As humanoid robot systems expand from research prototypes into practical applications such as service, inspection, and complex manipulation, the performance demands on their actuation units become increasingly stringent. The joint drive module, which typically integrates a frameless torque motor, a precision reducer, encoder systems, power electronics, and structural components, is the core electromechanical system determining the motion accuracy, dynamic response, and long-term reliability of a humanoid robot. In my research, I focus on the key engineering challenges arising from the high integration and high power density of humanoid robot joint modules. Among these challenges, harmonic drive performance, thermal management, and electromagnetic interference are the most critical because they directly affect the transmission accuracy, load capacity, and stable operation of the joint. I therefore carried out a systematic investigation combining tooth profile design, three-dimensional finite element simulation, experimental testing, and structural optimization. The main objective was to establish a practical design and analysis method for harmonic drives in humanoid robot joint modules and to explore effective structural measures for heat dissipation and electromagnetic shielding inside the compact joint package.
A humanoid robot has many degrees of freedom, and the joint actuators are often required to operate under frequent start-stop cycles, rapidly changing loads, and prolonged continuous work. These operating conditions impose contradictory requirements on the joint module. It must be compact and lightweight, yet powerful enough to deliver high torque; it must be precise and rigid, yet durable and thermally safe; and it must integrate motor and electronics in close proximity, yet remain immune to electromagnetic disturbances. In the existing joint module architecture, the harmonic reducer has become the preferred transmission component for small and medium torque joints because of its high reduction ratio, low lost motion, and compact structure. The performance of a harmonic drive is highly sensitive to tooth profile geometry, the matching relationship between the flexspline and the circular spline, and the shape of the wave generator. I therefore structured my work around these aspects, with the aim of providing a complete engineering methodology for the design and evaluation of harmonic drive transmission systems in humanoid robot joints.
Background and Research Objectives
In the evolution of humanoid robot joint modules, the earlier designs often used separate motor-reducer assemblies with relatively large envelopes. As humanoid robots moved toward modular architecture, advanced products began to use frameless torque motors directly embedded into the structural housing. Frameless torque motors eliminate the motor casing and allow the joint module to achieve a higher torque-to-weight ratio. However, these motors also create concentrated heat sources inside a space that has little ventilation and limited convective surface area. At the same time, the encoder, control printed circuit board, and power converter are placed close to the motor because of space constraints. This close arrangement makes the PCB sensitive to the time-varying magnetic field produced by the motor. Therefore, the thermal and electromagnetic design of the joint module cannot be separated from the mechanical transmission design. In this research, I attempted to treat the harmonic drive, the thermal path, and the electromagnetic propagation path as parts of a single integrated system. This is particularly important for humanoid robot joint modules that must deliver high performance in a very limited volume.
The research objectives can be summarized as follows. First, I proposed a composite tooth profile for the harmonic drive flexspline and analyzed the influence of key profile parameters on contact stress using finite element simulation. Second, I established a three-dimensional harmonic drive performance simulation model and investigated how circular spline adjustment and wave generator shape affect meshing clearance, tooth pair participation, contact area, lost motion, backlash, and stiffness. Third, I performed experimental accuracy tests to validate the simulation trends and to provide practical guidance for harmonic drive assembly. Fourth, I created a steady-state thermal model of the joint module and compared different housing structures to improve passive cooling. Finally, I constructed an electromagnetic interference model of the motor and PCB arrangement and evaluated the effect of a shielding layer on the magnetic flux density at critical chip locations and current density in PCB loops. In this way, the research covers the mechanical, thermal, and electromagnetic domains that are essential for the practical design of high-performance humanoid robot joint modules.
Composite Tooth Profile Design for the Harmonic Drive
The harmonic reducer in a humanoid robot joint must maintain smooth meshing despite the continuous elastic deformation of the flexspline. The conventional double-circular-arc tooth profile is widely used because of its mature manufacturing method, but its geometric discontinuity at the tooth tip and tooth root can lead to stress concentration and limited load-sharing behavior. To improve the contact state without substantially increasing manufacturing complexity, I developed a composite multi-curve tooth profile. The proposed profile consists of five segments: a tip transition arc, a convex circular arc, a tangent line segment, a concave circular arc, and a root transition arc. These segments are connected with positional and tangential continuity, giving a smoother geometric transition from the tooth tip to the root. This design is especially suitable for the demanding operating conditions of a humanoid robot joint, where high torque, high cycle rates, and compactness coexist.
The geometrical parameters of the flexspline tooth profile are summarized in the following table. These parameters were used to construct the profile model and to define the finite element analysis cases.
| Parameter | Symbol | Value | Parameter | Symbol | Value |
|---|---|---|---|---|---|
| Addendum | $h_a$ | 0.175 mm | Root transition radius | $r_3$ | 0.116 mm |
| Dedendum | $h_f$ | 0.275 mm | Tip center offset | $e_0$ | 0.025 mm |
| Tip transition radius | $r_0$ | 0.103 mm | Convex center offset | $e_1$ | 0.036 mm |
| Convex arc radius | $r_1$ | 0.208 mm | Concave center offset | $e_2$ | 0.024 mm |
| Concave arc radius | $r_2$ | 0.227 mm | Root center offset | $e_3$ | 0.029 mm |
| Tangent length | $L$ | 0.190 mm | Tangent angle | $\lambda$ | 12.6 deg |
| Convex center offset | $c_1$ | 0.145 mm | Concave center offset | $c_2$ | 0.169 mm |
To describe the profile mathematically, I introduced a local coordinate system in which the arc length $s$ is used as the independent variable. The tip transition arc EA is expressed as
$$
\begin{cases}
x_{r0} = r_0 \cos(\pi/2 – s/r_0) + x_{o0},\\[4pt]
y_{r0} = r_0 \sin(\pi/2 – s/r_0) + y_{o0},
\end{cases}
\quad s \in (0, l_0),
$$
where $l_0 = r_0 \beta_0$ and $\beta_0$ is the central angle of the tip arc. The convex circular arc AB is described by
$$
\begin{cases}
x_{r1} = r_1 \cos(\alpha – s/r_1) + x_{o1},\\[4pt]
y_{r1} = r_1 \sin(\alpha – s/r_1) + y_{o1},
\end{cases}
\quad s \in (l_0, l_1),
$$
where $\alpha$ is the pressure angle, $l_1 = l_0 + r_1(\alpha – \lambda)$, and $\alpha = \arcsin\bigl((h_a+e_1)/r_1\bigr)$. The tangent segment BC is defined as
$$
\begin{cases}
x_{r2} = r_1 \cos\lambda + (s-l_1)\sin\lambda + x_{o1},\\[4pt]
y_{r2} = r_1 \sin\lambda – (s-l_1)\cos\lambda + y_{o1},
\end{cases}
\quad s \in (l_1, l_2),
$$
with $l_2 = l_1 + h_f/\cos\lambda$. The concave circular arc CD is
$$
\begin{cases}
x_{r3} = -r_2 \cos\bigl(\lambda + (s-l_2)/r_2\bigr) + x_{o2},\\[4pt]
y_{r3} = -r_2 \sin\bigl(\lambda + (s-l_2)/r_2\bigr) + y_{o2},
\end{cases}
\quad s \in (l_2, l_3),
$$
and the root transition arc DF is
$$
\begin{cases}
x_{r4} = -r_3 \cos\bigl(\gamma + (s-l_3)/r_3\bigr) + x_{o3},\\[4pt]
y_{r4} = -r_3 \sin\bigl(\gamma + (s-l_3)/r_3\bigr) + y_{o3},
\end{cases}
\quad s \in (l_3, l_4),
$$
where $\gamma$ is the starting angle of the root transition arc. The complete tooth profile is therefore defined by the above parametric equations, and each segment is continuous with its neighbors. This model gives the flexibility to adjust individual geometric parameters while preserving the overall profile smoothness. For a humanoid robot joint harmonic drive, such parametric control is valuable because the tooth profile can be tailored to different load levels, accuracy classes, and manufacturing constraints.
Contact Mechanics Simulation of the Tooth Profile
To evaluate the influence of design parameters on the contact performance of the proposed profile, I used the finite element method with a two-dimensional plane strain model. The circular spline was modeled as a discrete rigid body, the wave generator was simplified as a line rigid body, and the flexspline was treated as a deformable body with material properties listed in the table below.
| Component | Material | Elastic modulus | Poisson ratio | Density |
|---|---|---|---|---|
| Flexspline | 40CrNiMoA | 209 GPa | 0.3 | 7850 kg/m3 |
In the simulation, the first analysis step established the interference contact between the wave generator and the flexspline. The second step applied rotation to the wave generator while a torque of 52 N·m was applied to the output reference point on the circular spline. I then used the control variable method to investigate the addendum coefficient $h_a^*$, the clearance coefficient $c^*$, and the pressure angle $\alpha$. The design cases are listed in the table.
| Case | $h_a^*$ | $c^*$ | $\alpha$ (deg) | Purpose |
|---|---|---|---|---|
| 1 | 0.8 | 0.25 | 18 | Baseline |
| 2 | 0.6 | 0.25 | 18 | Addendum effect |
| 3 | 1.0 | 0.25 | 18 | Addendum effect |
| 4 | 1.2 | 0.25 | 18 | Addendum effect |
| 5 | 0.8 | 0.15 | 18 | Clearance effect |
| 6 | 0.8 | 0.35 | 18 | Clearance effect |
| 7 | 0.8 | 0.45 | 18 | Clearance effect |
| 8 | 0.8 | 0.25 | 20 | Pressure angle effect |
| 9 | 0.8 | 0.25 | 22 | Pressure angle effect |
| 10 | 0.8 | 0.25 | 25 | Pressure angle effect |
In the addendum coefficient series, the maximum contact stress increased from 1445 MPa to 2564 MPa as $h_a^*$ increased from 0.6 to 1.2. The baseline case with $h_a^*=0.8$ produced a maximum contact stress of 1642 MPa. The increase was particularly steep between $h_a^*=1.0$ and $h_a^*=1.2$, indicating that excessive addendum height tends to concentrate stress near the tooth root and the tip region. For a humanoid robot joint, where the reducer must survive millions of load cycles, avoiding this stress peak is essential.
The clearance coefficient series showed a non-monotonic trend. The maximum contact stress was 1102 MPa at $c^*=0.15$, 1642 MPa at $c^*=0.25$, 2646 MPa at $c^*=0.35$, and 1273 MPa at $c^*=0.45$. This non-linear behavior arose because changing the clearance coefficient altered both the tooth height and the geometric relationship between the tooth tip and root. The worst stress concentration occurred at $c^*=0.35$, demonstrating that clearance coefficient must be selected carefully rather than following a simple rule.
The pressure angle had a clear effect on the stress distribution. The maximum contact stress decreased from 1642 MPa at $\alpha=18^\circ$ to 1037 MPa at $\alpha=22^\circ$, and then increased to 1468 MPa at $\alpha=25^\circ$. The reduction of about 36.8% between $\alpha=18^\circ$ and $\alpha=22^\circ$ was significant. This result can be explained by the fact that a larger pressure angle increases the root thickness and improves the bending resistance, but an excessively large pressure angle changes the direction of the contact force and worsens the local contact condition. Therefore, in the design of a humanoid robot joint harmonic drive, the pressure angle should be optimized within a moderate range rather than selected based solely on manufacturing experience.
The conclusions from the contact mechanics simulation are summarized in the following table.
| Parameter series | Trend of maximum contact stress | Best case |
|---|---|---|
| Addendum coefficient $h_a^*$ | Strong increasing trend for large values | $h_a^*=0.6$ gave the lowest stress, but $h_a^*=0.8$ is a practical compromise |
| Clearance coefficient $c^*$ | Non-monotonic, peak at $c^*=0.35$ | $c^*=0.15$ or $c^*=0.45$ |
| Pressure angle $\alpha$ | Decreases then increases | $\alpha=22^\circ$ |
These findings were important because they allowed me to identify a reasonable range for each tooth profile parameter before moving to the three-dimensional transmission performance simulation. In particular, the pressure angle near $22^\circ$ showed the best stress distribution, which I regarded as a valuable design guideline for humanoid robot joint harmonic drives.
Geometric Mapping of the Circular Spline Tooth Profile
After establishing the flexspline tooth profile, I needed to determine the circular spline tooth profile that satisfies the conjugate meshing relationship. The circular spline tooth profile cannot simply be copied from the flexspline profile because the flexspline undergoes a controlled periodic deformation in the harmonic drive. I therefore employed a geometric mapping method to derive the theoretical circular spline tooth profile. In this method, a point on the flexspline profile is expressed in the flexspline coordinate system $S_1$ and then transformed to the circular spline coordinate system $S_2$ using the homogeneous transformation matrix
$$
\begin{bmatrix}
x_2 \\[2pt] y_2 \\[2pt] 1
\end{bmatrix}
=
\begin{bmatrix}
\cos\xi & \sin\xi & \rho\sin\varepsilon \\[2pt]
-\sin\xi & \cos\xi & \rho\cos\varepsilon \\[2pt]
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\[2pt] y_1 \\[2pt] 1
\end{bmatrix},
$$
where $\xi = \varepsilon + \mu$, $\varepsilon$ is the rotation angle of the deformed flexspline relative to the circular spline, $\mu$ is the normal angular deformation, and $\rho$ is the radial displacement. The envelope condition ensures that the two profiles remain in continuous contact during meshing. The conjugate equation is
$$
\frac{\partial x_2(s,\theta)}{\partial s}
\frac{\partial y_2(s,\theta)}{\partial \theta}
–
\frac{\partial x_2(s,\theta)}{\partial \theta}
\frac{\partial y_2(s,\theta)}{\partial s}
= 0.
$$
By solving this equation together with the coordinate transformation, I obtained the theoretical circular spline profile. This geometric mapping method provided a rigorous basis for creating the three-dimensional harmonic drive model. It also made it possible to quantify how changes in the circular spline geometry affect the clearance and contact condition in the assembled harmonic drive. For humanoid robot joint modules, where assembly tolerances and transmission accuracy are tightly coupled, this theoretical foundation is essential.
Three-Dimensional Harmonic Drive Performance Simulation
I built a complete three-dimensional model of the harmonic reducer in SolidWorks and then simplified it for finite element analysis. The complete model included the circular spline, flexspline, wave generator cam, and flexible bearing. In the simplified model, the cam and the inner ring of the flexible bearing were merged into an equivalent elliptic cam; the bearing balls and bearing rings were combined into a simplified flexible bearing structure. This approach preserved the main loading and deformation mechanism while reducing the computational cost. The finite element mesh contained about 1.677 million elements, with the flexspline mesh refined in the tooth region and the thin-walled cup area.
In Abaqus, the circular spline and wave generator cam were treated as discrete rigid bodies, while the flexspline and flexible bearing were deformable bodies. Surface-to-surface contacts were defined between the cam and the bearing inner ring, the bearing balls and outer ring, the bearing outer ring and flexspline inner wall, and the flexspline teeth and circular spline teeth. The friction coefficient between the tooth surfaces was set to 0.3. A two-step static analysis was used. In the first step, the wave generator expanded the flexspline into contact with the circular spline. In the second step, the output torque was gradually increased to 52 N·m, reduced to zero, reversed to -52 N·m, and then reduced to zero again. This loading sequence allowed me to evaluate the hysteresis behavior, lost motion, and backlash.
Effect of Circular Spline and Flexspline Matching
To investigate the matching relationship between the circular spline and the flexspline, I adjusted the circular spline tooth dimensions artificially. Six cases were considered: the original unadjusted case and five cases in which the meshing clearance was reduced by 2.5, 5.0, 7.5, 10.0, and 12.5 μm. The contact clearance contours showed that the effective meshing area expanded as the clearance was reduced. The number of meshing tooth pairs increased, the contact stress distribution became more uniform, and the load was shared by a larger number of teeth.
The following table summarizes the number of meshing tooth pairs at different loading instants for each adjustment case.
| Adjustment (μm) | 0% torque (0 s) | +52 N·m (0.1 s) | 0% torque (0.2 s) | -52 N·m (0.3 s) | 0% torque (0.4 s) |
|---|---|---|---|---|---|
| 0 | 43 | 56 | 44 | 55 | 44 |
| 2.5 | 45 | 56 | 45 | 56 | 45 |
| 5.0 | 45 | 56 | 46 | 56 | 46 |
| 7.5 | 47 | 54 | 46 | 55 | 46 |
| 10.0 | 47 | 55 | 46 | 55 | 46 |
| 12.5 | 47 | 56 | 47 | 55 | 47 |
From the table, it can be seen that the 12.5 μm adjustment maintained a relatively high and stable number of meshing tooth pairs at all instants. The proportion of meshing teeth also increased with the adjustment level, and the contact area became larger. This behavior was directly reflected in the hysteresis curves. For the unadjusted case, the lost motion and backlash were 64.23 arcsec and 43.14 arcsec, respectively. As the adjustment increased, the lost motion and backlash both decreased gradually. At the largest adjustment of 12.5 μm, the lost motion and backlash reached the minimum values in the simulation, while the equivalent stiffness increased significantly.
It is important to note that the relationship between the circular spline adjustment and the backlash is not purely linear. In the simulation, small adjustments of 2.5 μm and 5 μm changed the lost motion and backlash only slightly. The improvement became noticeable when the adjustment reached 7.5 μm and continued to improve at 12.5 μm. This nonlinear trend was also observed in the stiffness results. The high-load stiffness $K_3$ was the most sensitive to the adjustment level. A reasonable reduction of the meshing clearance helped to increase the number of simultaneously meshing teeth and reduce the elastic deformation under load, which is favorable for the positioning accuracy of a humanoid robot joint.
Effect of Wave Generator Shape
I also analyzed the influence of the wave generator shape on harmonic drive performance. Instead of a standard elliptical cam, I adopted a composite curve cam consisting of a parabolic segment near the long axis, a transition arc, and a side circular arc. In this cam definition, the long-axis dimension and the transition radius remained unchanged, while the parabolic end angle $A$ was varied. The four cases are listed in the table.
| Case | Long axis (mm) | Parabolic coefficient | Transition radius (mm) | Parabolic end angle A (deg) |
|---|---|---|---|---|
| 1 | 36.1684 | -0.029 | 7 | 15 |
| 2 | 36.1684 | -0.029 | 7 | 25 |
| 3 | 36.1684 | -0.029 | 7 | 30 |
| 4 | 36.1684 | -0.029 | 7 | 35 |
The cam profile in the first quadrant can be described as a piecewise polar curve:
$$
\rho(\theta) =
\begin{cases}
a – k\theta^2, & 0 \le \theta \le A, \\[4pt]
\rho_A + r_b\left(\theta – A\right), & A < \theta \le B,
\end{cases}
$$
where $a$ is the long-axis radius, $k$ is the parabolic coefficient, $r_b$ is the transition arc radius, and $A$ and $B$ define the angular extents of the parabolic and side regions. When the parabolic end angle $A$ increased, the profile transition from the long axis toward the side became smoother. The finite element results showed that this smoother transition reduced the clearance gradient in the meshing zone and made the deformation of the flexspline more continuous.
| Case | Lost motion (arcsec) | Backlash (arcsec) | Stiffness K1 (N·m/rad) | Stiffness K2 (N·m/rad) | Stiffness K3 (N·m/rad) |
|---|---|---|---|---|---|
| 1 | 18.04 | 11.50 | 33717 | 51667 | 56050 |
| 2 | 8.23 | 5.02 | 56671 | 62821 | 87094 |
| 3 | 6.01 | 3.25 | 72184 | 75650 | 89467 |
| 4 | 4.47 | 2.60 | 89610 | 89689 | 143210 |
The results indicated that a larger parabolic end angle reduced the lost motion and backlash, increased the effective meshing ratio, and improved the contact stress distribution. The best overall performance was obtained for the case with $A=35^\circ$. I therefore selected this wave generator configuration as the basis for the experimental verification. For a humanoid robot joint module, a smooth wave generator profile helps to stabilize the load sharing among multiple teeth and reduces the localized deformation that causes transmission error under dynamic conditions.
Experimental Verification
To validate the simulation results, I performed accuracy tests on a harmonic reducer test bench. The tests were carried out for different circular spline adjustment conditions. The wave generator parameters were fixed to the best case from the simulation, namely the composite cam with parabolic end angle $A=35^\circ$. The key measured indicators were the lost motion, forward and reverse transmission error, positioning accuracy, and repeatability. The results are shown in the table.
| Circular spline adjustment (μm) | Lost motion (arcsec) | Forward error (arcsec) | Reverse error (arcsec) | Transmission error (arcsec) | Positioning accuracy (arcsec) | Repeatability (arcsec) |
|---|---|---|---|---|---|---|
| 5 | 60 | 94 | 92 | 101 | 98 | 67 |
| 7.5 | 11 | 35 | 35 | 37 | 40 | 32 |
| 10 | 15 | 52 | 55 | 54 | 38 | 32 |
| 12.5 | 5 | 52 | 53 | 52 | 30 | 17 |
The experimental data showed a clear overall trend. When the circular spline adjustment was too small, the lost motion and transmission error were large, and both the positioning accuracy and repeatability were poor. As the adjustment increased, the meshing state improved and the measured errors decreased. The best repeatability of 17 arcsec and the best positioning accuracy of 30 arcsec were observed at the 12.5 μm adjustment condition. This trend agreed well with the simulation results, which predicted that a larger adjustment would increase the number of meshing teeth and the contact area, thereby reducing the non-linear elastic deformation and lost motion.
The experiment also revealed that the adjustment amount should not be increased without limit. Some indicators, such as transmission error, improved at 12.5 μm but not as much as the lost motion. The relationship between the adjustment and the measured accuracy was not perfectly monotonic, which suggests that the matching between the circular spline and the flexspline must consider the combined effect of meshing stability, structural elasticity, and assembly tolerance. This is an important practical conclusion for humanoid robot joint module assembly because the optimal adjustment should be defined not only by the nominal backlash but also by the actual stiffness and contact behavior of the assembled harmonic drive.
Thermal Structure Optimization of the Joint Module
In a compact humanoid robot joint module, the frameless torque motor is the dominant heat source. Because the motor is embedded in the structure and surrounded by the housing and reducer, the heat is mainly removed by conduction through the metallic components and natural convection from the outer surface. The limited space makes active cooling methods such as fans or liquid cooling difficult to implement, so I chose to optimize the passive heat dissipation path through the housing structure.
I built a three-dimensional steady-state thermal model of the joint drive module. The model included the simplified frameless motor, the rotor, the harmonic reducer, and the aluminum alloy housing. The motor rotor was represented as a constant-temperature heat source at 130°C, while the ambient temperature was set to 25°C. The outer surfaces of the module exchanged heat with the environment through natural convection with a film coefficient of 10 W/(m²·K). The material thermal conductivities used in the model are summarized below.
| Component | Material | Thermal conductivity (W/(m·K)) |
|---|---|---|
| Housing | Aluminum alloy | 167 |
| Stator core | Silicon steel | 22 |
| Rotor magnet | Permanent magnet | 20 |
| Harmonic reducer | Alloy steel | 45 |
| Bearing | Bearing steel | 46 |
| PCB | FR-4 | 0.3 |
Four housing configurations were compared: the smooth housing without additional cooling features, an axially slotted housing, a housing with external cooling fins, and a housing with graphene heat-spreading patches on the outer surface. The steady-state temperature results are listed in the table.
| Configuration | Maximum temperature (°C) | Minimum temperature (°C) |
|---|---|---|
| Smooth housing (control) | 127.1 | 24.94 |
| Axially slotted housing | 127.1 | 25.0 |
| Cooling fin housing | 117.4 | 25.0 |
| Graphene patch housing | 127.1 | 24.74 |
The cooling fin housing provided the most significant improvement. Compared with the smooth housing, the maximum steady-state temperature was reduced by approximately 9.7°C. The temperature contours showed that the high-temperature zone near the motor was compressed, and the outer surface displayed a larger area of lower temperature. This result demonstrates that for a humanoid robot joint module, increasing the external surface area with fins is an effective passive cooling method that does not consume additional energy. The axially slotted housing increased the heat transfer area only slightly, so its effect on the maximum temperature was negligible. The graphene patch improved the in-plane spreading of heat but did not increase the convective heat transfer surface; therefore, it did not reduce the peak temperature significantly. However, the graphene patch could be useful as a supplementary approach to reduce local temperature gradients in a humanoid robot joint module.
Electromagnetic Interference Simulation and Shielding
In a compact humanoid robot joint module, the motor, power electronics, encoder, and MCU are located close to one another. The rotating permanent magnets and the stator currents generate a time-varying magnetic field that can propagate into the PCB region. Although the magnetic flux density decays with distance, the close spacing inside the joint module may still affect sensitive chip locations and signal loops. To investigate this problem, I established an electromagnetic simulation model in which the motor and the PCB were placed in their actual relative positions. Two configurations were compared: one with a shielding layer between the motor and the PCB, and one without the shielding layer.
The shielding layer was modeled as a low-conductivity, high-permeability material. The material parameters used in the simulation are given below.
| Property | Value |
|---|---|
| Relative permeability | 2000 |
| Electrical conductivity | 1×10-6 S/m |
| Relative permittivity | 1 |
The electromagnetic field distribution was computed by solving the magnetostatic approximation of Maxwell’s equations:
$$
\nabla \times \mathbf{H} = \mathbf{J}, \qquad
\nabla \cdot \mathbf{B} = 0, \qquad
\mathbf{B} = \mu_0 \mu_r \mathbf{H}.
$$
The simulation results are summarized in the following table. I compared the motor magnetic flux density, the overall PCB magnetic flux density, the magnetic flux density at the chip location, and the current density in the PCB loop.
| Metric | Without shielding | With shielding | Relative change |
|---|---|---|---|
| Maximum motor flux density | 6.58 T | 6.61 T | 0.46% |
| Maximum PCB flux density | 40.4 mT | 41.3 mT | 2.23% |
| Maximum flux density at chip location | 34.9 mT | 3.14 mT | -91.0% |
| Maximum current density in PCB loop | Reference | Reduced | -93.4% |
The shielding layer had only a small effect on the motor’s main magnetic circuit and on the overall magnetic flux density in the PCB region. The maximum motor flux density changed by only 0.46%, which indicates that the shielding layer did not disturb the normal operation of the motor. However, at the chip location, the maximum magnetic flux density was reduced by approximately 91.0%. The maximum current density in the PCB loop was also reduced by approximately 93.4%. This shows that the shielding layer played a more important role in local sensitive areas than in the global space around the PCB.
From the electromagnetic simulation, I concluded that the magnetic field from the motor had already attenuated significantly before reaching the PCB. However, the remaining field was still strong enough to produce a non-negligible effect on the chip location and the loop current density. The shielding layer did not need to reduce the entire magnetic field in the joint module; it only needed to redirect the magnetic flux away from the most sensitive components. For a humanoid robot joint module, this is a practical solution because it can be integrated into the structural housing without adding significant mass or volume.
Integrated Design Considerations for Humanoid Robot Joint Modules
The results of this research show that the mechanical, thermal, and electromagnetic performances of a humanoid robot joint module are interconnected. The harmonic drive tooth profile determines the stress state and transmission error. The thermal management strategy affects the operating temperature and the reliability of the motor and electronic components. The electromagnetic layout determines the noise level in the control circuits. These three aspects cannot be optimized independently when the available space is limited.
At the transmission level, I found that a smooth composite tooth profile and an appropriate circular spline adjustment can reduce lost motion and improve stiffness. At the thermal level, I found that a finned housing can lower the peak temperature of the joint module without active cooling. At the electromagnetic level, I found that a high-permeability shielding layer can significantly reduce the magnetic flux density at the chip location and the current density in the PCB loop. These conclusions are all relevant to humanoid robot joint module design because they provide quantitative guidelines that can be applied during the early design phase.
In order to achieve a balanced design, the tooth profile parameters should be selected from a range that gives both low contact stress and acceptable meshing stiffness. The circular spline adjustment should be controlled according to the measured assembly tolerance because an excessive or insufficient adjustment can degrade the performance. The housing should be designed with an external fin geometry to maximize the natural convection area, while maintaining the structural stiffness required by the joint. The shielding layer should be placed between the motor and the PCB, but its magnetic design should also be verified to avoid concentrating flux in other areas.
In my simulation model, the torque level used for the harmonic drive analysis was 52 N·m, which represents a common medium-load condition for a humanoid robot joint. The actual torque in a humanoid robot joint can vary widely during walking, lifting, or manipulation. Therefore, the tooth profile design and the circular spline adjustment should be re-evaluated for the specific torque envelope of the target humanoid robot joint module. The same approach can be used to examine other operating points, but the optimal parameters may shift when the load changes.
One of the important observations from the thermal and electromagnetic simulations is that the boundary conditions strongly influence the results. For the thermal model, the natural convection coefficient of 10 W/(m²·K) represents a nearly still-air environment. In a real humanoid robot, the joint module may be exposed to some air movement due to robot motion. The actual convection coefficient may be higher, which would improve the heat dissipation. The electromagnetic model was based on a static magnetostatic approximation, so the frequency-dependent effects of the PWM carrier and switching transients were not captured in detail. Nevertheless, the comparative study between the shielded and unshielded configurations provides a reliable indication of the shielding layer effectiveness.
For practical engineering, I recommend the following design workflow for humanoid robot joint modules. First, define the load envelope and the required positioning accuracy. Second, design the flexspline tooth profile using the composite curve method and evaluate the contact stress with finite element simulation. Third, derive the circular spline profile using the geometric mapping method and analyze the meshing clearance distribution. Fourth, optimize the wave generator profile to improve the multi-tooth contact condition. Fifth, verify the transmission performance with three-dimensional finite element simulation and prototype tests. Sixth, carry out a steady-state thermal simulation to determine the housing geometry and passive cooling strategy. Seventh, perform an electromagnetic simulation to evaluate the risk of interference on the PCB and to assess the effect of a shielding layer. This workflow can help to reduce the number of experimental iterations and to improve the reliability of the final design.
Conclusion and Outlook
In this research, I carried out a systematic study on the harmonic drive tooth profile design and the thermal and electromagnetic structure optimization for humanoid robot joint modules. The main conclusions are as follows.
First, the proposed composite multi-curve tooth profile provided a smooth geometric transition between the tooth tip and root, which helped to reduce the local stress concentration compared with the conventional double-circular-arc profile. The finite element parameter study showed that the addendum coefficient, clearance coefficient, and pressure angle all have significant but different effects on the contact stress. The addendum coefficient should not be too large, the clearance coefficient should be selected away from the critical value of 0.35, and the pressure angle should be chosen near the performance optimum of about 22 degrees.
Second, the three-dimensional harmonic drive simulation demonstrated that the matching relationship between the circular spline and the flexspline has a direct influence on the meshing tooth pairs, contact area, lost motion, backlash, and stiffness. A suitable reduction of the meshing clearance, such as the 12.5 μm adjustment in this study, improved the transmission accuracy and increased the stiffness. The experimental test results were consistent with the simulation trends. At the same time, the experimental results indicated that a larger adjustment does not always produce a better result because the assembly tolerance and the elastic deformation of the flexspline must be considered.
Third, the wave generator profile had a notable influence on the deformation of the flexspline and on the meshing performance of the harmonic drive. A larger parabolic end angle made the cam profile smoother, improved the meshing clearance distribution, increased the proportion of meshing teeth, and reduced the lost motion and backlash. The best performance was obtained with the composite cam having a parabolic end angle of 35 degrees.
Fourth, the steady-state thermal simulation of the joint drive module showed that the housing structure can be optimized to improve passive heat dissipation. The cooling fin housing reduced the maximum temperature by about 9.7°C compared with the smooth housing. This approach is practical for humanoid robot joint modules because it does not require additional power or moving parts.
Fifth, the electromagnetic simulation showed that a shielding layer between the motor and the PCB can reduce the magnetic flux density at the chip location and the current density in the PCB loop. The shielding layer had little effect on the motor main flux path and the overall PCB magnetic field, which is favorable because it does not interfere with motor operation. This makes the shielding layer a suitable structural measure for improving electromagnetic compatibility in a compact humanoid robot joint module.
There are several directions for future work. The dynamic behavior of the harmonic drive under time-varying loads and impact conditions should be investigated, because humanoid robot joints often experience abrupt torque changes during walking and manipulation. The interaction between the wave generator flexible bearing and the flexspline deformation should also be included in a more detailed model. For the thermal management, transient thermal cycles and the effect of heat on the lubricant and encoder performance deserve further attention. For the electromagnetic study, a more realistic model including the PWM waveform and the frequency-dependent material properties would provide a deeper understanding of the conducted and radiated interference. In addition, new motor topologies such as axial flux motors and new reducer concepts could be integrated into the joint module design to achieve even higher torque density and better controllability for future humanoid robot systems.
Overall, this research provides a comprehensive reference for the design and optimization of harmonic drives and joint modules for humanoid robots. The combination of tooth profile design, three-dimensional simulation, experimental verification, thermal analysis, and electromagnetic evaluation offers a practical engineering framework that can be used to improve the accuracy, reliability, and stability of humanoid robot joint modules.