1. Introduction and Research Motivation
The humanoid robot represents a pinnacle of modern mechatronic engineering, integrating advanced technologies in mechanical design, computer science, bionics, materials science, control theory, and sensor technology. Among the various components that constitute a humanoid robot, the joints serve as the fundamental carriers of functionality, and their performance directly determines the overall capability of the entire robotic system. The ankle joint, in particular, holds a unique position as both a stable and flexible load-bearing joint, serving as the critical hub connecting the humanoid robot with the ground. The ability of a humanoid robot to perform complex anthropomorphic movements, including squatting, stair climbing, walking, running, and jumping, is intimately connected to the structural configuration and kinematic performance of its ankle joint.
Traditional humanoid robot joints typically adopt serial structural configurations, which suffer from inherent limitations such as lower stiffness, cumulative errors, and significant deviation from the natural driving mechanisms observed in biological joints. These shortcomings hinder the bionic design process and limit the overall performance of humanoid robots. Parallel mechanisms, in contrast, offer numerous advantages including high stiffness, superior load-carrying capacity, minimal error accumulation, compact structure, and remarkable consistency with the actual driving patterns of biological joints. Therefore, the application of parallel mechanisms—particularly low-degree-of-freedom spherical parallel mechanisms—to humanoid robot joints represents a rational and feasible approach that aligns with the principle of bionic design.
This research was conducted under the sponsorship of the National Natural Science Foundation of China (Project No. 51275443), focusing on the configuration synthesis, bionic design, and experimental investigation of spherical parallel humanoid robot joints. The primary objective was to develop a high-performance bionic ankle joint for humanoid robots through systematic configuration design, kinematic analysis, dynamic modeling, prototype development, and control system implementation. This paper presents the complete research journey, from conceptual design through to physical prototyping and experimental validation.
2. Bionic Configuration Design
2.1 Original Configuration and Its Limitations
The original UP+R+S bionic ankle joint mechanism consisted of three primary branches: a spherical support branch, an arc branch, and a tie-rod branch. In the arc branch, an electric motor drove an external-gear-ring semi-circular arc fork to achieve the rolling motion of the mechanism. In the tie-rod branch, the motor drove the platform’s pitching motion through a slider-tie-rod transmission system. During the three-dimensional prototype design process, several significant problems emerged:
- The external-gear-ring structure of the arc fork presented substantial machining difficulty, making precision control problematic.
- The gear transmission mechanism was overly complex, creating installation and arrangement challenges within confined spaces, which adversely affected the bionic design.
- The revolute joint R connecting the upper and lower tie rods resulted in complicated structures with oversized dimensions, impeding overall size reduction.
2.2 New Configuration Proposals
To address these limitations while retaining the advantages of high stiffness and static unloading characteristic of the original UP+R+S configuration, the following modifications were implemented: the revolute joint R in the tie-rod branch was relocated downward until the lower tie-rod length reached zero, and the external gear ring structure of the arc fork was eliminated, replaced by a slider-driven tie-rod mechanism. These improvements led to the formulation of four novel bionic ankle joint configurations:
| Configuration | Equivalent Mechanism | Key Characteristics |
|---|---|---|
| Linear-driven bionic ankle joint | PRU-PURU-S | Both branches utilize linear driving; compact structure suitable for bionic design |
| Hydraulic-driven bionic ankle joint | RPS-UPS-S | Hydraulic actuation; high power density; flexible arrangement |
| Lead-screw redundant-driven | 2PRU-PURU-S | Redundant rolling branch; enhanced precision tolerance and stiffness |
| Hydraulic redundant-driven | 2RPS-UPS-S | Combines parallel, hydraulic, and redundant driving advantages |
Each of these configurations corresponds to a two-degree-of-freedom spherical parallel mechanism. After comprehensive consideration of manufacturing feasibility, technical maturity, and control complexity, the linear-driven configuration was selected for subsequent prototype development, as it represents the simplest structural form while fulfilling the essential performance requirements.
3. Kinematic Analysis
3.1 Mechanism Description and Coordinate Systems
Figure 2-6 in the original research presents the kinematic diagram of the linear-driven bionic ankle joint mechanism. The key structural parameters are defined as follows: the input displacement of the rolling driving slider is denoted as \(l_1\), and that of the pitching driving slider as \(l_2\). The length of the rolling branch tie-rod (P12-P13) is \(L_1\), while the pitching branch tie-rod (P22-P23) has length \(L_2\). The arc radius is \(r_1\), and the distance from point O to P23 is \(r_2\). The coordinate system is established with the origin at the center of the spherical joint. The fixed coordinate system \(OX_0Y_0Z_0\) is defined, and the moving coordinate system \(OXYZ\) coincides with the fixed system at the initial posture.
When the moving platform first rotates about the X-axis by angle \(\alpha_x\) and then about the Y-axis by angle \(\alpha_y\), the rotation matrix describing the orientation of the moving coordinate system relative to the fixed system is:
$$
\mathbf{R}_{DO} = \mathbf{R}(Y, \alpha_y) \mathbf{R}(X, \alpha_x) =
\begin{bmatrix}
\cos\alpha_y & 0 & \sin\alpha_y \\
\sin\alpha_x \sin\alpha_y & \cos\alpha_x & -\sin\alpha_x \cos\alpha_y \\
-\cos\alpha_x \sin\alpha_y & \sin\alpha_x & \cos\alpha_x \cos\alpha_y
\end{bmatrix}
\tag{1}$$
3.2 Inverse Position Solution
The coordinates of the key points in the rolling branch at the initial posture are:
$$
\mathbf{P}_{11} = \begin{bmatrix} 0 \\ e_{11} \\ -h_1 \end{bmatrix}, \quad
\mathbf{P}_{12} = \begin{bmatrix} 0 \\ e_{12} \\ -h_1 \end{bmatrix}, \quad
\mathbf{P}_{13} = \begin{bmatrix} 0 \\ r_1\cos\alpha_0 \\ r_1\sin\alpha_0 \end{bmatrix}
\tag{2}$$
Since the length of the tie-rod in the rolling branch remains constant during motion:
$$
L_1^2 = [r_1\cos(\alpha_x + \alpha_0)]^2 + [r_1\sin(\alpha_x + \alpha_0) – (e_{12} – h_1)]^2
\tag{3}$$
This yields the inverse solution for the rolling slider displacement:
$$
l_1 = r_1\sin(\alpha_x + \alpha_0) + \sqrt{L_1^2 – [r_1\cos(\alpha_x + \alpha_0) – e_{12}]^2} – h_1
\tag{4}$$
Similarly, for the pitching branch, the coordinates of point P23 in the fixed frame are:
$$
\mathbf{P}_{23} = \mathbf{R}_{DO} \cdot \begin{bmatrix} r_2 \\ 0 \\ 0 \end{bmatrix} =
\begin{bmatrix} r_2\cos\alpha_y \\ -r_2\sin\alpha_x \sin\alpha_y \\ r_2\cos\alpha_x \sin\alpha_y \end{bmatrix}
\tag{5}$$
Applying the constant length constraint for the pitching tie-rod:
$$
L_2^2 = (r_2\cos\alpha_y – e_{22})^2 + (r_2\sin\alpha_x \sin\alpha_y)^2 + (r_2\cos\alpha_x \sin\alpha_y – h_2 + l_2)^2
\tag{6}$$
Solving for the pitching slider displacement:
$$
l_2 = \sqrt{L_2^2 – (r_2\cos\alpha_y – e_{22})^2 – (r_2\sin\alpha_x\sin\alpha_y)^2} + r_2\cos\alpha_x\sin\alpha_y – h_2
\tag{7}$$
3.3 Forward Position Solution
By defining the tangent half-angle substitutions:
$$
\tan\left(\frac{\alpha_x + \alpha_0}{2}\right) = x, \quad \tan\left(\frac{\alpha_y}{2}\right) = y
\tag{8}$$
The forward position solution can be derived by solving the resulting quadratic equations. For the rolling branch:
$$
x = \frac{-B + \sqrt{B^2 – 4A(A – e_{12})}}{2(A + e_{12})}
\tag{9}$$
where \(A = \frac{L_1^2 – r_1^2 – (l_1 – h_1)^2 – e_{12}^2}{2r_1}\) and \(B = l_1 + h_1\).
For the pitching branch:
$$
y = \frac{D + \sqrt{D^2 – 4(C – e_{22})(C + e_{22})}}{2(C – e_{22})}
\tag{10}$$
where \(C = \frac{L_2^2 – r_2^2 – (l_2 – h_2)^2 – e_{22}^2}{2r_2}\) and \(D = (h_2 + l_2)\cos\alpha_x\).
3.4 Jacobian Matrix
Differentiating the inverse position equations with respect to time yields the velocity relationship between the actuator space and the task space:
$$
\mathbf{B}\dot{\boldsymbol{\alpha}} = \mathbf{A}\dot{\mathbf{l}}
\tag{11}$$
where \(\boldsymbol{\alpha} = [\alpha_x, \alpha_y]^T\) represents the orientation angles of the moving platform, and \(\mathbf{l} = [l_1, l_2]^T\) represents the actuator displacements. The Jacobian matrix that maps actuator velocities to platform angular velocities is:
$$
\dot{\boldsymbol{\alpha}} = \mathbf{J}\dot{\mathbf{l}}, \quad \mathbf{J} = \mathbf{B}^{-1}\mathbf{A}
\tag{12}$$
The analysis reveals an important characteristic of this mechanism: the rolling motion of the moving platform depends exclusively on the rolling slider input and is completely decoupled from the pitching slider input. Conversely, the pitching motion is influenced by both inputs, exhibiting partial coupling. This semi-decoupled property significantly simplifies the motion control strategy compared to fully coupled mechanisms.
3.5 Simulation Verification
To validate the theoretical inverse position solution, human ankle joint motion data during normal walking were collected using the Xsens MVN inertial motion capture system. The acquired attitude data were fitted through Fourier series expansion:
$$
\alpha_x = 0.004\sin(5.3t) + 0.066\sin(10.6t) + 0.1\cos(5.3t) – 0.02\cos(10.6t) – 0.044
\tag{13}$$
$$
\alpha_y = 0.009\sin(10.6t) + 0.073\sin(5.3t) + 0.1\cos(10.6t) – 0.058\cos(5.3t) + 0.035
\tag{14}$$
Using the structural parameters listed in Table 1, the inverse position solution was computed in MATLAB and compared with ADAMS simulation results. The comparison demonstrated excellent agreement with relative errors below 1%, thereby confirming the correctness of the inverse kinematic solution.
| Parameter | \(L_1\) | \(L_2\) | \(r_1\) | \(r_2\) | \(\alpha_0\) | \(\alpha_{10}\) | \(\alpha_{20}\) | \(e_{11}\) | \(e_{12}\) | \(e_{21}\) | \(e_{22}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Value | 142.17 | 163.0 | 47.8 | 50.0 | 154° | 94.3° | 90.88° | 8.5 | 32.3 | 26.7 | 52.5 |
3.6 Trajectory Planning
To ensure smooth and stable motion of the bionic ankle joint, a trajectory planning method employing acceleration as a symmetric continuous function was adopted. The trajectory function \(s(\tau)\) was designed with its second derivative (acceleration) specified as a piecewise continuous sine function:
$$
\ddot{s}(\tau) =
\begin{cases}
m\sin(n\pi\tau), & 0 \leq \tau < \tau_1 \\
m, & \tau_1 \leq \tau < \tau_2 \\
-m\sin(n\pi\tau), & \tau_2 \leq \tau < \tau_3 \\
-m, & \tau_3 \leq \tau < \tau_4 \\
m\sin(n\pi(\tau-1)), & \tau_4 \leq \tau \leq 1
\end{cases}
\tag{15}$$
The velocity and displacement functions were obtained through successive integrations, with the boundary conditions \(s(0) = 0\), \(\dot{s}(0) = 0\), \(\ddot{s}(0) = 0\) and \(s(1) = 1\), \(\dot{s}(1) = 0\), \(\ddot{s}(1) = 0\). This planning approach ensures continuous acceleration throughout the motion, thereby minimizing impact forces and vibration. Trajectory planning was applied to rolling motion, pitching motion, and compound motion of the ankle joint, utilizing the Xsens motion capture data to replicate natural human ankle movements.
4. Dynamic Modeling and Analysis
4.1 Theoretical Framework
The Lagrange equation provides a systematic approach for establishing the dynamic model of the bionic ankle joint mechanism:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) – \frac{\partial L}{\partial q_i} = Q_i, \quad (i = 1, 2)
\tag{16}$$
where \(L = V – T\) is the Lagrangian function representing the difference between total kinetic energy \(V\) and total potential energy \(T\), \(q_i\) represents the generalized coordinates (the platform orientation angles \(\alpha_x\) and \(\alpha_y\)), and \(Q_i\) represents the generalized forces.
Applying the principle of virtual work, the relationship between the actuator forces and the generalized forces on the moving platform can be established:
$$
\boldsymbol{\tau}^T \delta\mathbf{l} = \mathbf{Q}^T \delta\boldsymbol{\alpha}
\tag{17}$$
Combining this with the differential kinematic relationship \(\delta\boldsymbol{\alpha} = \mathbf{J}\delta\mathbf{l}\), the actuator forces are obtained as:
$$
\boldsymbol{\tau} = \mathbf{J}^T \mathbf{Q}
\tag{18}$$
4.2 Component Velocities
The mechanism comprises six moving components: the moving platform, the arc, the rolling tie-rod, the rolling slider, the pitching tie-rod, and the pitching slider. The angular velocity of the moving platform was determined using the XYZ Euler angle representation:
$$
\boldsymbol{\omega}_1 =
\begin{bmatrix}
\dot{\alpha}_x \cos\alpha_y \\
\dot{\alpha}_y \\
\dot{\alpha}_x \sin\alpha_y
\end{bmatrix}
\tag{19}$$
For the arc, which rotates solely about the \(X_0\)-axis:
$$
\boldsymbol{\omega}_2 = \begin{bmatrix} \dot{\alpha}_x \\ 0 \\ 0 \end{bmatrix}
\tag{20}$$
The tie-rod posture angles in the two branches were solved iteratively from the geometric constraint equations. For the rolling tie-rod, the posture angle \(\beta_1\) was determined as:
$$
\beta_1 = 2\arctan\left(\frac{\sin\alpha_{10} – \sqrt{E_1^2 – \sin^2\alpha_{10}}}{E_1 + \cos\alpha_{10}}\right)
\tag{21}$$
where \(E_1 = \frac{r_1\cos(\alpha_x + \alpha_0) + e_{12}}{L_1}\). The centroid linear velocities of all components were then computed through differentiation of their position vectors.
4.3 Inertia Properties
An efficient method was proposed for determining the mass and inertia tensor of each moving component. Within the SolidWorks assembly environment, selecting the target component and invoking the mass properties tool provides immediate access to the mass, centroid position, and the inertia tensor relative to a centroid-aligned coordinate frame. This approach proves particularly valuable for non-symmetric, complex-shaped components. The inertia tensor of component \(i\) at any arbitrary configuration is related to its initial-posture inertia tensor through the rotation transformation:
$$
\mathbf{I}_i = \mathbf{R}_i^O \mathbf{I}_{i0} (\mathbf{R}_i^O)^T
\tag{22}$$
The moving platform’s inertia tensor (units: kg·mm²) about its centroid frame at the initial posture was determined as:
$$
\mathbf{I}_{10} =
\begin{bmatrix}
2590.672 & 0 & -1963.799 \\
0 & 14660.97 & 0 \\
-1963.799 & 0 & 1548.008
\end{bmatrix}, \quad m_1 = 0.717 \text{ kg}
\tag{23}$$
4.4 Kinetic and Potential Energy
The kinetic energy of the moving platform combines both translational and rotational contributions:
$$
V_1 = \frac{1}{2}m_1\mathbf{v}_{c1}^T\mathbf{v}_{c1} + \frac{1}{2}\boldsymbol{\omega}_1^T\mathbf{I}_1\boldsymbol{\omega}_1
\tag{24}$$
Similarly, the kinetic energies of the arc, rolling tie-rod, rolling slider, pitching tie-rod, and pitching slider were computed. The potential energies were determined from the vertical displacements of the respective centroids. For instance, the potential energy of the moving platform is:
$$
T_1 = m_1 g (z_{c1} – z_{a1}) = m_1 g (y_{a1}\sin\alpha_x – z_{a1}\cos\alpha_x\sin\alpha_y + z_{a1})
\tag{25}$$
4.5 Dynamic Simulation and Validation
The complete dynamic model was established by summing the kinetic and potential energies of all six components, constructing the Lagrangian function, and applying the Lagrange equation to obtain the generalized forces. The actuator forces were then computed using the virtual work principle. The material density was set to 7800 kg/m³ with gravitational acceleration of 9.8 m/s². The Xsens motion capture data provided the prescribed platform motion.
The dynamic simulation was performed in ADAMS by importing the SolidWorks model and defining the two slider displacements as spline functions. The comparison between MATLAB numerical results and ADAMS simulation outputs for the rolling slider driving force and pitching slider driving force showed excellent agreement, with relative errors consistently below 5%. This confirmed the correctness of the established dynamic model, providing a solid foundation for subsequent controller design and dynamic performance optimization.
| Force Component | MATLAB (N) | ADAMS (N) | Relative Error |
|---|---|---|---|
| Rolling slider max driving force | 2.85 | 2.80 | 1.8% |
| Pitching slider max driving force | 31.2 | 30.5 | 2.3% |
5. Prototype Development
5.1 Actuation Unit Selection
Considering the ankle joint’s role as a load-bearing structure in humanoid robots, the actuation system must provide sufficient force while maintaining compact dimensions. After comparing various linear motion transmission alternatives, the ball-screw type linear module was selected due to its high positioning accuracy (±0.005 mm), suitability for vertical axis applications, high stiffness, and low friction.
The actuation unit consists of a Yaskawa Σ-7 series AC servo motor (model SGM7J-01A7C6S) coupled with a THK LM intelligent guide combination unit (model KR2602A-0060-H). The key specifications are:
| Parameter | Value |
|---|---|
| Rated power | 100 W |
| Rated torque | 0.318 N·m |
| Maximum instantaneous torque | 1.11 N·m |
| Rated speed | 3000 rpm |
| Maximum speed | 6000 rpm |
| Rotor moment of inertia | 0.0659 × 10⁻⁴ kg·m² |
| Parameter | Value |
|---|---|
| Lead | 2 mm |
| Stroke | 60 mm |
| Repeated positioning accuracy | ±0.005 mm |
| Positioning accuracy | 0.06 mm |
| Basic dynamic load rating | 7240 N |
| Basic static load rating | 12150 N |
The theoretical maximum driving force was calculated based on the motor torque and ball-screw lead:
$$
F_{max} = \frac{2\pi \cdot T_{max} \cdot \eta}{p} = \frac{2\pi \times 1.11 \times 0.9}{0.002} = 3136 \text{ N}
\tag{26}$$
Limited by the axial basic dynamic load rating of the bearing section (1380 N), the practical maximum driving force was capped at 1380 N to ensure adequate service life. The maximum available slider velocity was:
$$
v_{max} = \frac{n_{max} \cdot p}{60} = \frac{6000 \times 0.002}{60} = 0.2 \text{ m/s}
\tag{27}$$
which exceeds the required 0.08 m/s identified from human walking data.
5.2 Key Component Design
5.2.1 Spherical Support Joint and Moving Platform
The spherical support joint constitutes the core of the bionic ankle joint mechanism, providing static unloading and enhancing translational stiffness. The conventional ball joint cover was eliminated to achieve the required 45° maximum deflection angle. Instead, a support roller mechanism was designed below the arc’s inner surface to constrain the ball socket and ball head into continuous contact. The ball head rod connects to the fixed platform base through a connection bracket, with adjustment shims ensuring that the sphere center coincides with the arc’s inner surface center. The ball socket is secured to the ball hinge seat, which connects to the arc through symmetric arc shafts whose axes pass through the sphere center.
The moving platform assembly comprises the ball socket, ball hinge seat, foot plate, and rubber sole. This assembly can rotate about the arc shafts while also following the arc’s rotational motion, providing the required two degrees of freedom.
5.2.2 Arc Support and Clamping Mechanism
To accommodate machining errors and deformation of the open-loop arc structure, the support roller was designed as a crankshaft configuration. The shaft neck installs onto the fixed platform base, while the crank pin supports double-row deep groove ball bearings. After installation, rotating the adjustment plate drives the crankshaft rotation to eliminate clearance between the bearing outer surface and the arc’s inner surface, ensuring intimate contact between the ball socket and ball head. Flanged bearings were incorporated on both sides of the arc to reduce frictional resistance while restricting rotation about the Z-axis, thereby enhancing the stiffness of the rolling branch.
5.2.3 Side Branches
The pitching branch consists of the heel shaft, lower U-joint body, fork head, pitching tie-rod, rear bracket, and upper U-joint body. The rear bracket connects to the pitching module slider via screws, while the heel shaft connects to the ball hinge seat of the moving platform with a transition fit secured by set screws. The rolling branch comprises the side bracket, rolling tie-rod, arc, and arc shafts, with two symmetric arc shafts connecting to the ball hinge seat.
5.3 Assembly and Commissioning
The complete prototype comprises 227 components. Prior to final assembly, the core components including the fixed platform leg, ball hinge seat, arc, and arc support mechanisms were assembled and commissioned separately. The commissioning process involved adjusting shims and support rollers to eliminate clearance and testing the motion performance. Design improvements were implemented based on commissioning results, particularly the addition of flanged bearings to reduce friction. Following successful commissioning, the complete assembly was performed according to the planned sequence, progressing from inner to outer and from difficult to easy components. The initial posture was defined with the foot plate maintained horizontal, verified using a high-sensitivity bubble level.
6. Control System Development
6.1 Control Architecture Selection
The control system for the bionic ankle joint was designed using a PC + multi-axis motion control card + AC servo unit architecture to achieve precise position and velocity control. The UMAC (Universal Motion and Automation Controller) from Delta Tau was selected as the core motion controller, featuring an open-structure platform based on the Motorola DSP56303 processor. The control system adopted position control mode with absolute encoder feedback to ensure high positioning accuracy and robust anti-interference capability.
| Component | Model | Function |
|---|---|---|
| Motion controller | UMAC | Multi-axis motion control and interpolation |
| Axis expansion card | ACC-24E2S | Motor channel control |
| Digital I/O interface | ACC-65E | Input/output signal processing |
| Absolute encoder interface | ACC-57EY | Encoder feedback reception |
| Servo driver | SGD7S-R90A | Servo motor driving |
| Servo motor | SGM7J-01A7C6S | Actuation |
| Limit switches | APM-D3B1-003 | Over-travel protection |
6.2 Servo Parameter Setting
Before commissioning, key servo driver parameters were configured through the panel as specified in the Yaskawa manual:
| Parameter | Setting | Description |
|---|---|---|
| Pn000 = n.xx1x | Position control mode | Control mode selection |
| Pn001 = n.x0xx | AC power input | Power source |
| Pn00B = n.x1xx | Single-phase AC input | Phase configuration |
| Pn500 = n.xx0x | CCW positive | Rotation direction |
| Pn50A = n.2xxx | Over-travel protection enabled | Limit function |
| Pn001 = n.xx00 | Dynamic brake | Stop method |
| Pn20E = 64, Pn210 = 1 | Electronic gear 64:1 | Gear ratio |
| Pn200 = n.xxx0 | Pulse + sign | Command pulse train format |
| Pn212 = 2048 | Encoder output pulses | Feedback pulses |
For the UMAC controller, the I-variable settings were established for motor channel assignment, command output address, and feedback mode. The command output was configured as pulse + direction with CCW positive rotation. The servo loop parameters were tuned using the PMAC Tuning Pro2 software, beginning with automatic PID tuning followed by manual refinement to optimize performance.
| Gain Parameter | I-Variable | Function |
|---|---|---|
| Proportional gain (\(K_P\)) | Ixx30 | Minimize position error |
| Derivative gain (\(K_D\)) | Ixx31 | Damping effect |
| Velocity feedforward (\(K_{vff}\)) | Ixx32 | Reduce following error |
| Integral gain (\(K_I\)) | Ixx33 | Eliminate steady-state error |
| Integrator mode | Ixx34 | Integrator activation |
| Acceleration feedforward (\(K_{aff}\)) | Ixx35 | Compensate acceleration error |
| Notch filter parameters | Ixx36-Ixx39 | Resonance cancellation |
6.3 Communication and Testing
Communication between the PC and UMAC was established via Ethernet with IP address 192.6.94.5. The PEWIN32PRO2 software was installed to enable parameter configuration, status monitoring, and program development. Hardware state testing verified encoder feedback functionality, limit switch operation, servo unit activation, and motor rotation direction using open-loop tests. The open-loop test confirmed that the encoder feedback increased when the command direction was positive, validating the correct configuration of the direction-related parameters.
7. Experimental Studies
7.1 Workspace Verification
Following final assembly and control system commissioning, comprehensive experimental studies were conducted to verify the performance of the bionic ankle joint prototype. The workspace validation demonstrated that the prototype achieves the following extreme motion ranges, all meeting or exceeding the corresponding limits of the human ankle joint:
| Motion | Direction | Maximum Angle | Human Ankle Limit |
|---|---|---|---|
| Pitching | Plantarflexion | 45° | 45° |
| Pitching | Dorsiflexion | 30° | 30° |
| Rolling | Eversion | 25° | 25° |
| Rolling | Inversion | 45° | 45° |
The workspace was verified to be continuous without voids, confirming complete coverage of the human ankle joint motion envelope.
7.2 Motion Function Validation
With the absolute position encoder initialized at the level foot plate orientation, the prototype was programmed to execute rolling motion, pitching motion, and compound motions designed to replicate natural human ankle movements. The experimental observations are summarized as follows:
- The bionic ankle joint prototype exhibited smooth and continuous motion throughout all tested trajectories.
- The rolling motion demonstrated independence across the full range of pitch angles, maintaining a range of [-25°, 45°].
- The pitching motion exhibited partial coupling with the rolling motion; as the roll angle increased, the pitching motion range correspondingly decreased.
- This coupling-induced workspace reduction characteristic aligns with the natural behavior of the human ankle joint, further validating the bionic design approach.
The control system demonstrated excellent performance, achieving the intended precision in reproducing the prescribed motion trajectories. Motion programs and PLC control routines were developed using UMAC programming language, enabling autonomous execution of rolling, pitching, and compound motion sequences that resemble the motions of the human ankle joint during daily activities.
8. Conclusions
This research has systematically addressed the design, analysis, prototyping, and control of a parallel bionic ankle joint for humanoid robots. The following conclusions can be drawn from this comprehensive investigation:
- Four novel bionic ankle joint configurations were proposed, each corresponding to a two-degree-of-freedom spherical parallel mechanism. These configurations retain the advantages of high stiffness and static unloading while offering compact structures and convenient actuator placement, thereby enriching the configuration options for bionic humanoid robot joints.
- The kinematic analysis established the forward/inverse position solutions and Jacobian matrix for the selected linear-driven configuration. The mechanism exhibits semi-decoupled kinematic characteristics, wherein rolling motion depends solely on the rolling slider input. The inverse position solution was validated through MATLAB-ADAMS comparative simulation with relative errors below 1%.
- The dynamic model, established through the Lagrange formulation and virtual work principle, accurately predicts actuator forces with relative errors below 5% compared to ADAMS simulations. A practical method for determining mass and inertia properties of complex components within the SolidWorks assembly environment was proposed, offering simplicity and effectiveness.
- A functional bionic ankle joint prototype was successfully developed and assembled, incorporating a novel spherical support joint design that achieves 45° deflection without requiring bulky joint covers. The workspace and motion characteristics of the prototype fully satisfy the functional requirements of the human ankle joint, providing a physical platform for further research on control strategies and performance optimization.
- The PC + UMAC + AC servo control architecture proved effective for the bionic ankle joint system. Systematic parameter tuning, including PID gain adjustment and feedforward compensation, resulted in stable and smooth motion reproduction of natural human ankle movements.
The experimental results confirm that the parallel bionic ankle joint represents a viable and promising approach for advancing the state of the art in humanoid robot joint design. Future work should focus on error modeling and calibration to enhance positioning accuracy, implementation of dynamic model-based control for improved dynamic performance, and exploration of hydraulic and redundant actuation configurations to further increase the load-carrying capability of the bionic ankle joint.