Work with Q.-C. Pham & Y. Nakamura
Journées Nationales de la Robotique Humanoïde
June 23, 2016
The ZMP is traditionally defined as the point on the floor where the moment of the contact wrench is parallel to the surface normal (Sardain & Bessonnet, 2004).
The point at which the robot “applies its weight”.
The ZMP must lie inside the support polygon (convex hull of ground contact points).1
What wrenches are feasible under frictional contacts?
A motion or wrench is weak contact stable iff it can be realized by contact forces inside their friction cones.
Friction cones can be combined as Contact Wrench Cone (CWC) at the
COM—see e.g. (Caron et al.,
The ZMP is mathematically defined from a wrench (Sardain & Bessonnet, 2004). The ZMP in the plane Π(O,n) of normal n containing O is the point such that n×τZ=0, that is:xZ=n×τOn⋅f+xO.
We define the full support area S as the image of the CWC by this equation.
Good news! This area can be computed geometrically:
Bad news! It is not always a polygon:
The Newton-Euler equations of the system are:[m¨xG˙LG]=[mg0]+∑contacti[fi−−→GCi×fi]
They show how the motion of unactuated DOFs results from interactions with the environment.
The Newton equation can be written equivalently:¨xG=g+¨zGzG−zZ(xG−xZ)−˙LGxm(zG−zZ)
where x now denotes X-Y plane coordinates. The Linear Inverted Pendulum Mode (Kajita et al., 2001) is obtained by constraining:zG−zZ=h˙LG=0
The system dynamics become ¨xG=gh(xZ−xG).
The support area in the LIPM is smaller than the convex hull of contact points:
We provide an algorithm to compute the support area corresponding to the system:w∈CWCzG−zZ=h˙LG=0
We call it the pendular support area.
We can now consider the ZMP above the COM ⇒ Linear (non-inverted) Pendulum Mode:¨xG = gh(xG−xZ)
This is the dynamic equation of a spring.
The robot is driven from above, controlling its target position.
The shape of the pendular support area changes “conically” with the ZMP altitude: