## 2 Device and Method  
### Torsion Pendulum and Its Dynamics  
The torsion pendulum rotates in a horizontal plane perpendicular to the axis, supported by suspended threads or flexural pivots that provide the restoring torque. It is commonly used to measure micro thrust or micro impulse through angular deflection in the horizontal direction, due to the balance of gravity of the pendulum frame. In theory, the equation of motion of a torsional pendulum can be expressed as:  
\[J\theta^{\prime\prime}+\ \lambda\theta^{\prime}+\ K\theta\ =\ \tau(t), \tag{1}\]  
where \(J\) is the moment of inertia of the torsion pendulum, \(\theta\) is the angular displacement of the torsion pendulum, \(\lambda\) is the damping coefficient of the torsion pendulum, \(K\) is the torsional stiffness of the torsion pendulum, and \(\tau(t)\) is the external force acting on the torsion pendulum.  
In the actual measurement, the displacement sensor is usually used to measure the displacement of the endpoint of the torsion pendulum, and then the electrostatic force or electromagnetic force is used for calibration. When the torsional pendulum is statically balanced and coupled without gravity, the relationship is satisfied:  
\[\tau(t)=FL=K_{0}\theta, \tag{2}\]  
where \(F\) is the thrust to be measured, \(L\) is the torque arm, and \(K_{0}\) is the torsional stiffness of the pivot. Generally, when the thrust measurement period is short, the relationship between and is constant. However, when the measurement period is long, because of the thermal strain of the structure [17], the installation base of the torsion pendulum is no longer fixed, the tilt caused by thermal expansion and cold contraction, the center of mass of the pendulum frame deviates from the rotating axis, so that the quasi-static balance relationship of the torsion pendulum becomes:  
\[FL\ +Mgsin(\alpha)l_{cm}=K_{0}\theta, \tag{3}\]  
where \(M\) represents the mass of the pendulum frame, \(g\) denotes the acceleration due to gravity, \(l_{cm}\) signifies the distance between the center of mass of the pendulum frame and the axis of rotation, and \(\alpha\) represents the angle between the axis of rotation and the direction of gravity, as well as the angle of tilt of the horizontal plane. In the given equation, the force being measured is balanced by the gravitational force acting on the pendulum frame and the restorative force of the rotating shaft. Additionally, the reference tilt and the offset of the center of gravity contribute to the noise in the force.  
For the convenience of research, as shown in Figures 1 and 2, the length direction of the pendulum frame is defined as \(x\), and the vertical direction of the pendulum frame is defined as \(y\). The reference inclination \(\alpha\) can be decomposed into rotations around \(x\) and \(y\), corresponding to the roll and pitch of the pendulum frame, respectively. Similarly, \(l_{cm}\) can be decomposed into the horizontal offsets along the \(x\) and \(y\) directions, denoted as \(l_{cm,x}\) and \(l_{cm,y}\). In this paper, the inclination in the horizontal plane can be decoupled into roll or pitch, which correspond to the combinations of \(\phi_{x}\) and \(l_{cm,y}\), or \(\phi_{y}\) and \(l_{cm,x}\), respectively. Although it is not possible to completely decouple pitch and roll when discussing the sensitivity to inclination, it is still possible to discuss a single-dimensional coupled model and remove the influence of inclination on force measurement results.