# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods  
### Manipulability Ellipsoid  
The forward velocity kinematics of an \(n\) degree-of-freedom serial robotic manipulator, which we consider to be operating in an \(m\)-dimensional Euclidian operational space, describes the relation between the end-effector velocities \(t\), called twist, and joint velocities \(\hat{\theta}\):  
\[t=\begin{bmatrix}v\\ \omega\end{bmatrix}=J(\theta)\dot{\theta}, \tag{1}\]  
where \(t\) is defined as a set of the linear velocity vector \(v\in\mathbb{R}^{3}\) and the angular velocity vector \(\omega\in\mathbb{R}^{3}\); \(J(\theta)\in\mathbb{R}^{mxn}\) is a Jacobian matrix of the manipulator, with components of \(J_{ij}=\partial f_{i}(\theta)/\partial\theta_{j}\), where \(i=1\dots\ m,j=1\dots\ n\), and \(m=6\); \(\theta\in\mathbb{R}^{n}\) is the joint position vector, and \(\dot{\theta}\in\mathbb{R}^{n}\) defines the joint velocity vector. We assume rotational joints without a significant loss of generality.  
The most widely used kinematic performance measure, which is based on the Jacobian matrix, was introduced in [21]. The velocity manipulability ellipsoid of a given configuration provides an intuitive graphical representation of how efficient the mapping of velocities is from the joint space to the task-space. The set of joint velocities with the unit Euclidian norm associated with a unit sphere in the joint velocity space:  
\[E=\left\{\dot{\theta}\in\mathbb{R}^{n}\ \Big{|}\ \dot{\theta}^{T}\dot{\theta} \leq 1\right\} \tag{2}\]  
maps into the velocity manipulability ellipsoid \(\varepsilon\) in the task space:  
\[\varepsilon=\left\{t^{T}{(JJ^{T})}^{-1}t\leq 1\right\}. \tag{3}\]  
Note that \(\dot{\theta}^{T}\dot{\theta}\) is equal to the square of the Euclidian norm \(\left\|\dot{\theta}\right\|_{2}^{2}\) of a vector \(\dot{\theta}\). The volume of \(\varepsilon\), however, may be used to define the quantitative measure of manipulability \(w\):  
\[w=\sqrt{\det(JJ^{T})}, \tag{4}\]  
which is often used, although it suffers from a few limitations related to the physical inconsistency of the Jacobian matrix [19]. Furthermore, despite its popularity, the manipulability measure only gives a rough estimation of the robot's movement capability and closeness to the singularity, and it is not accurate enough to determine how fast the robot can move in a certain direction. The manipulability ellipsoid principal axes--the direction of which coincides with the eigenvectors of the square matrix \(JJ^{T}\), and whose lengths correspond to the singular values of \(J\)--represent the best and worst directions for the robot to perform a movement.  
A robot's movement capability in any other direction can be geometrically described as the distance from the center of the ellipsoid to the point where the line along the direction of interest intersects the ellipsoid surface. Let \(\hat{u}\) denote the unit vector in the task velocity direction of interest, and \(\mu_{2}\) be the distance from the ellipsoid center to the intersection point on the ellipsoid surface in the direction along the vector \(\hat{u}\). Thus, one can write \(t=\mu_{  
The velocity transmission ratio (6), also known as the manipulator velocity ratio [25] or directional manipulability measure [42], can also be derived from the ratio of the task velocity vector norm to the joint velocity vector norm:  
\[\mu_{2}=\|t\|/\left\|\hat{\theta}\right\|. \tag{7}\]  
The value of this index defines how effectively the robot's end-effector can move in the direction \(\hat{u}\) to satisfy condition \(\left\|\hat{\theta}\right\|_{2}\leq 1\) in the joint space. The direction in which the velocity transmission ratio is at its maximum is the optimal task direction for affecting velocity, as the lowest robot overall kinematic effort is needed. However, since a natural norm does not exist in the space used to represent the twist [29], which involves both linear and angular velocities, it leads to the physically non-consistent definition of the performance index with no clear physical meaning [19].