Robotics, kinematics and our existence | Combine

Robotics, kinematics and our existence

What makes you a human, and what made us progress during evolution? Within this article I will present an idea of how some nitty gritty mathematics is connected to your mere existence. Thrilled? Tag along. Over the years I’ve done a few projects on kinematics for robotic arms. Let’s dive straight in to what a typical problem within this context can be. Probably you are located close to a desk or table right now. Move your hand leftward over the table while keeping the distance between the palm and the table constant reaching a cup of coffee. For most of us lucky enough not to suffer from physical discomfort or disabilities, this is a trivial task. Try to do this with a robotic arm, and you will soon see that it is all but trivial. If you have ever operated an excavator or a similar machinery, you will know how hard it is to make the bucket move along a straight line.

1. Grabbing that cup of coffee

In the figure below, a two link robot arm is depicted. We say this depicts your arm. The upper
arm has the length r0 and the forearm the length r1. The position (x; y) of your wrist s described
by the two angles q0 and q1. Your wrist is currently located at the red dot. Say we want to the
grab a cup of coffee that is located at the blue dot. Pretend that you are a robot being controlled
by a computer giving commands to your upper- and forearm. What should the angles q1 and q2
be to achieve this? This type of problem is usually called the inverse kinematics problem. First,
we note that there should be two valid solutions, one with the “elbow” up and one with it down,
right? However, assuming this is us grabbing a cup of coffee, our arm is imposed to kinematical
constraints meaning that the elbow can only be down. I skip the equations for this, but they
can be found in for instance Robotics by Bruno Siciliano et.al.. The solution for this problem is
analytical, but however not completely trivial I would say.
Let’s just say we found the angles q0 and q1 and full of confidence we are to grab our cup of coffee.
Preferably we should move along a straight line, because we have a computer screen in front of
us. However, these angles say nothing about how we reach our cup of coffee. If we just first set q1
to our calculated target value, and then q0 the robot arm end point might move along the dashed
line or something similar. We would knock down our screen and maybe even that cup of coffee in
the process, which is all but a good start of the day.
So, we don’t just want to move to a certain point of interest, we usually want to reach that point
while being constrained to a specific trajectory. A common way in the digital world to do this
is to solve the inverse kinematics in very small steps, where all small steps adds up to a straight
line. This implies that the angular velocities q_1 and q_2 should be coordinated so we get a smooth
movement. A challenge here is that the dynamics of your arm is highly non-linear. Even if your
arm were actuated by ideal electrical motors, the equations of motion could well fill up a whole
page. In this case your arm are actuated by various muscles, which further adds complexity.
Then, we probably want the Cartesian velocity (x_ ; y_)T along the straight line to fulfill some
criterion. Probably we want to move a bit faster at first and slower when we reach the cup so we
can slow down in time and not spill coffee all over the place. This implies that we should want to
employ some trajectory planning. In the end we have a trajectory involving joint space variables
q1; q2; q_1; q_2 as well as Cartesian space variables x; y; x_ ; y_. So this starts to sound at least like a bit
of a hairy problem, eh? Then, add other challenges as that maybe we cannot even reach the blue
dot given that our arm is too short? Or that there is a bowl of cereals, a computer screen or some
other constraint in the way that we need to circumvent.
Then let’s say we add another link to your arm. Now we have three joint variables q1; q2; q3. Having
three input variables and two output variables (x; y) we can suddenly have infinite solutions to the
inverse kinematics problem. This is called a kinematic redundant manipulator. Off course, your
arm probably doesn’t have three links (would be quiet cool though?). But, your arm has far more
degrees of freedom than depicted here in 2D. You can tilt your forearm, upper arm and what not.
Each and every of your fingers are even more complex than the planar manipulator depicted. For
this 2D problem, your arm is kinematically redundant and we couldn’t find a closed form solution
for the inverse kinematics problem. We have to select some of the infinite solutions according to
criterion. A way to do this is to formulate some form of optimization problem. Machine learning
is also being employed sometimes.

 

Figure 1: A figure used to derive some basic ideas related to inverse kinematics

 

2 Kinematics and the evolution

Are you convinced now that grabbing a cup of coffee is maybe not as trivial as you might have
thought? Back to the philosophical question, what makes us human? For one, we are an animal
with relatively high intelligence compared to others creatures we know in this world. Without
dwelling into the subjective definition of what intelligence is, I think we can all agree on this. This
is what enabled us to develop tools helping us to gather, store and process foods among other
things. It encompasses all from weapons such as spears and archery, agriculture, taming fire and
so on. But what would these ideas concepts be without our ability to physically manipulate the
world? Agriculture and fire are merely theoretical pipe dreams when lacking some sort of ability to
achieve this things in practice. Surely, thinking might be existing. But you couldn’t think without
eating. Not in the way humankind exists today anyway.
This is where our advanced kinematics come into play. Advanced kinematics control is as crucial
to define humanity as we know it, as our intelligence I would say. You as a reader, might
not be dependent on your ability to make fire in order to survive in this world. Perhaps clacking
the keyboard of a computer is quiet enough. Which is applying kinematics control. But Stephen
Hawking could do without it you say then. Sure, he personally, yes. But those who built the first
computers? Those who scrabbled down the theorems necessary to build the first computer? Nope.
Then, say we all would communicate and operate like Hawking’s did. You would still at least
need to eat, right? Even if you could communicate with merely your mind, you would still need
to somehow run agriculture. Which again, means physically manipulating the world. It seems
inevitable that our existence is depending completely on our ability to physically manipulate the
world.
Let’s assume that the evolution made us interested in things that were beneficial for our survival
and reproduction in various ways. This is probably where sports come in. Apart form competition,
it is a way of refining our kinematic abilities. Spear-throwing, wrestling and boxing can probably
be directly related to the need of defense and hunting. A tribe being good at handball or a precursor
to it, would probably also be better at throwing stones at both prey and attacking enemies.
Those playing football would probably catch a hare better. Most culture employ dancing. While
dance can fulfill a variety of purposes, one is certainly displaying reproductional benefits. It could
be a way showing off genes being able to handle kinematics well. Do we have complex robots?
Yes. Does anybody of them dance well yet? Well, no. A robot that could dance well would thus
arguably be more complex and all robots so far known and would be able to perform many other
complex tasks then dancing. A person showing off some smooth moves at the dance floor basically
communicates “hey, my gene-pool is probably super-good for a variety of tasks in this world that
is beneficial for our existence. Good from all to hunting rabbits to climb trees”.
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So, smooth and advanced kinematics is not a of interest for engineers and such only. The interest
for it is probably even coded into the DNA of each and every human being. It is fascinating
how we as a humanity, consciously or not, many times happen to mimic features and stages of the
evolution.

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