TL;DR:
The reachable workspace of a planar two-link arm is a ring: its inner radius is |L1 − L2| and its outer radius is L1 + L2.
Joint limits, link collisions and the base itself carve pieces out of that ideal ring.
Place the task in the middle of the workspace, away from the edges where the arm stretches out and loses control.
You mount a robot arm on a table and put a cup on the far side. The arm stretches and stretches, and the gripper just misses the cup. Moving the cup 5 cm closer fixes it. That is the workspace at work: every arm can reach some places and not others, and knowing the shape of that region saves a lot of trial and error.
What the workspace is
The workspace is the set of points the hand can reach. The reachable workspace is every point that the hand can touch in at least one pose. The dexterous workspace is the smaller region where it can arrive with any orientation you like. For arms with few joints, the dexterous space can be small or empty.
Workspace is determined by link lengths, joint types, joint limits and the base position. It is worth knowing before you build, because moving a joint's mounting point by a few centimeters can fix a task that looked impossible.
The geometry of a two-link arm
Look at an arm with two links, of length L1 and L2, in a plane. The shoulder is at the origin. The hand is as far as possible when the arm is fully stretched out in a straight line, so the maximum distance from the shoulder is:
r_max = L1 + L2
The hand is as close as possible when the arm is folded back on itself, so link 2 lies along link 1 pointing back toward the shoulder:
r_min = |L1 − L2|
If the links are equal, r_min = 0, and the hand can reach the shoulder. If one link is longer, there is a hole in the middle. Every distance in between can be reached by choosing the elbow angle. Since the shoulder can rotate all the way round, the workspace is a ring (an annulus) with those two radii.
Worked example: a two-link arm reaches a ring between L1 − L2 and L1 + L2
Let L1 = 0.50 m and L2 = 0.30 m.
Outer radius:
0.50 + 0.30 = 0.80 m.Inner radius:
0.50 − 0.30 = 0.20 m.Ring area (ideal):
π(0.80² − 0.20²) = π(0.64 − 0.04) = π × 0.60 = 1.885 m².
Test three points:
Target (0.60, 0.40): its distance from the base is
√(0.36 + 0.16) = √0.52 ≈ 0.721 m, inside the 0.20 to 0.80 band, so the arm gets there.Target (0.10, 0.10):
√0.02 ≈ 0.141 mis under the 0.20 inner limit, so the point sits in the dead zone around the base and cannot be reached.Target (0.70, 0.50):
√(0.49 + 0.25) = √0.74 ≈ 0.860 mis past the 0.80 outer limit, so the arm falls short.
For a reachable point you can find the elbow angle from the law of cosines. For (0.60, 0.40), r² = 0.52. With the interior angle at the elbow called φ, cos φ = (L1² + L2² − r²) / (2·L1·L2) = (0.25 + 0.09 − 0.52) / (0.30) = −0.60, so φ = 126.9°, and the elbow bend is 180° − 126.9° = 53.1°. A valid angle between −1 and 1 for the cosine is exactly the condition for reachability. For the point (0.70, 0.50): (0.34 − 0.74) / 0.30 = −1.33, which is outside −1 to 1, so no pose exists.
What carves pieces out of the ring
The ideal ring assumes every joint can turn freely. Real arms have:
Joint limits. If the shoulder turns only 180°, the ring becomes a half ring. If the elbow cannot fold fully, the inner hole grows.
Self-collision. Links, motors and cables cannot pass through each other, so folding all the way back is rarely possible.
The base and table. The arm cannot go through its own mount or the surface it sits on.
Obstacles. A box on the table removes the reachable space behind it.
Payload. A heavy load might exceed the motor torque when the arm is stretched, so the usable reach is shorter than the geometry says. The torque at the shoulder grows with distance (see torque and levers).
Where it goes wrong
Working at the edge. Near the maximum reach the arm is almost straight. A small motion of the hand needs a large change in the angle, and the arm is at a singularity. Control gets fragile, and the load on the motors is highest.
Forgetting orientation. A point can be reachable but not with the gripper pointing the way you need.
Trusting measurements. Real link lengths, mount offsets and joint zero positions differ from the drawing. Check by measuring a few reachable points.
Assuming a symmetrical ring. With limits and an offset base, the shape is not a clean annulus.
Safety: The workspace is also the danger zone. Anything inside the reachable region can be hit by the arm, including at speeds you did not expect when a fault sends a joint to its limit. Mark the region on the table, keep people and loose items out while the arm is powered, and have an emergency stop that does not depend on the controller.
How real robots use it
Cell designers place an industrial arm so that every pick and place point sits well inside its workspace, and plan the layout in simulation first. Mobile manipulators (an arm on a wheeled base) fix the edge problem by driving the base closer, which makes the workspace as large as the room. The idea also applies to legs, camera gimbals and cranes. For picking an object up once it is in reach, see grippers: how robots pick things up.
Try it yourself
On a large sheet of paper, draw the ring for your own arm from its two link lengths, using a string and a pencil for the two circles. Mark the shoulder's range of motion and shade the part the arm can reach. Place a coin at one point inside and one outside, and see if the arm (or a card model) can touch each. Then place the coin on the edge and notice how awkward the arm gets.
Frequently asked questions
Why is there a hole in the middle for unequal links?
When the arm folds back, the hand is |L1 − L2| from the shoulder at the closest. If the links differ, that distance is above zero, so points nearer the shoulder cannot be reached.
Is the workspace the same as the places the arm can safely go?
No. Joint limits, collisions, payload and singularities mean the safe workspace is smaller than the geometric one.
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