## 4 Formulation of Optimization Problem  
Without loss of generality, and under the assumption that the zero failure life test time \(T_{n}\) required for the gearboxes used in each joint of the robot are the same, all acceleration factors must be the same to ensure that the acceleration life test time \(T_{accel}\) is the same for each joint. All of the acceleration factors of the drive components derived from Equations (8)-(10) for each joint of the robot have non-negative values, so an inequality as shown in Equation (12) holds for their arithmetic and geometric means. Here, the degrees of freedom of robot is designated by \(N\) to make the expression more general. The necessary and sufficient condition to make them equal is that acceleration factors for all drive components have the same value. \[\frac{AF_{1}+AF_{2}+\cdots+AF_{N}}{N}\geq\sqrt[N]{AF_{1}\cdot AF_{2}\cdots AF _{N}}. \tag{12}\]  
Therefore, in this study, a function J, which represents the ratio of the arithmetic mean of the acceleration factors to their geometric mean, is defined as Equation (13), and its logarithmic value as in Equation (14) is defined as the cost function of the optimization problem. This ensures that the acceleration factor of each drive component converges to the same value when the global minimum is reached, where the cost function is zero as in Equation (15). \[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]\[\log\mathrm{J}=\log\frac{1}{\mathrm{N}}\sum_{i=1}^{\mathrm{N}}AF_{i}-\frac{1}{ \mathrm{N}}\sum_{i=1}^{\mathrm{N}}\log AF_{i}\geq 0, \tag{14}\]  
\[\log\mathrm{J}=0\leftrightarrow AF_{1}=AF_{2}=\cdots=AF_{N}. \tag{15}\]  
As a result, the problem of obtaining a motion profile in which all acceleration factor are the same during the predefined cycle time \(t_{f}\) can be defined as a constrained optimization problem that minimizes the logarithm of AM-GM ratio (Equation (16)) while satisfying constraints on torque \(\tau_{i}(t)\), speed \(\dot{q}_{i}(t)\), and displacement \(q_{i}(t)\) of the robot drive components as in Equations (18)-(22). Here, the Equation (17) is the dynamic equation which is solved from the Equations (1)-(6) and \(\tau_{i}^{max},q_{i}^{max}\) is limit on joint torque and speed, respectively. The lower and upper bound of each joint displacement is designated by \(q_{i}^{min}\) and \(q_{i}^{max}\), and the initial and final value of the joint angles during the specific cycle time \(t_{f}\) is \(q_{0}\) and \(q_{f}\). \[\min_{q(t)}(\log\frac{1}{\mathrm{N}}\sum_{i=1}^{N}AF_{i}-\frac{1}{\mathrm{N}} \sum_{i=1}^{N}\log AF_{i}), \tag{16}\]  
\[\mathrm{M}(q)\ddot{q}+\mathcal{C}\big{(}q,\ \dot{q}\big{)}\dot{q}+\mathcal{G}(q)=\tau, \tag{17}\]  
\[-\tau_{i}^{max}\leq\tau_{i}(t)\leq\tau_{i}^{max}, \tag{18}\]  
\[-\dot{q}_{i}^{max}\leq\dot{q}_{i}(t)\leq\dot{q}_{i}^{max}, \tag{19}\]  
\[q_{i}^{min}\leq q_{i}(t)\leq q_{i}^{max}, \tag{20}\]  
\[q(0)=q_{0},\ q\big{(}t_{f}\big{)}=q_{f}, \tag{21}\]  
\[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]0  
Since the motion profile that causes the acceleration factor of the gearboxes of each joint to be the same is not unique the above optimization problem may contain multiple global minimum. Therefore, the solution of the above constrained optimization problem can converge to different global optimal solutions depending on the initial setting values of the optimization variable. Moreover, according to the definition of the acceleration factor in Equation (10), the acceleration factor can be maximized by making the drive torque \(\tau_{i}(t)\) of the individual gearbox have two states of \(\pm\tau_{i}^{max}\). However, the drive torque \(\tau_{i}(t)\) obtained through such two state controls can cause chattering. In addition, the stepwise rapid change of drive torque can stimulate unmodeled dynamics such as the stiffness of the links and joints of the robot. This can lead to oscillatory robot motion, and the acceleration factor will also vary as the robot may not move as intended.