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Project: Gait Pattern Generation
Gait pattern generation copes with the dynamics of the robot and generates feasible gaits trajectories which lead to dynamically stable biped motion. The robot is modeled as a "3D Linear Inverted Pendulum (3D-LIP)"

A 3D-LIP is an inverted pendulum that moves on a specific plane. In mathmatics, this physics model can be described according to the definition of ZMP, as follows

Online Sampling Search
Simplified walking is a theoretic walking process based on a series of assumptions. It requires low computation and provides us a new layer to plan biped motions. The walking process only consists a series of a series of single-support phases. Each single-support phase is named as a "Simplified Step". In a simplified step, we only use a single ZMP, named ZMP Decision, instead of a ZMP trajectory in real biped walking. Then the analytical solutions of the above equations when the height of the pendulum is constantly zp are given as follows

where is the initial state, is the final state at time t, I is a 2x2 identity matrix, is the ZMP Decision.

    Change Speed
      Based on simplified walking, we can achieve goal speed using two steps by solving a pair of ZMP Decisions. We insert a middle state to connect the initial state to the goal one.

     Online Sampling Search
      Given a footstep plan and an initial state of the CoM, we use a unidirectional search algorithm to search for a sequence of ZMP points that make the robot follow the footstep plan from the initial state.

        In the demo, the robot walks through three stairs smoothly.

Simultaneously Planning
To increase the flexibility of the robot, we modified simplified walking where ZMP Decision is substituted by a cubic polynomial (ZMP polynomial of Y axis is similar)

Then the analytical solutions of the system dynamics is given as follows (solution of y axis is similar)

      Only one step is needed to achieve the goal state as long as we solve the 8 coefficients in the two ZMP polynomials. We introduces a technique of simultaneously planning to solve these coefficients. This makes the robot very flexible. we achieved a fast walk with step duration 180ms (20ms for double support phase). The parameter that specifies the percentage of the user required speed is 0.8. The maximum forward speed is around 0.33m/s. The maximum backward speed is around 0.2m/s. The maximum sideways speed is around 0.11m/s. The maximum rotational speed is around 90\textdegree/s.

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.