This paper presents the kinematic and force analysis of a pantograph-based industrial gripper designed for transportation cylindrical parts in automated systems. The gripping mechanism was modeled and validated using numerical simulations in Mathematica and SolidWorks Motion. The results show that the gripper provides stable and accurate operation within the design operating range, applying a maximum gripping force of 45 N. Compared to conventional gripping mechanisms, the proposed design provides improved force distribution and efficient energy utilization. The study confirms the effectiveness of the pantograph-based gripper for industrial automation applications that require stable and controllable gripping forces.
Automated lines in factories significantly increase production efficiency, ensuring accuracy, speed
and minimizing human intervention in technological processes. Thanks to the integration of
robotic manipulators, conveyor systems and software control, modern production lines are able to
adapt to changes in production requirements, reducing costs and improving the quality of the final
product [
Industrial robots equipped with grippers are widely used in manufacturing, logistics, and
assembly lines to handle components, optimize production processes, and reduce human
intervention [
Robotic grippers can be classified into several categories based on their actuation mechanisms
and operating principles. Namely, they are mechanical grippers, pneumatic grippers, and
electromechanical grippers. They are key elements of modern industrial manipulators and are
widely used in manufacturing processes for automated gripping, moving and stacking of objects
[
Although existing gripping technologies have been successfully integrated into various
industrial applications, they have certain limitations and development prospects [
The choice of gripper design largely depends on the specifics of the tasks performed, the shape
of the object, the requirements for accuracy and stability of the gripper, as well as on energy
efficiency and production costs [
Modern developments of grippers are aimed at optimizing the kinematics of the mechanism,
improving controllability and increasing adaptability to different types of objects, which allows for
effective performance of "pick and place" tasks in production processes [
For example, pneumatic grippers demonstrate high speed but require additional compressor
systems and have limited grip force and position control capabilities [
Research [
Research work [
Despite significant progress in the development of robotic grippers, research in this field reveals
a number of problem areas that need to be further addressed. Precise control of gripping force
remains a challenge due to nonlinear effects such as friction and airflow dynamics in pneumatic
grippers. Furthermore, while finger position control has reached an acceptable level in electric
grippers, pneumatic mechanisms still suffer from poor controllability in both force and positioning,
which limits their effectiveness [
The development of robotic grippers requires the introduction of new design concepts, including flexible mechanisms, topology optimization, material selection, and drive methods, to ensure grip reliability and adaptability.
The main objective of this research is to develop a high-precision, energy-efficient and adaptive robotic gripper. The proposed design includes a pantograph-based mechanism combined with a lead screw drive, which provides precise force control and flexible adaptation to objects of different sizes. By integrating this design, we aim to overcome the limitations of traditional grippers and provide an effective solution for industrial automation applications.
The design and performance of robotic grippers is based on an effective combination of kinematic structure, force distribution, and energy efficiency. This section outlines the methodological framework used to design and analyze the proposed gripping mechanism. The study includes kinematic modeling, force analysis, and numerical verification to evaluate the adaptability and effectiveness of the gripper. Computational simulations were performed in Mathematica and SolidWorks Motion to verify the theoretical conclusions. 2.1.
For this research, a stationary robot (Figure 1) was designed to perform reloading operations based on a SCARA-type manipulator and a pantograph gripper. The robot in Figure 1 is mounted on a vertical column 1, which is attached to the foundation. A mounting plate 2, with several holes for bolted connections, is used to install the robot on the column. Four clamps 3 are also installed on plate 2 to fix the vertical guides 4 of the vertical lifting mechanism of the manipulator. The drive of this mechanism is carried out from a stepper motor 5, through a toothed belt transmission 6 to a screw shaft 7. For the upper fixation of guides 4 and shaft 7, plate 8 with a set of clamps 3 is used. The screw 7 is installed in the bearing supports and drives the nut 9 attached to the vertical lifting plate of the manipulator. Guide (linear) bearings 10 are also fixed on it, which provide vertical movement of the plate along four guides. To drive the first link 11 of the manipulator into rotational motion, stepper motor 12 and a gear-belt transmission 13 are used. Another electric motor 14 through a gear-belt transmission 15 ensures the rotation of the second link 16 of the manipulator relative to the first link 11 around the bearing units 17. At the free end of the link 16, mechanisms for rotation and vertical movement of the gripper are installed. The rotation of the gripper in the horizontal plane is carried out using a stepper motor 18 and a gear 19. The mechanism for vertical movement of the gripper 20 is driven by an electric motor 21 through a gear-belt transmission 22. At the same time, the gripper 20 can move along vertical guides 23.
The main object of design in this study is the gripper, the general view of which is shown in Figure 2. Through the mounting plate 1, it is connected to the rotary gear sector, which provides the possibility of rotation around the bearing assembly, which is installed on the free end of the second section of the SCARA-type manipulator. A stepper motor 2 is fixed on plate 1, which, through a system of one drive and two tension pulleys 3, drives the belt 4, which, in turn, ensures the vertical movement of the gripper plate 5 along the profile guide 6. The gripper itself is a lever pantograph (parallelogram) mechanism, the links of which are driven by the servomotor 7 through a screw-nut transmission 8. The gripper is designed to move cylindrical parts with a diameter of 10 to 40 mm and a length of 20 to 150 mm. The developed model of the gripper movement is based on the transmission of torque from the electric motor through the screw shaft to the nut A, which plays a key role in the functioning of the mechanism (Figure 3). Nut A is pivotally connected to four levers (DB, D1B1, GE, G1E1), which form a pantograph kinematic chain. This design ensures the vertical position of the gripper jaws, regardless of their degree of opening or closing. О2 С С1 D D1 B
OC – DB О2С1 – D1B1
DB – А
D1B1 – А Class 5 5 5 5 5 5 GE – А
G1E1 – А
GE – BEHK G1E1 – B1E1H1K1
where is the number of moving links of the mechanism, p5 is the number of single-moving kinematic pairs, p4 is the number of double-moving kinematic pairs.
Based on the structural model in Figure 3, it is established that the mechanism has 9 moving links and 13 single-moving kinematic pairs (Table 1). There are no higher kinematic pairs in the gripping mechanism, so p4 = 0. Thus, the DOF is W = 1. This means that the mechanism is characterized by one DOF.
To define class of the mechanism, we decompose the mechanism into mechanisms of the 1st class and structural groups in the reverse order of its assembly (see Figure 4).
To verify the class of structural groups of the gripper mechanism, its DOF was determined, taking into account 4 moving links (DG (A), DB, GE, BE) and 6 single-moving kinematic pairs (C, D, B, G, E, A) (see Figure 4) (W = 0).Thus, we can conclude that the gripper mechanism is indeed a class IV mechanism. Its structural formula is written as follows
І (O , C )← ІV ( BDGE ); І (O2 , C1)← IV ( B1 D1 G1 E1) . 5 5 5 5 5 (1) (2) (3)
Based on the structural analysis, it can be concluded that to uniquely determine the position of the gripper jaws (points K and K₁) at any given moment, it is sufficient to know the law of variation of a single generalized coordinate, specifically, the position of the lead screw nut. To analyze the kinematics of the gripper mechanism, we choose an inertial (stationary) reference frame, which is presented in the form of a flat Cartesian coordinate system with the center in the hinge O1, which is located on the axis of the screw of the gripper drive mechanism (Figure 5). The vertical axis Oy coincides with the axis of the drive screw. As a generalized coordinate, we will take the vertical position of the nut A is yA relative to the hinge O1. The initial data are the geometric parameters of all links and the coordinates of the hinges O and O2.
A kinematic analysis is conducted to establish the maximum and minimum distance between the gripper jaws depending on the movement of the drive mechanism nut. According to the adopted data of the study, the gripper should provide movement of cylindrical parts with a diameter of 10 to 40 mm.
Let's write the expressions for finding the coordinates of the hinges D and G (4) (5) (6) To find the coordinates of hinge C, we derive the following geometric relations (see Figure 5) x D=−l AL , y D= y A +lLD ; xG=−l AL , yG= y A−lLG . lOD=√ (( x D− xO)2+( yO− y D)2); α =arctg (( x D− xO)/( yO− y D)); γ =arccos (((lOC )2+(lOD)2−(lCD)2)/(2⋅ lOC⋅ lOD));
β =γ −α ; xC= xO−lOC⋅ sin β , yC= yO−lOC⋅ cos β .
Let's find the coordinates of points B, E, H, and K: xB= x D−( x D− xC )⋅ lDB / lCD , y B= y D−( y D− yC )⋅ lDB / lCD ; x E= xB , y E= y B−lBE ;
φ=π −θ ; x H = x E+lEH⋅ sin φ , y H = y E−lEH⋅ cos φ ;
x K = x H +lKH , y K = y H .
The last stage of the kinematic analysis is to determine the distance between the gripping jaws. Considering that the gripper consists of two symmetrically arranged pantograph mechanisms (Figure 4), we can state that the total distance between the jaws will be Δx K =2⋅ ∨ x K ∨. (7)
To confirm the theoretical assumptions and assess the performance of the developed gripping mechanism, computational and numerical modeling was carried out. 3.1.
Based on the developed solid-state model of the gripper (see Figure 2), designed in the SolidWorks software product, and analytical dependencies derived above, a simulation of the movement of the gripper jaws was performed in the Mathematica software product. For the simulation, the corresponding geometric parameters presented in Table 2 of the gripper links were set.
Based on the results of substituting the specified geometric parameters into the analytical expressions derived above, the trajectories of the gripper jaws were obtained (Figure 6) and a graph of the dependence of the distance between the jaws on the position of the nut of the drive screw mechanism are shown in Figure 7. section with a diameter of up to 40.8 mm when the gripper approaches from the end of the part (see Figure 8). The minimum distance between the jaws is zero when they are fully closed, with a distance from the nut to the mounting plate of about 21 mm. Considering the design of the gripper jaws, the results of solid-state simulation showed that when the jaws are fully closed, the minimum diameter of the part to be gripped is about 9.5 mm (see Figure 8). Thus, the results obtained confirm that the gripping mechanism provides the ability to work with cylindrical objects in a given range.
To verify the correctness of the derived analytical dependencies and the results of numerical modeling in the Mathematica software, virtual experiments of the movement of the gripper mechanism were conducted in the Motion application of the SolidWorks program, which is shown in Figure 9. As a result of moving the nut of the drive screw mechanism by a distance of 19 mm, each of the gripper jaws moved from their contact state by approximately 19.2 mm in the horizontal direction and by 15.1 mm in the vertical direction. The virtual trajectory of the jaws movement fully corresponds to the theoretically derived trajectory. The maximum distance between the jaws in the open state of the gripper is about 38.4 mm, which practically coincides with the data of mathematical modeling.
To conduct the force analysis, the required gripping force necessary, for the secure transportation of the object within the gripper jaws, is determined. This estimation is based on the assumed mass and weight (Eq. 8) of a cylindrical workpiece with maximum dimensions (dmax=40 mm, lmax=150 mm), manufactured from structural steel St3 (GOST 380-2005), which has similar characteristics to S235JR (EN 10025-2). The analysis aims to ensure reliable fixation and stability of the object during handling and transportation.
mmax=( π (dmax)2)/ 4 ∙ lmax ∙ ρ=3.14 / 4 ∙ 0.15 ∙ 7850 ≈ 1.5 kg ;
Gmax=mmax ∙ g=1.5 ∙ 9.81 ≈ 15 N , where ρ =7850 kg/m2 is steel density, g = 9.81 m/s2 is the acceleration due to gravity.
Knowing the material of the gripper sponges is polyurethane rubber, we can determine the
approximate coefficient of friction between the surfaces of the sponge and the part. The coefficient
of friction between polyurethane rubber and steel St. 3 can vary depending on specific conditions,
such as surface cleanliness, the presence of lubricant, temperature and other factors. In general, for
dry conditions the coefficient of friction between polyurethane rubber and steel is approximately f
= 0.5...0.7. [
Taking the coefficient Kn of possible overloads up to 150% in the process of accelerated lifting of parts, we will determine the necessary pressing force of the gripper jaws from the condition of ensuring the necessary friction force (8) (9) (10) (11) (12) (13)
N pr=( K n ∙ Gmax)/ f =(1.5 ∙ 15)/ 0.5=45 N .
To ensure the possibility of designing individual reactions in the links of the gripper mechanism, we will additionally determine the angles λ, χ and ψ using Eq. 10. (see Figure 10) λ=arctg (( yO− y D)/( x D− xO)); χ =arccos (((lCD)2+(lOD)2−(lOC )2)/(2 lCD lOD));
ψ = χ −ψ .
Let us write the equation of equilibrium for the lever BEHK (see Figure 10)
∑ Fkx=0 :− N pr + REG ∙ cosψ − X B=0 ; ∑ Fky=0 :−0.75 Gmax + REG ∙ sinψ +Y B=0 ;
∑ M B ( Fk )=0 : − N pr ∙(l(BE)+lEH ∙ cosφ +lKM )+ REG ∙ cosψ ∙ lBE−0.75 ∙ Gmax ∙(lEH ∙ sinφ +lKH )=0.
Solving this system of equations (Eq. 11), we can determine the unknown reactions in the hinge B and in the rod EG
REG=( N pr ((l(BE)+lEH ∙ cosφ +lKM )+0.75 Gmax ∙(lEH ∙ sinφ +lKH )))/(cosψ ∙ lBE);
X B=− N pr + REG ∙ cosψ ;
Y B=0.75 ∙ Gmax− REG ∙ sinψ .
Let us write the equilibrium equation for the lever BCD (see Figure 10) ∑ F kx=0 : X D+ RCO⋅ sin β + X 'B=0 ; ∑ F ky=0 : Y D+ RCO⋅ cos β −Y 'B=0 ; ∑ M D ( F ⃗ k )=0 : X 'B⋅ lDB⋅ cos ψ − RCO⋅ sin ( π −γ − χ )lCD+Y 'B⋅ lDB⋅ sin ψ =0.
Solving this system of equations (Eq. 13), we can determine the unknown reactions in the hinge D and in the rod OC:
RCO=( X 'B⋅ lDB⋅ cos ψ +Y 'B⋅ lDB⋅ sin ψ )/(sin ( π −γ − χ )⋅ lCD);
X D=− RCO⋅ sin β − X 'B ; Y D=− RCO⋅ cos β +Y 'B . (14) ∑ Fky=0 :−Y 'D− RG⋅ sin ψ +Y (O1)=0 ;
Y (O1)=Y 'D + RG⋅ sin ψ . (15)
In the above formulas, the reactions XB, YB, YD, RGE are equal in magnitude to the corresponding reactions XB, YB, YD, REG but opposite in direction.
Substituting the calculated values of the active forces GHmax and Npr=45N into the analytical expressions derived above, we can build graphical dependences of the reactions in the hinges and rods of the pantograph mechanism of the gripper on the position yA of the nut of the drive screw mechanism. For this, we will use the geometric parameters of the mechanism adopted in our research and the Mathematica software, and additionally the following size lKM=12.5mm will be specified.
The simulation results are shown in Figure 11. As can be seen, the horizontal reaction in the hinge B practically does not change when the adjusting nut is moving, and it is about 120 N. The vertical reaction in the hinge B reaches a maximum value of about 230 N. The reaction in the rod O1C during the jaws tightening process decreases from 300 N to –230 N. The rod EG is constantly under the action of a tensile force, which varies in the range from 165 N to 290 N. The horizontal reaction in the hinge D during the jaws movement is in the range of 120…200 N, while the vertical reaction reaches 280 N when the jaws are spread apart and goes to zero when they are brought together. The vertical force on the screw of the drive screw mechanism during the jaws tightening process changes from –280 N to 240 N, causing both the screw to stretch and compress (depending on the jaws position). The obtained dependencies were used in the process of designing the corresponding hinges and levers of the gripper in order to ensure its strength during operation.
Despite significant progress in the development of pantograph grippers for robotic manipulators, a number of challenges remain that need to be addressed to increase their efficiency, adaptability, and reliability. One key aspect is improving the control of the gripping force without the use of complex sensor systems. The introduction of intelligent control algorithms based on machine learning and adaptive models will allow automatically adjusting the gripping force depending on the shape, material, and mechanical properties of the object [15, 16,17, 18].
Another important area of development is optimizing the gripper design to achieve greater flexibility and versatility. In particular, the use of composite materials and additive manufacturing technologies opens up new opportunities for reducing the structure's weight and improving its mechanical characteristics.
In general, the future development of pantograph grippers will be aimed at combining advanced design solutions with flexible control methods, which will ensure their high performance, reliability, and adaptability to a wide range of tasks in industrial robotics.
In this study, robotic gripper design was developed, analyzed, and optimized, providing efficiency, adaptability, and ease of control. It was found that the position of the gripper jaws can be uniquely determined by a single generalized coordinate, which significantly simplifies mathematical modeling and control algorithms.
The results of kinematic and force analysis confirmed the stability and accuracy of the proposed design, which makes it promising for automated production systems and robotic manipulators in industries where reliable and controlled fixation of objects is required. The proposed approach can be used for further improvement of industrial robotic grippers, as well as the integration of intelligent control algorithms for adaptive manipulation of objects of complex shape.
Thus, the analysis and modeling confirmed the effectiveness of the proposed design solution, which provides stability, accuracy, and reliability of gripping in automated production systems.
The authors have not employed any Generative AI tools. [15] R. Kumar, U. Mehta, P. Chand, "A Low Cost Linear Force Feedback Control System for a Twofingered Parallel Configuration Gripper," Procedia Computer Science, vol. 105, 2017, pp. 264-269. doi: 10.1016/j.procs.2017.01.220. [16] X. Zhang, Q. Xu, "Design and testing of a novel 2-DOF compound constant-force parallel gripper," Precision Engineering, vol. 56, Mar. 2019, pp. 53-61. doi: 10.1016/j.precisioneng.2018.09.004. [17] P.-L. Chang, I.-T. Chi, N. D. K. Tran, D.-A. Wang, "Design and modeling of a compliant gripper with parallel movement of jaws," Mechanism and Machine Theory, vol. 152, Oct. 2020, Art. no. 103942. doi: 10.1016/j.mechmachtheory.2020.103942. [18] T. Takahashi, K. Yamazaki, S. Saito, H. Kobayashi, "Design and Control of Parallel Gripper with Linear and Curved Trajectory Consisting of Only Revolute Pairs," 2020 IEEE/SICE International Symposium on System Integration (SII), Honolulu, HI, USA, 2020, pp. 557-562. doi: 10.1109/SII46433.2020.9025997.