Robot arm with two joints Adding vectors looks geometrically reasonable. And although the formula is nice, I thought a better question was "why is this natural and important?" Why does this phenomenon occur in many places? One place it happens is in physics. So you learned that if two different forces are pushing on something, such as a mass on an inclined plane being pulled by gravity and the inclined plane itself being pulled by gravity, then the total force is simply the sum of those two vectors. This formula is the most accurate representation of how the real world functions. I will now introduce you to another problem where the vector addition process comes in quite naturally. This is a problem in sort of robot engineering that we will be working with for the next few weeks. We will examine the problem of controlling a robot arm with two joints. The robot's arm consists of a ground base (shown iheren the first image) and a joint at the origin (shown over there). The rod extends out of that joint. This rod has a length 2, and at the other end there is another joint, where another rod emerges. This other rod is shorter with a length 1. At the end of one rod is a tip that looks like the finger on a robot hand. Let us say that x comma y is the position of this point here, at the end. So I'm not sure if this drawing is clear. There is a joint here, so this piece can turn, and another joint here, so that piece can turn independently. The robot, therefore, can traverse the edge of this fingertip by moving along two angles: one here, theta 1, and one here, theta. Inside the robot, it directly controls theta 1 and theta 2. If you're writing a computer program to control this robot, you can tell it to increase theta 1 by 0.1, and the robot will do so. Our long-term goal with this robot is to determine how it functions. How can we program it to perform certain tasks. For example, we could have programmed the robot to lift its finger straight up. Someday it might be fancier and imagine that this robot arm is holding a piece of chalk. We wish to program the robot to take a piece of chalk and draw a parabola on the board. The robot's two degrees of freedom are theta 1 and theta 2. We want to program it so that it can adjust its angles to perform certain tasks. Now today we will discuss the first step of this project. The first step is determining the angle θ1 and θ2 of the tip of the finger. Thus, the question of finding x and y in terms of theta 1 and theta 2 remains. If I attempt to tackle this problem all at once, the task seems overwhelming. I even feel a little overwhelmed trying to find x or y, but this problem can remind us of this picture with its v and w vectors. So there are two vectors in this picture. A vector goes from here to here. And we'll call the vector from the first joint to the second joint w. So the vector that goes all the way from the origin to x, y is v plus w. x, y is the v plus w. In order to find x, y, we have to find v and w. And this is helpful, because each of those subproblems is easier to solve than the original problem of finding x and y. The vector concept, and the idea of vector addition, allows us to take a complicated problem and break it into two smaller problems—a theme in mathematical problem-solving. Smaller problems are easier. Thus, we can solve the system of equations by finding two vectors, v and w. To find v, we basically already found it. That's the same problem that we solved on the board. So we know this angle, theta 1, and this length of 2 instead of 3. Now, v is 2 cosine of theta 1 comma 2 sine of theta 1.