Rehabilitation robot joint performance evaluation of a zero-spin traction drive with non-Newtonian fluid considered

Abstract

In this study, a zero-spin cone-roller traction drive (CRTD) is presented for the joints transmission system in rehabilitation robots due to its high transmission performance and characteristics of overload protection. It can achieve safe interactions among humans, rehabilitation robots, and the environment, making it a potential substitute for traditional gear-based transmission systems. The performance of CRTD, especially efficiency, is studied in this paper based on an elastohydrodynamic lubrication (EHL) model with the considerations of the non-Newtonian effect. The results demonstrate that the overall efficiency differs in different stages, reaching a maximum value of 95%. The overload protection activates when there is a sharp drop in efficiency, and the overload threshold can be identified by the efficiency, which may provide guidance for operation and optimization.

Keywords

rehabilitation robot joint zero-spin traction drive elastohydrodynamic lubrication non-Newtonian efficiency

Introduction

With the development of the medical–industrial integration technology, rehabilitation robots, as shown in Figure 1, have become widely used to provide auxiliary power for the rehabilitation training of patients to help rebuild nervous tissue.^1,2 As the key components in rehabilitation robots, joints transmission system must have the following characteristics: small size, low torque ripple, high level of transmission precision, and overload protection.³ Traditionally, the gear-based transmission systems have been widely used in joint driving systems for the industry robots and achieve relatively favorable performance. However, this gear-based system is essentially a rigid transmission system without overload protection, which may be inappropriate for rehabilitation robots if the safety is in consideration. Traction drives have recently drawn more research attention in joint transmission systems due to their characteristics of small size, low vibration, high precision, and overload slipping,^4,5 making it potential to substitute traditional gear-based transmission systems to achieve safe interaction and integration among humans, rehabilitation robots, and the environment.

Figure 1.

Rehabilitation robot and its joint transmission system.

The power and motion in traction drives are transmitted by the film in the gap of the rolling element and disks, and recent developments have heightened the need for increasing the transmission performance, especially in terms of the transmission efficiency.^6–9 Florian et al.¹⁰ studied the trade-off behavior of half-toroidal CVT between the dynamics of ratio variation and efficiency performance. In a later study, the trade-off behaviors of the full-toroidal and half-toroidal CVT were compared, and the results showed that the half-toroidal configuration outperforms the full-toroidal one over the entire operating range.¹¹ Luo et al.¹² presented a type of adaptive loading ball traction drive reducer and compared the traction drive performance under constant and adaptive loading. An asymmetric loading multi-roller planetary traction drive and a refined analysis model were developed by Jiang et al.¹³ to investigate the traction capacity and transmission efficiency, achieving a maximum transmission efficiency of 98.5%.

The spin loss generally dominates the power loss and has a significant impact on the efficiency performance, which can be eliminated by proper structural design.^14,15 Li et al.¹⁶ and Jiang et al.¹⁷ experimentally examined the film thickness through interferometry measurements and found that the film shape was obviously skewed if spin was introduced. Cretu and Glovnea¹⁸ evaluated the effect of spin motion on traction curves and proposed an original method to reduce the spin loss. Novellis et al.¹⁹ demonstrated that the efficiency and traction capacity of the double-roller full-toroidal variator was enhanced when zero-spin was achieved. Later, Li et al.^20,21 developed a novel toroidal CVT with a logarithmic surface to eliminate the spin loss theoretically, which proves that the efficiency of the logarithmic CVT is higher than that of the half-toroidal CVT; several zero-spin schemes of the roller-disk type traction drive were provided based on the principle of zero-spin.^22,23

Generally, the maximum pressure in contact areas can reach the GPa level, and the lubricant exhibits non-Newtonian behavior under such high pressures.^24,25 Tevaarwerk and Johnson²⁶ studied the elastic effects with the proposed J-T model and indicated that the traction drive performance largely depended on the rheological properties. Kohshiro et al.²⁷ investigated the relationship between the maximum traction coefficient and temperature based on the determined rheological parameters. Liu et al.²⁸ developed a thermal EHL model to deal with the effective viscosity of the Eyring shear thinning fluid with spin motion involved. The performance of the NuVinci drive was also estimated by Tomaselli et al.²⁹ based on the simplified EHL model with considerations of the non-Newtonian effect.

Although much efforts have been conducted for traction drives, the simplified models based on the assumption of ellipsoidal distributed pressure may be unable to estimate the traction drive performance accurately. In this study, a zero-spin traction drive system is proposed herein to potentially substitute the traditional gear-based transmission for rehabilitation robots, and the performance of CRTD, especially efficiency, is studied based on EHL model with the considerations of the non-Newtonian effect. The rest of this paper is organized as follows: the numerical model is established in detail in section “Descriptions of CRTD,” the efficiency performance of CRTD is illustrated and discussed in section “Results and discussion,” and section “Conclusions” concludes the paper.

Descriptions of CRTD

Geometrical and kinematics description

The geometry of the CRTD mainly consists of two cones and rollers, as displayed in Figure 2, and the points $A$ and $B$ are the contact points formed by the driving/driven components with the rotating roller, respectively. The power transfers from the right cone a, which is the driving component, to the driven component b. The symbols $O_{a}$ and $O_{b}$ are the intersections of the axes of the driving and driven components with the tangent lines of the contact point, respectively, and the roller axis meets the tangent lines at $O_{s}$ . According to the geometry shown in Figure 2, the following can be inferred:

{\begin{matrix} r_{sa} = r_{s} \cos β \\ r_{sb} = r_{s} \cos β \\ r_{b} = r_{a} + 2 r_{s} \cos β \cos (α + β) \end{matrix}

(1)

Where $r_{a}$ and $r_{sa}$ are the rotation radii between the driving component and roller, respectively, and $r_{sb}$ and $r_{b}$ are the radii between the roller and driven component, respectively; $α$ and $β$ denote the half-cone angle of the driving component and roller, respectively.

Figure 2.

Schematic diagram of the CRTD.

Spin motion occurs when there is a relative angular velocity along the tangent line and leaded to a high level of spin loss. However, this spin motion can be weakened or even eliminated through proper design. Following the principle described in previous researches,^22,23 zero-spin is achieved for CRTD when the intersections formed by these axes coincide with each other, which means that the points $O_{a}$ , $O_{b}$ , and $O_{s}$ are assumed to coincide. Then, the following can be obtained:

\frac{r_{a}}{\sin α} = \frac{r_{s}}{\tan β}

(2)

The ideal speed ratio $S_{ID}$ of CRTD can be defined as the ratio of the driven component radius $r_{b}$ with that of driving component $r_{a}$ when no relative slip occurs in the contact areas. Then, the ideal speed ratio of CRTD with zero-spin can be obtained based on equation (2), and the variation of the ideal speed ratio $S_{ID}$ with the half-cone angle of the roller $β$ at different half-cone angles $α$ is illustrated in Figure 3.

S_{ID} = \frac{r_{b}}{r_{a}} = 1 + \frac{2 \sin β \cos (α + β)}{\sin α}

(3)

Figure 3.

The ideal speed ratio $S_{ID}$ varies with the half-cone angle of the roller $β$ .

Although the spin motion can be eliminated through proper design, relative slipping is required to produce a traction force and then transfer power. It is necessary to describe the slipping behavior of each contact pair to achieve a more accurate kinematics description of CRTD; the concept of creep is also introduced to estimate the relative slipping:

{\begin{matrix} C r_{in} = \frac{| ω_{a} | r_{a} - | ω_{s} | r_{sa}}{| ω_{a} | r_{a}} \\ C r_{out} = \frac{| ω_{s} | r_{sb} - | ω_{b} | r_{b}}{| ω_{s} | r_{sb}} \end{matrix}

(4)

Where $C r_{in}$ and $C r_{out}$ are the creep coefficients in the input and output contact pairs, respectively; $ω_{a}$ , $ω_{s}$ , and $ω_{b}$ are the angle velocities of the driving component, roller, and driven component, respectively.

Based on the definition, the actual speed ratio $S_{AC}$ can be obtained as

S_{AC} = \frac{| ω_{a} |}{| ω_{b} |} = \frac{r_{b}}{(1 - C r_{in}) (1 - C r_{out}) r_{a}} = \frac{S_{ID}}{1 - Cr}

(5)

Where $1 - Cr = (1 - C r_{in}) (1 - C r_{out})$ , and $Cr$ is the equivalent creep coefficient of the CRTD.

Equilibrium and efficiency description

The free-body diagram of the CRTD with zero-spin is presented in Figure 4, where an external axial bearing balances the roller. $F_{Na}$ and $F_{Nb}$ are the clamping forces applied on the driving/driven contact pairs, respectively, which can be obtained by integrating the fluid pressure over the contact areas. $F_{tsa}$ and $F_{tsb}$ are the traction forces induced by the lubricant shear behavior when the lubricant is injected into the gap. $F_{Da}$ and $F_{Db}$ represent the axial clamping forces of the driving and driven components, respectively. $T_{in}$ and $T_{out}$ are the input and output torque, respectively, while $N_{B}$ and $T_{B}$ are the normal force and torque loss introduced by the axial bearing, respectively. According to the forces or torques applied on the corresponding parts, the balance equations of each transmission part can be expressed as follows:

Figure 4.

Free-body diagrams of the transmission parts.

For the driving and driven component

{\begin{matrix} m_{s} F_{Na} \sin α = F_{Da} \\ m_{s} F_{ta} r_{s} \cos β = T_{in} \end{matrix}

(6)

{\begin{matrix} m_{s} F_{Nb} \cos | \frac{π}{2} - α - β | = F_{Db} \\ m_{s} F_{tb} r_{b} = T_{out} \end{matrix}

(7)

For the rotating roller

{\begin{matrix} F_{Na} \cos β = F_{Nb} \cos β \\ F_{Na} \sin β + F_{Nb} \sin β = N_{B} \\ F_{tsa} r_{s} \cos β - F_{tsb} r_{s} \cos β - T_{B} = 0 \end{matrix}

(8)

Where $m_{s}$ is the number of rotating rollers in the CRTD.

The empirical formula proposed in Carbone et al.¹⁵ is adopted to estimate the torque loss caused by the axial bearing:

T_{B} = 4.65 \times 10^{- 5} N_{B}^{1.03}

(9)

According to these equilibrium equations, the overall efficiency of the CRTD can be expressed as the product of the speed efficiency $ν_{speed}$ and the torque efficiency $ν_{torque}$ :

ν = \frac{| T_{out} ω_{b} |}{| T_{in} ω_{a} |} = (1 - Cr) \cdot \frac{| T_{out} |}{| T_{in} | S_{ID}} = ν_{speed} \cdot ν_{torque}

(10)

Where $ν_{speed} = 1 - Cr$ and $ν_{torque} = \frac{| T_{out} |}{| T_{in} | S_{ID}}$ .

Contact description

The traction force used to transfer power originates from the lubricant shear behavior under high pressure level, and a typical contact action of elastohydrodynamic lubrication is shown in Figure 5(a). The rolling action of disks 1 and 2 – with speeds of $u_{1}$ and $u_{2}$ , respectively – draw the lubricant into the narrow gap between the disks to establish a thin layer of lubrication film; then, the traction force is obtained by integrating the shear stress when relative slipping occurs. Generally, the pressure reaches an order of magnitude of GPa, and the lubricant in the narrow gap exhibits an obvious non-Newtonian effect, which may lead to an obvious divergence with Newtonian fluids. In the current study, the Eyring model expressed by equation (11) is adopted to describe the behavior of non-Newtonian fluids, and the difference in the dimensionless shear stress between Newtonian and non-Newtonian fluids is presented in Figure 5(b).

\frac{τ}{τ_{E}} = \sin h^{- 1} (\frac{η \overset{\cdot}{γ}}{τ_{E}})

(11)

Where $τ$ is the shear stress; $η$ is the viscosity; $\overset{\cdot}{γ}$ is the shear strain rate; $τ_{E}$ is the Eyring stress with a value of 10 MPa.

Figure 5.

Schematic of lubrication in the traction drive: (a) distributions of pressure and shear stress in typical EHL; (b) rheological behavior between a Newtonian and non-Newtonian fluid.

In the current study, the Reynolds equation in the cartesian coordinate system is used to determine the fluid pressure under steady-state condition, expressed as

\frac{\partial}{\partial x} (\frac{ρ h^{3}}{η} \frac{\partial p}{\partial x}) + \frac{\partial}{\partial y} (\frac{ρ h^{3}}{η} \frac{\partial p}{\partial y}) = 12 u_{m} \frac{\partial (ρ h)}{\partial x}

(12)

The boundary conditions for the Reynolds equation are as follows:

{\begin{matrix} p (x_{0}, y) = 0 \\ p (x_{out}, y) = 0 & \frac{\partial p (x_{out}, y)}{\partial x} = 0 \\ p (x, y_{0}) = p (x, y_{1}) = 0 \end{matrix}

(13)

Where $ρ$ is the lubricant density; $h$ is the film thickness; $p$ is the fluid pressure; $u_{m}$ is the entrainment speed; $x_{0}$ and $x_{out}$ are the starting position of the contact area and the outlet location of fluid flow along the $x$ direction, respectively; $y_{0}$ and $y_{1}$ are the starting and ending position of the contact area, respectively.

Previous studies have shown that the surface deformation induced by the fluid pressure has a significant impact on the film thickness, and the film thickness in equation (12) can be expressed as follows:

h (x, y) = h_{0} + \frac{x^{2}}{2 R_{x}} + \frac{y^{2}}{2 R_{y}} + w (x, y)

(14)

Where $h_{0}$ is the rigid displacement; $R_{x}$ and $R_{y}$ are the equivalent radii along the $x$ and $y$ direction, respectively; and $w (x, y)$ represents the surface deformation.

Additionally, the following equations are employed to estimate the relationship between the viscosity and density:

η = η_{0} \exp {(\ln η_{0} + 9.67) [- 1 + {(1 + 5.1 \times 10^{- 9} p)}^{Z_{0}}]}

(15)

ρ = ρ_{0} (1 + \frac{0.6 \times 10^{- 9} p}{1 + 1.7 \times 10^{- 9} p})

(16)

Where $η_{0}$ and $ρ_{0}$ are the reference viscosity and reference density, respectively; $Z_{0}$ is the viscosity–pressure index.

The normal force $F_{N}$ and traction force $F_{t}$ can be calculated by integrating the pressure and shear stress over the corresponding contact area $Ω$ :

{\begin{matrix} F_{N} = \int \int_{Ω} pdxdy \\ F_{t} = \int \int_{Ω} τ dxdy \end{matrix}

(17)

Results and discussion

As a key index of the traction drive performance, the transmission efficiency of CRTD is explored in this section based on the parameters listed in Table 1. During the simulation process, the creep coefficients $C r_{in}$ and $C r_{out}$ are iteratively identified according to the input torque $T_{in}$ based on the EHL model with consideration of the non-Newtonian effect, and the traction force and overall efficiency of CRTD can then be obtained using equations (10) and (17).

Table 1.

Simulation parameters for CRTD with zero-spin.

Parameters	Value (Unit)
Radius of driving part $r_{a}$	15 mm
Radius of roller $r_{s}$	35 mm
Half-cone angle of driving part $α$	20°
Number of rollers $m_{s}$	3
Clamping force $F_{Na}$	3 kN
Rotating speed $n_{a}$	5000 rpm
Lubrication viscosity $η_{0}$	0.02 Pa·s
Equivalent elastic modulus $E_{e}$	2.21 × 10¹¹ Pa
Viscosity–pressure index $Z_{0}$	0.68

The information on the overall efficiency and power loss of CRTD are presented in Figure 6(a). It can be observed that the overall efficiency quickly increases with the output torque in the initial period, and then increases smoothly, finally decreasing at higher output torques. A maximum efficiency of 95% is obtained during the simulation process, and such overall efficiency is due to the power loss induced by the slipping loss. The power loss is low and slightly increases when the output torque is less than about 22.5 Nm, resulting in the proportion of power loss decreasing gradually. High-level creep is required to generate the required output torque, and the power loss increases sharply at this stage.

Figure 6.

Plots of the efficiency and power loss as a function of the output torque: (a) overall efficiency and power loss; (b) torque efficiency and speed efficiency.

As shown in Figure 6(b), the defined efficiencies for the traction drive, namely the torque efficiency and speed efficiency, exhibit an opposing trend with the output torque. The torque efficiency behaves the same in terms of the early overall efficiency and keeps increasing with the output torque. The speed efficiency decreases progressively in the early period and decreases sharply when an enhanced traction force is required, which necessitates a high level of relative slipping in contact areas. The rapidly decreasing efficiency at high levels of output torque may be unsuitable to transfer power, however, it can effectively protect patients due to overload slipping and achieve safe interactions among humans, rehabilitation robots, and the environment when CRTD is utilized for the joint transmission systems in rehabilitation robot.

The distribution of shear stress in contact areas under different $C r_{in}$ values is further analyzed, as shown in Figure 7. The shear stresses in both cases have similar distributions, but the values of the creep coefficient vary, allowing the maximum shear stress to reaches about 229 MPa when $C r_{in} = 0.0176$ but achieves about 244 MPa when $C r_{in} = 0.0784$ . It can be seen that the increased shear stress is not proportional to the creep due to the non-Newtonian behavior at high pressures. However, the larger creep could lead to high-level slip loss for CRTD, implying that it is unwise to further deliver power by increasing the creep when the threshold is exceeded.

Figure 7.

Contours of shear stress at different output torques: (a) Cr_in=0.0176 and (b) Cr_in=0.0784.

The relationships of the overall efficiency and creep with the output torque are shown in Figure 8 to further showcase the effect of the rotating speed. As shown in Figure 8(a), the overall efficiencies in both cases are consistent with each other when the output torque reaches about 22.5 Nm, however, the overall efficiency declines when $n = 5000 rpm$ while grows smoothly when $n = 7000 rpm$ . It is found that the same creep in high-speed case is able to produce a higher traction force, which delivers higher power and simultaneously maintains a high transmission efficiency.

Figure 8.

Comparisons of CRTDs at different speeds: (a) overall efficiency; (b) the creeps at the input and output stage.

As shown in Figure 8(b), the creep coefficients in the input and output stages increase progressively in the early period and increase sharply when a further output torque is applied. The creep coefficients in the input stage, referring to the red lines, are almost the same for both cases. However, a sharply increasing behavior in the output stage is observed, and the turning points of the creep prematurely occur. This suggests that the traction drive may be in sliding failure due to the high-level creep in the input stage, and the overload protection automatically activates if CRTD is equipped in the joints transmission system.

The relationship between the maximum shear stress $τ_{\max}$ at the input stage and the creep coefficient $C r_{in}$ at different rotating speeds is further shown in Figure 9 to illustrate the non-Newtonian effect. The maximum shear stress is observed to be proportional to the creep coefficient when a relatively small creep coefficient occurs in the local area. However, nonlinear behavior is observed due to the non-Newtonian effect with further increases in the creep coefficient. Moreover, it can be seen that a relatively high maximum shear stress can always be achieved at higher rotating speeds, which may be beneficial for the operation of the traction drive system.

Figure 9.

Plot of the maximum shear stress of the input stage at different speeds.

Conclusions

In this paper, a new type of zero-spin traction drive (CRTD) is presented for the joints transmission system in rehabilitation robots to potentially substitute the traditional gear-based transmission. Its efficiency performance is primarily studied based on the EHL model with the considerations of the non-Newtonian effect, where the traction force is obtained by integrating the shear stress over the contact areas. The overall efficiency of CRTD can be divided into three stages, achieving a maximum value of 95%. The rapidly decreasing efficiency in the last stage may be unsuitable for transferring power, however, it can effectively protect patients due to the characteristic of overload protection and achieve safe interactions among humans, rehabilitation robots, and the environment if CRTD is equipped in the joints transmission system. The speed effect indicates that traction drive system is more suitable for high-speed scenarios, and sliding failure prematurely occurs at lower speeds, which may potentially provide guidance for the operation and optimization of zero-spin traction drives used in rehabilitation robot joints transmission system.

Footnotes

Acknowledgements

The authors would like to thank the National Nature Science Foundation of China, China Disabled Persons’ Federation, Chongqing Bureau of Science and Technology and Chongqing Electronic Engineering Vocational College. The authors also appreciate the reviewers and editor for their valuable comments on the manuscript.

Handling Editor: Aarthy Esakkiappan

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is funding by the National Nature Science Foundation of China (Grant No. U19A2082), Project of China Disabled Persons’ Federation (2021CDPFAT-25), Talents Program of Chongqing Bureau of Science and Technology (cstc2021ycjh-bgzxm0319), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJQN202103112, KJQN202303123, KJQN202303124, KJZD-K202203104) and Chongqing Electronic Engineering Vocational College College-level Project (XJWT202101).

ORCID iD

Zou Shuaidong

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