**Forward Kinematics 2D:**

Problem:

Given the angles and , find the X and Y coordinates of the end effecter of a robotic arm with segment lengths a_{1} and a_{2} as shown.

First write the segments x and y as a function of the specified coordinates: The x and y coordinates of segment a_{1} can be written as:

Since the angle is relative to the segment a_{1}, the angle can be defined as follows:

Then:

The x and y coordinates of segment a_{2} can be written as:

Therefore the total X and Y coordinates can be written as

**Forward Kinematics: 3D**

The arm also features base rotation so that there is some angle as shown which rotates the arm into the z axis. Building on the 2D example, we now need to rotate the arm. The base rotation will not change the definition of the y coordinate. However the x coordinate will be projected onto the z axis as follows:

Therefore by substituting rx into the above, the components can now be written as:

So given the angles , and along with the segment lengths a_{1} and a_{2}, the gripper will be positioned according to the above equations.