Linear Inverted Pendulum Model (LIPM)
It turns out that we can model a biped robot quite simply as a single point mass with two force vectors representing the feet. Modeling the robot this way allows for easy mathematical modeling as well as being easier to visualize than a multi-mass model with many joints.
One further simplification which is very popular at the moment is to restrict the robots motion to the X-Y plane and prevent motion in the up down direction. This is called the Linear Inverted pendulum model (LIPM) [1] and is used extensively in current biped robotics research.
The equation of motion for the LIPM is described as follows
![\left[\begin{array}{c}x_{i+1}\\ \dot{x}_{i+1}\end{array}\right]=\left[\begin{array}{cc}cosh(\omega t) & \frac{1}{\omega}sin(\omega t)\\ \omega sin(\omega t) & cosh(\omega t)\end{array} \right]\left[\begin{array}{c} x_{i}\\ \dot{x}_{i}\end{array}\right]](http://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_%7Bi%2B1%7D%5C%5C+%5Cdot%7Bx%7D_%7Bi%2B1%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Dcosh%28%5Comega+t%29+%26+%5Cfrac%7B1%7D%7B%5Comega%7Dsin%28%5Comega+t%29%5C%5C+%5Comega+sin%28%5Comega+t%29+%26+cosh%28%5Comega+t%29%5Cend%7Barray%7D+%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D+x_%7Bi%7D%5C%5C+%5Cdot%7Bx%7D_%7Bi%7D%5Cend%7Barray%7D%5Cright%5D+&bg=000000&fg=ffffff&s=1 "\left[\begin{array}{c}x_{i+1}\\ \dot{x}_{i+1}\end{array}\right]=\left[\begin{array}{cc}cosh(\omega t) & \frac{1}{\omega}sin(\omega t)\\ \omega sin(\omega t) & cosh(\omega t)\end{array} \right]\left[\begin{array}{c} x_{i}\\ \dot{x}_{i}\end{array}\right]")
where
and 
is the height of the COM which should be a constant.
Next: Trajectory Generation
Previous: Center Of Mass Jacobian
References
[1]. “The Linear Inverted Pendulum Model: A simple modeling for a biped walking pattern generation”, Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kazuhito Yokoi and Hirohisa Hirukawa. Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems Maui, Hawaii, USA, Oct. 29 - Nov. 03, 2001
Ehm, what is the t after w in the equations?
Also how does this can be applied in ZMP calculation?
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