1 00:00:00,210 --> 00:00:00,640 OK. 2 00:00:00,960 --> 00:00:07,260 In previous video, we have gained some insight about manipulator dynamic modelling before jumping directly 3 00:00:07,260 --> 00:00:08,070 to the concept. 4 00:00:08,100 --> 00:00:14,310 I want to give you some insight about the method and the related terminology so you can easily digest 5 00:00:14,310 --> 00:00:15,090 next lessons. 6 00:00:15,510 --> 00:00:19,140 Warning I will not teach everybody the scenes, which is not possible. 7 00:00:19,620 --> 00:00:25,200 Please, if you have any lack of basic terminologies here, make sure you have served and learned them. 8 00:00:25,650 --> 00:00:29,030 So let's get started before anything else. 9 00:00:29,040 --> 00:00:34,320 I would like to start with the notion of generalized coldness, which will be very useful for us. 10 00:00:34,320 --> 00:00:35,760 During dynamic modelling. 11 00:00:36,300 --> 00:00:42,970 A set of position and orientation variables that describe the configuration of a system is defined as 12 00:00:42,980 --> 00:00:44,910 a set of generalized coordinates. 13 00:00:45,270 --> 00:00:50,680 Generalized coordinates that we choose have to be able to describe the system completely. 14 00:00:51,210 --> 00:00:53,490 The set of genera last quarters is not unique. 15 00:00:54,930 --> 00:01:01,200 You can sometimes use rectangular coordinates x y to describe the system or cylindrical coordinates, 16 00:01:01,200 --> 00:01:02,870 such as rotation angle of Titan. 17 00:01:03,450 --> 00:01:04,530 It doesn't matter. 18 00:01:04,770 --> 00:01:11,010 The key thing is that you have to choose such coordinates that makes your equation of motions simpler. 19 00:01:11,610 --> 00:01:16,950 Additionally, the number of generalized coordinates has to be larger than the number of degree of freedom 20 00:01:16,950 --> 00:01:22,980 of the system, because, as I have said before, systems should be complete described by generalized 21 00:01:22,980 --> 00:01:23,640 coordinates. 22 00:01:24,150 --> 00:01:28,830 However, in general, the number of generalized coordinates is the same with the degree of freedom 23 00:01:28,830 --> 00:01:29,610 of the system. 24 00:01:30,150 --> 00:01:35,790 Let's take this situation as an example, which is nothing but a simple one degree of freedom pendulum. 25 00:01:36,240 --> 00:01:43,350 The system has three dimensions as you can see motion along x axis, motion along y axis and rotation 26 00:01:43,350 --> 00:01:44,520 about that access. 27 00:01:44,940 --> 00:01:51,600 However, we have two constraints that are indicated in equation, and one point mass can only move 28 00:01:51,600 --> 00:01:55,410 in circular fashion, which is indicated to be the first equation. 29 00:01:55,650 --> 00:01:58,230 And there is no motion along zip axes. 30 00:01:58,650 --> 00:02:05,060 So three dimensions for which two dimensions are constrained. 31 00:02:05,070 --> 00:02:11,250 We have one degree of freedom, so the number of generalized coordinates has to be either one or greater 32 00:02:11,250 --> 00:02:11,790 than one. 33 00:02:12,240 --> 00:02:15,570 In this case, we have two sets of generalized coordinates. 34 00:02:15,870 --> 00:02:23,160 The first is X and Y, but these are not independent corners because X and Y is related by Eq. one. 35 00:02:23,640 --> 00:02:28,740 So it is enough to know one of them and the other will be known automatically. 36 00:02:29,190 --> 00:02:31,860 The second set is rotation angle of TITA. 37 00:02:32,190 --> 00:02:36,200 You can see what about X anteater or Y anteater? 38 00:02:36,480 --> 00:02:42,420 But if you think carefully, you will see that X, Y and Z are related with each other, especially 39 00:02:42,420 --> 00:02:43,520 if we know Titov. 40 00:02:43,530 --> 00:02:49,050 We can know both X and Y because they are related as given in these equations. 41 00:02:49,620 --> 00:02:56,340 So one coordinate is enough to complete the described the position of rigid body, and this can be chosen 42 00:02:56,340 --> 00:02:59,640 as to which makes equations of motion easier. 43 00:03:01,560 --> 00:03:06,780 Let's not talk about the kinetic energy of a system which is given Equation 1.1. 44 00:03:07,320 --> 00:03:13,410 As you can see, it consists of two parts, namely kinetic energy due to translational motion and the 45 00:03:13,410 --> 00:03:15,210 one due to the rotational motion. 46 00:03:15,750 --> 00:03:21,480 Here is the interesting thing kinetic energy due to the translational motion, namely, the first part 47 00:03:21,480 --> 00:03:24,750 of the equation doesn't change with respect the reference frame. 48 00:03:25,200 --> 00:03:27,140 Let's analyze it here. 49 00:03:27,150 --> 00:03:29,940 The velocity is given with respect to the Baz's frame. 50 00:03:30,330 --> 00:03:37,530 Let's say we want to convert it to another reference frame, so apply a rotation matrix of R from the 51 00:03:37,530 --> 00:03:39,740 transportation transposition rule. 52 00:03:39,750 --> 00:03:42,750 Excuse me, we can write equation in this way. 53 00:03:43,020 --> 00:03:49,290 This equation is valid for all rotation mattresses of R as the transpose of rotation matrix is equal 54 00:03:49,290 --> 00:03:50,430 to its inverse. 55 00:03:50,760 --> 00:03:57,030 So the given product of rotation mattresses will be identity and we will get the same equation as before. 56 00:03:57,240 --> 00:04:00,930 Interestingly, this property is also valid for the second term. 57 00:04:01,500 --> 00:04:03,920 Let's analyze the second part of the equation. 58 00:04:03,930 --> 00:04:11,340 Also, let's say in the equation, both inertia matrix both ie and angular velocity of omega are given 59 00:04:11,340 --> 00:04:12,900 with respect to the base frame. 60 00:04:13,560 --> 00:04:20,250 We say that it is not important with respect to which reference frame we write this equation. 61 00:04:20,430 --> 00:04:21,480 It will not change. 62 00:04:21,870 --> 00:04:24,600 So then we can write this here. 63 00:04:24,600 --> 00:04:31,260 Angular velocity and inertia matrix are written with respect to F Dash reference frame. 64 00:04:31,770 --> 00:04:38,460 We know that the relation between the old inertia matrix and new inertia matrix is given by the rotation 65 00:04:38,460 --> 00:04:40,260 mattresses between the two friends. 66 00:04:40,770 --> 00:04:47,550 If you cannot remember these, please revise energy matrix concept so we can avoid the equation in this 67 00:04:47,550 --> 00:04:47,820 way. 68 00:04:48,240 --> 00:04:53,820 Here are is the rotation matrix between base frame and F Typekit reference frame. 69 00:04:54,750 --> 00:04:59,880 If we manipulate the equation by using the transposition rule, we will get. 70 00:04:59,990 --> 00:05:07,130 This new equation where our transports are is identity, and we can exactly we we get the exactly the 71 00:05:07,130 --> 00:05:08,750 same equation at the beginning. 72 00:05:09,320 --> 00:05:14,060 So we proved that the second part also is invariant with respect to the reference frame. 73 00:05:14,540 --> 00:05:18,440 So kinetic energy is invariant with respect to the reference frame we choose. 74 00:05:18,830 --> 00:05:20,270 So what is the benefit of this? 75 00:05:20,690 --> 00:05:26,600 This means that we can choose such a reference frame that makes our measurements and calculations easier. 76 00:05:27,140 --> 00:05:32,660 For example, we chose by the frame, which is placed in the center of mass of the region, the body 77 00:05:32,870 --> 00:05:38,210 and moves with it for inertia matrix right by the frame, but not based frame. 78 00:05:38,510 --> 00:05:44,480 Because if we adjust based frame as a reference frame for the inertia matrix, it will change as the 79 00:05:44,480 --> 00:05:45,500 body moves. 80 00:05:45,500 --> 00:05:51,350 And we don't want that because it will just make calculations complicated at every instant. 81 00:05:51,350 --> 00:05:53,540 We have to calculate new inertia matrix. 82 00:05:53,870 --> 00:06:00,140 However, if we chose by the frame, which moves with body as a reference frame, then inertia matrix 83 00:06:00,140 --> 00:06:02,730 will not change and calculations will be simpler. 84 00:06:03,110 --> 00:06:09,800 But be careful here if you change your reference frame for inertia matrix, we have to do the same for 85 00:06:09,800 --> 00:06:16,970 angular velocity as angular velocity is measured with respect to the base frame based reference frame. 86 00:06:17,150 --> 00:06:23,000 We have to multiply it with the rotation matrix of R to convert it to the body reference frame. 87 00:06:23,270 --> 00:06:26,510 So all in all, the kinetic energy form look becomes like that. 88 00:06:26,840 --> 00:06:34,620 Here I indicates rigid body I, in our case of manipulators, link I let's not these down. 89 00:06:34,670 --> 00:06:40,190 Also to clarify everything linear velocities and angular velocities are measured with respect to the 90 00:06:40,190 --> 00:06:46,790 base frame inertia matrix is defined with respect to the body reference frame R is the rotation matrix 91 00:06:46,790 --> 00:06:48,920 between base frame and body frame. 92 00:06:49,460 --> 00:06:54,140 Total kinetic energy is given by the sum of kinetic energies of each ring. 93 00:06:54,890 --> 00:07:03,060 OK, now let's see potential energy before we have said that before we have said that we will accept 94 00:07:03,060 --> 00:07:06,110 leaks as very rigid bodies, so no elasticity. 95 00:07:06,560 --> 00:07:11,780 Due to that assumption, the potential energy will be only due to the gravitational field. 96 00:07:12,230 --> 00:07:13,760 Here's the scenario We are rigid. 97 00:07:13,760 --> 00:07:20,960 The body is under the ground force of gravity and this is the potential energy of lint I am. 98 00:07:20,960 --> 00:07:24,130 I is nothing but the mass of Link I PC. 99 00:07:24,170 --> 00:07:31,340 I is position vector of the center of mass of Brigida by the eye given by coordinates of X Y and Z G 100 00:07:31,430 --> 00:07:32,720 is the gravity vector. 101 00:07:32,990 --> 00:07:41,630 Be careful it is vector, so only z component of it as value other than zero, because in X and Y directions 102 00:07:41,630 --> 00:07:42,890 there is no gravity force. 103 00:07:43,340 --> 00:07:48,980 Total potential energy is given by the sum of potential energies of each of the links. 104 00:07:50,870 --> 00:07:57,110 After we get the formulas for kinetic and potential energy are ready for getting dynamic model of the 105 00:07:57,110 --> 00:08:03,830 robot manipulator using the Lagrange formula, the Lagrange formula is given by this equation. 106 00:08:04,160 --> 00:08:11,120 As you can see, it contains partial derivatives and also derivatives with respect to time here. 107 00:08:11,480 --> 00:08:17,330 In the case, the number of rigid bodies or links in this case, you can ask What is it? 108 00:08:17,380 --> 00:08:21,240 Here it is the popular Lagrange variable. 109 00:08:21,260 --> 00:08:24,770 Nothing but the difference between kinetic and potential energies. 110 00:08:25,340 --> 00:08:30,530 If L is written in this way, then we can write electrons for us in Eq.. 111 00:08:30,530 --> 00:08:31,490 One point five. 112 00:08:32,030 --> 00:08:38,510 Be careful In the first term, partial derivative of potential energy with respect to joint velocity 113 00:08:38,510 --> 00:08:44,630 is zero because potential energy does not depend on joint velocities, but only joint positions. 114 00:08:45,680 --> 00:08:51,110 C k in this formula is nothing but that it calls conservative forces. 115 00:08:51,530 --> 00:08:57,200 These forces include friction forces, damping forces, contact forces between the robot gripper and 116 00:08:57,200 --> 00:09:00,080 the environment, and joint actuator talks. 117 00:09:00,710 --> 00:09:07,250 Let's take this example in order to clarify the notion of dynamic modeling where it will grant formula. 118 00:09:07,700 --> 00:09:15,020 As you can see, a rigid body of mass and moves along z axis under the gravity force and pulled by external 119 00:09:15,020 --> 00:09:21,830 force of F friction coefficient is given by D in order to model this dynamically. 120 00:09:21,860 --> 00:09:26,180 We need first to calculate kinetic energy, which is given by this formula. 121 00:09:26,480 --> 00:09:29,270 This is very simple and comes from basic physics. 122 00:09:29,840 --> 00:09:34,160 Potential energy is also very simple, which is due to only the gravity. 123 00:09:34,730 --> 00:09:40,550 After we calculate the potential energy and kinetic energy, we can write Lagrange variable. 124 00:09:40,760 --> 00:09:43,280 Let's remember the formula again. 125 00:09:43,910 --> 00:09:50,660 If we calculate the first part we will get, this is it will not have any contribution because it doesn't 126 00:09:50,660 --> 00:09:53,960 contain velocity term for the second part. 127 00:09:53,990 --> 00:09:55,100 We will get this. 128 00:09:56,240 --> 00:09:59,750 In this case, kinetic energy will not have any contribution because. 129 00:09:59,790 --> 00:10:01,680 It is on the velocity dependent. 130 00:10:02,430 --> 00:10:10,170 If we consider non conservative forces of ebb and friction force, which is given that thought, we 131 00:10:10,170 --> 00:10:13,350 can obtain dynamic model of the given system. 132 00:10:13,350 --> 00:10:20,280 As this, as you can see, this is the correct model and you can verify that by use just using Newton's 133 00:10:20,280 --> 00:10:21,090 formulation. 134 00:10:21,870 --> 00:10:28,530 So you can see that by the help of a little Legrand's formulation, we can get dynamic model of the 135 00:10:28,530 --> 00:10:31,650 system from potential and kinetic energies.