# Uncertainty Analysis of Spherical Joint Three-Dimensional Rotation Angle Measurement
## 1 Introduction  
In modern measurement systems, the multi-input multi-output measurement model is widely used. In practical work and learning, many factors usually affect the output of dependent variables, and many independent variables often interact with each other in the measurement system. A series of measurement systems, such as the geometrical deviation modeling of a 3D surface [1], fault detection, diagnosis in large-scale multi-machine power systems [2], wireless power combining and delivering systems [3], polarization angle estimation of radar [4], and the evaluation of alpha-synuclein aggregation states [5] are examples of multi-input multi-output measurement models. The quality of the measurement system can be understood by the uncertainty evaluation of the system. The uncertainty evaluation methods of multiple-input and multiple-output measurement systems usually include the Monte Carlo method (MCM), Bayesian probability, and indirect evaluation methods based on neural networks. [PERSON] et al. [6] used an uncertainty evaluation method of an adaptive neural network algorithm to obtain the uncertainty value of the fringe image center extraction measurement system in real time. [PERSON] et al. [7] studied the evaluation of the roundness error by comparing the Bayesian method with the method provided by ISO. [PERSON] et al. [8] proposed a probabilistic neural network with Gaussian mixture distribution parameters to improve the detection accuracy of manufactured porosity monitoring systems and obtain confidence intervals. [PERSON] et al. [9] established the weight change combined prediction model by using the genetic algorithm optimized back propagation method. [PERSON] et al. [10] established a Bayesian network to predict the corn yield in the United States and evaluate the uncertainty of the prediction. [PERSON] et al. [11] used the backpropagation neural network and least squares support vector machine to establish a mathematical model, verified the accuracy of themodel, and proved the effectiveness of MCM in the whiplash test model. [PERSON] et al. [12] evaluated the uncertainty of various influencing factors by using the Monte Carlo method through the uncertainty analysis of dissolved oxygen in Danube River water in Serbia. [PERSON] et al. [13] proposed an uncertainty evaluation method of a fiber optic gyroscope angle measuring system based on MCM for angle transmission and traceability technology. [PERSON] et al. [14] used MCM to evaluate and combine the uncertainty of obtained model parameters and verified its effectiveness in evaluating the uncertainty of a sensor dynamic system. [PERSON] [15] demonstrates how the posterior distribution can be used to build shape models that better generalize and visualize the uncertainty in the established correspondence. [PERSON] [16] presents a dynamic evaluation model of Markov chain Monte Carlo (MCMC) roundness error measurement uncertainty based on a stochastic process. [PERSON] [17] uses the MCMC method to calculate the uncertainty of the measurement reliability model and the optimal calibration interval. [PERSON] [18] uses the Monte Carlo method (MCM) and AMCM to evaluate the uncertainty of pose results. The above methods of uncertainty evaluation, based on neural networks and the Bayesian method, were too complicated, the technical threshold was high, and the sources of uncertainty were not considered comprehensively. For a dynamic measurement system with multiple inputs and multiple outputs, the uncertainty factors of measurement results were complex, and quantifying the influence of various error sources on measurement results, as well as and carrying out error transmission, is difficult. Therefore, MCM is more suitable to evaluate the quality of the measurement results of such systems. In the early stage, our team developed 2D or 3D rotation angle measurement systems based on the magnetic effect method and the combination of eddy current sensors and pseudorandom coding [19; 20; 21]. The magnetic effect method measures the value of the magnetic field.