Model-Based Estimation of Clamp Force in Bolted-Joint Tightening

Abstract

Bolted joints are widely used in mechanical assemblies, and their reliability depends strongly on the clamp force generated during tightening. Direct measurement of clamp force in industrial applications is often impractical. Conventional torquebased tightening methods estimate clamp force by assuming a constant coefficient of friction (CoF). However, the accuracy of this assumption is influenced by surface conditions, lubrication, tightening speed, and other factors. Therefore, accurate clamp force estimation remains a challenging problem. Previous Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF)-based approaches achieved estimation errors of approximately 3–5%, but were found to be sensitive to initial state assumptions. In addition, the coupling between clamp force and CoF can produce similar torque outputs for different state values, resulting in non-unique estimates. This thesis focuses on developing system formulations for tightening in the elastic region. The main states considered are clamp force and CoF, which are strongly coupled through the torque expression. To analyze the resulting non-uniqueness in state estimation, both local and global observability perspectives are considered. Multiple observer-based estimation approaches are then evaluated, including an LMI-based Luenberger observer using an LPV formulation, an output-inverted EKF, and a Rao-Blackwellized Particle Filter (RBPF). The EKF and RBPF formulations reduce the dimensionality of the directly estimated state space. The results indicate that the system is theoretically observable from the local observability analysis, but the near-marginal values of the observability matrix suggest limitations in state estimation. The output-trajectory-based observability study further shows trajectory-dependent regions of practical unobservability, making it difficult to generalize a single unobservable region across different tightenings. The LMI-based Luenberger observer was highly sensitive to initial-state errors and relied strongly on the motion model, whereas the output-inverted EKF was mainly limited by derivative-induced noise amplification in the measurement model. The RBPF provided improved estimations by maintaining multiple hypotheses, but its performance depended strongly on both initialization and the nonlinearity of the output trajectory. Overall, the study shows that reliable reconstruction of clamp force and CoF is fundamentally constrained by weak observability, low effective CoF contribution to the measured output, and model mismatch under nonlinear tightening conditions.