## 2 Preliminaries
### Kinematics Description of Robot Manipulator  
Generally speaking, two methods of describing the forward kinematics of open chains currently exist. The first method relies on the Denavit- Hartenberg parameters and the second relies on the PoE formula. Indeed, the latter method has advantages over the former (e.g., no link frames are necessary) [25], and it is, therefore, the preferred choice with respect to representing forward kinematics.  
The PoE formula used to describe the forward kinematics \((\mathbf{T}(\mathbf{\Theta}),\mathbf{\Theta}\in\mathbb{R}^{n})\) of an open chain with \(n\)-degrees of freedom follows Equation (1)  
\[\mathbf{T}(\mathbf{\Theta})=\begin{bmatrix}\mathbf{R}(\mathbf{\Theta})&\mathbf{P}(\mathbf{\Theta})\\ \mathbf{0}&1\end{bmatrix}=e^{\mathbf{S}_{1}\theta_{1}}\cdots e^{\mathbf{S}_{n-1}\theta_{n -1}}e^{\mathbf{S}_{n}\theta_{n}}\mathbf{M}. \tag{1}\]  
Specifically, Equation (1) is the space form of the PoE formula, referring to the fact that the screw axes are expressed in the fixed space frame, \(\mathbf{R}(\mathbf{\Theta})\in\text{{specialorthogonalgroup}}(\text{{SO}}(3))\), \(\mathbf{P}(\mathbf{\Theta})\in\mathbb{R}^{3\times 1}\), \(\mathbf{0}\) denotes the (\(1\times 3\)) null vector.  
To calculate the homogeneous transformation matrices \((\mathbf{T}(\mathbf{\Theta}))\), which represent both the position and orientation of the end-effector with respect to the base frame, the elements outlined below are required.  
(a) The end-effector configuration \(\mathbf{M}\in\text{{Special Euclidean Group }}(\text{{SE}}(3))\) is defined as  
\[\mathbf{M}=\begin{bmatrix}\mathbf{R}_{\mathbf{m}}&\mathbf{P}_{\mathbf{m}}\\ \mathbf{0}&1\end{bmatrix}, \tag{2}\]  
when the robot manipulator is at its home position.  
(b) The screw axes are  
\[\mathbf{s}_{i}=\begin{bmatrix}\mathbf{\omega}_{i}\\ \mathbf{v}_{i}\end{bmatrix}=[\omega_{i_{1}},\omega_{i_{2}},\omega_{i_{3}},v_{i_{1} },v_{i_{2}},v_{i_{3}}]^{T},\quad(i=1\cdots n), \tag{3}\]  
where  
\[\mathbf{v}_{i}=-\mathbf{\omega}_{i}\times\mathbf{q}_{i} \tag{4}\]  
with \(\mathbf{q}_{i}\) (any arbitrary point on the joint axis \(i\)) written in coordinates in the fixed base frame, corresponding to the joint motions when the robot manipulator is at the home position. Since the screw axis \(\mathbf{s}_{i}\) is just a normalized twist, \(\mathbf{S}_{i}\) represents Equation (5)  
\[\mathbf{S}_{i}=\begin{bmatrix}\mathbf{\Omega}_{i}&\mathbf{v}_{i}\\ \mathbf{0}&0\end{bmatrix}, \tag{5}\]  
where \(\mathbf{\Omega}_{i}\) is a \(3\times 3\) skew-symmetric matrix representation of \(\mathbf{\omega}_{i}\) and is defined as \[\mathbf{\Omega_{i}}=\begin{bmatrix}0&-\omega_{i_{3}}&\omega_{i_{2}}\\ \omega_{i_{3}}&0&-\omega_{i_{1}}\\ -\omega_{i_{2}}&\omega_{i_{1}}&0\end{bmatrix}. \tag{6}\]  
(c) The joint variables are represented as follows: \(\mathbf{\Theta}=[\theta_{1},\theta_{2},\cdots,\theta_{n}]^{T}\).  
(d) The matrix exponential \(e^{s_{i}\theta_{i}}\) can be calculated using Equation (7)  
\[e^{s_{i}\theta_{i}}=\begin{bmatrix}\mathbf{R}_{i}(\theta_{i})&\mathbf{P}_{i}(\theta_{ i})\\ \mathbf{0}&1\end{bmatrix}, \tag{7}\]  
where \(\mathbf{R}_{i}(\theta_{i})\) and \(\mathbf{P}_{i}(\theta_{i})\) are calculated as  
\[\mathbf{R}_{i}(\theta_{i}) =e^{\mathbf{\Omega}\theta_{i}}=\mathbf{I}+\mathbf{\Omega}_{i}\sin\theta_{i}+ \mathbf{\Omega}_{i}^{2}(1-\cos\theta_{i}), \tag{8}\] \[\mathbf{P}_{i}(\theta_{i}) =\left((\mathbf{I}+\mathbf{\Omega}_{i}^{2})\theta_{i}+\mathbf{\Omega}_{i}(1- \cos\theta_{i})-\mathbf{\Omega}_{i}^{2}\sin\theta_{i}\right)\mathbf{v}_{i}, \tag{9}\]  
where \(\mathbf{I}\) denotes the (\(3\times 3\)) identity matrix.  
The formulas listed in this section will be used for calculating the formulas in the NICCD method.