Experimental and numerical study on surface generated mechanism of robotic belt grinding process considering the dynamic deformation of elastic contact wheel

In the robotic belt grinding process, the elastic contact condition between the flexible tool and the workpiece is a critical issue which extremely influences the surface quality of the manufactured part. The existing analysis of elastic removal mechanism is based on the statistic contact condition but ignoring the dynamic removal phenomenon. In this paper, we discussed the dynamic contact pressure distribution caused by the non-unique removal depth in the grinding process. Based on the analysis of the removal depth of a single grit, we obtained the topology of a single groove. With the coupling analysis of the topology single groove and grit trajectories in elastic removing procedure, an elastic grinding surface topography model was established with the consideration of the dynamic contact condition in the removing process. Robotic belt grinding experiments were accomplished to validate the precision of this model, while the result showed that the surface roughness prediction error could be confined to 11.6%, which meant this model provided higher accuracy than the traditional predicting methods.


Abbreviations a s
The length of the contact area in the static contact condition a d The length of the contact area in the dynamic contact condition a p, total The totally removal depth a p,grit (x) The removal depth of a single grit a p,grit (x) The equivalent removal depth of a single grit a p,finial (x) The equivalent removal depth of final single grit b mn The offset of grain trajectory relative to the contact area center point B w The width of the workpiece B c The width of the contact wheel B The width of the contact area E Rub The Young's modulus of the contact wheel rubber outer layer E Al The Young's modulus of the contact wheel aluminum inner core E 1 The Young's modulus of the contact wheel E 2 The Young's modulus of the workpiece E* The equivalent Young's modulus e Ra The arithmetic average deviation error e Rt The maximum altitude error f 0 The frequency of the belt machine tool driving wheel F grit (x) The normal force of a single grain F n The normal force between the workpiece and the contact wheel h g The height of a single grain h s The height of the gullies on grinded surface The workpiece contour approximate equation in the contact area HV The Vicker's hardness of the workpiece H u The ungrinded surface height H g The grinded surface height k m The coefficient of grit morphology k p The coefficient of material removal k w The coefficient of grit wear k h The coefficient of surface topography k Ra The coefficient of arithmetic average deviation value caused by the stacking removal k Rt The coefficient of maximum altitude value caused by the stacking removal p s (x,y) The contact area pressure distribution in static contact condition p d (x,y) The contact area pressure distribution in dynamic contact condition R The radius of the contact wheel r The radius of the contact wheel inner core R 0 The radius of the driving wheel R a, exp The arithmetic average deviation value of the profile from experiments R t, exp The maximum altitude value of the profile from experiments R a, ide The ideal arithmetic average deviation value of the profile R t, ide The ideal maximum altitude value of contour R a, pre The arithmetic average deviation value of the profile from the prediction model R t,pre The maximum altitude value of contour from the prediction model S cont The area of wheel-workpiece contact space T The type of grain size v s The linear velocity of the belt v w The robotic feed speed. v f The robotic feed speed v m The equivalent cutting speed v 1 The Poisson ratio of the contact wheel v 2 The Poisson ratio of the workpiece δ max The maximum deformation value of the contact wheel in the contact area δ s (x,y) The contact area deformation in static contact condition δ d (x,y) The contact area deformation in dynamic contact condition θ The vertex angle of the grains ψ The angle between robotic feeding speed orientation and cutting velocity orientation φ The angle between grains arrangement orientation and robotic feed orientation Δl s The width of the gullies on grinded surface Δl x The horizontal spacing of grains Δl y The vertical spacing of grains

Introduction
The developments of advanced manufacturing industries, such as aerospace, new energy power generation, and new energy vehicles, promote the demands of manufacturing components with free-form surfaces, just as engine blades, fan blades, wheel hubs, and new energy vehicle bodies. In contemporary industrial productions, artificial works and special machine tools are the main craft for the preparation of parts with curved surfaces. These products fabricated by the manual grinding are sick in the rate of producing and qualities among different batches. While, manufacturing with special machine tools also takes disadvantages like the cost of equipment procurements. Robot-assisted abrasive belt grinding is one species of free-form surface fabricating method with enormous potentials. Because of its structural characteristics, the stiffness of the serial robots is much lower than one of the machine tools. While the elastic contact wheels of belt machine tools can provide passive flexibility which can reduce the deformation of robots in the machining process. Nevertheless, the elastic deformation of the contact wheel complicates the grain trajectories in the removal process. The removal process and surface forming mechanism of robot-assisted belt grinding are widely investigated [1,2]. For composite curved surface details with the requirement of dynamic characteristics, like blades, the surface roughness is an important property index of the qualities in service conditions. The threshold exceeded surface roughness causes air turbulent at the blade edge, resulting in the reduction of blade dynamic performance, and power losses of engine. Enormous surface roughness may cause cracks or even fracture due to the stress concentration effects and ruined the performances in high-temperature service conditions [3][4][5].
Being a flexible manufacturing method, the cutting parameters of robotic belt grinding should be real-time adjusting with the curvature feather changing of cutting point. So, the robotic belt grinding roughness predicting is integral to calculate the machining parameters of each machining point on the curved surface. At present, the universal method for predicting the rigidly grinding surface roughness is based on the undeformed chip thickness model. The maximum undeformed chip thickness (MUCT) is a comprehensive coefficient which integrates both the cutting parameters and the grain status [6][7][8][9][10]. While the MUCT method is unable to predict exact value for elastic contact grinding procedures, such as belt grinding. Though the error can be reduced by introducing some elastic contact coefficients, the MUCT method is deficient in theoretical basis and the trend of MUCTpredicted values is inconsistent with the experimental values [11]. Therefore, it is profitable to intensively investigate the removal mechanism of the elastic contact between abrasive grains and workpiece surfaces in elastic grinding process, which includes scratching, plowing, and cutting steps. Zhu et al. evaluated the mechanism of robotic belt grinding by the perspective of grinding force [12]. The accomplishment of their work is the settlement of the micro-cutting force model compensated with sliding, plowing, and cutting components. Based on the force model, the influence of force components on the machined surface roughness is discussed. Yang et al. instituted an energy-efficiency model based on the friction coefficient model of a single spherical grain from the perspective of abrasive geometry [13], which showed that the plowing energy took more proportion than the cutting energy and the scratching energy, while the grinding depth of a single particle was much smaller than the radius of grains. Based on the mechanism evaluation of robotic belt grinding, a novel grinding force model is accomplished by Xu et al. to predict the grinding removal depth and profit a quantificational machining process of robotic belt grinding [14]. Agustina et al. appraised the surface roughness obtained by robot-assisted polishing experiments with the analysis of the acoustic emission signal frequency domain features [15,16]. Cheng et al. established a belt wear online monitoring system with multi-sensor fusion of sound and current signals [17]. It was found that sound signals provided plentiful messages at high frequencies, while current signals contain abundant messages at low frequencies. Pandiyan et al. proposed an on-line multi-sensor integration system, which was composed with force sensors, acceleration sensors, and acoustic emission sensors, to predict the surface roughness formed by robotic belt grinding through the neural network method [18].
In this paper, a novel surface topography model for surface roughness prediction in robotic belt grinding procedure considering the dynamic deformation of the elastic contact wheel is provided. With the infrastructures of the elastic plane-cylinder contact model and Preston equation, an original distribution of the contact pressure considering the dynamically removal volume during the elastic grinding procedure is attained. According to the plastic contact theory, the motional removal depth of a single pyramidal grain is gained by its relationship with the normal force of the grain. On the basis of the study in the equivalent removal depth of a single grit and the trajectories of grits in elastic grinding procedure, an ideal surface topography model considering the effects of grain sizes is accomplished. Based on the topography model, the surface roughness obtained by robotic belt grinding can be predicted by introducing coefficients which means the effects of stacking removal phenomena. Through the robotic belt grinding experiments, the accuracy of the roughness predicting value is proved by comparing with the values in experiments.

The dynamic model and surface topography model involved in robotic belt grinding
The workpiece settled on the actuator at the end of the robot is grinded by the belt grinding machine tool. During the grinding procedure, caused by the contact force between the contact wheel and the workpiece, deformations happen on both of the contact wheel and the workpiece while the one of the contact wheels is greater. Since getting into the contact area, normally, the grains adhered on the belt suffer three steps: the scratching step, the plowing step, and the cutting step.

The dynamic model of contact pressure
The normal force which generates the material removal is administered by the contact between the workpiece and the contact wheel. The contact wheel is mainly composed of an aluminum inner core and a rubber outer layer.
In the research of the contact state, the deformation of the abrasive belt is ignored as for its thickness is much smaller than the radius of the contact wheel, while the diagrammatic sketch of the status of the contact interfaces is shown in Fig. 1, which means both of the aluminum inner core and rubber outer layer are compressed under stress [11]. Therefore, the composite Young's modulus of the contact wheel is crucial in establishing the contact model between the workpiece and the contact wheel. The combined Young's modulus of contact wheel with considering the deformations of the inner and the outer is shown as Eqs. (1) and (2).
In the robotic abrasive belt grinding procedure, both of the flexible contact wheel and the workpiece elastic take deformation. Though the elastic characteristic of rubber is non-linear, as for the elastic deformation is much smaller than the thickness of the outer layer, the linear elastic model can be used to calculate the rubber deformation. It is assumed that there is no relative motion between the wheel and the workpiece, which means that the contact is static. The contact wheel is abstracted as an elastic cylinder, while the workpiece is simplified as an elastic plane. According to the elastic cylinder-plane contact model [19], the normal contact force Fn is shown as Eq. (3).
The equivalent Young's modulus E* is shown in Eq. (4).
The length of the static contact area between the workpiece and the contact wheel is expressed as Eq. (5).
The static contact area is expressed as Eq. (6).
The pressure ps(x,y) in the static contact condition at an arbitrary point P(x,y) in the contact area is shown in Eq. (7).
The deformation at P(x,y) in the contact area is shown in Eq. (8).
At present, the static contact models, such as Hertz contact model or elastic cylinder-plane contact model, are generally used to predict the material removal volume in robotic belt grinding. The static contact models suppose that the contact parameters, such as contact area and contact stress distribution, are independent with material removal volume in grinding process. However, in the actual manufacturing, due to the grinding removal effect of the adhesive grits, the actual contact surface between the wheel and the workpiece is not an ideal horizontal plane. As shown in Fig. 2, the leading edge of the actual contact area is the ungrinded surface, while the training edge is the grinded surface. The height of workpiece in the contact area is analyzed. In the coordinate system of the contact area OXYZ, the orientation of x axis is the robotic feed direction, y axis is perpendicular to x axis, Fig. 1 The removing mechanism of the robotic belt grinding Fig. 2 The formation of contact area and plane XOY is parallel to ungrinded workpiece surface, while z axis is perpendicular to x and y axes. The origin point O of the coordinate system is the center point of the contact area in the static contact condition.
As x = x 1 , the height of the workpiece is the ungrinded surface height H u , while as x = x 2 is the grinded surface height H g , which is expressed as Eq. (10).
where a p,total is the totally removal depth, which can be obtained by Preston equation, as Eq. (11) [20].
In the grinding procedure, the length of the dynamic contact area ad is expressed as Eq. (12).
The workpiece contour in the contact area is fitted with a straight line, as shown in Fig. 3, and the approximate equation hd(x) is shown as Eq. (13), where the equivalent cutting speed v m is shown as Eq. (14), where, vs is the linear velocity of the belt, vw is the feed speed of the robot, '±' means that the linear velocity of contact wheel, and the workpiece feed speed takes the same/ opposite direction. Since the Young's modulus of the workpiece is much larger than that of the contact wheel, it can be considered that only the contact wheel takes the elastic deformation, while the deformation of the workpiece is ignored, which is shown as Eq. (15).
The removal volume coefficient kr,(x), expressed as Eq. (16), indicates the reduction of the contact wheel compression caused by material removal in the stable cutting procedure.
In the dynamic contact model, the dynamic pressure pd(x,y) at an arbitrary point P(x,y) in the contact area is shown in Eq. (17).

The single-grain grinding depth and material removal volume
The abrasive particles could be abstracted as pyramids. As displayed in Fig. 4, three characteristics of the abrasive particle morphology, which are abrasive grits height hgrit, horizontal spacing between two adjacent abrasive grits Δlx, and vertical spacing between two adjacent abrasive grits Δly, are attained.. The angle between the relative surfaces θ is shown in Eq. (18).
The abrasive particle trajectory equation on the contact area plane XOY is shown as Eq. (19) [21].
where ψ is the angle between the robotic feeding orientation and the cutting velocity orientation, and bmn is the off set of grain trajectory relative to the contact area center point.
The normal force of a single grit Fgrit(x) is expressed as Eq. (20).
Wang et al. used the material hardness to calculate the removal depth of a single grain [22]. Zhang et al. discovered that the grain wear condition influenced the material removal depth [23]. Asikuzun provided the (17) relationship between Vickers hardness, normal force, and indentation depth [24]. As the grain morphology is similar with the Vickers hardness testing penetrator, the removal depth of a single grain ap,grit(x) can be calculated as Eq. (21).
where kw is the coefficient of grain wear. km is the coefficient of abrasive grit morphology, as shown in Eq. (22).
When the pressure between the grain and the workpiece is less than 0.4 HB, only elastic deformation occurs [25]. As shown in Eq. (21), when the grains are abstracts as ideal pyramids, the pressure between the grain and the workpiece is p grit = HV > 0.4HB , so in this model, there is no scratching process.

The surface topography model
The removal width of a single grit is much smaller than the horizontal distance between two adjacent abrasive grits Δlx. Actually, the morphology of workpiece surface is formed by the grinding tracks of different grains at different times. In order to express the vertical positions of different grains reaching the cross section, the equivalent cutting depth of single abrasive particles is introduced as Eq. (23).
Since the removal depth of single grit is much smaller than the elastic contact deformation, the equivalent removal depth of a single abrasive grain a p,i (x) , shown as Eq. (24), can be regarded as a constant volume in the calculation of the robotic belt grinding surface roughness.
As for the angle φ between the grain arrangement direction and the robot feed direction, the horizontal distance between the locations which are trajectories of two adjacent grains in the same row in cross section is Δly × sin(φ), while the vertical one is −a p,finial (x) . The points in the area of {(xsection,ysection)| 0 < xsection ≤ Δlx, -ap,N(x) < ysection ≤ 0} can be expressed with (xm,ym) as Eq. (5). Fig. 4 The grit morphology characteristics where Rem() is the remainder function. The surface formed by the grain tracks is shown as Fig. 5. The grinded surface morphology, which is the blue line in Fig. 6, is the envelope of abrasive trajectories. Δls is the width of the gullies on the grinded surface, while hs is the height, expressed as Eq. (26).
where kh is the surface topography coefficient.
The grinding surface is the envelope of abrasive trajectories. Through their definitions, profile arithmetic average deviation Ra and profile maximum altitude Rt are calculated as Eq. (27).
The ideal surface roughness along the robot feed direction of belt grinding can be obtained as Eq. (28).
In the surface profile model establishing process, the relationship between different trajectories of grits is assumed to be the parallel removal mode, which means that the adjacent trajectory is approximately horizontally arranged. The adjacent removal mode is shown in Fig. 6. Nevertheless, in the actual surface forming process, besides the adjacent removal mode, the stacking removal mode also exists. The stacking mode means that the adjacent grinding tracks overlap in the vertical direction; therefore, the grinding depth stacking one by one makes the gully depth on the surface profile exceeding the ideal one hs. The stacking removal mode is also displayed in Fig. 6.
Through the analysis of the robotic belt grinding surface topography, it can be found that the adjacent removal mode is the main stream of material removing. Contrarily, the stacking removal mode occurs randomly, while the maximum removal depth formed by the stacking removal mode has a positive correlation with the single grit removal depth.
Above all, the robotic belt grinding surface roughness can be predicted as Eq. (29), where kRa is the coefficient of arithmetic average deviation value caused by the stacking removal, and kRt is one of maximum altitude values.

Experimental conditions
All the robotic belt grinding experiments were conducted on the robotic belt grinding system mainly composed of a 6-DOF robot with the type of Motorman DX100, as shown in Fig. 7. The Inconel 718 superalloy samples which were cut to 500 mm (length) × 15 mm (width) × 10 mm (thickness) pieces were fixed with the specific fixtures. The belt 237AA (3 M Company), covered with Al2O3 grains, was used in the experiments. The micro-morphologies of the grains were shown by KEYENCE VHX-1000C digital microscope as shown in Fig. 8, which verified the grain morphology model.
The abrasive grit morphology characteristics of each type of abrasive belts are shown in Table 1.
The grinding system characteristics in the belt grinding surface roughness model are shown in Table 2.

Experimental design
The experimental parameters were selected as shown in Table 3 After the tests, the surface morphologies of the workpieces were scanned by the laser confocal microscope,

Verification and discussion
For the sake of verifying the belt grinding surface roughness mode, Eq. (29) was used to calculate the surface roughness predicted values which are compared with the experimental values to characterize the accuracy of the surface roughness model. The arithmetic average deviation error eRa is 11.6% while the maximum altitude error eRt is 7.03%, respectively, which are calculated as Eqs. (30) and (31). Through error analysis, the efficiency of the surface roughness model is verified. Compared with the normal static contact surface roughness model, the improved dynamic contact surface roughness model provides higher precisions. Figure 9 shows the relationships between the surface roughness and the grinding parameters. As shown in Fig. 9a, when the belt linear velocity is under a threshold, the surface roughness model provides an accurate prediction. When the velocity is over the threshold, the surface roughness is much larger than the predicted value. This phenomenon is caused by the elastic recoil phenomenon of the rubber layer [27]. When cutting speed gets larger, the single-cutting time of single grit, which also means the rubber layer single deformation time, gets smaller. As for the rubber dynamic deforming characteristic, faster deforming makes no-linear increasing normal force. The increasing normal force makes the chatter of robot and single grit removal depth larger. Both of the will reduce the surface quality.
Both the simulations and the experiments suggest that the surface roughness gets deteriorated with the increasing of contact depth, as shown in Fig. 9b, while the varying of feeding speed only makes a slight effect on surface roughness as shown in Fig. 9c.
The robotic belt grinding system  It is observed that the grain size and contact depth have great influence on the surface quality, as shown in Fig. 9d. The abrasive particle size has the most significant effect on the surface roughness, and the surface roughness reaches over Ra0.6 when the type of grain is #100, while the surface roughness is reduced under Ra0.3 with #30. This conclusion is supported by Gorp et al. [28] that the most important factor affecting the workpiece roughness is the grain size.

Conclusion
In this study, a novel surface topography model of robotic belt grinding procedure considering the dynamic deformation of the elastic contact wheel is proposed. Based on the elastic planecylinder contact model and Preston equation, an original distribution of the contact pressure considering the dynamically removal of volume during the elastic grinding procedure is attained. Besides, on the basis of the study in the equivalent removal depth of a single grit and the trajectories of grits in elastic grinding procedure, an ideal surface topography model considering the effects of grain sizes is accomplished. The results indicated that the surface topography formed by elastic grinding related to the removal depth of a single grain, unlike the one formed by rigid grinding which is affected by the total removal depth. Moreover, the surface roughness forecast model is provided by considering the removal depth of a single grain and the trajectory of active grains in robotic belt grinding process, and the verified experimental results of robotic belt grinding disclosed that the prediction error can be confined to 11.6%, which can predict the surface topography information of robotic belt grinding procedure more accurately compared with the traditional predicted models.