# Region-Based Approach for Machining Time Improvement in Robot Surface Finishing
## 2 Preliminaries  
A free-form surface \(S\) in a three-dimensional space can be represented parametrically by the following vector function:  
\[r(u,v)=[x(u,v),y(u,v),z(u,v)] \tag{1}\]  
where \(u\) and \(v\) are parameters spanning the surface [37]. The explicit surface is examined, where the \(z\)-coordinate is represented as a function of the \(x\)- and \(y\)-coordinates:  
\[z=f(x,y) \tag{2}\]  
where \(f\) is a real-valued function that is at least twice continuously differentiable. The parametric form is obtained by setting \(x=u\) and \(y=v\), where \([x,y]\in D\), \(D\) being a connected domain in the \(xy\)-plane. The surface can be described by  
\[r=[x,y,f(x,y)] \tag{3}\]  
The unit normal vector at any point on the surface, necessary for many applications, including robot machining, is given by  
\[\hat{n}=\frac{r_{x}\times r_{y}}{\left\|r_{x}\times r_{y}\right\|} \tag{4}\]  
where vectors \(r_{x}\) and \(r_{y}\) span the tangent plane to the surface at a given point (\(x\),\(y\)) and are used to compute the surface normal. The first fundamental form, which encapsulates the surface's metric properties, is defined by the following matrix [37]:  
\[I= \begin{bmatrix}E&F\\ F&G\end{bmatrix}  \tag{5}\]  
with the first fundamental form coefficients \(E=r_{x}\circ r_{x}\), \(F=r_{x}\circ r_{y}\), \(G=r_{y}\circ r_{y}\). The second fundamental form, providing information crucial for understanding the surface curvature, is given by  
\[II= \begin{bmatrix}L&M\\ M&N\end{bmatrix}  \tag{6}\]  
The so-called second-form coefficients are derived from the directional derivatives of the normal vector \(L=r_{xx}\circ\hat{n}\), \(M=r_{xy}\circ\hat{n}\), \(G=r_{yy}\circ\hat{n}\). The relationship between the joint velocities and the end-effector's motion in the workspace is represented through the following velocity kinematics equation [34]:  
\[v=J(\theta)\dot{\theta} \tag{7}\]  
Here, \(J\in\mathbb{R}^{6\times n}\) denotes the geometric Jacobian matrix, which transforms the joint velocity vectors \(\dot{\theta}\) into operational velocity twist vector \(v\in\mathbb{R}^{6}\), encapsulating both the linear \(v\in\mathbb{R}^{3}\) and angular \(\omega\in\mathbb{R}^{3}\) velocities:  
\[v= \begin{bmatrix}v\\ \omega\end{bmatrix}  \tag{8}\]  
In robotic machining, especially in configurations where the robot manipulator is an open chain with six degrees of freedom (DOF), i.e., \(n=6\), kinematic analysis becomes crucial for effective operation. The manipulator's base is typically stationary, while the end-effector moves freely within the operational space to perform tasks accurately. The inverse kinematics problem must be solved to achieve the desired end-effector velocity. This involves determining the necessary joint velocities \(\dot{\theta}\) from given twist vector \(v\). The solution can be obtained by inverting the following full-rank, non-singular quadratic Jacobian matrix:  
\[\dot{\theta}=J^{-1}v \tag{9}\]  
The essence of manipulability in robotics is to understand how effectively a robot can change its position or orientation in space, which is quantified using the Jacobian matrix of the system. The manipulability measure is derived from the following condition:  
\[\dot{\theta}^{T}\dot{\theta}\leq 1 \tag{10}\]  
which maps the unit sphere in the joint velocity space onto an ellipsoid in the task space, described by  
\[v^{T}(JJ^{T})^{-1}v\leq 1 \tag{11}\]  
which is the equation of points on an ellipsoid's surface within the end-effector velocity space [38]. To analyze and simplify the manipulability condition, the Jacobian matrix \(J\) can be decomposed using the singular value decomposition (SVD) into  
\[J=U\sum V^{T} \tag{12}\]In this decomposition, \(U\) and \(V\) are unitary matrices representing the orthonormal bases of the Jacobian column and row space.