A small inverse kinematics problem

Control of the pelvis

Here is the problem:
We want to control this quadrilateral in order to be able to move the center of the pelvis where we want (ie: the goal is to move the CoG from on foot to the other).

As we don’t want the feet to slip or to lift of the ground we have to consider this linkage to be rigid.
Of course we also need an extra roll degree of freedom on the feet but let’s consider that we have it.
Then this is what we obtain:

From that we can “easily” get s given one angle:

Then as everything is rigid we have all the other angles:

and the coordinates of C (in particular the x coordinate which I am interested in):

So that’s ok for the forward kinematics but what I want is the inverse kinematics, find the angles given the C coordinates.
For now I just use a lazy numerical solution…
I think that this problem can be viewed as a Watt’s linkage.
Any idea?

Bonus question:
The problem is probably simpler if I don’t impose two equal leg length ie if I allow to bend the knees.
Any solution?

PS: please admin, can we have this MATHJAX stuff installed?

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Avoid trigonometry and model your problem using vector calculus, it is often easier.

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I have often heard people that tried to make Poppy walking, but I’ve never seen a Poppy walking by himself, is it only possible?

It could be but it is an EXTREMELY tricky thing…