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Polynomial Order Chooser

Run the Polynomial Order Chooser MicroSim Fullscreen

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Description

Compare a cubic, a quintic and a cubic spline: what each can control, and what the position and the acceleration of an example move look like. Then decide which kind each of eight trajectory requirements needs. Used in Chapter 19.

The rule is to count the conditions that the move must meet:

  • Cubic: four conditions, the position and the speed at both ends of one move
  • Quintic: six conditions, the position, the speed and the acceleration at both ends of one move
  • Cubic spline: several via points, joined so that the speed and the acceleration stay continuous

Lesson Plan

Grade level: Middle school and up (age 12+). Time: 10 to 15 minutes.

  1. Explore. Draw the cubic and look at its acceleration graph. Where does the acceleration jump? Now draw the quintic. What is different at the two ends?
  2. Draw the spline. It passes through four via points. Is there a jump in the acceleration at any via point in the middle?
  3. Eight requirements. Choose Eight requirements in the menu. Read each requirement, count its conditions, and press the kind it needs. Pressing a button commits your answer. You get one attempt at each, and 7 of 8 is mastery.
  4. Think about it. A higher degree is not always smoother. Why is a chain of cubics better than one high-degree polynomial through six via points?

The cubic and the quintic are drawn for a move from 0 to 1 in 2 seconds that starts and ends at rest. The spline passes through the chapter's via points 0.0, 0.5, 0.4 and 1.0 at 0, 1, 2 and 3 seconds, with zero speed at both ends. The graphs have no numbers on their axes because only their shapes matter here.

Specification

The learning objective, evidence of mastery and content for this MicroSim are specified in Chapter 19: Optional Advanced Chapter: The Mathematics of Arm Paths.