1 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 2 00:00:10,110 --> 00:00:17,110 a wobble? This wobble rotates with a different frequency than the disk itself spins. 3 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 4 00:00:23,790 --> 00:00:25,390 thrown disk. 5 00:00:25,390 --> 00:00:31,300 This video is part of the Linearity Video Series. Many complex systems are modeled or 6 00:00:31,300 --> 00:00:35,820 approximated linearly because of the mathematical advantages. 7 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 8 00:00:42,780 --> 00:00:49,780 at MIT. And I use rigid body kinematics myself when designing radio--controlled aircraft. 9 00:00:50,010 --> 00:00:56,600 Before watching this video, you should be familiar with eigenvalues and eigenvectors, 10 00:00:56,600 --> 00:01:03,600 the standard basis, e1, e2, e3 of R3, and orthogonal matrices. 11 00:01:04,509 --> 00:01:10,530 After watching this video, you will be able to: identify rotation matrices; decompose 12 00:01:10,530 --> 00:01:17,530 the motion of the badly thrown disk into translational and rotational components; and write the rotational 13 00:01:17,680 --> 00:01:24,680 motion of the disk as a product of rotation matrices. 14 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 15 00:01:33,170 --> 00:01:38,670 stretch, bend, or deform in any way when it is thrown. 16 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, 17 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 18 00:01:49,229 --> 00:01:56,229 describe the motion mathematically. This is a job for rigid body kinematics. We're going 19 00:01:56,490 --> 00:02:03,490 to describe the motion by decomposing it into translational and rotational components. 20 00:02:03,630 --> 00:02:07,630 We'll start with the mathematics of rotation matrices. 21 00:02:07,630 --> 00:02:11,560 This will allow us to build up to a description of the wobbly disk. 22 00:02:11,560 --> 00:02:17,129 Finally, we'll complete the description of the wobbly disk by adding in the translational 23 00:02:17,129 --> 00:02:18,169 component. 24 00:02:18,169 --> 00:02:24,060 Let's start with some linear algebra. A rotation is a mapping that takes any vector 25 00:02:24,060 --> 00:02:31,060 in R3 to some other vector in R3 via rotation about some axis by some angle. 26 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 27 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. 28 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 29 00:02:56,040 --> 00:03:01,469 you would get if you first rotate both vectors and then add them. 30 00:03:01,469 --> 00:03:07,510 These two properties together mean that Rotations act linearly on vectors. 31 00:03:07,510 --> 00:03:13,529 And by definition, linear operations can be represented by matrices. 32 00:03:13,529 --> 00:03:17,779 But what does a rotation matrix look like? 33 00:03:17,779 --> 00:03:24,779 We can learn a lot about a matrix by examining its eigenvalues and eigenvectors. Recall that 34 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 35 00:03:32,099 --> 00:03:35,168 when acted upon by the matrix. 36 00:03:35,168 --> 00:03:39,400 That scalar is the eigenvalue. 37 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 38 00:03:46,989 --> 00:03:53,989 video here and determine one eigenvalue and eigenvector. By the definition of an eigenvector, 39 00:04:01,029 --> 00:04:08,029 the vector e1+e2, which points along the axis of rotation, is an eigenvector with eigenvalue 40 00:04:08,150 --> 00:04:15,019 one. This is because this vector is UNCHANGED by the rotation matrix. 41 00:04:15,019 --> 00:04:22,019 Suppose you have a rotation matrix such that e1 and e2 are both eigenvectors with eigenvalue 42 00:04:23,150 --> 00:04:25,060 1. 43 00:04:25,060 --> 00:04:32,060 What would this mean about the rotation? Pause the video and think about this. The entire 44 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 45 00:04:43,280 --> 00:04:50,280 identity matrix! This is the null rotation... nothing happens! What are the eigenvalues 46 00:04:52,680 --> 00:04:59,680 and eigenvectors of a 180-degree rotation about the z-axis? 47 00:05:06,040 --> 00:05:12,560 This rotation matrix has one eigenvalue of 1, corresponding to the vector e3, which points 48 00:05:12,560 --> 00:05:15,310 along the axis of rotation. 49 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 50 00:05:22,280 --> 00:05:29,280 rotation, so any vector in the xy-plane is an eigenvector with eigenvalue -1. 51 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 52 00:05:39,370 --> 00:05:43,220 pi) clockwise about the z-axis. 53 00:05:43,220 --> 00:05:49,490 Write a matrix that represents such a rotation. 54 00:05:49,490 --> 00:05:55,870 Compute the eigenvalues of this matrix, and use the definition of an eigenvector to explain 55 00:05:55,870 --> 00:06:02,870 why this makes sense. 56 00:06:05,060 --> 00:06:12,060 You should have found 1 real eigenvalue equal to 1, and two complex conjugate eigenvalues. 57 00:06:12,980 --> 00:06:19,610 The real eigenvalue corresponds to the eigenvector e3, which is sent to itself by the rotation, 58 00:06:19,610 --> 00:06:22,450 hence the eigenvalue of 1. 59 00:06:22,450 --> 00:06:29,230 The fact that the other two eigenvalues are complex means that no other vector is sent 60 00:06:29,230 --> 00:06:34,110 to a REAL scalar multiple of itself. 61 00:06:34,110 --> 00:06:39,000 This makes sense geometrically because NO other vector points in the same direction 62 00:06:39,000 --> 00:06:46,000 it started in after being rotated. Now, how do we describe any general rotation about 63 00:06:48,370 --> 00:06:50,540 an arbitrary axis? 64 00:06:50,540 --> 00:06:57,540 Well, a matrix is completely defined by how it acts on basis vectors. 65 00:06:58,370 --> 00:07:04,090 Since a rotation doesn't change the lengths of vectors or the angles BETWEEN two vectors, 66 00:07:04,090 --> 00:07:09,980 a rotated basis will also be a basis for R3! 67 00:07:09,980 --> 00:07:16,980 This tells us that any rotation matrix can be described as an orthonormal matrix. 68 00:07:17,830 --> 00:07:23,490 The columns are the vectors each standard basis vector is sent to. 69 00:07:23,490 --> 00:07:30,490 Is it true that ALL orthonormal matrices rotate vectors? Pause the video. Nope, here's an 70 00:07:37,960 --> 00:07:42,650 orthonormal matrix that's not a rotation; it's a reflection. 71 00:07:42,650 --> 00:07:49,650 The rule is that only an orthonormal matrix whose determinant is positive 1 is a rotation. 72 00:07:56,370 --> 00:08:02,810 But let's get back to thinking about rigid body KINEMATICS. Remember, we want to describe 73 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 74 00:08:08,520 --> 00:08:15,520 important component: time! How will we describe time dependent rotation? 75 00:08:16,180 --> 00:08:23,180 That's right, time dependent matrices. Let's start by modeling a simple motion: the 76 00:08:25,010 --> 00:08:31,880 rotation of a disk as it spins clockwise about the positive z-axis. 77 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. 78 00:08:39,698 --> 00:08:46,699 How would you make this rotation time dependent? Pause the video and discuss. 79 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 80 00:08:57,709 --> 00:09:03,019 depend on time? To write an explicit function, we need to 81 00:09:03,019 --> 00:09:08,529 know the rate, omega, at which the disk is rotating. 82 00:09:08,529 --> 00:09:12,829 Assume the disk spins with constant angular velocity. 83 00:09:12,829 --> 00:09:18,009 We can easily calculate omega by counting the revolutions per second. 84 00:09:18,009 --> 00:09:24,889 And there's our matrix for a spinning—but not wobbling—disk. 85 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, 86 00:09:31,930 --> 00:09:36,329 spinning disk as it rotates on this stick. 87 00:09:36,329 --> 00:09:43,149 The disk is itself rotating clockwise about its center of mass when viewed from the positive 88 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 89 00:09:52,420 --> 00:09:59,420 the revolutions per second. Assume omega_D is constant. Now, observe the slight tilt 90 00:10:02,540 --> 00:10:05,980 of the disk off of horizontal. 91 00:10:05,980 --> 00:10:11,649 This tilt is created by a rotation about a tilt axis. 92 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 93 00:10:18,939 --> 00:10:25,939 angle theta, creating the tilt. The wobble is created because the tilt axis is rotating 94 00:10:27,230 --> 00:10:31,850 clockwise about the positive z-axis. 95 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 96 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 97 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 98 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 99 00:11:03,420 --> 00:11:08,149 rotation rates. For simplicity, we assume that the marked 100 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 101 00:11:16,920 --> 00:11:19,490 tilt angle theta. 102 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 103 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. 104 00:11:36,220 --> 00:11:41,410 To describe this motion, we are going to decompose the behavior into a sequence of rotations 105 00:11:41,410 --> 00:11:48,410 about e1, e2, and e3, which have the benefit of being easy to describe mathematically. 106 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 107 00:11:55,910 --> 00:11:58,199 to the ground. 108 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 109 00:12:05,929 --> 00:12:12,929 write down a matrix. We rotate by an angle negative theta about the positive x-axis, 110 00:12:18,610 --> 00:12:21,170 which is represented by this matrix. 111 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) 112 00:12:33,009 --> 00:12:34,920 times t. 113 00:12:34,920 --> 00:12:41,009 This matrix describes the angle difference traveled by the marked point relative to the 114 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 115 00:12:49,470 --> 00:12:52,079 we can describe the wobble. 116 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 117 00:12:58,709 --> 00:13:02,869 initial tilted position by theta degrees. 118 00:13:02,869 --> 00:13:09,869 This counterclockwise, time-independent rotation about the x-axis is represented by this matrix. 119 00:13:13,369 --> 00:13:19,069 Finally we must describe the wobble, created by the rotation of the tilt axis. The tilt 120 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 121 00:13:28,069 --> 00:13:35,069 t, it has rotated by omega_W t degrees, which is what this matrix does. 122 00:13:36,720 --> 00:13:41,920 How will we combine these matrices to describe the motion of the marked point? 123 00:13:41,920 --> 00:13:48,920 Pause and discuss. 124 00:13:49,660 --> 00:13:56,660 We multiply the matrices together. The order we apply each matrix matters. We must perform 125 00:13:57,989 --> 00:14:04,989 the rotations in the same order we decomposed the motion, because matrices do not multiply 126 00:14:05,379 --> 00:14:10,970 commutatively. 127 00:14:10,970 --> 00:14:17,809 Let's understand geometrically why this worked. The angle of the marker is changed in two 128 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 129 00:14:28,519 --> 00:14:32,110 by an angle omega_W times t. 130 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 131 00:14:40,230 --> 00:14:47,230 affects the tilt axis, rotating it by the angle omega-W times t. 132 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 133 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 134 00:15:01,649 --> 00:15:03,029 disk. 135 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 136 00:15:09,499 --> 00:15:16,499 vector on the initial disk. Now, let's go back to the badly thrown disk. 137 00:15:17,689 --> 00:15:21,519 We can apply the rotational transformation directly to our thrown disk. 138 00:15:21,519 --> 00:15:23,309 The only changes might be to the rotation rates and the initial position. You can think 139 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 140 00:15:25,809 --> 00:15:29,739 is the translation. 141 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 142 00:15:34,489 --> 00:15:40,429 on the translation of the center point of the disk. For the small time interval that 143 00:15:40,429 --> 00:15:46,329 we are interested in describing, the disk moves in a straight, horizontal path. So this 144 00:15:46,329 --> 00:15:51,709 vector equation approximates the translation. 145 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 146 00:15:56,149 --> 00:15:58,089 adding in the translation. 147 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 148 00:16:03,509 --> 00:16:06,100 on the disk. 149 00:16:06,100 --> 00:16:10,489 If you thought this problem was cool, you're not the only one. Richard Feynman studied 150 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 151 00:16:16,319 --> 00:16:22,769 rate of the special marked point, omega-D, was exactly twice the rotation rate, omega-W, 152 00:16:22,769 --> 00:16:29,769 of the tilt axis. This realization ultimately led to insights into the behavior of electrons. 153 00:16:34,059 --> 00:16:40,819 In this video, we saw that rotation matrices are orthogonal matrices with determinant equal 154 00:16:40,819 --> 00:16:43,059 to positive 1. 155 00:16:43,059 --> 00:16:49,910 The kinematics of rigid bodies involves breaking a problem into translation and rotation. 156 00:16:49,910 --> 00:16:55,709 The rotations may be decomposed into several time dependent rotation matrices that are 157 00:16:55,709 --> 00:16:58,160 multiplied together. 158 00:16:58,160 --> 00:17:04,030 The matrix product added to the translation describes the location at any time of all 159 00:17:04,030 --> 00:17:06,319 points on the rigid body. 160 00:17:06,319 --> 00:17:10,800 I hope you'll try to describe the motions of various rotating objects that you encounter. 161 00:17:10,800 --> 00:17:16,119 Have fun, and good luck!