Because accelerations are minimal in quasi-static walking,

it is typically quite slow.

Although quasi-static motions in general have there use,

we will focus only on dynamic motions in this lecture and

course due to their potential to make robots faster and more efficient.

Now that we have identified a mechanism of locomotion, how do we model it?

There are many ways to model locomotion,

from simulations involving thousands of variables to ones that are quite simple.

Overly complex models can fall victim to what is called Bellman's curse of

dimensionality, where the amount of computation time or data points needed for

analysis, increases exponentially with the number of variables.

On the other hand, extremely simple models can be too limited

to capture the full complexity of the desired behavior.

To address this tension between simplicity and complexity, roboticists and

biologists have developed a modeling framework called templates and

anchors to capture both the simplicity of the underlying mechanism of locomotion and

its complexity in a physical system.

Templates are the simplest model needed to describe the target behavior.

For example, with running and walking templates could be models representing

the underlying mechanisms observed in biology.

Because they have no more states or parameters than necessary, templates can

capture what is fundamentally going on in a movement in it's most simple form.

This simplicity makes templates very beneficial for analysis and control.

Anchors, on the other hand, are more elaborate models than incorporate

the important degrees of freedom of a physical system.

For example, when modeling a hopping kangaroo,

an anchor might take into account the leg joints.

The template then describes the motion of the kangaroo through the anchor.

By trimming away the degrees of freedom, a template

might consist of a simple bouncing mass on a spring to describe the kangaroo.

This relationship between templates and anchors is very important.

A template is only valid if it in some sense

is a version of the anchor with fewer degrees of freedom.

In which case, we say that the anchor embeds the template dynamics.

And then if the template is anchored to the physical system through the anchoring.

This relationship can also facilitate the control of robots as control policies at

the simple template level can then be mapped in a more complex anchor level.

For example, the control of the template can be used to guide the control of

the individual robot joints.

To illustrate how useful templates can be for low level robot control,

we're going to look at a model that we consider a template for

walking called the rimless wheel.

The rimless wheel has historical significance as it was

one of the models Tad McGeer, then a professor at Simon Treasure University,

used in analyzing his walkers powered entirely by gravity,

in his famous 1990 paper titled Passive Dynamic Walking.

The rimless wheel is a relatively simple dynamical system,

representing a spoked wheel with its rim removed.

This template also incorporates an important aspect of legged locomotion,

a power source.

Gravity pulls the rimless wheel down an incline ramp, accelerating the model

over the course of a stride and allowing it to walk on its spokes.

The fact that gravity is the power source instead of electricity

isn't that important as we will see that a small amount of constant work done by

gravity each stride can be replaced by motors to generate powered robot walking.

It might not be immediately obvious, however,

most wheels capture walking behavior.

A rimless wheel walking down a ramp is a lot like a wheel rolling down a ramp,

much like you probably studied in your first physics course.

In a rolling motion with a normal wheel,

gravity is accelerating the center of mass downwards, causing a movement

around whatever point the wheel is currently touching the ground with.