Sustainable Agriculture: Stable Robust Control in Presence of Uncertainties for Multi-Functional Indoor Transportation of Farm Products

: Currently, integrated trends play a key role in every aspect of automation applications. In particular, if the mechanization of agriculture becomes a competitive factor among farmers or nations, then the multi-functional transportation of agricultural products is inevitable in global trade. In sustainable transportation, the challenge of overcoming stable control in harsh environments, such as through imprecise parameters or varying loads, should be addressed. In this paper, a novel controller for a nonholonomic mechanical system able to adapt to uncertainties is proposed. Based on the multi-functional autonomous carrier (MAC), the system conﬁguration of the kinematic and dynamic model is launched in order to identify the unstable problems that arise when tracking the trajectory. To solve these troubles, the decoupled formation of a MAC system has been investigated by considering two second-order components, namely a linear speed-based sub-system and angular speed-based sub-system. To stabilize the whole system using the Lyapunov theory, the advanced control techniques are studied. To validate the proposed approach, a series of test scenarios have been carried out. From the superior performance of numerous trials, it is clear that our approach is e ﬀ ective, feasible, and reasonable for the advanced control of agricultural applications.


Introduction
Recent research efforts in mobile carriers have been devoted to the appearance of novel concept systems that allow vehicles to handle different missions with or without human supervision. One of the challenges in farming automation is the ability to perceive and adapt to uncertain factors existing in the farmhouse. Knowledge of advanced control strategies can be beneficial for autonomous carriers in order to deal with their environment more efficiently and to better support precise farming work. Moreover, developments in commerce and global trade are accelerating through technology. The revolution of agricultural robotics, which brings countless benefits, is currently occurring in every nation [1,2]. The perception of a mechanical carrier in agricultural transportation is not unfamiliar anymore. To achieve success in global competition, the optimization and automation of logistics processes is a way in which the efficiency of resource use in the environment of a distribution center was built, but it differed in the use of the backstepping technique. The limitation of this work is that it did not consider the fluctuation of a system mass.
Furthermore, combining the schemes is an alternative method for different developers in order to simultaneously lessen the tracking performance of the errors. In fact, perfect knowledge of the system information is unattainable. To deal with this, an additional controller with a guaranteed performance should be derived-namely, an artificial intelligence-based controller [23] consisting of conventional torque-control, neural network backstepping, and perfect velocity tracking controller.
A key point in this research is that the same control algorithm should work if the behavior or goal of the system is modified. In particular, a multi-layer feed-forward neural network together a with well-designed kinematic controller might enhance the performance of the mobile robot drastically.
To solve the problems of the dynamic characteristics of an autonomous robot with unknown coefficients, fuzzy control is used as a substitute tool [24]. An expert-related method does not actually need any information about the system state and freely controls it. In most situations, the input control signal is often a driving moment, while the voltage signal is fed to an actuator. In [25], a solution for the input voltage signal was implemented in a detailed procedure. In most previous studies, the term "perfect velocity tracking" seems to be unrealistic. In fact, the active wheels are generalized by actuators that are driven by voltage instead of torque. Thus, the wheel actuator input voltage was chosen as the control input for the torque controller. The fuzzy logic system was utilized to estimate the nonlinear robot functions involving unknown robot parameters, while the stability and error bound were proven using the Lyapunov theory. Several research articles (summarized in Table 1) were further reviewed in order to study their insights into mobile robot control.  From the above discussions, it can be seen easily that the sequential control in a kinematic-dynamic-based model approach is effective and less complicated than the combining control. In recent works, most researchers selected the center point of the autonomous robot to coincide with the middle point between two driving wheels. In some real situations, this would be inappropriate, owing to the inequivalent distribution of the load. In addition, the nonholonomic system-based model is considered as the fundamental theory for different feedback controllers in tracking the trajectory. The nonlinear feedback controllers could be explored through mathematical expressions. Nevertheless, these controllers might be too multifaceted to be embedded into a real platform. For several situations, the integrated trend among numerous schemes or the upgraded traditional control algorithm would be encountered.
In recent years, the wide range of applications for factory automation have caused an increasing number of studies on autonomous transportation. There are plenty of mobile robots that are capable of handling dissimilar tasks in a warehouse-for instance, a tractor-trailer using a feedback linearizing dynamic controller [26], automatic visual guidance for a forklift vehicle [27,28], enhanced control strategy for a mobile robot [29,30], and upper limits of forces for a freight-car truck [31]. Each of these is available for a specified working area. In the scope of this application, the autonomous transportation of medium-and small-size freights is demonstrated. In Figure 1, the topological map is potentially utilized to describe the environment map, as shown. The grid-based map is characterized by five parts, namely manual sorting space, operation center, charging space, driving line, and parking space. The role of each part is quite diverse. A worker accomplishes his mission for pickup and delivery in the manual sorting space. Then, carriers track the planning route from the start point to the target point in the driving line. The server computer and central control room are positioned in the operation center. When the power of a carrier is less, the battery needs to be replenished in the charging space. All of the carriers are collected in the parking space before the system is turned on.

Results
The initial physical architecture of the autonomous carrier is characterized in Figure 3. It consists of four wheels: two side driving wheels, one front passive wheel, and one rear passive one. There are some conventions-for example, the grounded carrier only moves on a two-dimensional flat map, the distribution of freight is uniformly weighted, and the central point of the carrier and origin of xOy are the same.  A newer platform [32] to transport cargo is explained in Figure 2. This hardware has more advantages, namely a compact size, flexibility, and multi-functions. It can carry materials in both the loading and unloading process via vertical and horizontal manipulation. In the vertical handling mode, the robot elevates materials on its back. Otherwise, it brings things on a trailer. In any case, the robot and cargo become a rigid body. Hereafter, the unknown loads could cause difficulties in robot control. Additionally, uncertain parameters always exist in a practical model. There is a need to develop a motion controller for the multi-function autonomous robot in order to adapt to uncertainties in the storehouse or distribution center.

Results
The initial physical architecture of the autonomous carrier is characterized in Figure 3. It consists of four wheels: two side driving wheels, one front passive wheel, and one rear passive one. There are some conventions-for example, the grounded carrier only moves on a two-dimensional flat map, the distribution of freight is uniformly weighted, and the central point of the carrier and origin of xOy are the same. Table 2 summarizes whole mathematical symbols in this section. Consider that the reference trajectory in reference coordinate and speed of autonomous carrier are , correspondingly. Consequently, the relationship between linear speed and angular speed is expressed as

Results
The initial physical architecture of the autonomous carrier is characterized in Figure 3. It consists of four wheels: two side driving wheels, one front passive wheel, and one rear passive one. There are some conventions-for example, the grounded carrier only moves on a two-dimensional flat map, the distribution of freight is uniformly weighted, and the central point of the carrier and origin of Agriculture 2020, 10, 523 6 of 18 xOy are the same. Table 2 summarizes whole mathematical symbols in this section. Consider that the reference trajectory in reference coordinate and speed of autonomous carrier are q P T = x P y P θ and ϑ T = v ω , correspondingly. Consequently, the relationship between linear speed and angular speed is expressed as The boundedness of d  ensures that the uncertainty factor is not too large and it is appropriate for taking into account the limitations of the physical hardware. The system parameters are chosen as follows: Driving torque of two active wheels, τ T = τ 1 τ 2 From Equation (1), the reference coordinate of the center point is determined. .
Agriculture 2020, 10, 523 7 of 18 Using the Lagrange approach, the dynamic characteristics, including the kinetic and potential energy, are illustrated [7].
The boundedness of τ d ensures that the uncertainty factor is not too large and it is appropriate for taking into account the limitations of the physical hardware. The system parameters are chosen as follows: Without loss of generality, the above parameters are simplified as Taking a derivative of (2), then substituting into (3) and multiplying with S T (q), we have Thus, we obtain Hypothesis 1 (H1). The autonomous carrier moves in three-dimensional space with generalized coordinate q C T = x C y C θ and without slipping phenomenon.
The driving wheels are designed to ensure that they are in touch with the floor at any time. Thanks to this assumption, the controller could drive the system efficiently.
Hypothesis 2 (H2). The external disturbance is unknown but bounded. With From (6) and (7), the dynamical equation becomes the following: Agriculture 2020, 10, 523 Forces which act on the system are derived from (12): 2 denotes factors of external disturbances and uncertainties; u i (t), i = 1, 2 means the control signal of two active wheels.
The purpose of the control scheme is to solve the tracking trajectory problem. In order to tend to zero in finite time, the reference path is defined as T ∈ 2 is the reference signals of linear speed and angular speed, respectively. The tracking error of system is identified as As a result, lim t→∞ q R − q C = 0 ⇒ lim t→∞ e = 0 , and T(q C ) is bounded (17) In the real world, several constraints related to hardware or physical phenomena still arise in the system. To meet these requirements, some theories are indicated as below.

Hypothesis 3 (H3).
The lower and upper limits of linear speed and angular speed exist correspondingly. . .
Obviously, the upper and lower values of linear speed help to maintain stable motion while the boundary values of angular speed could assist in avoiding collapse when it turns at a corner.

Hypothesis 4 (H4).
Although the external disturbances or uncertain signals are unknown, it is always bounded by a positive value so that d i (t) ≤ D.
Although the uncertain mass of cargo is unknown, however, the agricultural products are delivered in one package with max weight. Hence, it is reasonable to estimate the max value of total cargo. Later, the design of a dynamic controller is essential to secure that the system errors of Equation (14) tend to zero and are stable in finite time. v where k 1 , k 2 , k 3 > 0 are constant v f , ω f : linear velocity and angular velocity of a dynamic controller and d dt Lyapunov function V 0 is selected as Then, deriving (24) in the time domain and substituting (23) into the below result, In the theory of Lyapunov, it is concluded that the controller in (22) can guarantee the stability of the system in (16) and the tracking errors converge to zero in definite time.
Merging (13) and (22) into the below equations, At this stage, linear speed and angular speed play a role as target control when combined with the dynamic model. The speed errors are computed as e v = v − v f and e ω = ω − ω f . Far ahead, Equation (26) is transformed: To lessen the burden of computation, Equation (27) might be divided into two sub-systems: Agriculture 2020, 10, 523

of 18
For the second-order system in (28), the sliding surface is established as (k 4 e ω + k 5 |e ω | p sign(e ω ))dt (30) where 1 < p < 2, k 4 , k 5 > 0 is constant. Taking the first derivative of Equation (30) and replacing the speed errors into below expression, e ω + k 4 e ω + k 5 |e ω | p sign(e ω ) = To maintain the state variables on a sliding surface, it must be satisfied . s = 0; at that time, the equivalent controller is derived from (31).
In this case, the disturbance is rejected when the system state is on a sliding surface. Otherwise, the design controller must be able to adapt with the existing uncertainties. At this moment, the proposed controller can be rewritten as where 0 < q < 1 and σ 1 > 0 are conversion coefficientŝ η 1 > 0 is the estimation value of η 1 , η 1 ≥ D 1 s 1 q η 1 =η 1 − η 1 is the error of the estimation valueη 1 The update value forη 1 is calculated as where µ 1 > 0 is tuning coefficient Totally, the motion controller for autonomous carrier is . ω f + k 4 e ω + k 5 |e ω | p sign(e ω ) + σ 1 s 1 +η 1 |s 1 | q sign(s 1 ) The chosen Lyapunov function is illustrated as Moreover, taking the first derivative of Equation (36), we obtain .
with D ≤ η 1 |s 1 | q . By tuning J so that J − 1 ≥ 0, then . V 1 ≤ 0. As a result, the stability of the proposed controller (35) by the Lyapunov theory. As well, the second-order system (28) is stabilized by controller (35) with adaptive rule (34).
On the other hand, to certify that the error of speeds in (22) tends to zero in finite time, an integral terminal sliding mode surface for the second-order system (29) is built: The motion controller for the second-order system (29) is proposed as including 0 < q < 1, σ 2 > 0: conversion coefficientsη 2 > 0 is the estimation value of η 2 , η 2 ≥ D 1 s 2 q placing η 2 =η 2 − η 2 as the error of the estimation valueη 2 The update value forη 2 is calculated as where µ 2 > 0 is the tuning coefficient. The Lyapunov candidate function is performed as and .
where D ≤ η 2 |s 2 | q By tuning m so that m − 1 ≥ 0, then . V 2 ≤ 0. Accordingly, in the sense of Lyapunov theory, the stability of the proposed controller (40) is confirmed. In addition, the second-order system (29) is stabilized by controller (40) with adaptive rule (41).

Discussions
In this section, our purpose is to confirm the suitability and effectiveness of the proposed approach. Moreover, the improved performance of this system could be revealed by means of different kinds of reference paths. Specifically, a circular path and line path are introduced as reference tracking trajectories, as shown in Figure 4. Using the physical dimensions and capability, Table 3 shows the initial setup values. For convenient simulation, the external disturbances are estimated as d 1 = 0.3 . v + 0.2v + ω + cos t.e −t and d 2 = 0.28 . ω + v + 0.3ω + sin t. The primary reference position of nonholonomic carrier is q R = (x R (0), y R (0), θ R (0)) = (2, 1, 0.25π) while it starts at q C = (x C (0), y C (0), θ(0)) = (3, 0, 0.4). The reference signals of angular speed and linear speed are v R = 3 m/s, ω R = 1.5 rad/s correspondingly. reference tracking trajectories, as shown in Figure 4. Using the physical dimensions and capability, Table 3 shows the initial setup values. For convenient simulation, the external disturbances are   To match with the practical scenario, a closed trajectory including different continuous segments is suggested as Figure 5. The setting control process is recalled as Table 4 that the autonomous carrier initializes at , respectively. At this stage, the total mass of the autonomous system consists of a carrier and load. More specifically, the outcomes of the tracking performance are presented in Figures 6 and 7. As the carrier must follow the desired path uninterruptedly, the motion controller should generate the pulse train to servo actuators. The comments from these values are that the carrier tracks less errors when it stays in a straight path, and the tracking errors peak at the corner or contiguous segment between lines. This is reasonable, as the mechanical carrier does not know the reference map in advance.   To match with the practical scenario, a closed trajectory including different continuous segments is suggested as Figure 5. The setting control process is recalled as Table 4 that the autonomous carrier initializes at q C = (x C (0), y C (0), θ(0)) = (0, 0, 0), the reference signals of angular speed and linear speed are v R = 2 m/s, ω R = 2 rad/s, respectively. At this stage, the total mass of the autonomous system consists of a carrier and load. More specifically, the outcomes of the tracking performance are presented in Figures 6 and 7. As the carrier must follow the desired path uninterruptedly, the motion controller should generate the pulse train to servo actuators. The comments from these values are that the carrier tracks less errors when it stays in a straight path, and the tracking errors peak at the corner or contiguous segment between lines. This is reasonable, as the mechanical carrier does not know the reference map in advance.       Competitive performance is a must. Our approach originates from both kinematic and dynamic models so that control rules are created. The other control algorithm in the literature [34], which is based on the adaptive nonsingular fast terminal sliding mode control (ANFTSMC) technique to drive the wheeled mobile robot, was chosen for comparison. The same conditions were equally employed in the two controllers. To visualize the advanced achievements, the entire path was segmented into sub-routes-a circle route and line route. The tracking performance of the position using ANFTSMC in both the circle and line route is illustrated in Figure 8. Intuitively, the output results suffer unexpected effects from chattering on the system signals. The actual trajectory in the circular path does not track well, while it tends to diverge in the tracking line path. For more detail, several results of the errors and velocities are demonstrated in Figure 9. In a circular trajectory, the robot seems to struggle to track the reference profile as it starts at any point in the x/y plane. However, the circle curve requires the ANFTSMC controller to change the control signals continuously. Consequently, the tracking errors fluctuate slightly and are too large to drive the wheeled robot. In the linear trajectory, the control performance of the ANFTSMC scheme is not as good as its desired signal.  Competitive performance is a must. Our approach originates from both kinematic and dynamic models so that control rules are created. The other control algorithm in the literature [34], which is based on the adaptive nonsingular fast terminal sliding mode control (ANFTSMC) technique to drive the wheeled mobile robot, was chosen for comparison. The same conditions were equally employed in the two controllers. To visualize the advanced achievements, the entire path was segmented into sub-routes-a circle route and line route. The tracking performance of the position using ANFTSMC in both the circle and line route is illustrated in Figure 8. Intuitively, the output results suffer unexpected effects from chattering on the system signals. The actual trajectory in the circular path does not track well, while it tends to diverge in the tracking line path. For more detail, several results of the errors and velocities are demonstrated in Figure 9. In a circular trajectory, the robot seems to struggle to track the reference profile as it starts at any point in the x/y plane. However, the circle curve requires the ANFTSMC controller to change the control signals continuously. Consequently, the tracking errors fluctuate slightly and are too large to drive the wheeled robot. In the linear trajectory, the control performance of the ANFTSMC scheme is not as good as its desired signal. Competitive performance is a must. Our approach originates from both kinematic and dynamic models so that control rules are created. The other control algorithm in the literature [34], which is based on the adaptive nonsingular fast terminal sliding mode control (ANFTSMC) technique to drive the wheeled mobile robot, was chosen for comparison. The same conditions were equally employed in the two controllers. To visualize the advanced achievements, the entire path was segmented into sub-routes-a circle route and line route. The tracking performance of the position using ANFTSMC in both the circle and line route is illustrated in Figure 8. Intuitively, the output results suffer unexpected effects from chattering on the system signals. The actual trajectory in the circular path does not track well, while it tends to diverge in the tracking line path. For more detail, several results of the errors and velocities are demonstrated in Figure 9. In a circular trajectory, the robot seems to struggle to track the reference profile as it starts at any point in the x/y plane. However, the circle curve requires the ANFTSMC controller to change the control signals continuously. Consequently, the tracking errors fluctuate slightly and are too large to drive the wheeled robot. In the linear   For comparative purposes, the same settings were employed in our controller. The output performance of the proposed scheme for both cases is described in Figure 10. In the case of a circular track, better performance could be achieved by our controller, while the chattering phenomenon was perfectly discarded. Accordingly, the driving actuators could receive the correct command from the motion controller. Although the line path was easier to manipulate for the whole system, the controller needed to be sensitive to changes and was able to drive the system back to reference trajectory. The driving performance of the tracking errors and velocities is depicted in Figure 11. It is obviously recognized that even though unknown uncertainties exist in the system, the proposed control strategy perfectly generates the driving command. In a short period, the velocity performance of our controller did not possess the changes suddenly and continuously. It is also important to note that the terms of tracking errors are always crucial factors for discovering a superior response. The adaptive rule might enhance our controller so as to overcome the uncertain parameters. A list of comparative statistics is synthesized in Table 5. Three tracking values are mentioned-maximum, minimum, and root-mean-square (RMS) error. As the autonomous carrier starts far from the reference trajectory, in the cases of both paths, two controllers obtained a maximum tracking error at 0.1. Later, the control commands would drive the system to tend to the reference path. In the linear segments, the ANFTSMC controller did not track well, while our controller performed with excellent results. In the curved trajectory, two control schemes produced the same tracking performance.

Conclusions
In this paper, a novel controller for MAC in the presence of uncertainties was introduced. In our work, the critical problems from real-world hardware are pointed out after handling in its distribution center or storehouse. Later, the potential solutions are studied to overcome these drawbacks. The contributions in this research could be summarized as following: -Decoupling formation from kinematic-dynamic model of MAC is formed. Instead of solving the complicated system, it is advised to work with two second-order sub-systems.

Conclusions
In this paper, a novel controller for MAC in the presence of uncertainties was introduced. In our work, the critical problems from real-world hardware are pointed out after handling in its distribution center or storehouse. Later, the potential solutions are studied to overcome these drawbacks. The contributions in this research could be summarized as following: -Decoupling formation from kinematic-dynamic model of MAC is formed. Instead of solving the complicated system, it is advised to work with two second-order sub-systems. - The globally asymptotic stability of whole system is validated in the sense of Lyapunov theory. Therefore, the design controller and adaptive rules are derived. -Adapting with uncertain parameters, the proposed controller achieves excellent results in both the circular segment and linear segment. The competitive performance is also verified carefully to prove the effectiveness and correctness of our approach in agricultural warehouses. Funding: This research received no external funding.