## 7 Conclusions  
We have presented a new method for better monitoring and controlling the part deformation when an aero-engine casing is being machined. By integrating force sensors into the fixture, the deformation force on the part can be monitored. However, as the auxiliary supports of aero-engine casing fixture acting on the sidewalls can only provide a one-way support force, the part and fixture may be separated from each other during machining, making the monitoring of the deformation force impossible. As a solution, we have proposed a deformation force monitoring method based on applying a pre-support force. However, due to the introduction of pre-support force, the following three major concerns are introduced:  
1. Besides the deformation force, the clamping force is also affected by the stiffness of the part.
2. The deformation cannot be determined in advance, which leads to a low success rate in monitoring the deformation force.
3. The pre-support force must not be too large, as it would cause a large clamping deformation and hence affect the machining accuracy.  
We solve the above three problems through the following work. First, a set of sophisticated clamping devices with force sensors are developed. Second, we establish the conditions of auxiliary support failure and a method for calculating the deformation force based on the concept of equivalent flexibility. Third, we propose a probabilistic model to predict the deformation force of the future subsequent machining step by the use of deep autoregressive network and Kalman filtering. The probabilistic model predicts the subsequent deformation force based on the preceding deformation force, which can lower the risk of auxiliary support failure, while the Kalman filtering helps reduce the modeling and measurement errors. Ultimately, the support force is determined by the probabilistic model of deformation force prediction and the equivalent flexibility, whose use prevents the auxiliary support from failing (i.e., to guarantee the required deformation force).  
Both computer simulation and physical machining verification are carried out, and the experimental results confirm the following. (1) The proposed deformation force prediction probability model can reduce the risk of auxiliary support failure--when the confidence is 84.1%, the success rate of deformation force monitoring is increased by about 30% compared with the traditional approach. (2) The proposed pre-support force decision method can effectively control the clamping deformation--the final impact of clamping deformation of the proposed method is less than 0.003 mm. (3) The proposed methodology for calculating and monitoring the deformation force is highly accurate--the calculation error of deformation by the proposed method based on using the predicted deformation force is less than 0.008 mm.  
Our ongoing and further work is to incorporate the mechanics of elasticity into the predictive model to reduce the amount of training data required for the model and its accuracy. Furthermore, the problem of how to exert the support force to better control the deformation will also be investigated.  
Conceptualization, H.G. and C.L.; Funding acquisition, Y.L.; Methodology, C.L.; Resources, Y.L.; Supervision, Y.L.; Validation, H.G.; Writing--original draft, H.G.; Writing--review & editing, C.L., Y.N. and K.T. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China, grant number 52175467 and the National Science Fund of China for Distinguished Young Scholars, grant number 51925505.  
Not applicable.  
The authors declare no conflict of interest.