Just-in-time references. Print half-page, laminate, one set per table.
For anyone who missed the Day 0 interview or joined mid-unit. Pick one stand-in user — you'll have a user, a motion, and a constraint in five minutes and can start Day 2 with everyone else.
Use your stand-in's name and motion for the whole unit — the Brief (Day 5), FK Card (Day 10), IK Solve Card (Days 15–16), and Final Presentation (Day 20) all point to this person.
Interview a real person later? Only swap users while you are still early enough to redo the motion, the coordinates and all three criteria — realistically before your Brief is tagged. After that, every number you have solved belongs to the stand-in, and switching means starting the math again. Keep the stand-in and note the real interview in your reflection instead.
Name the sides relative to the angle θ you care about.
Distance formula — the straight-line reach to the target:
Use it when: you have a right triangle, you know two sides, and you want the angle θ.
1. Name the sides — always relative to θ:
2. Pick the ratio that uses the two sides you actually have:
3. Undo the ratio with inverse trig (sin⁻¹ / cos⁻¹ / tan⁻¹) to get the angle itself.
Worked example — target (10, 8). You know Opposite = 8 and Adjacent = 10, but not the hypotenuse → use tan:
tan θ = 8 / 10 = 0.8 → θ = tan⁻¹(0.8) ≈ 38.7°
Calculator: set it to DEGREES first — wrong mode, wrong answer. (TI-84: 2nd → TAN. Desmos: atan( ), in Degrees.) Inverse sin/cos only work when the ratio is between −1 and 1.
Check your mode first — DEGREES. Wrong mode = wrong answer.
You can only use sin⁻¹/cos⁻¹ when the ratio is between −1 and 1.
Arm sweeps 45°, r = 18 cm: 45° = 0.785 rad → arc = 18 × 0.785 ≈ 14.1 cm
Arm triangle: a = L₁ = 18, b = L₂ = 22, c = d (distance to target).
The one you want is θ₂′, the elbow bend used everywhere in this unit:
That is the whole card. BC6 puts it in the pipeline; BC7 turns it into what you type.
If a textbook disagrees with us: books solve for the triangle's interior angle, cos γ = (L₁² + L₂² − d²) / (2·L₁·L₂), which for (10, 8) gives γ = 35.6°. Our elbow is its supplement, θ₂′ = 180° − γ, and flipping the numerator gets there in one step instead of two. Same law, one step fewer — you never need to compute γ.
Given target (x, y), L₁ = 18, L₂ = 22:
Which of the two solutions this is. Every target has two: elbow-up and elbow-down. θ₁ = α + β is the elbow-up branch, and it is the one this whole unit uses. Elbow-down would be θ₁ = α − β. Keep the same branch at both endpoints of your motion. (The other branch, and why it often cannot run, is EC1.)
θ₂′ is a size, always positive. The signed joint angle is θ₂ = −θ₂′ for elbow-up, and only the signed one gets the hardware offset — see BC7. Or just press GO and skip the conversion entirely.
Running it: the easy path is the app's GO button — enter the target (x, y) and press GO; the app solves IK and sets both joints, offset handled.
Two different reaches, and the difference matters. Ideal two-link geometry allows 4 ≤ d ≤ 40 cm. This arm's elbow bends only to 160°, so its usable range is 8 ≤ d ≤ 40 cm. Solve against the math; commit against the hardware — a target at 6 cm has a perfectly good answer that this machine cannot hold.
Nine steps. Do them in order.
Why step 3 is on the list. Left on IN, a target typed as 10, 8 is read as inches and lands at (25.4, 20.3) cm. That is still inside the reach, so there is no warning of any kind — the arm just goes somewhere about 20 cm from where you meant, and on Day 17 it looks exactly like a calibration problem.
Reach: ideal two-link geometry gives 4–40 cm. This arm's elbow bends only to 160°, so its usable classroom range is 8–40 cm. Solve against the math, commit against the hardware.
Shoulder J₁ = θ₁ as solved. The elbow field is offset: J₂ = θ₂ + 90, using the signed θ₂ — and elbow-up means θ₂ = −θ₂′. At (10, 8): −144.4 + 90 = −54.4.
The field only catches impossible angles. Enter something outside a joint's travel and it flashes red and snaps back. It will not stop an angle the arm can reach that is simply the wrong one — and that is the mistake people make. Type θ₂′ raw, or add 90 to θ₂′ instead of to the signed θ₂, and the arm accepts it and puts the end-effector somewhere else entirely. GO avoids both.
Joint travel: shoulder J₁ 20° to 150° · elbow J₂ −70° to 250° · pen J₃ −50° to 35°. These decide which of your two IK solutions can actually run: a pose whose θ₁ falls outside 20–150° is correct mathematics the arm cannot hold. The arm also refuses any pose putting the tip below the shoulder line (y < 0).
Whose arm: your crew's arm is yours for the period. Four of you share it, and every crew runs at the same time, so you never wait on another crew.
Constraints are binary — a hard limit the design must satisfy, or it fails. (e.g. “fits on a 30 cm desk,” “clears the 8 cm rail.”)
Criteria are attributes you measure to judge success. Decide which are critical (must-have) and which are only desired (nice-to-have).
Make it a number. Attributes you can measure quantitatively:
Give each a threshold, not one exact value — then anything better than the threshold passes. Use a maximum when lower is better and a minimum when higher is better.
Avoid a band like “1–2 cm”: it fails a landing 0.4 cm out, which is the opposite of what accuracy means.
Subjective criterion? Build a User-Defined Scale (UDS). Turn an opinion into a number. For “easy to use”: Agree = 3, Neutral = 2, Difficult = 1. Now even a feeling has a score you can test.
Your robot is a 2-joint arm (shoulder + elbow) with an end-effector that opens and closes. No wrist, no twist — and it works in one flat plane. Pick a motion that fits it.
And it has a reach. Our arm's usable classroom range is the shaded band: no closer than 8 cm, no further than 40 cm. (Ideal two-link geometry would allow 4–40 cm; this hardware cannot use every pose the geometry permits, which is why the floor is 8.) Too far is obvious. Too close is the one that catches people — the elbow only folds so far, so a point almost on top of the base is out of reach too.
Check this before you solve anything. If your point is outside the band, no angles exist and the math will not tell you politely — the calculator just errors.
It CAN — move the end-effector from one reachable point to another in its plane, and grab and release. Holding a marker, it traces whatever curve the two joints make as they turn from the first pose to the second — enough to draw a checkmark. You choose the two points; the exact line between them comes out of the joints, so treat it as the journey you asked for rather than a path you specified.
It CAN'T — twist or rotate an object (no pouring, unscrewing a jar, turning a key), do fine dexterity (no buttoning, or neat handwriting), or move out of its plane (it's flat, not 3-D).
Keep it in reach: every point of the motion must sit 8–40 cm from the shoulder. Farther than 40 and the arm can't stretch there; closer than ~8 and the elbow can't fold that tight (it bends only to 160°). Check both your start and end points before you commit.
Good motions: carry a cup from a holder to a tray · slide a plate across a table · draw a checkmark on a whiteboard · press a button · move a block from bin A to bin B. If it's a path between two points, plus grab/release, your arm can do it.
For early finishers — self-guided. Your IK Solve Card gave one answer for the end point (10, 8): θ₁ = 128.2°, θ₂′ = 144.4° — an elbow-up pose. There is a second mathematical solution that lands the hand on the exact same point. Find it — and then find out whether this arm can actually hold it.
This card uses atan2, which is past the math this unit requires. You do not need it for your project.
Why two? The Law of Cosines gives one cosine for the elbow:
But arccos only gives you the size of the bend, not its direction — arccos(−0.813) = 144.4°. Both poses bend the elbow by the same 144.4°; what differs is which way it folds. That is the θ₂′ on your Solve Card, and it is the same number for both solutions.
Careful — two different symbols. θ₂′ is the magnitude from the Law of Cosines (always positive). The atan2 form of IK, which the app uses, needs a signed θ₂ — and the sign is exactly what picks the branch:
Run it both ways and you get the two poses — and the sign is the opposite of what you might guess:
Compute the elbow-down pose for (10, 8). Same reach, same 144.4° bend, opposite fold: β flips sign, so θ₁ = α − β = 38.7° − 89.5° = −50.8° (signed θ₂ = +144.4°). Verify with forward kinematics that this pose really does put the tip at (10, 8) — it does. The mathematics is sound.
Now the part that matters. Check each θ₁ against the shoulder's travel, 20° to 150° (BC7):
So for this target there is only one pose this arm can hold, and GO was never choosing between two. Try the elbow-down angles and the field will refuse them. Mathematically reachable is not the same as physically reachable — the model produces a valid answer the mechanism cannot execute, and noticing that is the engineering, not a disappointment.
A target where BOTH branches fit inside 20°–150° does exist — go looking for one. That is a better use of the aimer than trying to force this one.
Then answer: Why did you choose elbow-up for your project's motion? What would change if you needed to avoid an obstacle sitting between your start and end points — could the other solution route around it?
For early finishers — self-guided. Give the arm a third link — a wrist: L3 = 10 cm with its own angle θ₃. Everything stays in the plane; you just add one more segment.
Forward kinematics — just add a term. The tip is the sum of all three links:
Inverse kinematics — now it's underdetermined. Three unknowns (θ₁, θ₂, θ₃) but only two equations (x and y). There are infinitely many poses that reach the same point — the arm can hit a target with the wrist held at many different angles.
The trick: fix the wrist, then reuse what you know.
Now sweep φ and describe what changes: the hand stays on the target, but the whole arm reshapes and the wrist points differently.
Then answer: The REV arm has a pen joint (θpen). What would change about your motion if that joint could aim the pen at a specific angle — not just up or down — while still hitting your two endpoints?
Finished early? Push further. Pick one — each deepens the engineering, not just the busywork. Whatever you try, measure it and write down what you found — a strong result here is presentation gold on Day 20.
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