applied sciences

Article A New Tooth Profile Modification Method of Cycloidal Gears in Precision Reducers for Robots

Tianxing Li 1,* , Xiaotao An 1, Xiaozhong Deng 2, Jinfan Li 1 and Yulong Li 1

1 School of Mechatronics Engineering, Henan University of Science and Technology, 48 Xiyuan Road, Jianx District, Luoyang 471003, China; [email protected] (X.A.); [email protected] (J.L.); [email protected] (Y.L.) 2 Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing of Henan Province, 48 Xiyuan Road, Jianxi District, Luoyang 471003, China; [email protected] * Correspondence: [email protected]



Received: 9 January 2020; Accepted: 10 February 2020; Published: 13 February 2020


Featured Application: This method proposed in this paper is a new profile modification method of cycloid gears of precision reducers for robots, and provides the theoretical basis and technical support for improving the modification quality and motion accuracy of robot reducers.

Abstract: The tooth profile modification of cycloidal gears is important in the design and manufacture of precision reducers or rotary vector (RV) reducers for robots. The traditional modification design of cycloidal gears is mainly realized by setting various machining parameters, such as the size and center position of the grinding wheel. The traditional modification design has some disadvantages such as complex modification calculation, uncontrollable tooth profile curve shape and unstable meshing performance. Therefore, a new tooth profile modification method is proposed based on the consideration of the comprehensive influences of pressure angle distribution, meshing backlash, tooth tip and root clearance. Taking the pressure angle and modifications of tooth profile as the parameters of the modification function and the meshing backlash of gear teeth as constraints, the mathematical model for tooth profile modifications is built. The modifications are superimposed on the normal direction of the theoretical profile—the force transmission direction. The mathematical relationship between the modifications and the pressure angle distribution, which determines the force transmission performance, is established. Taking the straight line method, cycloid method and catenary method as examples, by means of the tooth contact analysis technology, the transmission error and minimum meshing backlash, which reflects the lost motion, of the newly modified profile are analyzed and verified. This proposed method can flexibly control the shape change of the modification profile and accurately pre-control the transmission accuracy of the cycloid-pin gear. It avoids the disadvantages of traditional modification methods, such as uncontrollable tooth profile shape and unstable meshing accuracy. The method allows good meshing characteristics, high force transmission performance and more precise tooth profile curve. The study provides a new design method of the modified profile of cycloidal gears.

Keywords: robot precision reducer; cycloidal gear; tooth profile modification; pressure angle distribution; transmission error; minimum backlash angle

1. Introduction The transmission performance of precision reducers for robots directly affects the kinematic accuracy and repetitive positioning accuracy of robots. The cycloid-pin gear pair is the key component of precision reducers for robots and its meshing performance is important in the evaluation of the

Appl. Sci. 2020, 10, 1266; doi:10.3390/app10041266 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1266 2 of 16 comprehensive performance of precision reducers for robots [1]. In cycloidal-pin gear planetary transmission, in order to compensate the assembly error and maintain reasonable backlash for the conveniences of lubrication and assembly, the theoretical tooth profile of the cycloidal gear is generally machined through the modification design and the tooth profile modification quality is the key to ensure the motion accuracy of precision reducers for robots [2]. Many experts and scholars have carried out in-depth studies on the profile curve, modification design and meshing characteristics of the cycloidal gear in the precision reducer and have achieved fruitful results. In the aspect of the profile curve characteristics of the cycloidal gear, Litvin and Feng developed a general method for the generation and design of cycloid tooth profile, and avoided the profile and surface singularities of cycloid gear through improving the design [3]. Yang and Blanche presented the computer-aided analysis and synthesis of cycloid drives and defined the kinematic relationships between tolerance drive parameters and performance indices in detail [4]. Li et al. proposed a new type of cycloid double-enveloping meshing pair with high transmission accuracy, low backlash and high torsional stiffness [5]. Sensinger and Jonathon provided a unified set of equations for optimizing the design of cycloid drives and introduced the sources and effects of various tolerances, such as profile reduction, backlash and torque ripple and maximum gear ratio [6]. Ye et al. established a translation-torsion coupled nonlinear dynamic model with multiple degrees of freedom for a planetary gear train system, and indicated that the load sharing coefficient varied more significantly with the increase in the pressure angle range of gear pair [7]. In the aspect of tooth profile modification of cycloidal gears, He et al. indicated the requirements of the modification of cycloidal gears used in robots and proposed the optimum mathematics model for the modification of cycloidal gears in order to achieve the high kinematic accuracy, little backlash, high-load capacity, high rigidity and high transmission efficiency of RV reducers [8]. Ren et al. presented a new method of cycloid disc tooth modification and designed the modification clearance curves by adjusting the positions of five key points according to different modification targets, and they found that the method could improve the carrying capability of cycloidal drive, eliminate noise and vibration and increase the transmission accuracy [9]. Lin et al. designed a new two-stage cycloidal speed reducer with tooth modifications, studied the profile generation and modifications of cycloidal gears, analyzed the kinematic errors with the tooth contact analysis method and presented the quantitative results of different modification combinations of the gear profile [10]. Lu et al. found that the cycloidal gear profile modification was sensitive to the backlash of RV reducer and they proposed a modification method of the cycloidal gear tooth profile based on the compensation of deformation, realized the backlash optimization of RV reducers and significantly reduced the gear side clearance caused by the cycloid gear modification [11]. Nie et al. proposed a new tooth surface topography modification method for spiral bevel gears and corrected the machine settings by establishing the modification mathematical model between the tooth surface deviations and machine settings [12]. Regarding the meshing characteristics of cycloidal-pin gear pairs, Demenego et al. studied the geometry modification of rotor profiles of a cycloidal pump and determined the virtual pair of contacting profiles, the backlash between the profiles that were not in mesh and the transmission errors in the rotation transformation at any time [13]. Cao et al. proposed a new method for the gear tooth contact analysis to avoid two considerable disadvantages of the numerical instability and computation process complexity [14]. Yu et al. proposed an analysis method, called non-Hertz flexibility matrix method (NHFMM), for the tooth contact of the modified profile of pin gear [15]. Jiang et al. proposed a new method for calculating the cutting force during processing hypoid gears [16]. Bahk and Parker investigated the impact of tooth profile modification on spur planetary gear vibration, proposed an analytical model and discussed the static transmission error and dynamic response [17]. From the above studies, it can be seen that the traditional modification design is mainly realized by changing the size and center position of the grinding wheel in the generation process. At present, the gear grinding technology and precision measurement technology are increasingly mature [18–20], so the modification methods of cycloidal gears are more flexible [21] and various tooth profile Appl.Appl. Sci. 2020, 10,, 1266x FOR PEER REVIEW 3 of 1615

optimization technologies are available. The profile shape of a cycloidal gear directly affects its optimizationtransmission performance technologies and are available.the pressure The angle profile distribution shape of on a the cycloidal cycloid gear profile directly determines affects the its transmissionforce transmission performance performance and the of pressure gear teeth angle in distributionmeshing. Therefore, on the cycloid in the profile profile determines modification the forcedesign transmission of a cycloid performance gear and its of meshing gear teeth performanc in meshing.e analysis, Therefore, the in pressure the profile angle, modification the backlash design of ofgear a cycloid teeth, the gear tooth and tip its meshingand root performanceclearance should analysis, be considered. the pressure angle, the backlash of gear teeth, the toothIn the tip study, and root based clearance on the comprehensive should be considered. consideration of the profile pressure angle, backlash, toothIn tip the and study, root based clearance, on the a comprehensivenew modification consideration method is prop of theosed. profile Based pressure on the angle, mathematical backlash, toothmodel tip for and the root profile clearance, modification a new modification of a cycloid method gear, isthe proposed. mathematical Based onrelationship the mathematical between model the formodifications the profile modificationand the pressure of a cycloidangle distribution gear, the mathematical is established relationship and the meshing between performance the modifications of the andmodified the pressure tooth profile angle distributionis discussed is in established depth. Th ande study the meshing provides performance a new way offor the the modified modification tooth profiledesign isof discussedcycloid gears in depth. in precision The study reducers provides for robots. a new way for the modification design of cycloid gears in precision reducers for robots. 2. Theoretical Tooth Profile and Pressure Angle of a Cycloid Gear 2. Theoretical Tooth Profile and Pressure Angle of a Cycloid Gear 2.1. Expression of the Theoretical Tooth Profile 2.1. Expression of the Theoretical Tooth Profile In the cycloid-pin gear planetary transmission of precision reducers for robots, the cycloidal gear Inrotates the cycloid-pin around its gear own planetary axis as well transmission as the ro oftation precision axis reducersof the pin for gear, robots, thus the generating cycloidal gear the rotatesconjugate around meshing its own motion axis between as well as the the tooth rotation profiles axis of of the the cycloid-pin pin gear, thus gear. generating Therefore, the the conjugate meshing meshing motion between the tooth profiles of the cycloid-pin gear. Therefore, the meshing motion motion relationship of a cycloid-pin gear can be established, as shown in Figure 1. Coordinate  relationship of a cycloid-pin gear can be established, as shown in Figure1. Coordinate system S f x f , y f system 𝑆𝑥,𝑦 is introduced and fixed on the frame. Coordinate systems 𝑆𝑥,𝑦 and ( ) is𝑆( introduced𝑥,𝑦) are respectively and fixed on defined the frame. for the Coordinate pin gear systemsand the Scycloidalp xp, yp andgear.S cThexc, pinyc aregear respectively acts as the defineddriving wheel for the and pin its gear axis and rotates the cycloidal around the gear. center The 𝑂 pin. The gear cycloidal acts as the gear driving acts as wheel the driven and its wheel axis rotatesand its aroundaxis rotates the centeraroundO pthe. The center cycloidal 𝑂. The gear rotation acts as center the driven 𝑂 of wheel the pin and gear its axis coincides rotates with around the O O O theorigin center 𝑂 ofc .coordinate The rotation system center 𝑆p. aof is the the pin eccentricity gear coincides of cycloidal with the origingear andf ofpin coordinate gear. 𝑟 and system 𝑟 Srepresentf . a is the the eccentricity roller radius of cycloidal and roller gear position, and pin respectively. gear. rp and Pointrrp represent K is the the current roller meshing radius and position roller position,and Point respectively. P is the instantaneous Point K is the center. current Line meshing PM is position the common and Point normalP is of the the instantaneous meshing point center. K. Line PM is the common normal of the meshing point K. Angles φ and φ are the rotation angles of the Angles 𝜙 and 𝜙 are the rotation angles of the pin gear and the1 cycloid2 gear, respectively. pin gear and the cycloid gear, respectively.

Yc Yp Yf rrp M ' γ M K

β X c φ 1 ϕ X p rp φ O 2

Oc a O O X f p f Figure 1. Meshing motion relationship of cycloidal-pin gear. Figure 1. Meshing motion relationship of cycloidal-pin gear.

In coordinate system Sp, without considering the tooth width, the pin tooth profile is a circle, with In coordinate system 𝑆 , without considering the tooth width, the pin tooth profile is a circle, Point M as the center and rrp as the radius. The tooth profile equation of the pin gear in Sp can be expressedwith Point as:M as the center and 𝑟 as the radius. The tooth profile equation of the pin gear in 𝑆 can be expressed as: h iT Rp(β) = rrp sin β rp rrp cos β 0 1 , (1) − − 𝐑 (𝛽) =−𝑟 sin 𝛽 𝑟 −𝑟 cos 𝛽 0 1 (1) where β is the parametric angle of the pin tooth. In order to facilitate the, analysis and calculation of gearwhere tooth 𝛽 is contact, the parametric Equation angle (1) is expressedof the pin intooth. the formIn order of homogeneous to facilitate the coordinates analysis and and calculation the following of equationsgear tooth are contact, also expressed Equation in (1) the is same expressed form. in the form of homogeneous coordinates and the following equations are also expressed in the same form. Appl. Sci. 2020, 10, 1266 4 of 16

According to the conjugate meshing relationship between tooth profiles of cycloid-pin gear, the theoretical tooth profile equation of cycloidal gear in coordinate system Sc can be obtained after the coordinate transformation of the pin gear tooth profiles:          rp rrp/S sin ϕ a k1rrp/S sin zpϕ    −  −  −      r r /S cos ϕ a k r /S cos z ϕ  ( ) =  p rp 1 rp p  Rc0 ϕ  − − − , (2)  0     1  where ϕ is the parametric angle of the tooth profile of the cycloidal gear; k is called short width q 1 2 coefficient, k = azp/rp; S = 1 + k 2k cos(zcϕ). 1 1 − 1 When the cycloidal gear and the pin gear engage in the conjugate mode, the direction of the common normal PM at the contact point K is the force direction of the gear teeth meshing, as shown in Figure1. The extension line of the common normal PM can be drawn and intersected with the Yc axis at Point M . In ∆M OM, ∠M OM = φ φ . The direction cosine of the common normal PM in 0 0 0 2 − 1 coordinate system Sc can be obtained as:      q   k sin z ϕ sin ϕ / 1 + k2 2k cos(z ϕ)   1 p 1 1 c   − q −       2   k1 cos zpϕ + cos ϕ / 1 + k 2k1 cos(zcϕ)  nc(ϕ) =  1 . (3)  − −   0     1 

2.2. Expression of the Modified Profile of a Cycloidal Gear Tooth profile modification is a necessary step in the design of cycloidal gear. The pressure angle distribution of the cycloidal profile determines the meshing force transmission state of the gear teeth and affects the motion accuracy of the reducer [7]. Therefore, in this study, the profile of the cycloidal gear is modified according to the distribution characteristics of the pressure angle of the tooth profile curve as well as the reference point, the tooth tip and root clearance defined by the transmission accuracy index. After superimposing the modifications ∆L into the theoretical tooth profile along the common normal direction, the modified tooth profile of the cycloidal gear can be expressed as:              rp rrp + ∆L /S sin ϕ a k1 rrp + ∆L /S sin zpϕ    −   −  −        r r + ∆L /S cos ϕ a k r + ∆L /S cos z ϕ  ( ) = L =  p rp 1 rp p  Rc ϕ Rc0 nc ∆  − − − . (4) − ·  0     1 

2.3. Pressure Angle Distribution of Cycloid Profile The pressure angle of tooth profile refers to the intersegment angle between the force direction at the meshing point and the velocity direction of this point without considering the friction force [7]. It is also the angle between the common normal direction at the meshing point and the rotation velocity direction of the cycloidal gear, as shown in Figure2. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 15

According to the conjugate meshing relationship between tooth profiles of cycloid-pin gear, the theoretical tooth profile equation of cycloidal gear in coordinate system 𝑆 can be obtained after the coordinate transformation of the pin gear tooth profiles: 𝑟 −𝑟 ⁄𝑆sin𝜑−(𝑎−𝑘 𝑟 ⁄𝑆)sin(𝑧 𝜑) ⎡ ⎤ 𝑟 −𝑟 ⁄𝑆 cos 𝜑 − (𝑎 −𝑘 𝑟 ⁄𝑆) cos(𝑧 𝜑) 𝐑(𝜑) = ⎢ ⎥, (2) ⎢ 0 ⎥ ⎣ 1 ⎦ where 𝜑 is the parametric angle of the tooth profile of the cycloidal gear; 𝑘 is called short width coefficient, 𝑘 =𝑎𝑧⁄𝑟; 𝑆=1+𝑘 −2𝑘 cos(𝑧𝜑). When the cycloidal gear and the pin gear engage in the conjugate mode, the direction of the common normal PM at the contact point K is the force direction of the gear teeth meshing, as shown in Figure 1. The extension line of the common normal PM can be drawn and intersected with the 𝑌 axis at Point 𝑀 . In Δ𝑀 𝑂𝑀, ∠𝑀 𝑂𝑀 =𝜙 −𝜙. The direction cosine of the common normal PM in coordinate system 𝑆 can be obtained as:

⎡ 𝑘 sin𝑧𝜑 − sin 𝜑1+𝑘 −2𝑘 cos(𝑧𝜑) ⎤ ⎢−𝑘 cos𝑧 𝜑 + cos 𝜑 1+𝑘 −2𝑘 cos(𝑧 𝜑)⎥ 𝐧(𝜑) = ⎢ ⎥. (3) ⎢ 0 ⎥ ⎣ 1 ⎦

2.2. Expression of the Modified Profile of a Cycloidal Gear Tooth profile modification is a necessary step in the design of cycloidal gear. The pressure angle distribution of the cycloidal profile determines the meshing force transmission state of the gear teeth and affects the motion accuracy of the reducer [7]. Therefore, in this study, the profile of the cycloidal gear is modified according to the distribution characteristics of the pressure angle of the tooth profile curve as well as the reference point, the tooth tip and root clearance defined by the transmission accuracy index. After superimposing the modifications Δ𝐿 into the theoretical tooth profile along the common normal direction, the modified tooth profile of the cycloidal gear can be expressed as: 𝑟 −𝑟 +∆𝐿⁄ sin𝜑−(𝑎−𝑘 𝑆 𝑟 +∆𝐿⁄ )sin(𝑧 𝑆 𝜑) ⎡ ⎤ 𝑟 −𝑟 +∆𝐿⁄ 𝑆 cos 𝜑 − (𝑎 −𝑘 𝑟 +∆𝐿⁄ ) 𝑆 cos(𝑧 𝜑) 𝐑(𝜑) =𝐑 −𝐧 ∙∆𝐿=⎢ ⎥. (4) ⎢ 0 ⎥ ⎣ 1 ⎦

2.3. Pressure Angle Distribution of Cycloid Profile The pressure angle of tooth profile refers to the intersegment angle between the force direction at the meshing point and the velocity direction of this point without considering the friction force [7]. It is also the angle between the common normal direction at the meshing point and the rotation Appl.velocity Sci. 2020 direction, 10, 1266 of the cycloidal gear, as shown in Figure 2. 5 of 16

Yc vc M α K Appl. Sci. 2020, 10, x FOR PEER REVIEW nc 5 of 15

Figure 2. Pressure angle of the cycloidal profile.

In Figure 2, Point K is a meshing contact point in coordinate system 𝑆 and the pressure angle 𝛼 of the point K is the angle between the normal direction 𝐧 and the velocity direction 𝐯. The pressure angle can be expressed as: Oc X c ⃗×𝐤 Figure𝛼 = 2.𝑎𝑟𝑐𝑐𝑜𝑠Pressure|(𝐯 ∙𝐧 angle)| = of 𝑎𝑟𝑐𝑐𝑜𝑠 the cycloidal profile.∙𝐧, (5) ⃗×𝐤

In Figure2, Point K is a meshing contact point in coordinate system Sc and the pressure angle α of where 𝐤=001 is perpendicular to the plane 𝑋𝑌. the pointThe pressureK is the angleangle between of each thepoint normal on the direction theoretincalc and tooth the profile velocity of directionthe cycloidalvc. The gear pressure can be calculatedangle can bewith expressed Equation as: (5). Then the distribution trend of the pressure angle on the tooth profile can be obtained. Figure 3 shows the pressure angle distribution  of a single tooth profile of the    →  cycloidal gear. The related geometric parameters of the cycloid-pin  OcK k gear are shown in Table 1. α = arccos (v n ) = arccos  × n  , (5) c c  c ·  → ·   OcK k  Table 1. Geometric parameters of the cycloid-pin× gear.

h iT Parameters Symbols Values where k = 0 0 1 is perpendicular to the plane X Y . Tooth number of cycloidal gear c c zc 39 The pressure angle of each point on the theoretical tooth profile of the cycloidal gear can be Tooth number of pin gear zp 40 calculated with Equation (5). Then the distribution trend of the pressure angle on the tooth profile can Roller radius rrp 3.5 mm be obtained. Figure3 shows the pressureEccentricity angle distribution ofa a single1.5 tooth mm profile of the cycloidal gear. The related geometric parameters of the cycloid-pin gear are shown in Table1. Roller position rp 82 mm

Figure 3. PressurePressure angle angle distribution of the single tooth profile. profile. Table 1. Geometric parameters of the cycloid-pin gear. Obviously, the pressure angle in Figure 3 generally shows a “high–low–high” distribution trend. The pressureParameters angle is the maximum at the Symbols tip and root of teeth respectively, Values and reaches the minimumTooth at numberPoint A of. From cycloidal the gear definition of the pressurezc angle and the force 39analysis, the larger pressure angleTooth of number tooth ofprofile pin gear indicates the larger anglezp between the common normal40 direction and the velocity directionRoller radiusat the meshing point and threrp poorer force transmission3.5 performance mm of the Eccentricity a 1.5 mm gear teeth. The smaller pressure angle indicates the better transmission performance. It can be seen Roller position rp 82 mm from Figure 3 that when the pin gear rotation angle is 137.49°, the pressure angle reaches the minimum value, 42.24°. Therefore, the transmission performance is optimal at this position. Point A Obviously, the pressure angle in Figure3 generally shows a “high–low–high” distribution trend. is used as a reference point in the tooth profile modification design. The pressure angle is the maximum at the tip and root of teeth respectively, and reaches the minimum 3.at Calculation Point A. From of thethe definitionModification of the of pressureTooth Profile angle and the force analysis, the larger pressure angle of tooth profile indicates the larger angle between the common normal direction and the velocity 3.1.direction General at Representation the meshing point of the andModification the poorer force transmission performance of the gear teeth. The The purpose of the tooth profile modification is to ensure that the modification in the meshing area (working segment) of cycloidal gear is as small as possible. The tooth profile in this working segment is as close as possible to the theoretical one, whereas the tooth profile in the non-working segment retains a reasonable backlash to meet lubrication and assembly requirements [3]. In this way, the profile modification strategy of a cycloidal gear can be established (Figure 4). Appl. Sci. 2020, 10, 1266 6 of 16 smaller pressure angle indicates the better transmission performance. It can be seen from Figure3 that when the pin gear rotation angle is 137.49◦, the pressure angle reaches the minimum value, 42.24◦. Therefore, the transmission performance is optimal at this position. Point A is used as a reference point in the tooth profile modification design.

Appl.3. Calculation Sci. 2020, 10, x of FOR the PEER Modification REVIEW of Tooth Profile 6 of 15

3.1. GeneralIn Figure Representation 4, the BC segment of the Modification of the tooth profile is the working segment and the BD and CE segmentsThe purposeare the ofnon-working the tooth profile segments. modification The modified is to ensure curve that in thethe modificationBC segment inis theclose meshing to the theoreticalarea (working conjugate segment) tooth of profile. cycloidal From gear the is asmodification small as possible. reference The point tooth (Point profile A) into thisthe tooth working tip (Pointsegment D) isor as tooth close root as possible(Point E), to the the modification theoretical one, value whereas gradually the increases tooth profile with in the the increase non-working in the pressuresegment retainsangle. Therefore, a reasonable Point backlash A is considered to meet lubrication as the reference and assembly point requirementsfor profile modification [3]. In this way,and the profilemodification modification value at strategy Point A of is aalso cycloidal the smallest. gear can be established (Figure4).

ΔL

Δ D Ltip

Δ A L0 α α 0 tip α α α α root 0 ΔL A 0

ΔL E root Δ L Figure 4. Figure 4. ToothTooth profile profile modification modification strategy of a cycloidal gear. In Figure4, the BC segment of the tooth profile is the working segment and the BD and CE The position selection of the reference point A is important in the modification design. segments are the non-working segments. The modified curve in the BC segment is close to the According to the variations of the pressure angle on the theoretical tooth profile of a cycloid gear, the theoretical conjugate tooth profile. From the modification reference point (Point A) to the tooth tip position of the reference point A is initially determined as the position with the smallest profile (Point D) or tooth root (Point E), the modification value gradually increases with the increase in the pressure angle to ensure that the reference point is in the meshing working segment of the tooth pressure angle. Therefore, Point A is considered as the reference point for profile modification and the profile. Moreover, it ensures that the force transmission performance of the meshing segment is modification value at Point A is also the smallest. optimal. It should be noted that the modifications of reference point A, the tooth tip point D and The position selection of the reference point A is important in the modification design. According tooth root point E should be reasonably determined according to the backlash, assembly accuracy, to the variations of the pressure angle on the theoretical tooth profile of a cycloid gear, the position of processing accuracy, lubrication conditions and machine size of the cycloid-pin gear drive. the reference point A is initially determined as the position with the smallest profile pressure angle According to the tooth profile modification strategy of cycloidal gear shown in Figure 4, the to ensure that the reference point is in the meshing working segment of the tooth profile. Moreover, mathematical model for tooth profile modification can be established with pressure angle 𝛼 as it ensures that the force transmission performance of the meshing segment is optimal. It should be independent variable, modification ∆𝐿 as dependent variable, and the meshing clearances between noted that the modifications of reference point A, the tooth tip point D and tooth root point E should be the reference point A, the tooth tip point D and tooth root point E as the constraints. The functional reasonably determined according to the backlash, assembly accuracy, processing accuracy, lubrication relationship between the modification and the pressure angle can be expressed as follows: conditions and machine size of the cycloid-pin gear drive. According to the tooth profile modification∆𝐿 = 𝑓(𝛼 strategy) of cycloidal gear shown in Figure4, ⎧𝑠. 𝑡. the mathematical model for tooth profile⎪ modification can be established with pressure angle α ∆𝐿 ≤𝐶 , (6) as independent variable, modification ∆L as⎨ dependent variable, and the meshing clearances between ⎪ ∆𝐿 ≤𝐶 ⎩ ∆𝐿 ≤𝐶 where ∆𝐿 = 𝑓(𝛼) is the objective function of tooth profile modification optimization, a piecewise function related to the distribution of the pressure angle; 𝐶, 𝐶 and 𝐶 are, respectively, the clearance index of reference point A, the tip point D and root point E; ∆𝐿, ∆𝐿 and ∆𝐿 are, respectively, the modification values of reference point A, the tip point D and root point E. Different methods of determining the piecewise function 𝑓(𝛼) correspond to different mathematical relations of modifications. The ideal profile curve and transmission accuracy can be obtained by the modification function 𝑓(𝛼) determined with mathematical curves such as straight line, cycloid line and catenary line. In this paper, the modification calculation of the cycloid method was adopted as an example.

3.2. Modification Calculation for the Cycloid Method Appl. Sci. 2020, 10, 1266 7 of 16 the reference point A, the tooth tip point D and tooth root point E as the constraints. The functional relationship between the modification and the pressure angle can be expressed as follows:   ∆L = f (α)   s.t.   ∆L C , (6)  tip tip  ≤  ∆Lroot Croot  ≤  ∆L C 0 ≤ 0 where ∆L = f (α) is the objective function of tooth profile modification optimization, a piecewise function related to the distribution of the pressure angle; C0, Ctip. and Croot are, respectively, the clearance index of reference point A, the tip point D and root point E; ∆L0, ∆Ltip and ∆Lroot are, respectively, the modification values of reference point A, the tip point D and root point E. Different methods of determining the piecewise function f (α) correspond to different mathematical relations of modifications. The ideal profile curve and transmission accuracy can be obtained by the modification function f (α) determined with mathematical curves such as straight line, cycloid line and catenary line. In this paper, the modification calculation of the cycloid method was adopted as an example. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 15 3.2. Modification Calculation for the Cycloid Method When a circle rolls along a straight line, the trajectory 𝑁𝑁𝑁" of Point 𝑁 on the circle forms a When a circle rolls along a straight line, the trajectory NN0N00 of Point N on the circle forms a cycloid,cycloid, as as shown shown in in Figure Figure 55..

Figure 5. Schematic diagram of forming a cycloid. Figure 5. Schematic diagram of forming a cycloid. The mathematical expression of the cycloid is: The mathematical expression of the cycloid is: ( x𝑥=𝜋𝑟−𝑟= πr r(θ(𝜃−𝑠𝑖𝑛𝜃sinθ) ) − − ,, (7) (7) y𝑦=𝐶−𝑟= C r((11−𝑐𝑜𝑠𝜃cosθ) ) − − (𝑥,) 𝑦 𝜃 wherewhere (x, y) isis thethe coordinatescoordinates of of any any point point on on the the cycloid; cycloid;r is ther is rollingthe rolling circle circle radius; radius;θ is the rollingis the rollingangle of angle the circle;of theC circle;is the C vertical is the vertical coordinate coordinate of the starting of the starting point N point. N. AccordingAccording toto EquationEquation (7), (7), there there are are two two methods methods to determine to determine the cycloid the cycloid modification modification function. 𝛼 ∆𝐿 function.In the first In method, the first the method, pressure the angle pressureα as the angleX-axis as and the the X modification-axis and the∆ modificationL as the Y-axis meet as the the Y-axis meet the cycloidal function relationship. In Figure 5, the coordinates (0, C − 2r) of the point 𝑁" cycloidal function relationship. In Figure5, the coordinates (0, C 2r) of the point N00 correspond to correspond to the coordinates (𝛼, ∆𝐿) of reference point A in the− modified coordinate system, and the coordinates (α0, ∆L0) of reference point A in the modified coordinate system, and the coordinates the coordinates (𝜋𝑟, C) of the point N correspond to the coordinates (𝛼, ∆𝐿), (𝛼, ∆𝐿) of (πr, C) of the point N correspond to the coordinates (αtip, ∆Ltip), (αroot, ∆Lroot) of the tooth tip D and the thetooth tooth root tipE, respectively.D and the tooth Then, root the cycloidalE , respectively. relationship Then, between the cycloidal the modification relationship and between the pressure the modificationangle can be established,and the pressure as shown angle in can Figure be established,6. as shown in Figure 6.

ΔL Δ Δ Lroot Ltip

Δ L0 α α α α α root 0 0 tip

Figure 6. First cycloid relationship between the modification and pressure angle.

As illustrated in Figure 6, the expression of the first cycloid modification curve from the reference point A to the tooth tip D can be obtained as: 𝛼=𝛼 −∆𝐿 −∆𝐿 (𝜃 − sin 𝜃)(𝛼 −𝛼 )(2𝜋𝑟)⁄ . (8) ∆𝐿 = ∆𝐿 −(∆𝐿 −∆𝐿)(1 − 𝑐𝑜𝑠𝜃)⁄ 2

Similarly, by replacing 𝛼 and ∆𝐿, respectively, with 𝛼 and ∆𝐿, the first cycloidal modification curve expression from the reference point A to the tooth root point E can be obtained. In the second method, with the pressure angle 𝛼 and the modification ∆𝐿 are, respectively, treated as the parameter angle 𝜃θ and the Y-axis. Thus, the function relationship between the modification and the pressure angle can be established (Figure 7). Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 15

When a circle rolls along a straight line, the trajectory 𝑁𝑁𝑁" of Point 𝑁 on the circle forms a cycloid, as shown in Figure 5.

Figure 5. Schematic diagram of forming a cycloid.

The mathematical expression of the cycloid is: 𝑥=𝜋𝑟−𝑟(𝜃−𝑠𝑖𝑛𝜃) , (7) 𝑦=𝐶−𝑟(1−𝑐𝑜𝑠𝜃) where (𝑥,) 𝑦 is the coordinates of any point on the cycloid; r is the rolling circle radius; 𝜃 is the rolling angle of the circle; C is the vertical coordinate of the starting point N. According to Equation (7), there are two methods to determine the cycloid modification function. In the first method, the pressure angle 𝛼 as the X-axis and the modification ∆𝐿 as the Y-axis meet the cycloidal function relationship. In Figure 5, the coordinates (0, C − 2r) of the point 𝑁" correspond to the coordinates (𝛼, ∆𝐿) of reference point A in the modified coordinate system, and the coordinates (𝜋𝑟, C) of the point N correspond to the coordinates (𝛼, ∆𝐿), (𝛼, ∆𝐿) of the tooth tip D and the tooth root E , respectively. Then, the cycloidal relationship between the modificationAppl. Sci. 2020, 10 and, 1266 the pressure angle can be established, as shown in Figure 6. 8 of 16

ΔL Δ Δ Lroot Ltip

Δ L0 α α α α α root 0 0 tip Figure 6. First cycloid relationship between the modification and pressure angle. Figure 6. First cycloid relationship between the modification and pressure angle. As illustrated in Figure6, the expression of the first cycloid modification curve from the reference pointAsA toillustrated the tooth in tip FigureD can be6, obtainedthe expression as: of the first cycloid modification curve from the reference point A to the tooth tip D can be obtained as:       α = αtip ∆Ltip ∆L0 (θ sin θ) αtip α0 /(2πr)  𝛼=𝛼−−∆𝐿 −−∆𝐿(𝜃− − sin 𝜃)(𝛼− −𝛼)(2𝜋𝑟)⁄ . (8)  . (8)  ∆L = ∆Ltip ∆Ltip ∆L0 (1 cosθ)/2⁄ ∆𝐿 = ∆𝐿 −−(∆𝐿−−∆𝐿)(1− − 𝑐𝑜𝑠𝜃) 2

Similarly, by by replacing replacing 𝛼αtip and ∆𝐿∆Ltip,, respectively, respectively, with with 𝛼αroot and ∆∆𝐿Lroot,, thethe firstfirst cycloidal modificationmodification curve expression from the reference point A toto thethe toothtooth rootroot pointpoint EE cancan bebe obtained.obtained. InIn thethe secondsecond method, method, with with the the pressure pressure angle angleα and 𝛼 and themodification the modification∆L are, ∆𝐿 respectively, are, respectively, treated treatedas the parameter as the parameter angle θθ angleand the 𝜃θY -axis.and the Thus, Y-axis. the function Thus, the relationship function relationship between the modificationbetween the modificationAppl.and theSci. 2020 pressure, 10 and, x FOR anglethe PEER pressure can REVIEW be establishedangl e can be (Figure established7). (Figure 7). 8 of 15

ΔL Δ Δ Lroot Ltip

Δ L0

α α α α α root 0 0 tip Figure 7. SecondSecond cycloid cycloid relationship relationship between between th thee modification modification and pressure angle.

As illustrated in Figure 77,, thethe secondsecond cycloidcycloid modificationmodification curvecurve expressionexpression fromfrom thethe referencereference point A to the tooth tip D can be determined as: ! ! ∆L∆∆∆L ∆𝐿 = tip 0 1 − cos α α0 𝜋 + ∆𝐿 . (9) ∆L = − 1 cos − π + ∆L . (9) 2 − α α 0 tip − 0 Through a similar modeling method, the modification equations of the straight line method and catenaryThrough method a similar can modeling also be obtained. method, the modification equations of the straight line method and catenary method can also be obtained. 4. Meshing Contact Characteristics of the Modified Profile 4. Meshing Contact Characteristics of the Modified Profile 4.1. Meshing Contact Model of a Cycloid-Pin Gear 4.1. Meshing Contact Model of a Cycloid-Pin Gear Through the spatial coordinate transformation, the position vector and the normal vector of the Through the spatial coordinate transformation, the position vector and the normal vector of the cycloidal gear and the pin gear can be, respectively, represented in the fixed coordinate system 𝑆. cycloidal gear and the pin gear can be, respectively, represented in the fixed coordinate system S f . According to the gear meshing principle and the continuous contact condition of the tooth surface, at According to the gear meshing principle and the continuous contact condition of the tooth surface, any time, the position coordinates of the contact points on the cycloidal surface should be consistent at any time, the position coordinates of the contact points on the cycloidal surface should be consistent with the position coordinates on the pin profile and the normal directions of the contact points with the position coordinates on the pin profile and the normal directions of the contact points should should also be the same [14]. Therefore, the meshing contact analysis model of a cycloid-pin gear can be established as follows:

𝐌(𝜙) ∙𝐑(𝜑) =𝐌(𝜙) ∙𝐑(𝛽) , (10) 𝐌(𝜙) ∙𝐧(𝜑) =𝐌(𝜙) ∙𝐧(𝛽) where 𝜑 and 𝛽 are the parametric angles of the cycloidal gear and the pin gear; 𝜙 and 𝜙 are, respectively, the rotation angles of the pin gear and the cycloidal gear. 𝐌 = cos 𝜙 −sin𝜙 00 cos 𝜙 −sin𝜙 00 sin 𝜙 cos 𝜙 0−𝑎 sin 𝜙 cos 𝜙 00 and 𝐌 = are the transformation matrices. 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 Equation (10) is the non-linear contact analysis system of equations for the cycloid-pin meshing. There are three equations and four unknown parameters in the system of equations. Given any unknown parameter, the remaining unknown parameters can be iteratively solved.

4.2. Transmission Error of the Modified Profile Theoretically speaking, in the cycloid-pin transmission, when the pin gear rotates by a certain angle 𝜙, the cycloidal gear rotates by a corresponding angle 𝜙. The ratio of 𝜙 to 𝜙 is called the theoretical gear ratio. However, the actual transmission ratio of cycloid-pin gear pair is not equal to the theoretical one because of the influences of various factors such as profile modification, manufacturing error and force deformation of the cycloidal gear. The difference between the actual and theoretical output angles of the cycloidal gear is called transmission error. The transmission error is an important index for evaluating the profile modification quality. Its existence directly affects the meshing performance of the cycloid-pin gear and causes certain shock and vibration [12,20,22]. However, the profile shape of the cycloidal gear has a great influence on the transmission error and the lost motion. Appl. Sci. 2020, 10, 1266 9 of 16 also be the same [14]. Therefore, the meshing contact analysis model of a cycloid-pin gear can be established as follows: ( M (φ ) Rc(ϕ) = M (φ ) Rp(β) f c 2 · f p 1 · , (10) M (φ ) nc(ϕ) = M (φ ) np(β) f c 2 · f p 1 · where ϕ and β are the parametric angles of the cycloidal gear and the pin gear; φ1 and φ2 are, respectively,  cos φ sin φ 0 0   2 2   −   sin φ2 cos φ2 0 a  the rotation angles of the pin gear and the cycloidal gear. M f c =  −   0 0 1 0     0 0 0 1   cos φ sin φ 0 0   1 1   −   sin φ1 cos φ1 0 0  and M f p =   are the transformation matrices.  0 0 1 0     0 0 0 1  Equation (10) is the non-linear contact analysis system of equations for the cycloid-pin meshing. There are three equations and four unknown parameters in the system of equations. Given any unknown parameter, the remaining unknown parameters can be iteratively solved.

4.2. Transmission Error of the Modified Profile Theoretically speaking, in the cycloid-pin transmission, when the pin gear rotates by a certain angle φ1, the cycloidal gear rotates by a corresponding angle φ2. The ratio of φ1 to φ2 is called the theoretical gear ratio. However, the actual transmission ratio of cycloid-pin gear pair is not equal to the theoretical one because of the influences of various factors such as profile modification, manufacturing error and force deformation of the cycloidal gear. The difference between the actual and theoretical output angles of the cycloidal gear is called transmission error. The transmission error is an important index for evaluating the profile modification quality. Its existence directly affects the meshing performance of the cycloid-pin gear and causes certain shock and vibration [12,20,22]. However, the profile shape of the cycloidal gear has a great influence on the transmission error and the lost motion. The transmission error can be expressed as:

δ = (φ φ ) zp/zc (φ φ ), (11) 2 − 20 − · 1 − 10 where φ10 and φ20 are the initial rotation angles of the pin gear and the cycloid gear, respectively. According to the pressure angle variations shown in Figure3, the modification at the position where the pressure angle is the smallest should also be the smallest. Generally, the position is used as the reference point for modification and its transmission ratio at this position is approximately equal to the theoretical transmission ratio. The process of solving transmission error is shown in Figure8. Firstly, according to the variation trend of the pressure angle, the parameter ϕ0 of the modification reference point of the cycloidal gear can be determined. By solving Equation (10), the initial engagement angles φ10 and φ20 of the pin gear and the cycloidal gear at the modification reference point can be obtained. Secondly, by changing the pin gear angle φ with a certain step size ∆φ (φ = φ n∆φ, where n is the number of iterations) 1 1 10 ± and substituting φ1 into Equation (10), a series of corresponding angles φ2 of the cycloidal gear can be calculated. Finally, the transmission error curve can be obtained by substituting meshing contact parameters φ10, φ20, φ1 and φ2 into Equation (11). Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 15

The transmission error can be expressed as:

𝛿=(𝜙 −𝜙) −𝑧⁄𝑧 ∙ (𝜙 −𝜙), (11)

where 𝜙 and 𝜙 are the initial rotation angles of the pin gear and the cycloid gear, respectively. According to the pressure angle variations shown in Figure 3, the modification at the position where the pressure angle is the smallest should also be the smallest. Generally, the position is used as the reference point for modification and its transmission ratio at this position is approximately equal to the theoretical transmission ratio. The process of solving transmission error is shown in Figure 8. Firstly, according to the

variation trend of the pressure angle, the parameter 𝜑 of the modification reference point of the cycloidal gear can be determined. By solving Equation (10), the initial engagement angles 𝜙 and 𝜙 of the pin gear and the cycloidal gear at the modification reference point can be obtained. Secondly, by changing the pin gear angle 𝜙 with a certain step size ∆𝜙 (𝜙 =𝜙 ±𝑛Δ𝜙, where n is the number of iterations) and substituting 𝜙 into Equation (10), a series of corresponding angles Appl.𝜙 Sci. of the2020 ,cycloidal10, 1266 gear can be calculated. Finally, the transmission error curve can be obtained10 of 16by substituting meshing contact parameters 𝜙, 𝜙, 𝜙 and 𝜙 into Equation (11).

N

Y

FigureFigure 8. 8.Process Process of of solving solving the the transmission transmission error. error.

4.3.4.3. Minimum Minimum Backlash Backlash Angle Angle of of the the Modified Modified Profile Profile LostLost motion motion determines determines the the positioning positioning accuracy accuracy of of a a precision precision reducer. reducer. It It is is also also one one of of the the importantimportant indicators indicators for for evaluating evaluating the the modification modification quality. quality. One One of of the the most most important important influencing influencing factorsfactors of of the the lost lost motion motion is the is minimum the minimum backlash backlash of the modifiedof the modified tooth profile. tooth In profile. order to In accurately order to evaluateaccurately the evaluate modification the modification quality, the minimum quality, the backlash minimum angle backlash of the modified angle of profilethe modified is discussed profile in is thisdiscussed study. The in this schematic study. diagramThe schematic of the minimumdiagram of backlash the minimum required backlash for the reverse required meshing for the contact reverse betweenAppl.meshing Sci. 2020 pin contact, 10 gear, x FOR andbetween PEER cycloidal REVIEW pin gear gear and is shown cycloidal in Figure gear 9is. shown in Figure 9. 10 of 15

Y f

X p φ 0 X 2 φ 0 c 1 X f a

Yp Y c Figure 9. Backlash of cycloid-pin gear in reverse rotation.

Under thethe non-loading non-loading condition, condition, only only one one pair pair of teeth of engageteeth engage in the transmissionin the transmission of the modified of the cycloidalmodified gearcycloidal and pin gear gear. and In pin Figure gear.9, In the Figure No. 0 tooth9, the isNo. the 0 current tooth is meshing the current tooth meshing pair and tooth the other pair toothand the has other no meshing tooth contact.has no meshing It can be seencontact. that It there can are be diseenfferent that backlashes there are between different the backlashes gear teeth. Thebetween tooth the pair gear with teeth. the smallestThe tooth backlash pair with can the be smal contactedlest backlash first when can thebe contacted pin gear is first reversely when rotated.the pin Therefore,gear is reversely it is necessary rotated. Therefore, to overcome it is the necessary smallest to backlash overcome before the smallest reverse backlash meshing before transmission. reverse Themeshing essence transmission. of the lost motionThe essence calculation of the is to lost obtain motion the angle calculation corresponding is to toobtain the firstthe pairangle of toothcorresponding contacts byto calculatingthe first pair the of minimumtooth contacts backlash by calculating at the moment the minimum of the reverse backlash rotation at the of moment the pin gear.of the The reverse flowchart rotation shown of the in Figurepin gear. 10 clearly The flow illustrateschart shown the process in Figure of determining 10 clearly theillustrates minimum the backlashprocess of angle. determining the minimum backlash angle.

Figure 10. Process of solving the minimum backlash angle.

In the calculation process of the backlash angles, it is necessary to firstly determine the initial rotation angles 𝜙 and 𝜙 of the pin gear and the cycloidal gear in the current meshing state. When the cycloidal gear remains fixed and the pin gear rotates in the opposite direction, the backlash angle 𝜙 of the pin gear in the meshing process is solved. In this way, a series of angles of the pin gear can be obtained and represented as 𝜙 (i = 1~𝑧). Thus, the backlash angle can be obtained with Δ𝜙 =𝜙 −𝜙 , which is the rotation angle of the i-th pin tooth required for eliminating the backlash and reverse meshing contact. The minimum backlash angle describes the lost motion and also determines the pin number of the first contact. With the pin tooth number as the abscissa and the rotation angle required for eliminating backlash as the ordinate, the minimum backlash angle curve can be plotted.

5. Examples and Discussion The essence of the modification method in this paper is to obtain different tooth profile shapes by establishing different modification piecewise functions 𝑓(𝛼). In order to compare the different modification piecewise functions 𝑓(𝛼), the function curves obtained by the straight line method, Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 15

Y f

X p φ 0 X 2 φ 0 c 1 X f a

Yp Y c Figure 9. Backlash of cycloid-pin gear in reverse rotation.

Under the non-loading condition, only one pair of teeth engage in the transmission of the modified cycloidal gear and pin gear. In Figure 9, the No. 0 tooth is the current meshing tooth pair and the other tooth has no meshing contact. It can be seen that there are different backlashes between the gear teeth. The tooth pair with the smallest backlash can be contacted first when the pin gear is reversely rotated. Therefore, it is necessary to overcome the smallest backlash before reverse meshing transmission. The essence of the lost motion calculation is to obtain the angle corresponding to the first pair of tooth contacts by calculating the minimum backlash at the moment

Appl.of the Sci. reverse2020, 10, 1266rotation of the pin gear. The flowchart shown in Figure 10 clearly illustrates11 ofthe 16 process of determining the minimum backlash angle.

Figure 10. Process of solving the minimum backlash angle.

In the calculation process of the backlash angles, it is necessary to firstlyfirstly determine the initial 0 0 rotation anglesangles φ𝜙1and andφ 2𝜙of of the the pin pin gear gear and and the cycloidalthe cycloidal gear ingear the in current the current meshing meshing state. Whenstate. theWhen cycloidal the cycloidal gear remains gear remains fixed and fixed the and pin the gear pi rotatesn gear inrotates the opposite in the opposite direction, direction, the backlash the anglebacklashφ1 ofangle the pin𝜙 of gear thein pin the gear meshing in the processmeshing is process solved. is Insolved. this way, In this a series way, ofa series angles ofof angles the pin of gear can be obtained and represented as φi (i = 1~z ). Thus, the backlash angle can be obtained the pin gear can be obtained and represented1 as 𝜙p (i = 1~𝑧). Thus, the backlash angle can be i i 0 withobtained∆φ with= φ Δ𝜙φ =𝜙 , which −𝜙 is the, which rotation is the angle rotation of the iangle-th pin of tooth the requiredi-th pin fortooth eliminating required thefor 1 1 − 1 backlasheliminating and the reverse backlash meshing and reverse contact. meshing The minimum contact. backlash The minimum angle describesbacklash theangle lost describes motion andthe alsolost motion determines and thealso pin determines number ofthe the pin first number contact. of the With first the contact. pin tooth With number the pin as tooth the abscissa number and as the rotationabscissa angleand the required rotation for angle eliminating required backlash for eliminating as the ordinate, backlash the minimumas the ordinate, backlash the angle minimum curve canbacklash be plotted. angle curve can be plotted.

5. Examples and Discussion The essence of the modification method in this paper is to obtain different tooth profile shapes Appl. TheSci. 2020 essence, 10, x FORof the PEER modification REVIEW method in this paper is to obtain different tooth profile shapes11 of 15 ( ) by establishing didifferentfferent modificationmodification piecewise functions f𝑓(α𝛼).. In order to compare the different different ( ) modificationmodificationcycloid method piecewisepiecewise and catenary functions functions methodf α𝑓(,𝛼 the), arethe function plottedfunction curves in curves Figure obtained obtained 11. byThe the bygeometric straight the straight line parameters method, line method, cycloid of the methodcycloidal and gear catenary are shown method in Table are plotted 1. According in Figure to 11 the. The requirements geometric parametersof the cycloid-pin of the cycloidaltransmission, gear arethe shownpredetermined in Table1. Accordingtooth tip and to the root requirements clearance ofare the 0.02 cycloid-pin mm and transmission, the modification the predetermined value of the toothreference tip and point root is clearance0.005 mm. are Through 0.02 mm calculation, and the modification the minimum value profile of the pressure reference angle point is isdetermined 0.005 mm. Throughto be 42.24°, calculation, which is the also minimum the pressure profile angle pressure at the angle modification is determined reference to be 42.24point.◦, Since which the is also change the pressuretrend of anglethe modification at the modification value from reference the reference point. Sincepoint theA to change the tooth trend tip of D the is similar modification to that value from fromthe reference the reference point point A to Atheto tooth the tooth root tipE, Donlyis similarthe function to that curve from thefrom reference the reference point Apointto the A to tooth the roottoothE ,tip only D is the plotted function in Figure curve 11. from Thro theugh reference substituting point theA to pressure the tooth angle tip andD is modification plotted in Figure value 11 at. Throughthe reference substituting point and the pressurethe tooth angle tip point and modification into Equations value (8) at and the (9), reference the modification point and the curves tooth are tip pointobtained into (Figure Equations 11). (8) and (9), the modification curves are obtained (Figure 11).

FigureFigure 11.11. ModificationModification curvescurves ofof didifferentfferent methods.methods.

The range of the working section of the cycloid-pin gear in the actual transmission process is relatively small. Similarly, the range of the profile pressure angle in the working section in Figure 11 is 42.24°~42.45°. In this working section, the modifications calculated by the new method are not much different, and their changes are also slow. Therefore, it can be preliminarily judged that the modified profile near the reference point A is the closest to the theoretical profile in the meshing area, and the modification effect should be reasonable. According to Equations (3) and (4), the modifications are superimposed on the normal direction of the cycloidal profile, and the modified tooth profile of the cycloid gear and pin tooth profile in coordinate system Sc are obtained. With the aid of Equation (10), the meshing contact analysis of the cycloid-pin gear after the modification is completed and the transmission error curve and the minimum backlash angle curve are obtained (Figures 12 and 13). The traditional modification method adopts the modification combination of the positive equidistance and the positive displacement. The equidistant modification value is 0.005 mm and the displacement modification is 0.015 mm. These settings can ensure that the initial conditions of the traditional and new modification methods are the same for the convenience of comparison and discussion.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 15 cycloid method and catenary method are plotted in Figure 11. The geometric parameters of the cycloidal gear are shown in Table 1. According to the requirements of the cycloid-pin transmission, the predetermined tooth tip and root clearance are 0.02 mm and the modification value of the reference point is 0.005 mm. Through calculation, the minimum profile pressure angle is determined to be 42.24°, which is also the pressure angle at the modification reference point. Since the change trend of the modification value from the reference point A to the tooth tip D is similar to that from the reference point A to the tooth root E, only the function curve from the reference point A to the tooth tip D is plotted in Figure 11. Through substituting the pressure angle and modification value at the reference point and the tooth tip point into Equations (8) and (9), the modification curves are obtained (Figure 11).

Appl. Sci. 2020, 10, 1266 12 of 16 Figure 11. Modification curves of different methods.

TheThe range range of of the the working working section section of of the the cycloid-pi cycloid-pinn gear gear in in the the actual actual transmission transmission process process is is relativelyrelatively small. small. Similarly, Similarly, the ra rangenge of the profileprofile pressurepressure angleangle inin thethe working working section section in in Figure Figure 11 11 is is42.24 42.24°~42.45°.◦~42.45◦. In In this this working working section, section, the the modifications modifications calculated calculated by theby newthe new method method are not are much not muchdifferent, different, and their and changes their changes are also are slow. also Therefore, slow. Ther itefore, can be it preliminarily can be preliminarily judged that judged the modifiedthat the modifiedprofile near profile the referencenear the reference point A is point the closestA is the to closest the theoretical to the theoretical profile in profile the meshing in the meshing area, and area, the andmodification the modification effect should effect beshould reasonable. be reasonable. AccordingAccording to to Equations Equations (3) (3) and and (4), (4), the the modifications modifications are are superimposed superimposed on on the the normal normal direction direction ofof the the cycloidal cycloidal profile, profile, and and the the modified modified tooth tooth pr profileofile of of the the cycloid cycloid gear gear and and pin pin tooth tooth profile profile in in coordinatecoordinate system system SSc careare obtained. obtained. With With the the aid aid of of Equation Equation (10), (10), the the meshing meshing contact contact analysis analysis of of the the cycloid-pincycloid-pin gear afterafter the the modification modification is completedis completed and theand transmission the transmission error curve error and curve the minimumand the minimumbacklash anglebacklash curve angle are obtained curve are (Figures obtained 12 and (Figures 13). The 12 traditional and 13). modification The traditional method modification adopts the methodmodification adopts combination the modification of the positive combination equidistance of the and positive the positive equidistance displacement. and The the equidistant positive displacement.modification valueThe equidistant is 0.005 mm modi andfication the displacement value is 0.005 modification mm and the is displacement 0.015 mm. These modification settings can is 0.015ensure mm. that theThese initial settings conditions can ofensure the traditional that the andinitial new conditions modification of methods the traditional are the same and fornew the modificationconvenience methods of comparison are the and same discussion. for the convenience of comparison and discussion.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 15

Figure 12. Transmission error curves of modified tooth profiles.

In Figure 12, the curve segment BiCi (i = 1, 2, 3, 4) indicates an engagement period of the modified profile and Points Bi and Ci, respectively, correspond to the meshing start and end points of the gear tooth. In the meshing area, the transmission error curves of the modified profiles are smooth and the symmetry is also good. The maximum transmission error of the traditional modification is −1.47 arcsec, which is significantly larger than that of the new method. The transmission error of the tooth profile modified by the second cycloid method is the smallest, which is −0.44 arcsec and indicates the transmission accuracy is higher. Figure 12. Transmission error curves of modified tooth profiles.

10 Traditional method 9 Straight line method 8 Catenary method 7 Second cycloid method 6 5 4 3 2 1 0 0 2 4 6 8 10121416182022242628303234363840 Pin tooth number Figure 13. Minimum backlash curves of modifiedmodified tooth profiles.profiles.

InUnder Figure the 12 same, the curveinitial segment clearanceBi Crequirements,i (i = 1, 2, 3, 4) the indicates minimum an engagement backlash angle period curve of the obtained modified by profilethe new and modification Points Bi and methodCi, respectively, is basically correspondthe same and to the the value meshing of the start minimum and end backlash points of angle the gear has tooth.little difference, In the meshing reaching area, approximately the transmission 0.6 arcmin error curves(Figure of 13). the The modified minimum profiles backlash are smoothangle of and the the symmetry is also good. The maximum transmission error of the traditional modification is 1.47 traditional modification method is significantly larger, reaching 0.99 arcmin. In addition,− the arcsec,minimum which backlash is significantly angle of largereach modification than that of themethod new method.corresponds The to transmission the No. 31 errortooth, of indicating the tooth profile modified by the second cycloid method is the smallest, which is 0.44 arcsec and indicates the that the No. 31 tooth first comes into contact with the cycloidal profile− in reverse rotation. It can be transmissiondetermined that accuracy the lost is higher. motions of the cycloid-pin gear after the modification by the traditional method and the proposed method are 0.6 and 0.99 arcmin, respectively. The above results indicate that under the same initial clearance requirement, the shape of the modified tooth profile had the greater influence on the transmission error and less influence on the backlash. For the traditional modification method, although the initial clearance condition is the same as that of the proposed method, the modifications of the meshing area cannot be determined in advance, thus resulting in the larger backlash in the meshing area, which indirectly affects the repeated positioning error of the precision reducer. Furthermore, a measurement experiment of the transmission error and the lost motion of the robot precision reducer were carried out on a RV reducer tester shown in Figure 14. The transmission error and lost motion were measured with the static measurement method [23]. Under low-speed and no-load conditions, the high-precision circular grating installed at the input and output shafts was used to acquire the angular changes in real time.

Figure 14. RV reducer tester. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 15

Figure 12. Transmission error curves of modified tooth profiles.

In Figure 12, the curve segment BiCi (i = 1, 2, 3, 4) indicates an engagement period of the modified profile and Points Bi and Ci, respectively, correspond to the meshing start and end points of the gear tooth. In the meshing area, the transmission error curves of the modified profiles are smooth and the symmetry is also good. The maximum transmission error of the traditional modification is −1.47 arcsec, which is significantly larger than that of the new method. The transmission error of the tooth profile modified by the second cycloid method is the smallest, which is −0.44 arcsec and indicates the transmission accuracy is higher.

10 Traditional method 9 Straight line method 8 Catenary method 7 Second cycloid method 6 5 4 3 2 1 0 0 2 4 6 8 10121416182022242628303234363840 Pin tooth number Appl. Sci. 2020, 10, 1266 13 of 16 Figure 13. Minimum backlash curves of modified tooth profiles.

Under the same initialinitial clearanceclearance requirements, the minimum backlash angle curve obtained by the new modification modification method is basically the same and the value of the minimum backlash angle has little didifference,fference, reaching approximately 0.6 arcmin (Figure 1313).). The minimum backlash angle of the traditional modificationmodification method method is significantlyis significantly larger, larger, reaching reaching 0.99 arcmin. 0.99 arcmin. In addition, In addition, the minimum the backlashminimum angle backlash of each angle modification of each modification method corresponds method correspo to the No.nds 31to tooth,the No. indicating 31 tooth, that indicating the No. 31that tooth the No. first 31 comes tooth into first contact comes withinto contact the cycloidal with the profile cycloidal in reverse profile rotation. in reverse It canrotation. be determined It can be thatdetermined the lost motionsthat the oflost the motions cycloid-pin of the gear cycloid-pin after the modificationgear after the by modification the traditional by method the traditional and the proposedmethod and method the proposed are 0.6 and method 0.99 arcmin,are 0.6 and respectively. 0.99 arcmin, respectively. The above results indicateindicate thatthat underunder thethe samesame initialinitial clearanceclearance requirement, the shape of the modifiedmodified tooth profile profile had the greater influenceinfluence on the transmission error and less influenceinfluence on the backlash. For For the the traditional traditional modification modification method, method, although although the the initial clearance condition is the same as that of the proposed method, the modificati modificationsons of the meshing area cannot be determined in advance, thusthus resulting resulting in in the the larger larger backlash backlash in the in meshingthe meshing area, whicharea, which indirectly indirectly affects the affects repeated the positioningrepeated positioning error of the error precision of the precision reducer. reducer. Furthermore, a measurement experiment of the tr transmissionansmission error and the lost motion of the robot precisionprecision reducer reducer were were carried carried out onout a RVon reducera RV testerreducer shown tester in Figureshown 14 in. TheFigure transmission 14. The errortransmission and lost error motion and were lostmeasured motion were with measured the static with measurement the static methodmeasurement [23]. Under method low-speed [23]. Under and no-loadlow-speed conditions, and no-load the high-precisionconditions, the circular high-precisi gratingon installedcircular atgrating the input installed and outputat the shaftsinput wasand usedoutput to shafts acquire was the used angular to acquire changes the in angular real time. changes in real time.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 15 Figure 14. RVRV reducer tester. The reducer used in the measurement had 40 pins. Therefore, within the angle range (360°) of The reducer used in the measurement had 40 pins. Therefore, within the angle range (360◦) of the the output shaft, the rotation angle of one point was measured every 9° to obtain the actual output output shaft, the rotation angle of one point was measured every 9◦ to obtain the actual output rotation rotation angles at 40 positions. The cycloidal gear in this reducer was respectively replaced by the angles at 40 positions. The cycloidal gear in this reducer was respectively replaced by the cycloidal cycloidal gears modified by the traditional method, straight line method, cycloid method and gears modified by the traditional method, straight line method, cycloid method and catenary method. catenary method. Their static transmission error curves and geometric lost motion curves were Their static transmission error curves and geometric lost motion curves were obtained (Figures 15 obtained (Figures 15 and 16). and 16).

Traditional method Straight line method 1 Catenary method Second cycloid method 0

-1

-2

-3 0 40 80 120 160 200 240 280 320 360 Output shaft angle(deg) Figure 15. Static transmission error curves based on actual measurements. Figure 15. Static transmission error curves based on actual measurements. Lost motion(arcmin) Lost

Figure 16. Geometric lost motion based on actual measurements.

It can be seen from Figures 12, 13, 15, and 16 that the error values obtained from actual measurements were greater than those obtained from theoretical modeling. The reasons might be as follows. The actual measurement process was affected by many influencing factors, such as installation errors, tooth profile errors, tooth pitch errors, and radial runout. The combined effects of these factors were complex and could not be accurately determined. However, before the test, a series of measures were taken to minimize these effects. For example, through the adjustment and detection of the tooling and precision instruments for many times, the high installation accuracy could be ensured. Through many times of precision grinding, the high processing accuracy could be ensured (the maximum profile and pitch errors of the cycloidal gear were respectively limited to 0.02 mm and 0.006 mm). However, we can see that the overall trends of transmission errors and backlashes measured in actual measurements are similar to the results of the theoretical modeling. The changing trend of the transmission error and lost motion of the new modification method is significantly smaller than that of the traditional modification. Among them, the transmission error after the second cycloid method is the least and the lost motions of the straight line method and the traditional method are basically the same. In summary, compared with the tooth profile obtained with the traditional modification method, the modified tooth profile obtained by the proposed method can reflect smaller transmission error and backlash, indicating the higher motion accuracy and repeated positioning accuracy. Based on the working conditions and transmission requirements of robot reducers in engineering practices, the appropriate modification function can be selected to carry out the modification design of the cycloidal gear. Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 15

The reducer used in the measurement had 40 pins. Therefore, within the angle range (360°) of the output shaft, the rotation angle of one point was measured every 9° to obtain the actual output rotation angles at 40 positions. The cycloidal gear in this reducer was respectively replaced by the cycloidal gears modified by the traditional method, straight line method, cycloid method and catenary method. Their static transmission error curves and geometric lost motion curves were obtained (Figures 15 and 16).

Traditional method Straight line method 1 Catenary method Second cycloid method 0

-1

-2

-3 0 40 80 120 160 200 240 280 320 360 Output shaft angle(deg) Appl. Sci. 2020, 10, 1266 14 of 16 Figure 15. Static transmission error curves based on actual measurements. Lost motion(arcmin) Lost

Figure 16. Geometric lost motion based on actual measurements. Figure 16. Geometric lost motion based on actual measurements. It can be seen from Figures 12, 13, 15 and 16 that the error values obtained from actual measurements It can be seen from Figures 12, 13, 15, and 16 that the error values obtained from actual were greater than those obtained from theoretical modeling. The reasons might be as follows. The measurements were greater than those obtained from theoretical modeling. The reasons might be as actual measurement process was affected by many influencing factors, such as installation errors, tooth follows. The actual measurement process was affected by many influencing factors, such as profile errors, tooth pitch errors, and radial runout. The combined effects of these factors were complex installation errors, tooth profile errors, tooth pitch errors, and radial runout. The combined effects and could not be accurately determined. However, before the test, a series of measures were taken to of these factors were complex and could not be accurately determined. However, before the test, a minimize these effects. For example, through the adjustment and detection of the tooling and precision series of measures were taken to minimize these effects. For example, through the adjustment and instruments for many times, the high installation accuracy could be ensured. Through many times of detection of the tooling and precision instruments for many times, the high installation accuracy precision grinding, the high processing accuracy could be ensured (the maximum profile and pitch could be ensured. Through many times of precision grinding, the high processing accuracy could errors of the cycloidal gear were respectively limited to 0.02 mm and 0.006 mm). However, we can be ensured (the maximum profile and pitch errors of the cycloidal gear were respectively limited to see that the overall trends of transmission errors and backlashes measured in actual measurements 0.02 mm and 0.006 mm). However, we can see that the overall trends of transmission errors and are similar to the results of the theoretical modeling. The changing trend of the transmission error backlashes measured in actual measurements are similar to the results of the theoretical modeling. and lost motion of the new modification method is significantly smaller than that of the traditional The changing trend of the transmission error and lost motion of the new modification method is modification. Among them, the transmission error after the second cycloid method is the least and the significantly smaller than that of the traditional modification. Among them, the transmission error lost motions of the straight line method and the traditional method are basically the same. after the second cycloid method is the least and the lost motions of the straight line method and the In summary, compared with the tooth profile obtained with the traditional modification method, traditional method are basically the same. the modified tooth profile obtained by the proposed method can reflect smaller transmission error In summary, compared with the tooth profile obtained with the traditional modification and backlash, indicating the higher motion accuracy and repeated positioning accuracy. Based on method, the modified tooth profile obtained by the proposed method can reflect smaller the working conditions and transmission requirements of robot reducers in engineering practices, transmission error and backlash, indicating the higher motion accuracy and repeated positioning the appropriate modification function can be selected to carry out the modification design of the accuracy. Based on the working conditions and transmission requirements of robot reducers in cycloidal gear. engineering practices, the appropriate modification function can be selected to carry out the modification6. Conclusions design of the cycloidal gear. In this paper, the new modification design method of the tooth profile of a cycloidal gear is proposed based on the comprehensive consideration of the pressure angle distribution characteristics, the meshing backlash, the tooth tip and root clearance. In this method, the modification value is superimposed on the normal direction of the theoretical profile, the force transmission direction. According to the mathematical relationship between the modifications and the pressure angle distribution, which determines the force transmission performance, this method can flexibly control the changing trend of the cycloidal profile by using different modification piecewise functions and realize the pre-control of the transmission performance of the cycloid-pin gear. This method avoids the disadvantages of traditional modification design, such as the uncontrollable tooth profile shape and the unstable meshing accuracy after modification. The modified cycloidal-pin gear pair has good meshing contact characteristics and the better force transmission performance. The method provides a new way for the modification design of the cycloidal gear of the precision reducer of a robot.

Author Contributions: Conceptualization, T.L. and X.D.; data curation, Y.L.; formal analysis, T.L. and X.A.; funding acquisition, T.L.; methodology, T.L. and X.A.; validation, T.L. and J.L.; writing—original draft, T.L.; writing—review and editing, T.L. and X.A. All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2020, 10, 1266 15 of 16

Funding: This research was funded by the National Natural Science Foundation of China (grant Nos. U1504522, 51705134 and 51775171), and key scientific research projects of higher education institutions in Henan Province of China (grant No. 20A460010). Conflicts of Interest: The authors declare no conflict of interest.

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