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Project: Footstep Revision
In most cases, we fail to generate the ZMP sample sets because the kinematic range is too narrow or we cannot find samples that satisfy search constraints. This usually means that the next step is too distant from the current step or has too much direction changes compared with the current step.

Revision for a single step
Our algorithm returns a revised gesture with the least cost in terms of modification, except that no constraint-satisfied solution can be found. As shown in the Figure (a) below, we fail to genearte the 4-step sample set according to the original footstep plan. To revise the fifth footstep, we generate a revision sample set in the CoM range and find a revision solution as show in Figure (b). We successfully obtain a ZMP solution by applying the ZMP sampling search on the revised footstep plan. The CoM trajectory with double support phase is shown in Figure (c).

Revision for a footstep plan
There are two main reasons for a failed single step revision. One reason is that the new kinematic range determined by the revised step, as shown in the above Figure (b), is still not sufficiently large for the ZMP sampling search to obtain a ZMP solution. This problem can be solved by rerunning revision algorithem of a single step. The other reason is that the failure is caused by the former steps. Thus, the robot cannot achieve the footstep plan. So, we need to revise the previous step, or earlier steps if the revison on the previous step does not work.

As shown in the Figure below, Steps A and B were revised because the contact surfaces of the supporting foot were detrimental to the robot's balance. Step C was revised to avoid collision with the revised Step B.

Footstep revision entails more computation than ZMP sampling search. It would be inefficient to revise a footstep plan repeatedly with no limitation. In this situation, it would be better to apply more constraints on the footstep planning to make the footsteps more achievable. We could then obtain the walking solution by ZMP sampling search with fewer revisions.

Related Publications
    Jinsu Liu, Feng Xue, Xiaoping Chen, A UNIVERSAL BIPED WALKING GENERATOR FOR COMPLEX ENVIRONMENTS WITH PATTERN FEASIBILITY CHECKING, International Journal of Humanoid Robotics. Vol. 8, No. 2 (2011) 323-357.
    Feng Xue, Xiaoping Chen, Jinsu Liu, and Daniele Nardi, Real Time Biped Walking Gait Pattern Generator for a Real Robot, In: Proceedings of RoboCup 2011 Symposium, Istanbul, Turkey,. June 5-11, 2011 (Also to be appeared in: RoboCup 2012: Robot Soccer World Cup XV, ser. Lecture Notes in Computer Science, 2012)..
    Jinsu Liu, XiaoPing Chen, and Manuela Veloso. Simplified Walking: A New Way to Generate Flexible Biped Patterns. Proceedings of the 12th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines (CLAWAR). Istanbul, Turkey, September 2009..
    Jinsu Liu and Manuela Veloso. Online ZMP Sampling Search for Biped Walking Planning. International Conference on Intelligent Robots and Systems.. 2008, 185 - 190.