Sliding Mode Control for Trajectory Tracking of a TurtleBot3 Mobile Robot in Obstacle Environments ^†

Abstract

The proposed work presents a Sliding Mode Controller (SMC) for trajectory tracking of a TurtleBot3 Burger mobile robot based on sliding mode. Trajectory tracking is performed in congested environments; therefore, an obstacle avoidance strategy is presented to prevent the robot from colliding with obstacles. A clear and detailed methodology is developed for the design of the SMC controller with a PID sliding surface for trajectory tracking that can be extrapolated to position control and posture control. The discontinuous part of the SMC is used to ensure that the robot tends to the desired reference whenever an obstacle appears. The stability analysis of the proposed controller is presented, and the experimental results are shown, demonstrating the good performance of the controller.

1. Introduction

Nowadays, robots have received remarkable consideration and application in various sectors, such as industrial, service, agriculture, and medicine, among others. The intelligent navigation of wheeled mobile robots is one of the important characteristics that have been of interest to researchers, together with their mobility capacity, their simple structure, their low cost, and their use in different environments that are difficult to access or that represent a risk to human beings [1,2]. This has enabled robots to play a crucial role in these fields, such as agriculture, industry, and medicine [3]. In recent years, research has increased into control strategies to address the challenges of controlling mobile robots, including trajectory tracking and path following, as well as comparative studies of controllers for tracking tasks.

In this context, several control strategies have been published concerning trajectory tracking of differential drive mobile robots. Classical techniques such as PID have been widely used for trajectory tracking control of mobile robots [4]. Advanced versions, such as PID fractional optimization employing particle swarm algorithms, have been developed [5,6]. Reference [7] proposes a retroactive PID to control both virtual velocity and angular velocity for trajectory tracking. Other approaches employ both the kinematic model combined with Lyapunov [8] for designing controllers that account for input saturation constraints. Additionally, ref. [9] employs a Lyapunov-based control for four-wheeled omnidirectional robots employing the dynamic model, and [10] a similar SMC is used for the four-wheeled robot at the simulation level. A proposed SMC aimed at suppressing the influence of electromechanical systems by feedback of the vortex and sliding mode algorithm is introduced in [11]. Meanwhile, ref. [12] considers a hybrid control approach that contemplates a virtual controller for the kinematic model to address stabilization and tracking, along with a fuzzy logic controller to handle uncertainties or uncertain parameters, combining the techniques to improve the robustness of the SMC and the flexibility of fuzzy logic to contribute to the control objective. Other control techniques have been widely implemented with an SMC due to their excellent results, such as the SMC controller presented in [13], which is designed for trajectory tracking for a finite time period, demonstrating its robustness in handling disturbances. Ref. [14] presents an SMC for trajectory tracking combined with an LQR algorithm to decrease the chattering. To achieve improved performance and effectiveness, the SMC has been combined with other control techniques. In [15], the combination of an SMC with neural networks to compensate for external disturbances in velocity tracking, along with a kinematic controller for pose tracking, allows for a significant improvement in handling variable conditions. Similarly, considering that obstacle avoidance plays an important role in mobile robot applications, several strategies have been developed for unstructured environments. Notable approaches include [16], which combines an online tuning of PID parameters without requiring a robot model for obstacle avoidance. Fuzzy logic has also been used for static obstacle avoidance [17], providing a flexible method to manage such events. Additionally, ref. [18] implements a detection algorithm based on YOLO-v4 combined with reinforcement learning for tuning PID controller parameters. Given its potential, the SMC has been considered alongside other algorithms for obstacle avoidance through bilateral teleoperation [19], denoting that the SMC can be effectively integrated with other control techniques to improve the results. It is worth noting that all the cited contributions have presented simulation-level results; thus, implementing a robust control algorithm, like the SMC, in real environments would allow for understanding the real behavior of these controllers, even more when environments with obstacles become a challenge, especially for the calibration and switching of the controller.

Therefore, this work presents the implementation of a Sliding Mode Controller for trajectory tracking of a TurtleBot3 Burger mobile robot in obstacle environments. Unlike previous studies, this work develops a clear and detailed methodology for the design of the SMC controller with a PID sliding surface for trajectory tracking, which can also be extrapolated to position control and posture control. Another contribution of this work is the use of the discontinuous part of the SMC to ensure that the robot tends to the desired reference every time an obstacle appears. A key challenge addressed in this study is the calibration of the controller parameters. The experimental implementation evidenced the criticality of the calibration of this controller; finally, the experimental results demonstrate the controller’s strong performance.

This document is structured as follows. Section 2 describes the kinematic model of the mobile robot; Section 3 presents the design and stability analysis of the SMC controller for trajectory tracking and the implemented obstacle avoidance strategy; and Section 4 presents the experimental results in an environment with obstacles. Finally, the conclusions of this work are given.

2. Modeling

The mobile robot used in this work is illustrated in Figure 1, corresponding to a TurtleBot3 Burger that has several features [20]. As shown in

Table 1. TurtleBot3 Burger mobile robot technical specifications.

Technical Feature	Description
Size (length × width × height)	138 mm × 178 mm × 192 mm
Weight (+SBC + battery + sensors)	1 kg
Maximum velocities	0.22 m/s, 2.84 rad/s
Single board computers	Raspberry Pi
Motors (2 units)	Dynamixel (XL430-W250-T)
Laser distance sensor (LDS)	360° LDS-1
Wheels (2 units)	Sprocket wheels for tire, diameter 66 mm
Battery	Lithium polymer 11.1 V, 1800 mAh
Operating time (battery)	2 h:30 min

, its use is for educational and research purposes. This mobile platform corresponds to a differential traction-type robot; its locomotion system is based on the use of two wheels.

The improved kinematic model of the mobile robot with a non-holonomic constraint is given by its compact form [21] as follows:

$$\dot{h}=J U$$

(1)

where $\dot{h}={\left[\dot{x} \dot{y}\right]}^T$ is the time variation of position in $x$ and $y$ and $U={\left[u \omega \right]}^T$ are the linear and angular velocities, respectively. $J$ is the Jacobian rotation matrix and is given by the following:

$$J=\left[\begin{array}{cc}\mathit{cos}\left(\psi \right)& -a\mathit{sin}\left(\psi \right)\\ \mathit{sin}\left(\psi \right)& a\mathit{cos}\left(\psi \right)\end{array}\right]$$

(2)

$\psi$ is the orientation of the mobile robot and $a$ is the distance between the point of interest and the center of the axis joining the wheels.

3. Controller

3.1. Sliding Mode Controller (SMC)

The schematic of the Sliding Mode Control strategy is presented in Figure 2, where $h_d={\left[x_d y_d\right]}^T$ is the desired position; $h={\left[x y\right]}^T$ is the position of the robot; $\tilde{p}=\left(h_d-h\right)=\left[\begin{array}{cc}\tilde{x}& \tilde{y}\end{array}\right]$ is the tracking error of $x$ and $y$, respectively; $\dot{\tilde{p}}$ corresponds to the first derivative of the position error; and $\ddot{\tilde{p}}$ represents the second derivative of the position error.

The Sliding Mode Controller is given by the following:

$$u_{SMC}=u_a+u_d$$

(3)

where $u_a,u_d$ correspond to the continuous and discontinuous part, respectively.

For the design of the controller, the sliding surface $s$ is set, which is given by the following:

$$s={\lambda }_p\tilde{p}+{\lambda }_i \int \tilde{p} dt +{\lambda }_d \dot{\tilde{p}}$$

(4)

where ${\lambda }_p, {\lambda }_i$, and ${\lambda }_d$, are the proportional, integral, and derivative constants, respectively. Deriving (4), we have the following:

$$\dot{s}={\lambda }_p\dot{\tilde{p}} +{\lambda }_i \tilde{p}+{\lambda }_d \ddot{\tilde{p}}$$

(5)

The velocity error is given by $\dot{\tilde{p}}={\dot{h}}_d-\dot{h}$, replacing (5) as follows:

$$\dot{s}={\lambda }_p\left({\dot{h}}_d-\dot{h}\right) +{\lambda }_i \tilde{p}+{\lambda }_d \ddot{\tilde{p}}$$

(6)

Replace (1) in (6).

$$\dot{s}={\lambda }_p\left({\dot{h}}_d-JU\right) +{\lambda }_i \tilde{p}+{\lambda }_d \ddot{\tilde{p}}$$

(7)

For the system to remain inside the sliding surface, we define $\dot{s}=0$, while to obtain $u_a$, we consider that $u_d=0$. In a closed loop, we have $u_{SMC}=u_a$. Replacing (7), one has the following:

$$0={\lambda }_p\left({\dot{h}}_d-J{u}_a\right) +{\lambda }_i \tilde{p}+{\lambda }_d \ddot{\tilde{p}}$$

(8)

By clearing $u_a$, we obtain the following:

$$u_a=J^{-1}\left[{\dot{h}}_d+{\displaystyle \frac{{\lambda }_i}{{\lambda }_p}}\tilde{p}+{\displaystyle \frac{{\lambda }_d}{{\lambda }_p}}\ddot{\tilde{p}}\right]$$

(9)

To calculate $u_d$, we consider the Lyapunov candidate function $V={\displaystyle \frac{1}{2}}{s}^{T}s$, where its derivative is $\dot{V}=s^T\dot{s}$; replacing (7) in $\dot{V}$, we have the following:

$$\dot{V}=s^T\left[{\lambda }_p\left({\dot{h}}_d-JU\right) +{\lambda }_i \tilde{p}+{\lambda }_d \ddot{\tilde{p}}\right]$$

(10)

Replace (3) and (9) in (10) and develop the following:

$$\dot{V}=-s^T J u_d$$

(11)

Therefore, $u_d$ can be defined as follows:

$$u_d=J^{-1}k sig\left(s\right); k>0$$

(12)

Replace (12) in (11).

$$\dot{V}=-s^T k sig\left(s\right)$$

(13)

where $\dot{V}<0$; hence, $s\to 0$ with $t\to \infty$. Analyzing (4) for $s=0$ and since we have a polynomial of two degrees, we have $\tilde{p}\to 0$. When the roots are $r={\displaystyle \frac{-{\lambda }_p-\sqrt{{\lambda }_p^2-4{\lambda }_d{\lambda }_i}}{2{\lambda }_d}}$, the following must be fulfilled: ${\lambda }_p^2>4{\lambda }_d{\lambda }_i$.

3.2. Obstacle Avoidance Strategy

Among the characteristics of the TurtleBot3 Burger is that it has a LiDar-type distance sensor capable of scanning 360° of its environment to perform a sweep for SLAM or navigation applications; it reaches a detection distance from 12 to 350 cm with a sampling rate of 1.8 kHz. For the implementation of the obstacle avoidance strategy, a 180° sweep seen from the front of the robot has been considered in order to avoid obstacles that may appear when the robot is executing the trajectory tracking.

Obstacle detection is performed when the LiDar sensor emits a laser light pulse and determines the distance ${\gamma }_i$ through the delay between emission and bounce off an object; in this case, the obstacle towards the sensor is shown in Figure 3.

The following consideration is established to determine the detection of the obstacle:

$$u_{SMC}=\left\{\begin{array}{cc}\hfill u_a+u_d,& {\gamma }_i<R_o\hfill \\ \hfill u=0.05\left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right], \omega =0.5\left[{\displaystyle \frac{\mathrm{rad}}{\mathrm{s}}}\right],& {\gamma }_i>\ge R_o\hfill \end{array}\right.$$

where $R_o$ represents the distance from the robot to the obstacle. In the case of obstacle detection, only the LiDar beams that are smaller than the radius $R_o$ are taken into consideration, which depends on the value that is set for the detection.

4. Test

In order to evaluate the trajectory tracking of the TurtleBot3 Burger mobile robot using the SMC controller, experimental tests were performed for a circular trajectory described by $h_d=\left[x_d y_d\right]$; $y_d=0.5\mathit{sin}\left({\displaystyle \frac{t}{10}}\right)+{\displaystyle \frac{1}{2}}$; and $x_d=0.5 \mathit{cos}\left({\displaystyle \frac{t}{10}}\right)+{\displaystyle \frac{1}{2}}$. Based on the expertise in the management of the SMC controller, the initial parameters that were considered were ${\lambda }_{po}=1, {\lambda }_{i0}=1$ and ${\lambda }_{d0}=0.1$. After the implementation of the tests, the respective calibration was carried out based on the index of the integral square error (ISE) of said parameters, and the values that were used in the tests were obtained as follows: ${\lambda }_p=1, {\lambda }_i=0.5$ and ${\lambda }_d=0.000001$. For obstacle avoidance, ${\gamma }_i=15 \mathrm{cm}.$ The experiment has a duration of 90 s and considers the incorporation of two obstacles within the desired trajectory. The physical obstacle used is a wooden box measuring 10 × 10 × 30 cm (width, length, and height) with a hollow structure that weighs 0.500 kg.

4.1. SMC Controller Test

This section presents the results of experimental tests for the trajectory tracking of the TurtleBot3 Burger mobile robot using an SMC controller considering obstacle avoidance.

4.1.1. Described Trajectory

Figure 4 shows the desired trajectory and the actual trajectory of the mobile robot during the experiment. With the results obtained, it is observed that at the beginning of the trajectory, the mobile robot is positioned to follow the circular trajectory. Once the robot is aligned with the trajectory, there is a perfect tracking of the trajectory. Likewise, in the presence of an obstacle, the robot executes the obstacle avoidance strategy, deviating its trajectory to avoid a collision; once it avoids the obstacle, the SMC control positions the robot again on the desired trajectory.

4.1.2. Mobile Robot Velocities

Figure 5 shows linear velocity $u$. At the beginning, it exhibits high values of 0.2 m/s as the robot tries to align itself with the desired trajectory from the initial position. Once the robot reaches the desired trajectory, the velocity remains constant at approximately 0.025 m/s. Between 19 and 31 s, obstacle 1 appears, during which the linear velocity drops to 0 m/s due to the detection of obstacle 1; this causes the robot to stop and start and turn to avoid a collision. Once the robot turns and the obstacle disappears from its path, the robot attempts to return to the trajectory, resulting in spikes in linear velocity as it strives to reach the reference. However, as the robot turns, it encounters obstacle 1 again, causing fluctuations in linear velocity until it completely avoids the obstacle. A similar pattern occurs between 57 and 66 s, where obstacle 2 is present. In other intervals, the robot continues along the trajectory at a velocity of approximately 0.025 m/s.

Figure 6 shows the angular velocity of the mobile robot. In the interval from 0 to 9 s, there are peaks in angular velocity as the TurtleBot3 Burger reaches the circular trajectory. During the interval from 9 to 19 s, the angular velocity fluctuates between 0.1 and 0.15 rad/s, indicating that the robot is following the circular trajectory. In the interval from 19 to 31 s, when obstacle 1 is present, the robot’s angular velocity increases to 0.5 rad/s to avoid the obstacle, causing the robot to temporarily deviate from the trajectory. Once the robot turns and the obstacle is no longer in its path, it attempts to return to the trajectory, resulting in spikes in angular velocity as it seeks to reacquire the trajectory. As the robot turns, it encounters obstacle 1 again, leading to further changes in angular velocity until it fully avoids the obstacle. A similar pattern is observed between 57 and 66 s, where obstacle 2 is present. In other intervals, the robot continues following the trajectory with an angular velocity of approximately 0.05 rad/s.

Figure 7 presents the position error, $\tilde{x}$. In the interval from 0 to 9 s, the mobile robot starts from the initial position and tries to align to the desired trajectory, generating an approximate maximum error of 0.9 m. In the interval from 9 to 19 s, the robot reaches the desired trajectory, which reduces the position error $\tilde{x}$ to approximately 0.05 m. However, at the instant from 19 s to 31 s, the robot detects obstacle 1, causing it to temporarily deviate from the circular trajectory, and the position error $\tilde{x}$ fluctuates until it returns to the desired trajectory. Similarly, this happens between 57 and 66 s, and the position error $\tilde{x}$ increases and shows oscillating behavior due to the avoidance of obstacle 2.

Figure 8 presents the position error, $\tilde{y}$. It is observed that in the interval from 0 to 9 s, the mobile robot is in search of the desired trajectory, which causes oscillations in the position error in $y$. During 9 to 19 s, the error gradually increases until it reaches a value of approximately 0.01 m. In the interval from 19 to 31 s, the $y$-position error presents oscillations due to the fact that the robot temporarily deviates from the trajectory while avoiding obstacle 1; the same happens in the interval from 57 to 66 s, where it detects obstacle 2. For the rest of the time, the error value is a maximum of approximately 0.02 m.

5. Discussion

The SMC controller proposed for the trajectory tracking of the mobile robot TurtleBot3 Burger presents good trajectory tracking without obstacles, and the linear and angular velocities do not present oscillations and tend to have values of 0.025 m/s and 0.15 rad/s, respectively. This allows the errors in $x$ and $y$ to have maximum values of 0.05 m, which are values expected by the odometry of the robot. When the obstacles are presented in the trajectory in the intervals from 19 to 31 s and 57 to 66 s, position errors in $x$ and $y$ are shown in Figure 7 and Figure 8, causing the robot to temporarily deviate from the trajectory to avoid colliding with the obstacle, generating an error 0.3 m and 0.15 m in $x$ and $y$, respectively, which is expected for the robot to avoid the obstacle. Likewise, the robot speeds in these intervals are oscillatory to avoid colliding with the obstacle.

6. Conclusions

This work presented the implementation of the Sliding Mode Controller for trajectory tracking of a TurtleBot3 Burger mobile robot in a congested environment. A clear and detailed methodology of the design of the SMC controller with a PID sliding surface for trajectory tracking is provided; this controller can be used for position control, and the methodology can be extrapolated to develop posture control. An obstacle avoidance strategy was implemented, which caused the robot to temporarily deviate from the trajectory. The discontinuous part of the SMC was utilized to ensure that the robot returns to the desired reference whenever it deviates from the intended trajectory to avoid obstacles. The calibration of the controller parameters was carried out based on experimental tests to achieve the smallest possible error. The experimental results demonstrate the controller’s strong performance. The main disadvantage of the implemented controller is its challenging calibration, which could be addressed in future work through optimization techniques for controller calibration.