Transport robots for logistics are used in warehouses and storage facilities to transport goods. In warehouses or storage facilities, robots organize and transport product in a process referred to as intralogistics. These robots offer far greater uptime than manual labor, leading to major productivity gains and profitability for those deploying logistics transport robots. Logistics transport robots operate in predefined pathways such as a magnetic guidelines and moving products for shipping and storage.

With low cost, ease of implementation, and reliable navigation technology owing to the stability of the feedback signal from the position sensor, transport robots based on trajectory-tracking control, which relies on the feedback signal from a position sensor located on the robot’s body to guide the robot closer to the center of the line [1], are becoming more widely used to improve the efficiency of transporting goods in chains that require high accuracy and difficulty for humans. Numerous studies have been conducted on trajectory-tracking robot systems. They can be divided into different problems, such as improving the movement algorithm [2], avoiding obstacles [2,3], improving or changing position sensing systems [4,5], and developing the following control techniques: proportional-integral-derivative (PID) [6,7], fuzzy logic control (FLS) [3], fuzzy PID [8], and sliding mode control [9], to better help the robot stick to the trajectory or complete the predefined path in a faster manner.

Among the aforementioned studies, the studies on the following controls are most interesting, and they are often contested in competitions. However, for the robots, besides the requirements of completion time and line position errors, another very important factor is safety. When transporting heavy loads around large turn corners or on tight curves along the movement path, moving quickly can cause the robot to make a higher line position error, and in the worst case, can cause the robot to run out of the way. In addition, at high speeds in the corners, the wheels are prone to slip and the robot shakes; as a result, the goods easily fall and break. Therefore, it was necessary to reduce the speed of the robot. However, until now, there have been almost no studies on the speed regulation of trajectory-tracking robots.

An intelligent controller using a type-1 fuzzy logic system (T1FLS) is known for its ability to supervise nonlinear systems [10–12]. However, type-1 fuzzy sets can neither model uncertainty nor function effectively when a higher degree of uncertainty is present in the system. This can be caused by a sudden change in the direction of the path. In the following control, the position of the robot relative to the reference path is a time-varying quantity that cannot be mathematically represented in advance. According to the recommendation of Wu, D. and M. Nie [13], the type-2 fuzzy logic system (T2FLS) [14,15], whose fuzzy sets can model the uncertainty, can be applied to improve the performance of the trajectory-tracking control system.

Therefore, in this study, we propose a new trajectory-tracking control with a speed regulation technique based on an interval T2FLS to help the robot change its speed and adapt to the changing characteristics of the moving segment of the path. The transport robot model is described in Section 2. In Section 3, kinematic control and velocity control of the transport robot for trajectory-tracking control are described. In Section 4, a new method using a speed regulator based on T2FLS for trajectory-tracking control is proposed. Section 5 presents the simulation results. Finally, the conclusions are presented in Section 6.

A model of a transport robot with a differential drive is shown in Figure 1.

The movement of the robot with dimensions W × L (width and length of the robot body) was carried out using two independent coaxial driving wheels with a diameter of 2R. The rotating velocities of the left and right wheels were φ̇_R and φ̇_L, respectively. Because the two driving wheels are located right on the horizontal axis of symmetry of the robot body, the arrangement of the four castor wheels located near the rear corner of the robot body helps the robot balance better and avoid rolling on the reference path and damaging it.

Transport robots are typically designed to operate in both automatic and manual control modes. Manual control with human beings is also equally important; for example, when the robot performs simple carrying tasks in short distances, or after the operation time, the robot should be sent for maintenance. Therefore, a simple way to switch from manual control to automatic control or vice versa is required. To control the robot while moving backward and turning left and right with reference to linear and angular velocity values, two methods are generally used. In the first method, the reference values are converted into two independent signals, each of which is used to individually control the velocity of each wheel based on its own PID controller with appropriate parameters [16]. In the second method, without any reference value conversion, the robot velocities were directly controlled by two linear and angular PID controllers [17]. However, using the first method, the transition from the manual mode to the automatic mode becomes inconvenient. To ensure the effectiveness of programming and control, this study used the second method. This method allows the system to switch from manual to trajectory-tracking mode by simply changing the PID controller of the angular velocity using the following PID controller.

3.1 Kinematic Control

The linear and angular velocities of the robot can be obtained as

The velocities of the robot can be controlled by a PID controller of linear velocity, denoted as PID_v, and a PID controller of angular velocity, denoted as PID_ω. The output control signals u_v of PID_v and u_ω of PID_ω can be obtained by:

Here, [K_pv, K_iv, K_dv] and [K_pω, K_iω, K_dω] are parameters of PID_v and PID_ω, respectively.

The control voltage signals (u_L, u_R) are used to send signals directly to the left and right motor drivers and can be calculated as

3.2 Velocity Control

The PID controller of the angular velocity was replaced by a PD controller of the line position in the velocity control. Thus, Eq. (2) can be written as

Here, K_pɛ and K_dɛ are the parameters of the PD controllers of the magnetic line following position, and ɛ and e_ɛ are the line position and position error (e_ɛ = −ɛ), respectively.

Therefore, Eq. (3) can be written as

Based on Eqs. (4) and (5), we obtained the control diagram shown in Figure 2.

4.1 Speed Regulation Technique

In this section, we propose a new technique to regulate the reference speed of the trajectory-tracking control of transport robots. To make the robot speed up or slow down according to the change in properties of the predefined path, a fuzzy-based speed regulator with feedback of the position error and change in the error was designed. This is responsible for generating the adjusting value of the speed. This method can be presented as a block-control diagram, as shown in Figure 3.

The fuzzy-based speed regulator includes two major parts: the fuzzy logic controller and conversion part. The architecture of the regulator is illustrated in Figure 4.

The output y_reg of the fuzzy-based speed regulator can be obtained using the proposed equation:

Here, Γ is the output of the fuzzy controller, δ and λ are the adjusting parameters of the speed regulator, and Ω is the conversion gain related to the reference value of the speed.

To prevent the speed from increasing too high or decreasing too low, the output y_reg should be limited.

Here, ϑ_U and ϑ_L are the upper and lower limit constraints of y_reg.

4.2 Fuzzy Logic Control

As shown in Figure 5, the fuzzy controller has two inputs: position error e_ɛ and change in the position error de_ɛ. The output signal of the fuzzy controller is Γ ∈ [−1, 1].

The fuzzy table rule is built with 25 rules, as shown in Table 1, in which DE = {VNDE, NDE, ZDE, PDE, VPDE}, E = {VNE, NE, ZE, PE, VPE}, and U = {N2, N1, Z, P1, P2} are the sets of linguistic variables of the two fuzzy inputs and outputs (e_ɛ, de_ɛ, Γ), respectively. Each rule is created as follows:

where n = 25 is the number of fuzzy control rules and C_i,j is the value of the cell corresponding to the ith row and jth column in Table 1.

The membership functions of the type-2 fuzzy inputs have a Gaussian shape, as shown in Figures 6 and 7.

4.3 Type Reduction Method for Type-2 Fuzzy Logic System

As shown in Figure 8, the considerable difference between T1FLS and T2FLS is the requirement of the type reduction step before the defuzzification step to find the final fuzzy output.

The output fuzzy set corresponding to each rule of the T2FLS has a type-2 set. The type-reducer is responsible for transforming the type-2 fuzzy set into an interval type-1 fuzzy set, called a type-1 reduced set, by computing the centroid of this type-2 fuzzy set.

The first type of reduction method, the KM algorithm, was developed by Karnik and Mendel [18]. Owing to the iterative nature of this algorithm, the type-reducer method incurs a high computational cost. Recently, to reduce the load on the calculation, some algorithms have been proposed to improve the type-reducer method, such as the enhanced KM (EKM) [19], iterative algorithm with a stop condition (IASC) [20], and enhanced iterative algorithm with a stop condition (EIASC) [21]. However, all the mentioned algorithms are still iterative. To avoid the iterative type-reducer method, the uncertainty-bound method was introduced [13]. However, this method is complex and unsuitable for control systems.

The heavy and complicated computation in the type-reducer step has become a challenge for the real-time applications of T2FLS. Recently, many researchers have been developing algorithms to bypass type-reducers, such as the Nie-Tan method (NT) [22] and the Begian-Melek-Mendel method (BMM) [23].

In this study, the T2FLSs using type-reducer methods and the T2FLSs bypassing the type-reducer step mentioned above are applied to the proposed speed regulator and used in the trajectory-tracking control system in Section 5 to evaluate the error and completion time.

In this section, the proposed speed regulator using an interval T2FLS is tested by simulation in the MATLAB/Simulink environment, as shown in Figure 9. We used the open-source MATLAB/Simulink toolbox for interval T2FLS [24]. To simulate the trajectory detecting sensor, the Line Sensor Simulation toolbox [25] has been modified to send a position signal with a measurement range of 14 cm and measurement resolution of 0.01 cm, instead of an on-off signal.

In this simulation, the width (W) and length (L) of the robot body was 0.56 m and 0.84 m, respectively. The diameter 2R of each wheel was 0.3 m, the weight of the robot was 125 kg, and the maximum torque of each wheel was 67 Nm. The shape of the test trajectory path for the transport robot is shown in Figure 10.

The parameters of the interval T2FLS was experimentally chosen as

With reference speed of 0.15 m/s, simulation testing was performed using different speed regulators. The simulation results obtained using the conventional method without a fuzzy logic system, T1FLS, and T2FLS, are presented in Figures 11 and 12. The dash lines show the time response of resulting values when the reference speed is limited to a maximum of 0.15 m/s, this is done by setting the limit constraint ϑ_L = 0.

Table 2 presents a performance comparison of the testing results of various speed regulators. Based on Figure 11 and Table 2, the proposed speed regulator based on the interval T2FLS helps the system to decrease the position error in terms of (AVMPE), root mean squared error (RMSE), and maximal peak-to-peak error ripple (in DS box) (MPPER), compared with conventional speed regulators. The control using the interval T2FLS with the KM method has better performance in tracking the position error but with a longer completion time compared with the control using the T1FLS. In the case where the robot’s reference speed was limited to 0.15 m/s, the position error when using the interval T2FLS was significantly reduced; however, the robot takes longer to complete the following path.

In this study, a new method of trajectory-tracking control with a speed regulator based on an interval T2FLS was proposed for automated transport robots to track complex predefined trajectory paths. The proposed method can enable the robot to regulate its speed to accommodate changes in the trajectory characteristics. The proposed control with the speed regulator using the T2FLS shows better performance compared with the conventional system in terms of tracking the position error. The RMSE of the tracking control using the T2FLS was 4.23% lower than that of the T1FLS. Because the proposed method can adapt to changes in trajectory properties and slow down the robot on the corners, it shows better trajectory tracking.

Table 1. Fuzzy rule table.

Table 2. Performance comparison according to various speed regulators.