Here[.pdf] is my ICRA 2015 paper with an errata (I highlighted the corrections in equation 23 and 30).
This paper was about variable Center-of-Mass height trajectory planning with reactive stepping by using non-linear model predictive control.
Here is an experiment using the proposed approach where a robot walking along a step while also following a variable COM height profile
Usually constant/predefined COM height or models that limit the robot to behaving as if it only has point feet is used when variable COM height is desired because the ZMP equation becomes non-linear when the COM height is a variable. However, it would be desirable to let the COM height be a free variable and let the ZMP move freely within the support polygon without limiting it to the centre of the foot.
In this work, the trajectory planning problem with variable COM height and reactive stepping was formulated as a Quadratically Constrained Quadratic Program (QCQP). This QCQP is then solved using Sequential Quadratic Programming (SQP).
The gradients and Hessian of a quadratic can be found analytically, thus time consuming numerical differentiation is not needed within the SQP solver, this helped to realize realtime performance (about 4 ms compute time on a 3.4 Ghz Quad-core) .
The main trick in this work was to take the linear relationship between the COM states and the COM jerks that is provided by the MPC literature for humanoid robots and substitute this into the expanded ZMP equation, after collecting the terms and cleaning up a little the quadratic ZMP equation was found.