M08 · optional contextual support
Turn an angle into local motion
Connect trigonometry, rotations, Jacobians, least-squares motion and singularities.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
A two-metre link at zero degrees ends at (2,0). A small angle of 0.1 radians predicts vertical motion about 0.2 m; the exact circle also moves slightly left.
2. Predict before changing a value
Compare small and large angle changes. Inspect exact versus linearized coordinates on equal axes. Set length zero and explain why the inverse direction becomes undefined. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Rotated point (m)
- 2.00000, 0.00000
- Jacobian (m/radian)
- 0.00000, 2.00000
- Linearized motion error (m)
- 0.00761
- Minimum-norm Δθ for dx=0.1,dy=0
- 0.00000
Vertical axis: World y metres. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below. Both coordinate axes use equal pixel scales so the displayed geometry matches the coordinate units.
| Coordinate | Current | Linearized next | Exact next |
|---|---|---|---|
| x | 2.00000 | 2.00000 | 1.99239 |
| y | 0.00000 | 0.17453 | 0.17431 |
Counterclockwise rotation in a fixed frame; angles convert degrees to radians before differentiation. A one-joint endpoint has only one local motion direction: a pseudoinverse projects a requested displacement and may leave residual error. At zero length the Jacobian and its Moore–Penrose pseudoinverse are zero: the minimum-norm least-squares angle update is zero, leaving the entire requested displacement as residual. An exact inverse motion is unavailable.
Compare small and large angle changes. Inspect exact versus linearized coordinates on equal axes. Set length zero and explain why the inverse direction becomes undefined.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
A two-metre link at zero degrees ends at (2,0). A small angle of 0.1 radians predicts vertical motion about 0.2 m; the exact circle also moves slightly left.
Connect trigonometry, rotations, Jacobians, least-squares motion and singularities. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
p=[l cos θ,l sin θ]; J=[−l sin θ,l cos θ]ᵀ; Δp≈J Δθ
l is link length in metres, θ is angle in radians, J measures metres per radian. The controls display degrees and convert them before calculation. ≈ marks a local approximation.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Compare small and large angle changes. Inspect exact versus linearized coordinates on equal axes. Set length zero and explain why the inverse direction becomes undefined. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C04, C11, C13. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
A pseudoinverse gives a minimum-norm least-squares update; it does not create an unavailable motion direction. For one nonzero column J, J⁺=Jᵀ/(JᵀJ). Near-singular systems amplify error; damping trades residual error against update size. Coordinate-frame and unit conventions must be checked before matrix operations.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Build an angle from a circle
On a unit circle, cos θ is the horizontal coordinate and sin θ the vertical coordinate. A full turn is 360 degrees or 2π radians, so radians=degrees×π/180. At zero degrees the unit point is (1,0); at 90 degrees it is (0,1). Multiplying by length scales the point to a link endpoint.
A two-dimensional rotation has rows [cos θ,−sin θ] and [sin θ,cos θ]. Apply it to local coordinates before adding the frame origin translation. Differentiating with degrees instead of radians requires an extra π/180 factor; ignoring that conversion changes the Jacobian’s units.
At zero link length, the Jacobian is zero. Its Moore–Penrose pseudoinverse is also zero, so the minimum-norm least-squares update is zero and leaves the requested displacement entirely unresolved. The nonzero-column shortcut Jᵀ/(JᵀJ) is undefined there; that does not make the general pseudoinverse undefined.
Optional implementation and research depth
A pseudoinverse gives a minimum-norm least-squares update; it does not create an unavailable motion direction. For one nonzero column J, J⁺=Jᵀ/(JᵀJ). Near-singular systems amplify error; damping trades residual error against update size. Coordinate-frame and unit conventions must be checked before matrix operations.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C04, C11, C13. Browse every support topic and course connection.