WEBVTT 00:00.260 --> 00:06.750 Using rotation matrices to represent the orientation of an object with the Euler angles. 00:06.770 --> 00:09.170 We need to store nine variables. 00:09.200 --> 00:12.500 The nine elements of the rotation matrix. 00:12.500 --> 00:19.970 And every time we need to compose consecutive rotations, we need to multiply two three by three matrices. 00:19.970 --> 00:25.280 And so we need to perform 27 multiplications and 18 sums. 00:26.670 --> 00:34.020 The quaternion is based on the real and imaginary numbers, and in fact it is an extension of it in 00:34.020 --> 00:37.080 the imaginary numbers we call a. 00:37.110 --> 00:41.160 The real part of the number and B the imaginary one. 00:41.670 --> 00:47.160 The quaternion is an extension of this concept and it is composed by four variables. 00:47.160 --> 00:55.560 So four coefficients that here I called A, B, C, and D, where A is the scalar part of the number. 00:55.650 --> 00:59.070 B, C and D are the vector part. 00:59.460 --> 01:05.580 Although this tool is less intuitive than the Euler angles and the rotation matrices to represent the 01:05.580 --> 01:12.570 orientation of an object, since it is not possible to draw in four dimensions, it offers several advantages 01:12.570 --> 01:17.390 that make it more efficient and therefore preferable to Euler angles. 01:17.400 --> 01:21.750 When it's time to express the orientation of an object in computer science. 01:22.910 --> 01:29.600 The first advantage is clear just by looking at the quaternion equation, it is fully described just 01:29.600 --> 01:31.020 by four components. 01:31.040 --> 01:33.620 The four coefficients of the quaternion. 01:33.620 --> 01:36.230 So the variables A, B, C, and D. 01:37.200 --> 01:38.970 One with the Euler angles. 01:38.970 --> 01:44.700 We needed nine components with the quaternion, we only need to store four variables. 01:44.940 --> 01:50.820 Furthermore, the quaternion also offered other properties that make it more efficient when it's time 01:50.820 --> 01:53.370 to manipulate and compose rotations. 01:54.000 --> 02:01.200 The first property is that the quaternion is unitary, namely, the sum of the squares of its coefficients 02:01.200 --> 02:02.430 is equal to one. 02:03.060 --> 02:09.150 The second property is that it is possible to combine two rotations, each of them represented with 02:09.180 --> 02:12.840 a quaternion simply by multiplying the two quaternions. 02:13.510 --> 02:19.600 Also, when we represent the orientation with Euler angles, it was enough to multiply two rotation 02:19.600 --> 02:22.990 matrices to compose two consecutive rotations. 02:23.170 --> 02:28.540 However, in that case we needed to calculate 27 products and 18 sums. 02:28.660 --> 02:36.610 In this case, when we multiply quaternions, we just need to calculate 16 products and nine sums. 02:37.830 --> 02:45.060 Another property of the quaternion is that the calculation of its inverse is extremely simple and actually 02:45.060 --> 02:46.640 is not even a calculation. 02:46.650 --> 02:51.330 It just needs a change of the sign of the vector part of the quaternion. 02:52.220 --> 02:57.140 The inverse of a quaternion also represents the inverse rotation of an object. 02:57.140 --> 03:00.770 And so the rotation that can sell the previous one. 03:01.680 --> 03:08.760 One can do this also with the rotation matrices and calculate the transpose of the matrix to calculate 03:08.760 --> 03:10.320 the inverse rotation. 03:10.320 --> 03:17.190 But of course, transposing a matrix is a far more expensive calculation than simply changing a sign. 03:18.160 --> 03:22.480 Because of these properties and because of its efficient representation. 03:22.480 --> 03:25.840 The quaternion is commonly used in computer science. 03:26.350 --> 03:32.890 However, as humans it is very complex for us to think about the orientation of an object in terms of 03:32.890 --> 03:39.910 quaternion, and instead it's much easier and intuitive to think in terms of Euler angles and rotation 03:39.910 --> 03:40.720 matrices. 03:41.600 --> 03:47.960 In the following laboratory lessons, we are going to use some useful tool that will let us think about 03:47.960 --> 03:54.230 the orientation in terms of Euler angles, and then we will convert this information in quaternion to 03:54.230 --> 03:55.520 be used in Ros2.