1. If a differential drive mobile robot travels from the...

1. If a differential drive mobile robot travels from the location shown in Figure 1 and then goes back to the same location in 40 seconds. The trajectory resembles the shape of circle with $R_T$ of 2m. 1) Calculate the linear velocity $V$ and angular velocity $\omega$ of the robot as well as the initial position ($X_0$, $Y_0$) of the robot as well as the initial angle $\theta_0$ of the linear velocity. 2) If the distance $L$ between two wheels is 15cm and the diameter $d$ of each wheel is 5cm, what are angular velocities ($\omega_L$ and $\omega_R$) of the left and right wheels with the linear and angular velocities ($V$, $\omega$)? (Please write the unit for each of the solutions and round to 4 decimal places.) Provided: $V = \frac{V_R + V_L}{2}$ $\omega = \frac{V_R - V_L}{L}$ $\dot{x} = V_x = V \cos \theta$ $\dot{y} = V_y = V \sin \theta$ $\dot{\theta} = \omega$ $\omega_R = 2V_R/d$ $\omega_L = 2V_L/d$

Transcript

00:01 In this video, i'm going to be looking at the relationship between linear and angular qualities.

00:07 Or quantities, sorry.

00:09 So what we have is a wheel.

00:11 Okay.

00:15 And we're going to be looking at three separate points on the wheel.

00:18 Okay, they're all on the same radial line.

00:21 We have point a, point b, and point c.

00:35 Okay.

00:36 So this will be a distance r, c.

00:38 From the center of the wheel.

00:41 Point b is up.

00:43 Point b is at a distance r sub b from the center, and point a is at a distance r sub a.

00:54 Okay, and i have the relationship that r sub a is greater than r sub c is greater than r sub b.

01:06 And we're told that this wheel is roe and it has a period of rotation of t equals two seconds.

01:16 And we want to find the angle theta, i'll call it theta 1, between the initial starting position and the final starting position at t equals 1 second.

01:33 Okay, so what it's made, one half of a revolution.

01:37 So let's say that at time t equals 0, we're in this initial configuration rotating counterclockwise.

01:46 So at one second, we're going to be a half a revolution away.

01:51 So point a is going to be here.

01:55 Point b is going to be here -ish.

02:01 And point c is going to be here.

02:06 Okay.

02:07 So let's look at those angles.

02:09 Angle a is one half of a circle.

02:13 Okay.

02:13 So so that is theta a equals 180 degrees or pi radian.

02:30 Okay, angle b, b goes from this position to this position.

02:35 Okay, so that's also 180 degrees or pi radians.

02:41 And then point c goes from here to here.

02:46 So again, equal to 180 degrees or a pi radian.

02:52 You want to find the rate of change, d theta d t, any point on the wheel.

03:01 Okay, so that's just going to be 180 radiance, or sorry, degrees per second, or pi radian per second.

03:16 And this quantity is known as angular velocity or angular speed, and it's denoted by omega.

03:24 And this is a vector quantity, so it's going to have a direction.

03:28 And the way we find this direction is we use something called the right -hand roll.

03:33 Now it works is you take your right hand, you curl your fingers in the direction of rotation, and your thumb will point in the direction of the angular velocity vector.

03:43 So for this wheel example, the velocity vector will be, or the angular velocity vector will be pointing directly out of the page or out of the screen...

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