Path planning of mechanical polishing process for freeform surface with a small polishing tool

Weiyang Lin; Peng Xu; Bing Li; Xiaojun Yang

doi:10.1186/s40638-014-0024-1

Research
Open Access
Published: 30 December 2014

Path planning of mechanical polishing process for freeform surface with a small polishing tool

Weiyang Lin^1,2,3,
Peng Xu^1,4,
Bing Li¹ &
Xiaojun Yang¹

Robotics and Biomimetics volume 1, Article number: 24 (2014) | Download Citation

3866 Accesses
5 Citations

Abstract

Products with freeform surface are widely applied in industries, and the surface quality plays an important role in order to fulfill the targeted functions. As polishing path of small polishing tool affects the polishing removal function considerably, it is highly necessary to study the polishing path of freeform surface for obtaining good polishing efficiency and well-proportioned surface quality. By combining the Preston polishing removal function, the material removal model of small polishing tool under the control of constant polishing force and pressure is established. Based on this model, the material removal functions of scan line, Archimedean spiral, and Hilbert fractal polishing path are derived. The simulation results show that the Hilbert fractal polishing path has the best comprehensive performance. By using the projection relation of differential geometry, the optimal path generation algorithm of the Bézier surface based on Hilbert fractal polishing path is established. The polishing experiments are conducted on a self-developed polishing machine which is based on a parallel manipulator. The experimental results demonstrate that the surface roughness is improved from level 9 to level 11.

Background

Product structure can be optimized by using freeform surfaces, which opens the door for solutions with improved performance, reduced complexity, lower mass, and smaller size. However, due to the geometric particularity, freeform surfaces face more challenges and difficulties in precision manufacturing. Polishing is usually one of the final processing steps of precision manufacturing, and the results directly affect the appearance and longevity of parts [1]. It is fundamentally different from other precision manufacturing technologies. Removal of polishing does not only depend on the position of the tool orthogonal to the workpiece, as for grinding and cutting processes but also proportional to the product of local pressure and relative-speed between tool and workpiece and the dwell time. In recent years, there were efforts to develop versatile polishing processes in order to achieve high accuracy. The current polishing methods mainly include electrochemical polishing [2], magnetorheological fluid polishing [3],[4], plasma polishing [5],[6], ultrasonic polishing [7],[8] and computer-controlled mechanical polishing [9],[10]. Nowadays, a lot of freeform components are still produced by final manual polishing. It not only heavily relies on the know-how and experience of technicians but also needs much attention for processing and testing. To achieve a given level of precision with high efficiency and reliability, process automation is clearly the way forward. As the computer-controlled mechanical polishing has high efficiency and can be controlled easily [11], it is the focus of this study. The mechanical polishing is a statistical ‘rubbing’ process that the microscopic loose-abrasive particles in the polishing liquid which is driven by high-speed rotational polishing tool could produce friction with the part surface. Protruding portions of the surface are removed to meet the roughness requirement. However, due to the various factors, the polishing process heavily relies on trial and experience, which leads to a slow development in the mechanical polishing process.

Compared with the large polishing tool, the small polishing tool used in this study has many advantages [12]. The small tool can follow the freeform surface with relatively large curvature, while large tool cannot polish the freeform surface with curvature smaller than its radius. The small tool can redress the error of local surface, while a large tool may polish the nearby surface when it polishes a local surface. The small can also operate at greater pressure and velocity than a large tool; therefore, it can remove the material in a rapid manner. In the case of manufacturing one single piece, small polishing tool has a higher polishing efficiency.

As the polishing surface is highly nondeterministic, one of the reasons is that the polishing path of the tool affects the removal considerably. A basic requirement for polishing paths is that the surface can be completely and uniformly covered during a polishing cycle. Therefore, the study of removal function of different polishing path has a very important significance. A lots of polishing paths in mechanical polishing process can be found in literatures [13]-[18]. The ultimate goal of the research about different polishing removal function is to select an optimal polishing path so as to avoid any under-polishing and over-polishing phenomenon and ensure the uniformity of the polishing process. Meanwhile, it can also present a clear understanding of the various factors that influence the polishing quality. The core issue of this research is to study the polishing removal function and the polishing performance in a given surface shape and a specified path, so that it can facilitate the subsequent polishing path control.

The rest of the paper is organized as follows. In the ‘Theoretical background’ section, some basic polishing assumptions and fundamental theories are introduced. In the ‘Methods’ section, the scan line path, Archimedean spiral path, and Hilbert fractal path are described and analyzed in details, while the modeling method of polishing path from plane surface to freeform surface is also given. The simulation and experiment are conducted, and the results are depicted in the ‘Results and discussion’ section. Conclusions are drawn in the last section.

Theoretical background

Removal rate of polishing

Removal rate of polishing, which depicts the properties of part surface’s removal amount at a local polishing position during every unit of time, is an important part of study about polishing process [19],[20]. Many factors affect the polishing removal rate, such as polishing tool and polishing environment [21]-[23]. For the polishing tool, the main factors are material property, shape, and surface roughness. While for the polishing part, the results are relevant to material property and friction coefficient between the part and the polishing tool. Additionally, for the polishing environment, the main factors include the particle size and viscosity of polishing paste and polishing liquid, the temperature and pressure during polishing and the chemical correction of part in the polishing liquid.

Preston presented theoretical basis for the prediction of material removal for mechanical polishing [18]. Although there are many other polishing formulations in polishing literatures, most of them are based on the Preston equation. Tseng and Wang [24] proposed a modification equation based on the Preston equation in chemical-mechanical polishing; Nanz [25] considered the bending of pad and flow of slurry into the Preston equation to obtain a new polishing formula. The formula of Preston polishing removal function is displayed as below:

f (q →, t) = d z d t = k · p (q →, t) · v (q →, t)

(1)

where $f (q →, t)$ is the material removal in unit time; $q →$ is the position vector of polishing point; t is the time; k is the Preston coefficient which is related to the material, polishing liquid, and environment; $p (q →, t)$ is the pressure at the polishing point; and $v (q →, t)$ is the instantaneous polishing velocity at the polishing point.

Actually, the problem of polishing removal rate is a very complex nonlinear problem. Preston polishing removal function is a linearization assumption, but it is approximately valid in many practical polishing experiments.

Removal amount of polishing path

In terms of mechanical polishing, the polishing velocity and polishing pressure can be controlled by computer numerical control (CNC) machine precisely. Therefore, during the polishing process, the Preston coefficient k is regarded as constant. Moreover, the polishing tool is usually made of relatively soft materials. At the micro level, the soft tool is adaptive to the shape of surface. It can keep the polishing tool and word-piece contact with each other throughout the polishing process. Thus, the polishing pressure applied by small polishing tool on freeform surface is assumed to be uniform; that is to say, the pressure of contact surface is equal everywhere at the same time.

At time τ and point $q →$ in the part coordinate system, the polishing velocity and pressure are set as $v (q →, τ)$ and n(τ), respectively. When the radius of polishing tool is R, the polishing removal amount is defined as follow:

z (x, y, t) = ∫ 0 t k · n (τ) 2 π · R 2 · v (q →, τ) d τ

(2)

The polishing pressure during the polishing process can be divided into the constant polishing pressure n_s and the time-varying polishing pressure n_v(τ), namely n(τ)=n_s+n_v(τ). However, in order to simplify the subsequent discussion, we assume that the polishing force remains constant during the polishing process. Then, the polishing removal function can be rewritten as:

z (x, y, t) = k · n s 2 π · R 2 · ∫ 0 t v (q →, τ) d τ

(3)

Equation 3 is a general polishing removal function. Given different polishing paths, the polishing velocity in Equation 3 will change correspondingly, and this will result in different integration results.

Removal function at a fixed point

Figure 1 is the diagram of polishing model at a fixed point. When the polishing tool is stationary and rotates with angular velocity ω_t, the polishing removal function of the point with radius r to the center of polishing tool is described as:

z (r, t) = k · n s 2 π · R 2 · r · ω t · t

(4)

The Equation 4 is a conical surface as shown in Figure 2. The velocity of polishing tool center equals zero which results in zero removal amount at this point.

In practice, the polishing velocity is the sum of feed velocity and rotation velocity; that is to say, v=v_t+r·ω_t. However, the angular velocity ω_t of the polishing tool is very large which can reach about thousands of revolutions per minute. As a result, v_t is very small and r·ω_t≫v_t, so it can be considered that v≈r·ω_t.

Methods

Tool paths are one of the key factors for the automatic mechanical surface polishing. Three polishing paths are covered in this study: scan line path, Archimedean spiral path, and Hilbert fractal path. How they affect the polishing results are investigated here.

Scan line polishing path

The scan line path, a conventional machining path, is frequently used in CNC machining. In the scan line polishing process, the polishing tool offsets an interval at the boundary and continues the machining until the processing is done.

Removal function of single line polishing path

In the modeling of single line polishing path along the X-axis, the removal function Equation 3 at point (x_p,y_p) on plane can be rewritten as:

z (x p, y p) = k · n s 2 π · R 2 · ∫ 0 ∞ v (q → (x (τ), y p), τ) dτ = k · n s 2 π · R 2 · ∫ − r 2 − y p 2 v t r 2 − y p 2 v t ω t · (v t τ) 2 + y p 2 dτ = k · n s 2 π · R 2 · ω t v t · R R 2 − y p 2 + y p 2 ln R + R 2 − y p 2 − ln (| y p |)

(5)

where y_p∈[−R,R].

The coefficient term of Equation 5 is proportional to the removal function, and it does not impact the shape of the removal function curve. Thus, let I represent the coefficient term:

I = k · n s 2 π · R 2 · ω t v t

(6)

Figure 3 is the removal function of single line polishing path in the XZ plane. The maximum removal value is not at the polishing tool center, although the center point has a longest valid polishing length.

Removal function of scan line polishing path

Figure 4 is the polishing model of scan line path. L denotes the interval of two adjacent lines. Equation 5 can be rewritten as the following piecewise function:

Z s (x, y) = I · R R 2 − y 2 + y 2 ln (R + R 2 − y 2) − ln (| y |), y ∈ [− R, R] 0, y ∉ [− R, R]

(7)

Let $Z y 0 (x, y) = Z s (x, y − y 0)$ denote removal function of the polishing line y=y₀. Calculating the sum of all the polishing lines with interval L, the removal function of scan line is:

Z (x, y) = ∑ i = − n n Z (i · L) (x, y)

(8)

Archimedean spiral polishing path

Archimedean spiral is an arithmetic spiral, which is one of the most useful machining paths. Due to the rotational motion of workpiece, with no need of reversing, the polishing machine can achieve smooth and steady motion.

Modeling of Archimedean spiral polishing path

Figure 5 is the model of Archimedean spiral polishing path. Let L be the constant separation distance and v_t be the feed velocity of the polishing tool. The Archimedean spiral formula in the polar coordinate system is ρ=(L·θ)/2π. The feed velocity of polishing tool is $v t = ρ ̇ 2 + ρ 2 ω 2$ . Therefore, the angular velocity of the Archimedean spiral $ω = θ ̇$ can be obtained:

θ ̇ = 2 π v t L · 1 1 + θ 2

(9)

With the initial condition θ(0)=0, the integral for Equation 9 is:

θ 1 + θ 2 + arcsinh (θ) = 4 π L · v t t

(10)

where $arcsinh (θ) = ln θ + 1 + θ 2$ is the inverse hyperbolic function. There is no analytical expression for θ in Equation 10. The value of θ should be calculated by numerical method and then the value of ρ can be obtained.

Figure 6 is the model of Archimedean spiral polishing path. R is the radius of the polishing tool and r is the distance between point (ρ₀,θ₀). (ρ,θ) denotes the polishing tool center. According to the cosine law $r (ρ, θ, ρ 0, θ 0) = ρ 2 + ρ 02 − 2 ρ ρ 0 cos (θ − θ 0)$ , the polishing velocity v_R of the polishing point (ρ,θ) can be expressed as:

v R (ρ, θ, ρ 0, θ 0) = σ R r (ρ, θ, ρ 0, θ 0) · ω t

(11)

where the piecewise function σ_R is defined as follows:

σ R = r, | r | ≤ R 0, | r | > R

(12)

For a given velocity v_t and a separation distance L, the variables ρ and θ are the function of t. With Equations 3 and 11, the integral of Archimedean spiral path can be obtained as:

Z (ρ 0, θ 0) = I · v t · ∫ 0 ∞ σ R × ρ (τ) 2 + ρ 02 − 2 ρ (τ) ρ 0 cos (θ (t) − θ 0) dτ

(13)

The polishing process of Archimedean spiral is the uniform motion polishing process, the natural coordinate is s=v_t·t. Thus, Equations 10 and 13 can be rewritten as:

θ 1 + θ 2 + arcsinh (θ) = 4 π L · s

(14)

Z (ρ 0, θ 0) = I · ∫ 0 ∞ σ R × ρ (s) 2 + ρ 02 − 2 ρ (s) ρ 0 cos (θ (s) − θ 0) ds

(15)

In the natural coordinate system, Archimedean spiral path is expressed by natural coordinate s rather than time t.

Removal function of Archimedean spiral polishing path

The expression of natural coordinate s can be obtained from Equation 14:

s = L 4 π θ 1 + θ 2 + arcsinh (θ)

(16)

Set the radius of the polishing tool R as a reference unit 1. The polar coordinate of polishing tool center is (ρ,θ). Let s_e=s(ρ₀+R); s_o=s(ρ₀−R) in condition ρ₀>R, and s_o=0 in condition ρ₀≤R. s_o and s_e are the new lower and upper limits of Equation 15. Thus, the removal function can be obtained with new limits:

Z (ρ 0, θ 0) = I · ∫ s o s e σ R × ρ (s) 2 + ρ 02 − 2 ρ (s) ρ 0 cos (θ (s) − θ 0) ds

(17)

Using numerical method and discrete method, the removal function of Equation 17 can be rewritten approximately as:

Z (ρ 0, θ 0) = I · ∑ s o s e σ R × ρ (s) 2 + ρ 02 − 2 ρ (s) ρ 0 cos [θ (s) − θ 0] Δs

(18)

The value of Equation 18 at polishing point (ρ₀,θ₀) can be calculated by Newton’s method.

Hilbert fractal polishing path

The scan line and Archimedean spiral polishing path discussed above are very directional. Therefore, the removal functions of them have strong anisotropy property. The anisotropy property destroys the polishing isotropy of the polished surface. In order to obtain high polishing quality, the removal function of designed polishing path with isotropy property is expected to apply into freeform surface polishing task.

Fractal curve

The fractal curve refers to the curve with fractal dimension. The local simple structure of fractal curve is suitable for CNC programming and controlling, while the global randomness property shows a global isotropy characteristic to remove the streaks of polished freeform surface and improves the polishing quality.

Fractal curves include snowflake fractal curve, triangle fractal curve, Hilbert fractal curve, and so on, but not every fractal curve is qualified for surface polishing. The qualified fractal curve needs to satisfy the following conditions: surface ergodic property, easy motion control for CNC, and strong isotropy property.

The surface ergodic property requires the fractal path with dimension greater than one, and the fractal curve can cover the polished surface when the order of the fractal curve is large enough. In mathematics, it means that the closure of the fractal curve is homeomorphism to two-dimensional plane.

The surface ergodic property requires that fractal curve can cover the whole higher-order polished surface. The snowflake fractal curve does not fulfill the surface ergodic property. For example, the cubic Koch snowflake cannot cover the two-dimensional surface. On the other hand, the easy motion control property requires the simple curve. The isometric segment is a good choice for this requirement because of easy planning, programming, and CNC machining. The snowflake fractal curve does not have well isotropic property globally, and it has a strong global orderliness with the high orders. Hence, it is not suitable for polishing purpose. Fulfilling the three requirements for freeform surface polishing, the Hilbert fractal curve is selected for freeform surface polishing here.

As shown in Figure 7, the Hilbert fractal path is organized with segments and traverses the whole two-dimensional plane. The segments of the Hilbert curve are parallel to X axis or Y axis. They can be easily controlled by CNC machine. Also, Hilbert fractal path has a good isotropy and no orderliness in global.

Modeling of Hilbert fractal polishing path

Figure 8 is the polishing model of the Hilbert curve. Similar to the scan line and Archimedean spiral polishing path, let ω_t and v_t denote the rotational velocity and the feed velocity in polishing process, respectively.

The basic unit of the Hilbert curve is segment. Therefore, Equation 3 with segment path can be applied for Hilbert polishing path. The segments of Hilbert curve are all parallel to X axis or Y axis.

For a given segment along X axis direction, the motion of polishing tool center is uniform linear motion in segment C_x[(x_s,y)−(x_e,y)]. The removal function is nonnegative. Hence, changing the integral variable from d τ to dx needs to use absolute function to guarantee the nonnegative value of Equation 3. The removal function on point (x_p,y_p) generated by the polishing segment C_x can be obtained:

Z C x (x p, y p, y) = I · ∫ x s x e σ R (x − x p) 2 + (y − y p) 2 dx

(19)

Similarly, C_y [ (x,y_s)−(x,y_e)] denotes the uniform linear segment of polishing tool center along Y axis direction, the removal function on point (x_p,y_p) generated by the polishing segment C_y can be obtained as:

Z C y (x p, y p, x) = I · ∫ y s y e σ R (x − x p) 2 + (y − y p) 2 dy

(20)

The σ_R in Equations 19 and 20 has a mutation at the point where |r|=R. Therefore, the mutation is not convenient for solving the integration of Equations 19 and 20. x₋ and x₊ are the x-coordinate of intersection points between the line and the circle. Let X_min= min(x_s,x_e) and X_max= max(x_s,x_e); let $x − = x p − R 2 − (y − y p) 2$ and $x + = x p + R 2 − (y − y p) 2$ under condition |y−y_p|≤R. The valid lower and upper limits $x s ′ ≤ x e ′$ of integral function along X axis direction the polishing segment can be defined as:

x s ′ = x −, X min ≤ x − ≤ X max, and | y − y p | ≤ R X min, x − ≤ X min ≤ x +, and | y − y p | ≤ R not defined, other

(21)

x e ′ = x +, X min ≤ x + ≤ X max, and | y − y p | ≤ R X max, x − ≤ X max ≤ x +, and | y − y p | ≤ R not defined, other

(22)

Similarly, let Y_min= min(y_s,y_e) and Y_max= max(y_s,y_e); let $y − = y p − R 2 − (x − x p) 2$ and $y + = y p + R 2 − (x − x p) 2$ under condition |x−x_p|≤R. The valid lower and upper limits $y s ′ ≤ y e ′$ of integral function along Y axis direction polishing segment can be defined as:

y s ′ = y −, Y min ≤ y − ≤ Y max, and | x − x p | ≤ R Y min, y − ≤ Y min ≤ y +, and | x − x p | ≤ R not defined, other

(23)

y e ′ = y +, Y min ≤ y + ≤ Y max, and | x − x p | ≤ R Y max, y − ≤ Y max ≤ y +, and | x − x p | ≤ R not defined, other

(24)

The integrands of Equations 19 and 20 are given, respectively:

f x p, y p (x, y) = δx δ x 2 + δ y 2 + δ y 2 ln δx + δ x 2 + δ y 2 2

(25)

g x p, y p (x, y) = δy δ x 2 + δ y 2 + δ x 2 ln δy + δ x 2 + δ y 2 2

(26)

where δ x=x−x_p, δ y=y−y_p.

When x in $f x p, y p (x, y)$ is not defined, $f x p, y p (x, y) = 0$ ; when y in $g x p, y p (x, y)$ is not defined, $g x p, y p (x, y) = 0$ . With new defined lower and upper limits, the absolute function can be removed from the integration. The single uniform linear polishing segment can be rewritten as:

Z X x p, y p, x s, x e, y = I · f x p, y p x e ′, y − f x p, y p x s ′, y

(27)

Z Y x p, y p, x, y s, y e = I · g x p, y p x, y s ′ − g x p, y p x, y s ′

(28)

where $x s ′$ , $x e ′$ , $y s ′$ , and $y e ′$ are defined in Equations 21, 22, 23, and 24.

Applying uniform linear motion segment, the removal functions of Equations 27 and 28 with the Hilbert fractal curve are the segments set H, the removal function of Hilbert fractal curve at point (x_p,y_p) is the sum of all the segments removal functions:

Z x p, y p = ∑ C X i ∈ H Z X i x p, y p, x si, x ei, y i + ∑ C Y j ∈ H Z Y j x p, y p, x j, y sj, y ej

(29)

Freeform surface polishing modeling

The previous polishing paths are all discussed in the planer plane. Referring to differential geometry method, the planer plane can be mapped on a freeform surface. The following discusses the general mapping method.

Natural coordinate expression of freeform surface path

The freeform curve expressed by natural coordinate s can be directly used in path planning. Freeform surface refers to a smooth two-dimensional surface with finite or infinite degrees of freedom. Because the order of freeform surface is unknown, it can hardly use an unified expression to describe the surface exactly. The freeform surface is expressed by linear combining of several basic functions to approach the freeform surface itself. The polynomial is one of the most useful basic functions due to its simpleness and easy calculation. According to the approximation theory, piecewise polynomial surfaces can approximate a smooth freeform surface in arbitrary precision.

Let C_uv be polishing path in parameter u-v plane and t be the parameter of path C_uv. Mapping the plane to the freeform surface obtains the project path C_f. The space path C_f can be expressed as follows:

x = x (u (t), v (t)), y = y (u (t), v (t)), z = z (u (t), v (t))

(30)

In application, x(t) and y(t) are selected as the parameter variables of u(t) and v(t). The path C_f on freeform surface is described as z=z(x(t), y(t)).

The natural coordinate parameter s which is the length of curve is used to replace the parameter variable t to describe the space path C_f. Referring to formula $d s = d x 2 + d y 2 + d z 2$ , natural coordinate parameter s can be written as:

s = ∫ 0 t 1 + z x 2 x t 2 + 2 z x z y x t y t + 1 + z y 2 y t 2 · d τ

(31)

where z_x=dz/dx.

With Equation 31, the interpolation algorithm of the polishing path on freeform surface can be easily designed.

Modeling of polishing path on the Bézier freeform surface

The traditional surface fitting methods include the Hermite method, Bézier method, B-spline method, and NURBS method. The Bézier surface can approximate any freeform surface and has fewer parameters which make it easier to use in practice. Thus, the Bézier method is studied in this paper for freeform surface polishing.

Bézier method is polynomial-based approximation algorithm. The degree n of the polynomial is related to the number of the knots. Polynomial can approximate a smooth function in arbitrary precision. Hence, the polynomial-based Bézier surface can approximate the smooth freeform surface as well in arbitrary precision.

Given m rows and n columns Bézier control points P_ij= $P ij x, P ij y, P ij z$ . Let $B k − 1 (α) = B k − 1 0 (α), B k − 1 1 (α), ⋯, B k − 1 k − 1 (α) T$ , $P mn x = P ij x$ , $P mn y = P ij y$ , and $P mn z = P ij z$ . Thus, the Bézier surface can be expressed as:

x (u, v) = B m − 1 T (u) · P mn x · B n − 1 (v) y (u, v) = B m − 1 T (u) · P mn y · B n − 1 (v) z (u, v) = B m − 1 T (u) · P mn z · B n − 1 (v)

(32)

where u∈[0,1],v∈[0,1].

The Bernstein basic function $B n i (α)$ is defined as:

B n i (α) = ∑ i = 0 n n! i! (n − i)! (1 − α) n − i α i

(33)

Set the Bézier surface with the range of [X_s,X_e]× [Y_s,Y_e]. The affine transformation from u-v to x-y is defined as:

u = x − X s / X e − X s, v = (y − Y s) / (Y e − Y s)

(34)

The surface is evenly divided into m×n matrix points. Therefore, $P ij x = X s + i − 1 m − 1 · (X e − X s)$ and $P ij y = Y s + j − 1 n − 1 · (Y e − Y s)$ . Bernstein polynomial is endowed with a binomial expansion $∑ i = 0 n B n i (α) = 1$ . The identical relation for the x(u,v) and y(u,v) in Equation 32 is:

x x − X s X e − X s, t = x y t, y − Y s Y e − Y s = y

(35)

The Bézier surface can be simplified as follows:

z (x, y) = B m − 1 T x − X s X e − X s · P mn z · B n − 1 y − Y s Y e − Y s

(36)

Thus, z_x and z_y in Equation 31 are expressed as:

z x = B m − 1 ′ T x − X s X e − X s · P mn z · B n − 1 y − Y s Y e − Y s

(37)

z y = B m − 1 T x − X s X e − X s · P mn z · B n − 1 ′ y − Y s Y e − Y s

(38)

The first order derivative of Bernstein function is $B n i ′ (t) = n B n − 1 i − 1 (t) − B n − 1 i (t)$ . Hence, $B m − 1 ′ T x − X s X e − X s$ and $B n − 1 ′ y − Y s Y e − Y s$ can be calculated by a series of simple algebraic operation. With the natural coordinate parameter s, the interpolation algorithm of freeform surface curve can be directly obtained.

Results and discussion

Different parameters for the same type of polishing path may cause different quality. The following will discuss removal function of the different polishing paths and optimize the parameters of the polishing path.