Generalised Inverses

A generalised inverse extends matrix inversion to rectangular or singular matrices.

What is $A$?

Start with the generic linear system:

$$Ax=b$$

where:

• $A\in {\mathbb{R}}^{m\times n}$ is the matrix that maps the unknown $x$ to the required output $b$
• $x\in {\mathbb{R}}^n$ is the unknown vector
• $b\in {\mathbb{R}}^m$ is the required output vector

The symbol $A$ does not represent a special robot matrix. In the robot application, $A$ is replaced by the tool-configuration Jacobian $V(q)$.

Choose the inverse

flowchart LR
    S["Linear system<br/>Ax = b"] --> Q{"Is A square<br/>and full rank?"}
    Q -- "YES" --> I["Use ordinary inverse<br/>x = A-inverse b"]
    Q -- "NO" --> P["Use pseudoinverse<br/>x-zero = A-plus b"]

If $A$ is square and full rank:

$$m=n,\mathrm{rank}(A)=n$$

then $A$ is nonsingular and:

$$x=A^{-1}b$$

For a square matrix:

$$\det (A)\ne 0⟺A\text{ is nonsingular}⟺A\text{ has full rank}$$

A rectangular matrix has no ordinary inverse. Describe it as full rank or rank-deficient, rather than singular or nonsingular.

Moore-Penrose pseudoinverse

The Moore-Penrose pseudoinverse is written as $A^{+}$. It:

• Always exists
• Is unique
• Reduces to $A^{-1}$ when $A$ is square and nonsingular

For a full-rank matrix $A\in {\mathbb{R}}^{m\times n}$:

$$\boxed{A^{+}=\{\begin{array}{ll}{A}^T(A{A}^T{)}^{-1},& m<n\text{wide matrix}\\ A^{-1},& m=n\text{square matrix}\\ (A^{T}A{)}^{-1}{A}^T,& m>n\text{tall matrix}\end{array}}$$

Important

These explicit formulas require the relevant full-rank condition. The pseudoinverse still exists when $A$ is rank-deficient, but these formulas cannot be used directly.

What solution does $A^{+}$ give?

Define the residual:

$$r(x)=Ax-b$$

The pseudoinverse gives:

$$\boxed{x^0=A^{+}b}$$

• If exact solutions exist, $x^0$ is the solution with the smallest $\Vert x\Vert$
• If no exact solution exists, $x^0$ minimizes the residual $\Vert Ax-b\Vert$

Thus, $x^0$ is the minimum-norm least-squares solution.

Example

Given:

$$A=\left[\begin{array}{ccc}1& 0& 2\\ 1& -1& 0\end{array}\right]$$

$A$ is a $2\times 3$ wide matrix with full row rank because $\mathrm{rank}(A)=2$. Therefore:

$$A^{+}=A^T(A{A}^T{)}^{-1}$$ $$A^{+}=\frac{1}{9}\left[\begin{array}{cc}1& 4\\ 1& -5\\ 4& -2\end{array}\right]$$

Robot motion

The robot differential-motion equation is:

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

Match it to the generic equation $Ax=b$:

$$A=V(q),x=\dot{q},b=\dot{x}$$

Therefore:

$$\boxed{\dot{q}=V^{+}(q)\dot{x}}$$

Warning

In the generic equation $Ax=b$, $x$ means the unknown vector. In the robot equation, this unknown becomes $\dot{q}$. The symbol $\dot{x}$ is the required tool-configuration velocity.

This relation is the basis of Resolved-motion Rate Control. Near a singularity, small tool velocities can require very large joint velocities.

Formal definition

Let $A$ be an $m\times n$ matrix. An $n\times m$ matrix $X$ is a generalised inverse of $A$ when it satisfies at least property 1 or property 2:

1. $AXA=A$
2. $XAX=X$
3. $(AX{)}^T=AX$
4. $(XA{)}^T=XA$

The Moore-Penrose pseudoinverse $A^{+}$ satisfies all four properties.