WEBVTT 00:00.300 --> 00:07.440 Before diving into the details of the forward and inverse kinematics equation of a robot manipulator. 00:07.440 --> 00:13.800 And before studying the motion of the robot in space based on the position and angle of its motors, 00:13.950 --> 00:20.850 let's clarify what does it mean, the term pose of a robot manipulator and how to mathematically describe 00:20.850 --> 00:21.270 it? 00:21.920 --> 00:28.700 We can represent the pose of our manipulator robot by focusing uniquely on the position of the center 00:28.700 --> 00:30.320 of the robot's gripper. 00:30.890 --> 00:36.950 This point of the robot can freely move in the three dimensional space as it joint rotate. 00:37.490 --> 00:43.940 We define a fixed reference frame, which we will call W that stands for word. 00:43.940 --> 00:51.500 And therefore the problem of defining the robot's pose translate into the problem of defining the position 00:51.500 --> 00:57.350 and orientation of the robot in the reference frame w of the word. 00:58.210 --> 01:05.230 To do this, we can introduce also a new reference frame, this time a mobile reference frame that is 01:05.230 --> 01:06.850 attached to the robot. 01:06.850 --> 01:09.250 And we are going to call this one R. 01:10.000 --> 01:17.170 This reference frame is mobile because it moves with the robot and so therefore is called inertial reference 01:17.170 --> 01:17.770 frame. 01:18.390 --> 01:22.230 As we can see, while the word reference frame remains fixed. 01:22.230 --> 01:25.980 And so it is called Non-inertial reference frame. 01:26.190 --> 01:34.110 The R reference frame instead moves along with the robot and is therefore called an inertial reference 01:34.110 --> 01:34.500 frame. 01:34.950 --> 01:42.090 Thus, the problem of determining the robot's pose can be reformulated as finding the position of the 01:42.090 --> 01:50.730 reference frame R, which is mobile relative to the reference frame W, which is fixed by significantly 01:50.730 --> 01:53.200 simplifying the way we express our problem. 01:53.220 --> 01:59.970 We can now intuitively say that three variables are enough to define the pose of the reference frame 02:00.000 --> 02:03.030 r relative to the reference frame w. 02:03.330 --> 02:10.740 These are the X, y and z coordinates that express the position of the center of the reference frame 02:10.740 --> 02:15.240 R with respect to the center of the reference frame w. 02:15.870 --> 02:22.630 Together, these three variables express the pose of one reference frame relative to another. 02:22.690 --> 02:28.320 Therefore, since the reference frame R is attached to the robot and so it moves with it. 02:28.330 --> 02:35.200 These three variables also represents the pose of the robot as it moves in the three dimensional space. 02:35.560 --> 02:40.150 However, with this we are intentionally omitting something. 02:40.990 --> 02:44.350 With the variables X, Y, and Z. 02:44.350 --> 02:50.800 We are only expressing the position of the reference frame R with respect to the reference frame W of 02:50.800 --> 02:54.370 the world and we are not expressing its orientation. 02:54.850 --> 03:02.740 In fact, in addition of being translated by a certain amount X, Y, and z, the two reference frames 03:02.740 --> 03:08.890 so R and W are also rotated by a certain angle around each axis. 03:09.600 --> 03:17.430 Therefore, for completeness, the pose of a manipulator robot is defined by six variables, three of 03:17.430 --> 03:22.650 which refer to its position and the other three that refers to its orientation. 03:23.700 --> 03:31.170 Pay attention to the number of variables that make up the robot's pose and to the number of motors and 03:31.170 --> 03:34.140 so to the number of joints that our robot has. 03:34.930 --> 03:38.980 Our robot has only three motors which can activate three joints. 03:39.010 --> 03:45.340 Meaning that our robot only has three degrees of freedom to completely define the robot's pose. 03:45.370 --> 03:46.330 As we can see. 03:46.330 --> 03:50.410 And so including its position and orientation, we need six variables. 03:50.410 --> 03:53.800 And so therefore we need six degrees of freedom. 03:54.520 --> 04:01.210 This means that with our robotic arm, we cannot reach any position and orientation of the gripper within 04:01.210 --> 04:02.590 the robot's workspace. 04:03.130 --> 04:09.820 Specifically, if we fix a position for the robot, for example, if we want the robot to reach a specific 04:09.820 --> 04:16.990 position in space, let's say a certain position, X, Y, Z, we cannot also determine the orientation 04:16.990 --> 04:19.510 with which the robot will reach that position. 04:19.630 --> 04:23.460 The orientation will be automatically determined by its position. 04:23.470 --> 04:29.860 In other words, our robot has no control over the orientation and we cannot change it. 04:29.890 --> 04:36.190 We don't have enough number of degrees of freedom, and so we don't have enough motors to define also 04:36.190 --> 04:37.240 the orientation. 04:38.220 --> 04:45.240 For this reason, especially for industrial applications, manipulator robots usually have six or even 04:45.240 --> 04:46.680 seven degrees of freedom. 04:46.950 --> 04:53.070 This allows reaching any position within the robot workspace also with any orientation. 04:53.850 --> 04:59.310 Moreover, manipulator robots that have more than six degrees of freedom as they have more degrees of 04:59.310 --> 05:05.700 freedom than are actually needed to define the robot's position and orientation can reach the same position 05:05.700 --> 05:11.390 with the same orientation, with different configuration of the arms and with different trajectories. 05:11.400 --> 05:14.070 And for this reason they are more flexible.