Joint-space Singularities

A joint-space singularity is a joint configuration $q$ where the tool-configuration Jacobian $V(q)$ loses rank.

flowchart LR
    Q["Joint configuration<br/>q"] --> R{"Is V(q) full rank?"}
    R -- "YES" --> N["Normal motion control"]
    R -- "NO" --> S["Singularity<br/>motion direction lost"]
    S --> W{"Where is the tool?"}
    W -- "Work-envelope surface" --> B["Boundary singularity"]
    W -- "Inside work envelope" --> I["Interior singularity"]

The velocity relation is:

$$\dot{x}=V(q)\dot{q}$$

For $V(q)\in {\mathbb{R}}^{6\times n}$, the universal singularity condition is:

$$\boxed{\mathrm{rank}[V(q)]<\min \{6,n\}}$$

Rank is the number of independent instantaneous motion directions represented by $V(q)$. When the rank decreases, the robot loses at least one tool-motion direction.

Workspace singularity

Let the direct kinematics relation be $x=w(q)$. A point $x$ in tool-configuration space is a workspace singularity if:

$$x=w(q_s)$$

for at least one joint-space singularity $q_s$.

Dexterity test

For the common case $n\le 6$:

$$\boxed{\mathrm{dex}(q)=\det \left[V^T(q)V(q)\right]}$$ $$\boxed{\mathrm{dex}(q)=0⟺q\text{ is singular}}$$

For a redundant robot with $n>6$, use:

$$\boxed{\mathrm{dex}(q)=\det \left[V(q)V^T(q)\right]}$$

The symbol $\Delta$ has no universal singularity formula. A robot example can define $\Delta$ as a convenient determinant. The rank condition above is the universal test.

Choose the correct singularity test

Start with rank. The determinant formulas are convenient tests for the same rank loss.

Matrix used by the problem	Matrix size	Singularity test
Complete tool-configuration Jacobian $V(q)$	$6\times n$, $n\le 6$	$\det [V^T(q)V(q)]=0$
Linear position block $A(q)=\partial p(q)/\partial q$	$3\times n$, $n\le 3$	$\det [A^T(q)A(q)]=0$
Square active submatrix $M(q)$ with the same rank as the selected Jacobian	$r\times r$	$\det [M(q)]=0$

Important

A rectangular matrix has no determinant. Use its rank, form a square product such as $V^{T}V$ or $A^{T}A$, or use a square active submatrix after confirming that it has the same rank as the selected Jacobian.

The matrix selected by the task also determines the meaning of the singularity:

• Rank loss of $V(q)$ is a complete tool-configuration singularity
• Rank loss of $A(q)$ is a position singularity

Two-link planar RR position singularity

For the planar RR robot, the linear position block is:

$$A(q)=\left[\begin{array}{cc}-a_1\sin {\theta }_1-a_2\sin ({\theta }_1+{\theta }_2)& -a_2\sin ({\theta }_1+{\theta }_2)\\ a_1\cos {\theta }_1+a_2\cos ({\theta }_1+{\theta }_2)& a_2\cos ({\theta }_1+{\theta }_2)\\ 0& 0\end{array}\right]$$

Define its active $x$-$y$ submatrix:

$$A_{xy}(q)=\left[\begin{array}{cc}-a_1\sin {\theta }_1-a_2\sin ({\theta }_1+{\theta }_2)& -a_2\sin ({\theta }_1+{\theta }_2)\\ a_1\cos {\theta }_1+a_2\cos ({\theta }_1+{\theta }_2)& a_2\cos ({\theta }_1+{\theta }_2)\end{array}\right]$$

The zero third row gives:

$$\mathrm{rank}[A(q)]=\mathrm{rank}[A_{xy}(q)]$$

Therefore, either determinant test can be used:

$$\det [A^T(q)A(q)]=0⟺\det [A_{xy}(q)]=0$$

The equivalence follows from:

$$\begin{array}{rl}\det [A^T(q)A(q)]& =\det [A_{xy}^T(q)A_{xy}(q)]\\ & =\det [A_{xy}(q){]}^2\end{array}$$

For this robot:

$$\boxed{\det [A_{xy}(q)]=a_1{a}_2\sin {\theta }_2}$$

Thus:

$$\boxed{\det [A_{xy}(q)]=0⟺\det [A^T(q)A(q)]=0⟺\sin {\theta }_2=0}$$

The two links are collinear at these singular configurations.

2 Different Types of Joint-space Singularity

Type	Tool location	Cause
Boundary singularity	On the surface of the work envelope	The robot reaches a geometric limit, such as a fully extended arm
Interior singularity	Inside the work envelope	Two or more joint axes become aligned, so their motion effects become dependent

Boundary singularity

A boundary singularity occurs when:

1. $V(q)$ loses rank
2. The tool tip is on the surface of the work envelope

The robot loses an instantaneous tool-motion direction at this boundary. Keep the manipulation away from the work-envelope surface to avoid this singularity.

Interior singularity

An interior singularity occurs when $V(q)$ loses rank while the tool is inside the work envelope. It can occur when two or more robot axes become collinear.

One joint motion can then cancel another joint motion. The joints move while the tool configuration stays fixed. This is a self-motion.

Important

Near a singularity, a small required tool velocity can demand very large joint velocities. Plan the trajectory away from singular configurations.

See Tool-configuration Jacobian Matrix, Manipulator Jacobian, and Resolved-motion Rate Control.