Research of a gantry robot for pushing bulk materials loaded in portions into a container

Abstract

Introduction:

Loading bulk materials into a container through a small upper hatch results in the formation of a cone-shaped pile that must be redistributed after each portion to ensure uniform filling. This paper presents the design, kinematic modeling, and experimental validation of a 4-DoF gantry robot developed for pushing and redistributing bulk materials inside a container under severe spatial constraints. A specialized end-effector, implemented as a rotatable rod with forward-facing blades, enables planar redistribution of the deposited material.

Methods:

A complete kinematic model based on the Denavit–Hartenberg method was derived for reliable control of the end-effector position and orientation. Analytical relationships were derived that determine the redistributed layers using geometric methods. Based on these analytical relationships, a redistribution algorithm was developed that takes into account the cone height, layer thickness, material density, and container geometry.

Results:

Analytical relationships describing the geometry of the redistributed layers were obtained, enabling the development of an algorithm for pushing material parallel to the container bottom. A physical prototype was fabricated, and experimental studies were conducted to measure the normal stresses acting on the pusher rod during material displacement.

Discussion:

The results confirm the effectiveness of the proposed robot design and control approach in achieving uniform redistribution of bulk material loaded in portions within a constrained container environment.

1 Introduction

In industrial enterprises that involve loading bulk materials into containers, there is a growing need for robotic systems to automate this process (Kang and Kim, 2019). Typically, loading is performed from above through a small hatch in several successive portions. Each portion of bulk material entering the hatch forms a cone-shaped pile, which must be redistributed after every load to ensure uniform filling of the container. Usually, after the container is fully loaded with several portions, sampling is carried out, which should include bulk material from all portions. Thus, the loaded portions are required to be evenly distributed parallel to the bottom of the container. The main difficulty arises from the fact that the hatch is located in the center of the container’s top surface, has a limited size, and restricts the motion of the robot’s end-effector inside the container. Therefore, the automation of bulk material loading through the upper hatch requires the development of a special robot capable of pushing and redistributing the cone-shaped bulk material under severe motion constraints.

The most suitable design choice to perform the technological operation of pushing through the hatch of the container is the gantry robot. One of the advantages of a gantry robot is that the container has the form of a rectangular parallelepiped, which coincides with the working area of the gantry robot. This makes it easier to plan the trajectories of the gantry robot’s end-effector inside the container. In addition, the bulk material pushing requires significant horizontal efforts, the gantry robot’s frame provides constant high rigidity throughout the working area.

Gantry robots and Cartesian manipulators are widely used in industrial automation due to their structural simplicity, high rigidity, and suitability for large workspaces. Their kinematic properties and control strategies have been extensively studied. Ibrahim et al. developed and evaluated a construction-oriented gantry robot for bricklaying, demonstrating the benefits of gantry architectures for precision material placement (Ibrahim et al., 2020). Kamargaonkar and Tiwari reviewed the design of X–Y gantry mechanisms and highlighted challenges in implementing accurate motion control with low-cost components (Kamargaonkar and Tiwari, 2019).

Significant attention has been devoted to modeling the geometric and kinematic accuracy of gantry systems. Marinescu and Nicolescu proposed an analytical method combined with FEM to estimate volumetric errors, emphasizing the importance of structural stiffness in Cartesian manipulators (Marinescu and Nicolescu, 2016). Sánchez and Cortés developed a Cartesian control scheme for robotic manipulators capable of ensuring stable translational motion under external disturbances (Sánchez-Sánchez and Reyes-Cortés, 2021). The use of the Denavit–Hartenberg (D–H) method for gantry robot modeling, as presented in (Perez et al., 2019; Yu and Zhang, 2021), is now recognized as a standard approach for describing link transformations and computing end-effector positions.

Beyond classical modeling, alternative navigation and perception-based control approaches have been explored. Beites et al. implemented a computer vision system for gantry robot positioning, suitable for tasks where objects lie within a planar workspace (Beites et al., 2019). Hauschultz et al. adapted a portal milling machine into a low-cost robot system, illustrating the potential of gantry platforms for laboratory automation (Hauschultz et al., 2020).

While these studies provide important insights, they predominantly assume an unrestricted manipulator workspace and constant external loads acting on the end-effector. In contrast, robotic systems interacting with bulk or granular materials face highly variable resistive forces and require dedicated mechanical solutions. Several works have explored material pushing and redistribution mechanisms. Tripathi et al. investigated robotic leveling of granular media with controlled blade motion (Tripathi et al., 2020; Tripathi and Dhar, 2022), while Foster and Yamada analyzed stress peaks during scraper–granular interaction (Foster and Yamada, 2018). Similar conclusions were reached by Alavi and Schmidt, who examined energy-efficient scraping mechanisms and highlighted stick–slip effects under varying load conditions (Alavi and Schmidt, 2020). Research related to the distribution of material in confined spaces is limited. Kutter et al. proposed a linear robotic actuator for redistribution in constrained environments (Kutter et al., 2021), and Bortolini et al. studied automated filling strategies for industrial containers, though neither addressed redistribution through a small hatch (Bortolini et al., 2019). Chen et al. investigated motion planning for robots interacting with granular media, reinforcing the need for accurate modeling of resistive forces during immersion (Chen et al., 2021). Yu and Zhang presented kinematic optimization strategies for large-workspace Cartesian robots, which inform the design of manipulators operating under geometric limitations (Yu and Zhang, 2021).

Despite these developments, previous designs do not account for the strong kinematic constraints and variable resistive forces encountered when a robot operates inside a closed transport container through a small hatch. Previous studies have not considered the application of a gantry robot for top-hatch containers. Moreover, existing studies do not provide a robotic solution capable of redistributing bulk material portions in a confined internal volume while maintaining reliable control of the end-effector and minimizing stress loads. To address this, the authors have developed a gantry robot equipped with a specialized end-effector designed to push bulk material within such containers.

Thus, the literature reveals a clear gap: there are no studies proposing a gantry robot capable of redistributing bulk materials loaded in portions inside a container with restricted workspace and variable material resistance. The present work closes this gap by introducing a 4-DoF gantry robot with a rotatable blade-type end-effector, developing its D–H-based kinematic model, and experimentally validating its ability to redistribute bulk material portions under constrained conditions.

In this paper, we investigate a gantry robot whose end-effector operates directly inside a container and is therefore subject to strict kinematic limitations. Moreover, the load applied to the end-effector is not constant but varies depending on the degree of immersion in the bulk material. These conditions considerably complicate the problem and necessitate the development of a dedicated software control system. Automation of the bulk material loading process eliminates the need for manual redistribution of material after each loading cycle, making it a pressing task for industrial enterprises. To address this, a rotating end-effector was selected, capable of performing pushing motions both longitudinally and transversely within the container. The designed end-effector is unique: it consists of several horizontally mounted blades attached to a vertical shaft, which rotates around its axis. This configuration enables the pushing of bulk material in different planar directions, ensuring uniform and complete filling of the container.

The novelty of this work lies in the development of a control algorithm that governs the pushing of bulk material portions parallel to the bottom of the container, tailored specifically for the control system of a gantry robot performing operations under spatial restrictions. A new design of the end-effector in the form of a rotary blade is proposed.

2 Methods and materials

2.1 Kinematic analysis of a gantry robot for pushing bulk materials

A gantry robot is a type of Cartesian robot distinguished by its suspended structure with a movable bridge (or beam) that enables movement along the X, Y, and Z-axes. Typically, gantry robots are installed above the workspace, which is why they are referred to as “portal” robots. Since the gantry robot employs two parallel X-axes (base axes), the load moments generated by the Y and Z-axes, as well as the working payload, are transferred as forces acting on the X-axis. This design greatly increases system rigidity and, in most cases, allows the axes to achieve longer stroke lengths and higher speeds compared to conventional Cartesian robots.

To design a gantry robot for pushing bulk materials, we consider a configuration that combines a three-degree-of-freedom Cartesian manipulator with an additional rotational degree of freedom. In this scheme, the Cartesian manipulator provides reliable positioning of the end-effector along the X, Y, and Z-axes, while the rotary joint enables rotation of the end-effector about the Z-axis. The resulting kinematic structure can be described as a 3P1R system (three prismatic joints and one revolute joint), illustrated in Figure 1. The mechanism includes three linear links (1, 2, 3) and one rotary link (4), mounted on a rectangular base frame (0). The end-effector (5) is rigidly fixed to the rotary link (4).

FIGURE 1

The proposed gantry robot has four degrees of freedom (4-DoF). The first three prismatic links provide linear motion of the end-effector along the X, Y, and Z-axes, while the revolute link adjusts its orientation about the Z-axis. The end-effector (5) is designed as a rectangular pusher with three blades. Such a 3P1R structural scheme allows the end-effector to be positioned at any point inside the container and rotated as required to redistribute or push aside the bulk material.

The gantry robot for pushing bulk materials is a 4-DoF system capable of executing three translational motions and one rotational motion. Its kinematic diagram is presented in Figure 2.

FIGURE 2

This kinematic scheme makes it possible to describe both the positioning of the end-effector point P inside the container and the orientation adjustment of the end-effector, which is necessary for pushing and redistributing bulk materials.

Structural constants: — the base spans of the guides, the horizontal link , the vertical link of the end effector (pointing down) and the end effector span .

To strictly follow the rules for constructing a standard D-H table (where the axis is the direction of movement of the ith link), we introduce the basic transition matrix from the world system to the zero-system D-H (Equation 2):

This aligns the axis with the direction.

In standard D-H notation, the transition from the system to is defined by four parameters: . The complete set of D–H parameters for the 4-DoF gantry robot is summarized in Table 1.

2.1.1 Homogeneous transformation matrices

The General form of the matrix D-H (Equation 3):

Let’s substitute the parameters from the table for each link:

Matrix (transition ) (Equation 4)

Matrix (transition ) (Equation 5)

The product returns the orientation of the axes to their original world position (Equation 6):

Matrix (transition ) (Equation 7)

Matrix (transition , end effector) (Equation 8)

2.1.2 A direct kinematics problem

The resulting matrix of end effector position and orientation in the absolute system (Equation 9):

From this we obtain the final equations of the direct kinematics problem for the end effector coordinates (Equation 10):

Below is an interactive kinematics calculator where you can change the coordinates of the hinges and visualize the position of the end effector in real time:

2.1.3 The inverse problem of kinematics

Since robot type is gantry, the problems are decoupled along the axes, and the system of equations has a single analytical solution for any given rotation angle of the end effector .

2.2 3D model of a prototype gantry robot for pushing bulk materials

To develop a 3D model of a prototype 4-DoF gantry robot designed for pushing bulk materials, the following rectangular frame dimensions were adopted: length – 0.765 m, width – 0.7 m, and height – 0.45 m. The ranges and rates of angular and translational motions of the end-effector are summarized in Table 2.

The end-effector of the prototype 4-DoF gantry robot is capable of smooth motion throughout the workspace. The linear drives along the Z, X, and Y-axes provide the required translational movements. The rotary link of the end-effector, aligned parallel to the Z-axis, is mounted at the base of its linear guide.

Figure 3 illustrates the 3D model of the prototype 4-DoF gantry robot for pushing bulk materials, developed in Autodesk Inventor. The following designations are used: 1 – frame, 2 – stepper motors, 3 – linear guide with X-axis ball screw, 4 – linear guide with Y-axis ball screw, 5 – linear guide with Z-axis ball screw, 6 – rotary link for rotating the end-effector around the Z-axis, 7 – end-effector, 8 – container.

FIGURE 3

The prototype of the 4-DoF gantry robot for pushing bulk materials operates as follows (see Figure 3). To perform the pushing task within container 8, the trajectory of the end-effector 7 is predefined in the computer. Control signals are transmitted from the computer to the drivers of the three stepper motors 2, enabling the end-effector 7 to execute translational motions along the X, Y, and Z-axes via the linear guides with ball screws 3, 4, and 5. In addition, according to the control commands from the computer, the rotary link 6—driven by a stepper motor 2—provides angular orientation of the end-effector 7 around the Z-axis.

2.3 Prototype of a gantry robot for pushing bulk materials

Based on the developed 3D model of the gantry robot for pushing bulk materials, a physical prototype was manufactured. The main structure of the 4-DoF prototype follows a classical scheme with three translational motions along three mutually perpendicular axes. The end-effector is designed as three parallel blades rigidly connected by rods. This configuration was chosen to enable efficient displacement of bulk materials away from the hatch under restricted movement conditions. Since the operation requires pushing bulk materials both longitudinally and transversely within the container to achieve uniform loading, the end-effector is mounted on a rod capable of rotation about the vertical axis.

The prototype frame measures 0.765 × 0.7 × 0.45 m and is fabricated from metal (see Figure 4). To realize the translational motions, ball-screw drives are employed. Each ball-screw has a shaft diameter of 0.012 m and a screw pitch of 0.01 m. The ball-screws are actuated by NEMA 23 stepper motors with gearboxes. The NEMA 23 motor provides a torque of up to 0.101 Nm, offers reliable long-term operation under industrial conditions, and allows reliable positioning and speed control without the need for feedback sensors, if torque and speed remain within the rated limits. The control system consists of stepper motor drivers, a power supply unit, and an Arduino MEGA 2560 controller housed in a control cabinet. Two non-contact inductive sensors of type SN04-P are installed on each axis for calibration and travel limitation. Since the end-effector of the 4-DoF gantry robot operates inside the container under strict spatial constraints, a rotating end-effector design was implemented to enable pushing of bulk material both along and across the container.

FIGURE 4

Communication between the sensors and the actuators is implemented via an Arduino board. Six limit switches (two per axis) are connected to the Arduino to define the start and end positions of each axis for coordinate referencing. The stepper motors are driven through dedicated drivers: the Arduino transmits control signals to the drivers, which in turn generate the motor actuation. Operator commands are sent from the computer to the Arduino through the COM port, using a custom interface developed in C# within the Visual Studio environment. Figure 5 presents the wiring diagram of the control system of the prototype gantry robot for pushing bulk materials.

FIGURE 5

Feedback is implemented using limit switches. Each axis is equipped with two limit switches, which define the start and end of the coordinate reference range.

2.4 Algorithm for pushing the cone of bulk material poured into the container

FIGURE 6

To develop an algorithm for pushing material portions parallel to the bottom of the container, the dependencies between the thickness of the redistributed portion, its height from the container bottom, the cone height formed after filling, the geometric parameters of the container, and the mass and density of the material were established.

The angle of inclination of the end walls of the container is given by:

The length along the front wall of the line X^’ for a layer located at a distance h from the container bottom is expressed as:

The volume of a filled layer of thickness h is given by:

The volume of the filled material with mass m and density ρ is defined as:

Equations 14 and 15 yield:

By substituting Equation 14 into Equation 17, we obtain a quadratic equation for determining the thickness of the redistributed layer after filling:

The cone height is calculated using the cone volume equation:where is the angle of repose. The gantry robot was designed to handle a specific material with a predetermined density. If the density varies, this is accounted for in Equation 17. With constant mass input, a decrease or increase in density results in a corresponding increase or decrease in cone volume, directly affecting the robot’s productivity.

The container capacity is 80 kg of bulk material. In practice, the container is not completely filled, leaving room for pushing the top layer. For this purpose, a filling factor of 0,9 is selected. Thus, the maximum filling is 72 kg. In production, 5 portions of bulk material are usually filled in. For container sizes , the total container volume is . During the experiment, five portions of bulk material (sand, density ρ = 1,500 kg/m³) were sequentially loaded into the container, while the weight of each portion was 14.4 kg.

Using the above equations, the thickness of each redistributed layer, the height of the redistributed layer from the bottom of the container, and the height of the cone tip were calculated. The height of the cone obtained by Equation 19 was 0.143 m.

FIGURE 7

FIGURE 8

The control system of the prototype gantry robot designed for pushing bulk materials was implemented in the Arduino development environment. The program code is written in C++. The movement of the end-effector is provided by stepper motors, the control of which is carried out according to the calculations based on Equations 13–19. A block diagram of the algorithm for pushing bulk material inside the container is presented in Figure 9.

FIGURE 9

3 Results

3.1 Algorithm experimental determination of the force of bulk material on the blade during pushing

In the prototype of the gantry robot, the pusher is designed in the form of three blades fixed on the pusher rod (Figure 10a). The rod can be rotated around its axis using a stepper motor, which allows the blades to push both along and across the container. The bulk material poured into the hatch of the container forms a conical pile. To distribute it evenly, the gantry robot begins leveling the cone by moving the pusher along the X- and Y-axes parallel to the container bottom, starting from the upper layers of the cone.

FIGURE 10

To determine the force acting on the pusher blades during operation, the ZET 058 strain gauge system (ZETLAB Company, 2026) together with strain gauges was applied. A strain gauge was glued at the upper point of the pusher rod, where the maximum stress occurs during contact with the material (Figure 10a).

The strain gauge was connected to the ZET 058 strain gauge station, which, together with the ZETLAB TENZO software (version 2020.11.30) (ZETLAB Company, 2026), enabled real-time data acquisition from the sensor.

The experimental studies were conducted according to the methodology recommended by the ZETLAB company. Strain gauge calibration was also carried out following the ZETLAB procedure. In the process of using strain gauges, it is required to complete a calibration table, an example of which is shown in Figure 10b.

The calibration of the strain gauge was performed through the settings of the calibration field in the ZETLAB software. In the Calibration File field, the path to the previously saved configuration file and calibration table must be specified. By clicking the button next to this field, a standard file selection window opens, where the user can select the required calibration file. By default, calibration files are stored in the directory C:\ZETLab\config\ and have the extension *.clb. The Calibration Table block provides tools for creating, editing, and saving calibrations to be used in experimental studies.

To complete the calibration table for the strain gauge, reference loads were applied to the pusher using a digital scale (model X046, Guangzhou Huayuan Electronics Co., Ltd.; resolution 0.01 kg), as shown in Figure 10c. The strain gauge had a nominal resistance of 350 Ω and a sensitivity of 2.003 mV/V, and was connected in a full-bridge configuration with a 10 V supply voltage. The load was applied in increments of 0.05 kg up to a maximum of 0.5 kg. The calibration data yielded the exact linear regression formula , where is the applied weight in kg and is the output signal in mV (Table 3).

Quantitative comparisons were made using the strain gauge calibration tables obtained with software ZETLAB. The results demonstrated the absence of significant errors. According to the ZETLAB methodology, when measuring the normal stress in a rod using a properly calibrated strain gauge, the deviation between repeated measurements remains small. Table 4 presents the peak of normal stress depending on the pusher speed and the depth of blade immersion in the bulk material. The data presented in Table 4, based on three replicates (n = 3) for each condition, represent descriptive repeatability statistics of the steady-state peaks—excluding the initial high-stress acceleration phase—as no inferential statistical analysis was performed.

Figures 11–13 present the graphs of normal stress variations over time, depending on the pusher speed and the depth of blade immersion in the bulk material.

FIGURE 11

FIGURE 12

FIGURE 13

The developed robot was designed specifically for working with dry sand. In the future, it is planned to investigate the influence of moisture content on the pressure force acting on the pusher blades, as well as to extend the experiments to other bulk materials.

As can be seen from Figures 11–13, the experimental curves of the normal stresses recorded at the frame rod show a peak at the beginning of motion, during motor acceleration to the required speed. This effect occurs because the blades must initially shift the entire mass of material in front of them. With continued motion, the load decreases, as the bulk of the material is displaced mainly by the first blade, resulting in lower stress values after the initial peak. Periodic stress fluctuations are also observed in Figures 11–13, indicating that the pusher advances in jerks. This behavior is explained by friction between the blades and the bulk material, which causes the pusher rod to bend under resistance forces. When the accumulated bending energy of the rod, combined with the drive energy, exceeds the resistance of the material, a sudden forward movement occurs.

All the initial stress data shown in graphs 11–13 have been processed. The initial peak was excluded, and the steady-state oscillation frequency and average peak-to-peak amplitude were calculated for each graph. Table 5 shows a quantitative assessment of the slip dynamics at various operating parameters. The data presented in Table 5, based on three replicates (n = 3) for each condition, represent descriptive repeatability statistics of the steady-state peaks—excluding the initial high-stress acceleration phase—as no inferential statistical analysis was performed.

The results confirm the dynamics of grip and slippage.

Frequency and speed observed trend: as the pusher speed increases, the oscillation frequency increases and the stress amplitude decreases. Higher speeds provide less time for elastic bending energy to accumulate in the rod before it overcomes the static friction of the bulk material.

Amplitude to penetration depth ratio: Greater penetration depth increases the resistance force, requiring more accumulated bending energy to initiate slippage, resulting in higher stress amplitudes.

Experimental tests on pushing bulk material in the container have shown that both greater penetration depth of the blades and higher pusher speed lead to an increase in normal stresses.

To determine the maximum total force acting on the blades, the pusher was modeled as a cantilever beam rigidly fixed at the top of the rod and subjected to a transverse maximum force . In this analysis, deformations of the rods holding the blades and of the blades themselves were neglected, as the deformation of the supporting frame is small and follows Hooke’s law.

The maximum force can be calculated by the following formulas:where – elastic bending section modulus of the circular section of the pusher rod;

is the diameter of the circular section of the pusher rod.

The bending force of the pusher rod is determined by the following formula:where is the length of the pusher rod.

Then the maximum force acting on the pusher when pushing the bulk material is determined by the following formula:where is the maximum normal stress, is the elastic bending section modulus of the pusher rod, and is the rod length.

4 Discussion

This work presented the development, kinematic modeling, and experimental evaluation of a 4-DoF gantry robot designed for redistributing bulk materials loaded in portions into a container through a confined upper hatch. A D–H method based kinematic model was established to ensure reliable end-effector positioning, and analytical dependencies governing the geometry of redistributed layers were derived. A redistribution algorithm accounting for cone height, layer thickness, material density, and container geometry was formulated.

To check the kinematics of forward movement and evaluate the accuracy of the gantry robot positioning in real conditions, point-to-point motion in free space was tested. The linear axes of the gantry are driven by stepper motors connected to ballscrews with 10 mm pitch. Considering the motor step angle of 1.8° (200 steps/revolution), a direct drive configuration (1:1 gearbox ratio), and a full-step driving mode (no microstepping), the theoretical mechanical resolution of the system is 0.05 mm per step. To quantify the actual positioning error, we moved to five different spatial coordinates (X, Y, Z) covering different areas of the workspace. The actual physical positions were measured at idle to evaluate the observed positioning error without structural deflection caused by material resistance. The absolute errors are shown in Table 6.

As can be seen from Table 6, the maximum Euclidean positioning error is 0.37 mm. For the targeted alignment and redistribution of granular bulk materials, this observed precision ensures that the specified movement trajectories are adequately executed without significant spatial displacement.

A physical prototype was fabricated and tested. Experimental results showed that the highest normal stresses occur at the beginning of blade motion during initial material displacement, while subsequent motion exhibits characteristic stick–slip behavior caused by friction and elastic deformation of the pusher rod. The results of the pusher steady-state movement analysis confirm the dynamics of grip and slippage. Frequency and speed observed trend: as the pusher speed increases, the steady-state stress amplitude increases. Higher movement speeds cause a greater dynamic impact force against the bulk material before the static friction yields, leading to higher peak-to-peak bending stresses on the rod. The oscillation frequency generally increases or remains high at greater speeds. Amplitude to penetration depth ratio: Greater penetration depth increases the resistance force, requiring more accumulated bending energy to initiate slippage, resulting in higher stress amplitudes.

The results confirm that the proposed robot design and control strategy ensure effective leveling and redistribution of bulk materials under constrained workspace conditions.

Automated pushing primarily serves to protect human operators from potential harm and to accelerate the overall process. Table 7 outlines further advantages in detail.

The experiments were conducted using only dry sand, with a density of 1,500 kg/m³, as required by the future customer. The geometry of the container is also related to the requirements of the enterprise where we intend to implement this robotic complex.

Future work will focus on incorporating sensor feedback, analyzing the influence of material moisture, and optimizing the pushing trajectory to reduce peak loads and energy consumption during operation.

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