1 00:00:00,180 --> 00:00:04,170 In previous lessons, we have seen stiffness control for one day. 2 00:00:04,560 --> 00:00:07,950 But in practical case, we use more degree of freedom. 3 00:00:08,280 --> 00:00:09,540 Robotic manipulators. 4 00:00:09,840 --> 00:00:14,610 So let's develop the stiffness control for any degree of freedom robotic manipulators. 5 00:00:15,300 --> 00:00:21,960 Let's first see the formula for city-states interaction force and equal stiffness formula for any degree 6 00:00:21,960 --> 00:00:23,790 of freedom robotics manipulators. 7 00:00:24,180 --> 00:00:30,870 We don't have only forces, but also talks, so we have both linear stiffness due to the translational 8 00:00:30,870 --> 00:00:35,120 forces and rotational or torsional stiffness due to the talks. 9 00:00:35,760 --> 00:00:42,390 So we will utilize range in our formulations instead of just force, as you can see here. 10 00:00:42,690 --> 00:00:53,130 If it is nothing, but it is nothing but the linear spring can see here is nothing but linear spin constant 11 00:00:53,430 --> 00:01:00,570 and Kate and is rotational spring constant as the second diagonal of The Matrix is 0s. 12 00:01:00,750 --> 00:01:04,740 We don't have coupling between translation and rotation. 13 00:01:05,310 --> 00:01:10,710 PD and Omega D.T. are just infinitesimal translation and rotation. 14 00:01:10,710 --> 00:01:17,880 Accordingly, Eq. one point one is in workspace coordinates because interactions happen in operational 15 00:01:17,880 --> 00:01:22,710 space, so it's logical to represent formula in workspace coordinates. 16 00:01:23,160 --> 00:01:30,840 This creates no problem in terms of translation, but rotational part creates a problem because it's 17 00:01:30,840 --> 00:01:34,410 represented either in R, P, Y or all angles. 18 00:01:34,620 --> 00:01:40,140 And as you know, we cannot find angular velocity by just finding simply their derivatives. 19 00:01:40,200 --> 00:01:41,370 With respect to time. 20 00:01:41,940 --> 00:01:46,380 So Omega D is not valid for RPI or rectangles. 21 00:01:46,830 --> 00:01:51,540 In order to get valid form, we need to multiply the transformation matrix. 22 00:01:52,380 --> 00:01:53,850 Here is the corrected formula. 23 00:01:54,090 --> 00:02:01,260 As you can see it, transformation metrics consist of identity matrix, which corresponds to translational 24 00:02:01,260 --> 00:02:05,940 motion and T transformation matrix, which is configuration dependent. 25 00:02:06,660 --> 00:02:13,560 Now we have one additional problem translational and rotational coordinates are about x y z axis in 26 00:02:13,560 --> 00:02:16,740 terms of W, but all orange. 27 00:02:17,220 --> 00:02:22,050 But on the other side, translational coordinates are about x y z. 28 00:02:22,050 --> 00:02:24,690 In terms of x one. 29 00:02:24,690 --> 00:02:32,820 Rotational angles are r p y or earlier, as we have said before, so we need to make them compatible 30 00:02:33,120 --> 00:02:34,590 in order to do that. 31 00:02:34,800 --> 00:02:41,970 We use a game transformation matrix, but we transpose in that respect matrix dimensions by multiplying 32 00:02:42,960 --> 00:02:46,050 final formula is given in Equation 1.3. 33 00:02:46,650 --> 00:02:51,270 As you can see, the equation has the same structure as in one DB. 34 00:02:51,630 --> 00:02:59,010 We have seen before also, stiffness Matrix K is positive somewhere definite because the environment 35 00:02:59,010 --> 00:03:02,580 may not be able to generate forces on all directions. 36 00:03:03,090 --> 00:03:09,600 Additionally, by using green space of Matrix K, you can have information about the directions in the 37 00:03:09,600 --> 00:03:12,210 workspace along which forces may be applied. 38 00:03:13,200 --> 00:03:18,660 Because along some directions, we have only position control while along some directions on the force 39 00:03:18,660 --> 00:03:19,200 control. 40 00:03:20,010 --> 00:03:26,130 Now we want to find steady state position and range values during the robot's interaction with the environment. 41 00:03:27,780 --> 00:03:34,110 First, let's ride dynamics of robot dynamic equations robot manipulator, which we have seen many times 42 00:03:34,530 --> 00:03:34,830 here. 43 00:03:34,840 --> 00:03:41,880 G Transpose W is interaction forces and u is nothing but input force by the robot manipulator. 44 00:03:42,750 --> 00:03:48,060 This input force is controlled by PD control and gravity compensation. 45 00:03:48,540 --> 00:03:49,500 You may be. 46 00:03:49,500 --> 00:03:53,130 This formula seems intimidating, but let's see how it is formed. 47 00:03:53,640 --> 00:04:02,340 So as we have said before, force control is in workspace coordinates because interactions happen in 48 00:04:02,340 --> 00:04:03,360 operational space. 49 00:04:03,810 --> 00:04:09,810 However, dynamic model of the robot is in joint space, as you can see from Eq. one point four. 50 00:04:10,050 --> 00:04:12,650 So we need to convert input control. 51 00:04:12,660 --> 00:04:16,390 You also took the joint space coordinates first. 52 00:04:16,390 --> 00:04:23,160 Let's see PD control and gravity compensation formula in workspace coordinates, which is easier and 53 00:04:23,160 --> 00:04:24,420 more familiar to us. 54 00:04:25,020 --> 00:04:31,350 No, we need to convert this force vector into joint space by using Jacobean transpose because as you 55 00:04:31,350 --> 00:04:36,660 remember, Jacobean transports convert workspace wrench into joint space. 56 00:04:37,380 --> 00:04:41,310 However, we have one small issue here in this formulation. 57 00:04:41,310 --> 00:04:47,550 X2 term is workspace velocity, but we don't have sensors to measure workspace velocities. 58 00:04:48,210 --> 00:04:51,450 We have sensors in joints to get joint velocities. 59 00:04:51,810 --> 00:04:56,610 So we need to convert this term in the German space velocity using Jacobean matrix. 60 00:04:57,060 --> 00:04:59,580 So this is how we get Eq. one point. 61 00:04:59,670 --> 00:05:06,450 Five, let's analyze what will happen instead is state of the system in steady state acceleration and 62 00:05:06,450 --> 00:05:11,640 velocities will be zero and gravity turns will cancel each other in perfect assumptions. 63 00:05:12,990 --> 00:05:20,670 And we can get formulation for steady state error by just manipulating Equation 1.6 here in Red Circle. 64 00:05:20,700 --> 00:05:26,820 You can see compliance metrics, which is diagonal based on the structure of the transformation matrix. 65 00:05:26,820 --> 00:05:34,500 We can see that compliance matrix is linear and configuration independently in terms of forces, while 66 00:05:34,500 --> 00:05:38,580 nonlinear and configuration dependent in terms of talks. 67 00:05:39,600 --> 00:05:45,420 By combining equations 1.3 and 1.7, we will get below Eq.. 68 00:05:45,690 --> 00:05:51,570 From this equation, we can obtain steady state and end effector position and range. 69 00:05:52,020 --> 00:06:00,270 You can get Equation 1.9 by just manipulating Eq. one point eight As this of the vehicle equation one 70 00:06:00,270 --> 00:06:08,310 point nine, we can get Equation 1.10 through Equation one point three from equations one point nine 71 00:06:08,310 --> 00:06:09,570 and one point ten. 72 00:06:09,900 --> 00:06:15,900 We can see that steady state values depend on environment position, and these aren't in the vector 73 00:06:15,900 --> 00:06:18,810 position equation. 74 00:06:18,810 --> 00:06:24,390 One point is interesting in terms of control of the behavior of the robot manipulator. 75 00:06:25,170 --> 00:06:29,280 Interaction forces and talks depend on K and CP. 76 00:06:29,280 --> 00:06:36,720 As we have said before, we want to get rid of B to get rid of behavior of the robot in position control. 77 00:06:38,580 --> 00:06:45,360 Excuse me, as we have said before, we want to get rid of the behavior of the robot in position control 78 00:06:45,360 --> 00:06:46,050 directions. 79 00:06:46,470 --> 00:06:52,830 We can get this behavior by increasing the value of CP and to get compliant behavior of the robot. 80 00:06:52,830 --> 00:06:56,610 Enforce control actions by decreasing value of KP. 81 00:06:57,690 --> 00:06:59,360 Now let's see this example. 82 00:06:59,850 --> 00:07:07,230 As you can see, this is two degrees of freedom robot manipulator moving in x y plane desired position 83 00:07:07,230 --> 00:07:10,200 coordinates are specified by X and Y. 84 00:07:10,200 --> 00:07:13,500 Then there is no stiffness in the right direction. 85 00:07:14,010 --> 00:07:18,780 You can see that by analyzing a mattress, which is system stiffness. 86 00:07:19,860 --> 00:07:22,500 You can see that, um by. 87 00:07:22,740 --> 00:07:30,540 You can see that you can see this by analyzing key metrics, which is system stiffness or stiffness 88 00:07:30,540 --> 00:07:32,520 is given by CP metrics. 89 00:07:33,450 --> 00:07:42,120 As we have said before, we choose CP x low because environment stiffness is high in X direction, so 90 00:07:42,120 --> 00:07:44,370 we need compliant robot manipulator. 91 00:07:44,730 --> 00:07:52,740 We choose high volume for CP y stiffness in y direction, which makes robot manipulator stiff because 92 00:07:52,740 --> 00:07:54,390 environment is compliant. 93 00:07:55,320 --> 00:08:01,170 These are the steady state and the effector position and interaction forces. 94 00:08:01,920 --> 00:08:08,370 As you can see, a robot manipulator reaches desired position in y direction because in wide reaction, 95 00:08:08,370 --> 00:08:13,080 there is no stiffness and we don't need force control to get this position. 96 00:08:14,060 --> 00:08:19,250 Here are the statistics position on, first of all these again, let's analyze two cases. 97 00:08:20,150 --> 00:08:27,260 If CP X is much higher than K X, this means that environment is more compliant with respect to the 98 00:08:27,260 --> 00:08:27,740 robot. 99 00:08:28,100 --> 00:08:33,810 So our environment is compressed up to XD because there is almost no resistance. 100 00:08:33,830 --> 00:08:40,340 The robot the compliance is passive because environments acts like a spring and spring first is generated 101 00:08:40,340 --> 00:08:42,260 because of environment stiffness. 102 00:08:43,260 --> 00:08:50,630 Otherwise, escape is much less than K X. This means that the robot is more compliant with respect to 103 00:08:50,630 --> 00:08:52,520 the environment and environment. 104 00:08:52,520 --> 00:08:53,960 Almost not compressed. 105 00:08:54,830 --> 00:09:00,050 Robot acts like a spring and spring force is generated due to the robot. 106 00:09:00,530 --> 00:09:03,560 So this is active compliance.