How to invert D-H parameters

Question

I currently have a working kinematic chain made by a set of ten links in D-H convention (with usual parameters [ $A_i, D_i, \alpha_i, \theta_i$]).

But my task currently requires the inversion of some of them. Basically, I would have a part of the chain that is read from the end-effector to the origin, using the same links (and thus the same parameters). Is it possible? How to do so?

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Please notice that this is not related to the inversion of the kinematic chain. It's more basic: I want to find the dh parameters of the inverted forward kinematic chain.

Let's put it simple: I have dh parameters for a 2 link planar chain from joint 0 to joint 1, so I can compute its direct kinematics. But what if I want to compute the direct kinematics from joint 1 to joint 0?

Given DH parameters [ $A_i, D_i, \alpha_i, \theta_i$], I can retrieve the transform matrix with this formula:

$G = \left[\begin{matrix} cos(\theta) & -sin(\theta)*cos(\alpha) & sin(\theta)*sin(\alpha) & cos(\theta)*a \\ sin(\theta) & cos(\theta)*cos(\alpha) & -cos(\theta)*sin(\alpha) & sin(\theta)*a \\ 0 & sin(\alpha) & cos(\alpha) & d \\ 0 & 0 & 0 & 1 \end{matrix} \right]$

This is the transform matrix from the i-th link to the (i+1)-th link. Thus, I can invert it to obtain the transform matrix from the (i+1)-th link to the i-th link, but the problem is that this is not working. I believe that the reason is related to the fact that the DH convention doesn't work any more as it is.

Any help?

Answer 1

As you correctly say, if you want to compute the transform from link $(n+1)$ to link $n$, you should simply invert the matrix $G$. The inverse of a homogenous transformation matrix

$G = \left[\begin{matrix} R & p \\ 0 & 1 \end{matrix}\right]$

is

$G^{-1} = \left[\begin{matrix} R^T & -R^T p \\ 0 & 1 \end{matrix}\right]$.

If $G$ is already available, this is probably the fastest way of computing $G^{-1}$. Otherwise, if only the DH parameters are known, you can insert the expression for $G$ to find this simplified expression for the translation of $G^{-1}$:

$-R^T p = \left[\begin{matrix} -a\\ -d \sin(\alpha)\\ -d \cos(\alpha) \end{matrix}\right]$

Answer 2

Eventually, @alecive came up with an elegant solution he published here: http://dx.doi.org/10.1109/ICRA.2014.6907178