How a station robot arm reaches a target

A station arm has seven joints, but most of its reach comes from two long booms joined at an elbow. The shoulder, the elbow and the target form a triangle whose three sides are known, so the law of cosines gives the elbow angle and a little more trigonometry gives the shoulder angle. If the target is farther than both booms laid end to end, or closer than their difference, no angle works.

Controls

Labels

Animation speed

Target distance from shoulder

Target angle above the truss

Outer boom length

The inner boom stays at 7 m.

Step through it

Seven joints, one planeThe arm has three joints at the shoulder, one at the elbow and three at the wrist. With the shoulder roll and yaw set, the shoulder pitch and elbow do the long distance work in one flat plane.
A triangle of known sidesInner boom, outer boom and the straight line to the target make a triangle. All three side lengths are known, so the law of cosines gives the elbow angle directly.
Shoulder angleThe shoulder points at the target, then tilts up by the angle the outer boom makes in the triangle. That keeps the elbow above the line, the elbow up solution.
Beyond full stretchTwo 7 m booms laid straight reach 14 m. Ask for 16 m and the cosine falls outside the range from minus 1 to 1: there is no triangle, so the arm stretches out and stops short.
Too close to reachWith a 3 m outer boom, folding it fully back still leaves the wrist 4 m from the shoulder. Targets closer than the difference of the booms are out of reach too.

The law of cosines solves the elbow

For a triangle with sides L1, L2 and d, the law of cosines says d squared equals L1 squared plus L2 squared minus 2 L1 L2 times the cosine of the angle between the booms. Rearranged, the elbow bend from straight is the arccosine of (d squared minus L1 squared minus L2 squared) divided by (2 L1 L2).

The shoulder angle is the target direction plus the angle whose tangent is L2 sin(bend) divided by (L1 plus L2 cos(bend)). Flip that sign and you get the elbow down answer, which reaches the same point with the elbow below the line.

Try it: set the outer boom to 7 m and slide the target distance from 10 m up to 14 m. The elbow bend shrinks toward zero as the arm straightens, and at 14.5 m the target ring turns red because no triangle exists.

Why seven joints, if two do the reaching

Two planar joints fix where the wrist goes. Three wrist joints then set the payload's orientation, and the extra seventh joint gives a free choice of which plane the booms swing in, which helps the arm avoid hitting structure and avoid singular poses. Operators often lock one joint and solve the remaining six.

This is a teaching model: it treats the arm as two booms in one plane, measures the target from the shoulder pitch joint to the wrist, and keeps the payload level by setting the wrist pitch to cancel the other two angles.

Where Avid uses this

Solving joint angles from geometry is the same core step in the agricultural robotics Avid works on, where an arm or end effector has to reach a plant it has located.

Sources

A kinematic analysis of the Space Station remote manipulator system (SSRMS)NASA Technical Reports Server (Journal of Robotic Systems, University of Florida and NASA Goddard)
Introduction to Robotics, Chapter 4: Planar KinematicsMIT OpenCourseWare, Massachusetts Institute of Technology

How to cite this model

Avid Solutions International. (2026, November 11). How a station robot arm reaches a target [Interactive 3D model]. Avid Field Lab. https://avidsolutionsintl.com/explore/space-robotic-arm

Last updated November 11, 2026

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