Physically accurate hybrid dynamical systems can be very complex, and

typically are much more difficult to analyze than non-hybrid dynamical systems.

But despite their complexity, hybrid systems offer a natural choice for

modelling leg of locomotion, due to the changing dynamics depending on if a leg is

in contact with the ground or not.

Unfortunately, no analytic solution existS for

the equations in motion describing dynamics of SLIP.

So, it's not possible to get a close form expression

describing the position of the mass center as a function of time.

This is because the class of differential equations,

which can be solved analytically, is very small.

And no such solution exists for the stance dynamics here.

We can, however, perform numerical analysis of SLIP and

apply mathematical tools to qualitatively describe the behavior of this system,

which can be quite insightful.

If we as roboticists need analytic solutions governing the equations of

motion, often we can make approximations to the system to achieve our goals.

Several techniques, such as making small angle approximations to linearize

the sines and cosines, assuming that gravity acts radially instead of downwards

in stance, or outright ignoring gravity during stance,

can be used to get closed form expressions of the state from stride to stride.

Interestingly even though we are, in effect,

making an approximation to win approximation,

this can work quiet well and practice use for controlling robots to run.

As long as the state stays within the operating regime or

this assumption are approximately valid.

The body and legs of the Rex Robot in many ways embodied the slip template.

Rex has six C shaped springy legs that can

interact with the ground in unison to make the center of mass trajectory