1 00:00:00,180 --> 00:00:00,570 OK. 2 00:00:00,870 --> 00:00:07,230 This is the last force control technique that we will see, and as you can see from the title of the 3 00:00:07,230 --> 00:00:10,770 video, this is hybrid position, force control technique. 4 00:00:11,340 --> 00:00:18,360 Here, we will not specify a new method of control formulation for either position or force, but instead 5 00:00:18,450 --> 00:00:25,860 we will use the ones we have seen before in the way everything will be clear to you further in the presentation. 6 00:00:27,510 --> 00:00:33,540 Manipulation of an object includes both control of position, velocity and force torque. 7 00:00:33,870 --> 00:00:40,200 It can be complex to develop a control technique that will help us in controlling position and force 8 00:00:40,410 --> 00:00:41,250 precisely. 9 00:00:41,610 --> 00:00:48,420 Also, until this lesson, we have seen either pure position, velocity control and force control. 10 00:00:48,750 --> 00:00:55,950 But what we can do in order to simplify the difficulty of control is to decouple the directions in which 11 00:00:55,950 --> 00:01:02,970 position velocity control is required and in which force total control is required with respect to some 12 00:01:02,970 --> 00:01:03,900 reference frame. 13 00:01:04,500 --> 00:01:11,910 After this decoupling process, we can apply pure position and force control in appropriate directions. 14 00:01:12,330 --> 00:01:17,850 This will be much simpler for us to control both position and the force simultaneously. 15 00:01:18,690 --> 00:01:21,150 Let's take this scenario into consideration. 16 00:01:21,420 --> 00:01:25,710 A robot arm slides a rigid object or the friction this table. 17 00:01:26,250 --> 00:01:30,900 Let's see what we mean by decoupling position and force control directions. 18 00:01:31,330 --> 00:01:37,380 However, first, we have to define suitable a reference frame, and we take this reference frame in 19 00:01:37,380 --> 00:01:40,420 the contact point, which is given by a red color. 20 00:01:42,220 --> 00:01:47,610 You can control position in X and Y directions, which is obvious. 21 00:01:48,090 --> 00:01:50,820 We can rotate the object in that direction only. 22 00:01:51,420 --> 00:01:54,750 Also, we can control force only in that direction. 23 00:01:55,470 --> 00:02:00,600 This can be confusing for you, but don't worry, we will clarify it further in the presentation. 24 00:02:01,080 --> 00:02:07,650 And finally, we can control talking on the X and Y directions, which will be clear for you during 25 00:02:07,650 --> 00:02:13,620 the presentation after we have seen what we mean by decoupling generally. 26 00:02:13,860 --> 00:02:16,500 Let's go into deeper in this issue. 27 00:02:16,950 --> 00:02:23,520 Before doing that, we have to consider constraints as you know, whatever we do with the robots, which 28 00:02:23,520 --> 00:02:25,230 also includes manipulation. 29 00:02:25,590 --> 00:02:29,250 We have constraints either due to the robot or the environment. 30 00:02:29,670 --> 00:02:35,370 In this section, we will ignore the constraints due to the robot, but only consider the ones due to 31 00:02:35,370 --> 00:02:36,360 the environment. 32 00:02:36,990 --> 00:02:42,720 OK, we have two types of constraints, namely the natural ones and artificial ones. 33 00:02:43,560 --> 00:02:49,920 Natural constraints are due to the geometry of the environment, so they are environment dependent or 34 00:02:49,920 --> 00:02:51,330 also task dependent. 35 00:02:51,750 --> 00:02:53,160 We cannot control them. 36 00:02:53,430 --> 00:02:59,220 They are already there and we have to obey these constraints if we don't want problems. 37 00:03:00,090 --> 00:03:03,390 However, artificial constraints can be controlled by us. 38 00:03:03,660 --> 00:03:10,200 For example, if you want to have zero point one meters per second and the fact that a velocity in external 39 00:03:10,200 --> 00:03:18,210 action, this is the constraint that we are setting and we can change it as we want, as in a suitable 40 00:03:18,210 --> 00:03:18,660 range. 41 00:03:19,200 --> 00:03:26,190 So artificial constraints are set by us through our control algorithm and we can control either position 42 00:03:26,190 --> 00:03:30,240 or velocity and force or torque as we want. 43 00:03:31,230 --> 00:03:38,220 Now the important thing is that we define natural and artificial constraints, not with respect to the 44 00:03:38,220 --> 00:03:43,980 base frame or end the vector frame, but constraint or task frame. 45 00:03:44,580 --> 00:03:46,200 They have the same meaning. 46 00:03:46,620 --> 00:03:53,310 This task frame is task dependent because based on task, we change in which directions we have control 47 00:03:53,520 --> 00:03:55,200 and in which direction not. 48 00:03:55,770 --> 00:03:59,160 Also, this frame is not fixed like inertial frame. 49 00:03:59,460 --> 00:04:04,140 It changes its orientation and position depending on where it's defined. 50 00:04:05,100 --> 00:04:12,030 As, you know, all of the general, we can specify 12 variables that we can't control if we don't consider 51 00:04:12,030 --> 00:04:18,180 any kind of constraints, namely six linear variables and six rotational variables. 52 00:04:18,540 --> 00:04:24,780 But surely we will not be able to control all of them because we have constraints. 53 00:04:24,780 --> 00:04:32,490 So we want if we want to define which ones we can control and which ones we can not in order to clarify 54 00:04:32,490 --> 00:04:33,310 this concept. 55 00:04:33,330 --> 00:04:38,000 Let's take some real world scenarios in this task. 56 00:04:38,010 --> 00:04:41,250 We want to slide the rigid object or frictionless table. 57 00:04:41,640 --> 00:04:49,080 The frame indicated the C in contact point is our task frame based on which will specified the constraints. 58 00:04:49,770 --> 00:04:56,430 OK, now this is the velocity space where the linear and angular velocities have been specified. 59 00:04:56,760 --> 00:04:57,570 Be careful. 60 00:04:57,810 --> 00:04:59,810 These are specified with respect to. 61 00:04:59,880 --> 00:05:07,050 The see frame or contact frame or tusk frame, excuse me, not on the frame, but constrained frame. 62 00:05:07,560 --> 00:05:14,310 We have to express everything on tusk frame because decoupling happens with respect to it and control 63 00:05:14,310 --> 00:05:16,830 will be specified with respect to it. 64 00:05:16,830 --> 00:05:24,360 Also in the same way we define for space, which includes forces and talks in tusk frame. 65 00:05:25,140 --> 00:05:28,740 Now let's define natural and artificial constraints. 66 00:05:29,010 --> 00:05:34,350 Natural constraints will be written in red color while the artificial ones in blue color. 67 00:05:35,160 --> 00:05:36,780 Here is the natural constraints. 68 00:05:37,050 --> 00:05:39,240 Let's look at them one by one. 69 00:05:39,600 --> 00:05:47,130 We cannot control velocity in that direction because we have an obstacle, namely a rigid table in front. 70 00:05:47,640 --> 00:05:50,640 So there will be no motion in that direction. 71 00:05:51,090 --> 00:05:53,060 Don't forget, please analyze. 72 00:05:53,090 --> 00:06:00,450 With respect to the tusk frame, we have no obstacle in the directions of X and Y, so there will be 73 00:06:00,450 --> 00:06:04,320 motion in that direction and we can control it. 74 00:06:04,770 --> 00:06:09,840 In terms of rotational velocities, we are able to rotate only in that direction. 75 00:06:10,290 --> 00:06:16,050 We can not rotate in either X or Y again due to the obstacle we have. 76 00:06:16,620 --> 00:06:21,150 So because of that, omega X and Y are natural constraints. 77 00:06:21,600 --> 00:06:30,030 Now, in terms of forces, we can apply desired force only in that direction, but not in X or Y directions 78 00:06:30,360 --> 00:06:31,980 y look carefully. 79 00:06:32,280 --> 00:06:39,390 We have rigid obstacle in front if you want to apply force in that direction so we can apply desired 80 00:06:39,390 --> 00:06:40,080 force on it. 81 00:06:40,470 --> 00:06:44,760 However, object is free to move in directions of X and Y. 82 00:06:45,060 --> 00:06:48,330 Additionally, don't forget that table is frictionless. 83 00:06:48,720 --> 00:06:56,490 So when we try to apply force in X or Y directions, object starts to move and we cannot apply force 84 00:06:56,940 --> 00:06:58,270 sync it in this way. 85 00:06:58,500 --> 00:07:01,620 Can you apply force on space without an obstacle? 86 00:07:01,980 --> 00:07:05,610 Can you apply force on air if we ignore air resistance? 87 00:07:05,940 --> 00:07:13,830 No, as we cannot control force, as we want interruptions X and Y forces in these directions are natural 88 00:07:13,830 --> 00:07:14,520 constraints. 89 00:07:15,090 --> 00:07:20,650 Talks are the same as forces as we can rotate in that direction freely. 90 00:07:20,700 --> 00:07:23,970 We will not be able to generate torque in that direction. 91 00:07:24,330 --> 00:07:28,950 But the same is not true in terms of talks in X and Y directions. 92 00:07:29,880 --> 00:07:31,650 We can control them as we want. 93 00:07:32,130 --> 00:07:32,530 OK. 94 00:07:32,880 --> 00:07:34,740 We have seen natural constraints. 95 00:07:35,190 --> 00:07:37,110 Now let's see the artificial ones. 96 00:07:37,410 --> 00:07:42,930 Indeed, after determining the natural constraints, it's very easy to determine the artificial ones. 97 00:07:43,350 --> 00:07:48,990 First, compare each element of artificial constraints with each elements of natural ones. 98 00:07:49,500 --> 00:07:52,440 You will see there is a relation between them. 99 00:07:52,740 --> 00:07:59,730 So named linear velocities in natural constraints will correspond to forces in artificial ones in the 100 00:07:59,730 --> 00:08:00,520 same way. 101 00:08:00,540 --> 00:08:07,200 Angela velocities will correspond to talks, forces will correspond to linear velocities and talks will 102 00:08:07,200 --> 00:08:09,030 correspond to angular velocities. 103 00:08:09,510 --> 00:08:14,280 First, observe that please also note that accesses don't change. 104 00:08:14,580 --> 00:08:18,780 Now let's understand that the logic axis don't change. 105 00:08:19,740 --> 00:08:21,660 By this, I mean the reference frames. 106 00:08:22,680 --> 00:08:30,600 Now let's understand the logic of that correspondence as we don't have linear motion in x axis due to 107 00:08:30,600 --> 00:08:31,470 the obstacle. 108 00:08:31,650 --> 00:08:34,710 We can upload desired force to this object. 109 00:08:35,160 --> 00:08:42,600 So force is artificially constrained in extraction also as we cannot rotate in X and Y directions. 110 00:08:42,840 --> 00:08:45,690 We can upload talks in those directions. 111 00:08:46,170 --> 00:08:53,860 Furthermore, as V cannot apply force in X and Y direction because motion is free in those partitions. 112 00:08:54,120 --> 00:08:58,020 We can control velocity as we want in those directions. 113 00:08:58,500 --> 00:09:06,480 So linear velocities in X and Y directions can be controlled and they are considered as artificial constraint. 114 00:09:06,840 --> 00:09:14,370 Finally, as we cannot apply desire to move in that direction, then the rotation is free in that direction 115 00:09:14,370 --> 00:09:18,570 and we can control amount of this rotation before continuing. 116 00:09:18,570 --> 00:09:20,280 I want to add something here. 117 00:09:20,730 --> 00:09:28,710 Neutral constraints constrain us to have either zero velocities or zero forces and torus, and these 118 00:09:28,710 --> 00:09:29,400 are on the way. 119 00:09:30,180 --> 00:09:35,820 However, when we say that we can assign any desired value to the artificial constraints. 120 00:09:36,030 --> 00:09:37,530 This is not totally correct. 121 00:09:37,980 --> 00:09:42,750 We won't sometimes assign zero to also these values as desired value. 122 00:09:43,200 --> 00:09:47,130 For example, force in that direction is artificial constraint. 123 00:09:47,370 --> 00:09:54,240 However, we want to assign it as almost zero because both object and the table is rigid and design. 124 00:09:54,240 --> 00:09:58,950 Assigning higher values to these force can cause serious issues. 125 00:10:00,050 --> 00:10:06,590 You can generalize this not about other artificial constraints also, though, as you may notice, we 126 00:10:06,590 --> 00:10:09,890 want artificial constraints, not the natural constraints. 127 00:10:10,160 --> 00:10:10,670 Why? 128 00:10:10,970 --> 00:10:15,410 Because we have a control over artificial walls, not on natural ones. 129 00:10:15,710 --> 00:10:22,970 So we need to select artificial constraints from given general six by one velocity vectors and six by 130 00:10:22,970 --> 00:10:29,390 one force vector now based on a determination of natural and artificial constraints. 131 00:10:29,660 --> 00:10:37,820 We can write this here if tower are the general force and talk values that are defined with respect 132 00:10:37,820 --> 00:10:38,850 to the base frame. 133 00:10:39,230 --> 00:10:43,250 The dimension of the left hand side vector is six by one. 134 00:10:43,630 --> 00:10:50,750 OK, so we need to select artificial constraints from given general six by one velocity vectors and 135 00:10:50,750 --> 00:10:52,880 six by one force vector. 136 00:10:53,360 --> 00:11:01,340 Now, based on the termination of natural and artificial constraints, we can write this here Eiffel 137 00:11:01,520 --> 00:11:07,910 Tower or the general force and talk values that are defined with respect to the base frame. 138 00:11:08,310 --> 00:11:15,950 The dimension of left hand side vector is six by one, while the far right side vector is designed force 139 00:11:15,950 --> 00:11:18,290 and talk values we want to achieve. 140 00:11:18,740 --> 00:11:19,550 Be careful. 141 00:11:19,790 --> 00:11:21,530 They are in tusk frame. 142 00:11:22,400 --> 00:11:25,730 And this is the same love for velocity subspace. 143 00:11:26,630 --> 00:11:27,380 Be careful. 144 00:11:27,410 --> 00:11:28,730 They are in the tusk frame. 145 00:11:29,320 --> 00:11:32,210 And this is the same formula for velocity subspace. 146 00:11:32,510 --> 00:11:39,980 And this a a f and a V mattresses are called seleccion mattresses. 147 00:11:42,620 --> 00:11:44,570 Why they are called seleccion mattresses. 148 00:11:44,810 --> 00:11:52,340 Because we can get the variables that we have control on them or artificial constraints by applying 149 00:11:52,340 --> 00:11:56,540 these selection mattresses to the velocity and for sub spaces. 150 00:11:57,230 --> 00:12:03,980 Here's another scenario which is insertion of a pig into a hole first noticed the tusk frame, which 151 00:12:03,980 --> 00:12:06,140 is defined fixed in object. 152 00:12:06,710 --> 00:12:09,680 Here is a general velocity and force vectors. 153 00:12:10,070 --> 00:12:12,770 Let's first define natural constraints. 154 00:12:13,100 --> 00:12:17,990 I will summarize here shortly because I have explained carefully in previous example. 155 00:12:18,320 --> 00:12:23,270 So we cannot move in X and Y direction because of object. 156 00:12:23,570 --> 00:12:31,730 So zero linear motion cannot rotate around X and Y, so zero rotational motion as V can move freely 157 00:12:31,730 --> 00:12:32,980 in z direction. 158 00:12:33,050 --> 00:12:41,870 We cannot apply force, so zero force in the same way we have three rotation or set axis, so we cannot 159 00:12:41,870 --> 00:12:43,760 apply torque or this axis. 160 00:12:44,030 --> 00:12:45,080 So zero torque. 161 00:12:45,560 --> 00:12:51,440 Now let's see artificial constraints as we don't have motion in X and Y directions. 162 00:12:51,710 --> 00:12:55,410 We can apply force as we want in these directions. 163 00:12:55,790 --> 00:13:04,820 Then, as we cannot rotate or x and y axis, we can apply desired tools in these directions as we cannot 164 00:13:04,820 --> 00:13:07,330 apply force or torque insert dilutions. 165 00:13:07,340 --> 00:13:12,410 We can control linear and angular velocities as we want in these tight actions. 166 00:13:13,910 --> 00:13:19,880 As we have said before, we want to choose artificial constraints from general velocity and force vectors 167 00:13:19,880 --> 00:13:21,440 so we can control them. 168 00:13:22,190 --> 00:13:25,220 And here you can see again the selection mattresses. 169 00:13:25,640 --> 00:13:32,090 Please be sure that you understand them because they will be essential for us further in that control 170 00:13:32,090 --> 00:13:32,540 method. 171 00:13:34,400 --> 00:13:40,190 As we have said before, selection mattresses are used to get the artificial constraints, namely the 172 00:13:40,190 --> 00:13:47,720 variables that we can control from general velocity and force vectors mode that also artificial and 173 00:13:47,720 --> 00:13:50,570 natural constraints are complementary to each other. 174 00:13:50,840 --> 00:13:57,170 You can see that easily by analyzing them from previous examples so we can decompose. 175 00:13:57,170 --> 00:14:03,110 Generalized velocity vector as artificial was our initial ones, and their dot product is zero. 176 00:14:03,710 --> 00:14:05,750 The same is valid for the force vector. 177 00:14:06,410 --> 00:14:10,970 Based on that, we can say that the product selection matrices are also zero. 178 00:14:11,330 --> 00:14:17,570 You can also check that from previous examples, this has physical meaning of reaction forces. 179 00:14:17,570 --> 00:14:24,770 Talks don't perform and useful work on feasible motions, which is obvious to us from basic physics. 180 00:14:26,870 --> 00:14:33,590 I know that you are tired now, but let's set this example also, which is nothing new but will help 181 00:14:33,590 --> 00:14:35,810 you to understand some basic concepts. 182 00:14:36,320 --> 00:14:39,530 Here we have a robot arm tries to turn crank. 183 00:14:40,280 --> 00:14:47,990 We defined two axis name, the base frame, which is indicated by f zero and constraint or task frame 184 00:14:47,990 --> 00:14:52,940 designated by f c crank mousa in an arc trajectory. 185 00:14:53,240 --> 00:14:59,510 As you can see why C is in the direction of tangential velocity the orientation. 186 00:14:59,760 --> 00:15:07,060 Difference between base frame and tusk frame is angle alpha, which changes as Chrome rotates. 187 00:15:07,840 --> 00:15:10,780 Here is the general velocity and force vectors. 188 00:15:11,770 --> 00:15:13,810 Let's determine natural constraints. 189 00:15:13,930 --> 00:15:20,680 If we don't have motion in X and Z directions because of geometry of the environment, we don't have 190 00:15:20,680 --> 00:15:22,940 rotation in X and Y direction. 191 00:15:22,960 --> 00:15:29,980 Also, again, due to the geometry of the environment, we cannot apply force in y direction because 192 00:15:29,980 --> 00:15:32,470 we have three motion in that direction. 193 00:15:32,950 --> 00:15:39,520 Also, we cannot apply torque over the z axis because we have the rotation in that direction. 194 00:15:40,420 --> 00:15:42,310 Here is the artificial constraints. 195 00:15:42,730 --> 00:15:44,050 I want you yourself. 196 00:15:44,050 --> 00:15:44,770 Analyze it. 197 00:15:45,040 --> 00:15:46,230 There is nothing new. 198 00:15:46,480 --> 00:15:48,130 You can analyze it easily. 199 00:15:48,760 --> 00:15:50,230 Here's the important thing. 200 00:15:50,530 --> 00:15:54,010 Don't fear we don't have anything new concept. 201 00:15:54,520 --> 00:16:00,790 We have stated before that constraints are defined in tusk frame because it's easy for us. 202 00:16:01,360 --> 00:16:03,010 OK, that's perfect. 203 00:16:03,250 --> 00:16:08,320 But general velocity and force vectors are defined with respect to base frame. 204 00:16:08,740 --> 00:16:14,950 So after the selection, we need to convert selected velocities UN forces to base frame. 205 00:16:15,520 --> 00:16:19,900 This will be done, as you know, through the rotation matrix of R. 206 00:16:20,770 --> 00:16:25,280 Here's the same concept for Force Vector, as you can see. 207 00:16:25,510 --> 00:16:30,010 Now the selection mattresses are angled dependent due to the rotation mattresses. 208 00:16:30,700 --> 00:16:34,840 However, again, dot product of these selection mattresses are zero. 209 00:16:35,500 --> 00:16:42,790 OK, now we know how to get the variables that we can control from the general force and velocity vectors. 210 00:16:43,300 --> 00:16:46,210 Now it's time to control these variables.