Module 2-24: Velocity and Force Transformations

Added: 2026-05-25

[00:00:00]hi i'm scott knockley in this video we're going to look at velocity and force transformations so by the end of this video you should be able to implement velocity and force transformations so we previously saw how we can find the jacobian for our robot manipulators we had two approaches we could use joint directions and derivatives and this was the formulation as well we could use joint directions and cross products and this was the formulation now when we did this we could choose a point on our end effector and then we had a reference frame now that point that we used to calculate so in the case of joint directions and derivatives we use the point from the origin of the base frame to the order of the end effector or for the cross product method we use the point from the origin of frame i to the original vector it could be any point on our end effector now some points make derivations and computational complexities simpler and more efficient so if we choose this point appropriately on our end effector we can actually make the calculation of our jacobian simpler now let's look at an example that suppose we take a point attached to our end effector and make it coincident with the center of our spherical wrist all right so it's a point on the end effector but it's located at our wrist center so our jacobian of our end effector is according to the jacobian of the wrist all right and if you were to work out that jacobian you would get a jacobian of this form where you have this matrix of three by three matrix here of zeros so this would make calculation of your your jacobian a lot simpler because you have a you know a three by three matrix of zeros here and things like calculation of our singular configurations which we're going to talk more about in a subsequent video but if we wanted to calculate the singular configurations we need to take the determinant of this matrix so we're to take the determinant of jw well because of that formulation it would just simply be the determinant of a times the determinant of c all right so that would make calculating the singular configurations easier our velocity would be of the wrist center now times our jacobian times our vector of joint rates q dot where vw is our translational velocity of our wrist and our angular velocity of our wrist so we can make a note vw is the translational velocity at a point coincident with the wrist center okay so now we could use this formulation now often we have the velocity of our end effector at say the tool tip that we want to move that so we need a way to actually convert our velocity at the tooltip to the be the velocity at say the wrist center all right and to do this we need a velocity transformation so suppose we have points a and b and two respective reference frames with orientation c and d so suppose we have a rigid body here and we have frame c over here and suppose we have frame d over here we have two points we have point b and we have point a and our rigid body has angular velocity omega and say velocity a here and suppose we know this vector r b to a so let's write what's known is the velocity of a with respect to frame c which is the translational velocity and the angular velocity and what we're trying to find is the velocity at point b with respect to frame d so that's the translational velocity of point b and the angular velocity with respect to frame d all right so our angular velocity let's deal with that first because it's simpler the angular velocity of d is just simply going to be the rotation matrix describing frame c with respect to frame d's orientation times omega c okay now the velocity the translational velocity of b with respect to d it's going to be the velocity of a with respect to c plus the angular velocity omega with respect to c cross our position vector r a to b with respect to c and then this all has to be multiplied by r c with respect to d so we can write this out okay and now this cross product okay we can actually multiply in by the r and then we can flip the direction of the cross product so we can change the direction of this vector all right and so if we do that we're going to have d frame d b to a that's r cross dcr our angular velocity all right and so that's the translational component so we can then write this as a matrix equation where we now have a velocity transform so t subscript v will be our velocity transform a to b with respect to c to d so in essence all we're doing is putting these two equations into a matrix form here and now obviously we have to know what this t is so that's our velocity transform it has this form here and then in the upper corner here we have r tilde which is a skew symmetric matrix that allows us to do that cross product so i'm just going to write that over here r tilde b to a is just simply the skew symmetric matrix of this form all right with rb to a just simply rx ry and rz all right so this is a skew symmetric matrix that allows us to do the cross product all right and so this is a six by six matrix that converts our velocity from one point to another point and from one frame of reference to another frame of reference all right similarly we can do a force transformation and so for the force transform what would be known is our wrench at point a with respect to frame c so it's our trans our force and our moment and what we're trying to find is our force at frame or sorry a wrench at frame point b with respect to frame d so our wrench at b with respect to frame d is going to be a force transformation now so we use a subscript f for force transformation c to d a to b here all right where the force transformation has a similar form as the velocity transform but it's a little different our skew-symmetric matrix is in this corner now just by the way we defined our wrench all right and again this is a six by six matrix so don't get confused with the force transformation and the velocity transformation with the t we use for our homogeneous transforms all right so this will convert the force or sorry our ranch here that's a this should be lowercase a sorry a at point a with respect to frame c to point b with respect to frame d all right and similar we had our velocity transformation all right and so with that we can then use our jacobian with respect to any point on our end effector you