## 3 Accelerated Life Testing of Robot  
Among the drive components of the high-precision robot dealt with in this study, the high-precision gearbox is the most prone to deterioration due to friction and wear, and thus can be seen as a component that has the greatest influence on the durability of the robot. The life of the high-precision gearbox is determined by the life of the bearing supporting the input-side rotary shaft. Therefore, the zero failure test time \(T_{n}\) to verify the durability of the gearbox is derived from Equation (7) [3].  
\[T_{n}=B_{100p}\left[\frac{\ln(1-CL)}{n\cdot\ln(1-p)}\right]^{\frac{1}{\beta}}. \tag{7}\]  
where \(B_{100p}\) is the guaranteed lifetime according to the conditions of use, \(CL\) is the required confidence level, \(n\) is the number of samples, \(p\) is the cumulative probability of failure, and \(\beta\) is the shape parameter.  
The guaranteed lifetime for the rated use conditions of the gearbox is usually given by component-level life test results from the manufacturer. The precision gearbox used in a variable load environment can then be guaranteed in service life only when used within the permissible torque presented by the manufacturer, which is related to the service life of the bearings of the precision gearbox components.  
Ball bearings (e.g., in harmonic gearboxes) or roller bearings (e.g., in planocentric type 2-stage precision gearboxes) are used in precision gearboxes depending on the load class and characteristics [1; 2]. Bearings have a well-established theory and wealth of verifications, and the lifetimes of precision reducers are also calculated according to this theory.  
First, from the load torque \(\tau\) and speed \(\omega\) of the gearbox over a fixed cycle time of \(t_{f}\), equivalent load torque \(\tau_{eq}\) and equivalent speed \(\omega_{eq}\) are defined as Equations (8) and (9). Here, the index \(\alpha\) is determined by the type of bearings used: \(\alpha=3\) when ball bearings are used and \(\alpha=10/3\) when roller bearings are used as input rotary parts [20].  
\[\tau_{eq}=\left(\frac{\int_{0}^{t_{f}}\lvert\omega\rvert\tau\rvert^{\alpha}dt }{\int_{0}^{t_{f}}\lvert\omega\rvert dt}\right)^{1/\alpha}, \tag{8}\]  
\[\omega_{eq}=\frac{1}{t_{f}}\int_{0}^{t_{f}}\lvert\omega\rvert dt. \tag{9}\]  
Then, the acceleration factor \(AF\) of the motion profile having the equivalent load torque \(\tau_{eq}\) and the equivalent speed \(\omega_{eq}\) based on the rated torque \(\tau_{nte}\) and the rated speed \(\omega_{nte}\) of the robot's precision gearbox can be defined as the Equation (10).  
Consequently, according to the definition of zero failure test time \(T_{n}\) and the acceleration factor, the accelerated life test time \(T_{accel}\) can be calculated as in Equation (11) [21].  
\[AF=\left(\frac{\tau_{eq}}{\tau_{rate}}\right)^{\alpha}\left(\frac{\omega_{eq }}{\omega_{rate}}\right)=\frac{1}{t_{f}}\frac{\int_{0}^{t_{f}}\lvert\omega \rvert\tau\rvert^{\alpha}dt}{\tau_{nted}^{mt}\omega_{nted}}, \tag{10}\]  
\[T_{accel}=\frac{T_{n}}{AF}. \tag{11}\]  
The purpose of the present study is to design a test mode in which the system-level life test of precision gearboxes for robots is completed in the shortest time possible, and the durability of all gearboxes can be verified in one test without replacing samples during the life test.  
In other words, the goal is to design a motion profile that allows the accelerated life test time \(T_{accel}\) required by the precision gearbox mounted to each joint of the robot to be the same, and makes \(T_{accel}\) as small as possible.