# IK_LM.step

IK_LM.step(ets, Tep, q)[source]

Performs a single iteration of the Levenberg-Marquadt optimisation

The operation is defined by the choice of method when instantiating the class.

The next step is deined as

$\begin{split}\vec{q}_{k+1} &= \vec{q}_k + \left( \mat{A}_k \right)^{-1} \bf{g}_k \\ % \mat{A}_k &= {\mat{J}(\vec{q}_k)}^\top \mat{W}_e \ {\mat{J}(\vec{q}_k)} + \mat{W}_n\end{split}$

where $$\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})$$ is a diagonal damping matrix. The damping matrix ensures that $$\mat{A}_k$$ is non-singular and positive definite. The performance of the LM method largely depends on the choice of $$\mat{W}_n$$.

Chan’s Method

Chan proposed

$\mat{W}_n = λ E_k \mat{1}_n$

where λ is a constant which reportedly does not have much influence on performance. Use the kwarg k to adjust the weighting term λ.

Sugihara’s Method

Sugihara proposed

$\mat{W}_n = E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)$

where $$\hat{\vec{w}}_n \in \mathbb{R}^n$$, $$\hat{w}_{n_i} = l^2 \sim 0.01 l^2$$, and $$l$$ is the length of a typical link within the manipulator. We provide the variable k as a kwarg to adjust the value of $$w_n$$.

Wampler’s Method

Wampler proposed $$\vec{w_n}$$ to be a constant. This is set through the k kwarg.

Parameters:
Raises:

numpy.LinAlgError – If a step is impossible due to a linear algebra error

Returns:

• E – The new error value

• q – The new joint coordinate vector