U10.1 A path and a trajectory worksheet

The same geometry with and without a clock attached, and why the clock changes what you can check.

Questions 6 marks in all

[1 mark]What is the practical reason for keeping the path and the time law separate?

AThe route can be re-timed for a low battery or a delicate payload without planning the geometry again
BA trajectory cannot be checked for collisions, so the path must be kept for that
CArc length only exists for a path, not for a trajectory
DThe planner needs the time law before it can search the map

[1 mark]What does this print? It is the lesson's route with a constant speed time law.

import math
WAYPOINTS = [(40, 40), (40, 140), (140, 140), (140, 60)]
CRUISE = 12.0
length = 0.0
for (ax, ay), (bx, by) in zip(WAYPOINTS, WAYPOINTS[1:]):
    length += math.hypot(bx - ax, by - ay)
print(round(length, 1), round(length / CRUISE, 1))

[1 mark]Where does the reference say the robot should be 10 s into the route?

import math
WAYPOINTS = [(40, 40), (40, 140), (140, 140), (140, 60)]
CRUISE = 12.0
legs, length = [], 0.0
for (ax, ay), (bx, by) in zip(WAYPOINTS, WAYPOINTS[1:]):
    d = math.hypot(bx - ax, by - ay)
    legs.append((ax, ay, bx, by, d))
    length += d

def at(t):
    s = max(0.0, min(length, CRUISE * t))
    for ax, ay, bx, by, d in legs:
        if s <= d:
            u = s / d
            return ax + (bx - ax) * u, ay + (by - ay) * u
        s -= d
    return WAYPOINTS[-1]

print(at(10.0))

[1 mark]A robot drives at waypoint 1 until it is close, then at waypoint 2, and so on. Why does it stop and start at every waypoint?

AIts demand is proportional to an error that collapses to nothing as it arrives at each waypoint
BThe planner inserted a pause at every waypoint
CThe drive cannot turn a corner while moving
DThe arc length is reset to zero at each waypoint

[1 mark]Which of these can be checked on a trajectory but not on the path alone?

Tick every answer that is true.

AWhether any part demands more speed than the robot has
BWhether it demands more acceleration than the drive can produce
CHow long the whole route will take
DWhether the route passes through an obstacle
EHow long the route is in centimetres

[1 mark]Why does the lesson call the constant speed time law s(t) = v t a lie?

AIt jumps from rest to cruise speed instantly, which would need infinite acceleration
BIt gives the wrong total duration for the route
CArc length cannot be proportional to time
DIt only works on straight legs

The task: put a clock on a path

Print length:, the total length of the path through the four waypoints, and duration:, how long it takes at 12 cm/s. Then sample the trajectory every 0.1 s and plot ref x and ref y, with a wait(0.1) between samples so the chart spreads them out in time. The robot does not move in this task.

from bugbot import *
import math
connect()

WAYPOINTS = [(40, 40), (40, 140), (140, 140), (140, 60)]
CRUISE = 12.0

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

Do it on the robot
www.bugbotlab.com/learn/u10-1-path-and-trajectory/
The simulator checks it and tells you when it passes. Nothing to install, no account.

U10.1 A path and a trajectory

What this lesson is about

Questions 6 marks in all

The task: put a clock on a path

Challenges