In practical applications, the dynamic performance of six-axis force sensors is crucial, as it directly impacts the accuracy and responsiveness of force measurements in real-time systems. This article presents a comprehensive theoretical and experimental investigation into the dynamic characteristics of an orthogonal parallel six-axis force sensor. The study begins by detailing the sensor’s structural design and mathematical modeling, followed by an in-depth vibration analysis based on screw theory and multi-degree-of-freedom system dynamics. The derived equations of motion are solved to determine natural frequencies, and experimental validation is conducted through impulse response tests. By comparing theoretical predictions with experimental results, the feasibility and correctness of the analytical approach are confirmed, underscoring the importance of dynamic performance in six-axis force sensor applications.
The orthogonal parallel six-axis force sensor comprises three main components: a force-measuring platform, a fixed platform, and six force-sensing branches. Each platform features three support columns arranged symmetrically around the z-axis of a Cartesian coordinate system. The six force-sensing branches, which are S-type uniaxial force sensors, include three vertically oriented branches connecting the two platforms and three horizontally oriented branches attached to the support columns. All connections utilize elastic spherical hinges, allowing the branches to act as two-force elements that primarily withstand axial tension and compression. This orthogonal arrangement enhances measurement precision for forces and moments along different axes while minimizing inter-dimensional coupling. For instance, forces in the x and y directions and moments about the z-axis are predominantly measured by the horizontal branches, whereas forces in the z-direction and moments about the x and y axes are captured by the vertical branches. This design inherently improves the decoupling capability of the six-axis force sensor, making it suitable for high-precision applications such as robotic force feedback systems.
To establish a mathematical model for the orthogonal parallel six-axis force sensor, a reference coordinate system o-xyz is defined, with the origin o at the geometric center of the force-measuring platform’s lower surface. The z-axis is perpendicular to the fixed platform, pointing upward, and the x-axis is orthogonal to one of the horizontal branches. The positions and orientations of the branches are described using vectors: for each branch i, the direction vector $\mathbf{S}_i$ represents its axial direction in the coordinate system, and the position vector $\mathbf{r}_i$ denotes a point on its axis. The relationship between the branch displacements and the platform’s generalized coordinates is derived from screw theory. The generalized coordinate vector $\mathbf{q} = [q_x, q_y, q_z, q_{mx}, q_{my}, q_{mz}]^T$ defines the platform’s translational and rotational motions, where $\mathbf{q}_1 = [q_x, q_y, q_z]^T$ represents translations along the x, y, and z axes, and $\mathbf{q}_2 = [q_{mx}, q_{my}, q_{mz}]^T$ represents rotations about these axes. The linear displacement $l_i$ of the i-th branch is given by:
$$ l_i = \mathbf{S}_i \cdot (\mathbf{q}_1 + \mathbf{q}_2 \times \mathbf{r}_i) = [\mathbf{S}_i^T \quad (\mathbf{r}_i \times \mathbf{S}_i)^T] \mathbf{q} $$
For all six branches, this can be expressed in matrix form as:
$$ \mathbf{l} = \mathbf{G}^T \mathbf{q} $$
where $\mathbf{G}$ is the transformation matrix that encapsulates the geometric configuration of the six-axis force sensor. This matrix is constant under small displacements, facilitating the derivation of velocity and acceleration relationships: $\dot{\mathbf{l}} = \mathbf{G}^T \dot{\mathbf{q}}$ and $\ddot{\mathbf{l}} = \mathbf{G}^T \ddot{\mathbf{q}}$.
The vibration analysis begins with the individual branch dynamics. Each branch is modeled as a spring-damper-mass system, where $m_i$ is the equivalent mass, $c_i$ is the equivalent damping coefficient, $k_i$ is the equivalent stiffness, $l_i$ is the axial displacement, and $f_i$ is the axial force. The equation of motion for a single branch is:
$$ m_i \ddot{l_i} + c_i \dot{l_i} + k_i l_i = f_i $$
For the entire set of branches, the equations are combined into matrix form:
$$ [\mathbf{m}] \ddot{\mathbf{l}} + [\mathbf{c}] \dot{\mathbf{l}} + [\mathbf{k}] \mathbf{l} = \mathbf{f} $$
where $[\mathbf{m}] = \text{diag}[m_1, m_2, m_3, m_4, m_5, m_6]$, $[\mathbf{c}] = \text{diag}[c_1, c_2, c_3, c_4, c_5, c_6]$, $[\mathbf{k}] = \text{diag}[k_1, k_2, k_3, k_4, k_5, k_6]$, $\mathbf{l} = [l_1, l_2, l_3, l_4, l_5, l_6]^T$, and $\mathbf{f} = [f_1, f_2, f_3, f_4, f_5, f_6]^T$. Substituting the kinematic relationships yields:
$$ [\mathbf{m}] \mathbf{G}^T \ddot{\mathbf{q}} + [\mathbf{c}] \mathbf{G}^T \dot{\mathbf{q}} + [\mathbf{k}] \mathbf{G}^T \mathbf{q} = \mathbf{f} $$
Considering the force-measuring platform as a rigid body with mass matrix $[\mathbf{M}_0] = \text{diag}[M_0, M_0, M_0, I_{ox}, I_{oy}, I_{oz}]$, where $M_0$ is the platform mass and $I_{ox}$, $I_{oy}$, $I_{oz}$ are its moments of inertia, the equation of motion under an external wrench $\mathbf{F}_w(t)$ is:
$$ [\mathbf{M}_0] \ddot{\mathbf{q}} + \mathbf{G} \mathbf{f} = \mathbf{F}_w(t) $$
Substituting the branch dynamics into this equation results in the overall system equation:
$$ [\mathbf{M}_0] \ddot{\mathbf{q}} + \mathbf{G} \left( [\mathbf{m}] \mathbf{G}^T \ddot{\mathbf{q}} + [\mathbf{c}] \mathbf{G}^T \dot{\mathbf{q}} + [\mathbf{k}] \mathbf{G}^T \mathbf{q} \right) = \mathbf{F}_w(t) $$
Simplifying, we obtain the differential equation of motion for the six-axis force sensor:
$$ [\mathbf{M}] \ddot{\mathbf{q}}(t) + [\mathbf{C}] \dot{\mathbf{q}}(t) + [\mathbf{K}] \mathbf{q}(t) = \mathbf{F}_w(t) $$
where $[\mathbf{M}] = [\mathbf{M}_0] + \mathbf{G} [\mathbf{m}] \mathbf{G}^T$ is the total mass matrix, $[\mathbf{C}] = \mathbf{G} [\mathbf{c}] \mathbf{G}^T$ is the total damping matrix, and $[\mathbf{K}] = \mathbf{G} [\mathbf{k}] \mathbf{G}^T$ is the total stiffness matrix. To find the natural frequencies, damping is neglected, and the free vibration equation is considered:
$$ [\mathbf{M}] \ddot{\mathbf{q}}(t) + [\mathbf{K}] \mathbf{q}(t) = 0 $$
Assuming a harmonic solution of the form $\mathbf{q}(t) = \mathbf{A} \cos(\omega t + \phi)$, where $\mathbf{A}$ is the amplitude vector, $\omega$ is the angular frequency, and $\phi$ is the phase angle, substitution leads to the eigenvalue problem:
$$ ([\mathbf{K}] – \omega^2 [\mathbf{M}]) \mathbf{A} = 0 $$
For non-trivial solutions, the determinant must vanish:
$$ |[\mathbf{K}] – \omega^2 [\mathbf{M}]| = 0 $$
Solving this equation yields the first six natural frequencies and corresponding mode shapes of the six-axis force sensor, which are critical for assessing dynamic performance.
The dynamic characteristics of the six-axis force sensor were experimentally evaluated using the impulse response method. The sensor was rigidly mounted on a fixed platform, and an impulse hammer was used to apply excitation along various directions on the force-measuring platform. An IEPE accelerometer, attached to the platform, captured the response signals, which were processed by a dynamic signal analysis system. The time-domain data were transformed into frequency-domain spectra via Fast Fourier Transform (FFT), producing frequency response curves. Peaks in these curves indicate the natural frequencies, as the system exhibits amplified responses when excited at these frequencies. The experimental setup ensured accurate measurement of the sensor’s dynamic behavior, providing data for comparison with theoretical predictions.
The theoretical natural frequencies were computed using numerical values derived from simulation software. The equivalent mass and stiffness for each branch were $m_i = 4.5312 \times 10^{-2} \, \text{kg}$ and $k_i = 4.06 \times 10^5 \, \text{N/m}$, respectively. The force-measuring platform had a mass $M_0 = 0.2350 \, \text{kg}$ and moments of inertia $I_{ox} = I_{oy} = 1.8084 \times 10^{-4} \, \text{kg} \cdot \text{m}^2$ and $I_{oz} = 3.4644 \times 10^{-4} \, \text{kg} \cdot \text{m}^2$. The mass, stiffness, and platform matrices were constructed as follows:
$$ [\mathbf{m}] = \begin{bmatrix}
4.5312 & 0 & 0 & 0 & 0 & 0 \\
0 & 4.5312 & 0 & 0 & 0 & 0 \\
0 & 0 & 4.5312 & 0 & 0 & 0 \\
0 & 0 & 0 & 4.5312 & 0 & 0 \\
0 & 0 & 0 & 0 & 4.5312 & 0 \\
0 & 0 & 0 & 0 & 0 & 4.5312
\end{bmatrix} \times 10^{-2} $$
$$ [\mathbf{k}] = \begin{bmatrix}
4.06 & 0 & 0 & 0 & 0 & 0 \\
0 & 4.06 & 0 & 0 & 0 & 0 \\
0 & 0 & 4.06 & 0 & 0 & 0 \\
0 & 0 & 0 & 4.06 & 0 & 0 \\
0 & 0 & 0 & 0 & 4.06 & 0 \\
0 & 0 & 0 & 0 & 0 & 4.06
\end{bmatrix} \times 10^5 $$
$$ [\mathbf{M}_0] = \begin{bmatrix}
0.2350 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.2350 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.2350 & 0 & 0 & 0 \\
0 & 0 & 0 & 1.8084 & 0 & 0 \\
0 & 0 & 0 & 0 & 1.8084 & 0 \\
0 & 0 & 0 & 0 & 0 & 3.4644
\end{bmatrix} \times 10^{-3} $$
After computing the total mass and stiffness matrices $[\mathbf{M}]$ and $[\mathbf{K}]$, the eigenvalue problem was solved using MATLAB. The results, along with experimental values, are summarized in the table below.
| Mode | Theoretical Natural Frequency (Hz) | Experimental Natural Frequency (Hz) | Vibration Characteristic |
|---|---|---|---|
| 1 | 1262.26 | 1156.25 | Translation along x-axis |
| 2 | 1262.26 | 1187.75 | Translation along y-axis |
| 3 | 1693.12 | 1450.25 | Rotation about z-axis |
| 4 | 1812.12 | 1729.16 | Rotation about x-axis |
| 5 | 1812.12 | 1762.50 | Rotation about y-axis |
| 6 | 1850.60 | 2016.67 | Translation along z-axis |
The comparison reveals close agreement between theoretical and experimental natural frequencies, with consistent trends across modes. The slight discrepancies arise from factors such as model simplifications, ignored damping effects, and experimental uncertainties. The symmetry of the six-axis force sensor design is reflected in the nearly identical frequencies for x and y translations and x and y rotations, validating the theoretical model. This analysis demonstrates the robustness of the orthogonal parallel six-axis force sensor in dynamic applications and confirms the efficacy of the derived mathematical framework.
In conclusion, this study provides a thorough theoretical and experimental examination of the dynamic characteristics of an orthogonal parallel six-axis force sensor. The vibration models, based on screw theory and multi-degree-of-freedom dynamics, accurately predict the sensor’s natural frequencies, as evidenced by experimental validation. The findings highlight the importance of dynamic performance in six-axis force sensor design and offer valuable insights for future optimizations in robotics and other precision engineering fields. The repeated emphasis on the six-axis force sensor throughout this work underscores its critical role in advanced force measurement systems.