The predictable errors, where each one bites, and the fastest way to surface it in the room. Ordered roughly by how much it costs when it goes unnoticed.
They thinkA bigger joint angle means the tip reaches farther.
Reach is set by the ELBOW. Bending it more (a larger θ₂′) folds the arm and pulls the tip CLOSER; the farthest reach is a straight arm. The SHOULDER rotates that reach toward the target direction — it changes where the tip is, not how far out it sits.
Catch it: Hold the shoulder fixed and compare a straight arm with a bent elbow. Ask which tip is farther from the base BEFORE measuring, then measure.
Days 9–14 (forward & inverse kinematics)
On this arm: Demonstrating this on the panel is easy to get backwards, because the elbow field is not θ₂′. Type 90 for a straight arm and 234 for a 144° bend, and the reach drops from about 40 cm to 12.8 cm. Entering 0 and 144 raw shows 28 cm then 36 cm — the opposite of the point you are making. Use GO with two targets instead if you are not sure.
They thinkIf my calculator gives an angle, that is what I type into the robot.
A correct mathematical angle is not automatically the command the machine takes. The math angle and the joint field can use different zero positions, directions and ranges, so one has to be converted into the other. This is the single most expensive error in the unit: the arm accepts the wrong number without complaint and goes somewhere else entirely.
Catch it: Before any angle is typed, ask: “Is that a MATH angle or a ROBOT field value?” Make them say which, out loud, every time.
Days 9–17 (every time an angle reaches the arm)
On this arm: The elbow field takes the SIGNED θ₂ plus 90, and elbow-up means θ₂ = −θ₂′. At (10, 8): −144.4 + 90 = −54.4. Pressing GO avoids the conversion entirely, which is why GO is the default route.
They thinkCorrect math means the arm lands exactly where I predicted.
It means the arm lands close. A physical machine has servo tolerance, backlash, flex and a base that may not be square, and your ruler has its own error. Predicting (24, 18) and measuring (23, 19) is not a failed calculation — it is what a real mechanism does. What tells you something IS wrong is the size and the repeatability of the gap, judged against the criterion you set beforehand.
Catch it: When a crew reports a small miss and starts re-deriving the math, stop them: “Does it clear your criterion? Does it repeat?” Those two questions come before any change.
Days 10, 16, 17 (every measured run)
They thinkCosine only works in right triangles.
SOHCAHTOA needs a right triangle, because it reads a ratio off two named sides. The functions themselves do not: sine and cosine are coordinates on the unit circle, defined for any angle. The Law of Cosines extends that to ANY triangle, which is exactly why the arm needs it — the elbow triangle is not generally right, so SOHCAHTOA cannot reach its angle.
Catch it: Watch for students forcing SOHCAHTOA onto the arm triangle. Point at the missing right angle at the elbow.
Days 12–13 (Law of Cosines)
They thinkInverse trig just reverses multiplication — arctan(y/x) is basically dividing.
Two steps, not one. y/x gives you a RATIO. arctan turns that ratio into an ANGLE. The division is real and necessary; the mistake is stopping after it and calling the ratio an angle.
Catch it: Catch the student who computes y/x and stops. Ask “y/x is a ratio — where is the angle?” and have them hit the arctan key.
Days 7–8, 14 (finding angles)
They thinkIf I know how far away the target is, I know which way to aim.
Distance says how FAR, never which DIRECTION. (12, 16) and (−12, 16) and (16, 12) are all 20 cm from the base and need three different shoulder angles. Direction comes from the two coordinates and their SIGNS — which is why arctan alone is not enough when x is negative, and why the quadrant has to be checked.
Catch it: Put two targets the same distance from the base in different quadrants and ask: “Same r — same shoulder angle?” Then have them compute both.
Days 7–8, 14–15 (aiming, and every IK solve)
They thinkEvery target has two solutions — elbow-up and elbow-down.
In the ideal two-link model, targets strictly between the minimum and maximum reach have two: elbow-up and elbow-down. At either boundary the two collapse into one. And the real arm can rule one out entirely — its shoulder travels only 20° to 150°, so at (10, 8) the elbow-down solution needs θ₁ = −50.8° and simply cannot run. Two solutions on paper is not two poses on the machine.
Catch it: Push an edge case — “solve for a target exactly 40 cm out, how many elbow choices now?” — then a limit case: “this one has two on paper; check each θ₁ against 20°–150°.”
Days 11, 15–16 (two solutions & branch choice)
They thinkThe robot can always reach the target.
Only if the target lies in the ring 8 ≤ r ≤ 40 cm, and a motion needs BOTH endpoints in range. Geometry alone would allow 4 cm, but the elbow bends only to 160°, so 8 cm is the real floor. Passing that check means the point is worth solving — the shoulder still has to be able to aim there.
Catch it: Ask the physical question first — “is this point inside the ring?” — and make it Step 0, before any solving. In the arithmetic, the same fact shows up as a cosine outside −1 to 1, which means no elbow angle exists that closes the triangle.
Days 15–16 (reachability · Step 0, and the gate)
They thinkA motion is a single point — solve it once.
A motion has a START and an END; you solve inverse kinematics TWICE, once per endpoint, then sweep the joints between the two poses.
Catch it: Any brief with one point: “A motion needs a start AND an end — give me both, in cm.”
Day 11 onward (path planning)
They thinkDegrees and radians are interchangeable in the formulas.
Arc length s = r·θ requires θ in RADIANS — that is what a radian is for, tying an angle to a distance around a circle. Other things want degrees: this arm's joint fields take them, and so does every angle students read off a protractor. So the rule is not “always radians”, it is CHECK WHAT THIS FORMULA OR FIELD EXPECTS. Feed a degree count into s = r·θ as though it were radians and the answer comes out 57.3 times too large.
Catch it: Make “which unit does this one want?” the reflex before any s = r·θ. Note that an arc CAN legitimately be longer than the arm — an 18 cm link sweeping 150° traces about 47 cm — so length alone proves nothing. What gives the degree error away is its size: an answer tens of times bigger than the sweep could produce.
Days 3–4 (radians, arc length), Day 25 (why the angle inside cosine is in radians)
Emphasis, not deficiency — these are things most courses do less of, not things they do wrong.
| Traditional trig memorizes identities | RoboTrig uses them to position a robot arm |
| Traditional assessment grades a worksheet | RoboTrig proves the arm reaches the target |
| Traditional engineering builds, then explains | RoboTrig derives the mathematics before building |
| Traditional robotics codes first | RoboTrig does the mathematics first |
| Traditional math solves for an abstract x | RoboTrig solves for a real person’s motion |