1 00:00:00,060 --> 00:00:03,480 From this lesson, we will start to analyze stiffness control. 2 00:00:03,600 --> 00:00:09,030 It will be divided in several parts in order to grasp the concept much more better. 3 00:00:09,540 --> 00:00:15,540 The name is stiffness or compliance control because, as you know, compliance is nothing but inverse 4 00:00:15,540 --> 00:00:16,200 of stiffness. 5 00:00:16,680 --> 00:00:21,000 So if you control stiffness, then you control compliance and vice versa. 6 00:00:21,450 --> 00:00:23,850 Anyway, let's jump to the core concept. 7 00:00:24,390 --> 00:00:25,770 Let's consider this simple. 8 00:00:26,220 --> 00:00:32,460 On the left, the robot arm is given while on the right is the object or more generally environment 9 00:00:32,460 --> 00:00:34,260 that the robot is interacting with. 10 00:00:34,770 --> 00:00:38,040 Here is the input force that's applied by the robot. 11 00:00:38,370 --> 00:00:40,770 Next is the position of the environment. 12 00:00:41,220 --> 00:00:44,310 This can be obtained from the model of the environment. 13 00:00:44,730 --> 00:00:47,700 X is a steady state position of the end effector. 14 00:00:48,150 --> 00:00:51,450 It is nothing but the desired position of the end effector. 15 00:00:51,840 --> 00:00:55,350 Don't forget that the robot and the environment have some stiffness. 16 00:00:56,160 --> 00:01:02,070 Based on these given variables, we can get the interaction forces between robot arm and the environment. 17 00:01:02,730 --> 00:01:09,480 This is simple springform forced formula because as robot and the environment has some compliance, 18 00:01:09,720 --> 00:01:13,230 we can accept them as springs. 19 00:01:13,860 --> 00:01:17,310 Additionally, we can ride the system dynamics easily like that. 20 00:01:18,030 --> 00:01:22,920 Now, let's assume that our force input is nothing but simple PD controller. 21 00:01:23,400 --> 00:01:25,860 Then we can write Equation 1.1. 22 00:01:26,460 --> 00:01:31,770 After doing some algebraic manipulation, we can obtain Eq. one point two. 23 00:01:32,340 --> 00:01:37,500 We are interested with what will be the steady state position of the end effector and what will be the 24 00:01:37,500 --> 00:01:39,330 steady state interaction forces. 25 00:01:40,140 --> 00:01:47,370 A steady state acceleration and velocity will be zero clearly, so we can easily get equation one points 26 00:01:47,370 --> 00:01:55,110 three from Equation one point two and based on Eq. one, we can get steady state interaction force between 27 00:01:55,110 --> 00:01:59,160 the robot and the environment, which is given by Eq. one point four. 28 00:01:59,760 --> 00:02:07,170 If you observe a bit carefully, you will see that equation 1.4 is simple spring force equation with 29 00:02:07,170 --> 00:02:10,020 equal and brigance of key e q. 30 00:02:12,240 --> 00:02:17,660 Let's see the equations of 1.4 and equal and spinning this again from equals. 31 00:02:17,670 --> 00:02:19,140 Bring this equation. 32 00:02:19,140 --> 00:02:26,160 You can see that the robot's spring and in warm spring is in serious because, as we have said before, 33 00:02:26,160 --> 00:02:29,700 we accept the robot and the environment as two springs. 34 00:02:30,090 --> 00:02:35,670 The robot's stiffness can be controlled by adjusting key, and this will surely affect the interaction 35 00:02:35,670 --> 00:02:36,040 force. 36 00:02:36,100 --> 00:02:42,740 Also, here's the important thing if you analyze equation one point carefully, you will see that when 37 00:02:42,750 --> 00:02:49,470 k the name environment stiffness is much higher, the robot's stiffness will get stiff environment but 38 00:02:49,470 --> 00:02:50,460 compliant robot. 39 00:02:50,670 --> 00:02:56,850 So interaction force is generated mainly due to the robot because it is the elastic and deformable one. 40 00:02:57,240 --> 00:03:02,690 However, when the robot's stiffness is much higher than the environment stiffness, then increment 41 00:03:02,700 --> 00:03:09,420 is compliant and interaction force is generated mainly due to the environment because it is the compliant 42 00:03:09,420 --> 00:03:10,380 and deformable work. 43 00:03:10,920 --> 00:03:17,700 In practice, general condition is correct because robot is more compliant with some kind of damages 44 00:03:17,910 --> 00:03:19,680 while the environment is stiff. 45 00:03:20,400 --> 00:03:27,420 So based on a book considerations, we can see that we increase the stiffness of robot on directions 46 00:03:27,420 --> 00:03:29,190 where position has to be controlled. 47 00:03:29,190 --> 00:03:36,150 Precisely because when we choose keep a high robot will be stiff and trajectory will be tracked precisely. 48 00:03:36,510 --> 00:03:39,000 KP is proportional control term. 49 00:03:39,180 --> 00:03:46,920 So by choosing higher vehicle for trajectory errors, we decrease CP in the directions where there is 50 00:03:46,920 --> 00:03:48,090 interaction forces. 51 00:03:48,090 --> 00:03:55,170 And so these contact forces have to be controlled, so we decrease keeping in them the robot's stiffness 52 00:03:55,170 --> 00:03:57,570 and so increase robot compliance. 53 00:03:58,140 --> 00:04:05,350 This will increase effectiveness of interaction force control while decreased trajectory control preciseness. 54 00:04:05,370 --> 00:04:12,060 So we had to choose KP at first peak until we reached the point that interactions happen. 55 00:04:12,630 --> 00:04:16,260 At this point, we decrease KP to get compliant robot. 56 00:04:17,820 --> 00:04:21,660 In general, our robotic system is stable because of friction forces. 57 00:04:21,990 --> 00:04:30,510 However, if it is not the how or if it is not, we can provide stability by adjusting KP and Kiwi terms, 58 00:04:30,510 --> 00:04:31,560 which are positive. 59 00:04:31,980 --> 00:04:36,210 So our system will be stable and we will have steady state. 60 00:04:36,960 --> 00:04:41,680 As we have seen before, the steady state interaction force is given by this formula. 61 00:04:41,970 --> 00:04:50,040 And if environment is much more stiff than the robot, then this equation 1.4 can be approximated like 62 00:04:50,040 --> 00:04:50,340 that. 63 00:04:52,080 --> 00:04:58,980 As you can see, in order to control interaction forces, the first fix the robot stiffness keep based 64 00:04:58,980 --> 00:04:59,820 on environment. 65 00:04:59,960 --> 00:05:06,230 Stiffness, which is known approximately how do you have to know environment position precisely. 66 00:05:06,620 --> 00:05:07,340 Be careful. 67 00:05:07,550 --> 00:05:10,790 Choose Cape is such that it is much less than key. 68 00:05:11,300 --> 00:05:17,360 Then we adjust to and based on it, we can control interaction forces. 69 00:05:17,870 --> 00:05:22,910 So as you can see, first control is obtained by means of positional control. 70 00:05:23,360 --> 00:05:25,880 Namely, this is indirect force control.