Intelligent Motion Control Design for an Omnidirectional Conveyor System

Nowadays, an omnidirectional conveyor system has been introduced as a new means of package transportation. The aim of this study is to achieve trajectory-tracking and collision avoidance of multiple packages which has not been done before on an omnidirectional conveyor platform. Despite the kinematic similarity to omnidirectional mobile robots which have built-in sensors to measure velocities and accelerations, this system only measures the position of the transported package via external sensors which makes it a unique and intriguing area of research. To tackle this challenge, this study employs the proposed Fuzzy Sliding-mode Tracking Control (FSTC) and Fuzzy Inter-package Collision Avoidance (FICA) schemes. The FSTC has been enhanced with fuzzy sliding surfaces that take tracking errors as control inputs and linear forces as control outputs. Furthermore, using a fuzzified package distance and collision angle as inputs, the inference engine of FICA is designed to generate the deflection angle and force gain as outputs. Additionally, the conveyor platform is modeled and built by a multiple modular omnidirectional wheel system including the conveyor and actuator dynamics. To determine its desired motion trajectory, a planned 4th-order Bèzier curve is utilized. Then, to assess the effectiveness and robustness of the proposed methods, simulations are conducted under diverse conditions. The results indicate that the package position converges within a finite time frame, highlighting the superior trajectory-tracking capabilities of FSTC in spite of any disturbances injected into position feedback signals. Meanwhile, the proposed FICA has demonstrated its effectiveness by enabling the package to navigate through the omnidirectional conveyor platform while avoiding both stationary and moving obstacles.


I. INTRODUCTION
An omnidirectional conveyor system is a type of conveyor system that can move in any direction, including diagonal movements. It is commonly used in the manufacturing and logistics industries for the transportation of goods and materials. The use of omnidirectional conveyor systems has many advantages, including package classification and sorting, flexible route reconfiguration, and improved efficiency in material handling processes [1], [2], [3], [4]. So far, modular design of omnidirectional conveyors has been The associate editor coordinating the review of this manuscript and approving it for publication was Kathiravan Srinivasan . thoroughly developed in [5], [6], [7], [8], [9], [10], and [11]. Subsequently, a more advanced control such as trajectory-tracking has been made possible. This can be achieved through control of a multiple modular omnidirectional conveyor system. The trajectory control combined with collision avoidance is essential and significant, particularly as a sorting system.
Although the omnidirectional conveyor design is not new, the research on trajectory control in this field is considerably few. Despite the lack of close loop control, some works attempted to put a predefined trajectory on an omnidirectional conveyor [8], [9], [10], [11]. Later on, a work was done by Sun et al. [12] who developed wheel speed control using PID control. The velocity feedback is done in the actuator level. This way, the entire conveyor platform has predefined velocity and direction. On the other hand, works in [13] and [14] used a different PID control approach that allowed an individual conveyor module to change velocity and correct the package position. Also, there is conducted research for an omnidirectional conveyor using reinforcement learning [15]. However, the proposed method did not produce continuous control inputs. The common challenge based on the previous works is it is hard to acquire the dynamic model of the platform. The dynamics form dictates the way control schemes used.
A single omnidirectional conveyor module shares kinematic similarities with omni-wheeled mobile robots. Trajectory-tracking research has been conducted on these robots, utilizing state-space control techniques (Refs. [16], [17], [18]) and sliding-mode control techniques (Refs. [19], [20], [21]). However, these approaches required numerous sensors including velocity, acceleration, torque, or current sensors to be installed in the robot for successful implementation. Referring to Refs. [1], [2], and [14], it can be observed that dissimilar to the prior approaches, the transported packages are not equipped with sensors. Instead, an omnidirectional conveyor system typically employs a camera or proximity sensor to externally determine the position of the package. Furthermore, the aforementioned model-based control techniques are optimized specifically for omni-wheeled mobile robots and may not be directly compatible with an omnidirectional conveyor system. Thus, further investigation is necessary to develop effective trajectory-tracking solutions for omnidirectional conveyor systems.
Further, there is little research on inter-package collision avoidance of omnidirectional conveyor systems. Despite the difference in terms of dynamics, some previous works addressing this matter are available in the field of mobile robots; for example, a study in [22] developed a complex if-then rules and the works in [23], [24], [25], and [26] developed fuzzy inference systems to avoid obstacles. The common issue (based on these previous works) is to design a smooth transition between the trajectory tracking and obstacles avoidance control.
This study proposes an intelligent control scheme with high-level robustness, i.e., Fuzzy Sliding-mode Tracking Control (FSTC) and Fuzzy Inter-package Collision Avoidance (FICA) techniques, to make a package track the desired trajectory on an omnidirectional conveyor and avoid possible collision between packages. The contributions are enumerated as follows: 1) A dynamic model is developed for an omnidirectional conveyor system. The omnidirectional platform is based on Refs. [12] and [13], however, the dynamic and actuator models in the research are introduced as they have not been studied in Refs. [12], [13], [14], abd [15]. Without the dynamic model, the control inputs become discretized as presented in Ref. [15]. Also, the conveyor module in Ref. [14] is modeled by using a sphere that rotates in two axes. The results are oversimplified to cause an inaccurate model since the sphere is assumed to have a single contact point to represent 5 wheels in a conveyor module. 2) A close-loop control with package position feedback is completely executed. A different control approach is used compared to the PID used in Refs. [12], [13], and [14]; FSTC is utilized in this research since it is more robust in the area of non-linear controllers.

3) A mission for collision avoidance between packages
is successfully achieved. The FICA technique has the advantage of fuzzy interpolative reasoning unlike the hard computing technique utilized by Ref. [22]. This also includes a seamless switching between trajectory tracking and obstacle avoidance by fuzzy inference engine contrasting with Refs. [23] and [25]. The inference engine has numerous simplicity in comparison with Refs. [23], [24], [25], and [26] leading to a reduction of computation power demands. Combined with FSTC, FICA makes a multiple intersecting trajectories tracking possible, dissimilar to predefined trajectories in works [8], [9], [10], [11], [12], [15].
The remaining sections of the paper are organized as follows: In Section II, the system modelling of the multiple modular omnidirectional conveyor module is described and the problem statement is presented. In Section III, the design of the proposed FSTC is explained. In addition, section IV describes the FICA design for collision avoidance. The simulation results and discussions are presented in Section V. Finally, concluding statements are given in Section VI.

II. SYSTEM MODELING AND PROBLEM FORMULATION
An omnidirectional conveyor system is typically used to overcome two overlapping conveyor belts from different directions. It can direct the flow on a conventional conveyor from one end to another. Also, it is used as a sorting bed before transferring goods from one conveyor belt to another as shown in Fig. 1. In both situations, the package must follow a prescribed trajectory. The omni-wheels on the conveyor are utilized to control the moving position of the packages according to the planned trajectory. Hence, a dynamic model of the system must be derived to  enable the velocity transferred from each omni-wheel to the package.
The conveyor system is driven by DC motors equipped with omni-wheels. A single module is designed as a hexagonal shape with three omni-wheels as shown in Fig. 2. In a polar coordinates relative to the ith module frame {C i }, the omni-wheels positions W ij (r j , θ j ) are at is the index of the omni-wheels. Together with the hexagonal chassis, there exist some properties in the module i.e. the length of the hexagonal edge is L = √ 3r and the width of the module is δ = 3r.
The force direction and magnitude of the package on the conveyor are determined by the force resultant of omni-wheels laying underneath the package. From the geometric relationship shown in Fig. 2, a package dynamics is given by where F Px and F Py are respectively the forces of the package along X and Y-axes; µ is a viscous friction constant of the package; m is the mass of the package. Here, k ⊂ i represents module index number under the package and F Wk1,2,3 are the corresponding omni-wheels touching the bottom of the package. To infer this law, the package is assumed to be bigger than 2δ×δ m 2 to ensure that there is a package transfer between 2 neighboring modules. By considering the inertia I W , viscous friction constant ζ , angular velocity ω W , and radius r W of the wheel, the dynamics of each wheel is defined as where τ Wij denotes generated torques from omni-wheels.
Here, the omni-wheels are actuated by DC motors in which mathematical model is given as where V Wij is the input voltage of ijth the motor; R a , i a , L a , and e a are respectively armature resistance, current, inductance, and back emf voltage; K b and K t are respectively back emf and torque constants.
The objective of the study is to transfer a package across an omnidirectional conveyor system from one end to the other. To achieve this goal, we propose the FSTC technology that tracks the package's movement along a prescribed trajectory. The tracking error between the actual position and intended trajectory is computed and then used in constructing sliding surfaces. A fuzzy inference engine analyzes these sliding surfaces to determine actuation voltage for the omni-wheels, aiming to adjust tracking errors towards zero. A detailed description about the calculation of tracking errors and sliding surfaces can be found in the subsequent section of this paper. Moreover, the FICA approach is proposed to prevent collisions between multiple packages on the omnidirectional conveyor system. For this purpose, it is essential to maintain a distance greater than zero between package boundaries. The proposed FICA infers the distance between colliding packages and the direction of trajectory tracking in order to produce magnitude and direction of the avoidance. The overall control scheme is shown in Fig. 3. Further mathematical descriptions of the proposed controller will be provided in the subsequent section.

III. FUZZY SLIDING-MODE TRACKING CONTROL DESIGN
At first, through the planned motion trajectory function B(t), a new reference pose (x ref , y ref ) is fed to the controller for the following of the package as t evolves. Then, by using the integration of the dynamic Eqs. (2) and (4), the position of the package (x P , y P ) with respect to the global coordinate can be obtained. In this way, tracking errors e x and e y can be calculated as follows: It is well-known that the robust controller, FSTC, can deal with the uncertain control systems very well. To begin with, sliding surfaces are designed based on the error information as shown in the following Eq. (6) σ x = a x e x + b xėx σ y = a y e y + b yėy (6) where the chosen constants a x,y , b x,y > 0 such that the sliding surfaces are stable. Then, the sliding surfaces as inputs are fuzzified and the input fuzzy membership functions (µ σ x , µ σ y ) are designed in Fig. 4a.  be normalized by gain values if needed. That is to say, the membership value ranges of the µ σ x and µ σ y are between [−1, 1]. This also applies to the membership value ranges of µ ux and µ uy which are estimated voltage outputs of the package that can be adjusted by gains. The designed output fuzzy membership functions (µ ux , µ uy ) are singleton type as shown in Fig. 4b that is proven for effective elimination of chattering effects [27].
Incorporating an expert's knowledge is the common approach to design fuzzy membership functions. However, a significant number of fuzzy membership functions result in more intricate computation and inference rules. Furthermore, as each package has its own FSTC, the computational load also increases with the number of packages. To simplify computation, this study selects only seven membership functions for FSTC scheme. Here, the most important memberships for the FSTC are Z , PL, and NL. The purpose of the Z membership function is to produce a fuzzy output value of 0 when the tracking error falls between ±0.05 m. For tracking errors exceeding ±1 m, fuzzy outputs are handled by both PL and NL with corresponding values of ±10. Besides, NM , NS, PM , and PS fuzzy membership functions are introduced to ensure smoother transitions between Z , PL, and NL and prevent overshoots in the tracking response.
Next, fuzzy inference rules are designed such that the designed sliding surfaces σ x → 0 and σ x → 0 are achieved as t → ∞. Subsequently, the tracking errors asymptotically converge to zero. To do so, the input fuzzy membership functions are paired with the output fuzzy membership functions accordingly to form inference rules as (7): The consequence functions (μ ux ,μ uy ) of the rules are defuzified using the center of gravity approach as The next section will focus on the approach of collision avoidance of multiple moving packages.

IV. FUZZY INTER-PACKAGE COLLISION AVOIDANCE DESIGN
As two packages (P and P cc ) simultaneously move on the conveyor system, one package needs to avoid another package in case another package is on the trajectory at the same time. Here, a fuzzy inference control technique, i.e. FICA, is proposed to attain the task. Fig. 5 illustrates the geometric relations for the package positions in the global frame {G}. In this section, the proposed FICA is applied only to a package P to avoid package P cc . The same FICA technique can be applied to another package. In this case, two moving packages are expected to avoid each other when the FICA techniques are applied to both packages. The proposed FSTC and FICA are applied to make the package P(x P , y P ) track the trajectory B(t) and avoid the package P cc (x Pcc , y Pcc ). Here, (x ref , y ref ) is a reference point on the planned trajectory B(t). In a polar coordinate relative to a point P, the Euclidean distance d pp from P to P cc is given by d pp = (x Pcc − x P ) 2 + (y Pcc − y P ) 2 (9) and the polar angle θ cx is given by where atan2(·) is an inverse tangent function that returns values in [−π, π] [28], [29], [30], [31]. By using the similar  principle, the polar angle of a reference point θ bx is calculated as follows: Using (10) and (11), a collision candidate angle θ cc can be obtained by θ cc = θ cx − θ bx θ cc = atan2(sin(θ cc ), cos(θ cc )) Here, the value θ cc is normalized between [−π, π] using atan2(·) function. Then, the collision candidate distance d cc can be obtained by where l cc is a preferred safe distance between packages. In this study, l cc is chosen to be equal to the width of package P cc . The collision avoidance goal can be achieved by inferring from d cc and θ cc .  Fig. 6. The collision avoidance inference engine is to use two inputs (X dc , X θ c ) and generate two outputs (Z sg , Z da ) with the following fuzzy rules listed in Table 1.  The accumulated outputs of the fired rules are defuzzified using the weighted average approach as follows: where µ R denotes the firing degree of rule R and (Z daR , Z sgR ) are singleton output values of a gain (g sg ) and deflection angle (θ da ) of rule R. To avoid collision while there is a package close by, the proposed FICA technique adjusts the magnitude and direction of the outputs (u * x , u * y ) generated from the proposed FSTC by the factor g sg and θ da respectively as Because the control inputs are represented in the global frame {G}, a transformation to the wheel frame {W ij } is needed. Hence, using the geometric relationship shown in Fig. 2, the following equation can be obtained as (16) where V uk1,2,3 are input voltages for the actuators under the package.

V. SIMULATION RESULTS AND DISCUSSIONS
To investigate the feasibility of the proposed control schemes, the trajectory-tracking and collision avoidance of a multiple modular omnidirectional conveyor system is simulated with the parameters listed in Table 2.
As illustrated by the use cases of the omnidirectional conveyor system in Fig. 1, the purpose of the omnidirectional conveyor system is to transfer a package from one end to the other. Unlike circular or lemniscate trajectories that form a loop, an open-end trajectory that connects two points is used for this task. Moreover, a trajectory may appear in various shapes in real applications. Therefore, the package is simulated to follow a trajectory generated from a Bézier curve in this study. A Bézier curve is one of the most intuitive parametric functions that can form any shape and has been widely used in computer-aided design software.
A 4th-order Bézier curve trajectory B 1 (s); s ∈ [0, 1] is planned as a motion trajectory of moving packages. The conveyor is arranged to move on an 10 × 10 m 2 floor. The prescribed trajectory starts at (1, 1) and ends at (10,9) giving out approximately 13.38 m of the path length. Two initial cases are performed independently in sequential order, namely case #1 and case #2 which have the initial positions of (0, 4) and (4, 0), respectively. Both packages must be transferred from one end to the other one within 20 s i.e. s = t 20 . First, under the given prescribed trajectory and initial conditions, a PID control method is applied as a baseline performance whose results are shown in Fig. 7. It can be seen that a bit of tracking error arises and further details are shown in Fig. 8. The results demonstrate that the package converges to the prescribed trajectory within 1.5 s with a maximum error of 0.32 m. 47356 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   Second, FSTC is applied under the same trajectory and initial conditions. The tracking response is shown in Fig. 9 where tracking errors are smaller compared to results of the PID method. The trajectory-tracking accuracy can be seen clearly in the error response shown in Fig. 10. As observed from the figure, the tracking responses are quite satisfactory. Furthermore, the proposed FSTC has a fast response as the errors approach to zero within 1 s. Aligned with the error response, the sliding surfaces also converge to zero within 1 s as shown in Fig. 11. The corresponding voltages generated by the FSTC are demonstrated in Fig. 12. From the figure, the chattering phenomenon is indeed reduced by means of the designed fuzzy singleton-type outputs. As stated in Ref. [27], fuzzy membership functions can improve the control system's performance to handle disturbances and uncertainties.   When utilizing fuzzy logic control, alterations to a crisp value may not necessarily impact the corresponding linguistic term. This results in the reduction of the chattering effect on the control output.
Third, to investigate the feasibility of the proposed FICA, a static package is placed at (6,4) in which the moving  package would collide and the moving package is initially placed at (4, 0). The response is shown in Fig. 13 where the red dash-dotted lines represent the movement trajectory of the package corners. We can see that collision of the moving package does not happen with the static package. In addition, the responses of g sg and θ da for the moving package are exhibited in Fig 14. It is observed that the moving package turns left corresponding to the positive value of θ da .
Fourth, to examine the performance of the proposed FSTC and FICA, an additional intersecting trajectory is designed as Responses of g sg and θ da for the moving package.
for a second moving package. The packages are initially located at {(0, 4), (7, 0)} respectively for the first and the second packages. The proposed FSTC and FICA are applied to both of the moving packages. The movement history of both packages is shown in Fig. 15. It can be seen that the second package avoids the first package by turning northward at t = 9 and then followed by the turning of the first package eastward at t = 10. This shows that both packages can avoid collision with each other. Moreover, between t = 9 and t = 13, both packages successfully avoid each other without collision and then return to the individual prescribed trajectories once the path is clear from a possible collision. Fifth, a Gaussian noise is injected into the position feedback signals of the packages (see Fig. 3) to examine the robustness of the proposed control schemes. Usually, the position feedback signals of the packages are contaminated, either caused by uncertainties in the dynamics, such as friction, or some noises in the measurement. For these reasons, to show the robustness of the proposed FSTC and FICA, a Gaussian noise with the mean of 0 m and variance of 0.1 m is added to the position feedback of the two packages. The initial conditions and planned trajectories of the packages are the same with the fourth simulation. The result is demonstrated in Fig. 16. As observed from the movement history, both packages successfully avoid each other and both packages converge to the individual prescribed trajectories between 12 < t < 20 s. Thus, the trajectory tracking and collision avoidance performances are satisfactory even under disturbances injected into position feedback signals.  Sixth, four packages are placed at the same time on the omnidirectional conveyor system. Their initial positions, as displayed in Fig. 17, are {(0, 4), (0, 0), (7, 0), (10, 0)}. It is worth noting that a practical scenario could involve collision risks between the front and back of packages. To demonstrate VOLUME 11, 2023 47359 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
how FSTC and FICA handle this situation effectively, one trajectory is set to be used by two packages. At t = 7 s in the simulation, all the packages approach the trajectories to bring about a possible collision event between the first and third packages. By passing through the front of the third package at t = 8 s, the first package is able to resolve the situation. Simultaneously, while sensing a potential collision between its front and back, the third package efficiently maintains its distance to prevent any collision. The first package then proceeds to cross over successfully and return to its dedicated trajectory by t = 12. At this same time point, in order to steer clear from each other, both the fourth and second packages adjusted their positions accordingly. Finally, all packages successfully avoid each other by t = 20 and safely return to their respective planned trajectories.

VI. CONCLUSION
In this study, the FSTC technique is proposed to track the planned trajectory for a package on an omnidirectional conveyor platform. The motion trajectory is planned using the 4th-order Bézier curve. Two cases in different initial positions are investigated under the proposed FSTC. Moreover, the FICA approach is presented to achieve collision avoidance of multiple moving packages. Furthermore, two intersecting trajectories for two packages are designed to investigate the feasibility and robustness of the proposed FSTC and FICA under disturbances injected into position feedback signals. The simulation results validate the effectiveness and feasibility of the proposed control schemes.
In our future work, we aim to develop a vision-based system to recognize package size and position. This will involve implementing an object detection algorithm that has been enhanced through a deep learning technology. Furthermore, proximity sensors will be placed on each wheel to assist with package recognition. Additionally, the proximity sensor will aid in reducing energy consumption by deactivating the wheel that is not in contact with the package. This research direction will make the implementation of the proposed FSTC and FICA more feasible in real-world scenarios.