WEBVTT 00:00:03.959 --> 00:00:09.040 How did MIT undergraduates design a robot to lift a small model police car and place 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 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 00:00:20.179 --> 00:00:22.839 of robotics. 00:00:22.839 --> 00:00:27.550 This video is part of the Derivatives and Integrals video series. Derivatives and integrals 00:00:27.550 --> 00:00:33.329 are used to analyze the properties of a system. Derivatives describe local properties of systems, 00:00:33.329 --> 00:00:36.429 and integrals quantify their cumulative properties. 00:00:36.429 --> 00:00:41.870 Hello. My name is Dan Frey. I am a professor in the Mechanical Engineering department at 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 00:00:48.309 --> 00:00:50.190 and rotation. 00:00:50.190 --> 00:00:56.170 In order to understand this video you should be very comfortable with linear motion, including 00:00:56.170 --> 00:01:02.039 position, velocity, and acceleration. You will want to know how to turn measurements 00:01:02.039 --> 00:01:07.180 in polar coordinates into measurements in Cartesian coordinates. You should also know 00:01:07.180 --> 00:01:11.479 enough introductory calculus to apply the chain rule. 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 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 00:01:23.180 --> 00:01:28.619 equations. Finally, you should be able to use derivatives and integrals to connect different 00:01:28.619 --> 00:01:33.100 mathematical descriptions of rigid body motion. 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 00:01:39.640 --> 00:01:45.350 used in the field of robotics. Here you can see some footage from a robotics competition 00:01:45.350 --> 00:01:52.350 at MIT. The competition is part of a Mechanical Engineering course, number 2.007. 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 00:01:59.590 --> 00:02:05.210 robot uses a gripper, at the end of the arm, to pick up objects. 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 00:02:11.030 --> 00:02:16.610 of where that gripper is. There's usually no convenient sort of "position meter" that 00:02:16.610 --> 00:02:21.910 could tell its location. Instead we might determine the location for the base of the 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 00:02:28.410 --> 00:02:35.410 that we can make in order to determine the location of the gripper. 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, 00:02:56.820 --> 00:03:03.820 but they do not bend, stretch, or twist. In mathematical terms, the distance between any 00:03:03.900 --> 00:03:09.010 two points in the object does not change. 00:03:09.010 --> 00:03:14.730 Rigid bodies are idealizations -- to simplify our work, we imagine that we are working with 00:03:14.730 --> 00:03:21.260 objects that do not deform. This idealization works best when the object only experiences 00:03:21.260 --> 00:03:27.060 low amounts of force. Higher amounts of force can lead to objects deforming or breaking, 00:03:27.060 --> 00:03:31.620 depending on the object. The robot we saw earlier is a good example 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 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 00:03:45.220 --> 00:03:51.440 definition of a rigid body to help us determine the location of points on that robot. 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 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 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" 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, 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. 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. 00:04:30.430 --> 00:04:37.349 First, set the origin for your coordinate system. Then, write an expression for the 00:04:37.349 --> 00:04:42.490 x and y position of the gripper in terms of the quantities shown here. 00:04:42.490 --> 00:04:47.610 Once you have that position, use derivatives to find the velocity and the acceleration 00:04:47.610 --> 00:04:50.050 of the gripper. 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 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 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 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 00:05:20.080 --> 00:05:25.110 our origin to the joint, and y sub j will be the vertical distance. 00:05:25.110 --> 00:05:29.800 We can use derivatives of these functions to describe any relative movement that the 00:05:29.800 --> 00:05:35.620 joint has when compared with our coordinate system, such as velocity or acceleration. 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 00:05:41.889 --> 00:05:47.139 the location of the joint. You can see that the expressions can easily become complicated 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 00:05:54.740 --> 00:06:00.080 is because some objects respond poorly to a high acceleration. Here you can see a different 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 00:06:05.849 --> 00:06:10.969 design. Their arms must move very gently, especially as they slow down, or the car will 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 00:06:18.090 --> 00:06:21.310 it to use a lower acceleration. 00:06:21.310 --> 00:06:27.080 This leads us to a discussion of constraints. This section will have a few examples, as 00:06:27.080 --> 00:06:33.949 well as several opportunities for you to practice. Be ready to pause the video. 00:06:33.949 --> 00:06:39.349 Constraints are any sort of restriction on a situation. When they can be expressed mathematically, 00:06:39.349 --> 00:06:45.490 we refer to Equations of Constraint. These describe the physical connection between two 00:06:45.490 --> 00:06:47.779 or more rigid bodies. 00:06:47.779 --> 00:06:52.259 Constraint equations are useful because they link one variable to another in a way that 00:06:52.259 --> 00:06:57.779 reduces the total number of variables in a problem. This helps to make otherwise impossible 00:06:57.779 --> 00:07:04.069 problems solvable. Constraint equations are used throughout physics and mechanical engineering. 00:07:04.069 --> 00:07:10.240 Some fields refer to a similar idea called "degree of freedom analysis." 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 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 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 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 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 00:07:41.009 --> 00:07:46.029 or another sensible measurement for the situation we are investigating. 00:07:46.029 --> 00:07:51.620 It's important to note that constraints mean giving up some freedom in our variables. In 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 00:07:59.689 --> 00:08:03.669 are only two y values that will work. 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 00:08:09.279 --> 00:08:14.889 across a flat surface, and the wheels turn without slipping—effectively, the wheel 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, 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, 00:08:28.309 --> 00:08:32.010 to theta, the amount that the wheels have turned? 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 00:08:43.600 --> 00:08:49.820 has many joints. Pause the video and write down the variables and constants you would 00:08:49.820 --> 00:08:56.820 use for this robot. 00:08:57.730 --> 00:09:02.780 There are three separate angles that must be recorded, as well as the extension of the 00:09:02.780 --> 00:09:09.310 arm. There are also three pieces of constant length. 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 00:09:16.120 --> 00:09:22.200 example, to constrain the motion of the arm to just the horizontal direction. Because 00:09:22.200 --> 00:09:27.540 there are many variables in this situation, there are many possible ways to satisfy the 00:09:27.540 --> 00:09:28.820 constraint. 00:09:28.820 --> 00:09:32.890 Write an expression for just the vertical position of the gripper in terms of the quantities 00:09:32.890 --> 00:09:39.550 shown. Once you have done that, answer this question: how might we move the gripper in 00:09:39.550 --> 00:09:42.310 just the horizontal direction? 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. 00:09:48.870 --> 00:09:53.470 Your teacher will then lead the class in a discussion of your answers. Pause the video 00:09:53.470 --> 00:10:00.470 here to do this. Here is a simulation of a "hydrabot" doing 00:10:02.060 --> 00:10:08.190 exactly what you just calculated: moving one end horizontally. You can see that it matches 00:10:08.190 --> 00:10:15.190 up quite well with our hypothetical robot. Examine its motion closely--is this one of 00:10:16.280 --> 00:10:21.410 the motions you described? Are there extra constraints present here? What freedom of 00:10:21.410 --> 00:10:27.030 motion did we give up by making our choice? 00:10:27.030 --> 00:10:28.630 Let's review. 00:10:28.630 --> 00:10:33.520 Today you used derivatives to find the velocity and acceleration of an object based on its 00:10:33.520 --> 00:10:34.200 position. 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 00:10:39.300 --> 00:10:40.600 force is applied. 00:10:40.600 --> 00:10:45.170 Finally, you saw that constraint equations can reduce the total number of equations in 00:10:45.170 --> 00:10:51.640 a system, thus making problems easier to solve. I hope you enjoyed seeing some applications 00:10:51.640 --> 00:10:56.190 of basic motion concepts. Good luck in your further investigation of physics and engineering!