The most robotics enthusiasts lack the necessary
theoretical knowledge. The lack of this knowledge mainly argumented by the
complexity of these theories.
It is for reason that enthusiasm is not
supported by the knowledge of theory the result work looks like a child's play.
Huge efforts and waste of time, most of the initiatives ends regular structure,
similar to hundreds of other similar structures and does not have any
different.
I would like to start a series of articles that
demonstrate the possibility of studying the necessary theories at the own
example. A month ago, I began to study the subject "Under-actuated
robotics", presented by the Massachusetts Institute of Technology and is
open to the public (MIT, Underactuated Robotics, OpenCourceWare).
Although the subject of my articles will be
"walking robots", it does not mean that the theory cannot be applied
to other areas of robotics.
As my study of the necessary material and
retention of knowledge in test labs with the help of MATLAB, I will make out it
in the summary form. The emphasis is not on the details of specific projects, but
the simplicity of the theories themselves.
If someone will be interested in the details,
which I did not mention in the articles, I will gladly try to answer all your
questions.
Before the beginning of articles, I would like
to add that the Russian version of the articles could be found at the web page of Students Design Bureau at the Kharkov Institute of Radio Electronics, where
I am honored to be as the enthusiast.
Dynamics
The first question on the way to understanding
of the theory of walking robots that take in account its own weight and
inertia, is the dynamics. Dynamics - the complex task, where were written a tons
of books, which includes the items "Theoretical Mechanics",
"Structural mechanics & Strength of Materials", some aspects of
the physics, mathematics, etc.
Although the theory of dynamics is a complex
problem, Lagrange dynamics would be enough to start. The formulation of the
dynamics by Lagrange simple enough to be understood quite ordinary student who
has a basic knowledge of mathematics, and able to distinguish the difference
between the kinetic energy and the potential energy.
The next example shows a system of two bodies
(cart and pendulum with famous name: Cart-Pole), which has two degrees of
freedom. The first one - cart horizontally, the second - rotation of the
pendulum. For detailed information about the concept of "degree of freedom",
refer to Lagrange Dynamics. System performs free oscillations of the pendulum.
You may notice the inertial deviation of the cart.
Although it was not difficult to accomplish, this
system does not take into account the loss of energy. Therefore, with
compliance the law of conservation of energy, there is no damping of the
pendulum.
Stability
The next question in the understanding of the
theory of walking robots is stability. On this subject, as for dynamics were
written many books, and this science continues to evolve to this day. Pleased
to note that the key works in this subject were made by Russian and Soviet
scientists, such as, for example: Lyapunov A.M., Pontryagin, etc.
In robotics, stability theory gained great
development. For the first steps is enough to use so-called "Linear
Quadric Regulator" (LQR) and "Energy Shaping" tools. The
combination of these two strategies in the control system is sufficient for the
creation and implementation of the necessary conditions for the stability of
dynamical systems, where number of degrees of freedom is greater than the
number of actuators.
The next example shows the earlier mentioned
Cart-Pole system. In the horizontal direction the controlled force applied to
the cart, force value determined by the automated control systems (ACS). ACS
configured for swinging the pendulum to the top vertical position and bringing
the cart to the origin.
Below is another example of a dynamic system,
which consists of two interconnected pendulums, one of which is attached to a
fixed base. The controlled force is a moment between two pendulums in their
attachment to each other. ACS configured to bring both pendulums into the top
upright.
Animation below shows the double pendulum, the
control torque applied to the different degrees of freedom of the system.
In all of the examples systems use a common
algorithm with implemented Energy Shaping controller to swing system to area
near stable-point and stabilizing it with help LQR. The only difference in the
implementation of these tasks - the matrix representing the dynamics of the
system and the procedure of drawing. Differences between implementations of a
double pendulum system with different control torque is to change the
corresponding matrix [0,1] to [1,0].
Thus in order to realize the movement of any
other system in its balanced position (even if not resistant), will be enough
to change the matrix dynamics and the procedure for rendering.
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