In this investigation, I focus on the operational requirements of embodied robot joints, which demand compact actuators with high load capacity, high precision, and integrated electromechanical design. The novel inverted recirculating planetary roller screw mechanism (IRPRSM) offers a promising solution because it combines the integratability of an inverted planetary roller screw with the fine lead and manufacturing simplicity of a recirculating design. However, machining errors inherent in real manufacturing processes alter the contact geometry and axial clearance in ways that are difficult to predict. I therefore develop a unified three-body helical-surface formulation for the screw, roller, and nut, including eccentricity, pitch-diameter error, thread-angle error, and thread-start indexing error. Based on spatial meshing constraints, I derive the actual meshing position, meshing radius, deflection angle, and axial clearance as functions of these errors. The results show that eccentricity error introduces periodic fluctuations in the meshing radius and axial clearance, with the fluctuation frequency increasing with the number of screw starts. For pitch-diameter errors within ±0.03 mm, the difference in axial clearance induced by screw and roller pitch-diameter errors remains below 2 μm. When only the screw or only the roller has a thread-angle error, the axial clearance reaches a maximum of 24.58 μm at 0° and decreases to 10–12 μm at ±3°. If the screw and roller thread-angle errors change together in the same direction, the axial clearance only grows from 22.87 μm to 26.11 μm, indicating a much weaker effect. Thread-start indexing errors do not change the axial contact position but redistribute the clearance between upper and lower helical flanks. These findings provide a theoretical basis for tolerance design, error-complementary pairing, and high-precision assembly of IRPRSMs in embodied robot joint actuators.
Keywords: embodied robot; planetary roller screw mechanism; meshing characteristics; machining error; axial clearance
1 Introduction
Embodied robots, especially humanoid platforms, require joint actuators that can perform high-frequency start-stop cycles, bidirectional reversing, and precise position output. The joint-level load capacity, transmission accuracy, and dynamic response directly affect the whole-body motion stability and control quality. A planetary roller screw mechanism (PRSM) converts rotation into linear motion with high efficiency, high stiffness, low backlash, and excellent load-carrying capability. These properties make PRSMs suitable for integrated linear actuators in embodied robot joints. Several configurations exist: standard, inverted, recirculating, and the newly developed inverted recirculating planetary roller screw mechanism (IRPRSM).
The IRPRSM is particularly attractive for embodied robot joints because the nut can be integrated with the motor rotor while the recirculating threads allow small leads and simplified manufacturing. It inherits the compactness of inverted roller screws and the small-lead advantage of recirculating roller screws. However, the meshing behavior of this new mechanism under practical manufacturing tolerances has not been studied systematically. Existing studies of PRSM mostly assume ideal geometries, or they focus on standard configurations. Manufacturing errors such as eccentricity, pitch-diameter deviation, thread-angle deviation, and thread-start indexing errors are known to affect load sharing, stroke accuracy, and transmission precision, but their influence on IRPRSM meshing position and axial clearance remains unclear.
In this paper, I present a comprehensive geometric model that simultaneously incorporates four types of machining errors in the screw, roller, and nut. I derive the unified helical-surface equations and apply spatial meshing conditions to obtain the actual meshing point and axial clearance. I then analyze the sensitivity of meshing characteristics to each error type. The results are intended to support the manufacturing and precision assembly of IRPRSMs for high-performance embodied robot joint actuators.
2 Structure and Geometric Constraints of the IRPRSM
The IRPRSM consists of a screw, multiple rollers, a nut, a cage, and two cam rings. The screw and nut have triangular thread profiles with one or multiple starts. The rollers are annular grooves with no helix angle, meaning that the roller threads do not advance along the roller axis. The rollers are arranged circumferentially around the screw and are retained in slots of the cage. The cam rings limit the axial movement of the cage; each cam ring has a raised platform with a height equal to one pitch, which assists roller reset when the roller enters the unthreaded zone of the screw.
During operation, the nut thread engages the roller groove and transmits torque. The roller rotates about its own axis while revolving around the screw axis. The roller simultaneously engages the screw thread, causing the screw to translate axially. When a roller moves into the unthreaded screw region, the cam-ring platform pushes it back axially and re-engages it with the nut thread.
For proper kinematic operation, the following geometric constraints must hold:
$$p_S = p_R = p_N = p$$
$$n_S = n_N, \quad n_R = 0$$
$$r_{N0} = r_{S0} + r_{R0}$$
where \(p_i\) is the pitch, \(n_i\) is the number of thread starts, and \(r_{i0}\) is the nominal radius of the screw (\(S\)), roller (\(R\)), and nut (\(N\)). The leads are:
$$L_S = 2\pi r_{S0}\tan\lambda_S$$
$$L_N = 2\pi r_{N0}\tan\lambda_N$$
$$L_R = 0$$
In the equations above, \(\lambda_S\) and \(\lambda_N\) are the helix angles of the screw and nut at the nominal pitch radius. Because the roller has no helix angle, \(\lambda_R=0\). These constraints guarantee that the screw and nut have the same number of contact counts and that the roller can mesh with both simultaneously.
3 Error Modeling
3.1 Eccentricity Error
Eccentricity error occurs when the actual axis of the thread helix deviates from the theoretical axis. I denote the eccentricity vector of component \(i \in \{S,R,N\}\) as \(e_i\). In the cross-sectional plane, the actual center \(o_i^{\mathrm{real}}\) is displaced by \(e_i\) from the theoretical center \(o_i\). The influence of eccentricity is not isotropic: the component along the line connecting the two mating centers changes the center distance directly, while the transverse component mainly changes the contact angle. I will analyze both directions later.
3.2 Pitch-Diameter Error
Pitch-diameter error is a deviation between the actual pitch diameter and the theoretical pitch diameter. On one flank, the deviation contributes \(\Delta d_i/2\). This error changes the radial position of the thread flank and therefore affects the meshing point and the clearance between mating parts.
3.3 Thread-Angle Error
Thread-angle error is defined as the deviation of the actual flank angle from the nominal flank angle \(\beta_i\). I denote the error by \(\Delta \beta_i\). The thread profile is modified accordingly. For the arched roller profile, the center of the circular arc is also shifted in the axial and radial directions.
3.4 Thread-Start Indexing Error
For multi-start threads, the angular position of the \(j\)-th thread start may deviate from its ideal position. I denote this deviation as \(\theta_i^j\). If the \(j\)-th thread start is advanced along the positive rotational direction of the helix, I define the sign variable \(\zeta_i^j=-1\); otherwise \(\zeta_i^j=+1\). In general, the phase angle of the \(j\)-th start is
$$\theta_i^j = \theta_i + \zeta_i^j \Delta\theta_i^j$$
The indexing error changes the axial phase of each helical flank and therefore alters which roller threads come into contact.
3.5 Unified Helical-Surface Equations
I establish a local cross-sectional coordinate system \(O_i-u_i v_i w_i\) for each component, where the origin is at the intersection of the pitch line and the measured section. The thread profile in this coordinate system is described by \(u_i\) (radial direction), \(v_i\) (tangential direction), and \(w_i\) (axial direction). For the screw and nut, the profiles are straight-sided triangles; for the roller, the profile is a circular arc.
With the included errors, the position vector of a point on the thread surface of component \(i\) is expressed in the global coordinate system as
$$
\mathbf{r}_i(\varphi_i, u_i) =
\begin{bmatrix}
x_i \\
y_i \\
z_i
\end{bmatrix}
$$
Using the coordinate transformation from the cross-sectional frame to the global frame, and incorporating the eccentricity displacement, I obtain a unified vector expression:
$$
\mathbf{r}_i = \mathbf{T}_i(\theta_i, q_i)
\begin{bmatrix}
u_i(\psi_i) \\
v_i(\psi_i) \\
w_i(\psi_i)
\end{bmatrix}
+ \mathbf{e}_i
$$
where \(q_i\) is the rotation angle of the component in the global frame, \(\psi_i\) denotes the thread-profile parameter, and \(\mathbf{T}_i\) is the rotation and translation matrix. The detailed matrix depends on whether the component is the screw, roller, or nut. For the roller, the helix angle is zero, so the axial position along the roller axis does not change with rotation.
I can write the explicit surface equations for the roller as an example. Let \(u_R’\) be the radial coordinate in the roller cross-section. The roller groove has a circular arc of radius \(r_{PR}\) and center coordinates \((u_{PR0}, w_{PR0})\). The meshing point \((u_R’, w_R)\) on the roller flank is:
$$
u_R = u_R’
$$
$$
v_R = 0
$$
$$
w_R = w_{PR0} + \sqrt{r_{PR}^2 – (u_R’ – u_{PR0})^2}
$$
When the thread-angle error \(\Delta\beta_R\) is present, the center coordinates become:
$$
u_{PR0} = r_{PR}\sin(\beta_R + \xi_R\Delta\beta_R)
$$
$$
w_{PR0} = -r_{PR}\cos(\beta_R + \xi_R\Delta\beta_R) + c_R
$$
where \(c_R\) is the half-thickness of the roller tooth, and \(\xi_R\) is a sign coefficient that depends on the error direction.
After transforming to the global frame and adding eccentricity, the roller position vector is:
$$
\begin{bmatrix}
x_R \\
y_R \\
z_R
\end{bmatrix}
=
\begin{bmatrix}
\cos\gamma_R & -\sin\gamma_R & 0 \\
\sin\gamma_R & \cos\gamma_R & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
u_R\cos\theta_R – v_R\sin\theta_R + r_{R0} + e_{Rx}\\
u_R\sin\theta_R + v_R\cos\theta_R + e_{Ry}\\
w_R – \frac{L_R\theta_R}{2\pi}
\end{bmatrix}
$$
Because \(L_R=0\), the axial term simplifies. Similar, although more complex, equations are derived for the screw and nut with helix angles.
4 Spatial Meshing Conditions and Axial Clearance
For two mating surfaces \(\Pi_m\) and \(\Pi_n\) to be in contact, the position vectors at the common point must satisfy:
$$
\mathbf{r}_m(\theta_m, u_m) – \mathbf{r}_n(\theta_n, u_n) = \delta_{mn} \mathbf{e}_{mn}
$$
where \(\delta_{mn}\) is the clearance vector (zero in perfect contact), and the normal vectors must be collinear:
$$
\mathbf{n}_m(\theta_m, u_m) = \xi_{mn} \mathbf{n}_n(\theta_n, u_n)
$$
where \(\xi_{mn}\) is either \(+1\) or \(-1\) depending on the orientation of the surfaces. By combining the surface equations of the screw and roller and solving the five algebraic equations, I obtain the unknown variables \(u_S, \theta_S, u_R, \theta_R, \delta_{SR}\). The axial component of \(\delta_{SR}\) is the axial clearance between the screw and roller flanks.
I define the axial clearance as:
$$
\delta_{SR} = \delta_{SR,z} = \mathbf{e}_z^{\mathrm{T}} \delta_{SR}
$$
Similarly, for the nut-roller pair I obtain \(\delta_{NR}\).
The global meshing position can be described by the meshing radii and deflection angles. In the end view of the IRPRSM, the instantaneous meshing point between the screw and the roller is characterized by the screw meshing radius \(r_{CS}\), the roller-side radius on the screw side \(r_{CRS}\), and the corresponding deflection angles \(\phi_{CS}\) and \(\phi_{CRS}\). These quantities are obtained from the projected coordinates of the contact point.
The numerical solution of the meshing equations requires the structural data of the IRPRSM. Table 1 lists the parameters used in this study.
| Parameter | Symbol | Screw | Roller | Nut |
|---|---|---|---|---|
| Nominal radius | \(r_{i0}\) / mm | 30 | 5 | 40 |
| Addendum | \(a_i\) / mm | 0.4 | 0.4 | 0.4 |
| Dedendum | \(b_i\) / mm | 0.55 | 0.55 | 0.55 |
| Half tooth thickness | \(c_i\) / mm | 0.48 | 0.48 | 0.50 |
| Number of starts | \(n_i\) | 8 | — | 8 |
| Flank angle | \(\beta_i\) / ° | 45 | 45 | 45 |
| Pitch | \(p\) / mm | 2 | ||
| Arc radius of roller thread | \(r_{PR}\) / mm | — | 7.071 | — |
5 Model Verification
To validate the developed model, I first set all machining errors to zero. The model then reduces to an ideal meshing model, and I compare the resulting meshing radii and axial clearance with values available in the open literature for a similar annular-tooth roller screw mechanism. The comparison is shown in Table 2. Since the screw and roller geometries in the literature are identical to those considered here, the screw-roller side should agree closely. The nut in the literature has an annular thread, whereas my nut has a helical thread, so the nut-roller side may show differences in the deflection angles and the axial clearance.
| Parameter | Present model | Reference model | Difference (relative) |
|---|---|---|---|
| Screw meshing radius \(r_{CS}\) / mm | 30.0064 | 30.006 | 0.01‰ |
| Screw deflection angle \(\phi_{CS}\) / ° | 0.6934 | 0.693 | 0.58‰ |
| Roller meshing radius on screw side \(r_{CRS}\) / mm | 5.0090 | 5.009 | 0 |
| Roller deflection angle on screw side \(\phi_{CRS}\) / ° | 4.1574 | 4.157 | 0.10‰ |
| Screw-roller axial clearance \(\delta_{SR}\) / mm | 0.0246 | 0.025 | 1.60% |
| Nut meshing radius \(r_{CN}\) / mm | 39.9958 | 40.000 | 0.10‰ |
| Nut deflection angle \(\phi_{CN}\) / ° | −0.4052 | 0 | — |
| Roller meshing radius on nut side \(r_{CRN}\) / mm | 5.0028 | 5.000 | 0.56‰ |
| Roller deflection angle on nut side \(\phi_{CRN}\) / ° | −3.2408 | 0 | — |
| Roller-nut axial clearance \(\delta_{NR}\) / mm | 0.0090 | 0.020 | 55.00% |
The large difference in \(\delta_{NR}\) arises because the reference nut is an annular thread with zero helix angle, whereas in the IRPRSM the nut has a helix angle. The screw-roller side matches well, confirming that the zero-error limit of my model is correct. The additional error terms are introduced by modifying the surface equations in the same mathematical framework, so the model is also valid for nonzero errors.
6 Results and Discussion
6.1 Effect of Eccentricity Error
I first analyze the sensitivity of the axial clearance to the direction of eccentricity. The eccentricity can be in the \(X\) direction, which lies along the line joining the screw center and the considered roller center, or in the \(Y\) direction, which is perpendicular to that line in the end plane. Table 3 summarizes the trends observed from the numerical solution.
| Error case | Axial clearance trend | Sensitivity |
|---|---|---|
| Screw eccentricity in \(X\) | Approximately linear negative | High |
| Screw eccentricity in \(Y\) | Very small change | Low |
| Roller eccentricity in \(X\) | Strong positive | High |
| Roller eccentricity in \(Y\) | Very small change | Low |
Because the \(X\) direction changes the distance between the two mating pitch cylinders, it directly alters the radial penetration and therefore the axial clearance. I therefore use \(X\)-direction eccentricity for the rest of the analysis.
With a roller eccentricity of \(-10\,\mu\text{m}\) and a screw eccentricity of \(+10\,\mu\text{m}\), I compute the meshing radii over one full revolution of the roller around the screw. The calculation shows that the meshing radii on both the screw and roller fluctuate periodically. The fluctuations are more pronounced on the roller side. For an 8-start screw, the meshing radius oscillation repeats eight times per roller revolution. I also evaluate the axial clearance for screws with different numbers of starts. The results in Table 4 give the peak-to-peak magnitude of axial clearance variation.
| Number of screw starts | Periodicity per roller revolution | Observed clearance fluctuation |
|---|---|---|
| 3 | 3 cycles | Moderate |
| 4 | 4 cycles | Moderate |
| 5 | 5 cycles | Higher frequency |
| 6 | 6 cycles | Higher frequency |
| 7 | 7 cycles | High frequency |
| 8 | 8 cycles | High frequency |
These periodic fluctuations can directly affect the contact establishment during reversing, thereby increasing the complexity of the backlash and dynamic response. In an embodied robot joint actuator, such periodic clearance variations would lead to position-dependent stiffness and additional vibration excitation.
6.2 Effect of Pitch-Diameter Error
I now consider the pitch-diameter error of the screw and the roller. For the roller, the circular-arc profile is radially shifted. The meshing point moves along a path in the end view. As the roller pitch-diameter error increases from \(-0.03\) mm to \(+0.03\) mm, the meshing point moves toward the screw side (along the negative \(X\) direction), approaching the root circle of the screw thread. For the screw pitch-diameter error, the screw flank is shifted radially, but the contact point remains at the same radial coordinate because the roller arc re-establishes contact at the same relative position.
Table 5 lists the computed axial clearances for selected values of the pitch-diameter error.
| \(\Delta d_i\) / mm | Error in roller \(\delta_{SR}\) / μm | Error in screw \(\delta_{SR}\) / μm |
|---|---|---|
| −0.03 | 41.8 | 40.2 |
| −0.02 | 35.1 | 34.0 |
| −0.01 | 28.4 | 27.6 |
| 0 | 24.6 | 24.6 |
| +0.01 | 18.9 | 19.7 |
| +0.02 | 12.2 | 13.3 |
| +0.03 | 5.6 | 7.0 |
The trend is monotonic: as the pitch diameter increases, the radial interference becomes larger, which suppresses the axial clearance. The difference between the screw- and roller-induced clearances remains below 2 μm over the whole ±0.03 mm range. This near symmetry suggests that a complementary compensation strategy can be used in manufacturing: a positive pitch-diameter error on the screw can be paired with a negative error on the roller, and vice versa, to keep the axial clearance within a controlled range without requiring tighter tolerances on each component.
6.3 Effect of Thread-Angle Error
Thread-angle errors alter the slope of the flanks. When only the screw has a thread-angle error, the contact point shifts radially as the flank shape changes. When only the roller has a thread-angle error, the circular arc center moves axially and radially, causing another shift of the contact point. If both components have thread-angle errors, the relative deviation between the two flank angles determines the actual contact position. The meshing point moves toward the screw root when the screw flank angle is larger than the roller flank angle; the opposite occurs when the roller flank angle is larger.
Figure and table data show the axial clearance as a function of the thread-angle error. In the first scenario, only the screw has the error. In the second, only the roller has the error. In the third, both errors are equal and have the same direction. Table 6 summarizes the axial clearance at selected error magnitudes.
| \(\Delta\beta\) / ° | Only screw \(\delta_{SR}\) / μm | Only roller \(\delta_{SR}\) / μm | Both same \(\delta_{SR}\) / μm |
|---|---|---|---|
| −3.0 | 10.3 | 11.8 | 22.87 |
| −2.0 | 14.6 | 15.9 | 23.44 |
| −1.0 | 19.8 | 20.7 | 24.02 |
| 0 | 24.58 | 24.58 | 24.58 |
| +1.0 | 19.5 | 20.4 | 25.14 |
| +2.0 | 14.2 | 15.5 | 25.72 |
| +3.0 | 10.1 | 11.5 | 26.11 |
The first two columns follow an approximately symmetric convex curve with a maximum of 24.58 μm at zero error. As the error magnitude increases, the clearance decreases rapidly to about 10–12 μm at ±3°. This behavior indicates that an individual flank-angle error increases the radial interference or modifies the effective profile mismatch, thus reducing clearance. In contrast, when the screw and roller errors change simultaneously, the clearance increases slowly from 22.87 μm to 26.11 μm over the same range. Therefore, if both components share the same flank-angle error, the relative profile geometry remains similar and the clearance is much less sensitive.
From a manufacturing perspective, the most important conclusion is not to control the absolute flank angle of each part, but to control the difference between the flank angles of the screw and roller. A small relative mismatch guarantees stable contact and predictable clearance for the embodied robot joint actuator.
6.4 Effect of Thread-Start Indexing Error
I next analyze the effect of thread-start indexing error on the axial clearance between individual thread flanks. I number the threads of the screw, roller, and nut as shown in the model definitions. For a three-start screw, I apply the following errors:
$$\Delta\theta_S^1 = 0^\circ, \quad \Delta\theta_S^2 = 0.05^\circ, \quad \Delta\theta_S^3 = -0.1^\circ$$
$$\Delta\theta_N^1 = -0.06^\circ, \quad \Delta\theta_N^2 = 0.04^\circ, \quad \Delta\theta_N^3 = 0.12^\circ$$
With these input errors, I compute the axial clearance associated with the upper and lower flanks of each roller thread. The results indicate that the indexing error does not alter the axial contact position; instead, it redistributes the clearance between the upper and lower helical flanks. For a positive indexing error, the clearance on the upper flank (roller top surface against the mating lower surface) decreases, while the clearance on the lower flank increases. For a negative indexing error, the trend reverses. This redistribution is shown qualitatively in Table 7 for three selected roller thread positions.
| Roller thread number | Upper flank clearance | Lower flank clearance |
|---|---|---|
| 1 | Decreased | Increased |
| 2 | Increased (depends on combined sign) | Decreased |
| 3 | Decreased | Increased |
When the mechanism is loaded, axial deformation compatibility must be satisfied over all contacting thread pairs. The flanks with smaller initial clearance will engage first and carry a larger share of the load. Conversely, flanks with larger clearance may not be engaged at all under light loads. Therefore, thread-start indexing errors directly affect the load-sharing pattern among the threads. In an embodied robot joint that experiences frequent bidirectional loading, this can lead to uneven wear and accelerated fatigue of certain threads.
6.5 Combined Discussion for Embodied Robot Actuators
The results demonstrate that IRPRSM meshing behavior is highly sensitive to manufacturing errors, but not all errors have equal impact. Table 8 summarizes the main effects and engineering implications.
| Error type | Effect on meshing position | Effect on axial clearance | Key engineering concern |
|---|---|---|---|
| Eccentricity | Periodic radial shift | Periodic fluctuation, frequency increases with starts | Clearance variation during rotation, dynamic excitation |
| Pitch-diameter | Roller error shifts point; screw error does not | Monotonic decrease with increasing diameter | Complementary pairing possible |
| Thread-angle | Radial and lateral shift | Convex peak for single-side error; weak effect for simultaneous same-direction error | Control relative difference between screw and roller |
| Thread-start indexing | No change in axial contact position | Redistributes clearance between upper/lower flanks | Affects load sharing and wear distribution |
For a compact embodied robot joint actuator, the axial clearance directly determines the lost motion, positioning repeatability, and torsional stiffness. The periodic variation caused by eccentricity errors is particularly problematic because it introduces position-dependent stiffness that cannot be compensated by a constant feedforward controller. The near equivalence of screw and roller pitch-diameter errors suggests that a low-cost compensation strategy can be implemented through selective assembly. The thread-angle error analysis strongly recommends that quality control should focus on the mismatch between the screw and roller flank angles rather than on each individual part. Finally, the indexing-error analysis highlights the need to control the repeating phase of multi-start threads so that the load is distributed evenly among all thread starts.
7 Conclusions
In this paper, I have studied how four types of machining errors affect the meshing behavior of the novel inverted recirculating planetary roller screw mechanism intended for embodied robot joints. I established a unified helical-surface model for the screw, roller, and nut, which includes eccentricity, pitch-diameter error, thread-angle error, and thread-start indexing error. By applying the spatial meshing conditions, I derived the actual meshing positions and axial clearances. The main conclusions are:
- The model correctly degenerates to the ideal meshing model when all errors are zero. The screw-roller meshing quantities agree with the existing literature within 1.6% error, while the nut-roller differences are attributed to the helical nut thread in the IRPRSM.
- Eccentricity error in the direction of the center line has the strongest effect. It causes the meshing radius and axial clearance to fluctuate periodically as the rollers revolve around the screw. The fluctuation frequency equals the number of screw starts. Higher start numbers produce higher-frequency clearance oscillations, which must be considered in the dynamic design of embodied robot joint actuators.
- Pitch-diameter error affects the axial clearance monotonically. A pitch-diameter error on the roller shifts the meshing point in the end view, while an error on the screw does not. Within ±0.03 mm, the maximum difference in clearance caused by screw versus roller error is below 2 μm, so complementary pairing can be used to maintain assembly consistency.
- Thread-angle error on only one component produces an approximately symmetric convex clearance curve with a maximum of 24.58 μm at zero error. At ±3°, the clearance drops to 10–12 μm. When the screw and roller have equal and same-direction thread-angle errors, the clearance changes only slightly from 22.87 μm to 26.11 μm. Therefore, the relative thread-angle error is the dominant factor.
- Thread-start indexing error does not change the axial contact position or the total clearance, but it redistributes the clearance between upper and lower flanks. This redistribution changes the contact sequence and load sharing among threads, which can affect wear and fatigue in a bidirectional-load joint.
These findings provide a theoretical foundation for tolerance design, error-compensation pairing, and high-precision assembly of IRPRSMs. In future work, I will extend the model to include thermal deformation, elastic deflection under load, and multi-source error coupling, and I will validate the results through experiments on an integrated embodied robot joint actuator.