# Numerical Quantification of Controllability in the Null Space for Redundant Manipulators
## 2 Method
### Null Space Quality  
The 'null space quality' is the index that numerically quantifies the manipulating ability of the robot in the null projected task space. In the null space, the controllable direction is changed not only in the joint space, but also in the task space. For enhanced task space control in the null space, it is necessary to identify the controllable direction in the null projected task space. For identification process, the components about the constrained joint motion by null space projector and the manipulating ability of the controlled body should be considered. The null projected Jacobian transpose with respect to the null space projector, \(N_{P}^{T}\), and Jacobian of the controlled body, \(J_{S}^{T}\), can be expressed as follows:  
\[J_{N}^{T}=N_{P}^{T}J_{S}^{T} \tag{7}\]  
The null projected Jacobian contains the relationship between the joint and the task motion in the null space. The relationship can be obtained from the singular values and vectors by the singular value decomposition (SVD) as follow:  
\[J_{N}^{T}=USV^{T} \tag{8}\]where \(U\in R^{n\times n}\) and \(V\in R^{m\times m}\) are orthogonal unitary matrices involving singular vectors of \(J_{N}^{T}\), and \(S\in R^{n\times m}\) includes singular values, \(\sigma_{i}\), in the diagonal. The rank of the \(J_{N}^{T}\) can be determined as follows:  
\[k=\begin{cases}n-m_{H}&\text{if }(n-m_{H})\leq m_{L}\\ m_{L}&\text{if }(n-m_{H})>m_{L}\end{cases} \tag{9}\]  
The relationship between \(\delta x\), the infinitesimal displacements of the controlled body, and \(\delta q\), the infinitesimal joint angular displacement is defined by \(\delta x=J\delta q\). Next, by letting the SVD of Jacobian be \(J=V_{J}S_{J}U_{J}^{T}\) and substituting it into the equation \(\delta x=J\delta q\), the following equation can be derived:  
\[\begin{split}\delta x&=J\delta q\\ &=V_{J}S_{J}U_{J}^{T}\delta q\\ &=V_{J}\delta q^{\prime}\qquad\qquad(where\quad\delta q^{\prime}=S_{ J}U_{J}^{T}\delta q^{\prime})\end{split} \tag{10}\]  
In this scope, the matrix \(V_{J}\) and \(\delta x\) is expressed in the form of \(V_{J}=[V_{J_{1}}\dots V_{J_{N}}]\) and \(\delta x=\delta q_{1}^{\prime}V_{J_{1}}+\dots+\delta q_{m}^{\prime}V_{J_{m}}\). So, the \(\delta x\) lies in the column space of \(V_{J}\).  
\[\begin{split}\delta q&=J^{+}\delta x\qquad\qquad( where\quad J^{+}=U_{J}S_{J}^{+}V_{J}^{T})\\ &=U_{J}S_{J}^{+}V_{J}^{T}\delta x\\ &=U_{J}\delta x^{\prime}\qquad\qquad(where\quad\delta x^{\prime}= S_{J}^{+}V_{J}^{T}\delta x^{\prime})\end{split} \tag{11}\]  
By repeating the derivation process above, the \(\delta q\) lies in the column space of \(U_{J}\). Based on the equations above, one can note that the \(U\) and \(V\) in Equation (8) are the basis vectors of the joint space and the task space, respectively. Furthermore, the singular values, \(\sigma_{i}\), represent the manipulating abilities of a robot in each direction of the null projected space. Finally, the null space quality can be expressed as:  
\[Q=\prod_{i=1}^{k}\sigma_{i} \tag{12}\]  
The null space quality is the dimensionless index. Since we have used Moore-Penrose pseudoinverse in our work, the null space quality ranges between 0 and 1.0. Generally, high null space quality value interprets the better controllability of the robot manipulator within the null space.  
By evaluating the null space quality, another type of singularity can be verified, which cannot simply be verified by evaluating the manipulability. As the null space quality includes the Jacobian matrix, the singular state of the null space quality exhibits similar aspects with the manipulability's singular state (Type 1 singularity). However, there is a difference due to the null space projector which causes another type of singularity. As mentioned previously, the null space projector restricts the joint motion of a excess Dof, causing singular state. When the constrained joint motion in the null space is orthogonal with the necessary joint motion by the Jacobian, another singular state is caused (Type 2 singularity).