WEBVTT - autoGenerated
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Great, let's continue. Hope you still enjoy the word of the extra parameters.
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In this video, we learned the assignment of the joint coordinate system.
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We will use the right-handed coordinate system. Mostly, you use the second configuration in this figure.
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Please use your right hand and align the sum of your right hand with the rotation axis.
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The positive rotation is the direction that your fingers crew.
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The counterclockwise direction is positive and the clockwise direction is negative.
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In two minutes, you know why we always use the second configuration.
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Let's start with the definition of joint coordinate system by this KUKA example.
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Let's see the video of this robot first. It has four relative joints.
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In here, the first joint has the jaw movement.
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This is the second joint. You can see the pitch movement.
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This is the third joint, the pitch movement and the fourth joint rotate in here.
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Then go back to here. These four dashed lines are the rotation axis of each joint.
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Here, here, here and here.
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It seems the rotation axis are quite important right when we explain the four DHT parameters.
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If we want to decide the joint coordinate system, then which typical staffs we should concern.
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We should think about the origin and the XYZ axis, the rotation of the frame.
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We need to define a role about how we select the origin and how we decide the directions of the axis.
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If you see the right figure in here, do you see which direction of the axis is aligned with this rotation axis?
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Z. If you see Z0 is parallel to the first rotation axis and Z1 is parallel to the second rotation axis and Z2 is parallel to the third rotation axis and Z3 in here is parallel to the fourth rotation axis.
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Then the first convention is here.
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Axis Zi-1 is set along the axis of motion of I joint.
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Here, because this is a regular joint, the Zi-1 is the rotation axis of I joint.
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If here is a prismatic joint, then axis Zi-1 is along the axis of the translational motion.
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Now we have Z. If we know the direction of X or Y axis, then we can get the origin.
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We need to know where the intersection happens.
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Then we can know the origin.
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Let's go back to this figure.
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Where are the intersections?
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Here, here, here.
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Always the rotation axis, the Z axis have intersection with their common normal.
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The common normal in here.
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For the joint coordinate system, we define axis Xi parallel to the common normal of Zi-1 and Zi.
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This also means Xi is parallel to the cross product of Zi-1 and Zi.
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Let's see Xi as an example.
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Xi is pointing out of the plane and Zi and Z0 are here.
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So the common normal of Zi and Z0 is along this direction.
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But Xi, X1 can be pointing out of the plane or pointing inside of the plane.
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Actually, Xi has two choices.
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But in here, we choose out of the plane.
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There are two reasons in here.
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The first one is definitely in this case, we choose out of the plane.
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It's very easy to draw. It's clear to see.
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And the second result is for this robot in here, the n-effector is in this side.
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So the approach direction of TCP is out of the plane.
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Therefore, in here, we choose Xi.
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It's pointing to the approach direction of the TCP.
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Therefore, we also choose out of the plane.
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Therefore, we have Xi, but you have to know these two choices are both correct.
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In here, if you choose Xi inside of the plane, then based on the right-handed convention,
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the Y also has two choices.
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And then the sign of alpha, the link twist, will also have two choices.
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So this is how we get Z, X, and Y.
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Therefore, we will have CS1, CS2, CS3, and CS4.
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The number of the coordinate systems, CS, is based on the link number.
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For the first link, we will have the coordinate system 1.
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And at the base of the manipulator, we will have CS0.
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So CS0 is the stationary origin at the base of the manipulator.
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So where is the base of the robot, and where is the CS0?
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And Z0, we know it depends on the rotation axis, the motion of the eye joint.
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And we choose X0, usually we choose X0 parallel to X1.
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Then we can find the Y0.
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And now this is how we define the joint frames of a robot.
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If we know these frames, then let's see these four different parameters again.
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This is like a new definition of this.
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I hope based on these frames, you can understand these four parameters better.
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Let's only see one parameter as an example here, maybe.
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Just look, joint angle theta.
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In here, joint angle theta i, in here.
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This is the angle from xi-1 to xi around zi-1 axis.
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And just similar to joint angle, I hope you can explain or understand these three,
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the other three parameters by yourself based on these frames.
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Okay, I hope you can do it by yourself.
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Then let's see if we have these delta parameters, how can we get the transformation matrix
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from, of this i-link based on, with respect to i-1 link.
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And there are two predictions I want to mention.
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Why we have these two preconditions?
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Actually they have a very logical, mathematical background.
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If we want to know how we derive these two preconditions, I can send you the slides,
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but in this lecture I won't mention this mathematical part.
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If you look at these two preconditions, you can understand.
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The first one is xi is perpendicular to zi-1 because we also talk about xi is the common
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normal of zi-1 and zi.
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For the second precondition is xi is always in sex xz i-1.
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So if we want to get the common normal of zi-1 and zi, so xi must intersect with xi-1.
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But these two preconditions are also very important.
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If you can always remember these two preconditions, this will be helpful to build the frames,
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to assign the frames for each joint.
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Now we have these four parameters and how to calculate the transformation matrix.
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You can see classic parameters.
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Some students maybe have noticed that I always put classic and non-classic in here
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because their two parameters have two versions.
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One is classic version and the other one is the modified version in here.
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I won't explain the modified version in detail in this lecture, but I put the slides in here.
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After I explain the way of the classic parameters, I can understand the modified parameters by yourself.
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And in this lecture, all the examples I give is based on the classic parameters.
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But in the exercise, in the assignment sheet, we are really welcome.
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You can try to use the modified way to finish the tasks.
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Let's see the transformation order.
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We want to get the transformation of T of i, link i, in relation to i-1 link.
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Let's see in this figure where is T i-1 here.
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The longer axis, the longer red axis, blue and green axis are T i-1.
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If we see this equation from left to right, then this is right-hand modification.
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The following three transformations are based on the newly defined frame.
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Let's see from the origin frame in here.
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The first thing, we want to rotate around Z i-1 or Z i.
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After the rotation, we can get the new frame, the short red, short blue and the short green.
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And then we translate this frame D i along Z i-1, so we get a new frame again.
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Then we translate this frame along X i-axis A i, then we can get the new frame.
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Then we rotate around alpha i around the X i-axis, then we can get the new frame of T i.
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Therefore, the transformation between link i and link i-1 is these four parts.
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This is the way how we can get the transformation.
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There are four steps based on the four parameters.
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If you cannot catch up with me, I hope you can read these slides again to understand the better of these four rotations and translations.
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Therefore, based on these four steps, we get this equation and if we extend the rotation matrix in here,
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then we can get the calculation results.
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The first one is to rotate theta i around Z i-1 and translate D i along Z i-1, translate A i along X i and rotate alpha i around X i.
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Then we get this, so this is the result of T i with respect to i-1.
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If we go back to the beginning of this lecture, the way we calculate T 6 is based on T 1 in relation to 0, T 2 in relation to 1.
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Based on the equation we explore, then we can calculate T 6, then we can get an effector of the robot.
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We can know where is the robot, so we can let the robot grasp some objects on the table.
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This is the way we can calculate the forward dynamics of a robot.
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This also explains why we need theta parameters.
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We can get this transformation, we can know where is the robot.
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This is the modified version, the rotation transformation order is also different.
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In the exam, we will only ask about the transformation order about the classic version.
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At the end of the video, we want to talk about some exceptions because the convention is ambiguous.
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Therefore, we want to talk about these special situations.
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When this ambiguous happens, how should you decide?
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When the i-1 is parallel to the i, so two rotation axes or translation actions are parallel.
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Therefore, there are multiple common normals.
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The way you choose where xi should be is we want di equals to 0, so we want more 0 in all dr parameters.
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This will make the calculation, the transformation, make this part easier to calculate.
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For this situation, we really choose di equals to 0, and if zi minus 1 intersects zi,
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so the two rotation axes or the translation axes, they are intersection,
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then usually we will choose the origin of position of the x-axis,
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make ai equals to 0 to get a 0 value in the dr parameters.
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Sometimes the orientation of the coordinate system n is ambiguous because there is no joint n minus 1.
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For example, in here, we only have four joints, but for z4, we don't have the joint 5.
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How can we get z5?
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There are ways to teach you how to get z5.
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Usually, zn is chosen to point in the direction of the approach vector a of the TCP.
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Usually, zn is parallel to the approach vector of TCP.
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And xn must be normal to zn minus 1.
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Yeah, it must be because xn is the common normal of zn minus 1, right?
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And zn, so it's perpendicular to the zn minus 1.
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And in the assignment sheet 2, the second one, you will find several tasks about the frame assignment
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and the dr parameter tables. I hope you will enjoy that.