1 00:00:00,640 --> 00:00:01,450 ‫Welcome back. 2 00:00:02,020 --> 00:00:08,660 ‫In this lecture, we're going to start looking at a drone from control systems point of view. 3 00:00:09,040 --> 00:00:11,620 ‫Let's look at the UAE more closely. 4 00:00:12,070 --> 00:00:14,730 ‫This is a typical quadcopter here. 5 00:00:15,340 --> 00:00:20,170 ‫It has four propellers or let's call them motors. 6 00:00:20,720 --> 00:00:25,390 ‫They are located on the edges of their respective beams. 7 00:00:25,900 --> 00:00:29,860 ‫And all of them are at an equal distance from the center. 8 00:00:30,070 --> 00:00:36,890 ‫And the distances here, the M and the two beams themselves are perpendicular to each other. 9 00:00:37,750 --> 00:00:47,860 ‫The first thing that we're going to do, we're going to attach an X, Y and Z body frame to it. 10 00:00:48,640 --> 00:00:51,510 ‫This frame is attached to the drone. 11 00:00:51,550 --> 00:00:54,360 ‫Always it moves with it. 12 00:00:55,150 --> 00:01:06,190 ‫It's important to note that this is a body frame around with small X, Y and Z axis and not the inertial 13 00:01:06,190 --> 00:01:10,990 ‫frame with big X, Y and Z axis. 14 00:01:11,890 --> 00:01:14,410 ‫And the inertia frame is fixed to the ground. 15 00:01:14,870 --> 00:01:23,290 ‫So we fixed the initial frame to the ground and we fix the body frame to the drone and to determine 16 00:01:23,740 --> 00:01:27,720 ‫the position of the body frame within the inertia frame. 17 00:01:28,090 --> 00:01:34,360 ‫We can have a position vector here that points to the origin of the body frame from the origin of the 18 00:01:34,360 --> 00:01:41,350 ‫inertia frame, and we can call it P vector, the position vector, the body frame can translate and 19 00:01:41,350 --> 00:01:43,270 ‫rotate within the inertia frame. 20 00:01:43,570 --> 00:01:50,170 ‫In fact, that is how we describe the drones position and orientation in our 3D space. 21 00:01:50,660 --> 00:01:59,140 ‫And now we're going to call the propeller that is on the positive small side, more one on the positive 22 00:01:59,140 --> 00:02:00,550 ‫small Y side. 23 00:02:00,580 --> 00:02:08,980 ‫We have more to negative small Exide model three and negative small Y side motor four. 24 00:02:09,910 --> 00:02:18,970 ‫Let's now try to identify the inputs and outputs of the drone using the plant block model. 25 00:02:19,660 --> 00:02:27,400 ‫Now, obviously, you need to measure or estimate the UVs position in the initial frame and for that 26 00:02:28,060 --> 00:02:38,800 ‫you need the big X dimension, the big Y dimension and the big Z dimension, because you have to know 27 00:02:38,800 --> 00:02:42,280 ‫the position of the drone in the inertia frame. 28 00:02:42,430 --> 00:02:47,850 ‫So you need to know these axes here, the big X, Y and Z axis. 29 00:02:47,890 --> 00:02:50,290 ‫However, there is more to that. 30 00:02:50,830 --> 00:02:56,470 ‫You also need to know the orientation of the drone in the inertial frame. 31 00:02:57,040 --> 00:03:08,020 ‫You can describe the drones orientation in the inertia frame using three angles roll pitch and your 32 00:03:08,620 --> 00:03:14,020 ‫which are Phi Theta and PSI respectively. 33 00:03:14,230 --> 00:03:20,860 ‫I will treat that extensively in the next section when we learn about 3D rotation matrices. 34 00:03:21,430 --> 00:03:30,220 ‫And so these triangles are to describe the drones body frame orientation in the inertial frame and you 35 00:03:30,220 --> 00:03:38,650 ‫can call it body frame orientation in the inertia frame or angular position in the inertial frame or 36 00:03:38,650 --> 00:03:41,790 ‫the drones attitude in the inertia frame. 37 00:03:42,550 --> 00:03:45,220 ‫So different terminology, but the same meaning. 38 00:03:45,730 --> 00:03:57,460 ‫But in addition, we can also measure the drones angular velocity about its own body frame, X, Y and 39 00:03:57,460 --> 00:03:59,190 ‫Z axis. 40 00:04:00,010 --> 00:04:09,100 ‫We're going to use the variables P, Q and R to describe the rotational velocity of the drone about 41 00:04:09,100 --> 00:04:11,740 ‫the body frame X, Y, Z axis. 42 00:04:12,190 --> 00:04:22,330 ‫Since they are rotational velocities, their units are radians per second, so peak are radians per 43 00:04:22,330 --> 00:04:22,930 ‫second. 44 00:04:23,080 --> 00:04:28,750 ‫OK, but how do we now determine if the rotation is positive or negative? 45 00:04:28,930 --> 00:04:35,830 ‫I mean, let's say that your drone is rotating about this small x axis. 46 00:04:35,980 --> 00:04:42,870 ‫Is it rotating, let's say, at two radians per second or at minus two radians per second? 47 00:04:42,910 --> 00:04:49,690 ‫How do you determine that the convention is to use the right hand rule to determine if the direction 48 00:04:49,690 --> 00:04:59,140 ‫of rotation is positive or negative if you wrap your right hand around a body X? 49 00:04:59,890 --> 00:05:10,300 ‫Y and Z axes in the same way like the rotation happens around those axis and if your thumb points in 50 00:05:10,300 --> 00:05:12,800 ‫the positive direction of that axis. 51 00:05:13,000 --> 00:05:20,250 ‫So in this case, it's in the positive direction, in the positive y direction and in the positive Z 52 00:05:20,260 --> 00:05:20,950 ‫direction. 53 00:05:21,200 --> 00:05:24,520 ‫If that's the case, then the rotation is positive. 54 00:05:24,730 --> 00:05:32,620 ‫If your thumb, however, points in the opposite direction, then the angles have negative radians and 55 00:05:32,620 --> 00:05:34,510 ‫therefore the rotation is negative. 56 00:05:34,720 --> 00:05:44,050 ‫So now, for example, if this is your body frame X and now you rotate about the axis like this, then 57 00:05:44,050 --> 00:05:50,530 ‫you would grab this axis like this where you can see the knuckles here and now you see that your thumb 58 00:05:50,530 --> 00:05:55,620 ‫points in the opposite direction to the positive x axis. 59 00:05:55,630 --> 00:05:56,040 ‫Right. 60 00:05:56,530 --> 00:06:01,780 ‫So that means that now your rotation would be negative. 61 00:06:02,020 --> 00:06:09,220 ‫So you would say that your P is, let's say, minus four radians per second.