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Jacobian Pseudoinverse

Let's break down Singular Value Decomposition (SVD) and the pseudoinverse with a focus on the mathematical steps and handling of singularities.

SVD: Singular Value Decomposition

Given a real \(m \times n\) matrix \(A\), the SVD is:

\[ A = U \Sigma V^T \]

Where:

  • \(U \in \mathbb{R}^{m \times m}\): orthogonal matrix, \(U^T U = I_m\)
  • \(V \in \mathbb{R}^{n \times n}\): orthogonal matrix, \(V^T V = I_n\)
  • \(\Sigma \in \mathbb{R}^{m \times n}\): diagonal matrix with non-negative real entries \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_r > 0\), \(r = \text{rank}(A)\)

To compute the singular values and U, V matrices of a matrix \(A\), you're performing Singular Value Decomposition (SVD). The SVD of a matrix \(A \in \mathbb{R}^{m \times n}\) is:

\[ A = U \Sigma V^T \]

Where:

  • \(U \in \mathbb{R}^{m \times m}\) is an orthogonal matrix (left singular vectors).
  • \(\Sigma \in \mathbb{R}^{m \times n}\) is a diagonal matrix with non-negative real numbers (singular values).
  • \(V \in \mathbb{R}^{n \times n}\) is an orthogonal matrix (right singular vectors).
Mathematical Process

You can compute the SVD in the following steps:

Compute Eigenvalues and Eigenvectors
  • Find eigenvalues \(\lambda_i\) and eigenvectors of \(A^TA\): these give the right singular vectors \(v_i\) (columns of \(V\)).
  • Find eigenvalues and eigenvectors of \(AA^T\): these give the left singular vectors \(u_i\) (columns of \(U\)).
Compute Singular Values
  • Singular values \(\sigma_i = \sqrt{\lambda_i}\), where \(\lambda_i\) are the non-zero eigenvalues of \(A^TA\) (or \(AA^T\)).
Interpretation
  • Columns of \(U\): directions in domain space (input).
  • Columns of \(V\): directions in codomain space (output).
  • Singular values \(\sigma_i\): how much stretching happens along each axis.
Structure of \(\Sigma\)
\[ \Sigma = \begin{bmatrix} D & 0 \\ 0 & 0 \end{bmatrix}, \quad D = \text{diag}(\sigma_1, \ldots, \sigma_r), \quad \sigma_i > 0 \]

The zero blocks correspond to singularities—directions in which \(A\) loses invertibility.

Moore-Penrose Pseudoinverse \(A^+\)

Given \(A = U \Sigma V^T\), the pseudoinverse is:

\[ A^+ = V \Sigma^+ U^T \]

Where \(\Sigma^+ \in \mathbb{R}^{n \times m}\) is defined by:

\[ \Sigma^+ = \begin{bmatrix} D^+ & 0 \\ 0 & 0 \end{bmatrix} \]

And:

\[ D^+ = \text{diag}\left(\frac{1}{\sigma_1}, \ldots, \frac{1}{\sigma_r}\right) \]

Behavior at Singularities

  • If \(\sigma_i = 0\), then \(\frac{1}{\sigma_i}\) is undefined.
  • The pseudoinverse explicitly sets \(\frac{1}{\sigma_i} = 0\) when \(\sigma_i = 0\).
  • This truncation is essential for numerical stability and defining the pseudoinverse in non-invertible or rank-deficient cases.

The pseudoinverse is defined via the SVD by inverting only the non-zero singular values. At singularities (zero singular values), the inversion is avoided by setting those components to zero. This yields a well-defined, unique, and stable linear operator even when \(A\) is not invertible.

Task-space control

In robotic control, mapping task-space velocities \(\dot{x}\) to joint-space velocities \(\dot{q}\) using the Jacobian pseudoinverse \(J^+\) is a standard approach:

\[ \dot{q} = J^+ \dot{x} \]

Now, if we do not properly handle small singular values in the Jacobian’s SVD, catastrophic behavior can result — both numerically and physically.

Let \(J \in \mathbb{R}^{m \times n}\) (e.g., end-effector velocity as a function of joint velocity), with SVD:

\[ J = U \Sigma V^T \quad \Rightarrow \quad J^+ = V \Sigma^+ U^T \]

Where \(\Sigma^+ = \text{diag}\left(\frac{1}{\sigma_1}, \ldots, \frac{1}{\sigma_r}, 0, \ldots, 0\right)\) in the standard definition. But if you do not truncate, and use:

\[ \Sigma^+ = \text{diag}\left(\frac{1}{\sigma_1}, \ldots, \frac{1}{\sigma_n} \right) \quad \text{with some } \sigma_i \approx 0 \]

If any \(\sigma_i \to 0\), then \(\frac{1}{\sigma_i} \to \infty\). So:

\[ \dot{q} = V \Sigma^+ U^T \dot{x} \]

will contain terms like:

\[ \frac{1}{\sigma_i} v_i (u_i^T \dot{x}) \]

These small singular directions correspond to poorly observable or weakly actuated movements (i.e., directions in which the robot has little control or effect). Amplifying these yields huge joint velocities in response to small end-effector demands — often impossible for the physical system to follow.

When the Jacobian becomes near-singular (e.g., robot arm is fully extended, or aligned in a singular configuration), at least one \(\sigma_i \approx 0\). Then:

  • Numerical instability in calculating \(J^+\)
  • Large errors in joint velocities \(\dot{q}\)
  • Violent, erratic, or unsafe motion if executed on hardware

Proper Treatment: Regularization or Truncation

To avoid this:

Truncated SVD

Set a threshold \(\epsilon\), and discard singular values \(\sigma_i < \epsilon\):

\[ \frac{1}{\sigma_i} := 0 \quad \text{if } \sigma_i < \epsilon \]
Damped Least Squares (Tikhonov regularization)

Use the damped pseudoinverse:

\[ J^+ = V \left( \frac{\Sigma}{\Sigma^2 + \lambda^2 I} \right) U^T \]

Where \(\lambda\) is a small positive constant. This smooths out \(\frac{1}{\sigma_i}\) near zero:

\[ \frac{\sigma_i}{\sigma_i^2 + \lambda^2} \to \frac{\sigma_i}{\lambda^2} \quad \text{as } \sigma_i \to 0 \]

Which prevents divergence and gives a well-behaved inverse even near singularities.

Conclusion

Failing to handle small singular values in Jacobian pseudoinversion causes dangerous and unstable joint behavior. It leads to:

  • Large, unrealistic joint velocities
  • Numerical blowup
  • Physical damage in real robots
  • Total control failure near singularities

Always apply regularization or truncation in inverse kinematics and velocity control.