WEBVTT - autoGenerated
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So let's see the general dynamic equations of a manipulator.
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If you look at these three terms, can you guess the possible meanings of these three terms?
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m, v, and g.
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At least from the equation in here, we can know m is a function of theta, the joint angle,
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and v is dependent on theta and theta dot, so the joint angle and also the velocity.
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And g is a function of theta, also depends on the robot configuration, the joint angles.
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Then think about the possible meaning of them.
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So I will start from an example, let's see here.
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Now I'm scratching my arm, and I know the force, all the strings to hold my arm.
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And now if I put a book on my hand, I need to increase the force.
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If the objects are heavier, I put an iPad in here, and now it's really hard to stretch my hand to extend my arm anymore,
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but I still increase the maximum force of my arm to hold my arm in the same position.
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And now if I just take one object away, actually my arm rolls up a bit when the weight is lighter.
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And if the object is heavier, so my arm will fall down a bit.
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And this is really similar to a robot, and since the robot has several links,
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it seems that the mass influences the stability of this system and the required force.
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So maybe this link is heavier than this link, and the effector is also very heavy,
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or somehow from a heavy and effective, heavy gripper, we change it to a very light, soft gripper, somehow like this.
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So the mass influences the stability of this system, and it will influence the required force, required torque should be generated by the motor.
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And then this is maybe a guess, so for the equation, you can think about mass will influence dynamics, influence the force,
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and then maybe mass goes to here.
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And also, we can think more about this rotation.
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If we go and analyze the inertial will from just one axis of a robot, let's see the first axis of rotation.
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Consider this is a robot arm.
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Then let's see the first joint, this is a railroad joint, it is rotating.
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And if my arm is here, then the robot has a big inertia.
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And if I go back, then for this joint, here is a small inertia.
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And by putting the masses away from the axis, your increase the inertia perceived about this axis.
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So the inertia then depends on the mass distribution, the mass of this link, the mass of this link, and the mass of the end effector in here.
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And also depends on the mass of center, so where is the mass of center.
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Therefore, also the length is different and the inertia is different.
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So that is associated with the other lengths, the mass distribution, the load associated with the manipulator.
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And if we go here, so not this joint anymore, we go to this joint.
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Then we also have changes in the inertia perceived on this joint, but it's independent of the previous length, previous link.
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So there is a structure we need to present this mass distribution to present this inertia, which is affected by the motion of the robot, the motion of the configuration of the robot.
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And because this will change the value of the inertia that the axis of the robot are perceiving.
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Therefore, we can go through the first term, M.
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So M post-multiply theta double dot.
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And the theta double dot is the acceleration.
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If the robot has only one degree of freedom and M is mass, so mass multiply the acceleration, then this is false, right?
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But because the robot is a multi-body system, so it's going to be a mass matrix.
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Therefore, so the mass matrix depends on how many joints the robot has.
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And if we think about we have a robot that is a prismatic robot, has three prismatic joints that essentially along the diagonal, along the diagonal of the matrix, you will see the element you are presenting the masses, the total masses of the robot reflected in the diagonal.
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And it will be a diagonal matrix.
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So this is for the prismatic robot.
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But for articulating the body with the railroad joints, you are going to have the off-diagonal terms that represent the coupling.
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That is the motion of one joint will accelerate the other joints, right, for the railroad joints.
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Okay, so this is M, and what is V?
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Maybe velocity, maybe vector.
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But in here we already have theta and theta dot.
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So theta dot is the joint velocity.
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I only can say maybe V is a vector, N times one vector.
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And what would be the sense that will depend on the velocity?
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See that dot. So what would be the sense that will depend on the velocity?
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Yeah, I already showed the results in here, centrifugal and Coriolis coefficients, the force.
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Let's see the other example.
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So this is a small tanda.
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And let's see, I rotate an object.
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You see?
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So when I rotate it, there seems there is a force pulling there, pulling forward, upward.
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And make the robot, make the object fall down.
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And if I stop moving, the object will fall, right?
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So the force pulling there is called centrifugal force, right?
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So this is the term when the object is moving, rotate, then this term, centrifugal force, involved in here.
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And it is really related to the velocity.
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If the velocity is zero, then this term will be zero.
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And also, now if you have a multiple body that are moving in addition to centrifugal, you will have a Coriolis force.
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So that is the product of velocities that will be involved, which will cause both centrifugal and Coriolis forces.
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So this I have the centrifugal Coriolis force, and we will see actually that this force disappears when velocity was zero.
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Or if the mass matrix was constant.
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So if the mass matrix was constant, this means the mass matrix is independent on the configuration.
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So then the derivative of this mass matrix is also zero.
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Therefore V is zero.
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So I mean the V term is zero.
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And then we go to the third term, g.
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What is g?
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Gravity, yeah, right? Like gravity.
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So gravity is dependent on the configuration.
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For instance, if we like this joint, for this joint, for the center joint in here, the joint in here.
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And if I put a weight, maybe here, if I put a weight in here, so you're going to feel a torque, right?
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You're going to feel a torque for this joint.
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And if I move and move, you see the same force, but if the configuration is changed, if I move this joint upward, so 90 degrees upward,
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therefore what is the torque due to the gravity for this joint?
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So the torque due to the gravity of this joint, it could be zero, right?
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Because the direction of gravity is point down, right?
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And so the torque actually on this axis will be zero due to the gravity.
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Therefore, I give this example just let you know, okay, the gravity term in here is dependent on the configuration.
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And then these are the three terms for the general dynamic equation.
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So therefore, when you design a robot or you want to use a robot, you want to know the dynamics system, then this is the basic three parts you need to start to think about.
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To get M, to get V, to get G in here, then you can calculate the torque, the required torque, then we can tell the motor to generate the required torque, right?
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And also, it is really important to realize that the dynamic equations we have introduced here do not encompass all the effects acting on the manipulator.
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So as the first slide in this lecture, I also talk about friction, right?
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In here, these three terms, these three forces are arising from rigid body mechanics.
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And the other most important sources of force, friction is not included here.
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But all mechanisms are of course affected by frictional forces. So usually the gears, the gear drive is used in robots.
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So the gears need to take the transmission.
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And commonly the forces due to friction can actually be quite large, perhaps equaling 25% of the torque required to move the manipulator in typical simulations.
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You see, 25% of the torque used to compensate the friction. And of course, for the industry robot we bought, they already considered the frictions in there, but depends on the different type of the robot, so the friction can be really different.
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So let's think about underwater robots. Instead of the general friction, you need to consider the other, like the buoyancy, which is an upward force, right, exerted by the fluid.
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So for different, and also some robots are the articulated robots, the motor driven robots, and also like a shadow hand, there are tendon driven robots.
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So there are two tendons for each joint, this tendon pole and this tendon will go up.
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So this tendon driven robots, the friction is also a big issue for the dynamics that really need to compensate for the friction term.
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And, of course, so I already said in this slide, I add first term in here, but for the friction term, the whole, what is the specific equation for f, how to calculate f, it really depends on the type of the robot.
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So why the friction is so, I mean, so tricky. From a simple explanation away from the kinetic and the static friction, we can, from this aspect, I can explain a bit.
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So we know when two surfaces are in contact and are in relative motion, we have a kinetic friction in which the torque due to the friction is proportional to the velocity of the object, right.
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And when two surfaces are not moving relative to each other, like here, the area from, so we have the static friction, so we know kinetic friction and static friction, the value, the magnitude are different.
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And usually, static friction are more than kinetic friction for the same materials, especially when we want the robot moves a really tiny motion.
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This is an area from static friction to the kinetic friction. So then the friction terms in here in this area for this tiny motion is really tricky.
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And also on the other hand, in many manipulate joints, friction also displays a dependence on the joint position. So a major cause of this, in fact, might be the gears that are not perfectly round.
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This is from the manufacturer. Therefore, if the gears are not perfectly round, then they're eccentricity.
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You get me? So their eccentricity would cause friction to change according to joint position.
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So therefore, in here we said the frictional term is dependent on the velocity and also on the joint position. So it is a function of theta and theta dot.
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Here I give the viscous friction and the columbia friction. And actually this term, now I'm thinking whether should I change it to kinetic friction and static friction.
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It's more easier to understand. If you look at the equations of these two friction, these two kinds of friction for the viscous friction is related to the velocity, is proportional to the velocity.
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And the columbia friction, sometimes when the velocity is fixed, then the velocity is zero, then it really depends on the static friction, the coefficient.
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And also, here is also to explain the friction as a dependency on the joint position.
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And so in here we also talk about, okay, so different robots have different kind of friction and big or small, they are different.
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There is a very popular type of robot, soft robots. In the Moodle course, I think last week I posted a video about the soft robot to show the difference between the soft robot and the rich robot hand.
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If you see, for the soft robot, the bodies, they are bending or stretching all the time, so the mass of center is hard to calculate and the friction is non-trivial.
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So the incomplete or inaccurate dynamic model is also the reason why it's easier to see a soft robot move really slow.
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Until now, I think still it's hard to quickly, to control a soft robot move quickly and stably.
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This is really, now all the demos of the robot move really small to keep stable.
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So great, then until now, we understand the factors affect dynamics and know we need an accurate dynamic model for the robots.
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And then how can we do that, how to get to calculate these terms for me, for the general M, V, G, these three terms.
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So next video, we will look at how to calculate dynamic equations and then see you there.