In modern industrial assembly processes, particularly for large-scale structures, precise measurement of contact forces is crucial for achieving active compliant assembly. Traditional methods often rely on geometric measurements, which can lead to forced assembly due to insufficient force control. This paper presents a novel approach using distributed three-dimensional force sensors to measure assembly contact forces, addressing the limitations of conventional six-axis force sensors. I will discuss the kinematic and dynamic modeling, gravity compensation strategies, error analysis, and experimental validation, emphasizing the advantages over six-axis force sensor-based systems.
The assembly positioning mechanism, typically a redundantly driven parallel system, consists of multiple three-coordinate numerical control positioners, a cradle, and distributed three-dimensional force sensors. These sensors measure the forces exerted by the positioners on the cradle, enabling real-time calculation of contact forces during assembly. Unlike a single six-axis force sensor, which suffers from complex inter-dimensional coupling and calibration challenges, distributed sensors offer improved accuracy and stability. The system’s kinematics relate the spatial pose of the assembly to the joint displacements, forming the basis for dynamic force computation.
To model the system, I define a base coordinate system \( O_b-xyz \) and an end-effector coordinate system \( O_t-xyz \). The position of spherical joint centers in these frames is given by:
$$ \mathbf{r}_b^i = \mathbf{R}_b^t \mathbf{r}_t^i + \mathbf{T}_b^t $$
where \( \mathbf{R}_b^t \) is the rotation matrix and \( \mathbf{T}_b^t \) is the translation vector. The end-effector pose is represented as \( \mathbf{w}_b^t = [x, y, z, \alpha, \beta, \gamma]^T \). The dynamics derive from Newton-Euler equations, accounting for inertial forces, gravity, and contact forces. For a system with three positioners, the equations in the base frame are:
$$ \sum_{i=1}^3 \mathbf{F}_b^i + \mathbf{G}_m + \mathbf{f}_e = m \mathbf{a}_b^t $$
and in the end-effector frame:
$$ \sum_{i=1}^3 \mathbf{r}_t^i \times \mathbf{F}_t^i + \mathbf{r}_t^m \times (\mathbf{R}_b^t)^T \mathbf{G}_m + \mathbf{M}_e = \mathbf{I}^t \dot{\boldsymbol{\omega}}^t + \boldsymbol{\omega}^t \times (\mathbf{I}^t \boldsymbol{\omega}^t) $$
Here, \( \mathbf{F}_b^i \) and \( \mathbf{F}_t^i \) are forces in base and end-effector frames, \( \mathbf{G}_m \) is gravity, \( \mathbf{f}_e \) and \( \mathbf{M}_e \) are contact force and moment, \( m \) is mass, \( \mathbf{a}_b^t \) is acceleration, \( \mathbf{I}^t \) is inertia tensor, and \( \boldsymbol{\omega}^t \) is angular velocity. Combining these, the contact force \( \mathbf{F}_e = [\mathbf{f}_e, \mathbf{M}_e]^T \) is computed as:
$$ \mathbf{I}_N – \mathbf{J}^T \mathbf{F}_B – \mathbf{R}_G \mathbf{G}_m = \mathbf{F}_e $$
where \( \mathbf{I}_N \) is the inertia matrix, \( \mathbf{J} \) is the Jacobian, \( \mathbf{F}_B \) is the force matrix, and \( \mathbf{R}_G \) is the gravity direction matrix. This formulation allows real-time force estimation without direct measurement, overcoming issues typical of six-axis force sensors.
Gravity compensation requires accurate knowledge of the end-effector’s center of mass. I propose a self-calibration method using multiple poses. When stationary, the moment balance gives:
$$ \sum_{i=1}^3 \mathbf{r}_b^i \times \mathbf{F}_b^i – \mathbf{r}_b^m \times \mathbf{G}_m = 0 $$
Expanding and transforming to the end-effector frame yields a linear system:
$$ \mathbf{H} \mathbf{r}_t^m = \mathbf{P} $$
where \( \mathbf{H} \) and \( \mathbf{P} \) are constructed from pose and force data. With \( n \) measurements, the least-squares solution is:
$$ \hat{\mathbf{r}}_t^m = (\mathbf{H}^T \mathbf{H})^{-1} \mathbf{H}^T \mathbf{P} $$
This calibration reduces errors in contact force calculation, especially compared to systems relying on six-axis force sensors, which require more complex calibration due to coupling effects.
Error analysis considers sensor measurement inaccuracies, installation angle deviations, and center-of-mass errors. For distributed three-dimensional force sensors, the force transformation accounts for installation angles:
$$ \mathbf{F}_b^i = \mathbf{R}_{s,i}^b \mathbf{F}_s^i $$
where \( \mathbf{R}_{s,i}^b \) is the rotation matrix for sensor \( i \). The contact force error is simulated numerically, assuming sensor errors of 0.5% FS and angle deviations under 0.01 rad. In contrast, a six-axis force sensor typically has lower accuracy (e.g., 1% FS) and higher sensitivity to calibration errors. The following table compares key parameters:
| Parameter | Distributed 3D Sensors | Six-Axis Force Sensor |
|---|---|---|
| Accuracy | 0.5% FS | 1% FS |
| Calibration Complexity | Moderate | High |
| Error Sources | Measurement, installation angles | Coupling, gravity compensation |
Simulations show that distributed sensors achieve lower standard deviations in force and moment errors. For instance, with optimal calibration (large pose variations and multiple weighings), force errors decrease by over 40% compared to six-axis force sensors. The standard deviation of contact force reduces from 2.34 N to 1.42 N, and moment from \( 1.630 \times 10^{-3} \) N·m to \( 0.862 \times 10^{-3} \) N·m. This highlights the superiority of distributed sensing for high-precision applications.
Experimental validation involved a helicopter swashplate ring assembly system. The setup included three-coordinate positioners, a cradle, ring components, and both distributed three-dimensional force sensors and a six-axis force sensor for comparison. The goal was to measure contact forces in free space, where actual forces are zero, to assess measurement noise and bias. The center-of-mass calibration used poses with maximum angles of 0.4 rad and six weighings, minimizing errors. Results confirmed that distributed sensors yielded more stable force readings, with smaller deviations in end-effector positioning. For example, position errors averaged 0.355 mm versus 0.553 mm with a six-axis force sensor, and orientation errors reduced by 30.9%.
The integration of distributed three-dimensional force sensors enables active compliant control by providing accurate force feedback. In assembly tasks, this allows real-time trajectory adjustments, reducing impact forces and structural damage. The method’s robustness stems from decentralized force measurement, which mitigates issues like moment amplification in large structures—a common problem for six-axis force sensors. Furthermore, the calibration process is efficient, requiring fewer pose changes than the six-or-more typically needed for six-axis sensor systems.
In conclusion, the distributed three-dimensional force sensor approach significantly enhances assembly contact force measurement precision. By leveraging dynamic modeling and multi-pose gravity compensation, it outperforms traditional six-axis force sensors in terms of error reduction and stability. This advancement supports the development of smarter, more adaptive assembly systems for large-scale industrial applications.