1. Introduction
In recent years, the development of modern machinery has been driven toward intelligence, high precision, high speed, high reliability, and high integration. Parallel mechanisms, with their compact structure, high stiffness, excellent kinematic performance, and strong load-bearing capacity, have attracted increasing attention from scholars worldwide. A humanoid robot is a complex electromechanical system that mimics the human body structure and motion mechanism. The joint mechanism is the fundamental carrier of motion functions. Traditional humanoid robot joints are mostly implemented by serially connected single-degree-of-freedom revolute joints, which suffer from low stiffness, large space occupation, and poor dynamic performance. To overcome these drawbacks, the use of parallel mechanisms as joint architectures for humanoid robots has become a promising research direction.
Among all parallel mechanisms, spherical parallel mechanisms (SPMs) can produce pure rotations about a fixed center point, which perfectly matches the motion of human hip joints. The hip joint is the largest weight-bearing joint in the human body. It enables three-dimensional rotation of the lower limb around the geometric center of the acetabulum, including flexion/extension, abduction/adduction, and internal/external rotation. Therefore, designing a humanoid robot hip joint based on a spherical parallel mechanism is both anatomically and mechanically reasonable.
This thesis focuses on several fundamental theoretical aspects of the bionic design of a humanoid robot hip joint based on a 3-SPS/S spherical parallel mechanism. The research includes kinematic analysis, workspace optimization, singularity analysis, and dynamic simulation. The results provide theoretical foundations for the development of high-performance humanoid robot joints.
2. Human Hip Joint Characteristics and Mechanism Selection
2.1 Structure and Motion of the Human Hip Joint
From the anatomical point of view, the human hip joint consists of the femoral head, acetabulum, articular capsule, ligaments, and muscle groups. The femoral head is connected to the femoral shaft by the femoral neck, which forms an angle of about 125° with the femoral shaft. The hip joint is capable of three rotational motions about three mutually perpendicular axes: the coronal axis, the sagittal axis, and the vertical axis. The main motion ranges are summarized in Table 1.
| Motion type | Range (active) | Range (passive) |
|---|---|---|
| Flexion | 90°–120° | 120°–145° |
| Extension | 15°–20° | 30° |
| Internal rotation | 30°–40° | — |
| External rotation | up to 60° | — |
| Abduction | 30°–40° | 75°–90° |
| Adduction | up to 30° | — |
The driving mechanism of the hip joint is realized by multiple muscle groups working in coordination. This is analogous to the coupled driving of parallel mechanisms, where multiple actuators jointly determine the motion of the moving platform. Therefore, the spherical parallel mechanism is a natural choice for hip joint bionic design.
2.2 Comparison of Spherical Parallel Mechanisms
Spherical parallel mechanisms can be classified into two main categories: the 3-RRR type and the S+X type. The 3-RRR spherical parallel mechanism has all revolute axes intersecting at one common point. Although it has been successfully used in the Agile Eye and bionic shoulder joints, its manufacturing and assembly precision requirements are extremely high. Moreover, in the 3-RRR mechanism, the branch chains must not only balance the external torque on the moving platform but also bear internal forces to balance external forces. This leads to uneven force distribution and limits the motion flexibility.
The S+X type spherical parallel mechanism inserts a passive spherical joint at the geometric center of a parallel mechanism. This central spherical joint can unload external forces, while the active branch chains only need to balance external torques. Among these, the 3-SPS/S mechanism has three symmetric SPS branches and one central passive S joint. Its structure is simple, and there is no overconstraint. Therefore, the 3-SPS/S spherical parallel mechanism was selected as the prototype mechanism for the bionic hip joint of a humanoid robot. The schematic representation is shown in Figure 1 (not reproduced here for brevity).
3. Kinematic Analysis of the 3-SPS/S Spherical Parallel Mechanism
3.1 Description of the Mechanism
The 3-SPS/S mechanism consists of a fixed base, a moving platform, three identical SPS limbs, and a central passive S joint. The SPS limb is composed of two spherical joints connected by a prismatic actuator. The spherical joints attached to the fixed base are denoted \(A_i\), and those attached to the moving platform are denoted \(B_i\) \((i=1,2,3)\). The three \(A_i\) points are uniformly distributed on a circle of radius \(r_A\), and the three \(B_i\) points are uniformly distributed on a circle of radius \(r_B\). The central spherical joint center is chosen as the origin of both the fixed coordinate system \(\{G\}\) and the moving coordinate system \(\{D\}\).
3.2 Inverse Position Solution
Using the \(z\)-\(y\)-\(x\) Euler angles \((\alpha,\beta,\gamma)\) to describe the orientation of the moving platform relative to the fixed frame, the rotation matrix is:
$$
\mathbf{R}_D^G =
\begin{bmatrix}
c\alpha c\beta & c\alpha s\beta s\gamma – s\alpha c\gamma & c\alpha s\beta c\gamma + s\alpha s\gamma \\
s\alpha c\beta & s\alpha s\beta s\gamma + c\alpha c\gamma & s\alpha s\beta c\gamma – c\alpha s\gamma \\
-s\beta & c\beta s\gamma & c\beta c\gamma
\end{bmatrix}
$$
where \(c\alpha = \cos\alpha\), \(s\alpha = \sin\alpha\), and similarly for the other angles. The position vectors of \(B_i\) in the moving frame are:
$$
\mathbf{b}_1^D = \begin{bmatrix} r_B/2 \\ -\sqrt{3}r_B/2 \\ a \end{bmatrix}, \quad
\mathbf{b}_2^D = \begin{bmatrix} r_B/2 \\ \sqrt{3}r_B/2 \\ a \end{bmatrix}, \quad
\mathbf{b}_3^D = \begin{bmatrix} -r_B \\ 0 \\ a \end{bmatrix}
$$
The position vectors of \(A_i\) in the fixed frame are:
$$
\mathbf{a}_1^G = \begin{bmatrix} r_A/2 \\ -\sqrt{3}r_A/2 \\ -b \end{bmatrix}, \quad
\mathbf{a}_2^G = \begin{bmatrix} r_A/2 \\ \sqrt{3}r_A/2 \\ -b \end{bmatrix}, \quad
\mathbf{a}_3^G = \begin{bmatrix} -r_A \\ 0 \\ -b \end{bmatrix}
$$
where \(a\) is the distance from the central spherical joint center to the moving platform center, and \(b\) is the distance from the central spherical joint center to the fixed base center. The inverse position solution is obtained by calculating the distances between \(A_i\) and \(B_i\):
$$
l_i = \left\| \mathbf{R}_D^G \mathbf{b}_i^D + \mathbf{p} – \mathbf{a}_i^G \right\|, \quad i=1,2,3
$$
where \(\mathbf{p} = [0,0,0]^T\) because the origins of the fixed and moving frames coincide. The explicit expressions of \(l_i\) are lengthy but can be expanded symbolically. They provide the actuator lengths for any given orientation \((\alpha,\beta,\gamma)\).
3.3 Velocity and Acceleration Analysis Using Influence Coefficients
The influence coefficient method is an efficient tool for kinematic and dynamic analysis of parallel mechanisms. For the 3-SPS/S mechanism, the complex spherical joints are replaced by equivalent combinations of revolute joints. For limb \(i\), the joint axes are denoted \(\mathbf{S}_{i1}\), \(\mathbf{S}_{i2}\), \(\mathbf{S}_{i3}\) for the lower spherical joint, \(\mathbf{S}_{i4}\) for the prismatic joint, and \(\mathbf{S}_{i5}\), \(\mathbf{S}_{i6}\), \(\mathbf{S}_{i7}\) for the upper spherical joint. The limb Jacobian matrix is:
$$
\mathbf{G}_{\phi}^{H(i)} =
\begin{bmatrix}
\mathbf{S}_{i1} & \mathbf{S}_{i2} & \mathbf{S}_{i5} & \mathbf{S}_{i6} & \mathbf{S}_{i7} \\
\mathbf{S}_{i1}\times(\mathbf{P}-\mathbf{R}_{i1}) & \mathbf{S}_{i2}\times(\mathbf{P}-\mathbf{R}_{i2}) & \mathbf{S}_{i5}\times(\mathbf{P}-\mathbf{R}_{i5}) & \mathbf{S}_{i6}\times(\mathbf{P}-\mathbf{R}_{i6}) & \mathbf{S}_{i7}\times(\mathbf{P}-\mathbf{R}_{i7})
\end{bmatrix}
$$
where \(\mathbf{P}\) is the position vector of the moving platform reference point \(O_B\). Since the moving platform only rotates about the central spherical joint, the velocity of \(O_B\) is:
$$
\mathbf{v}_{O_B} = \boldsymbol{\omega} \times \mathbf{r}_{O_B}
$$
$$
\mathbf{a}_{O_B} = \boldsymbol{\varepsilon} \times \mathbf{r}_{O_B} + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}_{O_B})
$$
where \(\boldsymbol{\omega}\) and \(\boldsymbol{\varepsilon}\) are the angular velocity and angular acceleration of the moving platform, and \(\mathbf{r}_{O_B}\) is the position vector of \(O_B\) in the fixed frame. By extracting the rows corresponding to the prismatic actuators, the mapping between the actuator velocities and the output angular velocity is:
$$
\mathbf{q} = \mathbf{G}_{\omega}^{q} \boldsymbol{\omega}
$$
where \(\mathbf{q} = [\dot{l}_1, \dot{l}_2, \dot{l}_3]^T\) and \(\mathbf{G}_{\omega}^{q}\) is the \(3\times3\) first-order influence coefficient matrix. The first-order influence coefficient matrix of the moving platform with respect to actuator inputs is:
$$
\boldsymbol{\omega} = \mathbf{G}_{q}^{\omega} \mathbf{q}
$$
Similarly, the second-order influence coefficient matrix \(\mathbf{H}_{q}^{\varepsilon}\) is defined by:
$$
\boldsymbol{\varepsilon} = \mathbf{G}_{q}^{\varepsilon} \mathbf{q} + \mathbf{q}^T \mathbf{H}_{q}^{\varepsilon} \mathbf{q}
$$
The explicit derivation of these matrices is based on the general influence coefficient formulation for spatial mechanisms. The correctness of the influence coefficient matrices was verified by comparing the computed actuator velocities and accelerations with the results from ADAMS simulation. The errors were less than 0.25%, confirming the validity of the kinematic model.
4. Workspace Optimization of the 3-SPS/S Mechanism
4.1 Workspace Evaluation Method
The workspace of the 3-SPS/S mechanism is defined as the set of all attainable orientations of the moving platform. A numerical grid search method is adopted. For each orientation \((\alpha,\beta,\gamma)\), the following constraints are checked:
- Actuator stroke limits: \(l_{\min} \le l_i \le l_{\max}\), for \(i=1,2,3\).
- Spherical joint angle limits: The angle between the limb axis and the normal of the platform at the spherical joint must not exceed the maximum allowable angle.
- Limb interference: The minimum distance between any two limbs must be greater than the limb diameter.
For a given set of structural parameters, the orientation workspace is obtained by discretizing the three orientation angles within specified ranges. The number of valid orientation points is used as a measure of workspace volume.
4.2 Structural Parameters and Genetic Algorithm Optimization
To maximize the workspace, the genetic algorithm (GA) is employed. The design variables are the moving platform radius \(r_B\), the distance \(a\), the distance \(b\), and the maximum spherical joint angle \(\theta_1\). The fixed base radius is set to \(r_A = 100\) mm, the limb diameter is \(d=10\) mm, the actuator stroke range is 120–280 mm, and the central spherical joint has a maximum rotation angle of 100°. The optimization ranges are:
$$
r_B \in [85, 105] \text{ mm}, \quad a \in [20, 60] \text{ mm}, \quad b \in [140, 180] \text{ mm}, \quad \theta_1 \in [80^\circ, 100^\circ]
$$
Table 2 lists the optimal parameters obtained by the genetic algorithm.
| Parameter | Value |
|---|---|
| \(r_B\) (mm) | 96.5 |
| \(a\) (mm) | 33.6 |
| \(b\) (mm) | 165.3 |
| \(\theta_1\) (°) | 87.0 |
The GA convergence curve showed that the fitness value converged after about 30 generations. Table 3 compares the orientation ranges before and after optimization.
| Orientation angle | Before optimization | After optimization |
|---|---|---|
| \(\alpha\) (around \(z\)-axis) | \(-70^\circ\) to \(+70^\circ\) | \(-70^\circ\) to \(+70^\circ\) |
| \(\beta\) (around \(y\)-axis) | \(-36^\circ\) to \(+30^\circ\) | \(-46^\circ\) to \(+44^\circ\) |
| \(\gamma\) (around \(x\)-axis) | \(-35^\circ\) to \(+35^\circ\) | \(-46^\circ\) to \(+46^\circ\) |
After optimization, the rotation range around the \(y\)-axis increased by 35.78%, and the rotation range around the \(x\)-axis increased by 31.01%. The resulting orientation workspace and position workspace are shown as three-dimensional point clouds in the original work.
4.3 Offset Output Design
To better match the asymmetric motion cone of the human hip joint, an offset output design is introduced. A bent output link is attached to the moving platform with an offset angle \(t\). The orientation of the offset coordinate frame \(\{P\}\) with respect to the moving frame \(\{D\}\) is:
$$
\mathbf{R}_P^D =
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos t & -\sin t \\
0 & \sin t & \cos t
\end{bmatrix}
$$
The overall rotation matrix of the offset frame with respect to the fixed frame is:
$$
\mathbf{R}_P^G = \mathbf{R}_D^G \mathbf{R}_P^D
$$
By comparing with a direct \(z\)-\(y\)-\(x\) Euler angle representation, the relationship between the original orientation angles and the offset output angles is:
$$
\alpha’ = \alpha, \quad \beta’ = \beta, \quad \gamma’ = \gamma + t
$$
Thus, the offset output effectively increases the rotation range around the \(z\)-axis. For \(t = 0^\circ, 30^\circ, 50^\circ\), the output motion spaces are shown in the original work. It is clear that the offset design makes the motion envelope closer to the natural shape of the human lower limb motion cone.
5. Singularity Analysis of the 3-SPS/S Mechanism
5.1 Singularity Conditions Based on Gosselin’s Method
Singular configurations of parallel mechanisms are critical to avoid in the design and control of a humanoid robot. According to Gosselin and Angeles, singularities can be classified into three types. For the 3-SPS/S mechanism, the Jacobian matrices \(\mathbf{I}\) and \(\mathbf{J}\) are defined by differentiating the inverse position equations:
$$
\mathbf{I} \mathbf{q} = \mathbf{J} \dot{\boldsymbol{\theta}}
$$
where \(\mathbf{q} = [\dot{l}_1, \dot{l}_2, \dot{l}_3]^T\) and \(\dot{\boldsymbol{\theta}} = [\dot{\alpha}, \dot{\beta}, \dot{\gamma}]^T\). The \(\mathbf{I}\) matrix is diagonal with entries \(2l_i\), which is always nonsingular since \(l_i \neq 0\). Therefore, only the second type of singularity exists, characterized by:
$$
\det(\mathbf{J}) = 0
$$
Using a numerical search in the orientation workspace, one singular configuration is found when:
$$
\alpha = 0, \quad \gamma = 0, \quad \beta \in [-46^\circ, 46^\circ]
$$
At this configuration, the three limb axes all intersect a common line in the \(o\)-\(xz\) plane, which produces an instantaneous rotational freedom of the moving platform about that line.
5.2 Other Singular Configurations by Symmetry
Because the 3-SPS/S mechanism has three-fold symmetry, there must be two additional singular configurations in the planes that are oriented at 120° to each other. By projecting the singular trajectory onto the \(xy\)-plane, the two additional lines are governed by:
$$
y = \sqrt{3} x, \quad y = -\sqrt{3} x
$$
The position of the moving platform center \(O_B\) is:
$$
\mathbf{P} = \begin{bmatrix} a \sin\beta \cos\gamma \\ a \sin\gamma \\ a \cos\beta \cos\gamma \end{bmatrix}
$$
Substituting the coordinates into the line equations yields the corresponding singular orientation angles:
$$
\gamma = \arctan(-\sqrt{3} \sin\beta) \quad \text{or} \quad \gamma = \arctan(\sqrt{3} \sin\beta)
$$
Both solutions satisfy \(\det(\mathbf{J}) = 0\), which was verified numerically. Thus, the 3-SPS/S spherical parallel mechanism possesses three singular configuration curves that divide the orientation workspace into three symmetrical regions. This information is crucial for trajectory planning of the humanoid robot hip joint to avoid singularities.
6. Dynamic Analysis and Simulation
6.1 Dynamic Model Based on Virtual Work
The dynamic analysis of the 3-SPS/S mechanism is performed by combining the influence coefficient method with the principle of virtual work. The external forces, inertial forces, and inertial torques acting on all moving components are considered. For a body \(k\), the six-dimensional inertial wrench is:
$$
\mathbf{F}_{Ic}^k =
\begin{bmatrix}
-\boldsymbol{\omega}_k \times (\mathbf{I}_k \boldsymbol{\omega}_k) \\
-m_k \mathbf{a}_k
\end{bmatrix}
$$
where \(\boldsymbol{\omega}_k\), \(\mathbf{a}_k\), \(m_k\), and \(\mathbf{I}_k\) are the angular velocity, acceleration of the center of mass, mass, and inertia tensor, respectively.
Using the influence coefficient matrices, the virtual displacements of the moving platform and the limbs can be expressed in terms of virtual actuator displacements. Applying the virtual work principle gives the actuator forces \(\boldsymbol{\tau}\):
$$
\boldsymbol{\tau} = – \mathbf{G}_q^{H\,T} \mathbf{F}_H’ – \sum_{i=1}^{3} \left( \mathbf{G}_{q}^{1i\,T} \mathbf{F}_{1i} + \mathbf{G}_{q}^{2i\,T} \mathbf{F}_{2i} \right)
$$
where \(\mathbf{F}_H’\) is the total wrench acting on the moving platform, and \(\mathbf{F}_{1i}\) and \(\mathbf{F}_{2i}\) are the wrenches of the lower and upper parts of limb \(i\). The matrix \(\mathbf{G}_{q}^{H}\) is the first-order influence coefficient of the moving platform, and \(\mathbf{G}_{q}^{1i}\), \(\mathbf{G}_{q}^{2i}\) are those of the limb components.
6.2 Simulation Setup and Results
A three-dimensional solid model of the 3-SPS/S mechanism was created in SolidWorks and then imported into ADAMS. The material properties of steel were assigned to all parts. A gravity field of \(9.8 \, \text{m/s}^2\) was applied. The moving platform was driven to rotate about the \(x\)-axis according to:
$$
\theta_x(t) = -10^\circ \sin(t) + 10^\circ
$$
The simulation time was 13 s with a step size of 0.01 s. The actuator length curves obtained from the inverse kinematic simulation were used as spline functions to drive the prismatic joints in the dynamic simulation. An external force of 650 N was applied to the moving platform along the negative \(z\)-axis to simulate the load from a human leg.
Figure 2 shows the actuator force curves over time. The three driving forces vary periodically and smoothly, with magnitudes ranging from 0 to approximately 68 N. At the instant when the moving platform is parallel to the fixed base, the forces become zero because the external force is fully balanced by the central spherical joint.
Figure 3 depicts the forces at the three upper spherical joints on the moving platform. The forces in the \(x\)- and \(y\)-directions are relatively small (maximum about 8 N), while the \(z\)-direction forces reach up to 68 N, consistent with the applied external load. Figure 4 shows the force at the central spherical joint. The central joint force in the \(z\)-direction varies between 711 N and 730 N, while the \(x\)- and \(y\)-direction forces are nearly zero.
A quantitative comparison is provided in Table 4.
| Quantity | Minimum \(F_z\) | Maximum \(F_z\) |
|---|---|---|
| Central spherical joint | 712 N | 730 N |
| Upper spherical joints | 0 N | 68 N |
| Ratio (central / upper) | 10.8 | 10.7 |
The simulation clearly demonstrates that the central passive spherical joint effectively unloads almost all the external force, while the three active limbs only need to balance the external torque. This is a major advantage of the 3-SPS/S configuration for the bionic hip joint of a humanoid robot, as it allows the actuators to be smaller and more responsive.
7. Conclusion
In this thesis, several fundamental theoretical issues regarding the bionic design of a humanoid robot hip joint based on a 3-SPS/S spherical parallel mechanism were systematically studied. The main conclusions are as follows:
- The 3-SPS/S spherical parallel mechanism was selected as the bionic prototype because its structure and force distribution characteristics closely mimic the human hip joint. The central spherical joint provides a natural load-bearing path, while the three active limbs coordinate to generate three-dimensional rotations.
- An influence coefficient method was successfully extended to a mechanism whose driving and constraint pairs are not in the same limb. The first-order and second-order influence coefficient matrices were established, and the correctness was verified by ADAMS simulations.
- The orientation workspace was maximized using genetic algorithm optimization. After optimization, the rotation ranges about the \(x\)- and \(y\)-axes increased by more than 30%. The offset output design was introduced to better approximate the asymmetric motion cone of the human lower limb.
- Singularity analysis based on Gosselin’s method identified one singular curve, and by exploiting the three-fold symmetry, the other two singular curves were deduced and verified. These results provide guidelines for singularity-free trajectory planning in future humanoid robot control.
- Dynamic simulation validated that the central spherical joint unloads external forces, while the driving limbs mainly balance external torques. The actuator forces remain small and smooth, which is beneficial for achieving flexible and energy-efficient motion in humanoid robots.
Future work will focus on the optimization of trajectory planning to avoid singularities, the development of control algorithms based on measured hip joint biomechanics, and the consideration of flexible components in the dynamic model. The ultimate goal is to build a fully functional bionic hip joint that can be integrated into an advanced humanoid robot with human-like agility and load-carrying capability.