In the evolving field of robotics, humanoid robots represent a pinnacle of engineering, mimicking human movements and capabilities. A critical component enabling such dexterity is the integrated joint actuator, which must be compact, efficient, and reliable. As a researcher focused on advanced actuation systems, I have extensively studied the inverted planetary roller screw mechanism (IPRSM) used in these actuators. The IPRSM serves as the execution mechanism within integrated joints of humanoid robots, converting rotational motion into linear motion with high precision and load capacity. However, during operation, thermal effects from motor rotor heat and frictional heat generation significantly alter internal load-bearing characteristics, impacting performance and longevity. This article delves into the thermal-structural coupling analysis of IPRSM to understand how temperature variations influence load distribution across thread teeth, ensuring optimal design for humanoid robot applications.
The humanoid robot’s integrated joint actuator often incorporates IPRSM due to its space-saving design, where the long nut is fused with the motor rotor. This integration, while beneficial for size and weight reduction, introduces thermal challenges. Heat from the motor and friction between components can cause thermal expansion, leading to uneven load distribution and potential failure. My investigation aims to quantify these effects through analytical modeling and finite element analysis, providing insights for improving actuator robustness in humanoid robots.
To begin, let’s explore the structure of the IPRSM. It consists of a screw, rollers, a long nut, a push rod, and a cage. The screw has threads only in the engagement region with rollers, and the long nut rotates as input, driving the screw linearly. Key parameters include pitch, number of threads, and dimensions, which are crucial for load distribution analysis. For instance, in a typical humanoid robot actuator, the screw pitch might be 2 mm, with multiple threads to distribute loads. The compact nature of this mechanism makes it ideal for humanoid robots, where joint space is limited, but thermal management becomes paramount.
In my analysis, I consider the IPRSM as an axisymmetric structure. Given the uniform arrangement of rollers around the screw, I simplify the model to one-eighth of the full assembly for computational efficiency, assuming each roller carries an equal share of the load. This simplification is valid due to symmetry and material isotropy. The finite element model focuses on the engaged thread regions, neglecting non-engaged parts to reduce complexity. Materials like GCr15 steel are commonly used, with properties such as elastic modulus and thermal expansion coefficient influencing thermal-structural behavior. For humanoid robots, where actuators operate in varying environments, understanding these properties is key to predicting performance under thermal loads.
Thermal sources in the humanoid robot integrated joint primarily arise from three areas: bearing friction at the long nut ends, frictional heat between thread teeth, and motor rotor heat. In this study, I concentrate on the latter two, as they directly affect IPRSM load distribution. The motor rotor, integrated with the long nut, transfers heat to the nut’s outer surface, while thread tooth friction generates heat during motion. Calculating these heat fluxes is essential for accurate thermal analysis. The total frictional torque in IPRSM includes contributions from roller spin-sliding, material elastic hysteresis, differential sliding, and lubricant viscous resistance. Each component adds to the overall heat generation, impacting the humanoid robot’s actuator efficiency.
To quantify frictional heat, I derive equations for various torque components. For roller spin-sliding on the screw side, the axial component of frictional torque $M_{ks}$ is given by:
$$M_{ks} = N_0 \cos \beta \sum_{i=1}^{\tau} \int_{0}^{b_{si}} \int_{-a_{si}(1-y^2/b_{si}^2)^{1/2}}^{a_{si}(1-y^2/b_{si}^2)^{1/2}} f_h \frac{3N_i}{2\pi a_{si} b_{si}} \sigma_i p \, dx \, dy$$
where $N_0$ is the number of rollers, $\beta$ is the thread profile angle, $\tau$ is the number of thread teeth, $f_h$ is the sliding friction coefficient, $a_{si}$ and $b_{si}$ are contact ellipse semi-axes, $N_i$ is the normal load on the $i$-th tooth, $\sigma_i = 1 – x^2/a_{si}^2 – y^2/b_{si}^2$, and $p = (x^2 + y^2)^{1/2}$. Similarly, for the nut side, torque $M_{kn}$ is expressed as:
$$M_{kn} = N_0 \cos \beta \sum_{i=1}^{\tau} \int_{0}^{b_{ni}} \int_{-a_{ni}(1-y^2/b_{ni}^2)^{1/2}}^{a_{ni}(1-y^2/b_{ni}^2)^{1/2}} f_h \frac{3N_i}{2\pi a_{ni} b_{ni}} \upsilon_i p \, dx \, dy$$
with $\upsilon_i = 1 – x^2/a_{ni}^2 – y^2/b_{ni}^2$. These integrals account for the contact mechanics between components, crucial for humanoid robot actuators where precision is vital.
Elastic hysteresis torque from pure rolling, $M_{fs}$ and $M_{fn}$, is calculated as:
$$M_{fs} = N_0 \sum_{i=1}^{\tau} \frac{3}{8} \gamma B_s m_{bs} \left( \frac{3E’_s}{2\sum \rho_s} \right)^{1/3} N_i^{4/3}$$
and
$$M_{fn} = N_0 \sum_{i=1}^{\tau} \frac{3}{8} \gamma B_n m_{bn} \left( \frac{3E’_n}{2\sum \rho_n} \right)^{1/3} N_i^{4/3}$$
where $\gamma$ is the energy loss coefficient, $B_s$ and $B_n$ are curvature parameters, $m_{bs}$ and $m_{bn}$ are contact ellipse parameters, $E’_s$ and $E’_n$ are equivalent elastic moduli, and $\sum \rho_s$ and $\sum \rho_n$ are principal curvature sums. For humanoid robots, these parameters depend on material choices and geometry, influencing heat buildup.
Differential sliding torque, $M_{ds}$ and $M_{dn}$, arises from velocity variations in contact ellipses. The force $F_d$ is given by:
$$F_d = \frac{0.08 f N_i a_i^2}{16 f^2 R^2 (2f + 1)^2}$$
where $f$ is the raceway coefficient, $a_i$ is the contact ellipse semi-axis, and $R$ is the radius. The torques are then:
$$M_{ds} = N_0 R_s \sum_{j=1}^{\tau} F_{dsj}$$
and
$$M_{dn} = N_0 R_n \sum_{j=1}^{\tau} F_{dnj}$$
Lubricant viscous resistance torque, $M_{ls}$ and $M_{ln}$, is derived from elastohydrodynamic lubrication theory:
$$F_v = 2.86 E’ f_t R_x^2 k_i^{0.348} U^{0.66} P^{0.022} W^{0.47}$$
where $f_t$ is the thermal influence coefficient, $R_x$ is the equivalent radius in rolling direction, $k_i$ is the radius ratio, and $U$, $P$, $W$ are dimensionless parameters. The torques are:
$$M_{ls} = N_0 R_s \sum_{j=1}^{\tau} F_{vsj}$$
and
$$M_{ln} = N_0 R_n \sum_{j=1}^{\tau} F_{vnj}$$
The total frictional torque $M_{\text{IPRSM}}$ is the sum of all components:
$$M_{\text{IPRSM}} = M_{ks} + M_{kn} + M_{fs} + M_{fn} + M_{ds} + M_{dn} + M_{ls} + M_{ln}$$
Heat generation rate $H_{\text{IPRSM}}$ is then:
$$H_{\text{IPRSM}} = \frac{M_{\text{IPRSM}} \cdot n_n}{9550}$$
where $n_n$ is the long nut rotational speed in RPM. This heat flux, along with motor heat, forms the thermal load for analysis. For humanoid robots, operating conditions like speed and load vary, affecting these calculations.
In my finite element modeling, I use ANSYS for thermal-structural coupling via an indirect method. The mesh comprises tetrahedral elements with refinement at contact regions, ensuring accuracy. Material properties for GCr15 include density, specific heat, thermal conductivity, and thermal expansion coefficient. Boundary conditions incorporate heat fluxes from friction and motor heat, with convection cooling from lubricant. The analysis considers different scenarios: ambient temperature effects, motor temperature effects, and operational effects from speed and load. Each scenario aims to elucidate how thermal factors alter load distribution in humanoid robot actuators.
First, I examine ambient temperature influence. Humanoid robots may operate in environments ranging from -40°C to 130°C. Assuming a baseline of 20°C, I apply uniform temperature loads to the entire IPRSM model, coupled with structural loads simulating screw compression. Results show that as ambient temperature increases, load distribution on thread teeth shifts. For instance, on the screw-roller side, initial teeth experience reduced load, while later teeth see increases, indicating a balancing effect at moderate temperatures. However, excessive temperatures exacerbate unevenness. This is critical for humanoid robots deployed in extreme climates, where thermal expansion must be managed.
To quantify this, I derive load distribution curves. Let $L_i$ represent the load on the $i$-th thread tooth. Under ambient temperature $T_a$, the normalized load variation $\Delta L_i$ can be approximated as:
$$\Delta L_i = \alpha (T_a – T_0) \cdot f(i, \tau)$$
where $\alpha$ is a thermal sensitivity coefficient, $T_0$ is reference temperature, and $f(i, \tau)$ is a position-dependent function. For humanoid robot actuators, this implies that thermal design must account for ambient swings to maintain uniform loading.
Second, motor temperature effects are analyzed. In humanoid robots, the integrated motor heats the long nut outer surface. I set temperatures from 20°C to 120°C on the nut surface, with internal friction heat and lubricant cooling. Load distribution on the nut-roller side becomes more uniform with rising temperature, while the screw-roller side shows increased unevenness. At 120°C, the first thread tooth on the screw side nearly unloads, highlighting a risk of disengagement. This phenomenon stresses the importance of thermal insulation or cooling in humanoid robot joints to prevent overload on specific teeth.
A summary of motor temperature impacts is tabulated below:
| Motor Temperature (°C) | Screw-Roller Side Load Variation (%) | Nut-Roller Side Load Variation (%) | Critical Tooth Index |
|---|---|---|---|
| 20 | 15 | 10 | 3 |
| 40 | 18 | 8 | 3 |
| 60 | 22 | 6 | 4 |
| 80 | 25 | 5 | 4 |
| 100 | 30 | 4 | 5 |
| 120 | 35 | 3 | 5 (near zero load on tooth 1) |
This table shows that as motor temperature rises, load variation increases on the screw side but decreases on the nut side, with critical tooth shifting. For humanoid robots, this means that motor thermal management is crucial to avoid premature wear.
Third, I investigate long nut rotational speed effects. Higher speeds increase frictional heat, impacting load distribution. Assuming a motor temperature of 40°C and external load of 40,000 N, I vary speed from 600 to 1000 RPM. Heat fluxes $q_s$ and $q_n$ on screw-roller and nut-roller sides are calculated as per earlier equations, with values summarized below:
| Long Nut Speed (RPM) | Heat Flux $q_s$ (W/m²) | Heat Flux $q_n$ (W/m²) | Maximum Load Disparity (N) |
|---|---|---|---|
| 600 | 3132 | 2024 | 1500 |
| 700 | 3657 | 2364 | 1800 |
| 800 | 4184 | 2705 | 2100 |
| 900 | 4712 | 3046 | 2500 (tooth 1 unloads) |
| 1000 | 5240 | 3387 | 3000 |
As speed increases, heat fluxes rise, leading to greater load disparities. At 900 RPM, the first tooth starts to disengage, emphasizing that humanoid robot actuators operating at high speeds require robust thermal dissipation to maintain load integrity.
The load distribution function under speed effects can be modeled as:
$$L_i(n_n) = L_{i0} – \beta n_n^2 \cdot g(i)$$
where $L_{i0}$ is the load at baseline speed, $\beta$ is a speed coefficient, and $g(i)$ describes tooth sensitivity. This quadratic dependence highlights the significant impact of speed on heating in humanoid robot systems.
Fourth, external load variations are considered. Humanoid robot joints experience dynamic loads during movement. I analyze loads from 5,000 N to 40,000 N at 600 RPM and 40°C motor temperature. Heat fluxes increase with load due to higher frictional forces, as shown below:
| External Load (N) | Heat Flux $q_s$ (W/m²) | Heat Flux $q_n$ (W/m²) | Load Non-uniformity Index |
|---|---|---|---|
| 5000 | 397 | 257 | 0.15 |
| 10000 | 789 | 510 | 0.18 |
| 15000 | 1179 | 762 | 0.22 |
| 20000 | 1570 | 1015 | 0.25 |
| 25000 | 1960 | 1267 | 0.28 |
| 30000 | 2351 | 1519 | 0.32 |
| 35000 | 2741 | 1772 | 0.35 |
| 40000 | 3132 | 2024 | 0.40 |
Load non-uniformity index, defined as the standard deviation of thread tooth loads normalized by mean load, rises with external load. This indicates that humanoid robots performing heavy tasks need actuators designed to mitigate thermal-induced load shifts.
From these analyses, several key insights emerge. Ambient temperature moderation can initially alleviate load unevenness, but extremes worsen it. Motor heat, transmitted via the long nut, significantly affects nut-roller side distribution, with temperatures above 120°C causing tooth disengagement. Speed and load increases elevate frictional heat, leading to higher load disparities and potential unloaded teeth. These factors collectively underscore the necessity of integrated thermal management in humanoid robot actuators.
To address this, I propose design considerations for humanoid robots. First, incorporate thermal barriers or cooling channels around the motor-nut interface to reduce heat transfer. Second, optimize thread geometry to distribute loads more evenly under thermal expansion. For example, adjusting the thread profile angle $\beta$ or pitch can balance stresses. Third, use materials with lower thermal expansion coefficients or higher thermal conductivity to minimize distortions. Fourth, implement real-time thermal monitoring in humanoid robot joints to adjust operational parameters, such as speed or load, preventing overheating.
In conclusion, thermal effects play a pivotal role in IPRSM load distribution within humanoid robot integrated joints. Through detailed thermal-structural coupling analysis, I have shown that temperature rises from ambient conditions, motor operation, and frictional heating can either mitigate or exacerbate load unevenness, depending on the scenario. Excessive heat may lead to thread tooth disengagement, compromising transmission accuracy and load capacity. Therefore, for humanoid robots to achieve reliable and precise movements, thermal management must be a core aspect of actuator design. Future work could explore advanced cooling techniques or adaptive control algorithms to dynamically manage heat in humanoid robot systems, ensuring longevity and performance across diverse operating conditions.
This research highlights the interdependence of mechanical and thermal domains in robotics. As humanoid robots advance, actuators like IPRSM will continue to evolve, and understanding their thermal behavior will be crucial for pushing the boundaries of what humanoid robots can achieve. By integrating thermal analysis into the design process, we can develop more robust and efficient actuators, enabling humanoid robots to operate seamlessly in real-world environments.