In recent years, industrial robots have become integral to numerous fields such as manufacturing, healthcare, and scientific research. A critical aspect of their operation involves the end effector, which often interacts with environments where collisions with objects or humans may occur. Accurate prediction of collision positions for the multi-degree-of-freedom (Multi-DOF) industrial robot end effector is paramount for enhancing safety, efficiency, and force perception capabilities. Traditional methods relying on optical or ultrasonic sensors are often costly and environmentally sensitive. In this study, I propose a novel contact position detection method based on a wrist force sensor mounted on the robotic arm. This method leverages dynamic mechanical data and a self-constraint model to predict collision locations with high precision and reliability, irrespective of the end effector姿态. Through rigorous experimentation, I demonstrate that this approach offers superior convergence and accuracy compared to conventional techniques.
The core innovation lies in the formulation of a dynamic mechanical data self-constraint collision localization model. When a collision occurs at the end effector, the wrist force sensor detects external forces and moments. I decompose these into components along three coordinate axes: forces \(F_x\), \(F_y\), \(F_z\) and moments \(M_x\), \(M_y\), \(M_z\). For a constant external force and moment during collision, the contact point \(P\) lies on a spatial curve known as the external force vector line, denoted as \(L_c(P)\). This line is defined by the following equation derived from equilibrium conditions:
$$ L_c(P): \quad \mathbf{r} \times \mathbf{F} + \mathbf{M} = 0 $$
where \(\mathbf{r}\) is the position vector from the sensor origin to the contact point, \(\mathbf{F} = (F_x, F_y, F_z)\) is the force vector, and \(\mathbf{M} = (M_x, M_y, M_z)\) is the moment vector. In practice, collisions are dynamic; the direction of force changes over time due to relative motion between the end effector and the object. Thus, at two distinct time instances \(t_1\) and \(t_2\), we obtain two non-parallel external force vector lines \(L_{c1}(P)\) and \(L_{c2}(P)\). The intersection of these lines, if it exists, yields the contact point \(P\):
$$ P \in L_{c1}(P) \cap L_{c2}(P) = \{(x, y, z) \mid \text{满足两组线性方程}\} $$
To solve for \(P\), I set up a system of linear equations based on the cross-product relation. For each time instance \(i\), the equation is:
$$ \begin{bmatrix} 0 & -F_z^i & F_y^i \\ F_z^i & 0 & -F_x^i \\ -F_y^i & F_x^i & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} M_x^i \\ M_y^i \\ M_z^i \end{bmatrix} = \mathbf{0} $$
This simplifies to:
$$ \mathbf{F}^i \times \mathbf{r} + \mathbf{M}^i = 0 $$
By combining equations from two time instances, I can solve for the coordinates \((x, y, z)\) of the contact point on the end effector. However, due to sensor noise and dynamic effects, the lines may not intersect perfectly in three-dimensional space, necessitating robust numerical methods.
In real-world scenarios, the force sensor data includes not only external collision forces but also internal forces (e.g., from joint friction) and dynamic forces (e.g., inertial effects). To isolate the external collision force crucial for localization, I perform data preprocessing. The raw sensor readings \(\mathbf{S}_{\text{raw}}\) are filtered using a high-pass Butterworth filter to remove low-frequency internal forces, followed by a window-based averaging to reduce noise. The processed force \(\mathbf{F}_{\text{ext}}\) and moment \(\mathbf{M}_{\text{ext}}\) are extracted as:
$$ \mathbf{F}_{\text{ext}} = \mathbf{S}_{\text{raw}} – \mathbf{F}_{\text{internal}} – \mathbf{F}_{\text{dynamic}} $$
where \(\mathbf{F}_{\text{internal}}\) is estimated via a pre-calibrated model of the end effector dynamics, and \(\mathbf{F}_{\text{dynamic}}\) is computed from the arm’s acceleration using Newton-Euler equations. This preprocessing ensures that the subsequent localization relies solely on collision-induced forces, enhancing accuracy. Table 1 summarizes the force components and their characteristics during collision at the end effector.
| Force Type | Symbol | Origin | Frequency Range | Processing Method |
|---|---|---|---|---|
| External Collision Force | \(\mathbf{F}_{\text{ext}}\) | Contact with object | Broadband (0-500 Hz) | High-pass filtering |
| Internal Force | \(\mathbf{F}_{\text{internal}}\) | Joint friction, gravity | Low (0-10 Hz) | Model-based subtraction |
| Dynamic Force | \(\mathbf{F}_{\text{dynamic}}\) | Arm acceleration | Medium (10-100 Hz) | Acceleration integration |
Due to measurement errors, the two external force vector lines \(L_{c1}\) and \(L_{c2}\) might not intersect in 3D space. To address this, I employ a projection method that solves for the contact point coordinate-wise by projecting onto coordinate planes. Let \(L_{c1}’\) and \(L_{c2}’\) be the projections of \(L_{c1}\) and \(L_{c2}\) onto the XY-plane. The intersection point \(P'(x, y)\) in this plane is found by solving the linear equations derived from the force-moment relations projected onto XY-plane. Similarly, I obtain projections onto YZ and ZX planes, yielding coordinates \(P”(y, z)\) and \(P”'(z, x)\). The final contact point \(P(x, y, z)\) is computed as the midpoint of the closest points between the lines in 3D, minimizing the perpendicular distance. Mathematically, for projections, I solve:
$$ \text{For XY-plane: } \frac{M_z}{F_z} = x F_y – y F_x \quad \text{(assuming \(F_z \neq 0\))} $$
This yields a linear equation in \(x\) and \(y\). By solving for two time instances, I get \(P’\). The 3D point \(P\) is then derived by combining results from all planes. The error in this projection method is bounded by sensor noise, and I quantify it using the root-mean-square error (RMSE) across multiple trials. The algorithmic steps for this projection-based solving are outlined in Table 2.
| Step | Action | Mathematical Formulation | Output |
|---|---|---|---|
| 1 | Project force-moment data onto XY-plane | Solve \(M_z^i = x F_y^i – y F_x^i\) for \(i=1,2\) | \(P'(x, y)\) |
| 2 | Project onto YZ-plane | Solve \(M_x^i = y F_z^i – z F_y^i\) for \(i=1,2\) | \(P”(y, z)\) |
| 3 | Project onto ZX-plane | Solve \(M_y^i = z F_x^i – x F_z^i\) for \(i=1,2\) | \(P”'(z, x)\) |
| 4 | Combine coordinates | \(x = \frac{x’ + x”’}{2}\), \(y = \frac{y’ + y”}{2}\), \(z = \frac{z” + z”’}{2}\) | \(P(x, y, z)\) |
| 5 | Compute error | \(\text{RMSE} = \sqrt{\frac{1}{N}\sum_{k=1}^N \| P_k – P_{\text{true}} \|^2}\) | Localization accuracy |
To enhance robustness across multiple data samples, I incorporate a minimum error search strategy. During a collision event, the wrist force sensor captures \(N\) sets of time-sampled data (where \(N \geq 2\)). For each pair of time instances \((i, j)\), I compute a candidate contact point \(P_{ij}\) using the projection method. The true contact point is identified by minimizing the error across all candidates. I define an error metric \(E_{ij}\) based on the consistency of force direction and moment equilibrium:
$$ E_{ij} = \| \mathbf{F}^i \times \mathbf{r}_{ij} + \mathbf{M}^i \| + \| \mathbf{F}^j \times \mathbf{r}_{ij} + \mathbf{M}^j \| + \lambda \| \mathbf{r}_{ij} – \mathbf{r}_{\text{prior}} \| $$
where \(\mathbf{r}_{ij}\) is the position vector for \(P_{ij}\), \(\lambda\) is a regularization parameter, and \(\mathbf{r}_{\text{prior}}\) is a prior estimate (e.g., from robot kinematics). The optimal contact point \(P_{\text{opt}}\) is selected as:
$$ P_{\text{opt}} = \arg \min_{P_{ij}} E_{ij} \quad \text{for all } i \neq j $$
This strategy effectively filters out outliers caused by sensor noise or transient dynamics, ensuring reliable localization for the end effector. The overall flowchart of the contact position detection process integrates data preprocessing, projection solving, and error minimization, as summarized in a later section.
To validate the proposed method, I designed and conducted two types of experiments: random point detection and deformation simulation. Both aimed to assess the accuracy, convergence, and robustness of the collision position prediction for the end effector under various conditions. The experimental setup involved a 6-DOF industrial robotic arm (simulated in a dynamic model) equipped with a wrist force/torque sensor. The end effector was a cylindrical probe, and collisions were induced with a cubic object of known geometry. Sensor data was sampled at 1 kHz, and processing was done in real-time using a custom algorithm.
In the random point experiment, I selected three orthogonal faces of the cube and marked two test points on each face, totaling six points. For each test point, the end effector was programmed to approach and collide at controlled velocities. At each point, 10 repeated trials were performed to gather statistical data. The traditional geometric constraint method (baseline) and my proposed force-based method were applied to estimate the contact point. The geometric method uses known end effector geometry and kinematics to triangulate position, whereas my method relies solely on force sensor data. Results were compared in terms of localization error (Euclidean distance from true point) and convergence time (iterations to stable solution). Table 3 presents the average errors and standard deviations across all trials.
| Test Point | True Position (mm) | Traditional Method Error (mm) | Proposed Method Error (mm) | Convergence Time (ms) |
|---|---|---|---|---|
| Face 1, Point A | (50, 30, 20) | 2.5 ± 0.8 | 0.8 ± 0.3 | 120 |
| Face 1, Point B | (-20, 40, 10) | 3.1 ± 1.0 | 0.9 ± 0.4 | 115 |
| Face 2, Point A | (10, -30, 50) | 2.8 ± 0.9 | 0.7 ± 0.2 | 130 |
| Face 2, Point B | (40, 10, -20) | 3.3 ± 1.2 | 1.0 ± 0.5 | 125 |
| Face 3, Point A | (-10, -20, -30) | 2.9 ± 1.1 | 0.8 ± 0.3 | 140 |
| Face 3, Point B | (30, -40, 0) | 3.0 ± 0.7 | 0.9 ± 0.4 | 135 |
The data clearly shows that my proposed method reduces average localization error by approximately 70% compared to the traditional method, with errors consistently below 1 mm. Convergence times are also faster, as the force-based approach requires fewer iterative adjustments. This demonstrates the high precision and efficiency of the method in predicting collision positions for the end effector under random contact scenarios.
For the deformation experiment, I simulated collisions where the object (cube) undergoes elastic deformation upon impact with the end effector. This is crucial because real-world objects may not be rigid, affecting force transmission and localization. I used a finite element model to simulate deformation, adjusting the Young’s modulus to represent materials like rubber or soft plastic. Ten test points were evenly distributed on the cube surface, and for each point, the end effector collided with varying force magnitudes. The force sensor data included components from deformation, which were accounted for in preprocessing by modeling the object’s stiffness matrix \(\mathbf{K}\). The contact force \(\mathbf{F}_{\text{contact}}\) is related to deformation \(\delta\) by:
$$ \mathbf{F}_{\text{contact}} = \mathbf{K} \delta $$
where \(\delta\) is computed from the end effector displacement. By incorporating this into the force balance, I modified the external force vector as \(\mathbf{F}_{\text{ext}}’ = \mathbf{F}_{\text{sensor}} – \mathbf{K} \delta\). The localization algorithm then proceeded as before. Results showed that even with up to 5 mm deformation, the method maintained an average error of 1.2 mm, compared to 3.5 mm for the traditional method which assumes rigidity. Table 4 summarizes the performance under deformation conditions.
| Deformation Level (mm) | Material Simulated | Proposed Method Error (mm) | Traditional Method Error (mm) | Force Variation (%) |
|---|---|---|---|---|
| 0 (Rigid) | Steel | 0.8 ± 0.3 | 2.9 ± 1.0 | 5 |
| 2 | Aluminum | 1.0 ± 0.4 | 3.2 ± 1.2 | 12 |
| 5 | Rubber | 1.2 ± 0.5 | 3.5 ± 1.3 | 25 |
| 10 | Soft Polymer | 1.5 ± 0.6 | 4.0 ± 1.5 | 40 |
The results indicate that my method is robust to object deformation, with errors increasing only marginally as deformation grows. This is attributed to the accurate modeling of force-deformation relationships and the adaptive preprocessing of sensor data. The end effector collision position prediction thus remains reliable even in non-ideal contact scenarios.
Further analysis involved evaluating the method’s performance across different end effector姿态s. I varied the robotic arm’s joint angles to achieve 20 distinct orientations and repeated the random point experiments. The localization error showed no significant correlation with orientation, with an overall RMSE of 0.9 mm and standard deviation of 0.2 mm. This confirms the method’s invariance to arm姿态, a key advantage for multi-DOF systems where the end effector orientation changes frequently. The error distribution across coordinate axes was also uniform, with X, Y, and Z errors averaging 0.85 mm, 0.88 mm, and 0.87 mm respectively, indicating balanced precision in all spatial dimensions.
To illustrate the computational efficiency, I measured the algorithm’s runtime on a standard industrial PC. For a single collision event with \(N=10\) data samples, the average processing time was 150 ms, which includes data preprocessing (50 ms), projection solving (70 ms), and error minimization (30 ms). This meets real-time requirements for industrial applications where the end effector operates at speeds up to 1 m/s. The algorithm’s scalability was tested by increasing the number of DOFs from 6 to 9; the error remained stable, though processing time rose linearly to 220 ms due to increased kinematic complexity.
In summary, the proposed force-based collision position prediction method for multi-DOF industrial robot end effector demonstrates high accuracy, robustness, and real-time capability. By leveraging wrist force sensor data, a self-constraint dynamic model, and innovative numerical techniques like projection solving and minimum error search, it overcomes limitations of traditional sensor-heavy approaches. The experiments validate its superiority in both rigid and deformable contact scenarios, with consistent performance across various end effector姿态s. Future work could integrate this method with machine learning for adaptive parameter tuning and extend it to collaborative robots where human-end effector interaction is critical. This research contributes significantly to advancing force perception in robotics, enhancing safety and precision in automated systems.