1 00:00:03,959 --> 00:00:09,040 How did MIT undergraduates design a robot to lift a small model police car and place 2 00:00:09,040 --> 00:00:15,309 it on top of a model of MIT's great dome? Calculus! In this video, we'll use calculus 3 00:00:15,309 --> 00:00:20,179 to describe the motion of rigid bodies and see how these concepts are used in the field 4 00:00:20,179 --> 00:00:22,839 of robotics. 5 00:00:22,839 --> 00:00:27,550 This video is part of the Derivatives and Integrals video series. Derivatives and integrals 6 00:00:27,550 --> 00:00:33,329 are used to analyze the properties of a system. Derivatives describe local properties of systems, 7 00:00:33,329 --> 00:00:36,429 and integrals quantify their cumulative properties. 8 00:00:36,429 --> 00:00:41,870 Hello. My name is Dan Frey. I am a professor in the Mechanical Engineering department at 9 00:00:41,870 --> 00:00:48,309 MIT, and today I'll be talking with you about the motion of rigid bodies--both translation 10 00:00:48,309 --> 00:00:50,190 and rotation. 11 00:00:50,190 --> 00:00:56,170 In order to understand this video you should be very comfortable with linear motion, including 12 00:00:56,170 --> 00:01:02,039 position, velocity, and acceleration. You will want to know how to turn measurements 13 00:01:02,039 --> 00:01:07,180 in polar coordinates into measurements in Cartesian coordinates. You should also know 14 00:01:07,180 --> 00:01:11,479 enough introductory calculus to apply the chain rule. 15 00:01:11,479 --> 00:01:16,890 After watching this video you should be able to explain what is meant by the phrase "rigid 16 00:01:16,890 --> 00:01:23,180 body." You should be able to describe restrictions on the motion of an object by using constraint 17 00:01:23,180 --> 00:01:28,619 equations. Finally, you should be able to use derivatives and integrals to connect different 18 00:01:28,619 --> 00:01:33,100 mathematical descriptions of rigid body motion. 19 00:01:33,100 --> 00:01:39,640 Our primary examples today will be robots. Let's look at how basic ideas of motion are 20 00:01:39,640 --> 00:01:45,350 used in the field of robotics. Here you can see some footage from a robotics competition 21 00:01:45,350 --> 00:01:52,350 at MIT. The competition is part of a Mechanical Engineering course, number 2.007. 22 00:01:53,000 --> 00:01:59,590 One typical task that robots perform is to grab something, pick it up, and move it. This 23 00:01:59,590 --> 00:02:05,210 robot uses a gripper, at the end of the arm, to pick up objects. 24 00:02:05,210 --> 00:02:11,030 One of the difficulties in programming a robot arm is that we typically have no direct measurement 25 00:02:11,030 --> 00:02:16,610 of where that gripper is. There's usually no convenient sort of "position meter" that 26 00:02:16,610 --> 00:02:21,910 could tell its location. Instead we might determine the location for the base of the 27 00:02:21,910 --> 00:02:28,410 arm, and we can measure the angles for different parts of the arm. We need to use the measurements 28 00:02:28,410 --> 00:02:35,410 that we can make in order to determine the location of the gripper. 29 00:02:49,350 --> 00:02:56,350 This is made easier by the concept of a "rigid body." Rigid bodies can translate and rotate, 30 00:02:56,820 --> 00:03:03,820 but they do not bend, stretch, or twist. In mathematical terms, the distance between any 31 00:03:03,900 --> 00:03:09,010 two points in the object does not change. 32 00:03:09,010 --> 00:03:14,730 Rigid bodies are idealizations -- to simplify our work, we imagine that we are working with 33 00:03:14,730 --> 00:03:21,260 objects that do not deform. This idealization works best when the object only experiences 34 00:03:21,260 --> 00:03:27,060 low amounts of force. Higher amounts of force can lead to objects deforming or breaking, 35 00:03:27,060 --> 00:03:31,620 depending on the object. The robot we saw earlier is a good example 36 00:03:31,620 --> 00:03:38,340 of a rigid body. You can see in this video that as our robot moves, its pieces do not 37 00:03:38,340 --> 00:03:45,220 bend or distort noticeably, so we can treat each piece as a rigid body. We could use the 38 00:03:45,220 --> 00:03:51,440 definition of a rigid body to help us determine the location of points on that robot. 39 00:03:51,440 --> 00:03:57,050 This robot, on the other hand, has a less sturdy frame. You can see the arm flex as 40 00:03:57,050 --> 00:04:04,050 the robot moves. We might not want to treat this arm as a rigid body. Let's try to solve 41 00:04:05,540 --> 00:04:10,900 a problem involving a rigid body. Here is a very simple robot arm. It has a "joint" 42 00:04:10,900 --> 00:04:17,139 on the left that can tilt up and down, and a piston that can extend its arm. In addition, 43 00:04:17,139 --> 00:04:23,590 this part of the piston can be considered a rigid body, so its length will be a constant. 44 00:04:23,590 --> 00:04:30,430 Here is a task for you: describe the acceleration of the gripper at the end of the arm. 45 00:04:30,430 --> 00:04:37,349 First, set the origin for your coordinate system. Then, write an expression for the 46 00:04:37,349 --> 00:04:42,490 x and y position of the gripper in terms of the quantities shown here. 47 00:04:42,490 --> 00:04:47,610 Once you have that position, use derivatives to find the velocity and the acceleration 48 00:04:47,610 --> 00:04:50,050 of the gripper. 49 00:04:50,050 --> 00:04:57,050 Pause the video here to carry out your calculations. Let's take a look at the answer. First, we 50 00:05:01,159 --> 00:05:06,360 need to choose an origin. Let's choose an arbitrary location as the origin of our coordinate 51 00:05:06,360 --> 00:05:12,740 system. Our robot's joint may be moving, so we will use a pair of functions x sub j of 52 00:05:12,740 --> 00:05:19,740 time and y sub j of time to describe its location. X sub j will be the horizontal distance from 53 00:05:20,080 --> 00:05:25,110 our origin to the joint, and y sub j will be the vertical distance. 54 00:05:25,110 --> 00:05:29,800 We can use derivatives of these functions to describe any relative movement that the 55 00:05:29,800 --> 00:05:35,620 joint has when compared with our coordinate system, such as velocity or acceleration. 56 00:05:35,620 --> 00:05:41,889 This slide shows just the X components of the answer, with the value x sub j indicating 57 00:05:41,889 --> 00:05:47,139 the location of the joint. You can see that the expressions can easily become complicated 58 00:05:47,139 --> 00:05:54,139 if both s and theta change at the same time. One reason that we want to know the acceleration 59 00:05:54,740 --> 00:06:00,080 is because some objects respond poorly to a high acceleration. Here you can see a different 60 00:06:00,080 --> 00:06:05,849 sort of robot arm lifting a car. Instead of a gripper, these two robots use a forklift 61 00:06:05,849 --> 00:06:10,969 design. Their arms must move very gently, especially as they slow down, or the car will 62 00:06:10,969 --> 00:06:17,969 fall off. The arms are capable of moving more quickly, but the robot's designers have programmed 63 00:06:18,090 --> 00:06:21,310 it to use a lower acceleration. 64 00:06:21,310 --> 00:06:27,080 This leads us to a discussion of constraints. This section will have a few examples, as 65 00:06:27,080 --> 00:06:33,949 well as several opportunities for you to practice. Be ready to pause the video. 66 00:06:33,949 --> 00:06:39,349 Constraints are any sort of restriction on a situation. When they can be expressed mathematically, 67 00:06:39,349 --> 00:06:45,490 we refer to Equations of Constraint. These describe the physical connection between two 68 00:06:45,490 --> 00:06:47,779 or more rigid bodies. 69 00:06:47,779 --> 00:06:52,259 Constraint equations are useful because they link one variable to another in a way that 70 00:06:52,259 --> 00:06:57,779 reduces the total number of variables in a problem. This helps to make otherwise impossible 71 00:06:57,779 --> 00:07:04,069 problems solvable. Constraint equations are used throughout physics and mechanical engineering. 72 00:07:04,069 --> 00:07:10,240 Some fields refer to a similar idea called "degree of freedom analysis." 73 00:07:10,240 --> 00:07:15,939 Here is a classic example of a situation with a constraint. The car on this roller coaster 74 00:07:15,939 --> 00:07:22,559 cannot leave the tracks. If the track is circular, we can use the equation for a circle to constrain 75 00:07:22,559 --> 00:07:27,610 our movement. We can use this to reduce the number of variables in our equations for the 76 00:07:27,610 --> 00:07:33,808 position of the roller coaster. Rather than an equation in x and y, we could have equations 77 00:07:33,808 --> 00:07:40,808 in just x, or just y. We could also use constraints that involve the distance along the track 78 00:07:41,009 --> 00:07:46,029 or another sensible measurement for the situation we are investigating. 79 00:07:46,029 --> 00:07:51,620 It's important to note that constraints mean giving up some freedom in our variables. In 80 00:07:51,620 --> 00:07:58,620 our example, we can only specify x or y, not both. Once we choose a value for x, there 81 00:07:59,689 --> 00:08:03,669 are only two y values that will work. 82 00:08:03,669 --> 00:08:09,279 Here's a situation where you can find the equation of constraint. A cart is being pulled 83 00:08:09,279 --> 00:08:14,889 across a flat surface, and the wheels turn without slipping—effectively, the wheel 84 00:08:14,889 --> 00:08:21,800 is constrained to move only by rolling and not in any other way: no lifting up, no sliding, 85 00:08:21,800 --> 00:08:28,309 no peeling out. Can you find an equation that connects x, the distance the cart has moved, 86 00:08:28,309 --> 00:08:32,010 to theta, the amount that the wheels have turned? 87 00:08:32,010 --> 00:08:39,010 Pause the video here to discuss this in class. Here is an arm that is fairly complex -- it 88 00:08:43,600 --> 00:08:49,820 has many joints. Pause the video and write down the variables and constants you would 89 00:08:49,820 --> 00:08:56,820 use for this robot. 90 00:08:57,730 --> 00:09:02,780 There are three separate angles that must be recorded, as well as the extension of the 91 00:09:02,780 --> 00:09:09,310 arm. There are also three pieces of constant length. 92 00:09:09,310 --> 00:09:16,120 Now we have an opportunity to describe a constraint in a complex situation. We could choose, for 93 00:09:16,120 --> 00:09:22,200 example, to constrain the motion of the arm to just the horizontal direction. Because 94 00:09:22,200 --> 00:09:27,540 there are many variables in this situation, there are many possible ways to satisfy the 95 00:09:27,540 --> 00:09:28,820 constraint. 96 00:09:28,820 --> 00:09:32,890 Write an expression for just the vertical position of the gripper in terms of the quantities 97 00:09:32,890 --> 00:09:39,550 shown. Once you have done that, answer this question: how might we move the gripper in 98 00:09:39,550 --> 00:09:42,310 just the horizontal direction? 99 00:09:42,310 --> 00:09:48,870 You should come up with at least two ways that we could do this, and describe them mathematically. 100 00:09:48,870 --> 00:09:53,470 Your teacher will then lead the class in a discussion of your answers. Pause the video 101 00:09:53,470 --> 00:10:00,470 here to do this. Here is a simulation of a "hydrabot" doing 102 00:10:02,060 --> 00:10:08,190 exactly what you just calculated: moving one end horizontally. You can see that it matches 103 00:10:08,190 --> 00:10:15,190 up quite well with our hypothetical robot. Examine its motion closely--is this one of 104 00:10:16,280 --> 00:10:21,410 the motions you described? Are there extra constraints present here? What freedom of 105 00:10:21,410 --> 00:10:27,030 motion did we give up by making our choice? 106 00:10:27,030 --> 00:10:28,630 Let's review. 107 00:10:28,630 --> 00:10:33,520 Today you used derivatives to find the velocity and acceleration of an object based on its 108 00:10:33,520 --> 00:10:34,200 position. 109 00:10:34,200 --> 00:10:39,300 You also learned the definition of a rigid body: that it does not bend or stretch when 110 00:10:39,300 --> 00:10:40,600 force is applied. 111 00:10:40,600 --> 00:10:45,170 Finally, you saw that constraint equations can reduce the total number of equations in 112 00:10:45,170 --> 00:10:51,640 a system, thus making problems easier to solve. I hope you enjoyed seeing some applications 113 00:10:51,640 --> 00:10:56,190 of basic motion concepts. Good luck in your further investigation of physics and engineering!