Analysis and Design of a Novel 3-DOF Translational and Rotational Elastomeric Bearing Positioning Stage

Positioning stage plays an important role in precision mechanical system. Fast steering mirror mainly comprised of a rotational positioning stage and it has been widely used in laser manufacturing related applications. In this article, a novel 3-DoF translational and rotational positioning stage is proposed and designed. It utilizes elastomeric bearing for providing stiffness and has the advantages of small size and structurally simple. This stage is driven by three voice coil motors and measured by three capacitive displacement sensors. In order to establish the system dynamics, the rubber stiffness is also characterized in different loading manners by both simulation and experiment using a biaxial material testing system and a high strain compression testing system. The characterized hyperelastic behavior is modelled via Arruda-Boyce model, for subsequent finite element simulation. Based on these stiffness characterization and additional vibrational analyses, a 3-DOF stage dynamics model is developed and is validated using FEM. Preliminary stage realization and maneuver with the associated actuators and sensors have been demonstrated in this article. Furthermore, controller design and positioning experiment of stage will go on in the future.


Introduction
Precision positioning stages have been widely used in various fields in inspection, manufacturing, and motion control [1].Traditionally, such stages are usually designed and realized based on compliant mechanisms [2].However, compliant stages usually have complicated shapes and relatively large size for specific mechanical designs.It also lacks of flexibility for adjusting their dynamics characteristics.Alternative design approaches should be worth to pursue for improving the future stage design.
Elastomeric bearings (aka rubber bearings) are rubber-metallic laminates structures utilizing the feature of incompressibility of elastomeric materials.With proper design under metallic motion constrains, its stiffness under compression can be much higher than that in shearing [3].As a result, they exhibit a strong stiffness anisotropy and the motion in compression direction can be effectively blocked while the motion in shearing direction can still be flexible.With this feature, it is possible to design more compact structures than traditional compliant mechanisms for providing necessary compliance in motion direction while keep sufficient bearing force to prevent motions in undesirable directions.
Previously, various stages have been designed based on the above mentioned rubber bearing approach [4,5].However, they were all limited in translational motion.For applications such as vibrational mirrors in optical communication of laser metal 3D printing, stage design with rotational degrees of freedom is required.Previously, Kluk designed a 2-DoF steering mirror based on rubber bearing used for optical communication related applications [6].
Unlike translational design where only the compression and shearing stiffness are considered, the torsional and bending stiffness also play important roles.However, researches for these two types of stiffness are limited in analytical level [6], whose assumptions could deviated from real scenarios significantly.As a result, additional efforts on characterizing both torsional and bending stiffness must be performed for achieving accurate system dynamics modeling.
Motivated by the above needs, the goal of this work is to realize a novel 3-DoF stage (one translational and two rotational DoFs) for laser scanning related applications and the associated control techniques.
Meanwhile, the associated material characterization and structural simulation during development would also be used for developing more accurate torsional and bending stiffness formulation.We believe that by the effort of this work, not only a better performed stage is developed, but more fundamental contributions on stiffness modeling in rubber bearing mechanics can also be achieved.Both should make significant contributions in both motion stage design and the fundamental mechanics.This article mainly presents the preliminary design analysis of the stage design, material characterization and mechanics constitutive relation for rubber bearing in different loading condition, and the system dynamics modeling and characterization, as well as the preliminary realization and testing of the stage system so far.In addition, essential discussion to address the near term future work are also provided for highlight the future perspective.

Analysis and Design
Fig. 1 is the research flow chart of this work and it presents the overall plan in stage analysis and design, rubber characterization, and the system control experiment.This article mainly focus on first and second parts to design a novel stage for positioning and third part is currently underway and will be complete in the near future.

Conceptual design
The stage is comprised of an aluminum block and four rubbers bearing attached on it, the outer surfaces of rubber are fixed as shown in Fig. 2(a).This stage is designed to be a 3-DOF positioning stage, thus it needs at least three actuators for driving.As the actuators provide different forces at the bottom of the stage, these actuators would induce both translation in y direction and rotation in x and z direction.It is reasonable to assume the stage a rigid body and thus it only requires to measure the displacement of three points on the top of stage to determine the translational displacement in y direction, y, and rotational angle in x and z direction, θ x , θ z (as shown in Fig. 2(a)).

Component design
For minimizing the stage weight and dimensions, an aluminum block with dimensions of 30x30x30 mm 3 is selected as the main body of the stage.Four rubber pads with dimensions of 15x15x1.5 mm 3 are attached for providing stiffness.Piezo actuators with large actuating force (800N) but small stroke (60 μm) are commonly used in precision mechanical system.However, the stroke is not sufficient for this design.Instead, three AVM40-20 voice coil motors with even larger stroke (20 mm) but with smaller actuation level (50 N) are hired to actuate the stage.Three ASP-50-CTA capacitive probes with a sensing range of 1250m are used to monitor the stage motion.Additional kinematic transform will convert the raw measurement data to the corresponding translation and rotational displacements.For increasing the measurement sensitivity and control authority, actuators and sensors are placed far away from the center of stage as shown in Fig. 2(b).

Mechanical design concerning
In this design, the stiffness of stage is mainly due to the deformation of the four rubber bearings in various manners, including compression, shearing, bending and torsion.Due to the incompressible nature of rubbers, their deformation may not be adequately described by elementary mechanics of materials and this increases the difficulty in machine design.In particular, only basic semi-analytical formulae in compression and shearing stiffness of rubbers are available, but not in bending and torsion.Meanwhile, the geometry and boundary conditions for deriving those semi-analytical formulae might possibly deviated from the current design significantly, therefore, it is necessary to individually determine the stiffness of rubber by simulation and experiment in this work.
Meanwhile, the stiffness could also be affected by the applied preload.As a result, it is required to monitor the preload level.However, the stiffness of the system will decease if load cells are included in the structural loop and additional undesirable vibrations will be generated on the weakened structure.
Consequently, the positioning performance will be degraded.In this work, instead of monitoring the preload force level, the preload deformation is chosen as the variable to be measured.As shown in Fig. 2(c), the rubber bearings are compressed by the compliant mechanism.By using laser sensors to measure the amount of compression, the compression preload can be determined by the rubber constitutive law.This design can maintain the stiffness of the structure and the whole system is shown in Fig. 2(d), the stage with rubbers is fixed in the compliant frame, actuated by those voice coil motors consists of stage and measured by the capacitive sensors.

Rubber bearing finite element analysis
The rubber bearings in stage provide stiffness during motion.In this design, they would carry compression, shearing, torsional, and bending loadings (as shown in Fig. 3), these four stiffness components of rubbers are denoted as, k c , k s , k t and k b , respectively.As mentioned in Section 3, they should be characterized by both FE simulation and experimental testing due to lack of precise model for design analysis.In order to estimate the characteristics of rubber bearing, here a 3D finite element model is constructed to simulate these four loading manners in rubber and to develop their corresponding stiffness models.
This 3D finite element model consists of two steel plates (Young's modulus E=200GPa, Poisson's ratio ν =0.33 and density ρ =7900Kg/m 3 ) and a rubber pad bounded between two plates.The rubber is modeled as a hyperelastic material using Arruda-Boyce model with parameters μ=0.79MPa, λ m =7, D=0.005GPa -1 , ρ=1900Kg/m 3 , and the equivalent linear elastic modules are E=2.4MPa and ν=0.499999.The stiffness obtained through this FE simulation is listed in Table 1, the stiffness of compression is much greater than shearing means the translation of stage in x and z direction are being fully constrained.The results of FEM (as shown in Fig. 4) and the comparison to experiment will be discussed in Section 4.

Rubber bearing material characterization
In order to realize the mechanical characterization of rubber, two mechanical material testing systems established in lab are used for validating the stiffness of rubber in different loading manners.Biaxial material testing system (BMT system) provides both translational and rotational testing in y direction as shown in Fig. 5(a).BMT system consists of a voice coil actuators, a load cell, and a laser displacement sensor.It provides testing in compression, shearing, torsion and bending through the design of fixture.
On the other hand, high strain compression testing system (HSC system, as shown in Fig. 5(b)) provide compression loading in x direction with a strain level one order of magnitude higher than that of BMT system.In HSC system, step motor drives lead screw to compress specimens and it measures the applied force and deformation by a strain-gauge-based load cell and a laser sensor.For testing shear stiffness of rubber under different pre-load levels, a shaker is also utilized to work with HSC and an extra load cell and a LVDT displacement sensor are also included to characterize the static and dynamic stiffness in lateral (i.e.z) direction.
The experiment results are then compared with those obtained from FEM simulation listed in Table 1 and it indicates that both shear and torsion stiffness are agree to each other but not for compression and bending.It is believe that this is due to the additional compliance from testing fixture.That is, because the fixture stiffness is not strong enough to support compression loading in experiment, the measurement of laser sensor contains deformations from both specimen and test fixture.On the other hand, the fixture for bending cannot be fixed as good as that in torsion and thus brings additional compliance.As a result, the experimental compression and bending stiffness are much smaller than those seen in FEM simulation.It needs to re-design fixture and perform more experiment in the future.
The results of shear stiffness with compression preload testing are shown in Fig. 6(a).Shear stiffness increases with compression preload.On the other hand, as shown in Fig. 6(b), the dynamic testing also shows that the shear compliance and resonant frequency varies with the applied compression pre-load.

Developing MDOF dynamic model and kinematics transformation
As shown in Fig. 2(a), the stage consists of 3-DoF including one translation in y direction and two rotation in x and z direction.As Fig. 7(a) shown, the stiffness of translation in y direction of stage is governed by shear stiffness of 4 rubbers in parallel.On the other hand, the rotational stiffness in x or z direction, denoted as k rot , is governed by torsional, bending, and shearing stiffness of rubbers and can be expressed as When the stage is pushed by three actuators, Refer to the deformation and applied force schematically shown in Fig. 7, the coupled force-displacement relationship between three applied force F 1 , F 2 and F 3 and the three static displacement y, θ x and θ z can be written in the following matrix form (Eq(2)) from Fig. 7(b).Determine the force actuators should apply by this relation to satisfy the input positioning command of y, θ x and θ z to the stage.Eq(2) represents that the action of any actuator will generate not only one degree of freedom motion, it's a couple motion design.
Finally, the equation of motion can be obtained by Lagrange's equation and is expressed as And the natural frequencies can be written as Furthermore, the transformation between the measured displacement and the required translational and rotational displacements should also be established.The kinematic transformation relations are listed below Where x i , y i and z i represent the displacement measured by i th sensor.

Finite element model of stage and validation
Based on the design shown in Section 2 and the mechanical model presented in Section 3, we develop a finite element model of stage to validate the dynamic model here.The model consists of an aluminum stage with four rubber bearings attaching on and set the surfaces of stage fixed as shown in Fig. 2(a).
The first issue is to characterize the stage stiffness by comparing the equilibrium positions under a given load.Figure 8 shows the FE model of the stage under various types of loading.The comparison results are listed in Table 2.It can be seen y-stiffness from the analytical model (with tested stiffness), the FE simulation, and the experimental characterization agree to each other very well.However, for the rotational stiffness, both analytical and FE predictions, although mutually agree, deviate from the experimental results considerably.Same situation can also be observed in natural frequency analysis.That is, the agreement on natural frequency in y-direction is significantly better than that in rotational direction.This implies that our rotational stiffness modelling is still far from satisfactory and must be refined.However, for proceeding the stage control work, it is still possible to use the established dynamic model (i.e., Eq.( 3)) with the experimentally characterized stiffness as the basis for designing controllers.

Preliminary step response testing of realized stage and future work
This section briefly addresses the current progress in system realization, which contains a stage with rubber bearings, three voice coil actuators, and three capacitive sensors as shown in Fig. 9.
Preliminary open loop characterization is performed, A typical step response testing result of the stage in y, θx and θz directions is as shown in Fig. 10.
These step responses can be used to characterize the system parameters, as shown in Table 2 and Table 3, the stiffness and natural frequencies are obtained.As mentioned earlier, the stiffness in y direction is similar to FEM and model but are not in θx and θz.Damping ratio can be determined by logarithmic decrement method and are listed in Table 4. Notice that although the design should be symmetrical in θx and θz directions, the results are different possibly due to imperfected rubber attachment during assembly.Meanwhile, this imperfection (and other possible assembly errors) also deteriorates the coupling between motion directions.It is hope that we shall be able to suppress this coupling by closed-loop control.
In the future, based on the realized system and the dynamic model, controllers will be designed for this stage.Preliminary, MIMO PID controller will be designed first to obtain a basis of performance evaluation.Other types of advanced controllers will be flowed for further improving the positioning performance.

Conclusions
I n this article, a novel 3-DoF rubber bearing-based positioning stage for laser scanning applications is designed.This stage consists of one translational and two rotational DoFs.In Compare with compliant mechanism, rubber bearing-based stage provides more flexibility and smaller size for achieving better performance.However, the stiffness model of rubber bearing is still primitive for torsional and bending.Without reliable stiffness models in and, it is hard to proceed accurate stage design.As a result, significant effort of this work is for establishing the associate rubber mechanics model by both FE simulations and experimental characterization.The results, although still primitive, should still be useful for related rubber engineering applications.
Meanwhile, a stage dynamic model is developed for guiding the mechanical and controller design.A primary system is also realized and the open loop characterization is performed.The close loop controller design is underway.It is believe that with adequate control, this novel 3-DoF stage design should be very useful for the applications in either manufacturing or metrologies.
Fig. 2. Schematic plots for system and stage design.
(a) Schematics plot for pure translation and pure rotation of stage with rubber bearings (b) of actuators and sensors on stage

Fig. 7 .
Fig. 7. Schematics plot for kinetic analysis of stage Fig. 8 Static equilibrium and mode shape of stage

Table 1 .
Stiffness of rubber bearing.

Table 2 .
Stiffness of stage.

Table 3 .
Natural frequency of stage.

Table 4 .
Damping ratio of stage by testing result.