In the field of embodied robot research, the development of humanoid robots represents one of the most active and challenging frontiers. As an important load-bearing and motion-transferring joint in the human lower limb, the ankle joint plays a crucial role in maintaining balance, adapting to terrain, and propelling the body during locomotion. However, existing bionic ankle joint mechanisms used in embodied robots often suffer from weak load capacity, unbalanced stiffness, limited workspace, and low bionic fidelity. To address these issues, this dissertation focuses on a novel three-branch spherical parallel bionic ankle joint mechanism UP+R+S with a central spherical joint branch, which is highly consistent with the human ankle joint in terms of morphological structure, motor function, balanced stiffness, and load capacity. This work systematically investigates the kinematics, static stiffness, and dynamics of this mechanism, aiming to provide theoretical foundations and technical support for the engineering application of embodied robot joints.
1. Kinematic Analysis of the 2-DOF Spherical Parallel Bionic Ankle Joint
Kinematic analysis is the foundation for performance evaluation and control of parallel mechanisms. For the novel three-branch spherical parallel bionic ankle joint mechanism UP+R+S, I first established its structural model and analyzed its mobility using screw theory.
1.1 Mechanism Description and Mobility Analysis
As shown in the mechanism schematic, the proposed ankle joint mechanism UP+R+S consists of three branches: a central leg rod with a spherical joint at its end (branch 1), an arc-shaped fork driven by a gear motor (branch 2), and a pull-rod branch composed of a slider, upper pull-rod, and lower pull-rod (branch 3). The central spherical joint is located at the center of the mechanism, which corresponds to the rotation center of the human ankle joint. The slider moves along the vertical guide rail, driving the pull-rod chain to achieve plantarflexion/dorsiflexion, while the gear motor drives the arc-shaped fork to achieve inversion/eversion.
Using screw theory, the motion screw systems of each branch are established. For the central leg rod branch (branch 1), the motion screw system is:
$$
\begin{aligned}
\$_1^1 &= (1,0,0; 0,0,0) \\
\$_1^2 &= (0,1,0; 0,0,0) \\
\$_1^3 &= (0,0,1; 0,0,0)
\end{aligned}
$$
The constraint screw system restricts translations along the X, Y, and Z axes. For the arc-shaped fork branch (branch 2), the motion screw system includes one rotation about the X-axis and one rotation about the axis passing through the center, providing constraints that limit translations and rotation about the Z-axis. For the pull-rod branch (branch 3), the motion screw system does not impose any constraint on the moving platform.
Based on the modified Kutzbach-Grübler formula, the degree of freedom (DOF) of the mechanism is calculated as:
$$
M = d(n – g – 1) + \sum_{i=1}^{g} f_i + \nu – \zeta = 6(6-7-1) + 11 + 3 – 0 = 2
$$
This confirms that the mechanism has exactly 2 DOFs, corresponding to rotation about the X-axis (inversion/eversion) and rotation about the Y-axis (plantarflexion/dorsiflexion), which matches the functional requirements of the human ankle joint. This makes the mechanism highly suitable for embodied robot applications where anthropomorphic motion is essential.
1.2 Position Analysis
For the position inverse analysis, the orientation of the moving platform is described by the rotation matrix:
$$
\mathbf{R}_1^0 = \text{Rot}(x, \theta_x) \text{Rot}(y, \theta_y) =
\begin{bmatrix}
\cos\theta_y & 0 & \sin\theta_y \\
\sin\theta_x \sin\theta_y & \cos\theta_x & -\sin\theta_x \cos\theta_y \\
-\cos\theta_x \sin\theta_y & \sin\theta_x & \cos\theta_x \cos\theta_y
\end{bmatrix}
$$
Given the output angles \( \theta_x \) and \( \theta_y \), the input parameters — the rotation angle \( \alpha \) of the gear motor and the displacement \( l \) of the slider — can be solved as:
$$
\begin{cases}
\alpha = \theta_x \\
l = L – \sqrt{ r^2(\sin\theta_x \sin\theta_y)^2 + (r\cos\theta_x – r_2)^2 + r^2\cos\theta_y \sin\theta_x – r^2\sin\theta_y \cos\theta_x}
\end{cases}
$$
For the position forward analysis, given the input parameters \( l \) and \( \alpha \), the output angles can be obtained as:
$$
\begin{cases}
\theta_x = \alpha \\
\theta_y = \arcsin\left(\frac{A – r}{B}\right) – \arctan\left(\frac{r}{B}\right)
\end{cases}
$$
where \( A = r^2 + l^2 + 2lL \) and \( B = r^2 + (l+L)^2\cos^2\alpha \).
1.3 Joint Displacement Analysis
For the pull-rod branch, the intermediate joint angles \( u_1 \), \( u_2 \), and \( u_3 \) are derived using Euler angle transformation. The rotation matrix of the upper pull-rod coordinate system relative to the base coordinate system is:
$$
\mathbf{R}_4^0 = \text{Rot}(x, u_1) \text{Rot}(y, u_2) =
\begin{bmatrix}
\cos u_2 & 0 & \sin u_2 \\
\sin u_1 \sin u_2 & \cos u_1 & -\sin u_1 \cos u_2 \\
-\cos u_1 \sin u_2 & \sin u_1 & \cos u_1 \cos u_2
\end{bmatrix}
$$
The joint angles are expressed as:
$$
\begin{cases}
u_1 = \arctan\left(\frac{r\sin\theta_x \sin\theta_y}{r\cos\theta_x \sin\theta_y + l + L}\right) \\
u_2 = \arcsin\left(\frac{r – r\cos\theta_y}{L}\right) \\
u_3 = \arctan\left(\frac{\sin\theta_y \sin(\theta_x – u_1)}{u_2 \sin\theta_y \sin\theta_x \sin(\theta_x – u_1) + \cos\theta_y}\right)
\end{cases}
$$
1.4 Jacobian Matrix
Differentiating the inverse position equations with respect to time yields the velocity mapping relationship between the joint space and task space:
$$
\mathbf{A}\dot{\boldsymbol{\theta}} = \mathbf{B}\dot{\mathbf{q}}
$$
where \( \dot{\boldsymbol{\theta}} = [\dot{\theta}_x, \dot{\theta}_y]^T \) and \( \dot{\mathbf{q}} = [\dot{\alpha}, \dot{l}]^T \). The Jacobian matrix is obtained as:
$$
\mathbf{J} = \mathbf{A}^{-1}\mathbf{B}
$$
1.5 Velocity Analysis of Components
For each moving component, I established the centroid position vector and derived the linear velocity and angular velocity. The centroid linear velocities of all components are calculated as:
$$
\mathbf{v}_i = \mathbf{A}_i \dot{\boldsymbol{\theta}} \quad (i = 1, 2, 3, 4, 5)
$$
where \( \mathbf{A}_i \) are coefficient matrices obtained by differentiating the centroid position equations. Similarly, the angular velocities are obtained using the Euler angle transformation:
$$
\boldsymbol{\omega}_i = \mathbf{E}_i \dot{\boldsymbol{\phi}}_i
$$
For the moving platform, the angular velocity is:
$$
\boldsymbol{\omega}_1 =
\begin{bmatrix}
\dot{\theta}_x \\
\dot{\theta}_y \cos\theta_x \\
-\dot{\theta}_y \sin\theta_x
\end{bmatrix}
$$
1.6 Numerical Analysis with Human Gait Data
To validate the mechanism’s bionic characteristics, I used the Xsens MVN inertial motion capture system to collect human ankle joint angle data during normal walking. The collected discrete data were fitted using Fourier series to obtain continuous functions:
$$
\begin{cases}
\theta_x(t) = -0.044 – 0.1\cos(5.3t) + 0.004\sin(5.3t) – 0.02\cos(10.6t) + 0.066\sin(10.6t) \\
\theta_y(t) = 0.035 – 0.058\cos(5.3t) + 0.073\sin(5.3t) + 0.1\cos(10.6t) + 0.009\sin(10.6t)
\end{cases}
$$
Using these fitted functions as the output trajectories of the mechanism, I performed numerical calculations for the inverse position solution. The simulation results demonstrate that the novel mechanism UP+R+S can effectively reproduce human ankle joint motion patterns, confirming its high bionic fidelity for embodied robot applications.
| Component | Velocity Component | Amplitude Range (m/s) | Periodic Pattern |
|---|---|---|---|
| Moving Platform | vx | -0.06 to 0.04 | Periodic |
| vy | -0.04 to 0.04 | Periodic | |
| vz | -0.03 to 0.03 | Periodic | |
| Arc Fork | vy | -0.02 to 0.03 | Clear periodic |
| vz | -0.01 to 0.01 | Weak variation | |
| Slider | vz | -0.06 to 0.04 | Same as pull-rod |
| vx | ≈0 | — | |
| vy | ≈0 | — | |
| Upper Pull-rod | vx | ≈0 | — |
| vy | ≈0 | — | |
| vz | -0.06 to 0.04 | Periodic | |
| Lower Pull-rod | vx | -0.02 to 0.02 | Periodic |
| vy | -0.03 to 0.03 | Periodic | |
| vz | -0.06 to 0.04 | Periodic |
For the angular velocities, the simulation results show that the moving platform angular velocity \( \omega_x \) is identical to that of the arc fork, confirming the partial decoupling property of the mechanism. The upper and lower pull-rods exhibit additional rotational components due to the intermediate joint \( R_3 \). All angular velocity components remain within reasonable ranges, satisfying the requirements for stable walking of an embodied robot.
2. Static Stiffness Analysis of the Bionic Ankle Joint Mechanism
Stiffness is a critical performance indicator for parallel mechanisms used in embodied robots, as it directly affects the positioning accuracy, motion stability, and load-carrying capacity. In this section, I present a systematic stiffness analysis of the UP+R+S mechanism based on the small-deformation linear superposition principle.
2.1 Static Equilibrium Equations
By isolating each moving component of the mechanism, I established the static equilibrium equations. For the moving platform, the equilibrium equations under external force \( \mathbf{F} \) and moment \( \mathbf{M} \) are:
$$
\begin{cases}
\sum_{j=1}^{2}(\mathbf{F}_{1j} + \mathbf{P}_{1j} + \mathbf{R}_{1j}) + \mathbf{R}_{21} + \sum_{i=1}^{3}\mathbf{F}_{3i} + \mathbf{F} = 0 \\
\sum_{j=1}^{2}(\mathbf{M}_{f1j} + \mathbf{M}_{r1j}) + \sum_{j=1}^{2}(\mathbf{v}_j \times \mathbf{F}_{1j} + \mathbf{R}_{1j}) + (\mathbf{u}_1 \times \mathbf{R}_{21}) + \mathbf{M} = 0
\end{cases}
$$
For the arc-shaped fork, the equilibrium equations are:
$$
\begin{cases}
\sum_{j=1}^{2}(\mathbf{F}’_{1j} + \mathbf{P}’_{1j} + \mathbf{R}’_{1j}) + \mathbf{F}_{13} + \mathbf{P}_{13} + \mathbf{R}_{13} = 0 \\
\sum_{j=1}^{2}(\mathbf{M}’_{f1j} + \mathbf{M}’_{r1j}) + \mathbf{M}_{f13} + \mathbf{M}_{p13} + \mathbf{M}_{r13} – \sum_{j=1}^{2}(\mathbf{v}_j \times \mathbf{F}’_{1j} + \mathbf{R}’_{1j}) + (\mathbf{w}_2 \times \mathbf{F}_{13} + \mathbf{P}_{13}) = 0
\end{cases}
$$
These equilibrium equations contain 28 unknown force/moment variables but provide only 20 independent equations, resulting in an 8-times statically indeterminate problem. Therefore, 8 additional deformation compatibility equations are needed.
2.2 Deformation Compatibility Equations
To establish the deformation compatibility equations, I made the following assumptions: (1) no deformation or clearance in the joints; (2) the links are continuous, homogeneous, and isotropic; (3) small deformations apply; (4) the moving platform is rigid; (5) link masses are neglected; and (6) the arc-shaped fork is equivalent to a straight beam for deformation analysis.
For the arc-shaped fork, I divided it into two segments and established local coordinate systems \( OSMN \) along the tangent, radial, and normal directions. The deformations at the connecting points can be written as:
$$
\begin{cases}
\Delta_i = \nu_{i1} \mathbf{v}_2 + \nu_{i2} \mathbf{w}_2 \\
\delta_i = \theta_{i1} \mathbf{u}_2 + \theta_{i2} \mathbf{w}_2
\end{cases} \quad (i = 1, 2)
$$
For the upper and lower pull-rods, the axial deformation caused by the axial forces is:
$$
\Delta_{21} = \frac{R’_{21} l_1}{EA_1} \mathbf{w}_4, \quad \Delta_{22} = \frac{R’_{22} l_2}{EA_2} \mathbf{w}_5
$$
Additionally, due to the small adjustment of the moving platform, the pull-rods experience convective angular displacements \( \delta_{21} \) and \( \delta_{22} \), which cause additional linear displacements at the joints.
For the central leg rod with the spherical joint, the deformation is:
$$
\Delta_3 = \nu_{311} \mathbf{u} + \nu_{312} \mathbf{v} + \nu_{313} \mathbf{w}
$$
According to the small-deformation linear superposition principle, the total deformation at the moving platform’s connection points must satisfy the compatibility conditions. By combining the static equilibrium equations (L1) and the deformation compatibility equations (L2), I obtained 28 independent equations, from which all unknown forces/moments can be determined.
2.3 Stiffness Matrix Formulation
By establishing the mapping between the external load \( \mathbf{F}_g = [\mathbf{F}^T, \mathbf{M}^T]^T \) and the elastic deformation of the moving platform \( \mathbf{D} = [\Delta^T, \delta^T]^T \), the flexibility matrix of the mechanism can be derived. Through the deformation analysis, the ball center linear displacement is:
$$
\Delta_{01-1} = \mathbf{G}_{11r} \mathbf{C}_{11r} \mathbf{F}_g
$$
and the angular displacement is:
$$
\delta_{01-1} = \mathbf{G}_{12r} \mathbf{C}_{12r} \mathbf{F}_g
$$
Combining these relationships, the flexibility matrix of the mechanism is:
$$
\mathbf{G} =
\begin{bmatrix}
\mathbf{G}_{11r} \mathbf{C}_{11r} \\
\mathbf{G}_{12r} \mathbf{C}_{12r}
\end{bmatrix}
$$
Finally, the stiffness matrix is obtained by matrix inversion:
$$
\mathbf{k} = \mathbf{G}^{-1}
$$
2.4 Principal Stiffness Analysis
To characterize the stiffness performance more intuitively, I applied orthogonal transformation to the stiffness matrix. Let \( \lambda_1 \) to \( \lambda_6 \) be the eigenvalues of the stiffness matrix \( \mathbf{k} \), and \( \mathbf{x}_1 \) to \( \mathbf{x}_6 \) be the corresponding eigenvectors. By orthogonalization, the transformation matrix is:
$$
\mathbf{P} = [\boldsymbol{\beta}_1, \boldsymbol{\beta}_2, \boldsymbol{\beta}_3, \boldsymbol{\beta}_4, \boldsymbol{\beta}_5, \boldsymbol{\beta}_6]
$$
After the orthogonal transformation, the stiffness matrix becomes diagonal:
$$
\boldsymbol{\Lambda} =
\begin{bmatrix}
k’_{11} & & & & & \\
& k’_{22} & & & & \\
& & k’_{33} & & & \\
& & & k’_{44} & & \\
& & & & k’_{55} & \\
& & & & & k’_{66}
\end{bmatrix}
$$
where \( k’_{11} \) to \( k’_{66} \) are the six principal stiffness values, and the corresponding principal directions are determined by the orthogonal vectors \( \boldsymbol{\beta}_i \).
2.5 Numerical Results
For the numerical analysis, I selected the following parameters: moving platform radius \( r = 35 \) mm, arc fork cross-section \( h = 10 \) mm, \( b = 10 \) mm, pull-rod lengths \( l_1 = l_2 = 50 \) mm, pull-rod diameter 10 mm, central leg rod length \( L_3 = 80 \) mm and diameter 20 mm. All links are made of 45 steel with shear modulus \( G = 80 \) GPa and elastic modulus \( E = 200 \) GPa.
Under a human single-leg stance load of 600 N, the maximum ball center linear displacement of UP+R+S is only 0.0006 mm, and the maximum angular displacement is \( 1.2 \times 10^{-4} \) rad. In comparison, the original UP+R mechanism exhibits a maximum linear displacement of 0.7 mm and angular displacement of 0.01 rad under the same load. This significant improvement confirms that the addition of the central spherical joint branch greatly enhances the stiffness, particularly the translational stiffness, making the mechanism highly suitable for load-bearing applications in embodied robots.
| Configuration | Load Condition | Max Linear Displacement (mm) | Max Angular Displacement (rad) |
|---|---|---|---|
| UP+R+S | 600 N | 0.0006 | 1.2 × 10⁻⁴ |
| UP+R | 600 N | 0.7 | 0.01 |
The principal stiffness values at three representative configurations are summarized in the following table:
| Configuration | Principal Stiffness Values (N/m, N/rad) |
|---|---|
| θx = π/12, θy = π/12 | 7.90×10¹⁴, 2.37×10¹⁵, 1.75×10⁸, 3.18×10⁵, 9.52×10³, 3.04×10⁵ |
| θx = π/6, θy = π/6 | 2.21×10¹⁵, 8.15×10¹⁵, 1.73×10⁸, 1.36×10⁵, 9.52×10³, 5.51×10⁵ |
| θx = π/3, θy = π/3 | 1.41×10¹⁵, 1.15×10¹⁶, 1.72×10⁸, 3.94×10⁴, 9.52×10³, 1.35×10⁶ |
Comparing the principal stiffness values of UP+R+S with those of UP+R, the translational stiffness is dramatically increased by orders of magnitude, and the overall stiffness distribution becomes more balanced. This balanced stiffness characteristic is essential for ensuring stable and accurate motion of the ankle joint in embodied robots, particularly during dynamic tasks such as walking and running.
3. Dynamic Modeling and Simplified Analysis
Dynamic modeling is crucial for the real-time control of parallel mechanisms in embodied robots. Due to the multi-loop closed-chain structure of the UP+R+S mechanism, its dynamic equations form a highly coupled nonlinear system. In this chapter, I established the dynamic model based on the Lagrange principle and further developed a simplified dynamic control model for real-time applications.
3.1 Lagrange Dynamic Model
The total kinetic energy of the system consists of five parts: the moving platform, the arc-shaped fork, the slider, the upper pull-rod, and the lower pull-rod. For each component, the kinetic energy is computed as:
$$
T_i = \frac{1}{2} m_i \mathbf{v}_i^T \mathbf{v}_i + \frac{1}{2} \boldsymbol{\omega}_i^T \mathbf{I}_i \boldsymbol{\omega}_i
$$
where \( m_i \), \( \mathbf{v}_i \), \( \boldsymbol{\omega}_i \), and \( \mathbf{I}_i \) are the mass, centroid linear velocity, angular velocity, and inertia tensor of the i-th component, respectively. The inertia tensor is transformed from the local coordinate system to the base coordinate system by:
$$
\mathbf{I}_i = \mathbf{R}_i^o \mathbf{I}_{i0} (\mathbf{R}_i^o)^T
$$
The potential energy of each component is:
$$
U_i = m_i g z_{ci}
$$
where \( z_{ci} \) is the Z-coordinate of the centroid in the base coordinate system. For the moving platform, the potential energy is expressed as:
$$
U_1 = m_1 g (-x_{a1}\sin\theta_y + y_{a1}\cos\theta_x \sin\theta_y + z_{a1}\cos\theta_x \cos\theta_y – z_{c1})
$$
The Lagrange function is constructed as:
$$
L(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) = V(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) – U(\boldsymbol{\theta})
$$
By applying the Lagrange equation:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\boldsymbol{\theta}}}\right) – \frac{\partial L}{\partial \boldsymbol{\theta}} = \mathbf{Q}
$$
and using the virtual work principle to relate the joint torques to the generalized forces:
$$
\boldsymbol{\tau}^T \dot{\mathbf{q}} = \mathbf{Q}^T \dot{\boldsymbol{\theta}}
$$
the driving forces are obtained as:
$$
\boldsymbol{\tau} = \mathbf{J}^{-T} [\mathbf{M}(\boldsymbol{\theta})\ddot{\boldsymbol{\theta}} + \mathbf{C}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}})\dot{\boldsymbol{\theta}} + \mathbf{G}(\boldsymbol{\theta})]
$$
where \( \mathbf{M}(\boldsymbol{\theta}) \) is the symmetric inertia matrix, \( \mathbf{C}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) \) contains Coriolis and centrifugal terms, and \( \mathbf{G}(\boldsymbol{\theta}) \) represents the gravity terms.
3.2 Simplified Dynamic Model for Real-Time Control
For real-time dynamic control, the computational efficiency of the driving force equations is critical. Traditional simplification methods often neglect the mass of the branches, which leads to significant errors. In this work, I adopted a more refined simplification approach based on the motion mechanism and equation structure of the UP+R+S mechanism.
Since the driving motor angle of the arc fork is equal to the output angle \( \theta_x \), the kinetic and potential energies of the arc fork are simple functions and do not require simplification. However, the pull-rod branch involves three intermediate joint angles \( u_1 \), \( u_2 \), and \( u_3 \), making its energy expressions complex and computationally expensive.
Based on the observation that the intermediate joint angles satisfy \( |u_i| \leq 0.15 \) rad within the workspace \( \theta_x, \theta_y \in [-0.25\pi, 0.25\pi] \), the small-angle approximation \( \sin(u_i) \approx u_i \), \( \cos(u_i) \approx 1 \) can be applied. The simplified joint angles are:
$$
u_{s1} \approx \frac{r\sin\theta_x \sin\theta_y}{L}, \quad u_{s2} \approx \frac{r – r\cos\theta_y}{L}, \quad u_{s3} \approx \frac{\sin\theta_y \sin\theta_x}{\cos\theta_y}
$$
The relative errors of these simplifications are within 0.01 for \( u_1 \) and \( u_2 \), and within 0.05 for \( u_3 \), which are acceptable for engineering applications.
Similarly, the slider displacement function contains a square root operation. Since the square root term is close to \( L \) within the workspace, the slider displacement is simplified as:
$$
l_s \approx L – r\cos\theta_x \sin\theta_y
$$
Substituting these simplified expressions into the Lagrange equations, the simplified driving forces are:
$$
\boldsymbol{\tau}_s = \mathbf{J}^{-T} [\mathbf{M}_s(\boldsymbol{\theta})\ddot{\boldsymbol{\theta}} + \mathbf{C}_s(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}})\dot{\boldsymbol{\theta}} + \mathbf{G}_s(\boldsymbol{\theta})]
$$
3.3 Comparative Analysis of Computational Efficiency
To evaluate the computational efficiency of the simplified model versus the theoretical model, I conducted comparative numerical experiments using the same gait trajectory data. The computational time and memory usage for both models are summarized below:
| Model | Computational Time (s) | Allocated Memory | Peak Memory | Relative Error |
|---|---|---|---|---|
| Theoretical Model | 118 | High | High | — |
| Simplified Model | 4 | Low | Low | ≤ 0.05% |
The simplified model achieves a nearly 30-fold reduction in computational time and approximately 7-fold reduction in memory usage, while maintaining high accuracy with relative errors below 0.05%. These improvements make the simplified model highly suitable for real-time dynamic control of the embodied robot ankle joint.
3.4 Dynamic Performance Indices
To evaluate the dynamic performance of the UP+R+S mechanism across its entire workspace, I derived global performance indices based on the condition numbers of the velocity and acceleration coefficient matrices.
The velocity global performance index is defined as:
$$
\eta_J = \frac{\int_W \frac{1}{k_J} dW}{\int_W dW}
$$
where \( k_J \) is the condition number of the Jacobian matrix and \( W \) is the reachable workspace. The value of \( \eta_J \) ranges from 0 to 1, with values closer to 1 indicating better motion transmission and controllability.
For the acceleration performance, considering the second-order influence coefficient matrix \( \mathbf{H} \), the acceleration error bound is related to both \( k_J \) and \( k_H \). The global acceleration performance index is defined similarly. The inertial force global performance index is based on the combined condition number \( k_{J+H} \).
Numerical calculations were performed with the platform radius \( r \) varying from 4 to 8 cm and the pull-rod length \( L \) varying from 8 to 12 cm. The results are illustrated in the performance charts:
| Performance Index | Range of Values | Optimal Region |
|---|---|---|
| Velocity Performance | 0.14 – 0.26 | r = 4 cm, L = 8–12 cm |
| Acceleration Performance | 0.06 – 0.105 | r = 4 cm, L = 8–12 cm |
| Inertial Force Performance | 0.69 – 0.71 | L = 8–9 cm, r = 7–8 cm |
The analysis reveals that when the platform radius \( r \) is relatively small, the velocity and acceleration performance indices are nearly independent of the pull-rod length \( L \). As \( r \) decreases, the velocity and acceleration performance improves, resulting in smaller output errors. For the inertial force performance, the mechanism exhibits better dynamic characteristics when \( L \) is between 8–9 cm and \( r \) is between 7–8 cm. These findings provide valuable guidance for the structural optimization of the ankle joint mechanism in embodied robots.
3.5 Dynamic Simulation Verification
To verify the correctness of the dynamic model, I established a virtual prototype of the UP+R+S mechanism in ADAMS software. The 3D solid model was created in SOLIDWORKS and imported into ADAMS in .x_t format. Material properties (density 7800 kg/m³) and gravitational acceleration (9.8 N/kg) were defined accordingly.
The simulation was performed with the same gait trajectory, and the driving forces and torques were measured. The comparison between the theoretical values and simulation results shows that the maximum relative errors are within 0.05%, confirming the correctness of the dynamic model.
| Quantity | Theoretical Value Range | Simulation Value Range | Relative Error |
|---|---|---|---|
| Translational Driving Force | -3.2 to -2.8 N | -3.18 to -2.82 N | < 0.05% |
| Rotational Driving Torque | -0.035 to 0.015 N·m | -0.0352 to 0.0148 N·m | < 0.05% |
The simulation results demonstrate that the driving force exhibits a periodic variation pattern consistent with the theoretical predictions. The slight deviation between the theoretical and simulation values is attributed to the simplifications made in the dynamic modeling process and the numerical integration errors in the simulation solver. These errors are well within the acceptable range for engineering applications.
4. Conclusions
In this dissertation, I conducted a systematic investigation of the novel three-branch spherical parallel bionic ankle joint mechanism UP+R+S for application in embodied robots. The main contributions and conclusions are summarized as follows:
(1) Kinematic analysis: Based on the closed-loop vector method, I derived the position forward and inverse solutions, joint angular displacements, centroid linear velocities, and angular velocities of all components. The Xsens MVN inertial motion capture system was used to acquire human ankle joint trajectories during normal walking, which were fitted with Fourier series and used as the mechanism’s output for numerical verification. The results confirm that the UP+R+S mechanism can effectively reproduce human ankle joint motion patterns, demonstrating high bionic fidelity.
(2) Static stiffness analysis: Based on the small-deformation linear superposition principle, I established the deformation compatibility equations and derived the flexibility and stiffness matrices of the mechanism. The orthogonal transformation method was employed to determine the six principal stiffness values and their directions. Compared with the original UP+R mechanism, the new UP+R+S mechanism exhibits significantly reduced linear and angular deformations and substantially increased translational stiffness. The overall stiffness becomes larger and more balanced, which is essential for load-bearing applications in embodied robots.
(3) Dynamic modeling and simplification: Using the Lagrange principle, I established the theoretical dynamic model of the UP+R+S mechanism. Global performance indices for velocity, acceleration, and inertial force were derived based on condition numbers and numerically evaluated. A simplified dynamic control model was developed using trigonometric function approximations for the pull-rod branch. The simplified model achieves high computational efficiency (30× faster and 7× less memory) while maintaining accuracy with relative errors below 0.05%, making it highly suitable for real-time dynamic control of embodied robot ankle joints.
(4) Simulation verification: Virtual prototype simulations in ADAMS confirmed the correctness of the kinematic and dynamic models, with all relative errors within the acceptable engineering range.
The research presented in this dissertation provides a solid theoretical foundation for the engineering design, performance optimization, and real-time control of the novel spherical parallel bionic ankle joint mechanism UP+R+S. The findings demonstrate that this mechanism is a promising candidate for ankle joint implementation in advanced embodied robots, offering excellent load capacity, balanced stiffness, high bionic fidelity, and superior dynamic performance.
Future work should focus on further simplifying the dynamic equations to meet the requirements of fast walking, running, and jumping motions. Additionally, the development of a physical prototype is necessary to validate the theoretical findings under practical operating conditions, particularly addressing the manufacturing and assembly challenges associated with the over-constrained spherical joint center coincidence requirement.