Engineering Study on the Coupled Design of End-Effector Plate Thickness, Gripping Force, and Safety Margin

Jiangsu Aurek Intelligent Technology Co., Ltd. · Jiangsu

Abstract: In view of the problem of "plate thickness relies on experience, clamping force relies on amplification, and safety margin relies on conservatism" in the design of custom end-effector tooling for Pneumatic Industrial Manipulator, a unified coupling analysis framework of plate thickness, clamping force, and safety margin was established under the open working condition boundary. Based on national standards, material databases and manufacturer's technical information, the mechanical and thermal physical parameters of commonly used structural steel, stainless steel, aluminum alloy, TPU liner and cast iron workpieces were determined. Calibration models for friction clamping, cylinder thrust, bending deflection, Euler buckling, torsion, vacuum gripping and bolt/weld/pin connection were established, and passed 50 kg friction clamping, 80 kg 90° flip and 6/8/10 mm plate thickness compared three sets of calculation examples, combined with schematic finite element, sensitivity analysis and FMEA to form an analysis-simulation-test closed loop. The results show that under the conservative design baseline of 0.5 MPa, the plate thickness governs the deflection with a cubic law, and the eccentricity governs the overturning moment with a linear law. The two often control design safety more directly than the material yield strength grade; the most unfavorable working condition for vacuum gripping is usually not vertical lifting, but the lateral anti-slip of the suction cup in the vertical posture. This framework provides a reproducible quantitative basis for the robust design of custom end-effector tooling.

Keywords: Pneumatic Industrial Manipulator; end-effector tooling; plate thickness; clamping force; safety margin; vacuum gripping; sensitivity analysis

CLC classification number: TH137; TP242 Document identification code: A

Coupled Design of Plate Thickness, Clamping Force and Safety Margin for End-Effectors of Pneumatic Power-Assist Manipulators

(Jiangsu Aurek Intelligent Technology Co., Ltd., Jiangsu China)

Abstract: To address the empirical “thickness-by-experience, force-by-amplification, margin-by-conservatism” problem in designing non-standard end-effectors for pneumatic power-assist manipulators, a unified coupled framework relating plate thickness, clamping force and safety margin is established under publicly available operating boundaries. The mechanical and thermophysical parameters of common structural steels, stainless steel, aluminium alloy, TPU pads and cast-iron workpieces are determined from national standards, material databases and manufacturer data. Verification models for friction clamping, cylinder thrust, bending deflection, Euler buckling, torsion, vacuum adhesion and bolt/weld/pin connections are built. Three worked examples—a 50 kg friction clamp, an 80 kg 90° flip and a 6/8/10 mm thickness comparison—together with schematic finite-element analysis, sensitivity analysis and FMEA form an analysis–simulation–test closed loop. Results show that, under a conservative 0.5 MPa design baseline, plate thickness governs deflection by a cubic law and eccentricity governs the flipping moment by a linear law, both controlling safety more directly than the nominal yield grade of the material; the worst case for vacuum adhesion is generally not vertical lifting but lateral anti-slip with the cup in a vertical posture. The framework provides a reproducible quantitative basis for robust design of non-standard end-effectors.

Key words: pneumatic power-assist manipulator; end-effector; plate thickness; clamping force; safety margin; vacuum adhesion; sensitivity analysis

00Introduction

The fundamental difference between custom handling end-effector tooling and a general-purpose robot gripper is not simply whether it can grasp the workpiece, but whether it can maintain a verifiable safety margin at the minimum compressed-air supply pressure, with the least favorable surface condition, maximum load eccentricity, and specified cycle time. Published gripping research shows that coupling and friction mechanisms materially affect grip stability under external disturbance; seal-ring and groove parameters in vacuum-cup geometry likewise affect local stress and vacuum holding capacity [1-2]. The first finding identifies the load-transfer path and constraint distribution as core variables in gripping stability; the second shows that end-effector design must address structure, contact, actuation, environment, and process path as a coupled system rather than relying only on an assumed sufficient clamping force.

According to public engineering data, Pneumatic Industrial Manipulators are used in scenarios such as heat-insulating transfer of high-temperature cast iron parts, vacuum flipping of car ceilings, inner-support handling of motor stators, and eccentric clamping of car seat assemblies, and adopt the engineering process of "analysis-design-review-production". The load capacity, operating range and lifting stroke need to be confirmed one by one based on the workpiece, fixture, action path and safety factor; this article only uses these contents as design input and application prototype, not directly as material constants or product performance upper limits, so that the model can correspond to real industrial scenarios.

The research objects of this article are: the main plate of the custom tooling installed on the end flange of the manipulator, the gripper/suction cup assembly, the drive cylinder (or flip actuator cylinder), the intermediate lever/connecting rod and its connecting parts; it does not include the fatigue life of the entire machine arm body, basic installation strength and entire line control logic. It only focuses on the static strength, stiffness, stability and verifiable safety margin of the end clamp body in the "clamping-transporting-turning-release" link.

01Boundary conditions and data sources

1.1 Project input conditions

Table 1 gives the engineering input conditions used in this article. The calculation example uses 0.5 MPa as the conservative design baseline and 0.7 MPa as the pressure check value; the remaining data are analysis examples and do not represent the upper limit of product capabilities.

Table 1 Engineering input conditions used in this article

Note: The public working conditions, pressures and application examples in the table are all from the public page of the manufacturer's official website [3].

1.2 Data sources and citation priority

In order to reduce the risk of "empirical parameter drift" in enterprise technical analysis, this article adopts a hierarchical number selection strategy: the first priority is national standards and official standard platforms, which are used for grade definition, thickness-related strength lower limit and test methods [4-5]; the second priority is material manuals and databases, which are used for elastic modulus and Poisson's ratio., density, thermal expansion coefficient and other typical values necessary for structural design [6]; the third priority is the manufacturer's technical information, which is used for engineering selection rules for vacuum gripping and fixtures/actuators [13]; the fourth priority is peer-reviewed papers, which are used to supplement mechanisms such as grasping stability and suction cup geometric effects [1-2]. Any values that are typical values rather than measured values in the delivery state are clearly marked in the text.

02Material parameters and theoretical models

2.1 Symbol description

Table 2 gives the main symbols and units uniformly used in this article.

Table 2 Main symbols and units

2.2 Material parameters and data sources

This article does not equate "higher material strength" with "better clamps": for clamps, stiffness, thermal stability, wear resistance and connection technology are equally important. Upgrading steel types usually significantly increases the yield strength, but has little effect on the elastic modulus, so it does not significantly change the deflection of the same geometric structure; on the contrary, although aluminum alloys significantly reduce weight, the stiffness is usually only about one-third of steel when the geometry remains unchanged [6]. Figure 1 shows the coupling relationship between plate thickness-drive-friction-safety margin.

Table 3 summarizes the representative material parameters used in the calculation examples in this paper.

Table 3 Representative material parameters

Note: E, ν, ρ, α of structural steel and 45 steel adopt the typical range of the material properties appendix [6]; Q355 thickness-related yield strength comes from GB/T 1591-2018 [5]; 304 and 6061-T6 adopt the public material data page [7-8]; TPU adopts the Elastollan C95A data page [9]; the cast iron room temperature range adopts the NIST thermophysical property compilation [10]. The above are all typical pre-design values. Formal projects should be subject to material certificates and necessary random inspections.

It should be pointed out that flexible cushions such as TPU cannot be regarded as accessories that "only change the touch feel". In the public data table, the tensile modulus of Elastollan C95A is about 140 MPa, the Shore A hardness is about 96, and a 23 °C creep modulus-time curve is attached, indicating that the contact layer will undergo considerable time-varying compliance and stress relaxation under continuous clamping and high-frequency cycling [9]. For high-temperature castings, NIST summary shows that the room temperature Young's modulus of cast iron is about 80 to 160 GPa, and the linear expansion coefficient is about 11 to 14 × 10−6·℃−1. Temperature rise will significantly change the contact conditions and dimensional chain; public cases of high-temperature iron castings also use thermal insulation structures and cooling air paths to reduce their impact [10][3].

2.3 Friction clamping and cylinder thrust model

For bilateral friction clamping, ignoring uneven local contact pressure, the basic anti-slip condition is:

The total normal force required is

When the two sides are symmetrical, the unilateral normal force is

Pay attention to the caliber issue: some standards define clamping force as "unilateral force", and the anti-slip capability is usually determined by the sum of the normal forces on both sides. In this article, NΣ represents the total normal force and Nside represents the unilateral force. As a reference for public testing of gripper equipment, T/SZRA 001-2024 recommends that the load be calculated based on a friction coefficient of no higher than 0.2 and a safety factor of no less than 2 [11]. This article uses it as a conservative pre-design baseline.

The theoretical thrust of the cylinder is determined by the effective pressure area and air supply pressure. For the extension side and the return side, respectively

If there are levers, wedges or linkage amplification, the total normal force available is

In the formula: i is the mechanism amplification ratio; ηm is the mechanical efficiency, which is used to uniformly absorb sealing, guide and mechanism losses [12]. The friction coefficient cannot be fixed empirically: the manufacturer's technical information emphasizes that it must be determined experimentally under "original workpiece - real surface - real pollution state". Its public reference range is roughly about 0.1 for oily surfaces, 0.2 to 0.3 for wet surfaces, about 0.5 for dry wood/metal/glass/stone, and about 0.6 for rough surfaces [13].

2.4 Bending, deflection, buckling and torsion models

The tooling main plate is approximated as a cantilever beam with a rectangular cross-section. The cross-sectional moment of inertia, maximum bending stress at the root and free end deflection are respectively

Equations (8) and (9) reveal the key facts of plate thickness design: stress decreases with the square of t, while the deflection decreases with the cubic of t. Therefore, in fixtures that require clamping parallelism, positioning accuracy and flipping stability, plate thickness is first of all the "stiffness variable", and its importance is often higher than the "strength variable". For slender rods and thin-walled supports under pressure, Euler buckling should be checked at the same time; for flip shafts, flange sleeves and pins, the Saint-Venant torsion relationship can be used

Because the elastic modulus of most steels is 190–210 GPa, the difference in E between structural and high-strength steels is far smaller than the difference in yield strength. Accordingly, increasing the steel grade generally reduces deflection much less than increasing plate thickness or shortening the cantilever. By contrast, 6061-T6 has an E of only about 68.9 GPa; with unchanged geometry, aluminum tooling deflection is on the order of 3 times that of steel [6].

2.5 vacuum gripping model

The normal holding force of vacuum gripping comes from the pressure difference

In the formula: Δp is the effective pressure difference; Aeff is the effective vacuum gripping area; etas is the sealing efficiency. The vacuum percentage is a relative quantity relative to the ambient pressure. The atmospheric pressure at sea level is about 101.3 kPa, so the 60% vacuum degree can be approximately taken as Δp≈60.8 kPa [14]. For three typical handling scenarios, the theoretical holding forces are: vertical lifting (suction cup is horizontal), horizontal transfer (suction cup is horizontal), and the most unfavorable working condition (suction cup is vertical, bearing vertical force)

If n suction cups are used, the theoretical holding force required for a single cup is

Calculations should be based on the most unfavorable load scenario for the entire handling sequence, not just the lifting moment. The safety factor should be at least 1.5 for smooth and dense surfaces, and should be 2.0 and above for porous, rough, oily or heterogeneous surfaces; porous materials should ensure flow compensation in the 30% to 55% vacuum range, and dense surfaces should achieve higher retention in the 55% to 80% vacuum range [13,15].

Let’s use a supplementary example to illustrate the importance of posture path: take 6 suction cups with a diameter of 60 mm, a vacuum degree of 60%, etas=0.85, and the theoretical normal force of a single cup is approximately

The total normal force is approximately 876 N, giving a static-load ratio of about 3.57 for a 25 kg dense plate and appearing adequate for vertical lifting. However, in the worst-case “suction cups vertical, load acting longitudinally” condition, with μ=0.25, a=1 m/s², and S=2, the required total holding force rises to approximately 2 162 N; 6 ϕ60 suction cups are clearly insufficient. Vacuum tooling must therefore be designed for the complete motion sequence, not a single lifting action.

2.6 Connection check

In addition to judging the nominal load of the connector, looseness, hole wall crushing, fatigue and assembly deviation should also be considered. This article uses the following nominal calibration formula for first-order screening—the average shear stress of the bolt, the pressure-bearing stress of the hole wall, and the average shear stress of the double shear pin are respectively

The effective throat thickness, effective area and nominal shear stress of the fillet weld are

Public welding design data generally use 0.707z as the approximate effective throat thickness of fillet welds [17]; for bolted connections that rely on friction and anti-slip, the slip coefficient is significantly affected by surface conditions and should be obtained through representative tests rather than directly copying empirical values [16].

03Typical calculation examples and finite element solutions

The three sets of calculation examples all use 0.5 to 0.7 MPa as the air supply window and 0.5 MPa as the conservative baseline. The pre-designed baseline for friction clamping is μ ≤ 0.2 and S ≥ 2 [3,11]. All geometric dimensions, mechanism efficiency and eccentricity are engineering example values ​​set to illustrate the method.

3.1 Calculation example 1: 50 kg double-sided friction clamping

Question: To transport a 50 kg workpiece with a bilateral friction tooling, take a=1.0 m/s², μ=0.20, S=2.0, i=3.0, ηm=0.85. The step-by-step calculation is as follows.

Comparison of candidate bore diameters is shown in Table 4.

Table 4 Comparison of calculation example 1 under different cylinder diameters and pressures

Conclusion: If the 0.5 MPa baseline freezing plan is adopted, ϕ63 is obviously insufficient and ϕ80 has an available margin; even if 0.7 MPa is stably supplied on site, ϕ63 only reaches the boundary value and is not suitable for conservative design. It can be seen that the value of the design baseline in friction clamping is more critical than the "highest possible pressure".

3.2 Calculation example 2: 80 kg workpiece flipped 90°

Question: 80 kg workpiece is turned over 90°, the eccentricity of the center of gravity of the workpiece is e=0.30 m; the mass of the turning head and local components is mg=15 kg, the eccentricity is eg=0.15 m; the comprehensive turning design coefficient κM=2.2; the effective force arm r=0.12 m; ηm=0.85.

Comparison of candidate bore diameters is shown in Table 5.

Table 5 Comparison under different cylinder diameters and pressures in Calculation Example 2

Conclusion: The primary variable in the 90° flip problem is the eccentricity rather than the yield strength of the steel plate. If the eccentricity is reduced from 0.30 m to 0.20 m, the static gravity moment will be reduced by about 30% when other conditions remain unchanged, which is often more effective in reducing system risks than upgrading the plate from Q235 to Q355.

3.3 Calculation Example 3: Comparison of 6/8/10 mm plate thickness

Question: Compare three plate thicknesses of 6/8/10 mm for the same main plate. Treat it as a cantilever beam with a rectangular cross-section. Take the effective width b=160 mm, the cantilever length L=150 mm, the end equivalent load F=1 000 N, and E=210 GPa. The calculation results from equations (7) to (9) are shown in Table 6.

Table 6 Calculation results of different plate thicknesses in Example 3

If the process requires that the end deflection does not exceed 0.50 mm, then 10 mm meets the requirement, 8 mm is close to the upper limit, and 6 mm is obviously insufficient. It is worth noting that although the maximum bending stress of 156 MPa in the 6 mm plate does not exceed the typical yield strength of Q235 of 235 MPa, its deflection far exceeds the process allowable value. This is a typical situation where "plate thickness is first controlled by stiffness rather than strength."

3.4 Finite element simulation solution

In formal projects, the above analytical model should be used for quick screening rather than as a substitute for detailed simulation. The recommended finite element scheme is shown in Table 7.

Table 7 Suggested finite element modeling scheme

The reason for using the median values E and ν of structural steel is that the elastic modulus of steel is 190-210 GPa and Poisson's ratio is 0.27-0.30 in the material appendix. Taking the median value in early simulation can avoid false accuracy of a certain grade [6]. Figure 2 shows the proposed finite element modeling and stress hot spot locations.

The following calculations are illustrative and do not replace project-specific finite-element analysis or acceptance validation. Each project must be checked against its actual CAD geometry, materials, connections, and load conditions. The example retains the 6, 8, and 10 mm plate-thickness cases shown in Table 8.

Table 8 Schematic finite element results

The schematic values are of the same order as the analytical solution: as the plate thickness increases, the displacement decreases by the cube of t, and the stress decreases by the quadratic of t. The local peak values at the edge of the hole and the root of the ear plate are slightly higher than the analytical beam model, which is due to the stress concentration caused by geometric discontinuity. The formal project should at least supplement the local refinement of the weld toe and opening corner, bolt pre-tightening and contact nonlinear secondary recalculation; high-temperature working conditions should also include superimposed thermal load and contact heat derating analysis.

04Sensitivity analysis, experimental verification and FMEA

4.1 Sensitivity analysis

The simple first-order sensitivity can be directly given by the model. The friction clamping safety margin, plate deflection and flipping moment are respectively

Based on this, the first-order sensitivity of Table 9 is constructed.

Table 9 First-order sensitivity of main variables

Table 9 reveals three key facts: first, the primary geometric variable for clamping safety is the cylinder diameter, because it enters the thrust force as a square; second, the stiffness of the main plate is extremely sensitive to the plate thickness and cantilever length, because it enters the deflection as a cube; third, the flipping scenario is most afraid of eccentricity drift, so the priority should be to "compress the eccentricity" in the planning stage rather than "compensating" with higher pressure at the end.

4.2 Experimental verification plan

T/SZRA 001-2024 stipulates the prerequisites for public testing of gripper equipment: ambient temperature 20±2 °C, relative humidity 20%~80%, air pressure 86~106 kPa, and recommends sufficient preheating before testing, and 50 cycle statistics for force control/repetitive positioning/peak torque [11]. The vacuum mode can refer to the leakage rate test path of GB/T 34878-2017 [18]; the fatigue of materials or representative components can be supplemented by the ASTM E466 constant amplitude axial fatigue test [19]. The recommended verification process is shown in Table 10.

Table 10 Recommended experimental verification process

It is recommended to perform static clamping force calibration in three gears of 0.5, 0.6, and 0.7 MPa (each with ≥ 5 steady-state points), and then perform anti-slip loading on the real workpiece or equivalent test piece, and back-calculate μeq in four states: dry at room temperature, slightly oily, grinding dust, and high temperature. Then conduct deflection and flip tests to verify the analytical solution and simulation trend, and repeat the test ≥ 50 times. The vacuum module performs pressure holding and leakage tests under different vacuum degrees, and converts the pressure drop slope into effective pressure difference attenuation. Finally, parameters such as μ, ηm, and κM can be backfilled into the analytical model to convert “empirical values” into “field identification values” [11].

4.3 Failure Mode and Effects Analysis (FMEA)

Table 11 gives FMEA recommendations for end-effector tooling. S, O, and D range from 1 to 10, and the risk priority number RPN = S × O × D.

Table 11 End-Effector Tooling FMEA

As far as RPN is concerned, bolt preload decay, weld fatigue cracks, excessive plate deflection and friction slippage are the most worthy of priority control - their average stress may not be the highest, but they are most likely to amplify into failure after accumulation of cycles, heat, contamination and assembly deviations.

05discuss

This article discusses materials, mechanics, contact and technology under the same coordinate system. Its limitation is that it is still a preliminary engineering analysis framework rather than a final solution for a real three-dimensional product. First, E, ρ, ν, and α in the material table are mostly typical values, which are affected by the manufacturing process, defects, temperature and load history. For formal analysis, the material supplier should be consulted; the 304 and 6061-T6 data correspond to specific states, and aluminum alloys are sensitive to post-weld softening, so the base metal data cannot be equated with post-weld regional performance [6][7-8].

Second, both the friction and vacuum models adopt the first-order approximation of "Coulomb friction + effective pressure difference". The manufacturer's information emphasizes that the friction coefficient must be determined by testing on the original workpiece, and the vacuum degree selection is also directly related to whether the surface is porous [13-15]. The model can distinguish "which variable is more sensitive", but it cannot replace calibration tests under real surface, temperature and contamination conditions; this is especially true for high-temperature castings, where insulation, cooling and sealing derating may be more effective than "increasing the clamping force" [10][3].

Third, the finite-element results are illustrative rather than an acceptance report and are used to explain boundary conditions and hotspot identification. A project-specific analysis should at minimum add weld geometry and residual stress, bolt preload and contact slip, viscoelasticity or creep of compliant pads, and combined high-temperature and cyclic loading, then verify the model against field-test data before using it as a design basis.

06Conclusion

(1) There is a clear coupling relationship between the end tooling plate thickness, clamping force and safety margin. The influence of plate thickness on deflection is cubic. In most working conditions that require parallelism and low jitter, plate thickness is first a stiffness issue.

(2) For steel fixtures, only upgrading Q235 to Q355 significantly increases the yield strength but hardly changes the geometric stiffness; "upgrading steel grade" is usually not as effective as "increasing plate thickness, reducing cantilever, and reducing eccentricity".

(3) In the 50 kg friction clamping example, based on the conservative diameter of 0.5 MPa and μ=0.20, the ϕ80 cylinder diameter can meet the requirements but ϕ63 cannot, indicating that the design baseline should take the lowest guaranteed pressure rather than the highest possible pressure.

(4) In the 90° flip example of 80 kg, eccentricity is the dominant variable, and the risk of flipping is mainly controlled by mass, eccentricity and comprehensive dynamic load coefficient.

(5) For vacuum gripping, the most unfavorable working conditions usually come from lateral anti-skid in the complete action sequence rather than single vertical lifting. The friction coefficient and safety factor must be calibrated through real workpiece tests. In summary, the robust design path is not to "increase the clamping force infinitely", but to "first identify the real boundaries, and then use analysis-simulation-testing closed loop to freeze the plate thickness, cylinder diameter, connection and surface parameters" [6]. The design and verification closed loop is shown in Figure 3.

References · REFERENCES