Research on cylindrical indexing cam’s unilateral machining

Abstract

The cylindrical cam ridge of the indexer is a spatial curved surface, which is difficult to design and machine. The cylindrical cam has some defects after machining because conventional machining methods have inaccuracies. This article aims at proposing a precise way to machine an indexing cam, using basic motion analysis and analytic geometry approach. Analytical methodology is first applied in the cam’s motion analysis, to obtain an error-free cam follower’s trajectory formula, and then separate the continuous trajectory curve by thousandth resolution, to create a three-dimensional discrete trajectory curve. Planar formulae and spherical formulae can be built on the loci. Based on the machine principle, the cutting cutter’s position and orientation will be taken into account. This article calculates the formula set as presented previously and obtains the ultimate cutter path coordinate value. The new error-free cutter path trajectory is called the unilateral machining trajectory. The earned results will compile into numerical control processing schedule. This processing methodology gives a convenient and precision way to manufacture a cylindrical indexing cam. Experimental results are also well supported.

Introduction

The cam-actuated indexing mechanism is very important in automated machinery. ¹ It is widely used in packing machinery, punching machinery, and pharmaceutical machinery, which has advantages of small size, compact structure, high stiffness, large transmission torque, and high transmission accuracy.^2,3 The cylindrical indexing cam ridge is a spatial curved surface which is very hard in design and to machine. The task has become the acknowledged problem, especially for the high-speed and high-precision occasion. ⁴ For the past few years, people have been thinking distinctive theories in different points of view. Bovisov ⁴ put forward an algorithm that considered the indexing cam contour surface as a linear surface development; Gonzalez-Palacios et al. built uniform equations for the spatial indexing cam contour surface using the spiral motion theory. In the meantime, Tsay and Lin ⁷ put forward a new theory: the geometry of cam contour surface was formed by the cutter’s sweep. After that, Hsieh ⁸ proposed a series of programs for the cam’s machining; Chen and Xin ⁹ presented the three-dimensional (3D) expansion method. These research achievements have provided methods for indexing cams’ design and manufacture but still have their shortcomings and imperfections. It is urgent to study in a geometric intuitiveness machining convenient and high machining precision method.

Indexer work situation analysis and error analysis

Indexer working situation analysis

The cylindrical cam and the dial with rollers constructed the indexer (Figure 1). As a driving member, the cylindrical cam rotates around the axis $\mathrm{OX}$ . And the dial dictated by the cam is doing index movement around axis $O\prime Z\prime$ . ¹⁰ The principal axis $O\prime Z\prime$ of the dial is parallel to the cylindrical roller’s central axis $\mathrm{cc}\prime$ and is perpendicular to the plane $\mathrm{XOY}$ . During the cam’s dwell phase, the cam confines the rollers to a rigid, stationary position. But during the remainder of the phase called index rotation, the cam causes the follower to move along its surface geometry. ¹¹

Figure 1.

Typical cam-actuated indexing.

Error analysis

Follower’s trajectory design error

To design a cam, commonly the first thing to do is to derive the index motion curve equation $\varphi =f(\phi )$ from the cam work requirements. This equation presents the relationship between the rotation angle of the cam and the index angle of the dial. The theoretical follower’s trajectory calculated by conventional method is not accurate. ¹⁰ This article uses fundamental motion analysis to fix the deviation (Figure 2).

Figure 2.

Trajectory error of the follower.

It assumes a situation as shown in Figure 2: The cam rotates from ${\psi }_0$ to ${\psi }_1$ during a process. This movement will force the dial’s diameter to change from $T$ to $T\prime$ theoretically. However, in fact, the dial’s geometry structure is unchangeable, and the dial is mounted on a fixed shaft, which means the shaft will be forced to move to $O_1^{\prime }$ point. The length of the segment $O\prime O_1^{\prime }$ is called the displacement error. In this article, the displacement error will be taken into consideration, to build the mended-trajectory equation of the follower.

Axis shifting error

When surface flatten method ¹² is used to design a trajectory of a follower, it overlooked an issue: the follower rollers’ central axis will shift from its original place during the cam’s rotation process (Figure 3). Point $C$ is not always in plane $\mathrm{XOZ}$ in an index cycle but will twice penetrate the plane. This slight movement is commonly ignored, as this deflection will certainly cause a considerable influence to build a cutter’s position and orientation formula.

Figure 3.

Rollers’ axis shifting error and angle deflection error.

Unilateral machining method error

For a long time, cylindrical indexing cam was machined by the bilateral machining method (Figure 4(a)). But for the efficiency and accuracy purpose, this article performed a new machining method called the unilateral machining (Figure 4(b)).

Figure 4.

Two machining methods: (a) bilateral machining method and (b) unilateral machining.

Cylindrical indexing cam’s ridge has dwell phase and index phase. The bilateral cutter path and the unilateral cutter path are different in these two phases (Figure 5). When a cam follower moves in the index phase at position $({\phi }_1,{\psi }_1)$ (Figure 5), the trajectory curve has its normal plane ${\mathrm{AA}}_1$ and the corresponding unilateral machining points also in this plane. In plane ${\mathrm{AA}}_1$ —a half diameter distance from $({\phi }_1,{\psi }_1)$ —there is point $({\phi }_2,{\psi }_2)$ , which is the required unilateral machining point. Each of the unilateral machining points has a new cam angle and a new indexing angle, other than simple curvilinear offset of these points. When we discuss the problem in a 3D space, it indeed searches the direction vector of each point. All of them start from the bilateral cutter path and end in the unilateral cutter path.

Figure 5.

Two different machining cutter paths.

Cutter’s position and orientation error

When a cutter mills a cam, its posture is fixed. This article uses a method to analyze the cam follower’s trajectory by fixing the cam and making the follower rotate, so that the cutter’s posture changes all the time. In a 3D space, the coordinate origin is put on the cutter’s geometry center and thereby derived a cutter posture formula.

Practical cutter’s posture will be different from the theoretical one (Figure 6(a)). The cutter is now at the bilateral machining point as shown in Figure 6(a), and its direction vector is presented as $\overrightarrow{n1}$ , and its cutting plane is $P1$ (Figure 6(b)). Relatively, the unilateral cutter is now at the unilateral machining point whose direction vector is $\overrightarrow{n3}$ , and the cutting plane is $P3$ . $\overrightarrow{n1}$ is same as the direction vector $\overrightarrow{n3}$ . At the unilateral machining position, the cutter’s theoretical directional vector is $\overrightarrow{n2}$ and the theoretical cutting plane is $P2$ . There is an angle between $\overrightarrow{n2}$ and $\overrightarrow{n3}$ . The extension of vectors $\overrightarrow{n1}$ and $\overrightarrow{n2}$ will go through the cam’s rotating shaft but not the extension of vector $\overrightarrow{n3}$ . This difference is called the unilateral cutter’s position and orientation error.

Figure 6.

Cutter’s position and orientation error: (a) cutters’ position with different tooling methods and (b) cutters’ orientation with different tooling methods.

Milling method

This article considered the follower’s trajectory spatial curvature to establish the osculation planar formula and the normal planar formula on the each locus and then combines the cutter’s position and orientation equation with a spherical equation to derive the unilateral machining trajectory equation. The obtained equation is still needed to verify whether there is a theoretical solution.

According to the above principles, this article has put forward an original cam milling method as shown in Figure 7: First is to use the cutter whose diameter is slightly smaller than the follower roller’s as the rough machining cutter ¹² and to choose the bilateral cutter path as the milling route to form the cam’s prototype. This procedure will form a slightly thicker cam ridge than the target one, and the redundant part is the finishing machining allowance in the next step. The second step is to use the small cutter or grinding wheels as the finishing machining cutter and to choose the unilateral cutter path as the milling route to form the cam’s accurate ridge. Using this method, the machining precision of a cylindrical indexing cam has significantly improved.

Figure 7.

Depiction of milling principle.

Unilateral cutter path equation and the cutter’s posture equation derivation

In order to improve the modeling efficiency, this article uses the formula below to represent the relationship between follower’s indexing angle and cam’s rotating angle

$$\psi =f\left(\phi \right)$$ (1)

where $\psi$ is the follower’s indexing angle and $\phi$ is the cylindrical cam’s rotating angle.

Unilateral cutter path equation derivation

During the process of a cycle, there are two actual movements: the cam’s rotation and the follower’s indexing. For the convenient of geometry intuition, the cam is assumed to be fixed, and the follower revolves both around the cam’s axis and its own axis as a consequence. The error-free cam follower’s trajectory is defined as follower’s geometry center route.

The solid curve $\mathrm{OBO}\prime$ shows the initial position of the cam follower, and the dashed curve $\mathrm{OBO}\prime$ is the imaginary position the cam follower will achieve at some time later (Figure 8). $R$ represents the cam’s radius, the length of segment $\mathrm{OB}$ is displayed as $c$ , $c\ge R$ . The length of segment $O\prime B$ is shown as $a$ , and the length of segment $O\prime C$ is presented as $T$ . $T$ will change as the cam follower moves, so that it has three possible states which are $T>a,T=a,\mathrm{and}T<a$ . $O$ is defined as the original point of the system whose coordinates are $O(0,0,0)$ .

Figure 8.

Cam follower’s movement analysis.

$\phi$ is the angle between the segment $\mathrm{OB}$ and the positive direction of Z-axis. According to different work demands, equation (1) has different expressions. In this article, $0\le \phi \le 4\pi$ is designed as the definition domain.

From Figure 8, the $O\prime$ trajectory matrix equation is formulated as

$$\left[\begin{array}{c}\hfill x_{O\prime }\hfill \\ \multicolumn{1}{c}{y_{O\prime }}\\ \multicolumn{1}{c}{z_{O\prime }}\end{array}\right]=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-\sin \phi }& \hfill -\cos \phi \hfill \\ \multicolumn{1}{c}{\cos \phi }& \hfill -\sin \phi \hfill \end{array}\right]\times \left[\begin{array}{c}\hfill c\hfill \\ \multicolumn{1}{c}{a}\end{array}\right]$$ (2)

The locus $C$ and the locus $O\prime$ are in the same rigid body system, so that it can obtain the relationship equation between those two as

$$\left[\begin{array}{c}\hfill x_C\hfill \\ \multicolumn{1}{c}{y_C}\\ \multicolumn{1}{c}{z_C}\end{array}\right]=\left[\begin{array}{c}\hfill x_{O\prime }\hfill \\ \multicolumn{1}{c}{y_{O\prime }}\\ \multicolumn{1}{c}{z_{O\prime }}\end{array}\right]+T\times \left[\begin{array}{c}\hfill \sin \psi \hfill \\ \multicolumn{1}{c}{\cos \psi \cos \phi }\\ \multicolumn{1}{c}{\cos \psi \sin \phi }\end{array}\right]$$ (3)

Substituting equation (2) into equation (3), we obtain the point $C$ trajectory equation

$$\left[\begin{array}{c}{x}_C\\ y_C\\ z_C\end{array}\right]=\left[\begin{array}{ll}0\hfill & 0\hfill \\ -\sin \phi \hfill & -\cos \phi \hfill \\ \cos \phi \hfill & -\sin \phi \hfill \end{array}\right]\times \left[\begin{array}{c}c\\ a\end{array}\right]+T\times \left[\begin{array}{c}\sin \psi \\ \cos \psi \cos \phi \\ \cos \psi \sin \phi \end{array}\right]$$ (4)

Equation (4) is a 3D curve which represents the locus C’s trajectory in the Cartesian space. Planar $P$ is defined on each locus of the curve (Figure 9), and each planar $P$ is perpendicular to the corresponding locus’ tangent vector. The $P$ planar equation is formulated as follows

$$x{\prime }_c(x-x_c)+y{\prime }_c(y-y_c)+z{\prime }_c(z-z_c)=0$$ (5)

Figure 9.

Equivalent cutter orthogonal plane and the P planar sketch.

At the same time, Figure 9 shows an equivalent cutter orthogonal plane drew on a follower roller, which means the cutter’s orthogonal plane is same as the follower roller’s cross plane. Moreover, for the particularity of this processing method, equation (4) not only means the follower’s trajectory but also refers the bilateral machining cutter path.

Cutter’s position and orientation equation

The error analysis explained above has not yet referred to two-axis-deviation error and two-angle-deviation error (Figure 10). It cannot machine the cam in a theoretical way for the existence of these errors.

Figure 10.

Cutter’s position and orientation.

For the particularity of this method, the cam follower’s posture is also the bilateral cutter’s posture and also the unilateral cutter’s posture. Once we obtain one, we can obtain the other two.

As the cam rotates at an angle $\phi$ , the bilateral cutter’s theoretical position is at $D$ . As this is influenced by the indexing movement, the cutter’s practical position is at $C$ , which is called two-axis-deviation error. Meanwhile, during such a process, the cutter will not change its axis direction. $P1$ , $P2$ , and $P3$ in Figure 10 are perpendicular to each roller’s directional vectors as $\overrightarrow{n1}$ , $\overrightarrow{n2}$ , and $\overrightarrow{n3}$ . $P1$ , $P2$ , and $P3$ are also called cutter’s orthogonal plane (Figure 10).

When we develop the cam’s surface into a flat plane—taking $C$ as the practical moving center—the cutter’s theoretical orthogonal plane would be $P2$ because vector $\overrightarrow{n2}$ must point to the original point. In fact, the cutter’s axis is at $\overrightarrow{n3}$ direction and the cutter’s practical orthogonal plane is $P3$ .

$P3$ has its equation as

$$\overrightarrow{n3}\times \{x-x_c,y-y_c,z-z_c\}=0$$ (6)

Since

$$\overrightarrow{n3}=\overrightarrow{n1}$$ (7)

Substituting equation (7) into equation (6), we have

$$\overrightarrow{n1}\times \{x-x_c,y-y_c,z-z_c\}=0$$ (8)

The cutter’s theoretical position $D$ has its coordinates in $\mathrm{YOZ}$ plane as

$$D(0,-c\sin \phi ,c\cos \phi )$$ (9)

The direction vector $\overrightarrow{n1}$ has its expression as

$$\overrightarrow{n1}=\frac{1}{\sqrt{x_D^2+y_D^2+z_D^2}}\{x_D,y_D,z_D\}$$

Substituting equation (8) into it and because of

$$\sqrt{x_D^2+y_D^2+z_D^2}=c$$

we can obtain $\overrightarrow{n1}$ as

$$\overrightarrow{n1}=\{0,-\sin \phi ,\cos \phi \}$$ (10)

Thus, $P3$ has its equation as follows

$$\{0,-\sin \phi ,\cos \phi \}·\{x-x_c,y-y_c,z-z_c\}=0$$ (11)

Equation (11) is the required cutter’s position and orientation equation at position $C$ . By the way, MATLAB software will be a proper cutter to analyze the trajectory and calculate those equations all through this article.

Unilateral cutter path trajectory equation derivation

Preceding part of this article has already derived the bilateral cutter path trajectory equation and cutter’s position and orientation equation. Now, we will use these two equations and combine some auxiliary equations to derive a new unilateral cutter path equation, which is the precise one.

Based on the machining theory, the unilateral cutter path is directional movement of bilateral cutter path loci. The directional movement vector is in cutter’s orthogonal plane. So the goal is to find a set of points that in a series of cutter’s orthogonal plane by equations (11), (12), and (14).

The desired cam ridge’s width is $B$ ; the micro-end milling cutter’s diameter is $d$ (Figure 11). At the contacting point of the cam and the cutter, cutter’s orthogonal plane will not change when the cam rotates, but the cam’s surface tangent plane will change all the time. This means the follower’s trajectory curve has its unique normal plane at each different locus, and the normal plane is $P$ . The cross line between the $P$ plane and the cutter’s orthogonal plane is different in each locus. We assume the cross line as a number axis and the follower’s geometry center is its origin. The positive $B/2$ and the negative $B/2$ represent two contacting points as mentioned previously. What is more, the unilateral machining points are at $(B-d)/2$ locus on the cross line, each side of the origin.

Figure 11.

Cam motion parameters diagram.

This methodology needs to calculate all the points first and then to form a curve called the unilateral cutter path using these data. Computer software can easily do these works for us.

We substitute equation (4) into equation (5)

$$\begin{array}{c}L\psi \prime \cos \psi (x-x_c)\hfill \\ +(-c\cos \phi +a\sin \phi -L\psi \prime \sin \psi \cos \phi -L\cos \psi \sin \phi )\hfill \\ \times (y-y_c)\hfill \\ +(-c\sin \phi -a\cos \phi -\psi \prime L\sin \psi \sin \phi +L\cos \psi \cos \phi )\hfill \\ (z-z_0)=0\hfill \end{array}$$

In the cutter’s orthogonal plane, equation (11) expands as

$$-(y-y_c)\sin \phi +(z-z_c)\cos \phi =0$$ (12)

Equations (11) and (12) will construct a space curve as

$$L{\psi }^{\prime }\cos \psi \left(x-x_c\right)-\left(c+L{\psi }^{\prime }\sin \psi \right)\left(\cos \phi +\sin \phi tan\phi \right)\left(y-y_c\right)=0$$ (13)

Spherical equation (14) presents as

$${(x-x_c)}^2+{(y-y_c)}^2+{(z-z_c)}^2={\left(\frac{B-d}{2}\right)}^2$$ (14)

Equation (11) and spherical equation (14) can construct another space curve as

$${(x-x_c)}^2+(1+{\tan }^2\phi ){(y-y_c)}^2={\left(\frac{B-d}{2}\right)}^2$$ (15)

From equations (13) and (15), we finally obtain the cutter path’s coordinate expressions

$$\{\begin{array}{c}\hfill x=\pm \left(\frac{B-d}{2}\right)\frac{c+L\psi \prime \sin \psi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+x_c\hfill \\ \multicolumn{1}{c}{y=\pm \left(\frac{B-d}{2}\right)\frac{L\psi \prime \cos \psi \cos \phi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+y_c}\\ \multicolumn{1}{c}{z=\pm \left(\tan \phi \right)\left(\frac{B-d}{2}\right)\frac{L\psi \prime \cos \psi \cos \phi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+z_c}\end{array}$$ (16)

In equation (16), $L$ is shown in Figure 11 and $\psi \prime$ is the first derivation of $\psi$ function.

Example experiment

In order to prove the correctness of the equation results, we have designed a machining experiment.

Example

The function relationship between $\phi$ and $\psi$ is designed as follows

$$\psi =\arcsin \left(\frac{{\mathrm{chord}}_i}{L}\right)$$ (17)

Moreover, the ${\mathrm{chord}}_i(i=0-8)$ inside is shown as follows

$$\{\begin{array}{cc}\hfill {\mathrm{chord}}_0=-{\mathrm{chord}}_{max}\hfill & \hfill 0\le \phi <\frac{\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_1=-\frac{{\mathrm{chord}}_{max}}{4}\cos (\phi -\frac{\pi }{3})-\frac{3\times {\mathrm{chord}}_{max}}{4}}& \hfill \frac{\pi }{3}\le \phi <\frac{2\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_2=-\frac{{\mathrm{chord}}_{max}}{2}}& \hfill \frac{2\pi }{3}\le \phi <\frac{4\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_3=-\frac{{\mathrm{chord}}_{max}}{4}\cos (\phi -\frac{4\pi }{3})-\frac{{\mathrm{chord}}_{max}}{4}}& \hfill \frac{4\pi }{3}\le \phi <\frac{5\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_4=0}& \hfill \frac{5\pi }{3}\le \phi <\frac{7\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_5=-\frac{{\mathrm{chord}}_{max}}{4}\cos (\phi -\frac{7\pi }{3})+\frac{{\mathrm{chord}}_{max}}{4}}& \hfill \frac{7\pi }{3}\le \phi <\frac{8\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_6=\frac{{\mathrm{chord}}_{max}}{2}}& \hfill \frac{8\pi }{3}\le \phi <\frac{10\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_7=-\frac{{\mathrm{chord}}_{max}}{4}\cos (\phi -\frac{10\pi }{3})+\frac{3\times {\mathrm{chord}}_{max}}{4}}& \hfill \frac{10\pi }{3}\le \phi <\frac{11\pi }{3}\hfill \\ \multicolumn{1}{c}{{\mathrm{chord}}_8={\mathrm{chord}}_{max}}& \hfill \frac{11\pi }{3}\le \phi \le \frac{12\pi }{3}\hfill \end{array}$$ (18)

Equation (18) shows the cam follower’s piecewise function; the curvilinear relationship chart of $\phi$ and $\psi$ has also been demonstrated (Figure 12).

Figure 12.

Curvilinear relationship chart of $\phi$ and $\psi$ .

These equation set can be programmed in MATLAB and can generate cutter path plot (Figure 13).

Figure 13.

Bilateral cutter path and two unilateral cutter paths.

Figure 13 demonstrates the obtained unilateral cutter paths. The blue one in the middle is the bilateral cutter path trajectory, and the red and green ones on each side are two unilateral cutter path trajectories.

Cam machining

We use MATLAB software to compute the equations, obtaining the coordinate set. But we cannot just put them into numerical control (NC) machining use; those coordinate set still need to be changed into machining coordinates. ¹³ The former coordinate system is fixed on the cam’s rotational center, but the machining coordinate center moves along with the cam’s rotation.

These two coordinate systems have coordinate relationship as

$$\left[\begin{array}{c}\hfill x_{\mathrm{machine}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{machine}}}\\ \multicolumn{1}{c}{z_{\mathrm{machine}}}\end{array}\right]=\left[\begin{array}{c}\hfill x\hfill \\ \multicolumn{1}{c}{y-R\sin \phi }\\ \multicolumn{1}{c}{z-R\cos \phi }\end{array}\right]$$ (19)

Indexing cam machining requires four-axis machining center, the required equation needs to obtain the parameter of A-axis rotating angle just as equation (19) presented previously.

We reformulate equation (16) as

$$\{\begin{array}{c}\hfill x_{\mathrm{machine}}=\pm \left(\frac{B-d}{2}\right)\frac{c+L\psi \prime \sin \psi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+x_c\hfill \\ \multicolumn{1}{c}{y_{\mathrm{machine}}=\pm \left(\frac{B-d}{2}\right)\frac{L\psi \prime \cos \psi \cos \phi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+y_c-R\sin \phi }\\ \multicolumn{1}{c}{z_{\mathrm{machine}}=\pm \left(\tan \phi \right)\left(\frac{B-d}{2}\right)\frac{L\psi \prime \cos \psi \cos \phi }{\sqrt{c^2+2\mathrm{cL}\psi \prime \sin \psi +L^2\psi {\prime }^2}}+z_c-R\cos \phi }\\ \multicolumn{1}{c}{a=\phi }\end{array}$$ (20)

According to equation (20) and using MATLAB software, we compute the results and compile the NC machining cutter path file. The computer simulation result is shown in Figure 14, which is the practical cutter path trajectory of four-axis milling machine. Figure 15 shows the processed cylindrical indexing cam with error-free cutter path in practical.

Figure 14.

NC machining cutter path’s simulation.

Figure 15.

Processed cylindrical indexing cam.

We measured the processed cylindrical indexing cam and compared the result with the theoretical one in computer. The practical cutter path meets the design requirement.

Conclusion

In this article, unilateral machining methodology was proposed to machine a cylindrical indexing cam, with the aim of highly precision. This article first analyzed the drawbacks of the existing machine methodologies, which is low in efficiency and in precision and built the error-free follower’s trajectory equation. Then, second, this article analyzed the cutter’s posture, just in order to derive the cutter’s position and orientation equation. Finally, this article proved the methodology can successfully solve the unilateral machining loci with equations mentioned above. This methodology separates the linear continuous rotating angle curve into discrete point and substitutes these discrete points into unilateral machining equation to obtain the objective cutter path.

This article also designed an experiment to verify the feasibility of the methodology. The unilateral machining cutter path equation may help to promote the cam’s manufacturing technology and to give a general way to design and machine an indexing cam.