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Manipulability Calculator

Run the Manipulability Calculator MicroSim Fullscreen

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Description

Bend the elbow of a two-link arm and watch its manipulability, its condition number and the ellipse at its tip. Then calculate the manipulability in six problems with the formula w = l1 × l2 × |sin(bend)|. Used in Chapter 19.

The links are 0.116 m and 0.135 m long unless a problem says otherwise. The manipulability of a two-link arm is

  • w = l1 × l2 × |sin(bend)|

where the bend is the elbow angle measured from a straight arm. The numbers are small, so the answers are typed as w × 1000: a w of 0.01566 is typed as 15.7.

Lesson Plan

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

  1. Explore the bend. Start with the elbow bent 90 degrees. Straighten it slowly toward 0. What happens to w, to the condition number and to the ellipse?
  2. Explore the shoulder. Now move only the shoulder slider. The ellipse turns. Does w change?
  3. Six problems. Choose Six problems in the menu. Work each one out with a calculator, type the number and press Check. An answer must be within 0.1. You get one attempt at each, and 5 of 6 is mastery.
  4. Think about it. Halving the bend from 90 to 45 degrees does not halve w. Why not? Use the shape of the sine curve in your answer.

The picture is a schematic, drawn with the shared robot-arm drawing library. The ellipse shows how fast the tip can move in each direction, and it is drawn at 0.4 of its true size so that it fits beside the arm.

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.