Module 2-7: Planar FV

Ajouté : 2026-05-25

[00:00:00]hi i'm scott knockley in this video we're going to look at the forward velocity solution for planar manipulators so by the end of this video you should be able to solve for velocity problems for planar manipulators so again we're going to use the planar 3r manipulator as an example so for the for the forward velocity problem we're going to be given the joint space variables so we're going to be given our joint rates we're going to be given theta 1 dot theta 2 dot and theta 3 dot and what we're trying to find is the velocity of our end effector so the translational velocity so x dot and y dot and the angular velocity which is alpha dot okay so previously we found the forward displacement solution all right what was x y and alpha given theta one theta two and theta three so in order to do our forward velocity solution we always first need to do the forward displacement so you go back to the previous videos and see how we solved this forward displacement all right now with the forward displacement we can easily find the forward velocity so we know that x dot is just simply going to be dx dt all right so if we take the time derivative of this equation here you're going to get minus l1 s1 theta 1 dot minus l2 s12 theta 1 dot plus theta 2 dot and then you get minus l three s one two three theta one dot plus theta two dot plus theta three dot and remember we use the shorthand notation c1 is cosine theta 1 and s1 is sine theta 1.

[00:02:14]we can do similar for y y dot is just going to be d y dt in this case we'll get l1 c1 theta 1 dot plus l2 c12 theta 1 dot plus theta 2 dot plus l three c one two three theta one dot plus theta two dot plus theta three dot and then lastly for alpha dot it's just going to d alpha dt and that's just simply theta 1 dot plus theta 2 dot plus theta 3 dot and that is our forward velocity solution given our three joint rates we can figure out the translational velocity and angular velocity of our ender factor now often in robotics we like to put things into matrix form so let's put these three equations into matrix form and see what we get so if we were to do that we'll get x dot y dot and alpha dot it's going to equal a matrix here and i'll just write it out quickly all right and what you see here are all the terms that multi the first column are all the terms that multiply our first joint rate theta one dot all right so that's everything associated with joint one our second column is everything associated with joint two and our last column of this matrix is everything associated with joint 3.

[00:04:24]all right and then this is times our vector of joint rates which is theta 1 dot theta 2 dot and theta 3 dot so that's those three equations on the previous slide put into matrix form this term here all right on the left here that's just simply going to be our velocity of our end effector which is our x dot our vector x in this case this matrix here we're going to call the jacobian all right and each column of the jacobian is uh one joint of the robot and then our last part here this is our vector of joint rates which is q dot so we have v is equal to our jacobian times q dot again where j is the jacobian and it maps our joint rates to our end effector velocity all right if you look at the formulation for j all right you'll see that each column is just the partial derivative of our x vector with respect to that joint and a joint value or joint very well i should say all right and so this jacobian plays an important role in robotics and you'll see how it's used in subsequent videos