# Numerical Quantification of Controllability in the Null Space for Redundant Manipulators
## 2 Method
### Null Space Quality Ellipsoid  
The null space quality ellipsoid, along with the singular values and singular vectors, represents the controllable direction of a robot in the null space. As the null space projected system has relatively low DoF due to the constrained joint motion, the controllable direction and the appropriate force input should be identified. The null space quality ellipsoid could be obtained with the singular values \(\sigma\) and the vectors \(V_{i}\) of \(J_{N}^{T}\) as shown in Figure 2.  
\[V=\begin{bmatrix}V_{1_{x}}&\cdots&V_{k_{x}}\\ V_{1_{y}}&\cdots&V_{k_{y}}\\ V_{1_{z}}&\cdots&V_{k_{z}}\\ V_{1_{rx}}&\cdots&V_{k_{rx}}\\ V_{1_{ry}}&\cdots&V_{k_{ry}}\\ V_{1_{rx}}&\cdots&V_{k_{rx}}\\ \end{bmatrix} \tag{13}\]  
The column vector \(V_{i}\) represents the controllable DoF in the null projected task space, while the singular values represent the axis length of the ellipsoid. The directional vectors \(V_{i_{x}},V_{i_{y}},\) and \(V_{i_{z}}\) can be normalized to express the unit vector \(V_{i_{X}}\). By using this ellipsoid, the local solution can be obtained to determine which direction the body may move along, and how the body moves well along that direction. Furthermore, by projecting the force vector to the ellipsoid, the appropriate force can be assigned to the null projected system to improve the control performance.  
\[F_{L}=K_{p}V_{i_{X}}\frac{V_{i_{X}}\cdot(x_{d}-x)}{|V_{i_{X}}\cdot(x_{d}-x)|} \left|x_{d}-x\right|-K_{p}\hat{x} \tag{14}\]  
Next, the null axis that the controlled body may rotate along can be obtained as shown in Figure 2. The articulated manipulator with the constraint can rotate their body along the specified axis. It is important to find the specified axis to predict the direction and the path of the controlled body within the null space. With a rotation term of the singular vector  
Figure 2: The comparison between the manipulability ellipsoid and the null space quality ellipsoid are shown in green and blue, respectively. The null space quality ellipsoid is displayed while controlling 3-DoF (x, y, and z) at the end-effector as a primary task. The controllable ellipsoid of the secondary task is changed from the manipulability ellipsoid to the null space quality ellipsoid by the null space projection. Furthermore, a 1-DoF of the null axis is shown in magenta.  
including \(V_{i_{rx}},V_{i_{ny}}\), and \(V_{i_{z}}\), the null axis that the body rotates along can be obtained. For example, the rotation matrix of the null axis \(R_{N}\) using the Roll-Pitch-Yaw method can be obtained from the z-axis as follow:  
\[Z^{\prime}=R_{N}Z=\begin{bmatrix}cos(\theta)&sin(\theta)sin(\phi)&sin(\theta) cos(\phi)\\ 0&cos(\phi)&-sin(\phi)\\ -sin(\theta)&cos(\theta)sin(\phi)&cos(\theta)cos(\phi)\end{bmatrix}\begin{bmatrix} 0\\ 0\\ 1\end{bmatrix}\]  
\[=\begin{bmatrix}sin(\theta)cos(\phi)\\ -sin(\phi)\\ cos(\theta)cos(\phi)\end{bmatrix}=\begin{bmatrix}V_{i_{rx}}\\ V_{i_{ny}}\\ V_{i_{rz}}\end{bmatrix}\]  
\[\phi=-asin(V_{i_{ny}})\]  
\[\theta=atan2(V_{i_{rx}},V_{i_{nz}}) \tag{15}\]  
\(\theta\) and \(\phi\) are the rotation angle in the pitch and yaw direction, respectively. By using (15), the rotation matrix of null axis \(R_{N}\) can be obtained. Based on this method, the null space motion that moves along the null axis can be predicted.  
Even though the controllable direction has been identified, the null space projector should be considered. Once the projecting method has been used, the motion of the controlled body remains unchanged, even with or without the null space projector. However, on the joint space level, different joints are activated, which is likely to cause interference to the primary task without the null space projection. Therefore, we have considered the motion of the robot not only in task space level, but also in the joint space level as well.