control | Yes | Lyapunov stability, CLFs | High | Design based on optimal control strategies | Moderate to high | Can address uncertainties through the optimization process | Can adapt to disturbances through real-time optimization | 87% |

Particle swarm optimization | Yes | Lyapunov stability, CLFs | High | Design based on particle swarm optimization | Moderate to high | Can address uncertainties through an optimization process | May not inherently handle disturbances | 81% |

Reinforcement learning | Yes | Lyapunov stability, CLFs | High | Design based on reward function optimization | Moderate to high | Can adapt to some uncertainties with robust training | Can adapt to disturbances through exploration and learning | 78% |

Robust control | Yes | Lyapunov stability, CLFs | High | Design based on robust control theory | Low to moderate | Specifically designed for handling uncertainties with robust control techniques | Can handle disturbances with designed robustness | 85% |

Sliding mode control | Yes | Lyapunov stability, CLFs | High | Design based on sliding mode control theory | Low to moderate | Specifically designed for handling uncertainties with sliding mode control techniques | Can handle disturbances with designed robustness | 80% |

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**MDPI and ACS Style**

Borkar, K.K.; Aljrees, T.; Pandey, S.K.; Kumar, A.; Singh, M.K.; Sinha, A.; Singh, K.U.; Sharma, V.
Stability Analysis and Navigational Techniques of Wheeled Mobile Robot: A Review. *Processes* **2023**, *11*, 3302.
https://doi.org/10.3390/pr11123302

**AMA Style**

Borkar KK, Aljrees T, Pandey SK, Kumar A, Singh MK, Sinha A, Singh KU, Sharma V.
Stability Analysis and Navigational Techniques of Wheeled Mobile Robot: A Review. *Processes*. 2023; 11(12):3302.
https://doi.org/10.3390/pr11123302

**Chicago/Turabian Style**

Borkar, Kailash Kumar, Turki Aljrees, Saroj Kumar Pandey, Ankit Kumar, Mukesh Kumar Singh, Anurag Sinha, Kamred Udham Singh, and Vandana Sharma.
2023. "Stability Analysis and Navigational Techniques of Wheeled Mobile Robot: A Review" *Processes* 11, no. 12: 3302.
https://doi.org/10.3390/pr11123302