The research presented in this dissertation focuses on the performance analysis and prototype development of a novel 3PSS/S parallel mechanism designed to serve as a shoulder joint for a humanoid robot. The humanoid robot shoulder joint is one of the most critical components, directly influencing the overall mobility, load-bearing capacity, and kinematic coordination of the entire robotic system. In contrast to traditional serial mechanisms commonly found in existing humanoid robots, parallel mechanisms offer superior stiffness, structural stability, and load capacity, which makes them an excellent candidate for bionic joint applications. This study systematically covers the kinematic analysis, static and dynamic force analyses, mechanical design, control system integration, and experimental validation of the proposed shoulder joint mechanism.
From a structural perspective, the human shoulder is a complex multi-degree-of-freedom (DOF) system driven by multiple muscles in parallel. Traditional series-driven robotic joints fail to replicate both the anatomical structure and functional characteristics of the human shoulder. Therefore, adopting a parallel mechanism as the fundamental architecture for a humanoid robot shoulder joint is both structurally and functionally more faithful to biological reality. This work addresses the growing need for high-performance, humanoid-compatible joints that can meet the stringent requirements of dynamic operation, compact packaging, and high load-to-weight ratios.
Mechanism Description and Coordinate System
The proposed 3PSS/S parallel mechanism consists of a fixed base platform, a moving platform, three symmetric PSS (Prismatic-Spherical-Spherical) kinematic chains, and a central spherical joint. The center of the central spherical joint coincides with the geometric center of the moving platform. The mechanism provides three rotational degrees of freedom about the center of the moving platform, mimicking the rotational capability of the human shoulder. A fixed coordinate frame H-XYZ is established at point H, which is the intersection of the axes of the three prismatic joints on the base. The Z-axis points vertically upward along the line HO₁. A moving coordinate frame O₁-uvw is attached to the moving platform, with its initial orientation aligned with the fixed frame.
The key geometric parameters of the mechanism include the angle ε between the inclined guide rails and the vertical direction, the circumradius l₂ of the equilateral triangle formed by the spherical joints on the moving platform, the fixed link length l of each side chain, and the vertical distance h between the center of the central spherical joint O₁ and point H. The position vectors of the base hinge points Aᵢ and the platform hinge points Bᵢ can be expressed in their respective coordinate frames as:
$$ \mathbf{A}_i^H = \begin{bmatrix} S_i \sin \epsilon \cos \eta_i \\ S_i \sin \epsilon \sin \eta_i \\ -S_i \cos \epsilon \end{bmatrix}, \quad (i=1,2,3) $$
where Sᵢ is the distance from the origin H to hinge point Aᵢ along the inclined guide rail, and ηᵢ = 2(i−1)π/3 is the azimuth angle of each chain. Similarly, the position vectors of the platform hinge points in the moving frame are:
$$ \mathbf{B}_{i}^{O_1} = \begin{bmatrix} l_2 \sin \theta_i’ \cos \eta_i \\ l_2 \sin \theta_i ‘ \sin \eta_i \\ l_2 \cos \theta_i ‘ \end{bmatrix}, \quad (i=1,2,3) $$
Since all spherical joints on the moving platform are coplanar, the angle θ′ᵢ equals π/2 for all chains.
Kinematic Analysis
The orientation of the moving platform is described using ZYX Euler angles (α, β, γ). The rotation matrix from the moving frame to the fixed frame is expressed as:
$$ \mathbf{R}_o^H = \mathbf{R}_Z(\alpha)\mathbf{R}_Y(\beta)\mathbf{R}_X(\gamma) $$
which expands to:
$$ \mathbf{R}_o^H = \begin{bmatrix} c\alpha c\beta & c\alpha s\beta s\gamma – s\alpha c\gamma & c\alpha s\beta c\gamma + s\alpha s\gamma \\ s\alpha c\beta & s\alpha s\beta s\gamma + c\alpha c\gamma & s\alpha s\beta c\gamma – c\alpha s\gamma \\ -s\beta & c\beta s\gamma & c\beta c\gamma \end{bmatrix} $$
where s and c denote sine and cosine functions respectively. By imposing the constraint that the length of each connecting link remains constant during motion, the inverse position solution can be derived. The constraint equation is given by |AᵢBᵢ| = l, which yields a quadratic equation in Sᵢ, leading to the inverse kinematics solution:
$$ S_i = \sqrt{a_i^2 + b_i^2 + e_i^2 – l^2} + \frac{a_i m_{i1} + b_i m_{i2} – e_i m_{i3}}{P_i} $$
where the intermediate parameters are defined as:
$$ P_i = \sqrt{(a_i m_{i1} + b_i m_{i2} – e_i m_{i3})^2 – (a_i^2 + b_i^2 + e_i^2 – l^2)} $$
with m₁ᵢ = sin ε cos ηᵢ, m₂ᵢ = sin ε sin ηᵢ, m₃ᵢ = cos ε, and eᵢ = cᵢ + h. The structural parameters used throughout this study are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Side chain link length | l | 96 mm |
| Moving platform circumradius | l₂ | 47.3 mm |
| Moving platform mass | m₀ | 0.21 kg |
| Side chain link mass | mₗ | 0.12 kg |
| Slider mass | mₕ | 0.05 kg |
| Distance O₁H | h | 9.6 mm |
| Structure angle | ε | π/6 rad |
Velocity Jacobian Analysis
Differentiating the inverse position equation with respect to time yields the velocity relationship between the actuated joint rates and the moving platform angular velocities. The velocity equation can be written in matrix form:
$$ \mathbf{D} \begin{bmatrix} \dot{S}_1 \\ \dot{S}_2 \\ \dot{S}_3 \end{bmatrix} = \mathbf{G} \begin{bmatrix} \dot{\alpha} \\ \dot{\beta} \\ \dot{\gamma} \end{bmatrix} $$
where D is a diagonal matrix with entries Dᵢᵢ = Pᵢ, and G is a full 3×3 matrix whose elements involve partial derivatives of the chain parameters with respect to the orientation angles. The overall Jacobian matrix J maps the moving platform angular velocity to the actuated joint rates:
$$ \mathbf{J} = \mathbf{D}^{-1}\mathbf{G} $$
To verify the correctness of the Jacobian derivation, a numerical example was conducted using MATLAB, while the same motion was simulated in ADAMS with the identical structural parameters. The moving platform was commanded to rotate with angular velocities of α̇ = 0, β̇ = 0, and γ̇ = 10π/180 rad/s from the initial pose. The resulting slider velocities from both analyses are compared with their corresponding time histories shown in agreement with each other, confirming the accuracy of the Jacobian formulation.
Acceleration Analysis
Differentiating the velocity equation with respect to time yields the acceleration relationship for each actuated slider:
$$ \ddot{S}_i = \frac{1}{P_i}\left(G_{i1}\ddot{\alpha} + G_{i2}\ddot{\beta} + G_{i3}\ddot{\gamma}\right) + \frac{1}{P_i}\left( \frac{dG_{i1}}{dt}\dot{\alpha} + \frac{dG_{i2}}{dt}\dot{\beta} + \frac{dG_{i3}}{dt}\dot{\gamma} \right) – \frac{\dot{P}_i}{P_i^2}\left(G_{i1}\dot{\alpha} + G_{i2}\dot{\beta} + G_{i3}\dot{\gamma}\right) $$
The angular velocity and angular acceleration of each side chain link can be obtained from the velocity and acceleration of the platform hinge points. For point Bᵢ, its velocity and acceleration can be expressed in terms of the moving platform motion or, alternatively, by propagating through the parallel chain:
$$ \mathbf{v}_{B_i} = \mathbf{v}_{A_i} + \boldsymbol{\omega}_{AB_i} \times \mathbf{AB}_i $$
$$ \mathbf{a}_{B_i} = \mathbf{a}_{A_i} + \boldsymbol{\alpha}_{AB_i} \times \mathbf{AB}_i + \boldsymbol{\omega}_{AB_i} \times (\boldsymbol{\omega}_{AB_i} \times \mathbf{AB}_i) $$
Solving these vector equations leads to the angular velocity and angular acceleration of each side link, which serve as critical inputs for the subsequent dynamic analysis. A numerical validation comparing MATLAB computation results with ADAMS simulations for a case where the platform accelerates with γ̈ = 5π/180 rad/s² confirmed that the derived acceleration equations are correct, as the slider acceleration profiles from both approaches overlap.
Singularity Analysis
Singularity analysis is essential for identifying configurations where the mechanism may lose controllability or exhibit degraded performance. Based on the classification proposed by Gosselin and Angeles, singularities are categorized into input singularities (det(D) = 0), output singularities (det(G) = 0), and combined structural singularities (both matrices singular). For the 3PSS/S mechanism, the matrix D is always non-singular within the design workspace, indicating the absence of input singularities. However, the determinant of matrix G can vanish under certain orientation conditions, leading to output singularities.
Using numerical searching techniques within an extended orientation range of α ∈ (−80°, 80°), β ∈ (−80°, 80°), and γ ∈ (−160°, 160°), the singularity locus was identified, as summarized in Table 2. The results show that the singularity occurs when α = 0, γ = 0, for specific values of β. Substituting these configuration parameters into the determinant of G verified that det(G) = 0, confirming that the search results are valid.
| Point 1 | Point 2 | Point 3 | Point 4 | Point 5 | |
|---|---|---|---|---|---|
| α | 0° | 0° | 0° | 0° | 0° |
| β | −32° | −14° | 0° | 20° | 26° |
| γ | 0° | 0° | 0° | 0° | 0° |
These singular configurations should be avoided during trajectory planning to ensure stable and reliable operation of the humanoid robot shoulder joint.
Static Force Analysis
Static analysis of the parallel shoulder joint provides crucial insights into its load-bearing capability and the force distribution among the various chains. The mechanism is designed such that the central passive chain can offload a significant portion of the external forces from the active side chains, thereby reducing the driving forces required and increasing the overall load capacity.
For the static analysis, all friction forces in the joints are neglected, and the external load is assumed to be an arbitrary six-dimensional force vector comprising a force F and a moment M applied at the center of the moving platform. The force balance equation for the moving platform is:
$$ \sum_{i=1}^{3} \mathbf{F}_{B_i} + \mathbf{F}_N + \mathbf{F}_w + \mathbf{G}_o = \mathbf{0} $$
and the moment balance equation about the platform center O₁ is:
$$ \sum_{i=1}^{3} \mathbf{O}_1\mathbf{B}_i \times \mathbf{F}_{B_i} + \mathbf{M}_w = \mathbf{0} $$
For each side chain link, the force equilibrium equation is:
$$ \mathbf{F}_{A_i}’ + \mathbf{F}_{B_i}’ + \mathbf{G}_l = \mathbf{0} $$
where the prime denotes the reaction force exerted on the link. Taking moments about point Aᵢ yields additional equations that can be simplified to express the constraint forces in terms of the link orientation and the applied loads.
For the actuated slider, the force balance equation projected along the direction of the prismatic joint provides the driving force Fₘᵢ:
$$ F_{A_i}’\cos\phi_i + F_{A_i}’\cos\varphi_i + F_{A_i}’\cos\psi_i + m_h g \cos\psi_i – F_{mi} = 0 $$
Solving the complete set of linear equations yields all constraint forces and driving forces. Numerical simulations were conducted for various loading scenarios, including pure force loading, pure moment loading, and combined loading. Table 3 summarizes the six loading cases used in the static analysis.
| Case | Orientation α | Force F (N) | Moment M (N·m) |
|---|---|---|---|
| 1 | −π/6 | (6, 4, 7) | (0, 0, 0) |
| 2 | π/3 | (6, 4, 7) | (0, 0, 0) |
| 3 | −π/6 | (0, 0, 0) | (6, 5, 3) |
| 4 | π/3 | (0, 0, 0) | (6, 5, 3) |
| 5 | −π/6 | (5, 4, 6) | (4, 5, 7) |
| 6 | π/3 | (5, 4, 6) | (4, 5, 7) |
The analysis results reveal that for pure force loading (Cases 1 and 2), the central chain force F′_N remains nearly constant at approximately 13.01 N regardless of orientation changes, while each side chain force F′_Bi stays close to 0.588 N. In this scenario, the central chain carries about 22 times the load of each side chain, demonstrating a significant force-offloading effect. For pure moment loading (Cases 3 and 4), the central chain force varies between 3.530 N and 4.389 N depending on orientation, while the side chain forces range from 0.330 N to 0.890 N, with the central chain still absorbing the majority of the load. For combined loading (Cases 5 and 6), the central chain force ranges from 11.296 N to 12.935 N, while side chain forces remain below 1.162 N, confirming that the central chain consistently offloads more than 90% of the external load across all tested configurations.
Dynamic Analysis Using Newton-Euler Method
The dynamic analysis of the 3PSS/S parallel shoulder joint is essential for trajectory optimization, motion control, and evaluating the dynamic performance of the humanoid robot shoulder. The Newton-Euler method is adopted in this work because it provides both the driving forces and the constraint forces between components, offering complete information for the mechanism design and control system development.
Inertia Tensors
Before establishing the dynamic equations, the inertia tensors of the side chain links and the moving platform are computed in the fixed reference frame. For each side chain link, the inertia tensor can be transformed from its local center-of-mass frame to the fixed frame using:
$$ \mathbf{I}_{AB_i}^H = \mathbf{R}_E^H \cdot \mathbf{I}_{AB_i}^E \cdot (\mathbf{R}_E^H)^T + m_l \, \tilde{\mathbf{E}}_i’ $$
where the local inertia tensor of the cylindrical link about its center of mass is:
$$ \mathbf{I}_{AB_i}^E = \text{diag}\left(\frac{m_l l^2}{12}, \frac{m_l l^2}{12}, \frac{m_l r^2}{2}\right) $$
Similarly, the inertia tensor of the moving platform about its geometric center in the fixed frame is:
$$ \mathbf{I}_o^H = \mathbf{R}_{O_1}^H \cdot \mathbf{I}_o^{O_1} \cdot (\mathbf{R}_{O_1}^H)^T + m_o \, \tilde{\mathbf{O}}_1′ $$
with the local inertia tensor of the disk-shaped platform given by:
$$ \mathbf{I}_o^{O_1} = \text{diag}\left(\frac{m_o r_o^2}{4}, \frac{m_o r_o^2}{4}, \frac{m_o r_o^2}{2}\right) $$
Dynamic Equilibrium Equations
Applying Newton’s second law to the moving platform yields the force balance equation:
$$ \sum_{i=1}^{3} \mathbf{F}_i + \mathbf{F}_N + \mathbf{F}_w + \mathbf{G}_o = m_o \mathbf{a}_o $$
where a₀ is the acceleration of the platform center of mass. The corresponding Euler equation for rotational motion about the center of mass is:
$$ \sum_{i=1}^{3} \mathbf{O}_1\mathbf{B}_i \times \mathbf{F}_i + \mathbf{M}_w = \mathbf{I}_o^H \cdot \dot{\boldsymbol{\omega}}_o $$
For each side chain link, Newton’s equation takes the form:
$$ \mathbf{F}_i” + \mathbf{F}_i’ + \mathbf{G}_l = m_l \dot{\mathbf{v}}_{AB_i} $$
and the Euler equation about the link center of mass is:
$$ \mathbf{E}_i\mathbf{B}_i \times \mathbf{F}_i’ + \mathbf{E}_i\mathbf{A}_i \times \mathbf{F}_i” = \mathbf{I}_{AB_i}^H \cdot \dot{\boldsymbol{\omega}}_{AB_i} $$
Finally, for the actuated slider along the prismatic joint direction:
$$ \mathbf{F}_i”’ \cdot \mathbf{S}\mathbf{A}_i + m_h \mathbf{G} \cdot \mathbf{S}\mathbf{A}_i – F_{mi} = m_h \dot{\mathbf{v}}_{A_i} \cdot \mathbf{S}\mathbf{A}_i $$
Numerical Results
The complete set of dynamic equations was solved numerically using MATLAB, with a prescribed external load of F = (−3, −4, −5) N and M = (−6, −4, 7) N·m. Three different motion patterns were analyzed:
- Rotation about the u-axis: α = π/3, β = π/4, γ = 0.2sin(πt) + 0.6
- Rotation about the v-axis: α = π/3, β = 0.2sin(πt) + 0.6, γ = π/4
- Rotation about the w-axis: α = 0.2sin(πt) + 0.6, β = π/3, γ = π/4
The simulation results demonstrate that the required driving forces and constraint forces vary periodically with time, tracking the oscillatory motion input. The central chain constraint force F′_N consistently remains at values above 9.3 N, which is more than ten times the side chain constraint forces. This confirms that the central chain provides a strong force-offloading effect even during dynamic operation, allowing the mechanism to achieve the desired rotational motion with relatively small driving forces.
Table 4 summarizes the ranges of forces observed for each motion pattern during the dynamic analysis.
| Rotation Axis | Driving Force Fₘ (N) | Side Chain Constraints F′ᵢ (N) | Central Chain Force F′_N (N) |
|---|---|---|---|
| u-axis | 1.02–1.18 | 0.34–0.84 | 9.65–9.76 |
| v-axis | 1.02–1.22 | 0.29–0.90 | 9.65–9.70 |
| w-axis | 0.80–2.20 | 0.31–1.80 | 9.31–9.72 |
Prototype Development
The mechanical design of the 3PSS/S parallel shoulder joint prototype involves careful selection of the actuation system, design of the central and side chains, and fabrication of the moving and fixed platforms. The prototype was developed with the goal of achieving a compact structure that can replicate the motion range and load capacity of the human shoulder while maintaining a lightweight design suitable for a humanoid robot.
Actuation System Selection
After comparing various linear actuation technologies, including gear-rack drives, belt drives, ball-screw assemblies, linear modules, and direct linear motors, the servo motor combined with a ball-screw linear module was selected as the most appropriate drive configuration. This choice was made based on considerations of precision, load capacity, compactness, and cost-effectiveness. Among ball-screw linear modules, the THK motor-side-mounted intelligent guide unit was chosen, as it provides a compact transverse configuration that minimizes the overall footprint of the shoulder joint mechanism.
The ball-screw selection process involved several verification steps:
- Precision class: C7 with ±0.05 mm/300 mm accuracy, satisfying the required ±0.02 mm positioning accuracy over a 50 mm stroke
- Lead selection: Pₕ ≥ 0.75 mm based on maximum speed of 0.05 m/s and motor rated speed of 4000 rpm
- Outer diameter: initially selected as 5 mm and 6 mm options based on axial clearance requirements
- Maximum axial load: Fₐ = 53.95 N as calculated from the worst-case acceleration condition
- Allowable axial load: P₁ = 2313.64 N for the 5 mm diameter screw, which is much larger than the actual load
- Critical speed: verified through both the screw critical speed and DN value criteria
- Nut load capacity: selected MIF0502-3.7 type with a maximum allowable axial load of 627 N
The servo motor was selected based on the required torque, speed, and inertia matching. The calculation process yielded:
- Total inertia referenced to the motor shaft: J_L = 9.89 × 10⁻⁷ kg·m²
- Peak torque required: T_K = 0.0259 N·m
- RMS torque required: T_rms = 0.0132 N·m
- Minimum encoder resolution: B ≥ 200 pulses/rev
Based on these calculations, the SGMMV series micro servo motor from Yaskawa was selected, which has a rated torque of 0.0637 N·m, maximum torque of 0.191 N·m, and rotor inertia of 4.66 × 10⁻⁷ kg·m², satisfying all the design requirements with sufficient safety margins.
Central Chain Design
The central spherical joint is the most critical component of the 3PSS/S mechanism, as it determines the rotational workspace of the moving platform. A novel multi-lobe spherical joint was designed, as shown in Figure 1, with improvements made to reduce the external dimensions while increasing the structural strength and rotational range. The improved central spherical joint consists of a ball seat, rotating socket, a cap with three radial slots, and a spherical head rod. The cap and the socket are connected via angular contact bearings, allowing the head rod to move through the slots without interference. This design achieves a maximum swing angle of 50°, significantly larger than the typical 30° of conventional spherical joints, thereby enabling a large rotational workspace for the humanoid robot shoulder joint.
Side Chain Design
Each PSS side chain includes a prismatic joint, a connecting rod, and two universal joints. To achieve the functionality of a spherical joint while maintaining a compact footprint and high load capability, each spherical joint is realized by combining a universal joint (Hooke joint) with an additional revolute joint. This composite configuration provides the required three rotational degrees of freedom while allowing a larger angular range and more robust bearing capacity. The lower universal joint connects to the slider via a bracket, and the upper universal joint connects to the moving platform.
Platform Design
The moving platform is manufactured from aluminum alloy LY12, which provides a good balance between strength, stiffness, and mass reduction. Its circular disk design includes three equally spaced mounting slots on the periphery for attachment of the upper universal joints, a central bore for connecting the central spherical joint, and additional mounting holes for potential end effectors.
The fixed platform features a frustum-shaped design with three inclined guide surfaces positioned at a 60° angle relative to the horizontal plane. The centerlines of these inclined surfaces intersect at a single point, which provides the correct geometric configuration for the parallel mechanism. The THK intelligent guide units are mounted directly onto these inclined surfaces, with the servo motors positioned laterally to minimize the overall height and footprint of the prototype. The material selected for the fixed platform is also LY12 aluminum alloy, and wire-cut electric discharge machining was used during fabrication to ensure that the three guide surface centerlines intersect precisely at a single point.
Using CATIA software, all components were modeled in 3D and assembled into a complete digital prototype of the shoulder joint mechanism. The assembly was then converted into engineering drawings and entrusted to a precision machining facility for manufacturing and assembly. During assembly, careful attention was paid to cleaning all components, ensuring smooth motion of the joints, correctly pairing the angular contact bearings, and verifying the proper installation of the cross blocks and pins in the universal joints. The completed physical prototype of the 3PSS/S parallel humanoid robot shoulder joint is shown in the associated figures.
Control System Development
The control system for the 3PSS/S parallel shoulder joint prototype is designed to provide precise position and velocity control of the three actuated prismatic joints, enabling the moving platform to perform coordinated rotational motions. Given the coupled nature of parallel mechanisms, the control system must be capable of synchronizing the three axes accurately to generate the desired orientation trajectories.
Control Architecture
The control system adopts a closed-loop position control architecture centered around the Yaskawa MP2300S machine controller. The MP2300S is a high-performance programmable controller that supports MECHATROLINK-II motion networks for synchronized multi-axis control. Key features of the control system include:
- Three Yaskawa SGDV-R90A11A servo units with MECHATROLINK-II communication
- Three SGMMV micro servo motors with 17-bit absolute encoders
- Position feedback via high-resolution encoders for precise orientation tracking
- Limit switches at each actuator end for safety protection
- Emergency stop and braking circuits for safe operation
- MPE720 ver.6 software for programming and monitoring
The overall control system architecture comprises a host computer, the machine controller, servo units, servo motors, and displacement sensors. Communication between the MP2300S and the servo units is achieved via MECHATROLINK-II, enabling real-time synchronous control of all three axes at communication cycle times as short as 250 μs.
Hardware Design
Hardware interconnections were carefully designed to ensure signal integrity and operational safety. The servo motor encoder cables use shielded twisted-pair wires to minimize noise interference, and the brake cables are connected through relay circuits controlled by the servo-ON signal. When the servo-ON signal is activated, the relay closes, supplying the 24 V DC power to release the electromagnetic brake, allowing motor motion. The connection between the machine controller and each servo unit includes pulse-train and direction signals for positioning control.
Each displacement sensor is a NPN-type three-wire photoelectric proximity sensor operating at 24 V DC. The normally closed sensor outputs are connected to the P-OT and N-OT inputs of the corresponding servo drive, enabling immediate forced stop of the motor when the slider moves beyond the safe travel range.
The control cabinet was designed to serve as both the motion platform and the housing for all electrical components. The cabinet layout, arranged over four vertical layers, includes a surge protector, main circuit breakers, power supply breakers, electromagnetic contactors, and a terminal block layer. The lower section contains the machine controller and the three servo units. The completed cabinet integrates all power and signal wiring in an organized and accessible manner, ensuring maintainability and ease of troubleshooting.
System Simulation and Integration
Before assembling the final system, a simulation setup was built to verify the correctness of the control system design. Using the SigmaWin+ parameter tuning tool, the servo unit parameters were adjusted, including station numbers, servo parameters, and JOG test settings. The execution of JOG and step operations confirmed that the servo motors respond correctly to commands and that the feedback signals are properly processed by the control electronics.
The servo parameter configuration steps included:
- Setting the SW2 switches on each servo unit to assign station numbers 1, 2, and 3
- Using the automatic configuration feature during power-up to establish initial communication
- Configuring the computer IP address and launching MPE720 software
- Setting Pn002 parameter bit 2 to 1 for single-phase 220 V input operation
- Assigning P-OT and N-OT input signals to limit switch connections
- Configuring JOG test parameters (Pn530–Pn536) for trial runs
Once the servo parameters were correctly configured, the prototype system was fully integrated with the control cabinet, and the functionality validation experiments were initiated.
Experimental Validation
Experimental validation of the 3PSS/S parallel humanoid robot shoulder joint prototype was conducted in two phases: functional verification and force feedback experiments. These experiments aimed to confirm the reliability of the mechanical system, the correctness of the control system, and the validity of the theoretical force analysis.
Functional Verification
The functional verification process began with setting the fixed axis parameters in the Module Configuration module of MPE720 software. The reference unit was configured for millimeters, the pitch was set to 20 mm, the electronic gear ratio to 1:1, and the axis type set to finite length. After completing the configuration and performing trial runs, the following sequential tests were executed:
- Axis parameter validation via three-axis synchronized test operations
- Limit switch calibration and verification of protective stop functions
- Homing program testing for each axis
- Point-to-point positioning program testing
- Electronic gearing program testing
- Finally, the trajectory plan of the moving platform was executed based on the inverse kinematic solution computed for a sequence of desired platform orientations
With the trajectory program downloaded to the MP2300S controller, the prototype successfully tracked the planned large-angle rotational motions of the moving platform, as recorded in the associated photographs. This confirms that the mechanical design, control system hardware, and software algorithms function together correctly and reliably.
Force Feedback Experiments
The force feedback experiment was designed to validate the static force analysis of the 3PSS/S parallel shoulder joint mechanism. The experimental setup consisted of the prototype itself, a high-precision push-pull force sensor (ZP-100N), a micro strain force sensor (LC1104) installed on the central chain, a dynamic signal testing and analysis system (PH5956D), and the MP2300S machine controller.
During the experiments, external loads were applied to the moving platform center using the push-pull force sensor, while the central chain force was measured directly through the micro strain gauge sensor and recorded by the dynamic signal analyzer. The side chain constraint forces were computed indirectly from the motor torque data obtained through the servo system’s current monitoring. The moving platform was oriented at α = −π/6, β = 0, γ = 0 during all tests.
For the pure force loading experiment, external forces of 5 N, 10 N, 15 N, 20 N, and 25 N were applied sequentially along the Z-axis. The resulting measured and theoretical forces are compared in Table 5.
| Load (N) | Central chain F′_N exp. (N) | Central chain F′_N theory (N) | Side chain 1 exp. (N) | Side chain 1 theory (N) |
|---|---|---|---|---|
| 5 | 4.6 | 5.31 | 0.019 | 0.015 |
| 10 | 9.2 | 10.31 | 0.017 | 0.015 |
| 15 | 13.9 | 15.30 | 0.017 | 0.015 |
| 20 | 19.4 | 20.31 | 0.018 | 0.015 |
| 25 | 23.5 | 25.30 | 0.015 | 0.015 |
The average absolute error for the central chain force under pure force loading was 0.98 N, while the side chain force errors were around 0.002–0.006 N. In the compound loading test, a constant Z-axis force of 2 N was combined with Z-axis moments increased in steps of 0.5 N·m from 0.5 to 2.5 N·m. The experimental and analytical results show better consistency, with the central chain mean absolute error of 0.44 N and the side chain errors in the range of 0.019–0.021 N.
The slight discrepancies between the experimental measurements and theoretical predictions can be attributed to friction at the joints, small deviations in the actual point of load application, sensor precision limitations, and measurement methodology. Nevertheless, the experimental results demonstrate that the central chain of the 3PSS/S parallel shoulder joint mechanism effectively carries a dominant share of the external loads, confirming the significant static force-offloading capability of the mechanism. This validates the theoretical force analysis and supports the applicability of this parallel mechanism as a high-performance shoulder joint for humanoid robots.
Conclusion
This dissertation comprehensively investigated the 3PSS/S parallel mechanism as a shoulder joint for a humanoid robot, covering kinematics, statics, dynamics, prototype development, control system integration, and experimental validation. The following key conclusions were drawn:
- The kinematic model of the 3PSS/S parallel mechanism was established, including inverse position solutions, velocity Jacobian matrices, and acceleration equations, all of which were verified through both MATLAB computations and ADAMS simulations. Singularity analysis revealed that the mechanism exhibits output singularities when α = 0 and γ = 0 with specific β values, which should be avoided in trajectory planning.
- The static analysis demonstrated that the central passive chain bears a dominant portion of the external loads across all tested loading scenarios, from pure forces to combined loads. The force-offloading ratio consistently exceeds 78% for pure loads, effectively reducing the driving force requirements of the active chains and enhancing the overall load capacity of the humanoid robot shoulder joint.
- The dynamic analysis using the Newton-Euler method confirmed that the central chain provides significant force-offloading during dynamic motions as well, with central chain forces exceeding 9.3 N while side chain forces remain below 1.8 N for the tested motion patterns. This favorable force distribution enables the mechanism to achieve desirable rotational motion with relatively low driving forces.
- A fully functional prototype of the 3PSS/S parallel humanoid robot shoulder joint was developed, integrating a ball-screw drive system, a customized central spherical joint with a 50° swing angle, and a closed-loop servo control system based on the Yaskawa MP2300S machine controller. Functional verification experiments confirmed the reliability and precision of both the mechanical system and the control system.
- Force feedback experiments validated the static analysis results, with the central chain force measurements matching theoretical predictions within acceptable error margins (average absolute errors between 0.44 N and 0.98 N across different loading conditions). These results further reinforce the structural advantage of the mechanism for high load-to-weight ratio applications.
Future work will focus on kinematic calibration experiments to further enhance motion accuracy, the planning of more biologically faithful trajectories based on actual human shoulder movement patterns, and the investigation of dynamic performance under high-speed and heavy-load conditions to provide comprehensive technical support for the practical application of this parallel bionic shoulder joint in advanced humanoid robot systems.