Worksheet

U2.5 Inverse kinematics

Kinematics and frames · University · about 30 min

What this lesson is about

The useful direction: a velocity in the world, a heading, and the command that produces it.

• The problem
• In code
• Saturation

Questions 6 marks in all

1. [1 mark]The robot faces east, heading 90, and you want it to move due north at 12 cm/s. With V_MAX = 20 and V_LAT = 15, what does this print?

   import math
   V_MAX, V_LAT = 20.0, 15.0
   h = math.radians(90)
   wx, wy = 0.0, 12.0
   vx = wx * math.cos(h) - wy * math.sin(h)
   vy = wx * math.sin(h) + wy * math.cos(h)
   print("fwd:", round(100 * vy / V_MAX), "lat:", round(100 * vx / V_LAT))

2. [1 mark]The robot is at heading 180, facing -y. You want a world velocity of 12 cm/s due east (+x), with V_LAT = 15. What lat command does the inverse kinematics give?

3. [1 mark]In which order does the inverse kinematics work?

   1. ARotate the world velocity into the body frame, then divide each component by that axis's full scale
   2. BDivide the world velocity by each axis's full scale, then rotate into the body frame
   3. CRotate with body_to_world, then divide by the full scales
   4. DDivide wx by V_LAT and wy by V_MAX, with no rotation

4. [1 mark]The inverse kinematics asks for fwd = 150 and lat = 60. This scales the command down the way the lesson does. What does it print?

   fwd, lat, rot = 150.0, 60.0, 0.0
   worst = max(abs(fwd), abs(lat), abs(rot), 100.0)
   fwd, lat, rot = fwd * 100 / worst, lat * 100 / worst, rot * 100 / worst
   print(fwd, lat, rot)

5. [1 mark]The same request, fwd = 150 and lat = 60, is instead clipped axis by axis to 100 and 60. What does the robot do?

   1. ATravels in the wrong direction as well as too slowly
   2. BTravels in the right direction, just more slowly
   3. CDoes exactly what was asked, because the robot clips internally anyway
   4. DStops, because an out-of-range command is rejected

6. [1 mark]Which request can a differential drive, with only a left and a right wheel speed, not satisfy?

   1. AA velocity straight sideways in its own frame
   2. BDriving forward while turning
   3. CTurning on the spot
   4. DDriving straight backwards

The task: north while facing east

The robot faces east. Get it into the green zone to the north without turning: heading must still be within 15 degrees of 90 at the end. Use the inverse kinematics.

from bugbot import *
import math
connect()

V_MAX, V_LAT = 20.0, 15.0

Plan your program here, then type it in and press Run.

Do it on the robot
www.bugbotlab.com/learn/u2-5-inverse-kinematics/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

1. Add saturation handling to world_drive() and ask for 40 cm/s. Does the robot still go in the right direction?
2. Ask for a world velocity and a spin at the same time, and confirm the world path is unaffected by the spinning.
3. Write the inverse kinematics for a differential drive (left and right wheel speeds). Which requests become impossible?