Forward Kinematics: Where Did the Hand Go?
An arm is trigonometry you can bolt together: each joint turns, each link carries the turn outward, and two lines of sine and cosine name the exact spot the hand ends up.
Builds on: 1.2 Servos & Steppers: Position Without Guesswork
Same idea, other course:Vector: splitting vectors with sin and cos
Links, joints, frames
Strip a robot arm to geometry and it is a chain: rigid links of fixed length, joined by joints that each contribute one angle. The two-link arm — shoulder θ₁, elbow θ₂, link lengths L₁ and L₂ — is the fruit fly of arm robotics: small enough to solve on paper, rich enough to contain every important idea.
One subtlety carries most of the machinery: the elbow angle θ₂ is measured relative to the upper arm, not to the table. The elbow rides on the shoulder’s rotation — each link works in the coordinate frame its parent hands it. Chains of frames, each riding the last, is precisely how professional software describes robots of any size; the two-link arm just keeps the chain short.
The forward map
Walk the chain and add up where each link carries you:
That (θ₁ + θ₂) is the frame idea wearing algebra: the forearm’s direction is the shoulder’s turn plus its own. Forward kinematics — angles to position — is the easy direction: plug in, read off, no ambiguity, works for six joints as surely as two (just with more stacking).
The workspace: where the hand can ever be
Sweep both angles through everything they can do and the hand paints an annulus — a ring with outer radius L₁ + L₂ (arm straight) and inner radius |L₁ − L₂| (arm folded). Everything the arm will ever touch lives in that ring; no cleverness reaches past it. Drawing the workspace is the first honest act of arm design: it tells you before any code whether the shelf is reachable.
Close your eyes and touch your nose: your brain aimed the fingertip without solving cosines consciously — and aiming is the reverse problem, position to angles. That direction is genuinely harder (multiple answers, unreachable spots) and is exactly the next lesson.
⚗️ Lab — The Two-Link Sandbox
Shoulder and elbow sliders, live hand coordinates, and a trace that remembers where the hand has been.
- Move only the shoulder: the whole arm sweeps, the elbow’s world moves with it — frames riding frames.
- Straighten (θ₂ = 0) then fold the arm fully: you have just drawn the workspace ring’s outer and inner edges.
- Sweep both sliders wildly and let the trace fill in the annulus. Anything unpainted is forever out of reach.
Work it by hand — 0 / 2
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±2% count.