Both fixed and mobile robots are usually modeled as rigid bodies connected by
actuated joints. The general equations of motion for such a system are:
where is the vector of actuated and unactuated coordinates.
Actuated coordinates correspond to joint angles directly controlled by motors.
Unactuated coordinates correspond to the six degrees of freedom for the
position and orientation of the floating base (a frame attached to any of the
robot's bodies) with respect to the inertial fame. The vector is
The first working assumption to simplify this model is (Assumption 1) that the
robot has enough joint torques to realize the actuated part of the equation,
and focus on the Newton-Euler equations that correspond to the six unactuated
where on the left-hand side is the position of the center
of mass (CoM) and is the net angular momentum around the CoM,
while on the right-hand side is the resultant of contact forces,
is the moment of contact forces around the CoM, is
the robot mass and is the gravity vector. This model is called
Angular momentum or height variations make centroidal dynamics nonlinear. This
means for instance that, to generate a trajectory for this system, one needs to
solve a nonlinear optimization. An alternative to linearize this system is to
make two assumptions:
- Assumption 2: there is no angular momentum around the center of mass
. This is why the Honda P2 walks with
- Assumption 3: the center of mass keeps a constant height. This is why the
Honda P2 walks with bent knees.
These two assumptions are used to derive linearized dynamics as follows.
Equations of motion
Let us consider the zero-tilting moment point (ZMP) of the contact wrench. It is a
point where the moment of contact forces is vertical:
with the unit upward vertical vector of the inertial frame. This
quantity defines an axis in general: to make a unique point, let us
take it on the ground with , where is the
constant height of the CoM. The moment of the contact wrench
at this ZMP is related to the moment at the CoM by:
Since (Assumption 2), we have:
Applying the vector triple product formula,
From Newton's equation, and we can rewrite
the equation above as:
Since (Assumption 3), this equation is a trivial identity in
the vertical direction while its horizontal coordinates are:
where , is the gravity constant and is the constant height of the center of mass. The constant
is called natural frequency of the linear inverted pendulum.
In this model, the robot can be seen as a point-mass concentrated at
resting on a mass-less leg in contact with the ground at .
Intuitively, the ZMP is the point where the robot applies its weight. As a
consequence, this point needs to lie inside the contact surface .