## 4 Formulation of Optimization Problem
Therefore, in this study, the displacement of the robot drive component \(q(t)\), which is an optimization variable, is limited to have a sufficiently smooth profile that is continuous up to the second derivative. This ensures the resulting load torque \(\tau_{i}(t)\) of the robot drive component is continuous function of the time. To this end, by parametrizing the displacement of each joint using B-spline as in Equation (23), we converted the constrained optimization problem defined in Equations (16)-(22) into a finite-dimensional constrained optimization problem [4; 5; 6; 7]. \[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]1  
where \(k\) is the order of the B-spline, \(m\) is the number of control points, \(p_{i}^{j}\) is the \(j\)-th control point for interpolation of the displacement of the \(i\)-th joint, and \(B_{j,k}(t)\) is the B-spline basis function. According to the definition of B-spline curve, the knot sequence was defined as a sequence \(\delta\) of \(m+k+1\) elements, and all elements in the sequence except for \(k+1\) overlapped elements at the beginning and end increase at equal intervals as the Equation (24)\[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]2  
When the displacement of the robot drive component is expressed using B-spline, the constraint Equations (21) and (22) on the displacement and speed of the optimization problem can be satisfied by setting four points of the B-spline control points as in Equations (25) and (26) [8]. \[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]3  
\[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]4  
Therefore, the optimal variable \(x\) is defined as in Equation (27) by augmenting \(m-4\) B-spline control points that describes the displacement of the drive components for each joint of the robot. \[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]5  
To summarize the above results, the constrained optimization problem of Equations (16)-(22) can be expressed as a finite-dimensional optimization problem of Equations (28)-(30) for the optimization variable \(x\). \[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]6  
\[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]7  
\[\text{J}=\frac{1}{N}\sum_{i=1}^{N}AF_{i}/(\prod_{i=1}^{N}AF_{i})^{\frac{1}{N}} \geq 1, \tag{13}\]8  
The linear inequality in Equation (29) represents the constraint in Equation (19) for the rotational speed of each joint in terms of the optimization variable \(x\), and the inequality in Equation (30) represents the constraint condition in Equation (20) for the displacement of each joint. The matrix \(A\) of Equation (29) is a block diagonal matrix and it is defined as a matrix having submatrix \(A_{i}\in\mathbb{R}^{(m-3)\times(m-4)}\) as elements. Submatrix \(A_{i}\) is a band diagonal matrix and is defined as Equation (31). \[A=diag\{A_{1},\ldots,\ A_{N}\},\ \ A_{i}(j,k)= \begin{cases}1&if\ \ j=k,\\ -1&if\ \ j=k+1,\\ 0&\text{otherwise}.\end{cases}  \tag{31}\]  
The vector \(b\) in Equation (29) is an augmentation of vector \(b_{i}\in R^{(m-3)}\), defined as in Equation (32). Then the deviation between adjacent control points of the B-spline was calculated by linear transformation of \(Ax+b\). \[b=\left[b_{1}^{T},\ \ldots,\ b_{N}^{T}\right]^{T},b_{i}=\left[-q_{i}(0),\ 0, \ldots,0,q_{i}\big{(}t_{f}\big{)}\right]^{T}. \tag{32}\]  
The vector \(c\) in the Equation (29) is an augmentation of vector \(c_{i}\in R^{(m-3)}\), defined as in Equation (33), and the vector \(c_{i}\) is derived by removing the first two and the last two elements from the knot sequence \(\delta\) of Equation (24) and using the \(k\) -th difference of thus formed sequence where \(k\) is the B-spline order.