WEBVTT

00:00.290 --> 00:08.030
Rotation matrices are a powerful tool, and their applications extend beyond robotics to include video

00:08.030 --> 00:13.010
games, development, space exploration, animation, filmmaking and much more.

00:13.800 --> 00:21.240
In this course, we will use rotation matrices to perform, coordinates, change and bring the position

00:21.240 --> 00:25.830
and the orientation of a point from one reference frame to another.

00:26.430 --> 00:32.640
For example, let's consider a point P located at the end of our robot arm.

00:33.000 --> 00:42.000
It has coordinates x, p, y, P and z P in the mobile reference frame, which is the yellow frame that

00:42.000 --> 00:45.120
is rotated and fixed to the robot arm.

00:46.080 --> 00:52.680
Suppose that we are dealing with an obstacle avoidance algorithm for our robot that continuously check

00:52.680 --> 00:56.610
whether any part of the robot is in contact with an obstacle.

00:56.940 --> 01:03.120
We may need to transform the coordinates of the point p into the fixed reference frame, which is the

01:03.120 --> 01:04.020
world frame.

01:04.020 --> 01:10.080
So the one in black to check for collisions between the robot and any potential obstacles.

01:10.880 --> 01:17.630
From the previous lesson, we remember that we can transform any point from one rotated reference frame

01:17.630 --> 01:20.530
to another using a rotation matrix.

01:20.600 --> 01:29.450
This matrix is done by three components, so three elemental rotations around the Z, y and x axis that

01:29.450 --> 01:32.870
are also known as roll pitch and yaw angles.

01:33.260 --> 01:40.040
Therefore, if we know the coordinates of the point P in the mobile reference frame, so the yellow

01:40.040 --> 01:48.560
frame that is attached to the robot which are x, P, y, P and z, P, multiplying them by the rotation

01:48.560 --> 01:54.380
matrix will give us the coordinates of the point P in the fixed reference frame of the world.

01:55.150 --> 02:00.790
Let's use some numerical values to make this example more practical and intuitive.

02:01.030 --> 02:07.600
Let's assume that we know the coordinates of the point P in the mobile reference frame, which are one,

02:07.600 --> 02:09.070
four and two.

02:09.460 --> 02:16.120
We also assume that we know the orientation angles of the mobile reference frames relative to the fixed

02:16.120 --> 02:21.430
reference frame, which are ten degrees, 30 degrees and 20 degrees.

02:22.260 --> 02:27.030
By rewriting the matrix equation with these numerical values.

02:27.030 --> 02:35.100
The rotation matrix is composed of three elemental rotation matrices around the Z axis by 20 degrees.

02:35.130 --> 02:41.090
The Y axis by 30 degrees and the x axis by ten degrees.

02:41.100 --> 02:48.990
And to this we need to multiply the pose of P in the mobile reference frame so the vector one, four

02:48.990 --> 02:49.710
and two.

02:50.100 --> 02:57.000
The result of this matrix multiplication gives us the coordinates of point P in the fixed reference

02:57.000 --> 02:57.510
frame.

02:58.250 --> 03:05.150
In general, a similar calculation can be performed to compute any coordinate transformation between

03:05.150 --> 03:09.080
two reference frame with the same origin but different orientation.