The acquisition of rotation angle and pose information of precision spherical joint is of great importance for error motion prediction analysis and motion control. In the early stage, an analytical measurement model based on the equivalent magnetic charge method has been purposed to realize the rotation direction identification and rotation angle measurement. However, several shortcomings were observed such as complicated calculations and time-consuming, and the solution accuracy of the theoretical model was decreasing with the expansion of measurement range. To improve this situation, new modeling methods based on artificial neural network have been researched. This paper employs radial basis function (RBF), extreme learning machine (ELM), and RBF–ELM hybrid neural networks to construct measurement models, then analyzes and compares their effectiveness to obtain the optimal algorithm. Analysis results show that the RBF–ELM hybrid neural network has a better calculation accuracy than the others. Finally, the experimental data are used to train and test the network model, and the error between output angle of the model and the actual measured rotation angle is calculated. The comparison results show that the measurement model based on the RBF–ELM hybrid neural network has a higher calculation accuracy and generalization capability. Within the range of ±20°, the maximum error of rotation angle around the X and Y axes are $9'~48''$ and $6'~55''$ , respectively, and the root mean squared error is $1' 59''$ .

Conventional high precision spherical joints provide three degrees of freedom for rotation, but lack the capability to measure the rotation angle by themselves. This limitation restricts their use in some high-precision applications. However, by utilizing the non-linear magnetic field of permanent magnets in conjunction with magnetic sensors, measurement of rotation angle becomes achievable. Based on previous research, the measurement of three-dimensional rotation angle of spherical joint was investigated. Unlike the mathematical models and optimization algorithms, the artificial neural network model was adopted to achieve the measurement. The influences of different factors such as the type of neural network, sampling interval, data partition, and the quantity of sensor on the measurement accuracy were explored. Some empirical equations of measurement accuracy were summarized. The distributions of measurement accuracy of 3D rotation angles were quantitatively known. The combined ELM-RBF neural network model was proposed to realize real-time and high-precision 3D angle measurement. The optimal measurement accuracies of three-dimensional rotation angles α, β, and γ of the spherical joint with the sampling interval of 0.1°, eight sensors, data partition of γ rotation angle, and combined ELM-RBF neural network reach 0.01°. It can provide more guidance for further research.

A precision spherical joint is a type of spherical motion pair that can realize three degrees of rotation freedom. In this paper, a specific method is used to assess the uncertainty of our measurement system. The measurement system uses the codes and eddy current sensor to measure the angle. Different codes are engraved on the surface of the spherical joint. Eddy current sensors are embedded in the spherical socket to detect and recognize the spherical code. The uncertainty of the measurement system hardware and an artificial neural network are studied. Based on the Monte Carlo method, the uncertainty components are evaluated and combined, and the comprehensive uncertainty of the measurement system is obtained. The evaluation results of system measurement uncertainty are verified from specific angles. In the three-axis measurement uncertainty, the minimum is about 2′, and the maximum uncertainty is about 1°. The proposed method can be used to evaluate the uncertainty of other multiple-input multiple-output measurement systems.

To improve the measurement accuracy of three-dimensional rotation angle of the spherical joint, a novel approach is proposed in this study, which combines the magnetic detection by Hall sensor and surface feature identification by eddy current sensor. Firstly, a permanent magnet is embedded in the ball head of spherical joint, and Hall sensors are set and distributed in the ball socket to measure the variation of magnetic flux density when spherical joint rotating, which are related to 3D rotation angle. In order to further improve measurement accuracy and robustness, we also set grooves on the ball head and use eddy current sensors to synchronously identify the rotation angle of the ball head. After the combination of two signals is adopted, a measurement model is established using the RBF neural network by training, and realized real-time measurement of the 3D rotation angle of spherical joint. The feasibility and superiority of this method are validated through experiments. The experimental results indicated that the measurement accuracy is promoted substantially compared to the preliminary measurement scheme based on spherical coding, the average measurement error of single axis is reduced by 9'9". The root mean square errors for the measurement of 3D rotation angles in this proposed method are as follows: the pitch angle α has an error of 1'8", the yaw angle β has an error of 2'15", and the roll angle γ has an error of 29'6".

To improve the measurement accuracy of the three-dimensional rotation angle of a spherical joint, a novel approach is proposed in this study, which combines magnetic detection by a Hall sensor and surface feature identification by an eddy current sensor. Firstly, a permanent magnet is embedded in the ball head of a spherical joint, and Hall sensors are set and distributed in the ball socket to measure the variation in the magnetic flux density when the spherical joint rotates, which are related to the 3D rotation angle. In order to further improve the measurement accuracy and robustness, we also set grooves on the ball head and use eddy current sensors to synchronously identify the rotation angle of the ball head. After the combination of two signals is performed, a measurement model is established using the RBF neural network by training, and the real-time measurement of the 3D rotation angle of the spherical joint is realized. The feasibility and superiority of this method are validated through experiments. The experimental results indicate that the measurement accuracy is substantially promoted compared to the preliminary measurement scheme based on spherical coding; the average measurement error of the single axis is reduced by 9′9″. The root mean square errors for the measurements of the 3D rotation angles in this proposed method are as follows: pitch angle α has an error of 1′8″, yaw angle β has an error of 2′15″, and roll angle γ has an error of 29′6″.