A curious observation of phenomena occurring during lapping/polishing processes

Abstract

We demonstrate that the kinematics of the polishing process is more intriguing than an idealized planetary movement as all the previous studies reported. In reality, the workpiece is pseudo-constrained by the planetary carrier and because of this its relative motion to the polishing pad also incurs a ‘parasitic’ movement that previously has not been observed. Here, we report and model this parasitic movement and quantify its effect upon the workpiece surface roughness. Using a motion capture system, the principal and ‘parasitic’ movements between the sample and polishing tool have been tracked and our models validated. It is proved that considering this parasitic movement the prediction of workpiece surface morphology can be significantly improved when compared with the idealized approach (i.e. planetary). Our observations and modelling framework open the avenue to carefully consider the compliance between the tools and workpiece in other manufacturing processes for accurate predictions of the process outcomes.

1. Introduction

Although the research topics related to manufacturing processes might be considered on the applicative side of engineering science, as sometimes they are made to work using empirical methods, some intriguing problems could emerge at their close observation (table 1).

Table 1.

Nomenclature.

$C_1$, $C_2$	number of revolutions of bodies 1 and 2
$d_{\text{se}}$	maximum deviations between simulated and experimental trajectories with clearance
$d_{\text{si}}$	maximum deviations between simulated and idealized trajectories without clearance
${\mathbf{\text{F}}}_{\mathrm{c}}$	required centripetal force of body 3
${\mathbf{\text{F}}}_{N1}$, ${\mathbf{\text{F}}}_{N2}$	reactions from bodies 1 and 2
${\mathbf{\text{F}}}_{r13}$	friction force between bodies 1 and 3
$\mathbf{\text{G}}$	gravity of body 3
$M_{r90}$	rotation matric for 90° in the counterclockwise direction
$O_1$, $O_2$, $O_3$, $O_4$	centres of bodies 1, 2, 3 and slots within body 2
${\mathbf{\text{O}}}_{12}$, ${\mathbf{\text{O}}}_{13}$, ${\mathbf{\text{O}}}_{14}$	vector from the body 1 centre to the body 2, body 3 and slots (within body 2) centres
${\mathbf{\text{O}}}_{23}$	vector from the body 2 centre to the body 3 centre
${\mathbf{\text{O}}}_{42}$	vector from the slot centre to the body 2 centre
${\mathbf{\text{O}}}_{1\mathrm{\_}3e\mathrm{\_}sa}$	vector from the body 1 centre to the points on body 3
$\mathbf{\text{P}}$	pressure applied on body 3
$r_3$, $r_4$	radius of body 3 and slots within body 2
$r_{3e\mathrm{\_}sa}$	distance of the calculated point on body 3 relative to the body 2 centre
$r_{\mathrm{s}}$, $r_{\mathrm{e}}$, $r_{\mathrm{i}}$	distance from the body 2 centre to the points on experimental trajectories with clearance, simulated trajectories with clearance and idealized trajectories without clearance
$T$	periodicity of scratch trajectories
$t_i$, $t_j$, $t_n$, $t_{i-j+1}$	ith, jth, nth and $(i-j+1)$th instants
${\mathbf{\text{u}}}_{\mathrm{v}}$	unit vector of the velocity of the body 3 centre relative to the body 1 centre
${\mathbf{\text{V}}}_{r13}$, ${\mathbf{\text{V}}}_{r32}$	velocity components of the body 3 centre from the bodies 1 and 2 rotations
$x_3$, $y_3$	positions of the body 3 centre on the $X_2$ and $Y_2$ axes
$x_4$, $y_4$	positions of the slot centre within body 2 on the $X_2$ and $Y_2$ axes
$x_{3e\mathrm{\_}sa}$, $y_{3e\mathrm{\_}sa}$	positions of the calculated point on body 3 on the $X_2$ and $Y_2$ axes
$x_{1\mathrm{\_}3e\mathrm{\_}sa}$, $y_{1\mathrm{\_}3e\mathrm{\_}sa}$	positions of the calculated point on body 3 on the $X_1$ and $Y_1$ axes
$\delta$	coverage angle of the ‘inflexion area’ in body 3 centre trajectories
${\theta }_{34}$	azimuth angle of the body 3 centre relative to the slot centre
${\theta }_{34\mathrm{\_}\max }$	maximum value of ${\theta }_{34}$ during the given time
${\theta }_{3e\mathrm{\_}sa}$	azimuth angle of the calculated point on body 3 relative to the body 2 centre
${\theta }_{3e\mathrm{\_}sa\mathrm{\_}1}$	azimuth angle of the calculated point on body 3 relative to the body 1 centre
${\theta }_{40}$	initial azimuth of the slot centre relative to the body 2 centre
${\mu }_{13}$	coefficient of the friction between bodies 1 and 3
${\psi }_2$	azimuth angle around the body 2 centre
${\omega }_1$, ${\omega }_2$	angular velocity of bodies 1 and 2 around their centres
$|\dots {|}_{\text{max}}$, $|\dots {|}_{\text{min}}$	maximum and minimum modulus of vectors

Surface roughness, containing combinations of high spatial frequencies, could be considered a key output of the machining operations [1,2] with implications on part functional performance (e.g. fatigue, wear) [3,4] and aesthetics.

In this respect, for material removal processes with deterministic kinematics and defined (e.g. milling, turning) or undefined (e.g. grinding) cutting edges, analytical models for prediction of surface morphology (e.g. roughness) exist [5,6]. These are usually based on time-dependent trajectories of the cutting edges transferring their profiles on the workpiece surface while considering various secondary effects (e.g. ploughing, rubbing, elastic recovery) [7,8] that can occur at the cutting edge–workpiece interfaces.

Polishing is a high-precision finish technique employed to improve the surface quality of high-value products including silicon wafers, camera lenses and freeform surfaces for electronics, optics, aerospace and medical industries. In the traditional polishing method (figure 1a), loose abrasives mixed into a polishing slurry roll and slide on the workpiece surface to remove materials while pressure is given by a polishing pad [9]. This principle leads to uncontrollable motions of the individual abrasives and consequently, unpredictable abrasives–part interactions, together with the high cost of consumables, i.e. slurry [10]. In order to overcome this, traditional polishing techniques are now migrating to fixed-abrasive polishing (FAP), figure 1b, for which the abrasives are embedded on pellets of various geometries/patterns (e.g. circular, annular, square, phyllotactic [11–14] using a bond matrix). Some abrasives in FAP are stochastic in terms of distribution, positions, shapes and dimensions. In these conditions, it is difficult to predict the workpiece surface topography with unknown cutting edges. Some authors relied on empirical [6,15,16] or analytical approaches [17–20] where a great number of assumptions were made, which led to their limited transferability for predicting surface roughness on other experimental set-ups. These approaches usually employed some statistical models to obtain the abrasives distribution and their interaction depth with the workpiece surface [18,19], and assumed average dimensions and regular shapes (e.g. spheres, pyramids) of the abrasives [15,17–19], without considering the effects of the real uncontrolled characteristics (e.g. real shapes, dimensions and number of active edges) of the abrasives.

Figure 1.

Schematics of a common polishing process. The loose abrasive polishing set-up which presents limited control on the supply of abrasives (a) and fixed abrasives polishing set-up on which the distribution of abrasives is closely controlled (b); nevertheless, in both cases, there is stochasticity of sizes, shapes and orientations of the abrasives when in contact with the workpiece. On both set-ups, there is a clearance between the ‘workpiece’ and the slot of the ‘carrier’ (c). (Online version in colour.)

For the clarity of the follow-up sections, we define here a generic polishing set-up (figure 1) that is composed of a rotating polishing pad (usually disc) on which a workpiece carrier is positioned eccentrically which rotates around its axis; the samples, i.e. workpieces, are mounted in dedicated slots of the carrier and rotate with the carrier under predetermined pressure to ensure material removal by the abrasives.

It is important to point out that, up to now, on all research related to polishing modelling the presence of the clearance between the workpiece and the carrier (figure 1c) has been neglected. Through the object of this research, what we will show is of importance in correctly predicting the surface roughness of the polished workpiece. As such, most published reports on the polishing process assumed that the workpiece totally synchronizes with the carrier as an ideal assembly (i.e. with no clearance).

The optimization of the rotation ratio (rotational speeds between the workpiece and the polishing pad) [21,22] and the patterns of the fixed diamond abrasives on the polishing pad [20] have been given attention to enable enhanced uniformity of abrasive trajectories left on the workpiece surface for achieving a fine finish surface. The material removal mechanism [23] and the evaluation of abrasive pads [24] for FAP have been studied based on empirical models without considering that trajectories of the abrasives could be influenced by the clearance between the carrier and the workpiece. Actually, up to now, no researchers commented on the existence of this clearance. Hence, as it will be demonstrated by us in the following, it is difficult (if not unrealistic) to reveal the real condition of workpiece surface roughness through theoretical approaches neglecting the clearance from assembling when the workpiece is assembled within the slot of the carrier.

Considering that inconsistencies in the relative motion between the workpiece and the polishing pad can occur on the actual set-up due to the clearance between them (figure 1c), it would be appropriate to propose an analytical modelling framework to describe the governing phenomena. This will be the base for predicting the traces (i.e. scratches) of the abrasives on workpieces and hence, allowing to determine the workpiece surface roughness.

In a typical set-up of FAP, the workpiece is placed in a rotating carrier which is positioned eccentric to the axis of a rotating polishing pad (figure 1b); thus, the principal movement of the workpiece appears to follow a circular trajectory around the carrier centre. In this respect, the FAP process can be schematically described as in figure 2 where body 1 is the polishing pad, body 3 is a workpiece to be polished which is mounted in the carrier body 2. If there is no relative movement between body 2 and body 3, i.e. there is ‘tight fit’ (no clearance) between the two, it means these two bodies can be considered as one (figure 2a); thus, body 3 always rotates around body 2 centre with the same rotation speed as body 2. This is the situation that all the previous researches considered, in our opinion, inadequately. However, this is not true as there is an inevitable clearance between bodies 2 and 3. Then, only the circular movement may have difficulty in fully describing the workpiece pseudo-deterministic trajectory which, as it will be proven later, is an important factor to influence the workpiece surface roughness. Thus, the centre of body 3, marked by a red cross in figure 2a, does not follow the red dash line as was previously assumed, but its locum is contained in an annular area (figure 2b) between the two red dash lines which are two extreme radial positions determined by the clearance with time in the real case.

Figure 2.

Example of a configuration for a common polishing set-up: body 1 is the lapping/polishing pad with some abrasives and also acts as a support for the whole set-up, body 2 is the workpiece carrier with two slots (does not contact with body 1, figure 1), body 3 is the workpiece placed in the dedicated slots of body 2. In the ideal case (a), when it is assumed there is no clearance between bodies 2 and 3, the red dash circle is the deterministic trajectory of the body 3 centre marked by a red cross. However, in reality (b), the trajectories of the body 3 centre are pseudo-deterministic and limited by the slot boundary in an annular area marked by two red dash lines when bodies 2 and 3 rotate actually with a clearance between them. (Online version in colour.)

To address this and adequately understand the effect of clearance upon the relative movement between bodies 1 and 3, we employed an inverted polishing test by switch bodies 1 and 3 positions compared with the typical set-up as it is easier to manufacture small abrasive pads. Meaning, an engineered polishing tool with regular abrasives acts as body 3 (instead of body 1 in the typical set-up) inserted in the circular slot of the carrier (body 2), and the workpiece is set as the body 1 mount below body 3 to be polished with pressure. Thus, in the following, we refer to the inverted polishing set-up.

A critical observation concerning a real polishing set-up is that the contact point between the carrier (body 2) and the polishing tool (body 3) changes with the rotation positions, due to the clearance between them. Thus, apart from the principal circular motion, a ‘parasitic’ movement, that results from the direction change of the friction force between bodies 1 and 3 with time, exists as it will be proved in the following. Hence, to the main circular movement of the abrasives, a ‘parasitic’ movement element needs to be added which affects the workpiece surface roughness accordingly. Up to now, no such in-depth analysis of the polishing process has been reported and therefore, no analytical models for defining the resultant workpiece surface topography (i.e. roughness) were proposed. Furthermore, it is even more difficult to prove that the resulting stochastic surface roughness is well linked with the motion law of abrasives in such complex set-ups because the abrasives on the conventional polishing tool (body 3) follow stochastic laws for shapes, sizes and distributions. Our approach is to use a polishing tool with defined abrasives (e.g. frustum with fixed sizes and known distributions) to better understand and quantify the influence of the parasitic movement, caused by the existence of clearance between bodies 2 and 3, upon the workpiece surface roughness generated under real polishing conditions.

We are reporting here, for the first time, on a detailed model that describes this ‘parasitic’ motion occurring during the polishing process in which the polishing tool and the workpiece have geometrical compliance. This paper used solid mechanics to analytically define the instantaneous relative motion between the workpiece and the polishing tool which is later validated experimentally using a motion capture system. The practicality of this work resides in the use of the models to predict the workpiece surface roughness considering real polishing conditions. In this context, we believe that understanding and modelling the relative motion of the sample/abrasive grit kinematics for polishing processes is not only an intriguing problem to solve but is of crucial importance for the development of the polishing process in order to improve the workpiece surface integrity and the design of critical characteristics of polishing tools (e.g. sizes, shapes and distributions of the abrasive grits).

2. Theoretical model

Body 3, here considered as an abrasive tool, has two movements in the polishing process (figure 2b). One is the circular movement around the body 2 centre $(O_2)$ at an angular velocity $({\omega }_2),$ and the other is the ‘parasitic’ movement occurring within the slot of body 2 and partly limited by slot boundaries since body 3 has some degree of freedom to move with the slot of body 2. In particular, the ‘parasitic’ movement is mainly dependent on (i) clearance $(r_4-r_3)$ between bodies 2 and 3; (ii) eccentricity $(|{\mathbf{\text{O}}}_{12}|)$ of body 2 relative to body 1; (iii) rotational velocities ${\omega }_1$ and ${\omega }_2$ of bodies 1 and 2, respectively; (iv) pressure (P) on body 3, gravity (G) of body 3 and friction coefficient $({\mu }_{13})$ between bodies 1 and 3. For the first factor (i), the clearance directly affects the ‘parasitic’ movement. If the clearance is bigger, this movement would be accelerated as the range of the ‘parasitic’ movement is extended. The second (ii) and third (iii) both have an effect on the relative velocity between bodies 1 and 3, this is corresponding to the direction of the friction between bodies 1 and 3 (see the following equations (2.6) and (2.7), and their quantitative effects are shown in figure 5). The last one (iv) is related to the value of the friction force between bodies 1 and 3. Here, to simplify this problem, we consider that the change of friction coefficient resulting from the surface roughness is quite small and the value of friction force is relatively unchanged.

Figure 5.

Simulated trajectories of the body 3 centre validated by experimental data considering clearance (4 mm) between bodies 2 and 3. All the trajectories of the body 3 centre when bodies 1 and 2 have direct (a–d) and inverse (e–h) rotational directions are relative to the coordinate system $(X_2,Y_2,Z_2)$ with an origin (O₂). An imperfect circle trajectory with an ‘inflexion area’, characterized by a coverage angle $(\delta ),$ was generated by the effects of clearance, instead of a perfect circle as it was idealized in previous studies. The ‘inflexion area’ is obviously generated in some medium angular velocities of body 1, when the ‘slippage’ motion of body 3 from the inner to the outer side of the body 2 slot is fast accomplished. (Online version in colour.)

We work in a Cartesian coordinate system $(X_2,Y_2,Z_2)$ with the origin point coinciding with the body 2 centre $(O_2)$ (figure 3a). We choose the counterclockwise as the positive direction for the rotating bodies. To implement the prediction of surface micro-topography (roughness) that the abrasive particles of body 3 would generate on body 1, we need a model to describe body 3 trajectories, namely, to find a set of position parameters of points on body 3 at any instant (t_i).

Figure 3.

Theoretical foundations for the occurrence of the parasitic movement of body 3 within the slot of the carrier (body 2). Based on geometrical relationships among three bodies (a), forces applied on body 3 are analysed in the plane A1 (b) and the plane $X_1{O}_1{Y}_1$ (c) during the polishing process. We choose two examples to present the direction change of friction force $({\mathit{F}}_{r13})$ between bodies 1 and 3, that is, when bodies 1 and 2 have the direct (d) and inverse (e) rotational directions. There is a mirror relationship of friction force directions $({\mathit{F}}_{r13})$ when comparing the two rotational directions at various circumferential positions. The direction of friction force is opposite to the velocity direction of body 3 relative to body 1 at any given time. The relative velocity originates from ${\mathit{V}}_{r32}$, marked by solid blue arrows (d,e) and ${\mathit{V}}_{r13}$, marked by dash black arrows (d,e), and is equal to $({\mathbf{\text{V}}}_{r32}-{\mathit{V}}_{r13})$. (Online version in colour.)

(a). Positions of body 3 in the coordinate system $(X_2,Y_2,Z_2)$

First, the position of the slot centre in body 2, $O_4(x_4,y_4),$ can be expressed as

$$x_4(t_i)=|{\mathbf{\text{O}}}_{42}(t_i)|\cos ({\theta }_{40}+|{\mathit{\omega }}_2|t_i)$$ 2.1

and

$$y_4(t_i)=|{\mathbf{\text{O}}}_{42}(t_i)|\sin ({\theta }_{40}+|{\mathit{\omega }}_2|t_i),$$ 2.2

where ${\mathbf{\text{O}}}_{42}$ is the vector from the slot centre (O₄) to the body 2 centre (O₂) and ${\theta }_{40}$ is the initial azimuth of the slot centre (O₄) relative to body 2.

Then, the position of the body 3 centre $O_3(x_3,y_3)$ is

$$x_3(t_i)=x_4(t_i)+(r_4-r_3)\sin ({\theta }_{34}(t_i)+|{\mathit{\omega }}_2|t_i)$$ 2.3

and

$$y_3(t_i)=y_4(t_i)-(r_4-r_3)\cos ({\theta }_{34}(t_i)+|{\mathit{\omega }}_2|t_i),$$ 2.4

where $r_4$ is the radius of the slot, $r_3$ is the radius of body 3, and ${\theta }_{34}$ is the azimuth angle of the body 3 centre relative to the slot centre (O₄).

This ‘parasitic’ movement, which we aim to model, is caused by the forces exerted on body 3 (figure 3a–c); pressure (P) applied on body 3 to enable the polishing process, gravity (G) of body 3 and reaction $({\mathbf{\text{F}}}_{N1})$ provided by body 1 are balanced in the plane A1 all the time (figure 3b). In the $X_1{O}_1{Y}_1$ plane, the required centripetal force $({\mathbf{\text{F}}}_{\mathrm{c}})$, oriented towards the centre (O₂) of body 2 for circular motion, is provided by the friction force $({\mathbf{\text{F}}}_{r13})$ between bodies 1 and 3 and the reaction $({\mathbf{\text{F}}}_{N2})$ from body 2 (figure 3c). The required centripetal force $({\mathbf{\text{F}}}_{\mathrm{c}})$ is

$${\mathbf{\text{F}}}_{\mathrm{c}}(t_i)=-m{\mathbf{\text{O}}}_{23}(t_i){\mathit{\omega }}_2^2,$$ 2.5

where ${\mathbf{\text{O}}}_{23}$ is the vector from the body 2 centre (O₂) to the body 3 centre (O₃) and m is the mass of body 3.

The friction force $({\mathbf{\text{F}}}_{r13})$ between bodies 1 and 3 plays a key role in this process. The value change of friction force $({\mathbf{\text{F}}}_{r13})$ is relatively small during polishing, but its direction, opposite to the velocity direction of body 3 relative to body 1 varies obviously (figure 3d,e). We group the changes in directions of the friction force $({\mathbf{\text{F}}}_{r13})$ into two cases: when bodies 1 and 2 have the same (clockwise) and inverse (body 1 is counterclockwise, body 2 is clockwise) rotational directions. These relative velocities $({\mathbf{\text{V}}}_{r32}-{\mathbf{\text{V}}}_{r13})$ consist of two components from body 1 rotation (${\mathbf{\text{V}}}_{r13}={\mathit{\omega }}_1\wedge {\mathbf{\text{O}}}_{13}$, refer to dash black arrows in figure 3d,e) and body 2 rotation (${\mathbf{\text{V}}}_{r32}={\mathit{\omega }}_2\wedge {\mathbf{\text{O}}}_{23}$, refer to solid blue arrows in figure 3d,e), respectively. They vary when body 3 rotates to different circumferential positions of body 2, and thus, attribute to different directions of the friction force $({\mathbf{\text{F}}}_{r13})$. Values of friction force $({\mathbf{\text{F}}}_{r13})$ are determined by the friction coefficient $({\mu }_{13})$ of the contact surface between bodies 1 and 3, and the reaction $({\mathbf{\text{F}}}_{N1})$ in the plane A1, so that

$$\begin{array}{rl}{\mathbf{\text{F}}}_{r13}& ={\mu }_{13}|{\mathbf{\text{F}}}_{N1}|(-{\mathbf{\text{u}}}_{\mathrm{v}}(t_i)),\end{array}$$ 2.6

$$\begin{array}{rl}{\mathbf{\text{u}}}_{\mathrm{v}}(t_i)& ={[M_{r90}{\mathbf{\text{O}}}_{23}{(t_i)}^T{\mathit{\omega }}_2-M_{r90}{\mathbf{\text{O}}}_{13}{(t_i)}^T{\mathit{\omega }}_1]}^T|{[M_{r90}{\mathbf{\text{O}}}_{23}{(t_i)}^T{\mathit{\omega }}_2-M_{r90}{\mathbf{\text{O}}}_{13}{(t_i)}^T{\mathit{\omega }}_1]}^T{|}^{-1}\end{array}$$ 2.7

$$\begin{array}{rl}\text{and}{M}_{r90}& =\left[\begin{array}{cc}\cos \left(\frac{\pi }{2}\right)& -\sin \left(\frac{\pi }{2}\right)\\ \sin \left(\frac{\pi }{2}\right)& \cos \left(\frac{\pi }{2}\right)\end{array}\right]\end{array}$$ 2.8

where ${\mathbf{\text{u}}}_{\mathrm{v}}$ is the unit vector of the velocity of the body 3 centre (O₃) relative to the body 1 centre (O₁); $M_{r90}$ is a rotation matrix for 90° in the counterclockwise direction; ${\mathbf{\text{O}}}_{13}$ is the vector from the body 1 centre (O₁) to the body 3 centre (O₃). The unit vector $({\mathbf{\text{u}}}_{\mathrm{v}})$ denotes the direction of the relative velocity between bodies 3 and 1 centres, opposite to the direction of friction force $({\mathbf{\text{F}}}_{r13})$ between them. The change of the unit vector proves our analysis concerning the friction direction changes.

The reaction $({\mathbf{\text{F}}}_{N2})$ from body 2 always points to the moving centre of the slot (O₄) in body 2. The continuous contact between bodies 2 and 3 is a moving one, due to the changes of the friction force $({\mathbf{\text{F}}}_{r13})$ direction (see the violet arrows in figure 3d,e); then, the direction of the reaction $({\mathbf{\text{F}}}_{N2})$ changes accordingly. The position of the body 3 centre determines the required centripetal force $({\mathbf{\text{F}}}_{\mathrm{c}})$ for its principal motion and the friction force $({\mathbf{\text{F}}}_{r13})$ direction between bodies 1 and 3. Then, the reaction $({\mathbf{\text{F}}}_{N2}),$ the difference between ${\mathbf{\text{F}}}_{\mathrm{c}}$ and ${\mathbf{\text{F}}}_{r13}$, is determined, which has an effect on the position of body 3 centre in return. In other words, for given fixed parameters, body 3 follows the same repetitive trajectory around the body 2 centre with time. In the formulation of the position of the body 3 centre (equations (2.3) and (2.4)), the azimuth angle $({\theta }_{34})$ of the body 3 centre (O₃) relative to the slot centre (O₄) is a key to depict its relative positions within the slot of body 2. The aim of our model is to find the azimuth angle $({\theta }_{34})$ for obtaining the positions of the body 3 centre at any instant (t_i). In our model, we neglect the friction force between body 3 side and body 2 boundary walls as it tends to be a rolling resistance with a small value which will not influence the nature of this phenomenon we studied (i.e. the clearance effects).

Owing to the existence of the clearance between bodies 2 and 3, a ‘parasitic’ movement of body 3 occurs with its principal motion that includes two components: a ‘slippage’ effect and a pseudo-rolling.

The ‘slippage’ effect induces changes in the position of the body 3 centre (O₃) relative to the slot centre (O₄) within body 2 (i.e. the azimuth angle ${\theta }_{34}$). When obvious changes in the direction of the friction force $({\mathit{F}}_{r13})$ occurs, this can result in an ‘inflexion area’ in the trajectory of the body 3 centre (i.e. sudden changes in the contact position between bodies 2 and 3): see, as examples, the areas in which ${\psi }_2$ is from 240° to 270° in figure 3d, and from 60° to 90° in figure 3e for our set-up and some fixed parameters. The ‘inflexion area’ happens at the moment $t_i$

$$\begin{array}{rlrl}{\theta }_{34}(t_i)& ={\theta }_{\text{34\_max}}& & \text{when}{\theta }_{34}(t_1,t_2,t_3,\dots ,t_n)<{180}^{\circ },\hspace{0.17em}\text{or}\\ {\theta }_{34}(t_i)& ={180}^{\circ }& & \text{when}{\theta }_{34}(t_1,t_2,t_3,\dots ,t_n)\ge {180}^{\circ }\end{array}\}$$ 2.9

where ${\theta }_{\text{34\_max}}$ is the maximum value of ${\theta }_{34}$ during the given time $t_n$.

The motion of body 3 relative to the slot in body 2 is a pseudo-rolling as body 3 does not roll along the boundary of the slot in body 2. In fact, it is because of the position of continuous contact between bodies 2 and 3 changes with time for the principal movement of body 3.

With the goal of starting this calculation of body 3 centre trajectories, the azimuth angle $({\theta }_{34})$ is set initially to zero and incremented up to 360° while using equations (2.3) and (2.4). Based on the trajectories of the body 3 centre $(O_3(x_3,y_3))$ relative to the body 2 centre (O₂), the trajectory of any point on body 3 can be expressed as following

$$x_{\text{3e\_sa}}(t_i)=r_{\text{3e\_sa}}\cos ({\theta }_{\text{3e\_sa}})$$ 2.10

and

$$y_{\text{3e\_sa}}(t_i)=r_{\text{3e\_sa}}\sin ({\theta }_{\text{3e\_sa}})$$ 2.11

where $r_{\text{3e\_sa}}$ and ${\theta }_{\text{3e\_sa}}$ are the distance and the azimuth angle of the calculated point on body 3 relative to the body 2 centre (O₂), respectively; sa is the index of the calculated points on body 3. Thus, the trajectories of the centre and any other points on body 3 relative to the body 2 centre can be expressed by equations (2.3), (2.4), (2.10) and (2.11).

(b). Positions of body 3 in the coordinate system $(X_1,Y_1,Z_1)$

With the equations for the movement of the body 3 centre verified, it is now possible to build the trajectories for the multiple abrasives of body 3 and derive the surface roughness that these generate on the workpiece, i.e. body 1. For the study of workpiece surfaces, we need to transfer these trajectories of points on body 3 from the coordinate system $(X_2,Y_2,Z_2)$ to the coordinate system $(X_1,Y_1,Z_1)$ with origin point coinciding with body 1 centre (O₁). Hence, the transferred positions of body 3 points are

$$x_{1\mathrm{\_}3\mathrm{e}\mathrm{\_}\mathrm{sa}}(t_j)=|{\mathbf{\text{O}}}_{1\mathrm{\_}3\mathrm{e}\mathrm{\_}\mathrm{sa}}(t_j)|\cos [{\theta }_{\text{3e\_sa\_1}}(t_j)+|{\mathit{\omega }}_1|t_{i-j+1}]$$ 2.12

and

$$y_{\text{1\_3e\_sa}}(t_j)=|{\mathbf{\text{O}}}_{1\mathrm{\_}3\mathrm{e}\mathrm{\_}\mathrm{sa}}(t_j)|\cos [{\theta }_{\text{3e\_sa}\mathrm{\_}1}(t_j)+|{\mathit{\omega }}_1|t_{i-j+1}]$$ 2.13

where $t_{i-j+1}(1\le i\le n,n=t/\mathrm{\Delta }t)$ denotes the (i−j + 1)th instant; $t_j$ denotes the jth instant; ${\mathbf{\text{O}}}_{1\mathrm{\_}3\text{e\_sa}}$ is the vector from the body 1 centre to body 3 points; ${\theta }_{\text{3e\_sa\_1}}$ is the azimuth angle of the body 3 points to body 1.

Note that the surface roughness derivation is based on the material removed (i.e. scratches) by the geometrically controlled abrasives and their trajectory distribution. Please consider that, here, we aimed to associate each scratch to the trajectory of a particular abrasive (i.e. to prove the validity of our model) and for this reason reduced the density of scratches were generated on the workpiece material; hence, common parameters of surface roughness are not relevant.

3. Material and methods

The experiments have been carried out on a 5-axis machine (Hurco VMX42SRTi, position repeatability: 0.005 mm, rotary repeatability: 7 arc seconds), figure 4a which allows the setting of the necessary rotational speed of a set-up and replicates, at high level of accuracy, the relative movements between the workpiece, the carrier and the polishing tool that can occur on a conventional polishing machine; additionally, the machine tool provides ample space around the set-up to install the motion capture system (to be described in the following). As we performed an inverted polishing test using such a polishing set-up (figure 4b), body 1 is the workpiece which is mounted on the machine worktable that provides a rotational movement around the z-axis. Body 2 is the carrier of the polishing tool with a shaft clamped in the machine spindle which is eccentric to the rotational axis of the worktable. The carrier has two circular posts in which the abrasive tools are inserted and pressed against the workpiece with a calibrated spring (max. 30 N) which is part of the body 2. Body 3 is the polishing tool placed in the slot of the carrier with the large clearance (4 mm) which replicates some values used on the real polishing set-ups in two directions with different angular velocities (direct rotation direction: 2.5, 3.0, 4.5 r.p.m. of workpiece and 11.0 r.p.m. of the carrier clockwise; inverse direction: 3.0, 7.0, 9.0 r.p.m. of workpiece counterclockwise and 11.0 r.p.m. of the carrier clockwise).

Figure 4.

Eight Vicon cameras were installed around the spindle of the 5-axis machine (a,e), with an aim to capture the markers (b) motion on the engineered abrasive tool (c,d) and the carrier. (Online version in colour.)

A plate of aluminium (Al) has been used as the workpiece (diameter 300 mm) as it provides only one cutting mode, i.e. ductile, and allows easy tracking of the scratches left by the abrasives of the polishing tool. To enable validation of the model for the prediction of the surface roughness while considering the ‘parasitic’ movement between bodies 1 and 3, we have chosen to use a polishing tool (figure 4c,d) that has controlled shapes (square frustum 0.45 × 0.45 × 0.7 mm in length, width and height) and spacing (8 mm) of the abrasives which has been realized by laser ablation a PolyCrystalline Diamond (diameter 22 mm, height 12 mm) plate. This is a reasonable way to conduct the experiments since a conventional polishing tool has random abrasives and it will be nearly impossible to trace and relate single scratch to particular abrasives and thus to validate the proposed model. Although some authors try to make this relationship, we believe that the approach is hardly reasonable for studying the minute ‘parasitic’ movements of the bodies 1 and 3. Hence, in this experiment, we used the ‘engineered’ polishing tool.

A Vicon motion capture system has been used to trace the movement of the polishing tool based on the optical signal acquisition in three dimensions. Eight Vicon MX-T40S cameras (2352 × 1728 QXGA, 400 million pixels, and 100 Hz sample frequency) are placed in such a way around the worktable inside and outside the 5-axis machine (figure 4a,c) to obtain a complete motion. Four Φ6.35 mm markers attached on the abrasive tool were used as the tracking target points (figure 4b), and two additional markers are placed on the workpiece and carrier of the polishing set-up to verify their motions, respectively. A specialized (Nexus) software is used to collect the motion data with 0.1 mm accuracy and analyse the trajectories of moving bodies. Then, the simulation results calculated by the Matlab are validated to trial data captured by the Vicon system.

A surface roughness instrument (Alicona) has been used to three-dimensional (3D) map the scratches obtained on the workpiece (body 1) when in contact with the engineered polishing tool (body 3) with controlled shapes and densities of abrasives. The workpiece surface topography obtained by the test with clearance between the polishing tool (body 3) and the carrier (body 2) has been 3D scanned to enable the validation of the proposed models for surface roughness. It is worth mentioning that the polishing tests have been carried out only for one rotation of the body 2 around its axis to enable accurate traceability of the scratches of the abrasives on body 1 and therefore, the validation of the ‘parasitic’ movement that is the object of our proposed study.

4. Results and discussion

To validate the model for the relative movement between abrasives and abraded bodies, an ‘inverted’ polishing set-up (i.e. abrasive tool as body 3; workpiece as body 1) has been installed on a machine tool, thus allowing precise control of velocities and positions of each element of the polishing system.

(a). ‘Slippage’ effects in the coordinate system $(X_2,Y_2,Z_2)$

The polishing set-up is supported by eight cameras of a motion capture system mounted in such a way that they can trace the trajectories of the markers installed on the three bodies (electronic supplementary material, Data file S1) and thus, to enable the validation of equations (2.3) and (2.4), i.e. body 3 trajectories.

Although our models allow the simulation of the relative movements of bodies 1 and 3, we have chosen here to verify our modelling framework on some relevant polishing parameters: clearance between bodies 2 and 3 (r₄–r₃ = 4 mm), eccentricity of body 2 relative to body 1 $(|{\mathbf{\text{O}}}_{12}|=90\hspace{0.17em}\text{mm}),$ rotational angular velocities of bodies 1 and 2 (ω₁ = −2.5, −3.0, −4.5, 3.0, 7.0, 9.0 r.p.m., ω₂ = −11 r.p.m.) contact pressure applied on body 3 $(|\mathbf{\text{P}}|=15\hspace{0.17em}\text{N}),$ gravity of body 3 $(|\mathbf{\text{G}}|=0.08\hspace{0.17em}\text{N}),$ coefficient of friction between bodies 1 and 3 $({\mu }_{13}=0.25)$.

Figure 5 shows some examples of the trajectories of the body 3 centre in the coordinate system $(X_2,Y_2,Z_2)$ when bodies 1 and 2 have direct (figure 5a–d, electronic supplementary material, Movie S1 and S2) and inverse (figure 5e–h, electronic supplementary material, Movie S3 and S4) directions of rotations. All simulated results (see solid red lines in figure 5) for different angular velocities of body 1 are presented against the experimental data (see solid blue lines in figure 5) acquired by the motion tracking system (Vicon — details in the Material and methods). The experimental and simulated trajectories are overlapped in figure 5a–c and e–g, as there are small errors between them. Details refer to the magnified inserts (figure 5d,h). It was found that the maximum deviations between the simulated and experimental trajectories, measured along the radius direction of body 2 are 0.49 and 0.97% in the direction of rotation of body 2 (table 2), where $r_e,r_s,r_i$ are the distances from the body 2 centre to the points on experimental trajectories with clearance, simulated trajectories with clearance and idealized trajectories without clearance, respectively.

Table 2.

Maximum deviations between the simulated versus experimental and simulated versus idealized results of the trajectory of the body 3 centre measured along the radial direction of the body 2.

		maximum deviations between simulated and experimental trajectories with clearance ($d_{\mathrm{se}}$)		maximum deviations between simulated and idealized trajectories without clearance ($d_{\mathrm{si}}$)
conditions ${\mathit{\omega }}_2=-11.0\hspace{0.17em}\mathrm{RPM}$		value (mm) $d_{\mathrm{se}}=|r_{\mathrm{s}}-r_{\mathrm{e}}|$	percentage $d_{\mathrm{se}}=|r_{\mathrm{s}}-r_{\mathrm{e}}|/r_{\mathrm{e}}\times 100\mathrm{\%}$	value (mm) $d_{\mathrm{si}}=|r_{\mathrm{s}}-r_{\mathrm{i}}|$	percentage $d_{\mathrm{si}}=|r_{\mathrm{s}}-r_{\mathrm{i}}|/r_{\mathrm{e}}\times 100\mathrm{\%}$
direct direction	${\mathit{\omega }}_1=-2.0\hspace{0.17em}\text{r.p.m.}$	0.15	0.47	3.58	13.41
	${\mathit{\omega }}_1=-3.0\hspace{0.17em}\text{r.p.m.}$	0.13	0.49	4.00	15.48
	${\mathit{\omega }}_1=-4.5\hspace{0.17em}\text{r.p.m.}$	0.11	0.33	4.00	15.41
inverse direction	${\mathit{\omega }}_1=3.0\hspace{0.17em}\text{r.p.m.}$	0.15	0.53	3.21	12.11
	${\mathit{\omega }}_1=7.0\hspace{0.17em}\text{r.p.m.}$	0.20	0.73	4.00	15.41
	${\mathit{\omega }}_1=9.0\hspace{0.17em}\text{r.p.m.}$	0.30	0.97	4.00	15.40

Also, our simulations and experimental validations of the trajectory of the abrasive tool (body 3) centre in the real polishing conditions (existence of clearance) and various angular velocities of body 1 show characteristics, i.e. deviation from roundness caused by the ‘parasitic’ movement previously described, that the idealized case (without clearance see dark dash lines in figure 5) simply is not able to capture. The shape of an imperfect circle trajectory presents an ‘inflexion area’ defined by a coverage angle $\delta$ (colour dark blue in figure 5) due to the existence of a clearance, which is the consequence of the abrasive tool ‘slippage’ when the friction force changes its direction; this leads the trajectory of the body 3 centre that is not a perfect circle around body 2 centre (see black dash line in figure 5). This ‘inflexion area’ in the trajectory starts from such a point when the unit velocity vector of the body 3 centre (O₃) relative to the body 1 centre (O₁) satisfies equation (2.9). The smaller angle $\delta$ indicates a higher ‘amplitude’ of the ‘inflexion area’ that is caused by a faster changing rate of the unit velocity vector $(|{\mathbf{\text{u}}}_v(t_{i+1})-{\mathbf{\text{u}}}_v(t_i)|/(t_{i+1}-t_i))$ of the body 3 centre (O₃) relative to body 1 (O₁). That is, the ‘slippage’ of body 3 from inner to the outer side of slot will be accomplished faster if the friction force $({\mathbf{\text{F}}}_{r13})$ direction changes rapidly in unit time (see the position change of body 3 within the body 2 slot from moment $t_i$ to $t_j$ in figure 5c,g).

With the scope to observe the effects of clearance upon the body 3 centre trajectories, we choose as a study case with a medium angular velocity of body 1 (${\omega }_1$ = −3.0 r.p.m.) with a more apparent ‘inflexion area’. It is similar to the results when bodies 1 and 2 rotate in inverse directions, that is, the ‘inflexion area’ of the $O_3$ trajectory is obtained also with a medium angular velocity of body 1 (ω₁ = 7.0 r.p.m., figure 5g,h). The ‘inflexion area’ does not remain stationary but moves along the rotational direction of body 2 (clockwise in our study case) when body 1 angular velocity increases (see the coverage angle $\delta$ in figure 5a–c and e–g). It was also observed that the ‘inflexion area’ presents an opposite orientation of occurrence when the rotational directions of bodies 1 and 2 are direct and inverse based on the comparison between figure 5c,g. Reasons for above two observations are the changes of relative velocities linked with friction force $({\mathbf{\text{F}}}_{r13})$ direction between bodies 1 and 3 due to the variation of body 1 angular velocities as it was depicted in figure 4d,e.

(b). Pseudo-rolling in the coordinate system $(X_1,Y_1,Z_1)$

As with randomly distributed abrasives expected on a polishing tool, it would be difficult to trace single trajectories and prove our surface roughness model, we have taken here the approach to verify the workpiece surface roughness (body 1) obtained by using a polishing tool (body 3) only for seven identical and ‘precisely placed’ abrasives (see in figure 4c,d). On this basis, we performed simulations and experiments (electronic supplementary material, Data file S2) of the scratches obtained by body 3 on the surface of the workpiece (body 1) with changes of some relevant parameters: coefficient of friction, ${\mu }_{13}=0.26$; gravity of body 3, $|\mathbf{\text{G}}|=0.36\hspace{0.17em}\text{N}$; pressure, $|\mathbf{\text{P}}|=2\hspace{0.17em}\text{N}$; body 1 angular velocity, ω₁ = −3.0, 7.0 r.p.m.; body 2 angular velocity, ω₂ = −11 r.p.m. Note that the mechanism of material removal (i.e. formation of scratches) is not the object of our models, and these observations aim to contribute to the understanding of workpiece surface roughness and morphology. Figure 6a,d show the outcomes of simulated seven scratches left on the workpiece surface considering the clearance after one rotation of the carrier (body 2) when bodies 1 and 2 have direct (figure 6a) and inverse (figure 6d) rotational directions. Each simulated scratch (marked by solid lines in figure 6a,d) was validated against experiments (marked by circles) obtained using our engineered abrasive tool on aluminium plates (diameter is 300 mm); these are also compared with the scratches that would have been obtained without clearance (marked by dash lines), as idealized by previous researchers. The average maximum deviation between our simulated and experimental scratches is around 0.35%.

Figure 6.

Based on the scope of scratches obtained by our abrasive tool (body 3) left on the workpiece (body 1) after one rotation of body 2 (a,d), the validation of simulated versus experimental results is performed for each abrasive scratch as well as the comparison with idealized scratches (without the clearance between bodies 2 and 3). We choose the scratch ‘sharp turns’ to (colour grey in a and d) to observe the apparent difference between the three scratches when bodies 1 and 2 have the direct (b) and inverse (e) rotational directions. Such crossings and turnings of scratches left on the workpiece caused by the clearance between bodies 2 and 3 are observed as we predicted in our surface model (c,f). (Online version in colour.)

Here again, for the experiments with the abrasive tools (figure 6), some obvious deviations from the ideal case (without clearance) occur in the trajectories each abrasive on the body 3 due to the existence of clearance between the abrasive tool (body 3) and carrier (body 2). We choose to carefully observe the scratch ‘sharp turns’ (a grey area in figure 6a,d) and magnify these zones (see in figure 6b,e) to put in evidence the detail that our model is able to capture.

The motion of body 3 centre relative to body 1 can be depicted as a rotation with a constant angular velocity (ω₁) but variable radius .. which are attributed to body 3 rotation around body 2 centre and ‘slippage’ within body 2 slots. Given angular velocities of bodies 1 and 2 in the same rotational direction, the scratch trajectories present ‘sharp turns’ (see the area colour grey in figure 6a,d) in which the slot centre (O₄) in body 2 rotates to the farthest position to the body 1 centre (O₁) (see $|{\mathbf{\text{O}}}_{14}{|}_{\text{max}}$ in the magnified figure 6a). For the inverse rotational direction of bodies 1 and 2, ‘sharp turns’ of scratch trajectories are presented in the area where the distance between the body 2 slot and body 1 centres is minimum (see $|{\mathbf{\text{O}}}_{14}{|}_{\text{min}}$ in the magnified figure 6e). Thus, when considering a real polishing case with clearance between bodies 2 and 3, scratches trajectories near the ‘sharp turns’ are more intricate. The distance between bodies 1 and 3 centres $(|{\mathbf{\text{O}}}_{13}|)$ is not equal to the distance between body 1 and the slot centre .. as idealized and therefore, the trajectory of the centre scratch in the real case shows a turning and a migrating in the position of the whole trajectory (compare dark red solid and dash lines in figure 6b,e). The centre scratch represents the general term of all abrasive trajectories on body 3, which means all eccentric abrasives scratches include the changing characteristic of the centre scratch.

Moreover, the effects of pseudo-rolling of body 3 (the abrasive tool) within the slot of body 2 (the carrier) in the condition of clearance make the eccentric scratches even more complex with elapsed time when these scratches cross due to the further rotation of body 1 (the workpiece). Some abrasive scratches (e.g. no. 1 colour orange and no. 6 colour violet eccentric abrasive scratches in figure 6b for our study case) cross with themselves (see in figure 6c), as a consequence of the ‘slippage’ motion of body 3 and rotations of bodies 1 and 2. The crossing areas indicate that the material is repeatedly removed by the same abrasive with a deeper indentation on the workpiece surface than non-crossing areas (figure 6c). Other scratches show such a turning in the trajectories far from the idealized scratches without clearance between bodies 2 and 3 (see in figure 6f). It can be observed that the intricate deviations of abrasive scratches between the case of clearance and no clearance (idealized) occur at the scratch ‘sharp turns’ where the slot centre rotates to extreme positions with the closest or farthest distance to the body 1 centre. Simultaneously, body 3 apparently slips from the inner side to the outside of body 2 slot boundaries due to the presence of the clearance between bodies 2 and 3. This coincidence between the scratch ‘sharp turns’ and body 3 ‘slippage’ motion is because both of them are highly dependent on the ratio of bodies 1 and 2 angular velocities and the eccentricity of body 2 relative to body 1. In other words, the position of the most significant scratch deviation between the case of clearance and no clearance is mainly determined by the ratio of bodies 1 and 2 angular velocities and the eccentricity of body 2 relative to body 1, while the magnitude of the scratch deviation rests with the value of the clearance between bodies 2 and 3. Thus, the clearance between bodies 2 and 3 is of importance to the prediction of workpiece surface roughness due to its non-negligible effects on abrasive scratches. Apart from providing an accurate prediction of the scratches of the abrasives on the workpiece, our model and approach allow the capture of intricate effects (e.g. scratches from a single abrasive crossing/overlapping themselves) that idealized models (i.e. without clearance) simply miss.

With the validation of the proposed model applied to each abrasive trajectory, we can now simulate the trajectories (scratches) of multiple abrasives after any required rotation of body 2 (time domain). Figure 7a shows examples of surface scratches obtained after five, ten and one hundred rotations of the carrier— body 2 (elapsed time: 27, 55, 546 s) on the workpiece surface employing the same tool with seven abrasives, in the condition of clearance and no clearance between bodies 2 and 3 and when body 1 has the direct rotation direction to body 2. It shows such a rule of periodicity in scratch trajectories when integer rotations of bodies 1 and 2, in an elapsed time, meet the requirement, that is, $C_1/{\mathit{\omega }}_1=C_2/{\mathit{\omega }}_2$, where C₁ and C₂ are the number of revolutions of body 1 and body 2, respectively, in the given time. Hence, we define that the periodicity of scratch trajectories (T) is the elapsed time when no new scratch trajectories are generated, i.e. scratch trajectories overlap. Apart from the angular velocities of bodies 1 and 2, this scratch periodicity is also influenced by the eccentricity of body 2 relative to body 1 which influences the scratch ‘sharp turns’ and body 3 ‘slippage’ motion within body 2. When the periodicity of trajectories (T) is longer, more evenly distributed trajectories can be obtained. In this case, after more rotations of body 2 (see last two figures in figure 7b), it is well observed there are two levels of deviations, i.e. ‘micro’ and ‘macro’, between the cases of clearance and no clearance.

Figure 7.

Examples of comparisons between scratch trajectories in conditions of clearance and no clearance with time when bodies 1 and 2 have the direct (a) and inverse (b) rotational directions. For given angular velocities of bodies 1 and 2 in the inverse direction (b), it shows a longer periodicity of trajectories (T) than the one in the direct direction (a). After a longer elapsed time (e.g. 100 rotations in the inverse direction), there is a ‘micro’ deviation between the simulated trajectories with clearance and idealized trajectories without clearance (c). It is because of the migrating, turning and crossing of abrasives caused by the clearance between bodies 2 and 3. The ‘micro’ deviations stimulate a ‘macro’ deviation occurring in the boundaries (inner and outer circles) of the whole trajectories, by changing the coverage area of the trajectories (d). (Online version in colour.)

At the ‘micro’ level (figure 7c), due to the migrating, turning and crossing of abrasives scratches caused by the clearance between bodies 2 and 3, the scratch trajectories will be changed in shape as well in their distribution on the workpiece surface. As example, we choose two zones to observe and count the numbers of scratches per unit area after a particular polishing time (546 s), and this resulted in a minimum 10% increase of scratch density when comparing the results of our simulation and the idealized (without clearance) one; of course, this difference is dependent on the area to be selected and polishing time. However, when comparing the results of the simulations based on our model and the idealized case, we could also observe ‘macro’ deviations which represent a relative variation on the scratches amplitude measured on the radial direction; here, the difference between the polishing with and without clearance was of min. 6% (figure 7d).

We acknowledge that the clearance is depending on the industry practice and our observations and models on the parasitic movement become more relevant at bigger values of the clearance. This paper does not replicate immediately the industrial set-ups, but it aims to highlight to the curious researchers that such a phenomenon exists and to appreciate its implications on the polishing/lapping set-ups. We proved this curious phenomenon using tools with abrasives of controlled distribution and shapes as it gives a certainty in model validation. While such abrasive tools exist [25–30], we appreciate that most of the tools present abrasives with stochastic distribution of shapes, densities and protrusions; nevertheless, our observations and models stand valid also in these circumstances.

Note that although these differences in surface morphology between the real (with clearance) and idealized (without clearance) polishing set-up might seem small, the reader should bear in mind that, here, we considered a very simplified polishing tool, which contains only seven abrasives, compared with the real ones that have hundredths of time more abrasive grits per considered active area. Hence, the difference in surface morphology that our model can capture relative to the idealized models is likely to be much higher.

5. Conclusion

Previous models of polishing processes have been simplistic and idealized, and for this reason, the prediction of resulting workpiece surface morphology still relies on empirical approaches. This is because the inevitable clearance between the sample (i.e. body 3, the abrasive tool in our study case) assembled within carrier slots and the carrier (body 2) has been neglected in previous studies.

Our study gives a good fundament of understanding how the clearance mentioned above is the source of a ‘parasitic’ relative movement between abrasives and workpiece. Here, we proposed a mathematical model to reveal the real polishing condition, i.e. trajectories of points on the moving bodies, considering the clearance effects to enable the prediction of workpiece surface roughness in polishing processes. Simulation results have been validated against experimental data using motion capture system, first against the centre of the moving bodies to verify the relative movements and then, against scratches obtained by engineered abrasive tools on the workpiece surface; low average relative errors (<1%) have been observed when comparing the experimental results with our model predictions.

Our model is able to capture intricate movements (e.g. self-crossings, turnings, trajectory migrations) of the abrasives against the workpiece surface which significantly affect the workpiece surface morphology; these effects are simply ‘non-existent’ for idealized (without clearance) polishing approaches. We validated (max. error of 0.35%) our models against an inverted polishing set-up where an engineered abrasive tool has well-controlled abrasives so that we could identify with certainty their trajectories. Further, we have proven the relevance of our modelling approach by simulating the workpiece surface morphology obtained after different polishing times and found, again, significant differences (e.g. for our set-up: min. micro-errors of 10%, e.g. min. macro-errors of 6%) when compared with the models based on the idealized set-up. Although the case study presented here is based on a simplified engineered polishing tool, our modelling framework which considers a pseudo-compliance between tool and workpiece, indicates that with the use of real abrasive tools (containing a multitude of abrasive grits) the workpiece micro and macro topology is likely to be much closer to the real process outputs when compared with the idealized case.

Data accessibility

The raw data for experimental results have been provided in electronic supplementary material ‘Data file S1.csv’ and ‘Data file S2.csv’. Videos for experimental and simulated results have been provided in electronic supplementary material ‘Movie S1.mp4’, ‘Movie S2.mp4’, ‘Movie S3.mp4’ and ‘Movie S4.mp4’. Codes/algorithms have been provided in electronic supplementary material ‘4.1 trajectories relative to body 2.m’ and ‘4.2 trajectories relative to body 1.m’.

Authors' contributions

D.A. identified the research idea; Y.Y., H.L., Z.L. and D.A. developed the model and designed the experiments; Y.Y. developed the codes/algorithms, performed the experiments (with assistance of W.Q. Zhu) and analysed the results; Y.Y., H.L., Z.L. and D.A. discussed the results; Y.Y., H.L. and D.A. prepared the figures; Y.Y. and D.A. prepared the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The authors acknowledge the supports from National Natural Science Foundation of China (grant nos 51805281 and 51975302), Ningbo 3315 Innovation Team Scheme (grant no. 2018A-08-C), University of Nottingham Ningbo China (grant nos I01180800099, I01190100001) and Research Project of State Key Laboratory of Mechanical System and Vibration (grant no. MSV201908).