Path Planning Method and Control of Mobile Robot with Uncertain Dynamics Based on Improved Artificial Potential Field and Its Application in Health Monitoring

Abstract

To enhance the navigation and control efficiency of mobile robots in the field of health monitoring, a novel path planning and control strategy for mobile robots with uncertain dynamics based on improved artificial potential fields is proposed in this paper. Specifically, we propose an attractive potential field rotation method to overcome the limitation that traditional artificial potential fields tend to fall into local minima. Then, we define a new class of attractive potential fields to address the goals non-reachable with obstacles nearby (GNRON) and collisions caused by excessive attractive force at long distances from the target point. Furthermore, a control law is proposed for the mobile robot with uncertain dynamics, and the stability of the closed-loop system is rigorously proven using the Lyapunov method. Finally, the feasibility and effectiveness of the proposed method are verified by simulations and experiments.

1. Introduction

Nowadays, mobile robots have been widely used in many fields such as industrial production, warehouse logistics, and security inspection [1,2,3]. The advent of rapidly developing healthcare service technology has led to a surge in the popularity of mobile robots in health monitoring. These robots provide convenient and beneficial solutions for individuals [4,5,6,7,8,9], improving their quality of life and promoting better health outcomes. As the global aging trend continues to intensify, the market for autonomous mobile robots for health monitoring will become increasingly widespread [10]. The safety and reliability of health monitoring is one of the most critical aspects of system design. This implies that mobile robots should plan their action paths in real time and accurately in unstructured environments such as homes and hospitals to avoid collisions and safety accidents and complete tasks efficiently. Therefore, path planning has become a key issue in the field of mobile robot research.

Artificial potential field (APF) [11] is one of the most commonly utilized local path planning methods. Its basic principle is to construct virtual attractive and repulsive fields in the robot’s motion space. The attractive force makes the robot converge toward the target, and the repulsive force serves to repel it from obstacles. This method is characterized by a straightforward structure, which enables real-time control of the lower layer [12]. The robot reproduces the therapist’s motion trajectory through a potential field function to provide rehabilitation and assisted exercises for the patient in [13]. The velocity differential potential field and acceleration differential potential field are proposed based on APF to ensure the overtaking safety of self-driving vehicles in [14]. The traditional attractive potential field is modified, and a localization accuracy field is introduced to achieve a balance between localization accuracy and obstacle avoidance in [15]. The repulsive potential field is made orthogonal to the attractive potential field to improve the navigation efficiency of the mobile robot in [16]. The APF method is combined with sliding mode control to achieve finite time convergence in the presence of bounded disturbances in [17].

However, in the above studies, the existence of a local minimum can result in the failure of the planning task. In addition, if there are obstacles near the destination, it will cause the mobile robot to fall into the local optimal solution before reaching the target point. This phenomenon is usually called the problem of goals non-reachable with obstacles nearby (GNRON) [18,19]. Furthermore, the robot may collide with obstacles when it is too far away from the target point due to excessive attraction. To solve the local minima problem, the potential field function is constructed by the harmonic function [20], but the computation time of this method increases linearly with the increase in the number of obstacles. Subsequent improvements to the algorithms have been proposed, and related work is categorized into four main areas according to their control strategies. The first method is to add virtual target points or obstacles to attract or repel the mobile robot to approach the target or move away from the local minima [21,22,23]. The second approach entails the implementation of suitable modifications to the attractive and repulsive functions [24,25,26,27]. The third approach introduces supplementary control within the potential field [28,29,30,31]. The fourth approach combines APF with other planning algorithms, including A* [32,33,34], RRT* (Rapidly exploring Random Tree Star) [35,36], MPC (Model Predictive Control) [37,38], and GA (Genetic Algorithm) [39].

An intriguing technique has been developed in the aforementioned research to overcome the limitations of the conventional artificial potential field by means of a rotation [19]. To address the issue of local minima, a safe potential field with vortex and repulsive potential switching is proposed in [40]. Nevertheless, existing techniques for rotating potential fields mainly focus on rotating the repulsive potential field [41], with limited attention given to the rotation of the attractive potential field. Moreover, most studies assume that the robot’s state can be acquired in real time. In hospital and home environments, mobile robots often face challenges related to uncertainties in kinematic and dynamic modeling, mechanical constraints, and measurement noise, making such assumptions potentially unrealistic. Therefore, this paper focuses on the path planning method and control of mobile robots with uncertain dynamics based on an improved artificial potential field and its application in health monitoring. The key contributions of this paper are threefold:

• A class of attractive potential fields is tailored to address the GNRON problem and to circumvent collisions caused by excessive attraction at considerable distances from the target point, thereby ensuring the safety of the intended trajectory.
• The issue of mobile robots falling into local minima is handled by the proposed attractive potential field rotation method.
• An adaptive controller is employed for a mobile robot with unknown inertial parameters and dynamic system perturbations. The stability of the closed-loop system and the boundedness of the estimation error are proven using the Lyapunov stability theory. The effectiveness and feasibility of the proposed planning and control strategy are then validated through simulations and experiments.

The rest of this paper is organized as follows. Section 2 covers the preliminaries and problem statement. Our methodology is elaborated in Section 3, followed by the simulations and experimental validation presented in Section 4. Section 5 discusses the findings of the study. Finally, Section 6 provides the conclusions of this paper.

$Notations:$ We use $\mathbb{R}$ to denote the set of real numbers, ${\mathbb{R}}^n$ to denote the n-dimensional Euclidean space, and ${\mathbb{R}}^{m\times n}$ to represent the set of $m\times n$ real matrices. $\left|\right|·\left|\right|$ denotes the Euclidean norm of a vector. The superscripts “$-1$” and “$\top$” denote the matrix inverse and transposition, respectively.

2. Preliminaries and Problem Statement

Kinematic and Dynamic Models

We consider a model of a nonholonomic mobile robot with its reference position point p located at the center of the axis connecting its wheels. The mobile robot has two rear wheels independently controlled by motors and a front caster wheel, as illustrated in Figure 1.

It is assumed that the mobile robot’s center of mass coincides with its geometric center, the wheels of the mobile robot are perpendicular to the ground, and there is only pure rolling between the wheels and the ground without lateral sliding. These assumptions imply that the robot’s lateral velocity is always zero. This condition is commonly referred to as the nonholonomic constraint and can be expressed as

$$\dot{x}sin\theta -\dot{y}cos\theta =0$$

where x and y represent the generalized coordinates of the point p in the global coordinate frame, and $\theta$ represents the heading angle of the mobile robot, i.e., the angle of the heading direction ${\mathrm{X}}_{\mathrm{h}}$ of the mobile robot relative to the $\mathrm{X}$-axis. The kinematics of the mobile robot can be described as

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}cos\theta & 0\\ sin\theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right]$$

(1)

where v and $\omega$ are the linear and angular velocities of the mobile robot, respectively. The system (1) can be rewritten as

$$\dot{\mathit{q}}=\mathbf{S}\left(\mathit{q}\right)\mathit{u}\left(t\right)$$

(2)

with

$$\mathbf{S}\left(\mathit{q}\right)=\left[\begin{array}{cc}cos\theta & 0\\ sin\theta & 0\\ 0& 1\end{array}\right]$$

where $\mathit{q}={[p,\theta ]}^{\top }\in {\mathbb{R}}^3$ denotes the position and the heading angle of the mobile robot, and $\mathit{u}\left(t\right)={[v,\omega ]}^{\top }\in {\mathbb{R}}^2$ denotes the velocity vector.

According to the Euler–Lagrangian formulation and [42,43], the dynamic model of the mobile robot can be demonstrated as

$$\mathbf{M}\left(\mathit{q}\right)\dot{\mathit{u}}+\mathbf{C}(\mathit{q},\dot{\mathit{q}})\mathit{u}+{\mathbf{\tau }}_{\mathrm{d}}=\mathbf{B}\left(\mathit{q}\right)\mathbf{\tau }$$

(3)

where $\mathbf{M}\left(\mathit{q}\right)\in {\mathbb{R}}^{2\times 2}$ is a positive definite symmetric inertial matrix, $\mathbf{C}(\mathit{q},\dot{\mathit{q}})\in {\mathbb{R}}^{2\times 2}$ represents centripetal and Coriolis torque, $\mathbf{B}\left(\mathit{q}\right)\in {\mathbb{R}}^{2\times 2}$ is an input transformation matrix, ${\mathbf{\tau }}_{\mathrm{d}}={[{\tau }_{d1},{\tau }_{d2}]}^{\top }\in {\mathbb{R}}^2$ represents unknown bounded disturbances including unstructured unmodelled dynamics, and $\mathbf{\tau }\in {\mathbb{R}}^2$ is the torque applied to the left and right wheels. It is assumed that $\mathbf{B}\left(\mathit{q}\right)$ is a matrix of full rank and the dynamic model (3) satisfies the following properties:

Property 1.

$\dot{\mathbf{M}}-2\mathbf{C}$ is a skew-symmetric matrix.

Property 2.

For any differentiable vector ξ, $\mathbf{M}\left(\mathit{q}\right)\dot{\mathit{\xi }}+\mathbf{C}(\mathit{q},\dot{\mathit{q}})\mathit{\xi }=\mathbf{Y}(\mathit{q},\dot{\mathit{q}},\mathit{\xi },\dot{\mathit{\xi }})\mathbf{g}$, where $\mathbf{Y}(\mathit{q},\dot{\mathit{q}},\mathit{\xi },\dot{\mathit{\xi }})$ is a known regressor matrix of $\mathit{q}$, $\dot{\mathit{q}}$, ξ and $\dot{\mathit{\xi }}$, and $\mathit{g}$ is an inertia parameter vector of the system.

3. Path Planning and Control Strategy Design for Mobile Robots with Uncertain Dynamics

3.1. Traditional Artificial Potential Field Method

In the 1980s, the APF method was proposed in [11]. This approach models the mobile robot as a positively charged particle that is attracted to a negatively charged target point while being repelled by positively charged obstacles. The robot’s movement is guided by the net force, which is the vector sum of these individual attractive and repulsive forces. The attractive potential field and corresponding attractive force are mathematically represented as follows:

$$U_{\mathrm{att}}\left(p\right)=k_{\mathrm{att}}{\displaystyle \frac{d^2(p,p_{\mathrm{t}})}{2}}$$

(4)

$${\mathit{F}}_{\mathrm{att}}\left(p\right)=-\nabla U_{\mathrm{att}}\left(p\right)=-k_{\mathrm{att}}·d(p,p_{\mathrm{t}})$$

(5)

where $k_{\mathrm{att}}$ is the positive gain coefficient of the attractive potential field, $p={[x,y]}^{\top }$ and $p_t={[x_t,y_t]}^{\top }$ are the current position of the mobile robot and target point, respectively, and $d(p,p_{\mathrm{t}})$ is the relative distance between the mobile robot and target point, i.e., $d(p,p_{\mathrm{t}})=\sqrt{{(x_t-x)}^2+{(y_t-y)}^2}$. The repulsive potential field and repulsive force can be expressed as

$$U_{\mathrm{rep}}\left(p\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}{k}_{\mathrm{rep}}{(\frac{1}{d(p,p_{\mathrm{ob}})}-\frac{1}{d^{\ast }})}^2,}\hfill & d(p,p_{\mathrm{ob}})\le d^{\ast }\hfill \\ 0,\hfill & d(p,p_{\mathrm{ob}})>d^{\ast }\hfill \end{array}\right.$$

(6)

$${\mathit{F}}_{\mathrm{rep}}\left(p\right)=-\nabla U_{\mathrm{rep}}\left(p\right)=\left\{\begin{array}{cc}{\displaystyle \frac{k_{\mathrm{rep}}}{d^2(p,p_{\mathrm{ob}})}}({\displaystyle \frac{1}{d(p,p_{\mathrm{ob}})}}-{\displaystyle \frac{1}{d^{\ast }}}){\displaystyle \frac{\partial d(p,p_{\mathrm{ob}})}{\partial p}},\hfill & d(p,p_{\mathrm{ob}})\le d^{\ast }\hfill \\ 0,\hfill & d(p,p_{\mathrm{ob}})>d^{\ast }\hfill \end{array}\right.$$

(7)

where $k_{\mathrm{rep}}$ is the positive gain coefficient of the repulsive potential field, $d(p,p_{\mathrm{ob}})$ is the shortest distance between the mobile robot and the obstacle, $d^{\ast }$ denotes the safe distance between the mobile robot and the obstacle, and it should be noted that the mobile robot is only affected by the repulsive force within this distance. The robot will move in the direction of the resultant force of attraction and repulsion, and the resultant force can be expressed as

$${\mathit{F}}_{\mathrm{total}}\left(p\right)={\mathit{F}}_{\mathrm{att}}\left(p\right)+\sum _{i=1}^n{\mathit{F}}_{\mathrm{rep}}^i\left(p\right)$$

where i represents the index of an obstacle exerting a repelling effect on the robot, and n denotes the total number of such obstacles. When a mobile robot enters the domain influenced by the repulsive forces of several obstacles, the cumulative repulsive force acting on it is the vector sum of the individual repulsive forces from each obstacle.

The APF method is known for its simple structure, which facilitates real-time control of the underlying layer. It exhibits reduced computational resources and time consumption, while also generating relatively smooth paths. Nevertheless, this approach exhibits certain limitations in practical path planning for mobile robots. A major challenge is the occurrence of local minima, which frequently leads to the failure of the planning task. Additionally, most previous studies have positioned the target point outside the repulsive potential field of obstacles. In contrast, when the target is located within this repulsive field, the mobile robot experiences a gradual reduction in attractive force as it approaches the target. This diminishing force makes it increasingly challenging for the robot to overcome the repulsive influence of obstacles, ultimately leading to the problem of GNRON, as illustrated in Figure 2.

3.2. Improved Attractive Potential Field

In addition to the previously outlined limitations, it is important to note that in traditional APF, if the mobile robot is situated at a considerable distance from its intended destination, the attraction of the destination to the robot may be excessive, potentially resulting in safety risks, such as collisions in unstructured scenarios, particularly in some large healthcare facilities. Therefore, these shortcomings of traditional methodologies warrant further investigation. In this paper, we assume that all obstacles are circular to simplify the Lyapunov-based analysis, allowing the direct use of Euclidean distance to represent the distance between the robot and the obstacles.

To address the issues of GNRON and the collision problem resulting from an excessive attractive force during long-distance planning, we propose a novel class of attractive potential fields, which can be expressed as

$$U_{\mathrm{att}}\left(p\right)=k_{\mathrm{att}}{\int }_0^{d(p,p_{\mathrm{t}})}\sigma \left(t\right)\mathrm{d}t$$

(8)

$${\mathit{F}}_{\mathrm{att}}\left(p\right)=-\nabla U_{\mathrm{att}}\left(p\right)=-k_{\mathrm{att}}·\sigma \left(d\right)$$

(9)

where $k_{\mathrm{att}}$ is the positive gain coefficient of the attractive potential field, and $\sigma \left(d\right)$ describes a smooth relationship between the distance of the robot from the target point with the following characteristics: $\left(\mathbf{i}\right)$ $\sigma \left(d\right)$> 0. $\left(\mathbf{ii}\right)$ $\sigma \left(d\right)$ strictly decreasing on (0, $d_{\lambda }$) and strictly increasing on ($d_{\lambda }$, +∞). $d_{\lambda }\in [0,d^{\ast }]$ is contingent upon the relative distance of the target point from nearby obstacles. $d^{\ast }$ indicates the safe distance that should be maintained between the mobile robot and any obstacles, within which the robot experiences the repulsive force. $\left(\mathbf{iii}\right)$ $\underset{d\to 0}{lim}\sigma \left(d\right)=+\infty$. And $\left(\mathbf{iv}\right)$ $\underset{d\to +\infty }{lim}\sigma \left(d\right)={\lambda }_1$, where ${\lambda }_1$ is a positive constant by considering that the attractive force exerted on the mobile robot before guidance cannot be infinite due to the limited capacity of mobile robots.

Remark 1.

In contrast to existing techniques that enhance APF-based repulsive potential fields, our approach preserves the traditional repulsive potential field construction methodology while introducing improvements to the attractive potential field for two key reasons. First, the most common solution to the GNRON problem is to incorporate the relative distance between the robot and the target into the repulsive potential field [18,26]. However, this increases the repulsive force as the distance between the robot and the target point increases, which is more likely to result in a local minimum. Second, rotating the repulsive potential field to avoid local minima represents a promising approach [19,40,41]. However, in practical mobile robot operations, it is necessary to consider the volume and geometry of both the robot and the obstacle. In these situations, the traditional repulsive potential field, which directs forces toward the robot, may be a safer and more effective choice. This is an avenue we plan to explore further in future research.

Remark 2.

To address the problem of excessive attractive force during long-distance planning, the attractive potential field needs to be modified. The attractive potential field function introduces a segmentation threshold in [25,27,40], but this approach causes a discontinuity at the segmentation point, leading to abrupt changes in the robot’s velocity or jitter at that point. Moreover, this threshold is empirically determined and lacks generalizability. In contrast, the method proposed in this paper overcomes these limitations by ensuring the attractive function remains smooth.

Remark 3.

Using nonlinear functions to reconstruct the attractive potential field can effectively mitigate the GNRON and the issue of excessive attraction caused by large distances in [24]. However, this method will cause the attractive force function to not be smooth enough at the inflection point, leading to the persistence of jitter and sudden velocity change problems. The enhanced attractive potential field function proposed in this paper, which avoids the need for segmentation and is generally built using exponential and power functions, provides a refined synthesis and improvement over similar methods.

3.3. Attractive Potential Field Rotation Method

The attractive potential rotation method proposed in this paper aims to enable the mobile robot to escape local minima, particularly in scenarios involving three-point collinear configurations. The attractive potential field rotation is activated when the robot is affected by the repulsive force of an obstacle during its movement toward the target point, as illustrated in Figure 3. The proposed new potential field can be described as

$$U\left(p\right)=U_{\mathrm{att}}^{\ast }\left(p\right)+\sum _{i=1}^n{U}_{\mathrm{rep}}^i\left(p\right)$$

(10)

where $U\left(p\right)$ represents the resultant potential field, which includes both attractive and repulsive potential fields, $U_{\mathrm{rep}}^i\left(p\right)$ denotes the repulsive potential generated by the i-th obstacle to the robot. $U_{\mathrm{att}}^{\ast }\left(p\right)$ is the newly introduced attractive potential field, specifically designed to address the issue of local minima, which can be expressed as

$$U_{\mathrm{att}}^{\ast }\left(p\right)=\mathbf{R}\left({\alpha }^{\ast }\right)U_{\mathrm{att}}\left(p\right)$$

(11)

where $U_{\mathrm{att}}\left(p\right)$ can take the form of either (4) or (8). To facilitate distinction and subsequent algorithmic comparison, we designate the process of rotating the traditional attractive potential field as RT-APF. Similarly, we designate the process of rotating our improved attractive potential field as RI-APF. $\mathbf{R}\left({\alpha }^{\ast }\right)$ represents the rotation matrix corresponding to the attractive potential field. This matrix enables a smooth transition between disparate potential states through the application of trigonometric functions and a continuous rotation angle ${\alpha }^{\ast }$. The relationship is as follows:

$$\mathbf{R}\left({\alpha }^{\ast }\right)=\left[\begin{array}{cc}cos{\alpha }^{\ast }& -sin{\alpha }^{\ast }\\ sin{\alpha }^{\ast }& cos{\alpha }^{\ast }\end{array}\right].$$

(12)

In practical applications, it is essential for the mobile robot to constrain $\alpha$ with a saturation function due to its nonholonomic constraints. We define the maximum rotation angle of the attractive potential field as ${\alpha }_{\max }$. The corresponding saturation function is defined as

$${\alpha }^{\ast }=\mathrm{sat}(\alpha ,{\alpha }_{\max })=\left\{\begin{array}{cc}{\displaystyle \alpha ,}\hfill & \left|\alpha \right|\le {\alpha }_{\max }\hfill \\ {\displaystyle {\alpha }_{\max }\frac{\alpha }{\left|\alpha \right|},}\hfill & \left|\alpha \right|>{\alpha }_{\max }.\hfill \end{array}\right.$$

(13)

Then, the rotation angle of the attractive potential field can be defined as

$$\alpha =D\left(\alpha \right)r\left(\beta \right)\pi$$

(14)

with

$$D\left(\alpha \right)=\left\{\begin{array}{cc}-1,\hfill & {\displaystyle \left|\right|{\mathit{F}}_{\mathrm{att}}\left(p\right)\times \sum _i^n{\mathit{F}}_{\mathrm{rep}}^i\left(p\right)\left|\right|\le 0}\hfill \\ 1,\hfill & {\displaystyle \left|\right|{\mathit{F}}_{\mathrm{att}}\left(p\right)\times \sum _i^n{\mathit{F}}_{\mathrm{rep}}^i\left(p\right)\left|\right|>0}\hfill \end{array}\right.$$

(15)

$$r\left(\beta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}(1-cos\beta ),}\hfill & {\displaystyle \left|\right|{\mathit{F}}_{\mathrm{att}}\left(p\right)\left|\right|·\left|\right|\sum _i^n{\mathit{F}}_{\mathrm{rep}}^i\left(p\right)\left|\right|\ne 0}\hfill \\ 0,\hfill & {\displaystyle \left|\right|{\mathit{F}}_{\mathrm{att}}\left(p\right)\left|\right|·\left|\right|\sum _i^n{\mathit{F}}_{\mathrm{rep}}^i\left(p\right)\left|\right|=0}\hfill \end{array}\right.$$

(16)

where $D\left(\alpha \right)$ is the directional function that determines whether the attractive potential field rotates clockwise or counterclockwise, based on the cross product of the attractive and repulsive forces. To ensure the viability of the generated path, we introduce $r\left(\beta \right)$ to smooth the change in the resultant force caused by the rotation of the attractive potential field when the mobile robot is subject to the repulsive force. Here, $\beta \in (-\pi ,\pi )$ denotes the angle between the attractive and repulsive forces.

By employing the method of smoothly rotating the attractive potential in this paper, we can prevent the mobile robot from becoming stuck in a situation where the resultant force is zero. This method ensures that the robot can continue its trajectory toward the target, even when faced with a local minimum.

3.4. Controller Design

We use the output of the APF as the reference linear velocity and direction angle and select the kinematic stabilizing control law ${\mathit{u}}_{\mathrm{d}}$ of the mobile robot (1) as

$${\mathit{u}}_{\mathrm{d}}=\left[\begin{array}{c}{v}_{\mathrm{d}}\\ {\omega }_{\mathrm{d}}\end{array}\right]=\left[\begin{array}{c}{k}_v\left|\right|\nabla U\left(p\right)\left|\right|\\ k_{\omega }\theta \end{array}\right]$$

(17)

with

$$\theta =\left\{\begin{array}{cc}\pi +arctan\left({\displaystyle \frac{\nabla U^y\left(p\right)}{\nabla U^x\left(p\right)}}\right),\hfill & \nabla U^x\left(p\right)<0\hfill \\ arctan\left({\displaystyle \frac{\nabla U^y\left(p\right)}{\nabla U^x\left(p\right)}}\right),\hfill & \nabla U^x\left(p\right)>0\hfill \\ -{\displaystyle \frac{\pi }{2}},\hfill & \nabla U^x\left(p\right)=0,\nabla U^y\left(p\right)<0\hfill \\ {\displaystyle \frac{\pi }{2}},\hfill & \nabla U^x\left(p\right)=0,\nabla U^y\left(p\right)>0\hfill \end{array}\right.$$

(18)

where $k_v$ and $k_{\omega }$ are the positive constant control gains, $U\left(p\right)$ is the resultant potential field defined in (10). $U^x\left(p\right)$ and $U^y\left(p\right)$ are the components of the resultant potential field and can be expressed as

$$\left\{\begin{array}{c}{U}^x\left(p\right)=U_{\mathrm{att}}^x\left(p\right)+U_{\mathrm{rep}}^x\left(p\right)\hfill \\ U^y\left(p\right)=U_{\mathrm{att}}^y\left(p\right)+U_{\mathrm{rep}}^y\left(p\right)\hfill \end{array}\right.$$

(19)

where $U_{\mathrm{att}}^x\left(p\right)$, $U_{\mathrm{att}}^y\left(p\right)$, $U_{\mathrm{rep}}^x\left(p\right)$, and $U_{\mathrm{rep}}^y\left(p\right)$ are the components of the attractive and repulsive potential fields, respectively.

We design the torque inputs $\mathbf{\tau }$ for the dynamic systems (1) and (3) such that $\mathit{u}\left(t\right)$ converges to the desired velocity ${\mathit{u}}_{\mathrm{d}}$. The auxiliary velocity tracking error signal $\mathit{e}\left(t\right)\in {\mathbb{R}}^2$ is defined as

$$\mathit{e}={[e_1,e_2]}^{\top }=\mathit{u}-{\mathit{u}}_{\mathrm{d}}.$$

(20)

By differentiating (20) and using Property 2, the dynamics of the mobile robot using the velocity tracking error can be reformulated as [42]

$$\mathbf{M}\left(\mathit{q}\right)\dot{\mathit{e}}+\mathbf{C}(\mathit{q},\dot{\mathit{q}})\mathit{e}+\mathbf{Y}\mathit{g}+{\mathbf{\tau }}_{\mathrm{d}}=\mathbf{B}\left(\mathit{q}\right)\mathbf{\tau }$$

(21)

where $\mathit{g}$ denotes the unknown inertia parameter vector of the system, $\mathbf{Y}$ represents the known desired regression matrix and $\mathbf{Y}\mathit{g}=\mathbf{M}\left(\mathit{q}\right){\dot{\mathit{u}}}_{\mathrm{d}}+\mathbf{C}(\mathit{q},\dot{\mathit{q}}){\mathit{u}}_{\mathrm{d}}$.

We apply the following continuous control torque input $\mathbf{\tau }\left(t\right)$:

$$\mathbf{\tau }\left(t\right)={\mathbf{B}}^{-1}\left(\mathit{q}\right)\overline{\mathbf{\tau }}$$

(22)

where $\overline{\mathbf{\tau }}\in {\mathbb{R}}^2$ is an auxiliary control signal defined as

$$\overline{\mathbf{\tau }}=\mathbf{Y}\widehat{\mathit{g}}-{\mathit{u}}_e-\left[\begin{array}{c}{k}_{d1}{e}_1\\ k_{d2}{e}_2\end{array}\right]$$

(23)

with

$${\mathit{u}}_e={\left[{\displaystyle \frac{{\widehat{b}}_1{e}_1}{\sqrt{e_1^2+{\varrho }^2\left(t\right)}}},{\displaystyle \frac{{\widehat{b}}_2{e}_2}{\sqrt{e_2^2+{\varrho }^2\left(t\right)}}}\right]}^{\top }$$

(24)

where $k_{d1}$ and $k_{d2}$ are positive control gains, and ${\widehat{b}}_1$ and ${\widehat{b}}_2$ represent estimates of the unknown boundaries $b_1>0$ and $b_2>0$ of ${\tau }_{d1}$ and ${\tau }_{d2}$, respectively, i.e., $|{\tau }_{d1}\left(t\right)|\le b_1$ and $|{\tau }_{d2}\left(t\right)|\le b_2$. The function $\varrho \left(t\right)>0$ is time-varying and satisfies ${\int }_0^{\infty }\varrho \left(t\right)\mathrm{d}t\le {\varrho }^{\ast }$, where ${\varrho }^{\ast }$ is a positive constant. The update laws for $\widehat{\mathit{g}}$, ${\widehat{b}}_1$, and ${\widehat{b}}_2$ are defined as follows:

$$\begin{array}{ccc}\hfill \dot{\widehat{\mathit{g}}}& =& -\mathbf{\Lambda }{\mathbf{Y}}^{\top }\mathit{e}\hfill \\ \hfill {\dot{\widehat{b}}}_1& =& {\displaystyle \frac{{\kappa }_1{e}_1^2}{\sqrt{e_1^2+{\varrho }^2\left(t\right)}}}\hfill \\ \hfill {\dot{\widehat{b}}}_2& =& {\displaystyle \frac{{\kappa }_2{e}_2^2}{\sqrt{e_2^2+{\varrho }^2\left(t\right)}}}\hfill \end{array}$$

(25)

where $\mathbf{\Lambda }$ is the positive definite gain matrix, and ${\kappa }_1$ and ${\kappa }_2$ are positive constant tuning gains. The initial values $\mathbf{Y}\left(0\right)$, ${\widehat{b}}_1\left(0\right)$, and ${\widehat{b}}_2\left(0\right)$ can be selected arbitrarily. By substituting (22)–(24) into (21), the closed-loop error dynamics for $\mathit{e}\left(t\right)$ can be expressed as

$$\mathbf{M}\dot{\mathit{e}}=-\mathbf{C}\mathit{e}-\mathbf{Y}\tilde{\mathit{g}}-{\mathbf{\tau }}_{\mathrm{d}}-{\mathit{u}}_e-\left[\begin{array}{c}{k}_{d1}{e}_1\\ k_{d2}{e}_2\end{array}\right]$$

(26)

where the parameter error $\tilde{\mathit{g}}$ is defined as $\tilde{\mathit{g}}=\mathit{g}-\widehat{\mathit{g}}$.

Theorem 1.

Utilizing Properties 1 and 2, for systems (1) and (3), the control inputs defined in (17), (20), and (22)–(25) guarantee that the position and orientation of the mobile robot’s coordinate system converge to the desired state, i.e.,

$$\underset{t\to \infty }{lim}{e}_1\left(t\right),e_2\left(t\right)=0.$$

(27)

Proof. 

We consider the Lyapunov function candidate

$$V={\displaystyle \frac{1}{2}}{\mathit{e}}^{\top }\mathbf{M}\mathit{e}+{\displaystyle \frac{1}{2}}{\tilde{\mathit{g}}}^{\top }{\mathbf{\Lambda }}^{-1}\tilde{\mathit{g}}+{\displaystyle \frac{{({\widehat{b}}_1-b_1)}^2}{2{\kappa }_1}}+{\displaystyle \frac{{({\widehat{b}}_2-b_2)}^2}{2{\kappa }_2}}.$$

(28)

Taking the time derivative of (28) and employing (25), (26), and Properties 1 and 2, we obtain

$$\begin{array}{cc}\hfill \dot{V}=& {\displaystyle \frac{1}{2}}{\mathit{e}}^{\top }\dot{\mathbf{M}}\mathit{e}+{\mathit{e}}^{\top }\mathbf{M}\dot{\mathit{e}}+{\tilde{\mathit{g}}}^{\top }{\mathbf{\Lambda }}^{-1}\dot{\tilde{\mathit{g}}}+{\displaystyle \frac{({\widehat{b}}_1-b_1){\dot{\widehat{b}}}_1}{{\kappa }_1}}+{\displaystyle \frac{({\widehat{b}}_2-b_2){\dot{\widehat{b}}}_2}{{\kappa }_2}}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\mathit{e}}^{\top }\dot{\mathbf{M}}\mathit{e}-{\mathit{e}}^{\top }\mathbf{C}\mathit{e}-{\mathit{e}}^{\top }\mathbf{Y}\tilde{\mathit{g}}-{\mathit{e}}^{\top }{\mathbf{\tau }}_{\mathrm{d}}-{\displaystyle \frac{{\widehat{b}}_1{e}_1^2}{\sqrt{e_1^2+{\varrho }^2\left(t\right)}}}-{\displaystyle \frac{{\widehat{b}}_2{e}_2^2}{\sqrt{e_2^2+{\varrho }^2\left(t\right)}}}\hfill \\ \hfill & -{\mathit{e}}^{\top }\left[\begin{array}{c}{k}_{d1}{e}_1\\ k_{d2}{e}_2\end{array}\right]+{\tilde{\mathit{g}}}^{\top }{\mathbf{\Lambda }}^{-1}\dot{\tilde{\mathit{g}}}\hfill \\ \hfill =& -k_{d1}{e}_1^2-k_{d2}{e}_2^2-e_1{\tau }_{d1}-e_2{\tau }_{d2}-{\displaystyle \frac{b_1{e}_1^2}{\sqrt{e_1^2+{\varrho }^2\left(t\right)}}}-{\displaystyle \frac{b_2{e}_2^2}{\sqrt{e_2^2+{\varrho }^2\left(t\right)}}}.\hfill \end{array}$$

(29)

Note that $|e_1{\tau }_{d1}|\le b_1\left|e_1\right|$ and $b_1\left|e_1\right|-{\displaystyle \frac{b_1{e}_1^2}{\sqrt{e_1^2+{\varrho }^2}}}\le b_1\varrho$ [44]. Integrating both sides of (29), we conclude that $V\left(t\right)$ is bounded, which implies that $e_1$, $e_2$, $\tilde{\mathit{g}}$, $\widehat{b_1}$, and ${\widehat{b}}_2$ are also bounded. Therefore, $\widehat{\mathit{g}}$ is also bounded. Furthermore, from (29), we obtain that $e_1$, $e_2\in {\mathcal{L}}_2$. By (26), it follows that ${\dot{e}}_1$ and ${\dot{e}}_2$ are both bounded. Applying Barbalat’s lemma, we conclude that $e_1\left(t\right),e_2\left(t\right)\to 0$ as $t\to \infty$, which completes the proof.    □

3.5. Flowchart and Pseudo-Code of the Method

This subsection shows the flowchart in Figure 4 and pseudo-code of the proposed method in Algorithm 1.

Algorithm 1 Pseudo-code of the attractive potential field rotation method

1: Initialize the current position of the mobile robot p, target point $p_t$, maximum number of iterations, step size, gain coefficients for attractive and repulsive forces;   2: while the mobile robot is affected by potential fields do   3:    if the robot affected by the repulsive potential field of obstacles then   4:      if the robot falls into a local minimum (${\mathit{F}}_{\mathrm{total}}=0)$ then   5:         Use the attractive potential field rotation method;   6:      else   7:         Calculate resultant force; (${\mathit{F}}_{\mathrm{total}}={\mathit{F}}_{\mathrm{att}}+{\mathit{F}}_{\mathrm{rep}}$)   8:         Calculate heading angle;   9:      end if 10:    else 11:      Calculate resultant force; (${\mathit{F}}_{\mathrm{total}}={\mathit{F}}_{\mathrm{att}}$) 12:      Calculate heading angle; 13:    end if 14:    Calculate the linear and angular velocities of the mobile robot; 15:    The mobile robot obtains the desired velocity variables according to the controller and moves towards the target point; 16:    if the robot has reached the target point then 17:      Break; 18:    else 19:      Calculate resultant force and update robot position and heading angle; 20:    end if 21: end while

4. Simulations and Experimental Validation

This section validates the path planning and control strategy of the proposed method for health-monitoring tasks in unstructured scenarios through the use of MATLAB simulations and Turtlebot3 experiments.

4.1. Simulation Results and Analysis

Mobile robots are often deployed in complex and varied environments, such as hospitals and nursing homes. To effectively navigate these areas and avoid obstacles, thereby ensuring the efficient completion of health-monitoring tasks, robust planning and navigation capabilities are essential. We evaluate our proposed method in a 120 m × 120 m simulation environment, where the robot navigates through the potential fields defined in (10). A high-density environment comprising 30 randomly generated obstacles was established, and the influence range of the obstacles was set to a sufficiently large value. This aims to facilitate the emergence of local minima and the GNRON problem to illustrate the advantages of the proposed method. The start and target coordinates were set to (0,0) and (100,100), respectively. When constructing the attractive potential field function, a nonlinear function is typically employed, such as an exponential or a logarithmic function. In cases where computational complexity is challenging, a polynomial can be utilized to fit the function. We set $\sigma \left(d\right)={\lambda }_1·e^{-{\displaystyle \frac{1}{d}}}+{({\lambda }_2·d)}^{-{\lambda }_3}$ with ${\lambda }_1=100$, ${\lambda }_2=1$, and ${\lambda }_3=5$. The remaining parameter settings are provided in

Table 1. Planning parameters in random dense obstacle environments.

Parameter	$k_{\mathrm{att}}$	$k_{\mathrm{rep}}$	$d^{\ast }$	${\alpha }_{\max }$
Value	0.3	10,000	20	${\displaystyle \frac{\pi }{12}}$

.

The parameters that play an important role are the attraction and repulsion scale factors and the repulsion influence range. The latter can be large enough to cover the trajectory of the robot moving to the target point. The former lacks a discernible size range and order of magnitude and needs to be selected based on specific map information. The selection principle is as follows. First, the attraction cannot increase indefinitely with the distance between the robot and the target point. Second, when there are obstacles near the target point, the attractive force can exceed the growth rate of the repulsive force such that the robot can reach the target point.

A total of 100 comparative simulations are conducted for the traditional artificial potential field (TAPF), the rotating traditional attractive potential field (RT-APF), and the rotating improved attractive potential field (RI-APF) as proposed in this paper, all under the previously described environmental and parameter configurations. The detailed simulation results are provided in

Table 2. One hundred comparative simulation results of three methods in random dense obstacle environments.

Method	Number of Successes	Number of Collisions and Timeouts	Average of Path Lengths for Successful Cases (Meters)	Standard Deviation of Path Lengths for Success Cases (Meters)
TAPF	30	70	149.42	4.91
RT-APF	44	56	158.96	5.92
RI-APF	91	9	156.24	5.66

.

The effectiveness of each simulation was assessed based on the number of successful instances in which the mobile robot reached the designated target point. Failures were categorized into two groups: those due to collisions with obstacles and those caused by the robot’s inability to reach the target point within 3000 control cycles. From the simulation results in Table 2, it can be seen that RI-APF significantly improves the success rate of the mobile robot reaching the target point, although the path length generated by this method increased slightly. Since the rotating attractive potential field is used to overcome local minima, it is difficult to avoid having to travel a longer distance to bypass obstacles. Furthermore, the standard deviations of the success cases of the three methods are not much different because they all calculate the attraction and repulsion based on the distance of the mobile robot from the target point and obstacles.

In numerous experiments, the TAPF method fails to identify the optimal path in the maximum number of iterations. This is due to the inherent limitations of the gradient descent method, which causes the robot to converge towards local minima. From the perspective of potential field theory, this can be interpreted as the robot falling into a pit during its descent and being unable to disengage from it. From the perspective of forces, it can be seen that the repulsive and attractive forces applied to the robot cancel each other out, thus preventing the robot from moving forward. When the map is large, the size of the attractive force is proportional to the distance between the robot and the target point. Therefore, excessive attractive force will cause the robot to easily collide with obstacles during the movement. Furthermore, most previous studies on potential fields place the target point in a position that is not affected by obstacles. If the obstacle is near the target point, the repulsive force will be significantly greater than the attractive force, and the robot will not be able to reach the target point. The above problems also appear quite frequently in our simulation results, as shown in Figure 5, Figure 6, Figure 7 and Figure 8.

As illustrated in Figure 5, RT-APF can effectively assist TAPF in overcoming local minima. Nevertheless, the GNRON problem remains difficult to solve. When the robot is situated at an excessive distance from the target point, the considerable attraction prevents TAPF and RT-APF from effectively avoiding obstacles near the starting point, as shown in Figure 6. Moreover, randomly generated obstacles near the target point present a significant challenge for both TAPF and RT-APF. These obstacles exert considerable repulsive forces while the attractive force toward the target point diminishes. This problem, known as GNRON, is illustrated in Figure 7.

RI-APF can be regarded as an improvement of RT-APF, which can effectively solve all the aforementioned problems. However, we observed failures in some simulations, as illustrated in Figure 8. These failures occur when random obstacles are very close to or overlap with the target point, a scenario that is uncommon in practical tasks.

To further validate the effectiveness of the three methods in obstacle avoidance, two distinct local minima were designed in a 6 m × 4 m simulation environment, with the target point situated at a reasonable distance from obstacles. The corresponding simulation results are presented in Figure 9.

The start and target points of the mobile robot are (0,2) and (5,2), respectively. The global coordinates of the five fixed obstacles are as follows: $O_1=(1,2.8)$, $O_2=(1,1.2)$, $O_3=(3,2)$, $O_4=(5.5,2.5)$, and $O_5=(5.5,1.5)$. For environments with small maps and simple obstacle configurations, the requirement that the attraction must remain bounded as the distance between the robot and the target point tends to infinity is not stringent. For simplicity, we set the $\sigma \left(d\right)={\lambda }_1·d^{{\lambda }_2}+{\lambda }_3·d^{-1}$. The remaining parameters are presented in

Table 3. Planning parameters under fixed obstacles and local minima.

Parameter	$k_{\mathrm{att}}$	$k_{\mathrm{rep}}$	$d^{\ast }$	${\alpha }_{\max }$	${\lambda }_1$	${\lambda }_2$	${\lambda }_3$
Value	3	100	1	${\displaystyle \frac{\pi }{12}}$	3	1	5

.

The attractive force can be combined with the repulsive forces of $O_1$, $O_2$, and $O_3$, leading the mobile robot to fall into local minima, as illustrated in Figure 9. However, rotating the attractive potential field can effectively overcome this issue. Furthermore, the RI-APF method proposed in this paper successfully addresses the GNRON problem. The specific changes in attractive and repulsive forces are illustrated in Figure 10.

4.2. Experimental Results and Analysis

To validate the effectiveness of our planning and control strategy in real-world health-monitoring scenarios, we perform experiments using a mobile robot within actual physical environments.

The experimental environment is configured similarly to the setup used in the fixed obstacles and local minima simulation, as illustrated in Figure 9. The mobile robot selected for this experiment is the Turtlebot3 Burger. The platform is a differential drive robot with a nonholonomic constraint on its velocity, rendering it unable to move in the direction of the wheel axis, i.e., in the ${\mathrm{Y}}_{\mathrm{h}}$ direction as in Figure 1. The estimation of the robot’s state based on sensor measurements may be influenced by uncertain dynamics, mechanical constraints, and measurement noise. To mitigate these challenges, a vision-based motion capture system is employed to directly obtain the Cartesian position information required by the controller. Additionally, an upper limit on the linear and angular velocity of the Turtlebot3 Burger is imposed, namely $v_{\max }<0.22$ m/s and ${\omega }_{\max }<2.84$ rad/s, based on the finite dynamics of the mobile robot.

The trajectory of the mobile robot with our proposed RI-APF method is shown in Figure 11. The results of multiple experimental paths show that their trajectories and path lengths are not significantly different from the simulation results in Figure 9. The entire experimental process is documented in Figure 12. The robot is attracted only by the attraction of the target point at the beginning, and it encounters two fixed local minima at $t=2$ s and $t=8$ s, respectively. However, it successfully overcomes the local minima by the rotation of the attractive potential field and maintains its forward progress at $t=2$ s and $t=9$ s. As the robot approaches the target point, the attraction decreases. Since the target point is set near obstacles, the GNRON problem begins to emerge when $t=18$ s. The improved attractive force solves the problem with a faster growth rate, allowing the robot to ultimately reach the target point.

5. Discussion

The simulation and experimental results of the traditional artificial potential field method and the two methods proposed in this paper show that there are significant differences in the performance of the three methods in mobile robot path planning and obstacle avoidance. As shown in Figure 5, Figure 6 and Figure 7, although TR-APF can effectively help TAPF overcome local minima, it performs poorly in long-range tasks and GNRON problems, and there is even the possibility of collision. RI-APF can effectively solve the above problems and generate fairly smooth paths. In contrast, in a slightly more complex environment, TAPF cannot reach the target point at all. This demonstrates the advantage of the RI-APF algorithm, which is to create a more efficient path through an improved attractive potential field.

This paper proposes a path planning and control strategy based on artificial potential fields, aiming to overcome the limitations of traditional potential fields and apply them to practical health monitoring. Overcoming local minima by rotating the attractive potential field typically requires circumventing obstacles, which means that the RI-APF method may require a slightly longer travel path. However, the significant improvement in ensuring task completion justifies this trade-off.

6. Conclusions and Future Work

The application of mobile robots to health monitoring presents significant challenges, particularly in navigation and control within unstructured environments. This paper proposes a new class of attractive potential fields based on the traditional artificial potential field to address the GNRON problem and the collision caused by excessive attraction triggered by a position that is too far away from the target point. In addition, the paper proposes a method for rotating the attractive potential field to effectively overcome local minima. Comparative simulations of the traditional attractive potential field and the improved attractive potential field, rotated in a dense random obstacle environment and a fixed local minima environment, were performed. The experimental results demonstrate that the rotation of the improved attractive potential field is an effective solution to the aforementioned problems. Finally, the effectiveness and feasibility of the method proposed in this paper are verified through experiments on the Turtlebot3 Burger mobile robot platform. Future work will focus on more complex environments, such as hospitals and nursing homes, taking into account the movement of people and dynamic obstacles to achieve safer and more efficient navigation and control.