1 00:00:00,090 --> 00:00:06,780 I hope that you have enough information about a positive trajectory until now, we analyze in detail 2 00:00:06,780 --> 00:00:10,260 how we can calculate the trajectory for single joint. 3 00:00:10,950 --> 00:00:17,460 But we know that robot arms don't consist of one joint, but instead of instead more than one joint. 4 00:00:18,000 --> 00:00:19,680 So what will happen in this case? 5 00:00:20,040 --> 00:00:21,930 Do we have to find new equations? 6 00:00:22,140 --> 00:00:22,890 Surely not. 7 00:00:23,370 --> 00:00:27,540 What we want is only what we want. 8 00:00:27,540 --> 00:00:34,440 Only is all the joints to start at the same time and finish at the same time because otherwise it would 9 00:00:34,440 --> 00:00:35,580 be very odd. 10 00:00:36,180 --> 00:00:40,290 Just think it like that you want to take a glass of water in front of you. 11 00:00:40,590 --> 00:00:45,810 So you first move your shoulder after you have finished, you move your elbow. 12 00:00:46,080 --> 00:00:51,510 After you have finished, you move your wrist, then your fingers and so on. 13 00:00:51,840 --> 00:00:53,880 You don't do it like that in real life. 14 00:00:54,180 --> 00:00:58,560 Instead, you move all of them together in a synchronized way. 15 00:00:59,700 --> 00:01:00,040 Yeah. 16 00:01:00,060 --> 00:01:01,860 This is called synchronization. 17 00:01:01,890 --> 00:01:04,710 So we want synchronization between joints. 18 00:01:04,980 --> 00:01:06,690 How will we achieve that? 19 00:01:07,020 --> 00:01:09,570 It's very easy and logical indeed. 20 00:01:09,750 --> 00:01:14,160 Namely, we have to determine first the joint that will move the mosque. 21 00:01:14,460 --> 00:01:16,440 So has the largest displacement. 22 00:01:16,890 --> 00:01:18,690 Let's say this is the kids joint. 23 00:01:19,200 --> 00:01:26,340 Then we will co-create acceleration, time and execution time for that joint base on the given acceleration 24 00:01:26,340 --> 00:01:27,570 and velocity values. 25 00:01:28,260 --> 00:01:34,320 Then we will use these time parameters for the whole trajectory and calculate corresponding velocity 26 00:01:34,320 --> 00:01:38,820 and acceleration values for the other joints, except Keith one. 27 00:01:39,480 --> 00:01:46,650 In this way, all the trajectories will start and end at the same time, so they will be synchronized 28 00:01:47,190 --> 00:01:54,870 after we have obtained velocity, acceleration values and also acceleration time and execution time. 29 00:01:55,110 --> 00:01:59,340 We can calculate the trajectory very easily for each joint. 30 00:01:59,850 --> 00:02:03,450 Let's jump to the math club and see how it will be done. 31 00:02:03,570 --> 00:02:03,960 OK. 32 00:02:05,430 --> 00:02:12,920 Here is how we will apply trapezoidal trajectory in six degrees of freedom robotic manipulator. 33 00:02:13,440 --> 00:02:16,890 Um, OK. 34 00:02:17,460 --> 00:02:20,910 We will just think that it's six degrees of freedom, Robert. 35 00:02:20,930 --> 00:02:21,450 Apparently there. 36 00:02:21,450 --> 00:02:28,610 Indeed, I have taken the example as you are, as you can see, you are five robotic manipulator. 37 00:02:28,620 --> 00:02:31,080 This is from Peter Cook his library. 38 00:02:31,350 --> 00:02:31,770 OK. 39 00:02:32,820 --> 00:02:34,860 So we just import this model. 40 00:02:35,410 --> 00:02:42,420 Then we are sitting for maximal velocity and maximal acceleration for each of the giants time, then 41 00:02:42,420 --> 00:02:43,910 maximal velocity. 42 00:02:43,920 --> 00:02:46,100 I found it from Datasheet Mart. 43 00:02:47,130 --> 00:02:50,100 I myself assigned for maximal acceleration. 44 00:02:50,370 --> 00:02:52,110 Anyway, it's just an example. 45 00:02:52,770 --> 00:03:01,950 Let's assume that we are given with maximum velocity and maximum acceleration that our robot's actuators 46 00:03:01,950 --> 00:03:02,790 can output. 47 00:03:03,520 --> 00:03:05,440 This is the initial OK. 48 00:03:05,790 --> 00:03:08,190 We have made it equal to the Q. 49 00:03:08,190 --> 00:03:13,080 Is it because this cuz it comes from this, then we import the model. 50 00:03:13,080 --> 00:03:13,800 You are five. 51 00:03:14,040 --> 00:03:14,580 You will see. 52 00:03:14,610 --> 00:03:17,730 Yes, this is cuz it is for each of the joints and this is zero. 53 00:03:17,730 --> 00:03:20,100 So every joint starts from zero orientation. 54 00:03:20,100 --> 00:03:23,640 OK, get your zero angle and the final is q r. 55 00:03:23,760 --> 00:03:26,790 We want to reach this final position or orientation. 56 00:03:26,820 --> 00:03:27,210 OK. 57 00:03:28,560 --> 00:03:29,040 OK. 58 00:03:29,220 --> 00:03:37,200 What we have see there in the lesson, we have said that as we have more than one joint, we have to 59 00:03:37,200 --> 00:03:38,220 do a synchronization. 60 00:03:38,220 --> 00:03:40,470 What was the first step for synchronization? 61 00:03:40,800 --> 00:03:48,420 We have to find the maximal displacement, OK and the join with the maximal displacement, how we will 62 00:03:48,420 --> 00:03:48,870 do that. 63 00:03:49,170 --> 00:03:54,840 We first find the difference between the final position, find the orientation and the initial orientation, 64 00:03:54,840 --> 00:03:58,540 and we think it's absolutely because displacement is needed for us. 65 00:03:58,560 --> 00:03:59,050 OK. 66 00:03:59,730 --> 00:04:06,960 And then what we are doing, we find it's the maximum displacement in that displacement vector. 67 00:04:07,200 --> 00:04:09,030 OK, and tackles its index. 68 00:04:09,810 --> 00:04:17,910 Then what we are doing, we are doing synchronization v calculating acceleration time and time duration 69 00:04:17,910 --> 00:04:19,530 to execute the trajectory. 70 00:04:19,830 --> 00:04:23,290 We are calculating it for their maximal displacement. 71 00:04:23,310 --> 00:04:30,690 OK, as you can see, L Max, we have seen it in the presentation then after we have obtained T A. Acceleration 72 00:04:30,690 --> 00:04:36,870 time and execution time, we can apply these two for the whole trajectory sort. 73 00:04:37,080 --> 00:04:38,760 So for the whole other joints. 74 00:04:38,790 --> 00:04:39,270 OK. 75 00:04:39,660 --> 00:04:42,020 So what we are doing, we are first. 76 00:04:42,070 --> 00:04:43,320 So this is for sampling. 77 00:04:43,320 --> 00:04:48,620 No, because as you know, our trajectory trapezoidal trajectory consists of what? 78 00:04:48,630 --> 00:04:51,350 Three phases? 79 00:04:51,360 --> 00:04:51,720 Yes. 80 00:04:51,990 --> 00:04:52,890 Acceleration. 81 00:04:52,890 --> 00:04:54,780 Constant velocity deceleration. 82 00:04:55,140 --> 00:04:55,590 OK. 83 00:04:55,620 --> 00:04:57,660 And for each. 84 00:04:59,810 --> 00:05:07,100 For each phase, we take the time, for example, this is the time from zero to a for the first phase, 85 00:05:07,310 --> 00:05:13,280 from T to T minus to for the second phase from T minus T a two T for the third phase. 86 00:05:13,310 --> 00:05:13,700 OK. 87 00:05:14,060 --> 00:05:18,050 And then we are creating by six by. 88 00:05:19,400 --> 00:05:23,270 This is the sampling number of matrix. 89 00:05:23,840 --> 00:05:31,880 This is three dimensional Matrix Y because in each phase, OK, we will talk with four different people, 90 00:05:31,880 --> 00:05:37,400 use different formulation to calculate the Q and the same for velocity and the same for acceleration. 91 00:05:37,610 --> 00:05:41,270 This six is because we have six degrees of freedom, a robot manipulator. 92 00:05:41,270 --> 00:05:42,350 I mean, six joint. 93 00:05:42,350 --> 00:05:50,990 OK, so this matrix will have all these mattresses are three dimensional, so they have six rows and 94 00:05:50,990 --> 00:05:52,220 some called arms. 95 00:05:52,580 --> 00:05:54,810 But three of these, OK. 96 00:05:56,830 --> 00:05:59,420 I mean the dimension or does it axis also? 97 00:05:59,480 --> 00:05:59,830 OK? 98 00:06:00,320 --> 00:06:09,620 Three of these three or six by s and two Matrix City of six by S and the Q Matrix and three of six by 99 00:06:09,620 --> 00:06:12,680 S and D D Q Matrics acceleration matrix. 100 00:06:12,710 --> 00:06:16,040 Then we are doing for each phase. 101 00:06:16,070 --> 00:06:16,670 OK. 102 00:06:17,710 --> 00:06:24,590 What we are doing for each phase we are calculate and for each this is for each joint. 103 00:06:24,620 --> 00:06:24,980 OK. 104 00:06:25,730 --> 00:06:31,550 We are looping in each phase and in each phase. 105 00:06:31,550 --> 00:06:37,370 We calculate for all the joints what except the joint keys joint. 106 00:06:37,670 --> 00:06:43,220 Do you remember the one with the maximum maximum 107 00:06:45,530 --> 00:06:46,210 displacement? 108 00:06:46,220 --> 00:06:48,200 OK, except this what we are doing. 109 00:06:48,200 --> 00:06:54,050 We are calculating for the others the velocity and acceleration from their calculated T. 110 00:06:54,050 --> 00:06:55,880 A. A. From here. 111 00:06:55,910 --> 00:06:56,420 OK. 112 00:06:56,720 --> 00:07:02,990 And if it is phase one, we apply the formulation for phase one velocity and acceleration. 113 00:07:02,990 --> 00:07:04,880 We have seen this from the presentation. 114 00:07:04,880 --> 00:07:07,190 You can revise it. 115 00:07:07,790 --> 00:07:08,540 You can check it. 116 00:07:08,540 --> 00:07:16,370 Or for the second phase, if you remember, this was the, um uh, for the cure and for this velocity 117 00:07:16,370 --> 00:07:17,930 and for this acceleration. 118 00:07:17,930 --> 00:07:21,950 As you can see, the velocity is constant and acceleration is zero in the first phase. 119 00:07:22,220 --> 00:07:27,080 This is the acceleration, OK, because it's positive in this case, as you can see this variation because 120 00:07:27,080 --> 00:07:29,610 it's negative or constant deceleration. 121 00:07:29,630 --> 00:07:29,960 OK. 122 00:07:30,560 --> 00:07:34,670 And uh, for the doing, this is the joint. 123 00:07:34,670 --> 00:07:37,160 This is the case joint key for case joint. 124 00:07:38,090 --> 00:07:48,320 We assign in the velocity and, um, acceleration as maximum as maximum. 125 00:07:48,380 --> 00:07:51,860 And then in the first phase, we are doing calculation. 126 00:07:52,040 --> 00:07:52,490 OK. 127 00:07:52,730 --> 00:07:56,510 In the second phase, we are doing calculation and for the third phase. 128 00:07:56,700 --> 00:07:58,100 OK, I think this is easy. 129 00:07:59,940 --> 00:08:03,160 The important thing is how we have done the synchronization. 130 00:08:03,170 --> 00:08:10,310 As you can see, if you want a maximal displacement joint, if you calculated for it, from it to A.. 131 00:08:10,550 --> 00:08:18,890 And we apply this to A. For all their joints except case joint OK, and find velocity and acceleration 132 00:08:18,890 --> 00:08:19,430 for them. 133 00:08:19,430 --> 00:08:26,210 And then what we have done, we calculated for maximal maximum acceleration for them and we have calculated 134 00:08:26,450 --> 00:08:27,740 the trajectory for them. 135 00:08:29,150 --> 00:08:29,510 OK. 136 00:08:30,830 --> 00:08:34,250 This is for concurrent creating the time vector. 137 00:08:34,250 --> 00:08:37,250 OK, let's do it in this way. 138 00:08:37,250 --> 00:08:38,750 So you can see first. 139 00:08:38,750 --> 00:08:40,070 Let's do that part. 140 00:08:40,570 --> 00:08:41,990 Uh, run on advanced. 141 00:08:42,020 --> 00:08:48,340 OK, let's if you run this again, if you run this, you will see. 142 00:08:48,920 --> 00:08:50,340 Indeed, I can do that. 143 00:08:50,360 --> 00:08:56,330 Let me just to evaluate in this way, in order to show you evaluate this section, this whole could, 144 00:08:56,330 --> 00:08:59,810 as you can see, what is the dimension of T? 145 00:08:59,810 --> 00:09:07,720 Let's let's see size of T size of T size of T is three by one end. 146 00:09:07,880 --> 00:09:08,200 Yeah. 147 00:09:08,540 --> 00:09:11,570 Well, what I want to do is I want to make it. 148 00:09:11,960 --> 00:09:15,720 I want to control the get all the time slices. 149 00:09:15,740 --> 00:09:21,710 OK, so our time thesis into one vector so we can plot all of them in one plot. 150 00:09:21,920 --> 00:09:27,810 So I want to make it, what, three thousand by one or one by three thousand? 151 00:09:27,830 --> 00:09:28,080 OK. 152 00:09:28,100 --> 00:09:30,920 And I'm doing it in this way first and transposing it. 153 00:09:31,220 --> 00:09:34,310 Then I'm doing to control communication. 154 00:09:35,720 --> 00:09:39,950 OK, let's run it again around the section, around this section also. 155 00:09:39,980 --> 00:09:40,340 OK. 156 00:09:40,550 --> 00:09:43,460 As you can see now, we can check the size of T. 157 00:09:44,510 --> 00:09:47,140 OK, as you can see one by three thousand. 158 00:09:47,150 --> 00:09:54,480 Yeah, then what we are doing this will make for us if this will make for us easier, uh, for plotting. 159 00:09:54,500 --> 00:09:59,690 OK, but then you are plotting the position velocity and. 160 00:09:59,950 --> 00:10:07,860 Acceleration for each axis, OK, for each of you will draw for old accesses, OK? 161 00:10:08,400 --> 00:10:12,420 Let's do this plotting around and advance. 162 00:10:12,450 --> 00:10:14,220 Okay. 163 00:10:14,490 --> 00:10:17,730 I think we have to see the plots, OK? 164 00:10:17,760 --> 00:10:19,740 As you can see, this is position plot. 165 00:10:20,910 --> 00:10:21,480 Yes. 166 00:10:21,750 --> 00:10:22,830 You can see that. 167 00:10:23,550 --> 00:10:29,160 First of all, let me show you that as you can see from Let Me Just We started from Q. 168 00:10:29,160 --> 00:10:33,600 Is it OK you initial was this one from zero zero zero zero? 169 00:10:33,610 --> 00:10:35,130 What the hell is that? 170 00:10:35,910 --> 00:10:38,130 Why does I dad queue initial? 171 00:10:38,250 --> 00:10:41,750 Is he OK? 172 00:10:41,760 --> 00:10:42,870 Excuse me, it's cute. 173 00:10:42,900 --> 00:10:44,100 I yeah. 174 00:10:44,130 --> 00:10:46,530 Four Q Initial what we have done. 175 00:10:47,070 --> 00:10:47,910 Q Zero. 176 00:10:48,000 --> 00:10:49,950 Oh yeah, it's Q zero. 177 00:10:51,090 --> 00:10:52,950 We will start from this configuration. 178 00:10:53,310 --> 00:11:00,060 And Q R, as you can see on the first and fifth joint, are moving. 179 00:11:00,060 --> 00:11:03,420 So that's why only we can see a position in first and fifth. 180 00:11:03,420 --> 00:11:06,180 Um, excuse me, first and fifth. 181 00:11:07,740 --> 00:11:08,360 Yeah. 182 00:11:08,370 --> 00:11:10,020 Which one was three point? 183 00:11:10,020 --> 00:11:12,170 OK, then this is the first one. 184 00:11:12,180 --> 00:11:14,920 This is the fifth one fifth. 185 00:11:14,940 --> 00:11:17,200 Stewart and all others are zero. 186 00:11:17,250 --> 00:11:18,630 As expected, OK? 187 00:11:18,870 --> 00:11:21,090 You can see the trapezoidal trajectory. 188 00:11:21,090 --> 00:11:24,570 You can see blends, OK? 189 00:11:24,780 --> 00:11:27,910 And you can see a linear segment and you can see deceleration. 190 00:11:27,930 --> 00:11:28,890 That's perfect. 191 00:11:29,280 --> 00:11:30,300 Then let's see it. 192 00:11:30,330 --> 00:11:35,460 As you can see, this is totally the trapezoidal velocity profile. 193 00:11:35,700 --> 00:11:37,460 You can see a linear segment. 194 00:11:37,470 --> 00:11:41,970 Excuse me, you can see, yeah, linear function. 195 00:11:41,970 --> 00:11:45,670 This is constant function and this is the deceleration phase. 196 00:11:45,690 --> 00:11:46,110 OK. 197 00:11:46,860 --> 00:11:49,920 And you can see the beautiful acceleration. 198 00:11:49,920 --> 00:11:57,450 Yeah, this is just because as as you can see, as you can see here, the synchronization, all of the 199 00:11:57,660 --> 00:12:01,930 acceleration time is the same for both of the joints. 200 00:12:01,930 --> 00:12:09,030 So for the first one and the fifth one, the same is deceleration phase as you can see as T A and T 201 00:12:09,300 --> 00:12:12,630 I mean s acceleration time and deceleration time are the same. 202 00:12:13,140 --> 00:12:14,850 The figures are symmetric. 203 00:12:14,920 --> 00:12:22,470 OK, and as you can see, the acceleration profile is discontinuous, which is not desired too much. 204 00:12:22,470 --> 00:12:26,100 But for that simple trajectory, we can accept it.