How do robot joints position a tool?

A robot arm reaches a position by combining the rotations of connected rigid links. A joint does not simply move the tool along one coordinate: changing a joint angle changes the direction of everything beyond that joint. Geometry translates these combined directions into a tool position.

The illustrated arm is a flat, two-joint example. Its first link is 2 meters long and its second is 1 meter long. The first angle, θ1, is measured counterclockwise from the horizontal. The second angle, θ2, is measured relative to the first link. Therefore the second link’s direction relative to the horizontal is θ1 + θ2, not θ2 alone.

Each link contributes a horizontal and vertical displacement. Adding them gives x = 2 cos θ1 + cos(θ1 + θ2) and y = 2 sin θ1 + sin(θ1 + θ2), measured in meters. This is forward kinematics: joint angles are the inputs, and tool coordinates are the result. At θ1 = 0° and θ2 = 90°, the first link contributes (2, 0), the second contributes (0, 1), and the tool arrives at (2, 1).

The reverse problem asks which joint angles place the tool at a chosen coordinate. This is inverse kinematics. For unrestricted joints, a reachable point commonly has two configurations, often described as elbow-up and elbow-down. Both can produce the same endpoint even though the links occupy different places. At fully stretched or fully folded boundaries, those alternatives can merge.

Link lengths also set a basic reach limit. With both links straight outward, the tool is 3 meters from the base. With the shorter link folded directly back along the longer one, it is 1 meter away. Allowing every joint angle fills the ring between those distances. The center is unreachable because the 1-meter link cannot cancel the full 2-meter displacement of the first.

That ring describes position reach, not a complete safe movement plan. An obstacle may block an otherwise reachable configuration, and actual joints may have restricted angles. A task can also require a particular tool orientation or a clear route from the previous position. Motors must supply suitable forces and control motion. The geometric calculation is valuable because it answers one precise question first: where do these link directions put the tool?