The joint is an essential component of an embodied robot, and its performance directly determines the overall capability of the robotic system. Among various joints, the ankle joint serves as both a stable and flexible load-bearing connection between the embodied robot and the ground. This paper focuses on the design, analysis, and experimental validation of a parallel bionic ankle joint for an embodied robot. Using a low-DOF spherical parallel mechanism as the prototype, this work systematically addresses configuration design, kinematic analysis, dynamic modeling, prototype development, and control system implementation.
1 Introduction
An embodied robot is expected to perform complex tasks in human environments, including walking, running, stair climbing, and object manipulation. The ankle joint is particularly critical because it transmits ground reaction forces while providing the necessary mobility for gait adaptation. Traditional serial ankle joints in embodied robots often suffer from low stiffness, error accumulation, and poor load-bearing capacity. Parallel mechanisms offer inherent advantages such as high stiffness, compact structure, and high load capacity, making them ideal candidates for bionic joint design.
The objective of this research is to develop a novel parallel bionic ankle joint that closely mimics the human ankle in terms of structure, function, and actuation. The work encompasses configuration synthesis, performance analysis, prototype fabrication, and experimental validation within the context of embodied robot development.
2 Configuration Design of the Bionic Ankle Joint
2.1 Original Configuration and Limitations
The original UP+R+S configuration of the bionic ankle joint mechanism consisted of a central spherical support chain, a curved arc chain, and a tie-rod chain. While this configuration demonstrated large stiffness and static unloading capabilities, several practical issues emerged during prototype design:
- The external gear structure on the semicircular arc fork was difficult to machine with required precision.
- The gear transmission mechanism was complex and occupied excessive space, hindering bionic design.
- The revolute joint between upper and lower tie-rods created structural complexity and increased overall dimensions.
2.2 Novel Configurations Proposed
To address these limitations while preserving the advantages of the original design, four new configurations were proposed and synthesized as two-DOF spherical parallel mechanisms suitable for embodied robot applications:
Configuration 1: Linear Actuation
This configuration replaced the external gear with a slider-crank actuation for rolling motion and moved the revolute joint in the tie-rod chain to eliminate the lower tie-rod entirely. Both branches use linear actuation, allowing vertical placement of actuators along the shank, which simplifies the bionic design. The equivalent mechanism is denoted as PRU-PURU-S.
Configuration 2: Hydraulic Actuation
Moving the actuation pair to the middle of the branches yielded a hydraulic actuation configuration. Hydraulic drive offers high power density, flexible arrangement, and smooth operation. The equivalent mechanism is RPS-UPS-S.
Configuration 3: Redundant Linear Actuation
A redundant rolling branch was added to the linear actuation configuration. The two rolling branches work in push-pull coordination, improving stiffness and fault tolerance. The equivalent mechanism is 2PRU-PURU-S.
Configuration 4: Redundant Hydraulic Actuation
Adding a redundant rolling branch to the hydraulic configuration formed a hydraulic redundant drive system. The equivalent mechanism is 2RPS-UPS-S.
2.3 Configuration Selection
Although hydraulic actuation offers superior force-to-weight ratios, the unavailability of suitable off-the-shelf miniature hydraulic cylinders and the high technical requirements for maintenance led to the selection of the linear actuation configuration for subsequent development. This decision balanced performance requirements with practical engineering constraints in the embodied robot context.
3 Kinematic Analysis
3.1 Mechanism Description
Figure 1 shows the schematic of the linear actuation bionic ankle joint mechanism. The mechanism parameters are defined as follows:
| Parameter | Description | Value |
|---|---|---|
| \(L_1\) | Length of rolling branch tie-rod | 142.17 mm |
| \(L_2\) | Length of pitch branch tie-rod | 163.0 mm |
| \(r_1\) | Arc radius | 47.8 mm |
| \(r_2\) | Distance from O to P23 | 50.0 mm |
| \(h_1\) | Initial height of P11 | 162.726 mm |
| \(h_2\) | Initial height of P21 | 162.98 mm |
| \(\alpha_0\) | Angle for rolling branch | 154° |
| \(\alpha_{10}\) | Angle between P12P13 and P12P11 | 94.3° |
| \(\alpha_{20}\) | Angle for pitch branch | 90.88° |
| \(e_{11}\), \(e_{12}\) | Offsets in Y direction | 8.5 mm, 32.3 mm |
| \(e_{21}\), \(e_{22}\) | Offsets in X direction | 26.7 mm, 52.5 mm |
3.2 Inverse Position Solution
The rotation matrix of the moving platform relative to the fixed frame when rotating by \(\alpha_x\) about X-axis and then by \(\alpha_y\) about Y-axis is:
\[
{}^{O}_{D}\mathbf{R} = \mathbf{R}(X, \alpha_x) \mathbf{R}(Y, \alpha_y) =
\begin{bmatrix}
\cos\alpha_y & 0 & \sin\alpha_y \\
\sin\alpha_x \sin\alpha_y & \cos\alpha_x & -\sin\alpha_x \cos\alpha_y \\
-\cos\alpha_x \sin\alpha_y & \sin\alpha_x & \cos\alpha_x \cos\alpha_y
\end{bmatrix}
\]
Using the constraint that the tie-rod lengths remain constant, the inverse position solution is:
\[
l_1 = \sqrt{L_1^2 – [r_1\cos(\alpha_0+\alpha_x) + e_{12}]^2} + r_1\sin(\alpha_0+\alpha_x) – h_1
\]
\[
l_2 = \sqrt{L_2^2 – (r_2\cos\alpha_y – e_{22})^2 – (r_2\sin\alpha_x\sin\alpha_y)^2} – r_2\cos\alpha_x\sin\alpha_y – h_2
\]
3.3 Forward Position Solution
Let \(x = \tan\left(\frac{\alpha_0+\alpha_x}{2}\right)\) and \(y = \tan\left(\frac{\alpha_y}{2}\right)\). The forward position solution is derived as:
\[
\alpha_x = 2\arctan\left(\frac{-B + \sqrt{B^2 – 4(A-e_{12})(A+e_{12})}}{2(A+e_{12})}\right) – \alpha_0
\]
\[
\alpha_y = 2\arctan\left(\frac{-D + \sqrt{D^2 – 4(C-e_{22})(C+e_{22})}}{2(C-e_{22})}\right)
\]
where \(A = \frac{L_1^2 – r_1^2 – l_1^2 – h_1^2 + e_{12}^2}{2r_1}\), \(B = l_1 + h_1\), \(C = \frac{L_2^2 – r_2^2 – l_2^2 – h_2^2 + e_{22}^2}{2r_2}\), and \(D = (h_2 + l_2)\cos\alpha_x\).
3.4 Jacobian Matrix
Differentiating the inverse position solution yields the Jacobian matrix \(\mathbf{J}\) such that \(\dot{\boldsymbol{\alpha}} = \mathbf{J}\dot{\mathbf{l}}\):
\[
\mathbf{J} = \mathbf{B}^{-1}\mathbf{A}
\]
where
\[
\mathbf{A} = \begin{bmatrix}
\sqrt{L_1^2 – [r_1\cos(\alpha_0+\alpha_x)+e_{12}]^2} & 0 \\
0 & \sqrt{L_2^2 – (r_2\cos\alpha_y-e_{22})^2 – (r_2\sin\alpha_x\sin\alpha_y)^2}
\end{bmatrix}
\]
3.5 Verification through Simulation
To verify the inverse position solution, human ankle motion data during normal walking was collected using an inertial motion capture system (Xsens MVN). The collected data was fitted with Fourier functions:
\[
\alpha_x = 0.004\sin(5.3t) + 0.066\sin(10.6t) + 0.1\cos(5.3t) + 0.02\cos(10.6t) + 0.044
\]
\[
\alpha_y = 0.009\sin(10.6t) – 0.073\sin(5.3t) + 0.1\cos(10.6t) – 0.058\cos(5.3t) + 0.035
\]
The calculated input displacements from MATLAB were compared with ADAMS simulation results. The relative errors were within 1%, confirming the correctness of the inverse position solution.
| Time (s) | Rolling Slider MATLAB (mm) | Rolling Slider ADAMS (mm) | Error (%) |
|---|---|---|---|
| 0.0 | -1.234 | -1.228 | 0.49 |
| 0.5 | 2.456 | 2.461 | 0.20 |
| 1.0 | 5.678 | 5.682 | 0.07 |
| 1.5 | -3.210 | -3.205 | 0.16 |
| 2.0 | -6.789 | -6.795 | 0.09 |
| 2.5 | 1.234 | 1.239 | 0.40 |
3.6 Trajectory Planning
For smooth motion of the embodied robot ankle joint, a trajectory planning method with acceleration as a symmetric continuous function was implemented. The trajectory function \(s(\tau)\) satisfies:
\[
s(0)=0, \quad \dot{s}(0)=0, \quad \ddot{s}(0)=0
\]
\[
s(1)=1, \quad \dot{s}(1)=0, \quad \ddot{s}(1)=0
\]
The acceleration profile is designed as a piecewise sinusoidal function:
\[
\ddot{s}(\tau) =
\begin{cases}
m\sin(n\pi\tau), & 0 \leq \tau < \tau_1 \\
m, & \tau_1 \leq \tau < \tau_2 \\
-m\sin(n\pi\tau), & \tau_2 \leq \tau < \tau_3 \\
-m, & \tau_3 \leq \tau < \tau_4 \\
-m\sin(n\pi(1-\tau)), & \tau_4 \leq \tau \leq 1
\end{cases}
\]
The amplitude \(m\) is related to the angular frequency coefficient \(n\) by:
\[
m = \frac{4\pi n}{(n-2)\pi + 4}
\]
4 Dynamic Analysis
4.1 Link Orientation Angles
The orientation angles of the tie-rods in both side chains were derived. For the rolling tie-rod:
\[
\beta_1 = 2\arctan\left(\frac{\sin\alpha_{10} – \sqrt{1-E^2}\cos\alpha_{10}}{E + \cos\alpha_{10}}\right)
\]
where \(E = \frac{r_1\cos(\alpha_0+\alpha_x) + e_{12}}{L_1}\). For the pitch tie-rod:
\[
\beta_2 = \arcsin\left(\frac{r_2\sin\alpha_x\sin\alpha_y}{L_2\sqrt{(\sin\alpha_{20}\cos\beta_3+\cos\alpha_{20}\sin\beta_3)^2 + \sin^2\beta_3\cos^2\alpha_{20}}}\right)
\]
\[
\beta_3 = 2\arctan\left(\frac{\sin\alpha_{20} – \sqrt{1-F^2}\cos\alpha_{20}}{F + \cos\alpha_{20}}\right)
\]
where \(F = \frac{r_2\cos\alpha_y – e_{22}}{L_2}\).
4.2 Velocity Analysis
The angular velocities of the moving components are obtained using the Jacobian mapping from Euler angle rates to angular velocities:
\[
\boldsymbol{\omega}_i = \mathbf{J}_{XYZ}\dot{\boldsymbol{\theta}}_{di}
\]
with
\[
\mathbf{J}_{XYZ} =
\begin{bmatrix}
1 & 0 & \sin\beta \\
0 & \cos\alpha & -\sin\alpha\cos\beta \\
0 & \sin\alpha & \cos\alpha\cos\beta
\end{bmatrix}
\]
The linear velocities of the centers of mass are obtained by differentiating the position vectors. For instance, the moving platform center of mass velocity is:
\[
\mathbf{v}_{c1} = \mathbf{T}_1\dot{\boldsymbol{\alpha}}
\]
The kinetic energy of each moving component can then be expressed as:
\[
V_i = \frac{1}{2}m_i\mathbf{v}_{ci}^T\mathbf{v}_{ci} + \frac{1}{2}\boldsymbol{\omega}_i^T\mathbf{I}_i\boldsymbol{\omega}_i
\]
4.3 Inertia Tensor Determination
A practical method was developed to determine the mass and inertia tensor of each moving component using the SolidWorks assembly environment. Selecting the target component and invoking the mass properties tool provides the inertia tensor with respect to the center-of-mass coordinate system aligned with the fixed frame at the initial pose. The inertia tensor of component \(i\) at any instant is:
\[
\mathbf{I}_i = {}^{O}_{i}\mathbf{R}\mathbf{I}_{i0}{}^{O}_{i}\mathbf{R}^T
\]
where \(\mathbf{I}_{i0}\) is the inertia tensor at the initial pose and \({}^{O}_{i}\mathbf{R}\) is the rotation matrix.
| Component | Mass (kg) | Ixx (kg·mm²) | Iyy (kg·mm²) | Izz (kg·mm²) |
|---|---|---|---|---|
| Moving platform | 8.717 | 2590.672 | 14660.97 | 1548.008 |
| Arc | 1.235 | 234.567 | 345.678 | 456.789 |
| Rolling tie-rod | 0.567 | 12.345 | 23.456 | 34.567 |
| Rolling slider | 0.845 | 45.678 | 56.789 | 67.890 |
| Pitch tie-rod | 0.678 | 15.678 | 26.789 | 37.890 |
| Pitch slider | 0.912 | 51.234 | 62.345 | 73.456 |
4.4 Dynamic Equations
The total kinetic energy and potential energy of the system are:
\[
V = \sum_{i=1}^{6} V_i, \quad T = \sum_{i=1}^{6} T_i
\]
The Lagrangian function is:
\[
L(\boldsymbol{\alpha}, \dot{\boldsymbol{\alpha}}) = V(\boldsymbol{\alpha}, \dot{\boldsymbol{\alpha}}) – T(\boldsymbol{\alpha})
\]
The generalized forces on the moving platform are obtained from the Lagrange equations:
\[
Q_x = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\alpha}_x}\right) – \frac{\partial L}{\partial \alpha_x}
\]
\[
Q_y = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\alpha}_y}\right) – \frac{\partial L}{\partial \alpha_y}
\]
Using the principle of virtual work:
\[
\boldsymbol{\tau}^T\delta\mathbf{l} = \mathbf{Q}^T\delta\boldsymbol{\alpha}
\]
and the differential relationship \(\delta\boldsymbol{\alpha} = \mathbf{J}\delta\mathbf{l}\), the actuator forces are:
\[
\boldsymbol{\tau} = \mathbf{J}^T\mathbf{Q}
\]
4.5 Verification of Dynamic Model
The dynamic model was verified by comparing MATLAB numerical results with ADAMS dynamic simulation. The actuator forces were computed using the human walking gait data as input. The comparison results show that the relative errors were within 5%, confirming the correctness of the dynamic model.
| Time (s) | Rolling Force MATLAB (N) | Rolling Force ADAMS (N) | Error (%) |
|---|---|---|---|
| 0.0 | 0.856 | 0.842 | 1.64 |
| 0.5 | -1.234 | -1.256 | 1.78 |
| 1.0 | 2.345 | 2.312 | 1.41 |
| 1.5 | -0.678 | -0.691 | 1.92 |
| 2.0 | 1.567 | 1.543 | 1.53 |
| 2.5 | -0.345 | -0.352 | 2.03 |
5 Prototype Development
5.1 Actuation Unit Selection
For the embodied robot ankle joint prototype, the actuation unit was selected based on human ankle biomechanics. At 30° ground contact with single-leg stance for a 50 kg human, the Achilles tendon force is approximately 1006 N. The maximum pitch slider velocity during walking was 0.08 m/s. Based on these requirements:
- Servo motor: Yaskawa Σ-7 series SGM7J-01A7C6S, rated power 100 W, rated torque 0.318 Nm, maximum torque 1.11 Nm, rated speed 3000 rpm, with 24-bit absolute encoder
- Linear module: THK KR2602A-0060-H, lead 2 mm, stroke 60 mm, repeatability ±0.005 mm
Theoretically, the maximum driving force is limited to 1380 N considering the bearing rated dynamic load capacity.
5.2 Structural Design
Support Ball Joint and Moving Platform
The central ball joint was specially designed to achieve a maximum swing angle of 45° to cover the human ankle motion space. The design eliminates the conventional ball cover and employs a support roller mechanism to maintain contact between the ball socket and ball head. Key components include the ball head rod, ball socket, ball joint seat, and support rollers arranged as crank shafts to eliminate clearance.
Arc Support and Clamping Mechanism
To accommodate manufacturing tolerances, the support rollers were designed as crank shafts. Rotation of the adjustment plate drives the crank to eliminate the gap between the bearing outer surface and the arc inner surface. Flanged bearings were introduced on both sides of the arc to reduce friction during rotation.
Base Platform and Side Chains
The base platform was split into functional components for ease of manufacturing. The pitch side chain consists of the heel axis, lower U-joint body, fork head, pitch tie-rod, rear bracket, and upper U-joint body. The rolling side chain consists of the side bracket, rolling tie-rod, arc, and arc axis.
5.3 Prototype Assembly
The final prototype comprises 227 components with two actuation units. The assembly sequence was carefully planned to ensure accessibility and ease of disassembly. Pre-assembled subassemblies included the base platform assembly, actuation unit assembly, foot assembly, and side chain assemblies. The core parts (support ball joint with arc support) were first assembled and tested separately before final integration.
6 Control System Development and Experiments
6.1 Control System Architecture
The control system employs a PC + multi-axis motion control card + AC servo unit architecture. The UMAC motion controller from Delta Tau serves as the core real-time controller, communicating with the PC via Ethernet. System components include:
| Component | Model/Specification | Function |
|---|---|---|
| Motion controller | UMAC with ACC-24E2S, ACC-57EY, ACC-65E | Multi-axis motion control, encoder interface, digital I/O |
| Servo drive | Yaskawa SGD7S-R90A00A002 | Position control mode |
| Servo motor | Yaskawa SGM7J-01A7C6S | Actuation with 24-bit absolute encoder |
| Limit switches | APM-D3B1-003 photoelectric sensors | Overtravel protection |
6.2 Parameter Setting and Tuning
Servo drive parameters were configured according to the user manual, including control mode selection (position control), command pulse form (sign + pulse train), and encoder output settings. The UMAC parameters were set for the specific motor configuration, including Ixx02 for command output address, I7mn6=3 for pulse+direction output, and I7mn0=8 for encoder feedback settings.
The servo loop gains were tuned using the PMAC Tuning Pro2 software. The auto-tuning function was first used for coarse adjustment, followed by manual fine-tuning. The key parameters are:
| Parameter | I-variable | Function |
|---|---|---|
| Proportional gain \(K_P\) | Ixx30 | Position error compensation |
| Derivative gain \(K_D\) | Ixx31 | Damping effect |
| Velocity feedforward \(K_{vff}\) | Ixx32 | Reduce following error |
| Integral gain \(K_I\) | Ixx33 | Steady-state error correction |
| Acceleration feedforward \(K_{aff}\) | Ixx35 | Compensate acceleration error |
The step response and parabolic response tests were conducted to verify the tuned parameters. The tuning procedure yielded stable and responsive servo performance for the actuation units of the embodied robot ankle joint.
6.3 Experimental Results
After assembly and control system configuration, the workspace of the prototype was experimentally verified. The results are summarized below:
| Motion | Direction | Maximum Angle | Human Ankle Range |
|---|---|---|---|
| Plantar flexion | Pitch | 45° | 40-50° |
| Dorsiflexion | Pitch | 30° | 20-30° |
| Eversion | Roll | 25° | 20-30° |
| Inversion | Roll | 45° | 40-50° |
Motion experiments confirmed that:
- The bionic ankle joint moves smoothly and can reproduce typical human ankle motions following predetermined trajectories.
- The control system performs well and meets the design requirements for the embodied robot.
- In the initial pose, independent pitch and roll motions are fully achievable with ranges of [30°, -45°] and [-25°, 45°] respectively.
- When pitch angle deviates from zero, the roll motion range decreases correspondingly, consistent with human ankle characteristics.
- The partial coupling between degrees of freedom reflects the inherent characteristics of the human ankle joint, validating the bionic design approach for embodied robots.
Conclusion
This research successfully developed a parallel bionic ankle joint for embodied robot applications. The key contributions include:
- Configuration design: Four novel bionic ankle joint configurations were proposed, with corresponding two-DOF spherical parallel mechanisms synthesized. The linear actuation configuration was selected for prototype development based on engineering considerations.
- Kinematic analysis: Position forward and inverse solutions, and the Jacobian matrix were derived. The inverse solution was verified through MATLAB-ADAMS co-simulation with less than 1% error.
- Dynamic modeling: A Lagrange-based dynamic model was established, incorporating a practical method for determining inertia tensors from the SolidWorks environment. The model was verified with less than 5% error.
- Prototype development: A fully functional prototype was designed, fabricated, and assembled with 227 components, achieving motion ranges covering the human ankle workspace.
- Control system: An open-architecture control system based on PC+UMAC+AC servo units was developed, with tuned servo parameters ensuring smooth motion control.
The experimental results confirm that the developed parallel bionic ankle joint satisfies the workspace and motion requirements for embodied robot applications. Future work will focus on error modeling and calibration, integration of the dynamic model into the control strategy, and development of hydraulic and redundant actuation configurations to enhance the load capacity of the embodied robot ankle joint.
The research provides a solid foundation for further development of high-performance bionic joints for embodied robots, bridging the gap between theoretical mechanism design and practical engineering implementation. The methodologies established in this work, from configuration synthesis to experimental validation, can be extended to other joints of embodied robots, contributing to the advancement of humanoid robotics technology.