The human shoulder represents one of the most complex and functionally versatile joints in the human body. For embodied robots, the design of a shoulder joint that truly mimics the natural range of motion, load-carrying capacity, and proprioceptive feedback of the biological shoulder remains a challenging task. Conventional serial-kinematic joints often suffer from low stiffness, limited payload, and poor dynamic performance. In this research, I combine the advantages of parallel mechanisms with the bionic requirements of an embodied robot shoulder. Specifically, I focus on a 3PSS/S parallel mechanism, which consists of three symmetric PSS limbs and a central spherical joint. This configuration provides three rotational degrees of freedom through a compact structure, making it highly suitable for an embodied robot shoulder joint. The present thesis systematically addresses kinematic analysis, static force analysis, dynamic modeling, prototype design, control-system development, and experimental validation. Throughout this work, the core objective is to establish a rigorous theoretical basis and practical engineering roadmap for the application of the 3PSS/S mechanism in future embodied robots.
1. Introduction and Research Background
The field of embodied robots has advanced rapidly in recent decades. Bipedal humanoids, prosthetic devices, exoskeletons, and service robots all rely on articulated joints that must provide both mobility and stability. Among all joints, the shoulder is arguably the most important for achieving dexterous manipulation and whole-body motion. The human shoulder is a multi-degree-of-freedom joint actuated by a complex set of muscles that act in parallel. This biological architecture naturally inspires the use of parallel mechanisms in robotics. Compared with serial mechanisms, parallel mechanisms offer high rigidity, high load capacity, low inertia, and superior accuracy. However, most existing humanoid robots still employ serial joints for simplicity. In this work I propose a parallel bionic shoulder based on the 3PSS/S topology. The mechanical structure consists of a fixed base plate, a moving platform, three identical PSS branches, and a central S branch. The central spherical joint constrains the moving platform to pure rotation about its center, while the three PSS limbs provide actuation. This design allows for a large orientation workspace and excellent force distribution characteristics, which are essential for an embodied robot interacting with its environment.
In this chapter I briefly review the state of the art in humanoid robot joints and parallel mechanism theory. Previous studies have shown that parallel mechanisms can be successfully used for wrists, ankles, hips, and waist joints. However, the shoulder joint imposes unique requirements in terms of workspace, singularity avoidance, and dynamic response. Therefore, a thorough investigation of the 3PSS/S mechanism is necessary. The main contributions of this thesis are: (1) deriving the forward and inverse kinematic models, including velocity and acceleration relations; (2) constructing a complete static model and verifying the load-unloading effect of the central limb; (3) establishing a Newton–Euler dynamic model to compute actuator forces and internal constraint forces; (4) designing and manufacturing a fully functional prototype; (5) developing a real-time control system based on an industrial motion controller; and (6) performing force feedback experiments to validate the theoretical models. The results demonstrate that the 3PSS/S parallel shoulder is a promising candidate for the next generation of embodied robots.
2. Kinematic Analysis of the 3PSS/S Parallel Shoulder Joint
2.1 Coordinate Systems and Position Inverse Solution
I first establish the geometric model of the 3PSS/S parallel shoulder. Figure in the beginning of this section illustrates the mechanism. The fixed coordinate frame \(H-XYZ\) is attached to the point \(H\), which is the intersection of the three actuator axes. The \(Z\)-axis points vertically upward along \(HO_1\), where \(O_1\) is the center of the moving platform and also the center of the central spherical joint. The moving coordinate frame \(O_1-uvw\) is rigidly attached to the moving platform. The initial orientation of the moving frame coincides with the fixed frame. The three actuators are placed on inclined guide rails at an angle \(\varepsilon\) relative to the vertical line. The distance from \(H\) to each actuator base point \(A_i\) is denoted by \(S_i\), which is the variable input of the \(i\)-th P joint.
The position vector of \(A_i\) in the fixed frame can be expressed as
$$
\mathbf{A}_i^H = \begin{bmatrix} S_i \sin\varepsilon \cos\eta_i \\ S_i \sin\varepsilon \sin\eta_i \\ -S_i \cos\varepsilon \end{bmatrix}, \quad i=1,2,3,
$$
where \(\eta_i = 2(i-1)\pi/3\). The points \(B_i\) are the spherical-joint centers on the moving platform. In the moving frame, their coordinates are
$$
\mathbf{B}_i^{O_1} = l_2 \begin{bmatrix} \sin\theta_i’ \cos\eta_i \\ \sin\theta_i’ \sin\eta_i \\ \cos\theta_i’ \end{bmatrix},
$$
where \(l_2\) is the radius of the circumcircle of the three \(B_i\) points, and \(\theta_i’=\pi/2\) for the present design.
The orientation of the moving platform is described by ZYX Euler angles \((\alpha,\beta,\gamma)\). The rotation matrix from the moving frame to the fixed frame is
$$
\mathbf{R} = \mathbf{R}_Z(\alpha)\mathbf{R}_Y(\beta)\mathbf{R}_X(\gamma),
$$
which expands to
$$
\mathbf{R} = \begin{bmatrix}
c_\alpha c_\beta & c_\alpha s_\beta s_\gamma – s_\alpha c_\gamma & c_\alpha s_\beta c_\gamma + s_\alpha s_\gamma \\
s_\alpha c_\beta & s_\alpha s_\beta s_\gamma + c_\alpha c_\gamma & s_\alpha s_\beta c_\gamma – c_\alpha s_\gamma \\
-s_\beta & c_\beta s_\gamma & c_\beta c_\gamma
\end{bmatrix}.
$$
Here \(s_\theta = \sin\theta\) and \(c_\theta = \cos\theta\). The position of \(B_i\) in the fixed frame is
$$
\mathbf{B}_i^H = \mathbf{O}_1^H + \mathbf{R} \mathbf{B}_i^{O_1},
$$
where \(\mathbf{O}_1^H = [0,0,h]^T\) and \(h\) is the length of \(HO_1\). Since each PSS limb has a constant link length \(l\), the constraint equation is
$$
\| \mathbf{B}_i^H – \mathbf{A}_i^H \| = l, \quad i=1,2,3.
$$
Expanding the above equations yields a quadratic equation in \(S_i\). Solving for \(S_i\), I obtain the inverse position solution:
$$
S_i = \frac{a_i m_1 + b_i m_2 – e_i m_3 \pm \sqrt{(a_i m_1 + b_i m_2 – e_i m_3)^2 – (a_i^2+b_i^2+e_i^2-l^2)}}{2},
$$
where \(m_1 = \sin\varepsilon \cos\eta_i\), \(m_2 = \sin\varepsilon \sin\eta_i\), \(m_3 = \cos\varepsilon\), and \(e_i = c_i + h\) with \(c_i\) being the \(Z\)-component of \(\mathbf{B}_i^H\). For the physical prototype only the branch with the positive sign before the square root satisfies the requirement \(S_i>0\). Thus the unique inverse solution is
$$
S_i = \frac{1}{2}\left( a_i m_1 + b_i m_2 – e_i m_3 + \sqrt{(a_i m_1 + b_i m_2 – e_i m_3)^2 – (a_i^2+b_i^2+e_i^2-l^2)} \right).
$$
2.2 Velocity Jacobian Matrix
Differentiating the inverse position solution with respect to time, I derive the relation between actuator velocities \(\dot{S}_i\) and angular velocities of the moving platform \(\dot{\boldsymbol{\omega}} = [\dot{\alpha},\dot{\beta},\dot{\gamma}]^T\). The result can be written in matrix form as
$$
\mathbf{D} \begin{bmatrix} \dot{S}_1 \\ \dot{S}_2 \\ \dot{S}_3 \end{bmatrix} = \mathbf{G} \begin{bmatrix} \dot{\alpha} \\ \dot{\beta} \\ \dot{\gamma} \end{bmatrix},
$$
where \(\mathbf{D}\) is a diagonal matrix with entries \(D_{ii} = P_i\), and \(\mathbf{G}\) is a \(3\times3\) matrix whose entries are functions of the orientation angles. The Jacobian matrix of the 3PSS/S mechanism is then defined as
$$
\mathbf{J} = \mathbf{D}^{-1} \mathbf{G}.
$$
For the numerical prototype, the following geometric parameters are used: \(l=96\) mm, \(l_2=47.3\) mm, \(h=9.6\) mm, and \(\varepsilon=\pi/6\). The mass properties are \(m_o=0.21\) kg for the moving platform, \(m_l=0.12\) kg for each limb, and \(m_h=0.05\) kg for each slider. Using MATLAB and ADAMS, I verified the correctness of the Jacobian matrix. For instance, when the moving platform performs a pure rotation with \(\dot{\gamma}=10\pi/180\) rad/s, the actuator velocities computed by both software tools match exactly. The Jacobian analysis is fundamental for singularity identification and subsequent dynamic modeling.
2.3 Acceleration Analysis
Acceleration analysis is essential for dynamic modeling using the Newton–Euler method. I differentiate the velocity equation once more with respect to time. The actuator acceleration is expressed as
$$
\ddot{S}_i = f_i(\alpha,\beta,\gamma,\dot{\alpha},\dot{\beta},\dot{\gamma},\ddot{\alpha},\ddot{\beta},\ddot{\gamma}),
$$
where \(f_i\) is a lengthy algebraic expression. The complete derivation is provided in the original thesis. I implemented this acceleration model in MATLAB and compared the results with ADAMS simulations. For example, when the moving platform begins from rest with angular acceleration \(\ddot{\gamma}=5\pi/180\) rad/s², the slider accelerations \(\ddot{S}_i\) obtained from MATLAB are in excellent agreement with ADAMS, confirming the correctness of the acceleration model.
2.4 Singularity Analysis
Singularity analysis is critical for safe operation and path planning. Based on the Jacobian matrices \(\mathbf{D}\) and \(\mathbf{G}\), I classify singular configurations according to Gosselin and Angeles:
- Input singularity occurs when \(\det(\mathbf{D})=0\) but \(\det(\mathbf{G})\neq0\). For the 3PSS/S mechanism, I found that \(\det(\mathbf{D})\neq0\) throughout the entire workspace, so no input singularities exist.
- Output singularity occurs when \(\det(\mathbf{D})\neq0\) but \(\det(\mathbf{G})=0\). I numerically searched for these configurations and found a singularity locus in the orientation workspace. The locus lies in the plane \(\alpha=0, \gamma=0\) with \(\beta\) varying between approximately \(-32^\circ\) and \(+26^\circ\). Table 2.1 lists some representative singular orientation points.
- Combined singularity would require both determinants to vanish, which does not happen for this mechanism.
| Point | \(\alpha\) | \(\beta\) | \(\gamma\) |
|---|---|---|---|
| 1 | 0° | -32° | 0° |
| 2 | 0° | -14° | 0° |
| 3 | 0° | 0° | 0° |
| 4 | 0° | 20° | 0° |
| 5 | 0° | 26° | 0° |
The orientation workspace of the prototype was found to be \(\alpha\in[-35.56°,67.05°]\), \(\beta\in[-32.73°,30.18°]\), and \(\gamma\in[-95.4°,50.34°]\). The singular locus lies inside the workspace; therefore, control trajectories must avoid the vicinity of this locus. This analysis provides an important safety constraint for the motion planning of the embodied robot.
3. Static Force Analysis
3.1 Static Equilibrium Equations
Static analysis aims at determining the force distribution among the central limb and the side limbs under a given external load. In this section I ignore friction in all joints. The external wrench acting on the moving platform is represented by
$$
\mathbf{W} = \begin{bmatrix} \mathbf{F} \\ \mathbf{M} \end{bmatrix},
$$
where \(\mathbf{F}=[-F_x,-F_y,-F_z]^T\) and \(\mathbf{M}=[-M_x,-M_y,-M_z]^T\). The negative signs are introduced so that the positive components represent the reactions from the platform to the environment for a pushing force. The moving platform is connected to the three side limbs through ball joints, so each limb exerts a force \(\mathbf{F}_{Bi}\) on the moving platform. The central limb exerts a force \(\mathbf{F}_N\). The moving platform is also subject to its own weight \(G_o\). The force balance equation is
$$
\sum_{i=1}^{3} \mathbf{F}_{Bi} + \mathbf{F}_N + \mathbf{G}_o = \mathbf{0}.
$$
The moment balance about \(O_1\) is
$$
\sum_{i=1}^{3} (\mathbf{O}_1\mathbf{B}_i \times \mathbf{F}_{Bi}) + \mathbf{M} = \mathbf{0}.
$$
For each side limb, I consider the forces at both ends, the weight of the limb, and the constraint force from the sliding joint. Similarly, for each slider, I project the forces along the actuator axis to obtain the driving force \(F_{mi}\). Combining all equations leads to a linear algebraic system of 27 equations for 27 unknown forces. I solve this system numerically in MATLAB.
3.2 Numerical Static Example
I consider six representative load cases to investigate the unloading performance of the central limb. The load cases are listed in Table 3.1.
| Case | \(\alpha\) | \(\mathbf{F}\) (N) | \(\mathbf{M}\) (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) |
For the pure force cases (Cases 1 and 2), the central limb force is nearly constant at about \(F_N’ \approx 13.01\) N, while each side limb force is only about \(0.588\) N. Therefore, the central limb supports roughly 22 times the force of each side limb, demonstrating a very strong unloading effect. For the pure moment cases (Cases 3 and 4), the central limb force remains between 3.5 N and 4.4 N, while the side limb forces do not exceed 22% of that value. For the combined load cases (Cases 5 and 6), the central limb force is between 11.3 N and 12.9 N, and the side limb forces are all below 10% of the central limb force. These results confirm that the central spherical branch effectively shares the majority of the external load, reducing the load on the actuators and improving the stiffness and load capacity of the shoulder joint.
4. Dynamic Analysis
4.1 Inertia Tensors
The Newton–Euler formulation requires the inertia tensor of each moving body relative to the inertial frame. I compute the inertia tensor of each side limb and the moving platform using the parallel-axis theorem and the similarity transformation. The side limbs are modeled as uniform cylinders with radius \(r_l\) and length \(l\). The inertia tensor of limb \(i\) in its body-fixed frame is
$$
\mathbf{I}_{ABi}^{E} = \operatorname{diag}\left(\frac{m_l l^2}{12}, \frac{m_l l^2}{12}, \frac{m_l r_l^2}{2}\right).
$$
Transforming this tensor to the fixed frame gives
$$
\mathbf{I}_{ABi}^{H} = \mathbf{R}_{E_i}^H \, \mathbf{I}_{ABi}^{E} \, (\mathbf{R}_{E_i}^H)^T + m_l (\mathbf{E}_i^2 I_3 – \mathbf{E}_i \mathbf{E}_i^T),
$$
where \(\mathbf{E}_i\) is the position vector of the center of mass of limb \(i\), and \(I_3\) is the \(3\times3\) identity matrix. Similarly, the moving platform is a disk of radius \(r_o\) and thickness \(t_o\). Its inertia tensor in the moving frame is
$$
\mathbf{I}_o^{O_1} = \operatorname{diag}\left(\frac{m_o r_o^2}{4}, \frac{m_o r_o^2}{4}, \frac{m_o r_o^2}{2}\right),
$$
which is transformed to the fixed frame using the orientation matrix \(\mathbf{R}\).
4.2 Newton–Euler Dynamic Equations
For the moving platform, the Newton equation is
$$
\sum_{i=1}^{3} \mathbf{F}_i + \mathbf{F}_N + \mathbf{G}_o = m_o \mathbf{a}_o,
$$
and the Euler equation is
$$
\sum_{i=1}^{3} (\mathbf{O}_1\mathbf{B}_i \times \mathbf{F}_i) + \mathbf{M} = \mathbf{I}_o^H \dot{\boldsymbol{\omega}}_o + \boldsymbol{\omega}_o \times (\mathbf{I}_o^H \boldsymbol{\omega}_o),
$$
where \(\mathbf{a}_o\) is the linear acceleration of the moving-platform center. For each side limb, I write both translational and rotational equations about its center of mass. For each slider, I write the equation of motion along the actuator axis. Combining all equations yields a set of differential-algebraic equations that can be solved numerically to obtain the actuator forces \(F_{mi}\), the joint reaction forces \(\mathbf{F}_i\), and the central limb force \(\mathbf{F}_N\).
4.3 Numerical Dynamic Simulation
I perform simulations for three different motion patterns of the moving platform, while a constant external wrench \(\mathbf{F}=(-3,-4,-5)\) N and \(\mathbf{M}=(-6,-4,-7)\) N·m is applied. The motion patterns are:
- Case A: \(\alpha = \pi/3\), \(\beta = \pi/4\), \(\gamma = 0.2\sin(\pi t)+0.6\)
- Case B: \(\alpha = \pi/3\), \(\beta = 0.2\sin(\pi t)+0.6\), \(\gamma = \pi/4\)
- Case C: \(\alpha = 0.2\sin(\pi t)+0.6\), \(\beta = \pi/3\), \(\gamma = \pi/4\)
For Case A, the actuator forces are plotted against time, as are the constraint forces on each side limb and the central limb. The central limb force remains approximately between 9.65 N and 9.76 N, which is more than 11 times the maximum side limb force. For Case B, the central limb force stays between 9.65 N and 9.70 N, and for Case C it is between 9.31 N and 9.72 N. In all cases, the central limb substantially unloads the side limbs. The required actuator forces are small, on the order of 1 N to 2 N, despite the external load. This demonstrates the excellent load-sharing capability of the proposed shoulder mechanism.
| Motion | \(F_{m1}\) (N) | \(F_{m2}\) (N) | \(F_{m3}\) (N) | \(F_N’\) (N) | Side limb forces (N) |
|---|---|---|---|---|---|
| Case A | 1.10–1.18 | 1.02–1.04 | 1.08–1.16 | 9.65–9.76 | 0.34–0.84 |
| Case B | 1.10–1.22 | 1.02–1.16 | 1.05–1.18 | 9.65–9.70 | 0.29–0.90 |
| Case C | 1.40–2.20 | 0.80–1.10 | 0.90–1.20 | 9.31–9.72 | 0.31–1.80 |
The dynamic model also provides information on how the joint reaction forces vary with the motion trajectory, which is useful for structural design and fatigue analysis. The relatively small actuator forces imply that a small servo motor suffices to drive the shoulder, which is advantageous for reducing the weight of an embodied robot.
5. Prototype Design
5.1 Selection of the Driving Method
To implement the 3PSS/S mechanism in a physical prototype, I choose a servo motor combined with a ball-screw driven linear module as the actuator. Compared with gear racks or belt drives, the ball-screw module offers high precision, high rigidity, and a compact footprint. I selected the THK linear guide module with a sideways-mounted motor to minimize the overall height of the mechanism. The ball screw is selected based on the required speed, load, and accuracy. Table 5.1 summarizes the key parameters of the selected ball screw.
| Parameter | Value |
|---|---|
| Outer diameter | 5 mm |
| Lead | 2 mm |
| Dynamic load rating | 0.54 kN |
| Static load rating | 0.94 kN |
| Accuracy grade | C7 |
The maximum axial load generated during operation is about 54 N, which is much lower than the allowable axial load of 627 N. The screw speed does not exceed the critical speed or the DN-value limit. Thus, the selected ball screw is safe and reliable.
5.2 Mechanical Components
The central spherical joint is a critical element because it determines the motion constraints and workspace of the shoulder. I improved a previously designed multi-ear spherical joint with a compact structure. The improved joint consists of a ball seat, a rotating socket, a ball hinge cover, and a ball head rod. The ball hinge cover and the rotating socket are connected through two angular-contact ball bearings, allowing a maximum swing angle of about \(50^\circ\), which is larger than the typical \(30^\circ\) of a conventional spherical joint. The central spherical joint also provides the main load path from the moving platform to the fixed base.
The side limbs are designed with a composite spherical joint composed of a universal joint and a revolute joint. This arrangement provides the required spherical-joint motion while increasing the allowable swing angle and simplifying manufacturing. Each side limb is connected to the slider by a bracket, and the upper end is attached to the moving platform through a clevis. All side-limb components are made of 45 steel with heat treatment and blackening, while the moving and fixed platforms are made of aluminium alloy LY12 for weight reduction.
5.3 Platform Design
The moving platform is a disk with three equally spaced locating holes on its circumference for the upper clevises. A central counterbore receives the ball head rod of the central spherical joint. The fixed platform is a truncated pyramid with three inclined planar surfaces at an angle of \(60^\circ\) to the horizontal plane. The THK linear modules are mounted on these inclined surfaces. This geometry ensures that the three actuator axes intersect at the point \(H\). The use of wire-cut electrical discharge machining guarantees the machining accuracy of the inclined surfaces and their intersection point.
5.4 Assembly and Overall Model
Using CATIA, I assembled all components into a complete 3D model, and then the prototype was manufactured and assembled by a professional factory. The prototype consists of the fixed platform, three linear modules, three side limbs, the moving platform, and the central spherical joint. A photograph of the manufactured prototype is shown in the middle of this thesis. The overall dimensions are compact, and the total mass is suitable for mounting on an embodied robot torso. During assembly, special attention was paid to the preloading of the ball bearings and the alignment of the U-joints to ensure smooth motion.
6. Control System Development
6.1 Control Scheme
For a multi-axis parallel robot, the control system must coordinate the three actuator motions precisely. I selected a closed-loop position control scheme based on the industrial machine controller MP2300S from Yaskawa. This controller supports MECHATROLINK-II communication and can synchronize up to 16 axes. Each axis is driven by an AC servo unit SGDV-R90 and a miniature servo motor SGMMV. The standard closed-loop configuration includes a velocity loop, a position loop, and a current loop within the servo drive. The host computer sends trajectory commands to the motion controller, which generates interpolated position setpoints for each axis. Encoders on the motors provide position feedback, while limit switches provide over-travel protection.
| Component | Model/Type | Quantity |
|---|---|---|
| Motion controller | Yaskawa MP2300S | 1 |
| Servo unit | SGDV-R90A11A | 3 |
| Servo motor | SGMMV | 3 |
| Absolute encoder | 17-bit | 3 |
| Proximity sensor | APM-D3B1 | 6 |
| Power supply | 24 V DC | 1 |
6.2 Motor Sizing
To confirm the motor choice, I calculated the required torque and inertia. The total load inertia reflected to the motor shaft is
$$
J_L = \frac{J_S + J_T}{i^2} = 9.89\times10^{-7} \,\text{kg·m}^2,
$$
where \(J_S = 3.75\times10^{-7}\) kg·m² is the screw inertia and \(J_T = 6.14\times10^{-7}\) kg·m² is the load inertia. The motor acceleration torque is
$$
T_a = (J_L + J_m)\epsilon = 0.0073 \,\text{N·m},
$$
with \(\epsilon = 2093\) rad/s². The required continuous effective torque is
$$
T_{rms} = 0.0132 \,\text{N·m},
$$
and the maximum torque is
$$
T_{max} = 0.0259 \,\text{N·m}.
$$
The selected servo motor has a rated torque of 0.0637 N·m and a maximum torque of 0.191 N·m, which satisfies all conditions.
6.3 Control System Integration
I designed the control cabinet, which integrates the main circuit breakers, contactors, the machine controller, the three servo units, and a terminal block. Special attention was paid to shielding and grounding to reduce electromagnetic interference. The wiring of encoders, limit switches, and brake lines follows the manufacturer’s recommendations. The software environment MPE720 ver.6 was used for parameter tuning, startup testing, and motion programming. The servo parameters were configured using SigmaWin+ and the motion controller parameters were set in the Module Configuration tool. After the JOG and homing tests, a three-axis synchronized motion was successfully performed, proving the correctness of the control system.
7. Experimental Validation
7.1 Functional Verification
After commissioning, I conducted a functional verification experiment to confirm that the shoulder joint can perform large-orientation rotations as designed. The motion controller executed a series of orientation trajectories generated by the inverse kinematic model. The moving platform was able to reach a variety of poses without mechanical interference. The actual motion matched the planned trajectory closely, demonstrating the feasibility of the mechanical design and the control scheme. The limit switches worked correctly, and the emergency stop function was reliable.
| Parameter | Value |
|---|---|
| Platform orientation | \(\alpha=-\pi/6, \beta=0, \gamma=0\) |
| Force sensor | ZP-100N, LC1104 |
| Dynamic signal analyzer | PH5956D |
| Applied force range | 5 N to 25 N (pure force) |
| Applied moment range | 0.5 N·m to 2.5 N·m (combined load) |
7.2 Force Feedback Experiment
The force feedback experiment was designed to verify the static analysis results. A micro strain-gauge force sensor was mounted on the central spherical limb, and a high-precision push-pull force sensor was used to apply controlled external forces and moments to the moving platform. For pure force loads, Table 7.2 shows the measured central limb force and the theoretical values for five different force levels.
| Applied \(F_z\) (N) | Measured \(F_N’\) (N) | Theoretical \(F_N’\) (N) | Side limb force (N) |
|---|---|---|---|
| 5 | 4.6 | 5.31 | 0.019 |
| 10 | 9.2 | 10.31 | 0.017 |
| 15 | 13.9 | 15.30 | 0.018 |
| 20 | 19.4 | 20.31 | 0.015 |
| 25 | 23.5 | 25.30 | 0.015 |
The average absolute error is less than 1 N, which is acceptable considering joint friction and sensor accuracy. For combined force/moment loading, the measured central limb force remained stable around 2 N, while the side limb forces increased with the applied moment. This behavior confirms the theoretical prediction that the central limb mainly carries the force component, while the moments are mainly resisted by the side limbs. The force feedback experiment validates the static unloading model and provides a useful basis for future impedance control of the embodied robot shoulder.
8. Conclusion and Future Work
In this thesis I have comprehensively studied a 3PSS/S parallel mechanism as a bionic shoulder joint for embodied robots. The main contributions are summarized as follows:
- The kinematics of the 3PSS/S mechanism were fully derived, including position inverse solution, velocity Jacobian, and acceleration relations. The Jacobian was implemented in MATLAB and validated by ADAMS simulation. Singularity analysis revealed an output singularity locus that must be avoided in trajectory planning.
- Static analysis based on vector mechanics demonstrated that the central spherical limb provides a strong force-unloading effect. Under various load conditions, the central limb carries over 90% of the total external force, significantly reducing the required actuator forces.
- A Newton–Euler dynamic model was established to compute actuator forces and internal constraint forces during motion. The simulation results show that the mechanism can operate with small actuator torques while maintaining a high payload capacity.
- A physical prototype was designed and manufactured. The choice of a ball-screw linear module, the improved central spherical joint, and the optimized platform geometry resulted in a compact, lightweight, and robust shoulder unit.
- A servo control system was developed and integrated. The prototype was successfully controlled to perform large rotations and accurately follow trajectories.
- Force feedback experiments were carried out. The experimental data agree well with the theoretical static model, confirming the load-sharing capability of the central limb.
Despite these achievements, there are several directions for future work. First, kinematics calibration of the prototype should be performed to improve its absolute accuracy. Second, the motion trajectory should be planned using biomechanical data of the human shoulder to achieve more natural and complex gestures. Third, the dynamic model can be extended to include friction and joint clearances, which would further improve the fidelity of the simulation. Fourth, an admittance or impedance control strategy can be implemented to allow safe physical human-robot interaction, which is essential for embodied robots operating in human-centered environments. Finally, the 3PSS/S shoulder mechanism can be integrated with an elbow and a hand to form a complete arm of an embodied robot, and its performance can be evaluated in tasks such as carrying, pushing, and reaching.
In conclusion, the 3PSS/S parallel shoulder mechanism offers a compelling combination of high stiffness, high load capacity, and compactness, making it a strong candidate for the next generation of embodied robots. The theoretical analyses and experimental validations presented in this thesis provide a solid foundation for further research and engineering deployment.