I was reading the paper, which you can find: https://hal.inria.fr/inria-00567668/document. I got so confused with equation (33) and (34), it seems like instead of an equal sign in eq.(33) on the left that should be a multiplication, which can be supported by the first author's thesis: https://pastel.archives-ouvertes.fr/pastel-01038025/document, eq.(2.62).

But I cannot understand what it means physically, can someone help me understand this!

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    $\begingroup$ Personally, I think it should be that they want to constraint the next foot step position to be within a feasible region, as shown in figure 1 of the "Walking without think about it paper". But the math just doesn't add up. $\endgroup$
    – RoboNoob
    Mar 3, 2021 at 15:08
  • $\begingroup$ Just a comment, I tried to implement the approach, the initial double support phase seems to be crucial, which is not highlighted in the paper. All you need to do for the initial double support phase is to alter the cost function and the constraints such as the CoP reference is at the center of the double support support polygon. $\endgroup$
    – RoboNoob
    Mar 15, 2021 at 23:06

1 Answer 1


I think you are correct that the equals sign in equation (33) should not be there, as reinforced by the first author's thesis.

The purpose of those equations are to constrain the MPC-generated footstep positions to ensure that each foot position is within the polygon shown in the following figure relative to the stance foot. In effect, the combination of those matrices will make sure that the generated steps are feasible, as you said.

The challenging part (to me at least) is that they have to do it for all steps from the current one to the end of the MPC, so they need to include details for all of those "future" steps in addition to the current step.

Hope that helps!

Cheers, Brandon

  • $\begingroup$ Thanks Brandon! $\endgroup$
    – RoboNoob
    Mar 11, 2021 at 2:27
  • $\begingroup$ Do you think the middle matrix might also be a typo? If you multiply it with the foot steps vector, you get (\Delta{x] + \Delta{y})'s, where you need [\Delta{x], Delta{y}]^T for calculating the inner product between the normal vector of the edges of the next foot step polygon and the planned next foot step to check if it is contained within. $\endgroup$
    – RoboNoob
    Mar 11, 2021 at 2:29
  • $\begingroup$ Given that there is an extra equals sign in the middle of the equation, I wouldn't be surprised if there was another typo in the same equation. My recommendation would be to get in touch with the author(s) and see if they have an updated/revised version of those equations in another paper (or in code they're using). I find that papers and theses are very much a snapshot of a moment in time, not a living document. $\endgroup$ Mar 14, 2021 at 22:54
  • $\begingroup$ Also: If you think I've answered your question as asked, please mark it as such so that this question doesn't stay on the "unanswered" list for the site. $\endgroup$ Mar 14, 2021 at 22:55

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