WEBVTT 00:00:04.290 --> 00:00:10.110 Here's a well-thrown disk. But have you ever thrown one badly, so that as it spins, there's 00:00:10.110 --> 00:00:17.110 a wobble? This wobble rotates with a different frequency than the disk itself spins. 00:00:17.660 --> 00:00:23.790 In this video, we're going to describe mathematically the motion of all of the points on this badly 00:00:23.790 --> 00:00:25.390 thrown disk. 00:00:25.390 --> 00:00:31.300 This video is part of the Linearity Video Series. Many complex systems are modeled or 00:00:31.300 --> 00:00:35.820 approximated linearly because of the mathematical advantages. 00:00:35.820 --> 00:00:42.780 Hi, my name is Dan Frey, and I am a professor of Engineering Systems and Mechanical Engineering 00:00:42.780 --> 00:00:49.780 at MIT. And I use rigid body kinematics myself when designing radio--controlled aircraft. 00:00:50.010 --> 00:00:56.600 Before watching this video, you should be familiar with eigenvalues and eigenvectors, 00:00:56.600 --> 00:01:03.600 the standard basis, e1, e2, e3 of R3, and orthogonal matrices. 00:01:04.509 --> 00:01:10.530 After watching this video, you will be able to: identify rotation matrices; decompose 00:01:10.530 --> 00:01:17.530 the motion of the badly thrown disk into translational and rotational components; and write the rotational 00:01:17.680 --> 00:01:24.680 motion of the disk as a product of rotation matrices. 00:01:27.220 --> 00:01:33.170 Our goal is to describe the motion of the disk. The disk is a rigid body; it doesn't 00:01:33.170 --> 00:01:38.670 stretch, bend, or deform in any way when it is thrown. 00:01:38.670 --> 00:01:44.479 In this video, we're not interested in why the disk moves the way it does—that is, 00:01:44.479 --> 00:01:49.229 we're not trying to describe torques and forces that govern the motion. We simply want to 00:01:49.229 --> 00:01:56.229 describe the motion mathematically. This is a job for rigid body kinematics. We're going 00:01:56.490 --> 00:02:03.490 to describe the motion by decomposing it into translational and rotational components. 00:02:03.630 --> 00:02:07.630 We'll start with the mathematics of rotation matrices. 00:02:07.630 --> 00:02:11.560 This will allow us to build up to a description of the wobbly disk. 00:02:11.560 --> 00:02:17.129 Finally, we'll complete the description of the wobbly disk by adding in the translational 00:02:17.129 --> 00:02:18.169 component. 00:02:18.169 --> 00:02:24.060 Let's start with some linear algebra. A rotation is a mapping that takes any vector 00:02:24.060 --> 00:02:31.060 in R3 to some other vector in R3 via rotation about some axis by some angle. 00:02:34.239 --> 00:02:39.930 Rotations don't change the length of a vector. So, if you scale a vector and then rotate 00:02:39.930 --> 00:02:46.790 it, you get the same thing as if you first rotate it, and then scale the vector. 00:02:46.790 --> 00:02:53.790 Also, if you take two vectors, sum them and then rotate the sum, this is equal to vector 00:02:56.040 --> 00:03:01.469 you would get if you first rotate both vectors and then add them. 00:03:01.469 --> 00:03:07.510 These two properties together mean that Rotations act linearly on vectors. 00:03:07.510 --> 00:03:13.529 And by definition, linear operations can be represented by matrices. 00:03:13.529 --> 00:03:17.779 But what does a rotation matrix look like? 00:03:17.779 --> 00:03:24.779 We can learn a lot about a matrix by examining its eigenvalues and eigenvectors. Recall that 00:03:25.129 --> 00:03:32.099 a vector v is an eigenvector of a matrix if it is sent to a scalar multiple of itself 00:03:32.099 --> 00:03:35.168 when acted upon by the matrix. 00:03:35.168 --> 00:03:39.400 That scalar is the eigenvalue. 00:03:39.400 --> 00:03:46.400 Consider a rotation of 60 degrees about the axis defined by the vector e1+e2. Pause the 00:03:46.989 --> 00:03:53.989 video here and determine one eigenvalue and eigenvector. By the definition of an eigenvector, 00:04:01.029 --> 00:04:08.029 the vector e1+e2, which points along the axis of rotation, is an eigenvector with eigenvalue 00:04:08.150 --> 00:04:15.019 one. This is because this vector is UNCHANGED by the rotation matrix. 00:04:15.019 --> 00:04:22.019 Suppose you have a rotation matrix such that e1 and e2 are both eigenvectors with eigenvalue 00:04:23.150 --> 00:04:25.060 1. 00:04:25.060 --> 00:04:32.060 What would this mean about the rotation? Pause the video and think about this. The entire 00:04:36.470 --> 00:04:43.280 xy-plane will be unchanged by this rotation. This is only possible if the matrix is the 00:04:43.280 --> 00:04:50.280 identity matrix! This is the null rotation... nothing happens! What are the eigenvalues 00:04:52.680 --> 00:04:59.680 and eigenvectors of a 180-degree rotation about the z-axis? 00:05:06.040 --> 00:05:12.560 This rotation matrix has one eigenvalue of 1, corresponding to the vector e3, which points 00:05:12.560 --> 00:05:15.310 along the axis of rotation. 00:05:15.310 --> 00:05:22.280 But it has more eigenvectors: any vector in the xy-plane is sent to its negative by the 00:05:22.280 --> 00:05:29.280 rotation, so any vector in the xy-plane is an eigenvector with eigenvalue -1. 00:05:32.740 --> 00:05:39.370 Now let's consider a rotation by some angle theta (that is not an integer multiple of 00:05:39.370 --> 00:05:43.220 pi) clockwise about the z-axis. 00:05:43.220 --> 00:05:49.490 Write a matrix that represents such a rotation. 00:05:49.490 --> 00:05:55.870 Compute the eigenvalues of this matrix, and use the definition of an eigenvector to explain 00:05:55.870 --> 00:06:02.870 why this makes sense. 00:06:05.060 --> 00:06:12.060 You should have found 1 real eigenvalue equal to 1, and two complex conjugate eigenvalues. 00:06:12.980 --> 00:06:19.610 The real eigenvalue corresponds to the eigenvector e3, which is sent to itself by the rotation, 00:06:19.610 --> 00:06:22.450 hence the eigenvalue of 1. 00:06:22.450 --> 00:06:29.230 The fact that the other two eigenvalues are complex means that no other vector is sent 00:06:29.230 --> 00:06:34.110 to a REAL scalar multiple of itself. 00:06:34.110 --> 00:06:39.000 This makes sense geometrically because NO other vector points in the same direction 00:06:39.000 --> 00:06:46.000 it started in after being rotated. Now, how do we describe any general rotation about 00:06:48.370 --> 00:06:50.540 an arbitrary axis? 00:06:50.540 --> 00:06:57.540 Well, a matrix is completely defined by how it acts on basis vectors. 00:06:58.370 --> 00:07:04.090 Since a rotation doesn't change the lengths of vectors or the angles BETWEEN two vectors, 00:07:04.090 --> 00:07:09.980 a rotated basis will also be a basis for R3! 00:07:09.980 --> 00:07:16.980 This tells us that any rotation matrix can be described as an orthonormal matrix. 00:07:17.830 --> 00:07:23.490 The columns are the vectors each standard basis vector is sent to. 00:07:23.490 --> 00:07:30.490 Is it true that ALL orthonormal matrices rotate vectors? Pause the video. Nope, here's an 00:07:37.960 --> 00:07:42.650 orthonormal matrix that's not a rotation; it's a reflection. 00:07:42.650 --> 00:07:49.650 The rule is that only an orthonormal matrix whose determinant is positive 1 is a rotation. 00:07:56.370 --> 00:08:02.810 But let's get back to thinking about rigid body KINEMATICS. Remember, we want to describe 00:08:02.810 --> 00:08:08.520 the motion of the disk. We've talked about rotation matrices, but we've left out a very 00:08:08.520 --> 00:08:15.520 important component: time! How will we describe time dependent rotation? 00:08:16.180 --> 00:08:23.180 That's right, time dependent matrices. Let's start by modeling a simple motion: the 00:08:25.010 --> 00:08:31.880 rotation of a disk as it spins clockwise about the positive z-axis. 00:08:31.880 --> 00:08:38.880 We know how to write a matrix that describes rotation by an angle theta about the z-axis. 00:08:39.698 --> 00:08:46.699 How would you make this rotation time dependent? Pause the video and discuss. 00:08:51.160 --> 00:08:57.709 The obvious choice here is to simply make theta a function of time! But how does it 00:08:57.709 --> 00:09:03.019 depend on time? To write an explicit function, we need to 00:09:03.019 --> 00:09:08.529 know the rate, omega, at which the disk is rotating. 00:09:08.529 --> 00:09:12.829 Assume the disk spins with constant angular velocity. 00:09:12.829 --> 00:09:18.009 We can easily calculate omega by counting the revolutions per second. 00:09:18.009 --> 00:09:24.889 And there's our matrix for a spinning—but not wobbling—disk. 00:09:24.889 --> 00:09:31.889 Now let's try a slightly more difficult example. Let's describe the motion of this wobbly, 00:09:31.930 --> 00:09:36.329 spinning disk as it rotates on this stick. 00:09:36.329 --> 00:09:43.149 The disk is itself rotating clockwise about its center of mass when viewed from the positive 00:09:43.149 --> 00:09:50.149 z-axis. As before, we can find the rotation rate, omega-D, of a marked point by counting 00:09:52.420 --> 00:09:59.420 the revolutions per second. Assume omega_D is constant. Now, observe the slight tilt 00:10:02.540 --> 00:10:05.980 of the disk off of horizontal. 00:10:05.980 --> 00:10:11.649 This tilt is created by a rotation about a tilt axis. 00:10:11.649 --> 00:10:18.649 The tilt axis is the vector in the xy-plane about which the disk is rotated by some small 00:10:18.939 --> 00:10:25.939 angle theta, creating the tilt. The wobble is created because the tilt axis is rotating 00:10:27.230 --> 00:10:31.850 clockwise about the positive z-axis. 00:10:31.850 --> 00:10:38.290 We can visualize this by observing that the normal vector to the disk rotates in a cone 00:10:38.290 --> 00:10:45.290 shape about the z-axis. By tracking the normal vector's revolutions per second, we can find 00:10:46.800 --> 00:10:53.800 the rotation rate of the wobble, omega-W, of the normal vector. This is also the rotation 00:10:54.089 --> 00:11:01.089 rate for tilt axis. We assume omega_W is constant. Notice that omega-D and omega-W are different 00:11:03.420 --> 00:11:08.149 rotation rates. For simplicity, we assume that the marked 00:11:08.149 --> 00:11:15.149 point begins along the x-axis; and the initial tilt axis aligns with the x-axis, with the 00:11:16.920 --> 00:11:19.490 tilt angle theta. 00:11:19.490 --> 00:11:25.449 Let's start by creating a sequence of rotations that rotates the marked point to the angle 00:11:25.449 --> 00:11:32.449 omega_D times t and the tilt axis to the angle omega_W times t for any time t. 00:11:36.220 --> 00:11:41.410 To describe this motion, we are going to decompose the behavior into a sequence of rotations 00:11:41.410 --> 00:11:48.410 about e1, e2, and e3, which have the benefit of being easy to describe mathematically. 00:11:49.369 --> 00:11:55.910 I want to start by eliminating the tilt of the disk, so we can imagine it spinning parallel 00:11:55.910 --> 00:11:58.199 to the ground. 00:11:58.199 --> 00:12:05.199 What is the matrix that undoes the tilt of theta degrees about the x-axis? Pause and 00:12:05.929 --> 00:12:12.929 write down a matrix. We rotate by an angle negative theta about the positive x-axis, 00:12:18.610 --> 00:12:21.170 which is represented by this matrix. 00:12:21.170 --> 00:12:28.170 Now, I rotate the marked point clockwise about the z-axis by the angle (omega_D minus omega_W) 00:12:33.009 --> 00:12:34.920 times t. 00:12:34.920 --> 00:12:41.009 This matrix describes the angle difference traveled by the marked point relative to the 00:12:41.009 --> 00:12:48.009 position of the tilt axis. Now, we need to make sure that we tilt the disk again so that 00:12:49.470 --> 00:12:52.079 we can describe the wobble. 00:12:52.079 --> 00:12:58.709 Since we assume the tilt axis begins along the x-axis, we rotate the disk back to the 00:12:58.709 --> 00:13:02.869 initial tilted position by theta degrees. 00:13:02.869 --> 00:13:09.869 This counterclockwise, time-independent rotation about the x-axis is represented by this matrix. 00:13:13.369 --> 00:13:19.069 Finally we must describe the wobble, created by the rotation of the tilt axis. The tilt 00:13:19.069 --> 00:13:26.069 axis is rotating clockwise about the positive z-axis with rotation rate omega-W, so at time 00:13:28.069 --> 00:13:35.069 t, it has rotated by omega_W t degrees, which is what this matrix does. 00:13:36.720 --> 00:13:41.920 How will we combine these matrices to describe the motion of the marked point? 00:13:41.920 --> 00:13:48.920 Pause and discuss. 00:13:49.660 --> 00:13:56.660 We multiply the matrices together. The order we apply each matrix matters. We must perform 00:13:57.989 --> 00:14:04.989 the rotations in the same order we decomposed the motion, because matrices do not multiply 00:14:05.379 --> 00:14:10.970 commutatively. 00:14:10.970 --> 00:14:17.809 Let's understand geometrically why this worked. The angle of the marker is changed in two 00:14:17.809 --> 00:14:24.809 steps of this process, first a rotation by angle omega-D minus omega-W times t, and then 00:14:28.519 --> 00:14:32.110 by an angle omega_W times t. 00:14:32.110 --> 00:14:39.110 In the end, it ends up exactly where it should, at omega-D times t. Only the final matrix 00:14:40.230 --> 00:14:47.230 affects the tilt axis, rotating it by the angle omega-W times t. 00:14:47.600 --> 00:14:54.149 Because the disk is a rigid object, by describing the position of the marked point and the tilt 00:14:54.149 --> 00:15:01.149 axis for all times with matrices, we've actually described the position of every point on the 00:15:01.649 --> 00:15:03.029 disk. 00:15:03.029 --> 00:15:09.499 We can find the location of any point at time t by applying this matrix operation to any 00:15:09.499 --> 00:15:16.499 vector on the initial disk. Now, let's go back to the badly thrown disk. 00:15:17.689 --> 00:15:21.519 We can apply the rotational transformation directly to our thrown disk. 00:15:21.519 --> 00:15:23.309 The only changes might be to the rotation rates and the initial position. You can think 00:15:23.309 --> 00:15:25.809 about how we might change the formula. We'll ignore that. So all that is left to consider 00:15:25.809 --> 00:15:29.739 is the translation. 00:15:29.739 --> 00:15:34.489 If you throw a disk and watch it from the side, we can ignore the rotations and focus 00:15:34.489 --> 00:15:40.429 on the translation of the center point of the disk. For the small time interval that 00:15:40.429 --> 00:15:46.329 we are interested in describing, the disk moves in a straight, horizontal path. So this 00:15:46.329 --> 00:15:51.709 vector equation approximates the translation. 00:15:51.709 --> 00:15:56.149 Because the disk is a rigid object, we get the full description of the motion by simply 00:15:56.149 --> 00:15:58.089 adding in the translation. 00:15:58.089 --> 00:16:03.509 To the rotation of the wobbly disk to obtain the following equation of motion for any point 00:16:03.509 --> 00:16:06.100 on the disk. 00:16:06.100 --> 00:16:10.489 If you thought this problem was cool, you're not the only one. Richard Feynman studied 00:16:10.489 --> 00:16:16.319 the kinematics AND the dynamics of the wobbly disk. He was able to show that the rotation 00:16:16.319 --> 00:16:22.769 rate of the special marked point, omega-D, was exactly twice the rotation rate, omega-W, 00:16:22.769 --> 00:16:29.769 of the tilt axis. This realization ultimately led to insights into the behavior of electrons. 00:16:34.059 --> 00:16:40.819 In this video, we saw that rotation matrices are orthogonal matrices with determinant equal 00:16:40.819 --> 00:16:43.059 to positive 1. 00:16:43.059 --> 00:16:49.910 The kinematics of rigid bodies involves breaking a problem into translation and rotation. 00:16:49.910 --> 00:16:55.709 The rotations may be decomposed into several time dependent rotation matrices that are 00:16:55.709 --> 00:16:58.160 multiplied together. 00:16:58.160 --> 00:17:04.030 The matrix product added to the translation describes the location at any time of all 00:17:04.030 --> 00:17:06.319 points on the rigid body. 00:17:06.319 --> 00:17:10.800 I hope you'll try to describe the motions of various rotating objects that you encounter. 00:17:10.800 --> 00:17:16.119 Have fun, and good luck!