## Abstract

To realize a no-backlash reducer for a small-size orthogonal-axis output, a novel design of a precession motion ball reducer is proposed. This reducer consists of a precession-motion-generating section, a motion direction converting section of the precession motion, and a reduced output-generating section. To avoid cutting tool interference, two types of distorted spatial circular grooves are located on the outer spherical surface of a precession rotor. Furthermore, the grooves of the reduced rotation generating section are profiled on the spherical surface at the output side of the precession motion rotor, which is different from the previous precession ball reducer. In this research, the equations for the groove loci of the precession motion generation, motion direction conversion, and reduced-rotation generation, which are located on the outer spherical surfaces, are proposed. The output rotating direction can be determined using the relationship between the input and output rotation angles of the precession motion direction converting groove. The prototype reducer is confirmed to smoothly rotate. Furthermore, at the hysteresis loss characteristic test, that is better than the backlash characteristic of a general orthogonal-axis output-type speed reducer with bevel gears.

## 1 Introduction

No-backlash characteristics are a typical and important performance of positioning mechanisms. The primary source of backlash arises from the back-side clearance in the normal direction of an involute gear. However, structural elimination of this characteristic proves challenging. Consequently, various gearless reducers have been developed [1–3]. Specifically, ball reducers exhibit a no-backlash characteristic attributed to their rolling transmission, based on a general spatial cam-mechanism design. Moreover, these reducers prove advantageous for positioning mechanisms with frequent forward and reverse rotations, given the easily adjustable preload. Various reducers have been proposed to satisfy such requirements [4–10]. On the other hand, the majority of developed reducers have been of the coaxial output type. Orthogonal-output reducers with a reduction ratio of approximately 10, devoid of backlash, have seldom found practical applications. In the eccentric motion, the radial component follows a linear locus. Consequently, extracting rotational motion in the orthogonal-axis direction directly from this form of motion proves challenging. To solve this limitation, in realizing an orthogonal-axis output reducer, we redirect our focus from eccentric to precession motion, which is an approach common in coaxial output-type reducers. This choice is motivated by the fact that, although precession motion may experience distortion in both axial and radial directions, it has a rotational periodic motion and is suitable for orthogonal-axis output. Precession motion represents a cone rotation around an imaginary base circle without slipping. Previous involute gear reducers that used precession motion have been developed [11–13]. Because eliminating backlash is difficult, previous involute gear reducers could not be used in a positioning mechanism. Terada et al. have developed a precession ball reducer to solve these problems [14]. However, each motion-generating groove needed to be located on the inner spherical surface, and interference between the rotor and cutting tool often occurred. Therefore, manufacturing a small reducer using the previous structure was impossible. Furthermore, a small-size reducer that could be embedded into a robot-arm wrist was required. Considering these factors, a design of an orthogonal-axis-type precession motion ball reducer with grooves on the outer spherical surfaces is proposed [15]. In the current research, the equations of the groove loci of the reducer, which are generated on the outer spherical surfaces, are proposed. Then, the prototype reducer is tested to evaluate the no-backlash performance.

## 2 Structure of a Newly Designed Precession Motion Ball Reducer

The proposed reducer consists of a precession-motion-generating section, the motion-converting section of the precession motion, and a reduced output rotation-generating section, as shown in Fig. 1. An inclined shaft that can freely rotate is rotated by an input shaft, and precession motion is generated by the constrained motion between the precession-motion-generating groove and a ball on a spherical pocket fixed on an input-side fixed plate. The groove is located on the spherical surface of a trunnion ball and can be located in any number of places. It differs from the previous reducer in that all spatial grooves are located on the outer surfaces of the reduced output-generating rotor and trunnion ball.

To generate the orthogonal-axis output rotation, a differential motion is generated by the constrained motion between the motion-converting groove and ball on a spherical pocket fixed at the output-side precession-motion-generating rotor. The inclined angle magnitude at the output side is different from that at the input-side. Then, this converting groove is located at the opposite-side spherical surface of the precession motion groove on the trunnion ball.

Finally, an output-side precession motion is generated by the constrained motion between the output-side precession-motion-generating rotor and fixed ball on an output-side fixed disk. This part is similar to the input-side precession-motion-generating section. Then, the output shaft is rotated at a constant reduced speed generated by the relative motion between a spatial trochoidal-wave groove and a fixed ball in the output shaft. Multiple balls are simultaneously installed at equal-pitch mesh on this groove.

## 3 Vector Analysis of Precession Motion Generation

When a cone rotates around an imaginary base circle without slipping, it generates a swing motion and rotates around the precession-disk center. This motion is defined as “precession motion.” Terada et al. analyzed the motion locus of precession motion for a coaxial output-type reducer [16]. To apply this method, the groove located on a spherical surface of a trunnion ball was analyzed.

**Q**, which rotates vector

_{f0}**P**with j-axis rotation

_{f0}**E**

^{jτ}^{0}. In this analysis, the

*x*,

*y*, and

*z*axes on the Cartesian space are defined as

*i*,

*j*, and

*k*axes, respectively, and the rotation matrices are defined

**E**

*,*

^{it}**E**

*, and*

^{jt}**E**

*as*

^{kt}**P**is defined as the initial position vector of the fixed ball center, which is a normal direction vector of inclined angle αin. The horizontal and vertical vectors are defined as

_{f0}**R**and

_{f0}**S**, as expressed in Eqs. (2) and (3). The values of the initial position from the horizontal and vertical lines are defined as

_{f0}*r*and

_{f0}*l*

_{0}, respectively.

This groove can be located in any number of places. Therefore, parameters *l*_{0}, *m*_{0}, and *η*_{0} in these equations can assume the length, ball numbers, and arbitrary angle.

*θ*

_{inp}around the arbitrary axis

*w*

_{inp}-axis is defined as follows:

**P**denotes the motion locus of the precession motion on the trunnion ball. Thus, Eq. (5) is substituted into Eq. (4) to obtain the motion locus, as expressed in Eq. (8).

_{fr0}*θ*

_{inp}around the

*i*-axis and angle

*ϕ*

_{inp}around inclined axis

*w*

_{inp}is required, as expressed in Eq. (9).

On the basis of the calculated ball-center locus, the grooves in the trunnion ball can be generated using an end-mill cutter with the same diameter as the ball.

## 4 Vector Analysis of Precession Motion Conversion

To generate the output-side precession motion, a differential motion must be generated by the constrained motion between the motion-converting groove and the ball on the spherical pocket.

*w*

_{out}as angle

*ϕ*

_{conv}. The initial position of the fixed ball is defined as

**P**, the same as that in the precession-motion-generating groove, which rotates vector

_{c1}**R**with

_{c1}*j*-axis rotation

**E**. Vector

^{jτ}^{1}**R**shows the initial position vector of the fixed ball center, as expressed in Eq. (10).

_{c1}In this equation, *r*_{1} denotes the initial position from the trunnion ball center.

*τ*

_{1}should be changed. In addition, the

*Z*

_{bc1}th balls are attached at equal angle pitch

*η*

_{1}, and offset angle

*ε*is an important parameter that creates the different motion phases of the balls. In previous research, the offset angle must be clearly selected from −30 deg to −65 deg. Therefore, the offset value is used under the same condition.

**E**

^{i}^{(−θinp)}and

**E**

^{w}^{(−}

^{ϕ}^{inp)}, as expressed in Eq. (13). In particular, to change the rotation direction of the output, the sign of each matrix can be selected to be in the positive direction as +

*θ*

_{inp}and +

*ϕ*

_{inp}.

*θ*

_{inp}, rotation angle around motion-converting angle

*θ*

_{conv}, input-side inclined rotating angle

*ϕ*

_{inp}, and output-side inclined angle

*ϕ*

_{out}is expressed in Eq. (14).

According to the calculated ball-center locus, converting grooves can be made similar to those at the input-side precession-motion-generating grooves.

## 5 Motion Analysis of a Reduced-Rotation Output

To use the precession motion to convert the motion direction, we need to generate a reduced rotation in this mechanism. The reduced-rotation output-generating section consists of precession motion and reduced-rotation generating parts. It is similar to the structure of a coaxial output-type precession ball reducer. In this research, the geometry of the reduced output-generating section is proposed as follows.

**R**and

_{f2}**S**, respectively, as expressed in Eqs. (15) and (16). The values of the initial position from the horizontal and vertical lines are defined as

_{f2}*r*

_{f}_{2}and

*l*

_{2}, respectively.

*Z*

_{bc2}th balls are attached at equal angle pitch

*η*

_{2}.

**P**represents the motion locus of the output-side precession motion. Hence, Eq. (18) is substituted into Eq. (17) to obtain the motion locus, as expressed in Eq. (19).

_{fr2}Furthermore, the geometry of the reduced-rotation part is defined using vector analysis, as shown in Fig. 5. The horizontal and vertical vectors are denoted as **R _{f3}** and

**S**, respectively, as expressed in Eqs. (20) and (21), similar to those in the other sections. The values of the initial position from the horizontal and vertical lines are denoted as

_{f3}*r*

_{r}_{3}and

*l*

_{3}, respectively.

The position of the fixed ball center on the output rotor is expressed in Eq. (22). The *Z*_{bc3}th balls are also attached at equal angle pitch *η*_{3}.

**P**represents the motion locus of the output motion. Therefore, Eq. (23) is substituted into Eq. (19) to obtain the motion locus, as expressed in Eq. (24).

_{r3}**Q**as Eq. (25).

_{ro3}## 6 Pressure Angle Conditions

In order to efficiently generate reduced motion, it is crucial to select appropriate values for the design parameters. Therefore, a method for determining the position of the precession-motion-converting grooves and the wave-number of the spatial trochoidal-curve groove is proposed. This method utilizes the pressure angle, which is a commonly employed parameter in cam mechanisms.

The pressure angle is defined as the angle between the common normal direction of the contact points of the driver and the follower in the cam mechanism and the motion direction of the follower [18]. Considering this definition, in this reducer, the pressure angle can be simply defined as the angle between the plane that includes the center of the rolling ball and the contact point on each groove, and the relative motion direction of the rolling ball, as shown in Fig. 6. In this case, the positive/negative normal direction at the point of contact between the groove and the rolling element is ascertained by the rotational direction of the follower. Generally, the absolute value of the pressure angle is used. Furthermore, a pressure angle of 45 deg or less is often employed in the case of an oscillating follower cam to ensure smooth driving.

Considering the aforementioned definition, the center locus of the rolling ball on the precession-motion-converting grooves, generated using the uniform radius, is depicted in Fig. 7, and their pressure angle variations are shown in Fig. 8. Although all the grooves mesh with the rolling elements simultaneously, the behavior of pressure angle fluctuations differs significantly due to variations in the shape of the converting grooves and the phase difference. As with cam mechanisms, rolling elements with a pressure angle of 45 deg or less are considered to be involved in motion transmission. Therefore, it is desirable to have a large number of grooves with such pressure angles. However, two grooves (G1 and G6), generated at approximately 90 deg and 180 deg to the trunnion axial direction, are excluded from the groove-generating conditions due to interference issues in this case. Consequently, it is possible to observe an angle condition with a smaller number of grooves resulting in a pressure angle of 45 deg or less. Upon examining the correlation between the input rotation angle and the count of rolling balls possessing a pressure angle of 45 deg or less (refer to Fig. 9), it becomes evident that the count diminishes to 3 at a specific input rotation angle. Essentially, this implies that a minimum of three points featuring a pressure angle of 45 deg or less are available for participating in motion transmission, thereby ensuring the three-point contact essential for a general object constraint.

Consequently, the risk of non-uniform meshing is avoided, affirming that the requisite conditions for meshing are met. Considering these findings, it is advisable to position the precession-motion-converting grooves not perpendicular to the trunnion axis but rather around G2, G3, and G4. These locations exhibit a relatively small variation in pressure angle and a maximum pressure angle of less than 60 deg. In regions characterized by the deterioration of pressure angles within these grooves, the pressure angles associated with grooves G8, G9, and G10 measure below 45 deg, with these sides being actively engaged in motion transmitting. Taking these considerations into account, it is deemed advantageous to position six precession-motion-converting grooves as G2, G3, G4, G8, G9 and G10 at least. This arrangement ensures stable rotation, as these grooves are actively engaged in motion transmission. Furthermore, an arbitrary number of grooves can be accommodated at any position, provided they do not interfere with each other.

On the other hand, in the reduced-rotation generating groove, multiple rolling balls mesh simultaneously within a single groove but are in different phases. Therefore, the determination of the pressure angle was carried out using the same method as for the precession-motion-converting grooves. Figure 10 shows the pressure angle variation when the number of balls meshing with the reduction groove is eight. Unlike the precession-motion-converting grooves, it is observed that the pressure angles of more than half of the balls are consistently below 45 deg, as shown in Fig. 11. This behavior remains unchanged even when the reduction ratio is altered. Thus, it can be concluded that there are no issues with motion transfer.

## 7 Simulation of Grooves

Based on the proposed groove calculation method, we conducted simulations for each groove locus. Figure 12 shows the center loci of a ball rolling on different grooves, including “precession-motion-generating grooves for the input side and output side,” “precession motion direction converting grooves,” and “reduced-rotation generating groove.” Notably, certain precession-motion-converting grooves were not generated to account for potential interference caused by the grooves themselves. Similarly, the precession-motion-generating grooves on the output side were not generated, considering the interference resulting from the motion of the input shaft. Furthermore, the spatial trochoidal-wave locus meshes with multiple balls simultaneously.

## 8 Prototype Reducer

A small-size prototype reducer is manufactured using stainless steel (SUS440C) with a hardness of HRc 34–35. The groove section of the reducer is machined with an accuracy of ±0.02 mm, following the proposed groove calculation method. The objective is to validate the no-backlash performance, as shown in Figs. 13 and 14, and in Table 1.

Signs | Terms | Specifications |
---|---|---|

α_{inp} | Input-side inclined angle αin | 10 deg |

r_{f}_{0} | Input-side precession motion standard radius | 20.5 mm |

l_{0} | Input-side initial position of a horizontal distance | 17.94 mm |

η_{0} | Input-side initial angle of an arrangement angle | 45 deg |

r_{1} | Motion-converting section standard radius | 20.5 mm |

Z_{bc1} | Number of balls on the converting section | 10 |

ɛ | Offset angle of a converting section | −65 deg |

α_{out} | Output-side inclined angle | 5.0 deg |

r_{f}_{2} | Output-side precession motion standard radius | 10.53 mm |

l_{2} | Output-side initial position of a horizontal distance | 26.48 mm |

Z_{bc2} | Number of balls of the output-side precession motion | 10 |

r_{f}_{3} | Reduced motion standard radius | 18.32 mm |

l_{3} | Reduced section side initial position of a horizontal distance | 17.74 mm |

Z_{bc3} | Number of balls of the reduced section | 8 |

i | Reduction ratio | 8 |

Signs | Terms | Specifications |
---|---|---|

α_{inp} | Input-side inclined angle αin | 10 deg |

r_{f}_{0} | Input-side precession motion standard radius | 20.5 mm |

l_{0} | Input-side initial position of a horizontal distance | 17.94 mm |

η_{0} | Input-side initial angle of an arrangement angle | 45 deg |

r_{1} | Motion-converting section standard radius | 20.5 mm |

Z_{bc1} | Number of balls on the converting section | 10 |

ɛ | Offset angle of a converting section | −65 deg |

α_{out} | Output-side inclined angle | 5.0 deg |

r_{f}_{2} | Output-side precession motion standard radius | 10.53 mm |

l_{2} | Output-side initial position of a horizontal distance | 26.48 mm |

Z_{bc2} | Number of balls of the output-side precession motion | 10 |

r_{f}_{3} | Reduced motion standard radius | 18.32 mm |

l_{3} | Reduced section side initial position of a horizontal distance | 17.74 mm |

Z_{bc3} | Number of balls of the reduced section | 8 |

i | Reduction ratio | 8 |

Specifically, the grooves near the center of the trunnion shaft are cut to account for interference from the input shaft bearing. Additionally, two of the precession converting grooves are removed to avoid interference between the grooves themselves. The volume of the prototype is less than 52% of the previous precession ball reducer. The rated torque is set at 2.0 Nm due to the relatively low material hardness of this prototype. While this prototype exhibits smooth rotation without backlash,^{2} non-linear characteristics resulting from friction and irregular meshing, friction, and irregular meshing due to insufficient preload may still exert influence. To evaluate the stiffness of the prototype reducer, a stiffness test has been conducted utilizing hysteresis loss and the spring constant. In this experiment, ±2.0 Nm served as the maximum twisting torque, a value experimentally confirmed to prevent permanent deformation in the grooves. This value was calculated from the twisting torque and twisting angle, taking into account normal operating conditions (25 °C).

The stiffness evaluation apparatus included driving-side and fix-side torque meters, driving-side and fix-side encoders, a twisting torque-adding arm, and an electromagnetic brake, as shown in Fig. 15 and Table 2. The experimental procedure for the stiffness test is outlined as follows: Initially, the precession input shaft is fixed, and a twisting torque of +2.0 Nm is applied to the orthogonal-output shaft in the forward direction. Subsequently, a twisting torque of −2.0 Nm is applied in the reverse direction. The twisting torque is then reapplied in the forward direction to +2.0 Nm. The twisting angle is measured during these loading cycles.

Items | Specifications |
---|---|

Driving-side/fix-side torque meter | Kyowa Co. TPS-A-10NM, Max. measured torque; 10 Nm |

Driving-side/fix-side encoders | MTL Co. MEH-30-10000PST2C, Resolution; 20,000 pulse/rev. |

Electromagnetic brake | Mitsubishi Electric Co. ZKB-0.6YN, Max. output torque; 6 Nm |

Items | Specifications |
---|---|

Driving-side/fix-side torque meter | Kyowa Co. TPS-A-10NM, Max. measured torque; 10 Nm |

Driving-side/fix-side encoders | MTL Co. MEH-30-10000PST2C, Resolution; 20,000 pulse/rev. |

Electromagnetic brake | Mitsubishi Electric Co. ZKB-0.6YN, Max. output torque; 6 Nm |

The hysteresis loss in the prototype reducer is measured to be 683 arc seconds (3.31 × 10^{−3} rad), and the average twisting spring constant is determined to be 2.36 × 10^{−3} Nm/arc second (or 487.23 Nm/rad), as shown in Fig. 16. It is noteworthy that this hysteresis loss characteristic surpasses the backlash characteristic typically observed in a conventional orthogonal-axis output-type speed reducer with bevel gears [19].

## 9 Conclusions

To realize an orthogonal-axis-type small-size reducer that can be embedded into a robot-arm wrist, a novel design of a precession motion ball reducer has been proposed. In particular, the proposed structure, which has grooves on the outer spherical surface of each precession motion part, can prevent interference between the rotor and the cutting tool. Furthermore, we have shown that the direction of the output rotation can be arbitrarily set as equations of the precession motion direction converting section. Subsequently, it is confirmed that the prototype reducer achieves smooth rotation without backlash. The volume of this reducer is 52% or less than that of the previous model, successfully fulfilling the objective of size reduction. However, the efficiency and strength characteristics of the reducer remain unclarified, primarily due to the low hardness of the groove elements in the prototype reducer and the inability to apply a high preload. In the evaluation of positioning performance under high-torque transmission conditions, a crucial step involves enhancing the surface hardness of the groove elements. Yet, it has been confirmed that the conventional machining method is inadequate for improving the machining accuracy of such spatial grooves due to the escape of the cutting tool. It was also confirmed that asymmetrically shaped groove elements have non-uniform strains when heat-treated to increase hardness. Therefore, future research should investigate methods of groove production in hard materials, methods to reduce machining errors (e.g., to ±0.01 mm or less), and heat-treatment methods that minimize strain. Then, the positioning performance under high-torque transmission conditions needs to be investigated.

## Footnote

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.