How can robots use their motors and sensors to move around in an unstructured environment? You will understand how to design robot bodies and behaviors that recruit limbs and more general appendages to apply physical forces that confer reliable mobility in a complex and dynamic world. We develop an approach to composing simple dynamical abstractions that partially automate the generation of complicated sensorimotor programs. Specific topics that will be covered include: mobility in animals and robots, kinematics and dynamics of legged machines, and design of dynamical behavior via energy landscapes.
2.1.1 Walking like a rimless wheel
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From the course by University of Pennsylvania
Robotics: Mobility
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From the lesson
Behavioral (Templates) & Physical (Bodies)
We’ll start with behavioral components that take the form of what we call “templates:” very simple mechanisms whose motions are fundamental to the more complex limbed strategies employed by animal and robot locomotors. We’ll focus on the “compass gait” (the motion of a two spoked rimless wheel) and the spring loaded inverted pendulum – the abbreviated versions of legged walkers and legged runners, respectively.We’ll then shift over to look at the physical components of mobility. We’ll start with the notion of physical scaling laws and then review useful materials properties and their associated figures of merit. We’ll end with a brief but crucial look at the science and technology of actuators – the all important sources of the driving forces and torques in our robots.
Link to bibliography: https://www.coursera.org/learn/robotics-mobility/resources/pqYOc
Meet the Instructors
Daniel E. KoditschekProfessor of Electrical and Systems Engineering
School of Engineering and Applied Science
Hello, today we will be talking about dynamically walking robots.
We'll go over what is walking, why is it useful for legged robots.
And introduce a simple walking model called the rimless wheel.
This rimless wheel is a special type of model called a template.
Templates are a special class of models that represent
the simplest models that are needed to describe a target behavior.
We'll show that templates can give design insights into building legged robots, and
then by building the rimless wheel template into our walking motion of
the mechanical system we can get a robot to walk robustly with minimal computation.
So before we get into walking let's take a step back and
talk about locomotion in general.
Robots and animals move through their environments by generating
forces on their environments.
For example, when you run, your leg pushes downwards and
backwards, creating a downwards and backwards force on the environment.
Through Newton's Third Law, a reaction force is created upwards and forwards.
Newton's Second Law then causes an acceleration which gives you movement.
Biologists studying legged locomotion have shown that the methods animals use to
generate forces on their environment are extremely diverse.
For example, the number of legs, and
the shapes of legs of the animals varies tremendously.
Humans have two legs, whereas millipedes, have approximately 200.
Hedgehogs have relatively short legs, while spiders have very long,
and thin legs.
But, biologists have also discovered that the actual forces generated
by these diverse strategies are remarkably similar.
For example, when ground reaction forces are measured in running
they generally follow the same basic force profile.
A human running on two legs and a dog trotting on four legs
can look very similar based on the ground reaction forces they generate.
They're even similar to cockroaches running with an alternating tripod gate
with six legs.
And even crabs running sideways with eight legs.
So biologists have proposed two basic mechanisms to describe these four
profiles generated during walking and running.
The forces generating during walking tend to appear
as a mass vaulting over a relatively stiff leg.
Forces generated during running tend to look like a mass bouncing on a spring
much like a pogo stick.
These mechanisms have important features that are relevant for robot design.
For example, both mechanisms involve some way of retaining a portion
of the mechanical energy of the previous stride to use in the next stride.
Also by analyzing simple models of these mechanisms,
we can make statements about their stability.
So let's take a closer look at walking, which is the focus of this lecture.
We can gain insight into the mechanism
behind walking by looking into its energetics.
So a distinctive feature of walking is the kinetic energy, which is the energy of
motion, oscillates out a phase with gravitational potential energy.
Which is the energy due to gravity.
When a leg touches down and
a body moves forwards in the beginning of a walking stance, the body uses it's
kinetic energy to vault upwards and over the leg in the first half of stance.
This process converts some kinetic energy into gravitational potential running
energy during the second half of stance.
Starting when the leg is vertical, this gravitational potential
energy is converted back into kinetic energy as the body is pulled forwards and
downwards by gravity, accelerating the body.
This process repeats itself when the next leg touches down.
This process of exchanging kinetic and
gravitational potential energy can be very efficient especially at low speeds.
Animals have been measured to conserve up to 70% of
their mechanical energy while walking.
Energy efficiency is important for legged robots because it allows us to
build robots that potentially move faster, operate for
longer durations, and don't have to carry as much fuel or battery weight.
The type of walking we have been talking about so far is called dynamic walking.
Where the robot is not perfectly balanced during stance.
As you may have seen in videos,
many robots also walk by a different mechanism called quasi-static walking.
In quasi-static walking, the robot is nearly balanced
throughout its entire walking stride, and undergoes minimal accelerations.
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.
This moment around the ground contact point accelerates the wheel down the ramp.
This ground contact points on the wheel touch the ground only for
an instant before lifting off again.
So what happens when we give the wheel spokes and
then take away the rim of the wheel?
Well, if there's an internet or
very large number of spokes then almost nothing changes.
We get rolling behavior or something very similar to it.
But as we take away the number of spokes, the center of mass has a more pronounced
vault over the point where the spoke contacts the ground, and the kinetic and
gravitational potential energy oscillate out of phase with greater amplitude.
We've arrived at a very simple model of walking.
In our model, the spoken contact with the ground
represents the leg in contact with the ground.
As one leg lifts off, another instantaneously touches down so
that the body represented by the center of mass always has a leg in ground contact.
Of course, a rimless wheel doesn't look entirely like a biped.
Maybe it reminds you more of a tumbleweed.
But if we choose to visually ignore the spokes not in contact with the ground, and
instead pretend that there's another leg recirculating to prepare for
the next step, we have a convincing picture of how this model
can represent a biped walking with what we call a compass gait.
The version of the rimless wheel, that we are discussing today,
only has one degree of freedom, which is the wheel angle.
And six parameters which are the mass,
how far the mass is distributed from the mass center, the leg length, the inter-leg
angle, the ramp angle, and finally, the vertical acceleration due to gravity.
The rimless wheel template is a simple enough model to analyze mathematically.
But we will forego the mathematical analysis, and
instead rely on physical intuition with a 3D printed wheel model.
So what does it take for our rimless wheel model to walk?
Over a level ground, in the absence of a source of energy,
the model quickly comes to a stop due to the legs impacting the ground.
Because the leg impacts involve mass coming sharply to a stop,
which results in a loss of kinetic energy.
This, of course, is in contrast to a wheel rolling down a ramp
which can accelerate for quite some time before being slowed down by friction.
This illustrates an important concept in legged locomotion which is that
legged locomotion is fundamentally and energetically a dissipative process.
As you can see, it only takes a small amount of gravitational potential energy,
provided by a slight ramp incline, for the model to walk at a steady speed.
From stride to stride, the mass center of the rimless wheel gets lower and
lower as it travels down the ramp.
This change in height is constant for each stride.
So, during each stride approximately the same amount of gravitational potential
energy is being added to the model which is converted into kinetic energy.
This kinetic energy is then lost when the leg impacts the ground.
This process then repeats giving us the repetitive behavior that we see.
If we tilt the ramp up more, more energy is added and the template walks faster.
If we lower the ramp less energy is added for stride and the template walks slower.
The fact that the model walks with such a small ramp angle, just that walking like
a rimless wheel is quite energetically efficient despite the losses involve.
And in a minute, we will see that using a modern of gravity to add the small
amount of fix energy per stride on a robot retains this efficiency property.
Aside from the efficiency of the motion,
there is another important feature at play here.
There is actually an advantage to legged locomotion being a lossy process, and
that is because it allows legged locomotion to be mechanically stable.
Notice, that if we roll a wheel down the ramp,
the speed of the wheel increases as it rolls.
But not so with our legged template.
This is because the energetic losses naturally regulate
the speed of the system.
Odic will go into more detail about this during his anchor lectures.
But the intuitive idea here is that the energetic losses from the leg impacts
increases with velocity, which slows down the template if it gets moving too fast.
However, if the template is to slow these impact losses are small and
the constant amount of energy added per stride by gravity speeds the system up.
Eventually the system reaches a periodic limit cycle, a steady rolling speed.
Stability is an incredibly important topic in leg and robots, and
we will continue to come back to it during this course.
So why is it important that our walking template is stable?
One reason is that if we build robots that passively anchored a template
through their mechanical design,
it allows us to build robots that walk stably with little to no computation.
This is Tad McGuire's famous passive dynamic walker.
The robot is purely mechanical, it has no computer, no motors, and no sensors.
And yet, it can walk down a ramp in a manner that is shockingly anthropomorphic.
Instead of being programmed on a computer, the control scheme is designed into
the mechanism, in much the same way as it is with the rimless wheel.
Because the underlying template behavior is stable and
has been mechanically encoded or anchored in the machine.
It is able to walk a non-trivial duration without falling over
at a nearly constant speed, despite the model being an imperfect representation
of the physical world, as all models are.
As we have said before, a robot can walk
like a rimless wheel using only a small amount of motor energy instead of gravity.
Andy Marina, a professor at Cornell University,
did just that with his robot called the Cornell ranger.
In 2011, the Cornell ranger was able to walk an ultra marathon of 40.5
miles non-stop on a single battery charge of less than 500 watt hours,
or about $0.05 worth of electricity.
By creating a robot that is close to a literal instantiation of an efficient
locomotion model, Angie Rerina was able to more than double the previous
distant record for robotic legged locomotion on a single battery charge.
Robots can also mechanically encode templates without literally
instantiating them.
For example, the REX robot walks with a motorized alternating tripod gait
that has rimless wheel dynamics in the forward direction, while the tripod and
support allows the robot to be very balanced in the other directions,
much like a three-legged chair.
This stability of both the template and
the anchor allows Rex to walk robustly over a wide variety of terrains,
even rocky slopes and mountains, with almost no control at the computer level.
Instead, the robot interacts with, and responds to changes in the environment at
the mechanical level, which is possible, because the primary dynamical effects can
be modeled as a mechanically stable template.
In this lecture,
we discussed what biologists have found to be a mechanism behind dynamic walking.
That walking can be approximated by the pendular dynamics
of a mass vaulting over its relatively stiff leg.
We've formulated a low degree of freedom model for
this mechanism, called the rimless wheel.
This model is simple enough to analyze and provide insight,
yet complex enough to capture general principles including efficiency and
self-stabilizing behavior in the presence of a power source.
And we saw that by embedding the template dynamics into a physical robot,
engineers can build machines that can walk without computers,
as well as make robot marathoners and mountain climbers.
Next lecture we'll focus on templates for running and
we'll see that running can be described using pendular dynamics as well.
When the relatively stiff leg is replaced with a spring leg.
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