WEBVTT - autoGenerated
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Hello, in this video we will learn homogeneous transformation matrix, and this is the last
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video in our first lecture.
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If you learn well about rotation matrix, so this video won't be a tough job for you.
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And from previous video, you know the rotation matrix, and you know how to present the configuration
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of a rigid body.
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So if we want to present the configuration of a body frame B, in a fixed frame or a
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reference frame A, then we can specify the position P of the frame B in A, and specify
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the orientation of B also in A. If we gather those together, gather the position and the
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orientation together, then we can get a single 4 times 4 matrix called T. So we use R to
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represent rotation matrix, then we use T to represent this homogeneous transformation
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matrix, or you can just call it transformation matrix for short.
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The bottom row, P means the perspective transformation, and S means the scaling.
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But in robotics, P is always 0, 0, 0, and S equals to 1.
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So other values are used for computer graphics, but in robotics, for the rotation, the transformation
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sense, this is the transformation matrix you will always use.
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And transformation matrices satisfy properties analogous to those 4 rotation matrix.
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For example, the product of two transformation matrices is also a transformation matrix.
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And a matrix multiplication is associative, but not communicative.
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And each transformation matrix has its inverse, also like rotation matrix.
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And you see here, we multiply transformation from B to A with transformation from C to
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B, then we can get transformation from C to A.
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So this is also the same of the usage of rotation matrix.
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We use this transformation matrix also can change the reference of one frame or one vector.
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Also analogous to rotation matrix, of course, they also have the other two common usages.
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The first one is to represent a rigid body configuration.
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So in the previous, the last video, we needed to use a position vector and then use a rotation
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matrix to represent the configuration.
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And now we use one transformation matrix, we can present the configuration of a rigid
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body.
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And then the second use is like this one, we can change the reference frame of a vector
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or a frame.
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And of course, the third one is to displace a vector or frame.
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So let's go to here.
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So this is the generalized format of a transformation matrix.
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And then this has 16 parameters.
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And if you see this slide, this is the inverse of a rotation matrix, it is simply transpose.
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So the rotation matrix part, this three plus times three rotation matrix inside of a transformation
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matrix, this three part is still the transpose of the rotation matrix.
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But for the position vector, make it to negative and then to the dot product of the pose of
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the position vector to the first column and second column and the third column of a rotation
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matrix.
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So this part just to show you, okay, rotation matrix part is still the same.
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And there's some change of the position vector of when you deal with the inverse of the transformation
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matrix.
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And you need to know how to calculate it, it is still dot product again.
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And this is showing if we only have a translation, so there is no rotation, only a translation.
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So how to express it?
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Therefore, the rotation matrix is an identity matrix and XYZ in here in the third column.
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So this is a translatory transformation.
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And regarding to rotatory transformation, so you have seen this already, do you remember?
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So if we rotate an object around the x-axis with angle phi, so this is the rotation matrix.
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And in the first column, 1, 0, 0, this means the orientation of the x-axis won't change,
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but there's something change, the projection of the unit vector will be changed.
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And the third column, the fourth column is 3, 0 in here.
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And similar to rotate around the y-axis and the z-axis, so this is the same for the rotatory
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transformation and that's it.
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So you will practice how to use this transformation matrix in your first assignment sheet of your
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exercise.
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So here I want to give you some specific calculation example.
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Now I will move to the real robot coordinate system.
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So this is more realistic and related to what you will do.
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So for a common robot state hub, they always have some typical frames, such as the base
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of the robot, the end-effector and the table, so they always have a table around the robot
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and which object the robot want to grasp.
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And also they have a camera to help the robot look at these objects.
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So for example, a camera always have at least 3 or 4, even 6 or 5 coordinate systems.
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So how to deal with these frames?
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Then we need to know the transformation between this coordinate system.
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Here we list 1, 2, 3 of 5, this 5 transformation, which is a very common transformation, and
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I hope you can understand this part, then this will help you to program and to play
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with the robot.
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So first is Z, Z is the transformation from word to manipulator base, and this also means
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the location of manipulator base in the word frame, right?
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And T6 means the translation from the base of the manipulator to the end of the manipulator.
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So the end of the manipulator also means the wrist of the manipulator.
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So do you remember?
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I will show you the wrist joints, the wrist link of the right arm of a PL2 robot.
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So the manipulator have the wrist, have the end.
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So T6 means from the base to the end.
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And then we also have the translation from the manipulator end to the end effector.
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So this is the transformation E, because we always mount an end effector or a gripper
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on the end of the manipulator.
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So like your file, we mount a three fingers gripper on the end of the manipulator.
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So we also need to know the translation between the end and the end effector.
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And then also here is the object.
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If we want to grasp something, then we need to know where is the object.
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So here is the transformation between word and the object.
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So this also means where is the object in the word frame.
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And also here also we need to know the transformation between the object and the end effector.
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You know, when we want the end effector manipulate the object, then first the end effector need
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more closer enough to the object.
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But we also cannot let the end effector crash or just break this object.
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So we also need to know the translation between object and the end effector.
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So you see, but in many cases, for a robot, we always focus on an effector.
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So because we need the end effector to interact with the word, with the environment.
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And then it depends on these transformations.
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How can we get the desired end effector position?
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Because the end effector position is quite important.
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We want the robot to manipulate the object, but we don't want the end effector crash
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on the table, crash with the objects.
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So here, have an illusion, if you see here.
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From z, what's z?
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Do you remember?
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So from word to manipulate the base.
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And then if we multiply the transformation from base to manipulator end, then we multiply
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the transformation from the end to the end effector.
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From z, multiply T6 and multiply E, then we can get from word to end effector.
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But if you see the two transformations in the bottom, we can also from transformation
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B, multiply G, we also can get the end effector location in word frame.
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But you see here, these two are different.
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So in the left side, this is in relation to the object, oh no, in the right side.
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In the right side, it is in the relation to the object, right?
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So B is the transformation between word and object.
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And the left side is in relation to the manipulator.
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So we can get the position or the configuration of the end effector from the word to the base
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of the robot, then to the end of the robot, and then to the end effector.
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But we also can get the post of the end effector from the object and then from the object
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to the end effector.
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So this equation, they are same, whatever you from z to E or you from B to G, both descriptions
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should be equal, right?
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But this is only for an effector, how you can get the transformation of an effector
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in word frame.
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But sometimes we also want to find the manipulator transformation, T6.
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So T6 means the location of an effector end in frame manipulator base.
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So we want to know with respect to the base of the manipulator, where is the end of the
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manipulator.
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Then it's very easy if you want to get the T6 from this fusion, then from the left side
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we multiply the inverse of transformation z, and then from the right side we multiply
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the inverse of transformation E. We can get this.
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And so it's similar if we want to get the position of the object, we can get the transformation
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matrix B, then from the both side we multiply the inverse of the transformation G, then
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we can get the transformation of the object in the word frame.
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So you see, if we know all the transformations, whatever which transformation matrix we want
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to calculate, we come from this kinematic chain to get them, right?
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So the definition of a kinematic chain is an assembly of rigid bodies connected by joints
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to provide a constraint of desired motion.
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So in this common robot setup, this whole system is a kinematic chain.
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We have word frame, then we have the base, and through different lengths connected by
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joints.
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Then we get the end of the manipulator, and then we mount an effect on the end, and then
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we can use the end effect to manipulate the objects.
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And then we based on the transformation between each other, so we can locate the object, we
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can locate the end effector, we also can locate the end of the manipulator.
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And now I want to show you a specific URL file, a grasp application.
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You have seen times URL file in the simulation visualization, right?
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And this is the real case.
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So we always put a camera close to URL file because we always put some objects on this
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table, then the robot can manipulate or grasp the object, but the robot cannot see the camera,
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so cannot see the objects.
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Then we use this camera to look at these objects.
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You see the camera always have multiple coordinate system, but in here I only choose one to simplify
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the problem.
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And then there have a frame on the top of the table, see this is the table top frame,
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and this is the end effector frame, the two frame.
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And this is the base of the robot, so I also have a frame.
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And here it's aprotech.
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I'm not sure you know about aprotech.
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So aprotech is the thing we can, so in Roast, they have a package called Roast aprotech.
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So then we can use this, if the robot look at the aprotech, then we based on this package,
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we can get the transformation between the camera and the aprotech.
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So because the aprotech is always fixed in one place, so in here, we always know the
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transformation between the base of the URL file and the location of the aprotech.
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So we know the transformation between these two, and based on the Roast aprotech package,
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we can get the transformation between camera and aprotech, and depends on this transformation,
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so we can know where is the camera.
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We can know the camera in the word frame or in the base frame of the robot.
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So now I show you a graceful video, you can see.
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So in this video, the blinking light is made by the camera.
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So the camera will generate point cloud of the objects, then send this point cloud to the robot,
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then the robot will use a grasp generation algorithm to generate different grasps, and
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then the robot need to judge which grasp is the best one, then the robot will plan and
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execute the best grasp and then to grasp the object.
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So in here, if the robot want to grasp the object, this means the robot need to know
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where is the object, right?
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But we only know the object, the robot get the information of the object is from the
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camera, because when the camera look at the object, then the camera can send some message
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about the transformation between the object and the camera to the robots.
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So therefore, now I close the video, therefore, we will know the transformation between the
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camera and the object.
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And then in here, we have a protect.
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So we also through this camera, we can know the transformation between the camera and
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the a protect.
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And also because the a protect is fixed, so we can get the transformation.
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Oh, no, the transformation between the base of the robot and a protect.
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Then now the question is how to calculate the transformation from the base to the object.
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So because when the robot want to grasp the object, we want to know where is the object.
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And here is also is like a real set up you need to figure out when you work with the
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robot.
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Okay, so now we want to calculate from base to object, then let's see the information
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in here.
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I accidentally choose the answer.
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So I hope you can calculate this by yourself.
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So this answer will be very similar to the transformation equation in here.
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We can use these three transformations to calculate this object transformation.
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Okay, so first from base, so first we know the transformation between the base and the
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a protect.
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Then we will multiply the inverse of the transformation between camera and a protect.
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So can any of you explain why here the multiplication is at the right hand, right side?
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Do you remember, we have talked about this in last video.
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If the rotations of the transformations are performed in relation to the current or newly
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defined coordinate system, then here should be right side a modification.
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You see the first transformation is in base frame and from base to a protect.
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And then the second transformation is the inverse and after the inverse is from a protect
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to camera.
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So the base frame will be a protect, it won't be base again.
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So this means the coordinate system will change.
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So here this means we need to use right hand a modification.
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And so then afterwards we need to multiply the transformation from camera to object.
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So it's similar.
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We also need to use right hand a modification because in the second transformation, the
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base frame is a protect and then in the third transformation, the base frame is camera.
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The coordinate system who performed this transformation are changed.
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So from these three transformations, we can get the transformation from base to object.
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You see, if you look at the video, it looks very simple and easy the robot just to grasp
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the objects.
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But if you really want to perform this on the robot, then at the beginning you need
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to figure out this coordinate transformation.
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And then you need to figure out the grasp generation algorithm, the grasp detection
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algorithm and how to plan and move the robot.
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Then let's see the summary of homogeneous transformation.
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A transformation matrix consists of the position and the orientation two parts.
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If the coordinate frame is defined in relation to a solid object or a rigid body, then this
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means the location of this body is specified, depends on this coordinate frame.
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And there also have three common usages of a transformation matrix.
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First is to represent the configuration of a rigid body and then to change the reference
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frame of a vector or a frame.
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The third one is to displace a vector or a frame.
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So this is very similar to rotation matrix.
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And the next is the very important thing about the order.
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So when we have multiple transformations, multiple translations and rotations, so which
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order you need to choose, right hand or left hand.
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This is related to which coordinate system you depend on.
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So if you are in relation to the current and newly defined coordinate system, then
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you need to choose right hand modification.
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If you are always in relation to the fixed coordinate system, like the row picture example,
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so you will use left hand modification.
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This is very important.
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Now we will finish all the transformation matrix, the rotation matrix part, but I still
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have one slide I want to show you.
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So I have shown you for common robot setup, they have the best frame and the effector
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frame and we use four times four matrix to describe these frames and they have a vector
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position and the rotation matrix.
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So this matrix has 16 parameters, but you see robots also have joints.
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We also can describe the robot by joint coordinates.
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So for example, your five has six degrees of freedom and has six joints.
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Then we can use the angles of each joint to describe a specific state of the robot.
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So this is also fixed if we already know the six angles of these joints.
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So joints are used to connect in two links.
00:23:47.000 --> 00:23:53.000
And also we can use joint coordinates to define a robot configuration.
00:23:53.000 --> 00:24:01.000
But if we use the joint coordinates, then we only know the angles between each link,
00:24:01.000 --> 00:24:03.000
but we don't know the x, y, z.
00:24:03.000 --> 00:24:10.000
We don't know exactly x, y, z, the coordinates of the end effector.
00:24:10.000 --> 00:24:15.000
So then this is something related to joint space and the Cartesian space, the end effector
00:24:15.000 --> 00:24:16.000
space.
00:24:16.000 --> 00:24:21.000
So then this is related to forward kinematics and inward kinematics.
00:24:21.000 --> 00:24:24.000
We will learn this later.
00:24:24.000 --> 00:24:29.000
The reason why I put joint coordinates in now is because I don't want you think, okay,
00:24:29.000 --> 00:24:35.000
we only can describe the configuration of manipulator by four times four transformation
00:24:35.000 --> 00:24:36.000
matrix.
00:24:36.000 --> 00:24:40.000
You also can use joint coordinates, but we will learn it later.
00:24:40.000 --> 00:24:44.000
And now we almost finish this lecture.
00:24:44.000 --> 00:24:51.000
And if you look at this PR2, this is the whole frames of PR2.
00:24:51.000 --> 00:24:53.000
There are so many frames.
00:24:53.000 --> 00:25:01.000
But if we use the transformation matrix or use the rotation matrix to describe the location
00:25:01.000 --> 00:25:06.000
of each frame, you at least use nine or 16 parameters.
00:25:06.000 --> 00:25:12.000
So can we use less of nine parameters to represent the orientation?
00:25:12.000 --> 00:25:19.000
I guess some of you must have been thinking about this at the beginning of this course
00:25:19.000 --> 00:25:22.000
when I start to introduce the rotation matrix.
00:25:22.000 --> 00:25:30.000
So some of you may be thinking, okay, I just want to use the three angles of the object
00:25:30.000 --> 00:25:32.000
around the x, y, z axis.
00:25:32.000 --> 00:25:33.000
This is pretty easy.
00:25:33.000 --> 00:25:35.000
This is Euler angles.
00:25:35.000 --> 00:25:41.000
And I will introduce three different representations of the orientation in the next lecture.
00:25:41.000 --> 00:25:45.000
So if you are interested in this, you can search it on Google first.
00:25:45.000 --> 00:25:49.000
And then the next thing is for this robot, like this robot.
00:25:49.000 --> 00:25:53.000
Each link has different lengths, has different orientation.
00:25:53.000 --> 00:25:57.000
And then how can we easily set up the transformation matrix?
00:25:57.000 --> 00:26:04.000
Is there some way you can easily construct the transformation matrix?
00:26:04.000 --> 00:26:05.000
Yes, of course.
00:26:05.000 --> 00:26:09.000
So we will learn this in the next video.
00:26:09.000 --> 00:26:13.000
And in the end of the video, this is the books I recommend for you.
00:26:13.000 --> 00:26:18.000
So all these books are available on Google or our library.
00:26:18.000 --> 00:26:23.000
You can borrow it from our library or not currently the digital library.
00:26:23.000 --> 00:26:30.000
And the last thing is to repeat your linear algebra knowledge if you think, okay, the
00:26:30.000 --> 00:26:34.000
course is not, you already think the course is too difficult for you.
00:26:34.000 --> 00:26:36.000
Okay, that's all.
00:26:36.000 --> 00:26:37.000
So see you in next week.