In recent years, the application of multi-robot systems in various fields has gained significant attention due to advancements in robot technology. Environmental sanitation tasks, such as street cleaning and waste collection, often require coordinated efforts across large areas, which can be efficiently handled by multiple robots working in formation. However, maintaining stable formations in complex environments remains a challenge. This paper addresses this issue by proposing a novel formation control method that integrates the leader-follower strategy with the artificial potential field algorithm. The approach enables a group of sanitation robots to navigate obstacles, pass through narrow passages, and maintain formation stability while performing collaborative tasks. Through simulations and real-world experiments, we demonstrate the effectiveness of our method in enhancing the robustness and flexibility of multi-robot systems in dynamic environments.
The core of our work lies in leveraging robot technology to model the kinematics of sanitation robots based on the leader-follower strategy. Each robot is equipped with sensors and actuators that allow it to perceive its surroundings and adjust its trajectory accordingly. The leader robot guides the formation, while follower robots maintain specific relative positions and orientations. To handle obstacles commonly encountered in urban settings, such as trash bins and walls, we incorporate an artificial potential field algorithm that generates attractive forces toward goals and repulsive forces away from obstacles. This combination ensures that the robot formation can adapt to changing conditions without compromising stability. Furthermore, we introduce a formation transformation strategy that allows the robots to reconfigure into a linear formation when traversing narrow spaces, such as alleys, thereby optimizing their operational efficiency.
In the following sections, we delve into the mathematical modeling of the robot formation, detailing the kinematic equations and control laws. We then describe the artificial potential field approach for obstacle avoidance and formation transformation. Simulation results using MATLAB are presented to validate the method under various scenarios, including environments with static obstacles and narrow passages. Additionally, we report on experimental tests conducted with physical robot prototypes in a real-world setting, highlighting the practical implications of our approach. The results show that our method achieves low tracking errors and stable formation maintenance, underscoring the potential of robot technology in advancing autonomous sanitation systems.
Kinematic Modeling of Robot Formation
To establish a reliable formation control system, we first develop a kinematic model for the sanitation robots. Each robot is modeled as a differential drive vehicle with a reference point at the center of its driving wheels. The position and orientation of the i-th robot are represented by coordinates (x_i, y_i) and a heading angle α_i. The linear velocity v_i and angular velocity ω_i are derived from the wheel speeds, as governed by the following equations:
$$ v_i = \frac{v_{L,i} + v_{R,i}}{2} = r \omega_{h,i} $$
$$ \omega_i = \frac{v_i}{d} \tan \psi_i $$
where v_{L,i} and v_{R,i} are the left and right wheel linear velocities, r is the wheel radius, ω_{h,i} is the motor angular velocity, d is the distance from the reference point to the steering axis, and ψ_i is the steering angle. The overall kinematic model for a single robot is expressed as:
$$ \begin{bmatrix} \dot{x}_i \\ \dot{y}_i \\ \dot{\alpha}_i \end{bmatrix} = \begin{bmatrix} \cos \alpha_i \\ \sin \alpha_i \\ 0 \end{bmatrix} v_i + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \omega_i $$
For the multi-robot formation, we adopt a leader-follower structure with one leader robot and two follower robots arranged in a triangular pattern. The relative positions of the followers with respect to the leader are defined by distances L_{ij} and angles φ_{ij}, where i denotes the leader and j the followers. The dynamics of the formation are captured by the following equations for follower j:
$$ x_j = x_i + L_{ij} \cos(\varphi_{ij} + \alpha_i) $$
$$ y_j = y_i + L_{ij} \sin(\varphi_{ij} + \alpha_i) $$
The rate of change of L_{ij} and φ_{ij} is derived as:
$$ \dot{L}_{ij} = v_j \cos \gamma_{ij} – v_i \cos \varphi_{ij} + d \omega_j \sin \gamma_{ij} $$
$$ \dot{\varphi}_{ij} = \frac{1}{L_{ij}} \left( v_i \sin \varphi_{ij} – v_j \sin \gamma_{ij} + d \omega_j \cos \gamma_{ij} – L_{ij} \omega_i \right) $$
where γ_{ij} = α_i + φ_{ij} – α_j. To maintain the desired formation, we apply a closed-loop control law based on the L-φ method:
$$ \dot{L} = k_1 (L_d – L) $$
$$ \dot{\varphi} = k_2 (\varphi_d – \varphi) $$
where k_1 and k_2 are proportional gains, and L_d and φ_d are the desired relative distance and angle. Solving for the control inputs yields the following expressions for the follower’s angular and linear velocities:
$$ \omega_j = \frac{\cos \gamma_{ij}}{d} \left[ k_2 L_{ij} (\varphi_{d,ij} – \varphi_{ij}) – v_i \sin \varphi_{ij} + L_{ij} \omega_i + k_1 (L_{d,ij} – L_{ij}) \tan \gamma_{ij} + v_i \cos \varphi_{ij} \tan \gamma_{ij} \right] $$
$$ v_j = \frac{k_1 (L_{d,ij} – L_{ij}) + v_i \cos \varphi_{ij}}{\cos \gamma_{ij}} – d \omega_j \tan \gamma_{ij} $$
This kinematic framework ensures that the followers accurately track the leader’s trajectory, enabling stable formation control in various environments. The integration of these equations into the robot technology platform allows for real-time adjustments based on sensor feedback, enhancing the system’s adaptability.
Obstacle Avoidance Using Artificial Potential Field
In complex environments, sanitation robots must navigate around obstacles while maintaining formation. We employ an artificial potential field algorithm to generate forces that guide the robots away from obstacles and toward their goals. The total potential field U_total at any point is the sum of an attractive field U_att toward the goal and a repulsive field U_rep from obstacles:
$$ U_{\text{total}} = U_{\text{att}} + U_{\text{rep}} $$
The attractive potential is defined as:
$$ U_{\text{att}} = \frac{1}{2} k_a l_{ig}^2 $$
where k_a is the attractive gain, and l_{ig} is the distance from robot i to the goal g. The repulsive potential is given by:
$$ U_{\text{rep}} = \begin{cases} \frac{1}{l_{ij}} \left( \frac{1}{l_{ij}} – \frac{1}{R_{\text{det}}} \right) k_r, & \text{if } l_{ij} \leq R_{\text{det}} \\ 0, & \text{if } l_{ij} > R_{\text{det}} \end{cases} $$
where k_r is the repulsive gain, l_{ij} is the distance from robot i to obstacle j, and R_det is the detection radius. The negative gradient of these potentials yields the forces acting on the robot:
$$ \vec{F}_{\text{att}} = -\nabla U_{\text{att}} = -k_a l_{ig} \hat{l}_{ig} $$
$$ \vec{F}_{\text{rep}} = -\nabla U_{\text{rep}} = \begin{cases} k_r \left( \frac{1}{l_{ij}^2} – \frac{1}{l_{ij} R_{\text{det}}} \right) \hat{l}_{ij}, & \text{if } l_{ij} \leq R_{\text{det}} \\ 0, & \text{if } l_{ij} > R_{\text{det}} \end{cases} $$
where \hat{l}_{ig} and \hat{l}_{ij} are unit vectors pointing from the robot to the goal and obstacle, respectively. The resultant force \vec{F}_{\text{total}} = \vec{F}_{\text{att}} + \vec{F}_{\text{rep}} dictates the robot’s acceleration and direction changes. To integrate this into the formation control, we modify the leader’s velocity commands based on the total force, ensuring that the entire formation avoids obstacles cohesively.
For narrow passages, we implement a formation transformation strategy. When the leader detects a passage narrower than the formation width, it triggers a reconfiguration into a linear formation. The followers reposition themselves behind the leader at a fixed spacing D, as described by:
$$ x_j = x_i – (j-1) D \cos \alpha_i $$
$$ y_j = y_i – (j-1) D \sin \alpha_i $$
where j is the follower index. This transformation minimizes the formation’s footprint, allowing safe passage through constricted areas. After clearing the passage, the robots revert to the triangular formation. This dynamic adjustment is a key aspect of our robot technology, enabling versatility in diverse operational scenarios.
Simulation Analysis
We conducted extensive simulations in MATLAB to evaluate the proposed formation control method. The simulation environment was set up as a 35 m × 35 m area with multiple static obstacles. The leader robot started at (2, 2) m with an initial velocity of 0.15 m/s, while the followers began at (-4, 3) m and (3, -4) m with heading angles of 0° and 90°, respectively. The desired relative distance was set to 3 m, and obstacles were placed at coordinates (4.5, 5) m, (4, 9) m, (9, 5) m, and (8, 10) m.
The robots successfully navigated around obstacles while maintaining the triangular formation, as shown in the trajectory plots. The following table summarizes key performance metrics during obstacle avoidance:
| Metric | Leader Robot | Follower 1 | Follower 2 |
|---|---|---|---|
| Average Velocity (m/s) | 0.40 | 0.39 | 0.41 |
| Maximum Tracking Error (m) | N/A | 0.35 | 0.34 |
| Average Angular Velocity (rad/s) | 0.60 | 0.59 | 0.61 |
In a second simulation, we introduced a narrow passage of 3 m width to test formation transformation. The robots initially formed a triangle, reconfigured into a line when approaching the passage, and then returned to the triangle formation afterward. The velocity and angular velocity profiles exhibited greater fluctuations during transformation, indicating the increased control effort required in constrained spaces. The tracking errors for followers remained below 0.35 m during transitions and dropped to under 0.1 m in stable phases, demonstrating the robustness of our approach.
The equations governing the simulation dynamics were solved numerically using ODE45 in MATLAB, with a time step of 0.1 s. The control gains were tuned to k_1 = 0.5 and k_2 = 0.3 for optimal performance. The artificial potential field parameters were set to k_a = 1.0, k_r = 0.5, and R_det = 2.0 m, ensuring effective obstacle avoidance without excessive repulsion.
Experimental Validation
To validate the simulation results, we deployed three custom-built sanitation robots in a real-world environment resembling a metro station square. Each robot was equipped with a single-line LiDAR, three 3D cameras, and ultrasonic sensors for perception. The robots communicated via a base station and ran the control algorithms on the Robot Operating System (ROS). The leader-follower formation was initialized with the leader at the front and followers in a triangular pattern.
During the experiments, the robots effectively avoided obstacles such as benches and poles while maintaining formation. In narrow alleyways, they transitioned to a linear formation with a spacing of 1.5 m, as predefined. The following table outlines the experimental results:
| Aspect | Performance |
|---|---|
| Formation Stability | Tracking error < 0.1 m in stable regions |
| Obstacle Avoidance | 100% success rate in avoiding static obstacles |
| Formation Transformation | Smooth reconfiguration within 5 s |
| Communication Latency | Average of 50 ms |
The robots’ velocities were controlled using PID regulators, with the linear velocity v_i and angular velocity ω_i updated at 10 Hz. The artificial potential field algorithm ran in real-time, processing sensor data to compute repulsive forces. The experimental data confirmed that the proposed method enhances the reliability of multi-robot systems in practical applications, showcasing the maturity of modern robot technology.
Conclusion
In this paper, we presented a formation control method for environmental sanitation robots that combines the leader-follower strategy with the artificial potential field algorithm. Our approach addresses the challenges of formation stability and obstacle avoidance in complex environments, leveraging advanced robot technology to enable dynamic reconfiguration for narrow passages. The kinematic modeling and control laws ensure accurate tracking, while the potential field method provides robust navigation around obstacles.
Simulations and real-world experiments demonstrated the effectiveness of our method, with followers maintaining tracking errors below 0.1 m in stable conditions and successfully adapting to environmental changes. The integration of formation transformation further enhances the system’s versatility, making it suitable for various sanitation tasks. Future work will focus on extending this approach to dynamic obstacles and larger robot swarms, continuing to push the boundaries of robot technology in autonomous systems.
The continuous evolution of robot technology promises to revolutionize fields like environmental sanitation, and our contributions lay a foundation for more intelligent and collaborative multi-robot applications. By refining these control strategies, we can achieve higher efficiency and reliability in real-world deployments, ultimately reducing human labor and improving urban maintenance.