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

$\left[\begin{array}{c}x_{i+1}\\ \dot{x}_{i+1}\end{array}\right]=\left[\begin{array}{cc}cosh(\omega t) & \frac{1}{\omega}sin(\omega t)\\ \omega sin(\omega t) & cosh(\omega t)\end{array} \right]\left[\begin{array}{c} x_{i}\\ \dot{x}_{i}\end{array}\right]$

where

$\omega=\sqrt{\frac{g}{z_{h}}}$

and $z_h$ is the height of the COM which should be a constant.

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

### 3 Responses to “Linear Inverted Pendulum Model (LIPM)”

1. Ehm, what is the t after w in the equations?

2. Also how does this can be applied in ZMP calculation?

3. I see a lot of interesting articles on your blog.

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