## 4 Joint Motion Planning Based on Modified Hermite Interpolation
### Arrangement for The Abscissas of Newly-Built Knots  
Given \(n\) initial knots to be interpolated denoted as \((x_{i-1},y_{i-1})\), \(I=1,2,\ldots,n\), \((n\geq 3)\). The abscissas of initial knots are \(x_{0}\), \(x_{1}\), \(\ldots\), \(x_{n-1}\). New knots are built for proposed interpolation, marked as \((r_{j-1}\), \(t_{j-1})\), \(j=1,2,\ldots,m\), \((m=2n-2)\). The abscissas of new knots are \(t_{0}\), \(t_{1}\), \(\ldots\), \(t_{m-1}\). We need to arrange \(t_{0}\), \(t_{1}\), \(\ldots\), \(t_{m-1}\) according to \(x_{0}\), \(x_{1}\), \(\ldots\), \(x_{n-1}\).  
The first idea is to average \(t_{0}\), \(t_{1}\), \(\ldots\), \(t_{m-1}\). Uniform abscissa knots are beneficial to the smoothness of the interpolation curves. However, averaging \(t_{0}\), \(t_{1}\), \(\ldots\), \(t_{m-1}\) may possibly lead to the situation that there are two initial knots existing between two new knots, seen in Figure 9. Since a piecewise curve must pass by the two initial points, too many constraints are imposed on this curve so that the curve function cannot be solved and interpolation functions cannot be found out.  
To avoid the trouble mentioned above, the following scheme is proposed on the basis of homogenization as much as possible. Take five initial knots for example. As can be seen in Figure 10, two newly-built knots \(R_{2}\) and \(R_{3}\) are built between the second and the third initial as described above.  
Based on the principle of average for new knots, the horizontal distance between the second and the third initial points is divided into three segments, so the abscissas \(t_{2}\) and \(t_{3}\) of the new knots \(R_{2}\) and \(R_{3}\) can be expressed as Equations (35) and (36) below. The same scheme is applied whenever there are two newly-built knots between every two adjacent initial points.  
Figure 10: Arrangement for abscissas of newly-built knots.  
Figure 9: Average for abscissas of newly-built knots.  
\[t_{2} =x_{1}+\frac{1}{4}(x_{2}-x_{1}) \tag{35}\] \[t_{3} =x_{1}+\frac{3}{4}(x_{2}-x_{1}) \tag{36}\]  
Between the first and the second initial points, is there only one newly-built knot \(R_{1}\) inserted, seen in Figure 10. The horizontal distance between the first and the second initial knots is divided by new point \(R_{1}\) into two segments with unequal length, so the golden ratio is a reasonable method. Consequently, the abscissa \(t_{1}\) of newly-built knot \(R_{1}\) can be expressed by Equation (37). The same is as for the last newly-built knot \(R_{6}\), and the abscissa \(t_{6}\) of the newly-built knot \(R_{6}\) can be expressed by Equation (38).  
\[t_{1} =x_{0}+0.618(x_{1}-x_{0}) \tag{37}\] \[t_{6} =x_{4}-0.618(x_{4}-x_{3}) \tag{38}\]  
Extended to the case that the number of the initial points is \(n\) (\(n\geq 4\)), formula (39) can be obtained. Abscissas of all newly-built knots \(\mathbf{t}\) are acquired from the abscissas of initial knots \(\mathbf{x}\). If there are only three initial points (\(n\)\(=3\)), take the first two Equations and the last two Equations in formula (39).  
\[\left\{\begin{array}{l}t_{0}=x_{0}\\ t_{1}=x_{0}+0.618(x_{1}-x_{0})\\ t_{j}=x_{i-1}+\frac{1}{4}(x_{i}-x_{i-1})\;i=2,3,\cdots,n-2,\;j=2,4,6,\cdots,m- 4\\ t_{j+1}=x_{i-1}+\frac{3}{4}(x_{i}-x_{i-1})\\ t_{m-2}=x_{n-1}-0.618(x_{n-1}-x_{n-2})\\ t_{m-1}=x_{n-1}\end{array}\right. \tag{39}\]  
where \(n\) and \(m\) (\(m\) = \(2n-2\)) are respectively the numbers of initial and newly-built knots. The subscript \(i\) is related to the initial knots while subscript \(j\) is related to the newly-built knots.