Based on the differential principle of thread transmission, an analytical model considering helical directions between screw and roller threads in planetary roller screw mechanism (PRSM) is presented in this work. The model is critical for the design of PRSM with a smaller lead and a bigger pitch to realize a higher transmission accuracy. The kinematic principle of planetary transmission is employed to analyze the PRSM with different screw thread and roller thread directions. In order to investigate the differences with different screw thread and roller thread directions, the numerical model is developed by using the software Adams to validate the analytical solutions calculated by the presented model. The results indicate, when the helical direction of screw thread is identical with the direction of roller thread, that the lead of PRSM is unaffected regardless of whether sliding between screw and rollers occurs or not. Only when the direction of screw thread is reverse to the direction of roller thread, the design of PRSM with a smaller lead can be realized under a bigger pitch. The presented models and numerical simulation method can be used to research the transmission accuracy of PRSM.
Planetary roller screw mechanism (PRSM) is used in various motiondelivery devices where power is transmitted by converting rotary motion to linear motion. The main components of PRSM are the nut, the screw, and the rollers, and the key components for transmission are the rollers. Compared to ball screw mechanism, PRSM has higher precision, higher speed, heavier load, and longer life, though the manufacture cost of PRSM is relatively high due to its complex structure. As such, the PRSM finds its applications as an actuator device in various machineries, such as machine tool [
The published research on the PRSM has been mainly dedicated in the following areas: efficiency and failure modes study [
To address the aforementioned problems and facilitate the PRSM design, a kinematic model and a comprehensive study on the helical directions between screw thread and roller thread based on differential principle of thread transmission are developed in this work. The mechanical structure of PRSM is introduced, followed by the motion analysis and computational modeling of the lead. Then, kinematics simulations of the PRSM are performed to validate the motion analysis which considers the helical directions between the screw thread and the roller screw. Finally, the models are examined in detail to explicitly show the relationships of the helical directions and the lead of the PRSM in the force analysis.
Figure
Configuration of PRSM.
The roller possesses a single start thread and two spur gears at each end of the roller and ring gears fixed at each end of the nut. The rollers roll around the inside of the nut as the screw turns, and one revolution of the screw causes the nut to advance one lead regardless of rolling or slipping between the components. The grooves on the nut and screw are helical with multistart. In order to ensure that the rollers do not become skewed or driven out of axial alignment between the screw and the nut, a planetary carrier and a gear pair are provided. The planetary carrier is located under each pivot end of the rollers and keeps the rollers spaced apart circumferentially around the screw. A gear pair includes a ring gear and spur gear on the roller. The ring gears time the spinning and orbit of the rollers about the screw axis by meshing gear teeth near the ends of the rollers. At the same, the planetary carriers float relative to the nut, axially secured by the spring rings which aligned with an axial groove in the wall of the nut.
Based on the principle of a planetary gear train, the kinematic principle of PRSM is shown in Figure
Kinematic principle of PRSM.
Axial view of PRSM.
According to the relationships of the movements and assuming no slip between the screw and the rollers, the linear velocity of contact point
Then the nut is fixed in the rotational direction, assuming that a roller travels from an initial point
Considering the relationships between the effective diameters of the screw and the roller and between the roller and the nut, (
Combining (
The relationships of helix angles of the screw, the roller, and the nut in terms of pitch, starts, and effective diameters are given in the following [
The helix angles of the roller and the nut are equal, that is,
Equation (
In order to ensure pure rolling of the rollers inside the nut, the angular velocity of the planetary carrier on the left side must be equal to that on the right side, which can be written as
The transmission ratios of components are shown as
Utilizing the relationship for the starts of nut, that is,
The relationships of angular velocities between the screw, the roller, and the planetary carrier are described as
As aforementioned, the roller rolls on the inner surface of the nut. The helix angles of the two components are identical. The roller gear meshes with the ring gear. No slip between the roller and the nut is allowed; however, there is always slip between the screw and the roller in the axial direction [
Because the roller and the nut have different leads and effective diameters, we assume that the axial displacement of roller relative to nut,
Based on the relative movement of the components,
As discussed above, the
Substituting (
It is well known, in the PRSM, that there is no relative axial displacement between the nut and the roller; that is,
Considering the slip between the screw and the roller, similarly, the angular motions of the PRSM can be decomposed into two components which are the motion without slip and motion with pure sliding [
Based on the geometry relationship, as shown in Figure
Substituting (
While the relative sliding occurs between the roller and the screw for one revolution of the screw, the pure sliding angle,
Because there is no relative axial movement between the nut and the rollers, the axial displacement of the roller relative to the screw is equal to the axial displacement of the nut,
Furthermore, the axial speed of the nut is calculated by differentiating the displacement of the nut with respect to time, as shown in the following:
Based on the analyses of Section
For the case in which the helical direction of screw thread is identical with that of roller thread, the lead of the PRSM can be written as
Substituting (
Equation (
For the case in which the helical direction of the screw thread is reversed to that of the roller thread, the lead of the PRSM can be expressed as
Similarly, substituting (
Also, the lead of PRSM is only determined by the starts and the pitch of the screw. Only when the helical direction of screw thread is reversed to the helical direction of the roller thread can the design of a bigger pitch and a smaller lead be realized with the same parameters of the starts and the pitch.
As indicated by (
The leads of the PRSM can be calculated by using (
Furthermore, the PRSM is an accuracy transmission which achieves the smallest lead by introduction of thread directions; however, compared to the conventional ball screw, the small lead is extremely difficult to reach due to the requirements of carrying capacity and transmission accuracy and the design difficulty of the return tube.
A model of kinematics simulation (as shown in Figure
Parameters of thread pair.
Parameter name  Symbol  Unit  Value 

Effective diameter of screw 

mm  39 
Starts 

5  
Pitch 

mm  5 
Effective diameter of roller 

mm  13 
Effective diameter of nut 

mm  65 
Parameters of gear pair.
Parameter name  Symbol  Unit  Value 

Module 

mm  1 
Tooth number of roller gear 

13  
Tooth number of ring gear 

65  
Pressure angle 

°  20 
Addendum coefficient 

0.8  
Clearance coefficient 

0.3  
Modification coefficients 

0  
Tooth width of roller gear 

mm  10 
Tooth width of ring gear 

mm  10 
The numerical model of PRSM.
Assume that all rollers have identical movements in the PRSM, and the steady state motion of the screw is considered in this paper.
Based on the relative movement shown in Figure
The connection relationships of components in PRSM.
In order to realize the kinematic transmission, the load constraints in the kinematics model are as follows: contact interactions are applied at the interfaces between the screw and the rollers and those between the rollers and the nut. Similarly, the contact interactions are also applied at the interfaces between the spur gear of the rollers and the ring gears.
The stiffness coefficient is set as
When the helical direction is identical between the screw thread and the roller thread, the relationships of helical direction in PRSM are as follows: screw is righthand, roller is righthand, and nut is righthand. The simulation results are shown in Figures
Axial displacement curve of the nut.
Angular velocity curve of the roller.
Angular velocity curve of planet carrier.
Axial speed curve of nut.
As Figure
Figure
The averaged angular velocity of the planet carrier is 260.6253°/s, that is, 4.5488 rad/s, as is demonstrated in Figure
As shown in Figure
Comparison of the analytical solutions with simulation results.
Displacement of nut at point 
Displacement of nut at point 
Angular velocity of roller ( 
Angular velocity of planet carrier ( 
Axial speed of nut ( 


Simulation results  24.8601 mm  49.8571 mm  22.6647 rad/s  4.5488 rad/s  49.7138 mm/s 
Analytical solutions  25 mm  50 mm  23.5620 rad/s  4.7124 rad/s  50 mm/s 
Relative error  0.5596%  0.2858%  3.8083%  3.4632%  0.5724% 
When the helical direction of the screw thread is reversed to that of the roller thread, the relationships of helical direction in PRSM are as follows: screw is righthand, roller is lefthand, and nut is lefthand. The simulation results are shown in Figures
Axial displacement curve of nut.
Angular speed curve of roller.
Angular velocity curve of planet carrier.
Axial speed curve of nut.
Compared to Figure
As shown in Figure
Comparisons of analytical solutions with simulation results.
Displacement of nut at point 
Displacement of nut at point 
Angular velocity of roller ( 
Angular velocity of planet carrier ( 
Axial speed of nut ( 


Simulation results  5.9200 mm  11.9506 mm  23.7979 rad/s  4.7630 rad/s  11.8169 mm/s 
Analytical solutions  6.25 mm  12.5 mm  23.5620 rad/s  4.7124 rad/s  12.5 mm/s 
Relative error  5.2800%  4.3952%  1.0012%  1.0738%  5.4648% 
The analytical solutions of angular velocity of planet carrier, angular velocity of roller, axial speed of nut, and displacement of nut can be obtained by (
As shown in Tables
According to the results of numerical simulation, the angular velocity and axial speed curves of the components generate a higher fluctuation. In addition to the influence of impact and clearance, the sliding is another important factor. Therefore, the analysis of the forces has been performed.
When the helical direction of screw thread is identical with that of the roller thread, as shown in Figure
Force analysis when thread direction is identical between the screw thread and the roller thread.
In Figure
When the helical direction of screw thread is reversed to that of the roller thread, the force analysis is shown in Figure
Force analysis when thread direction is reversed between the screw thread and the roller thread.
It is similar to Figure
Furthermore, the relative displacement errors shown in Table
Besides, the results of Table
This paper develops the kinematics by analytical modeling and numerical modeling of the PRSM considering helical directions between screw thread and roller thread to provide a method to support its design and application. The major findings are as follows:
The analytical modeling considering helical directions between the screw and the roller threads in PRSM is presented to realize the design of PRSM with a smaller lead under a bigger pitch based on the differential principle of thread transmission. Numerical modeling is developed by using Adams to validate the proposed analytical solutions. Besides, the kinematic models and simulation method considering helical directions of screw and roller threads are available to PRSM, which are beneficial to the further research of the PRSM.
The analytical solutions are close to the numerical results with errors less than 4% and 6% when the direction of screw thread is identical with or reversed to the direction of roller thread, respectively.
When the helical direction is identical between the screw thread and the roller thread, the friction force applied on the roller thread is in the helical direction of roller movement. However, the tangential force component is opposite to the movement direction. Therefore, such case has slip tendency and requires sufficient friction force to work properly.
When the helical direction of the screw thread is reversed to that of the roller thread, the PRSM is an accuracy transmission which achieves the smallest lead by introduction of a bigger pitch and a smaller lead as compared to the conventional ball screw where the small lead is extremely difficult to reach due to design difficulty of the return tube.
Tooth width of roller gear
Tooth width of ring gear
Clearance coefficient
Effective diameter of the screw
Denotes orbital diameter of roller
Effective diameter of the roller
Effective diameter of the nut
Axial force
Tangential force
Resultant force of
Addendum coefficient
Transmission ratio between the screw and the roller
Transmission ratio between the roller and the nut
Stiffness coefficient
Axial displacement of roller relative to nut
Axial displacement of roller relative to a rotating screw
Axial displacement of a rotating roller relative to a fixed nut
Axial displacement of the nut relative to roller
Axial displacement of the roller relative to the screw
Axial displacement component of the roller
Axial displacement component of the roller relative to the screw
Axial displacement of screw relative to roller
Axial displacement of the nut
Lead of the PRSM
Module of gear pair
Start of the screw
Start of the roller
Start of the nut
Pitch
Radius of rounded halfsection of roller thread
Operating time of the screw
Static slip velocity
Dynamic slip velocity
Linear velocity of the contact point
Linear speed of the roller center point
Axial speed of the nut
Modification coefficient
Tooth number of ring gears
Tooth number of gears near the ends of rollers
Pressure angle of gear pair
Contact angle
Helix angles of the screw
Helix angles of the roller
Helix angles of the nut
Static friction coefficient
The dynamic friction coefficient
Orbital angle of the roller
Rotational angle of the roller
Angular arc of contact of screw with roller
Pure sliding angle
Orbital speeds of the roller center point
Angular velocity of the screw
Rotational speed of the roller
Angular velocities of the planetary carrier on the left side
Angular velocities of the planetary carrier on the right side
Angular velocities of the planetary carriers
Angular velocity of the nut.
The authors declare that there is no known conflict of interests associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
The authors confirm that the paper has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. The authors further confirm that the order of authors listed in the paper has been approved by all of them.
The research was supported by the National Natural Science Foundation of China (no. 51275423), Specialized Research Fund for the Doctoral Program of Higher Education (no. 20126102110019), the 111 Project (no. B13044), and Fundamental Research Funds for the Central Universities (no. 3102015JCS05008).