Robotics: Locomotion Engineering Videos
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Robo4x - Video 01.2
Daniel Koditschek
We're going to start with principal bioinspiration.What we mean in by principal bioinspiration?Bioinspiration is very popular in many parts of engineering,and particularly popular in robotics.
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Robo4x - Video 01.3
Daniel Koditschek
Why do animals move? Humans have been thinking about this ever since they were humans. Aristotle introduced notions of the interplay between the animal's cognition, its appetites or desires and its ability to perceive changing world states and understood that the motive for mobility came about through this interplay of desires and and perception.
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Robo4x - Video 10.1
Daniel Koditschek
This second unit has to do with the building of the hybrid systems model, the crucial thing that we've been focusing on. And here we're going to introduce to the flows that we did in unit one, we're going to introduce the hybrid structures, the guards, and the resets, and the mode maps.
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Robo4x - Video 10.2a
Daniel Koditschek
In this segment, we're going to finally clean up and take advantage of
our carefully developed flows and guards and resets and hybrid systems model. -
Robo4x - Video 10.2b
Daniel Koditschek
Let's now pursue the more algebraic and the mathematical representation of this symmetry in the spring loaded inverted pendulum, by interpreting this symmetry and merging it with the notion of neutrality.
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Robo4x - Video 10.3
Daniel Koditschek
The return map analysis is the most important moment in the course. It's the point in the course where all those complicated things that we've been leading up to are going to turn out to be analyzable and they'll be analyzable in a way that gives us really, really important conclusions about the fore-aft motion of the spring loaded inverted pendulum.
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Robo4x - Video 10.4a
Daniel Koditschek
Remember that we had developed a model for the return map of the fore-aft
degree of freedom of this spring loaded inverted pendulum. And we had posited given a closed loop and return map, the imposition of a feedback law. And we would thereby close the loop, leading to a closed loop discreet dynamical system governed by the SLIP return map. The question now is what's the consequence of that feedback law? What sorts of feedback laws will have what sorts of fixed points, equilibrium states? -
Robo4x - Video 10.4b
Daniel Koditschek
Let's first come back to Raibert's stepping control, and let's
now analyze Raibert's stepping control, in light of this new analytical insight. -
Robo4x - Video 1.4
Daniel Koditschek
A lot of insight in locomotion engineering, particularly for legged locomotion, comes from bioinspiration. Bioinspiration is the importance of not copying, but rather seeking, extracting and using the principles.
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Robo4x - Video 1.5
Daniel Koditschek
We're going to try to explain how it is that the grand title of
Locomotion Engineering for this course gets narrowed down into what the course is really about which is all about legged mobility of a particular kind, of a dynamical kind. How do we do that ? How can we explain that? That's what we're gonna try to do in this segment. -
Robo4x - Video 2.1
Daniel Koditschek
We're going to start at the very beginning of things , namely the Prismatic one degree freedom physics. We'll introduce slightly different terminology and slightly different notation and so it's not a bad thing to review and start you on the escalator from the first step rather than the fifth step.
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Robo4x - Video 2.2a
Daniel Koditschek
We have a scalar 2nd order ordinary differential equation given by this acceleration field. We're going to continue solving for Vector 1st Order.
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Robo4x - Video 2.2b
Daniel Koditschek
By now you're starting to visualize geometrically what these solutions look like to these ODEs. We're going to plot various examples of the time trajectories from initial conditions and the flow orbits, as they're called on the plane for the same initial conditions.
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Robo4x - Video 2.3
Daniel Koditschek
Energy, which seems like a physical idea, is going to play the role of a geometric norm in much of what's to follow in the course. We'll calculate expressions for the energy, power and Basins Total Energy as the Norm.
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Robo4x - Video 3.1
Daniel Koditschek
We want to remind you of what you learned in, probably, college physics, or
what you will learn in college physics, if you haven't taken college physics yet. We discuss an example of a one degree of freedom revolute kinematic chain. -
Robo4x - Video 3.2
Daniel Koditschek
Let's think about the lossless version of the one degree of freedom
revolute vector field. We would like to write a vector field, mainly we want to
rewrite our 2nd order scalar ODE as a 1st order vector ODE. But this time we're gonna nonlinear vector field. -
Robo4x - Video 3.3a
Daniel Koditschek
It's time to start thinking about qualitative theory. The kind of theory that we're going to use in higher degree of freedom, more interesting robot systems, as the course goes on, where we just don't have any access to closed form solutions. Of course, we'll use the linear examples where we do have closed form solutions,
to get the ideas across. -
Robo4x - Video 3.3b
Daniel Koditschek
Let's talk about the stability properties of fixed points of the damped harmonic oscillator. To do that, we need to go back and remind ourselves what the vector field associated with that damped harmonic oscillator really is.
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Robo4x - Video 3.4
Daniel Koditschek
Let's look again at the pendulum that we introduced in the beginning part of this segment. This time we're going to add damping. We've explored the properties of damping in the exercises for the beginning of this unit. Recall that whereas without damping, we had the whatever total energy we started with we continued, the orbits continued on that same total energy shell.
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Robo4x - Video 5.1a
Daniel Koditschek
Let's remember what we're trying to do, where we're going. In this segment, we're going to address the problem of nonlinear vector fields generating flows that really we don't have any closed frame solutions to express. And what we're going to do is we're going to look for means of approximating them numerically
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Robo4x - Video 5.1b
Daniel Koditschek
In order to get this notion of sameness, to be mathematically formal, we have to now expand our notion of what we mean by a change of coordinates. So far all of our changes of coordinates have been linear. Now we're going to introduce nonlinear changes of coordinates.
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Robo4x - Video 5.2a
Daniel Koditschek
In the previous segment, we just introduced the notion of a normal form.
And the Flowbox theorem tells us that when we're not close to a fixed point,
then the vector field is conjugate to some constant velocity vector field.
We're now going to look and try to identify the normal form where the zeroth approximate fails, and instead we have to go to the first order, or the linearized vector field. -
Robo4x - Video 5.2b
Daniel Koditschek
That brings us to a crucial new idea, which is one of the most important
qualitative ideas that we'll use in the course. We're going to say that a vector field is hyperbolic at a fixed point, exactly when its linearization, that is it's Jacobian evaluated at that fixed point, has no purely imaginary eigenvalues. -
Robo4x - Video 5.3
Daniel Koditschek
Closing out this first week of background and practice with more of the technical ideas. We're going to introduce the, in some sense, the most important property
of dynamical systems that makes them so amenable to use not just in describing
the behavior of mechanical robot systems, but in trying to prescribe that behavior.
We're going to introduce the notion of Lyapunov functions, which generalizes physical energy. -
Robo4x - Video 5.4
Daniel Koditschek
We're now ready to talk about one of the central ideas in the course. In fact it has to do with one of the most fundamental ideas in dynamical systems theory and its applications, which is the notion of stability. It is this steady-state notion of persistence in state.