 Hello everyone, I am Mr. Ninath Kulkarni working as an assistant professor in department of mechanical engineering at Valtran Institute of Technology, Solapur. In today's session, I will be dealing with the velocity analysis of robots. At the end of the session, the students will be able to relate the joint velocities with the tool point velocities and they will be able to know what is Jacobian matrix. So, in velocity Jacobian of a two or manipulator, we are going to find out how the end effector velocity and joint velocities can be related with the Jacobian matrix. So, end effector velocity that is the velocity of the end effector p x p y can be represented as x dot that means the derivative of x with respect to time and y dot that means derivative of y with respect to time and similarly joint velocities can be represented as theta 1 dot for joint 1 and theta 2 dot for joint 2. Next, from the forward kinematics of two-link planar jointed configuration, we get we know this solution x is equal to a 1 c 1 plus a 2 c 1 2 and y is equal to a 1 s 1 plus a 2 s 1 2. This is the position we have got from forward kinematics of two-link planar jointed arm configuration. If we partially differentiate both the equations with respect to these angles theta 1 and theta 1 plus theta 2, we get x dot can be equal to the derivative of cos is minus sin. So, minus comes here minus a 1 sin theta 1. Similarly, minus a 2 sin theta 1 2 into the partial derivative of theta 1 dot plus theta 2 dot with respect to time. Similarly, it can be expanded as x dot is equal to minus a 1 s 1 and multiplying this bracket with theta 1 dot it becomes this and multiplying this bracket into theta 2 dot it comes to this. So, this is the solution of x dot that means the velocity of the end effector in x direction. Similarly, the solution of y dot can be got as a 1 c 1 plus a 2 c 1 2 theta 1 dot plus a 2 c 1 2 theta 2 dot same process following we can get this solution. Now, representing both these equations that means x dot and y dot in matrix form. So, we get x dot and y dot and whatever the factors of theta 1 dot that can be combined. So, as you can see a 1 c 1 and a 2 c 1 2. So, that can be combined here. Similarly, minus a 2 s 1 2 and a 2 c 1 2 that can be written in this cells and theta 1 dot and theta 2 dot they can be written here. So, this x dot y dot this is the end effector velocity that can be represented as capital V. This matrix 2 by 2 matrix is called as Jacobian matrix for velocity that is represented as J V its formula or matrix is minus a 1 s 1 minus a 2 s 1 2 minus a 2 s 1 2 a 1 c 1 plus a 2 c 1 2 and a 2 c 1 2. So, this is the Jacobian matrix for velocity of a 2 DOF jointed arm configuration. Similarly, this theta 1 dot theta 2 dot is the matrix for velocity angular velocity of joints joint 1 and joint 2 that is represented by omega. Next, suppose if the end effector velocity is given and we are required to find out the joint velocities that means this is given and this we have to calculate then we have to calculate the inverse of this Jacobian matrix denoted by J V inverse where J V inverse can be calculated as 1 upon determinant of J V into adjoint of J V. Now, if the determinant of J V it comes out to be 0 we call the configuration as a singular one that means singular configuration the robot it loses some degrees of freedom. So, to avoid singularity we have to take care of this Jacobian matrix of velocity such that the determinant should not come or anywhere close to 0 that care should be taken. Next, we will solve this problem the lengths of link l 1 that is the length of the link l 1 is 400 mm and this length is 500 mm a 2 and the end effector position is given 500 mm that means x is 500 mm and y is 600 mm. You have to find the angles made by the links that means we have to find out the angle theta 1 and theta 2 if the end effector travels at a speed of 100 mm per second the speed of end effector is given that means x dot is given 100 mm per second and y dot is given 100 mm per second. What speed the joint will rotate we have to calculate the angular speed of joints theta 1 and theta 1 dot and theta 2 dot. So, this is the given question a 1 can be 0.4 meter converting all the values into meter a 2 is 0.5 meter x is 0.5 meter y is 0.6 meter and x dot y dot can be represented in a matrix form 0.1 100 mm per second that is 0.1 meter per second and same 0.1 meter per second. We are required to find out the joint angles that is theta 1 and theta 2 and the joint speeds theta 1 dot and theta 2 dot. So, from inverse kinematics of a 2 R manipulator we know that theta 2 is equal to plus or minus cos inverse of x square plus y square minus a 1 square minus a 2 square divided by 2 a 1 a 2 this is the formula of theta 2 and by putting the values of x y a 1 and a 2 we can get theta 2 as plus or minus 60 degree. If we take any one orientation that is theta 2 is plus 60 degree we can calculate the value of theta 1 that is 16.53 degree by using this formula that we know from inverse kinematics of a 2 R manipulator. So, up to this it is joint angles are calculated. Now, we are interested in calculating the Jacobian matrix for velocity that is J v it is minus a 1 s 1 minus a 2 s 1 2 minus a 2 s 1 2 a 1 c 1 plus a 2 c 1 2 and a 2 c 1 2. So, substituting the values of a 1 as 0.4 and theta 1 as 16.53 a 2 as 0.5 and theta 1 2 that means theta 1 plus theta 2 16.53 plus positive 60. So, substituting these values in this matrix we get this matrix 0.6 minus 0.49 0.5 plus 0.5 plus 0.5 and 0.12. So, this is the 2 by 2 matrix of J v that we have got. After that we are required to find out the J v inverse. So, J v inverse can be calculated as determinant 1 upon determinant of J v into adjoint of J v. So, determinant of J v can be calculated as multiplying the diagonal elements minus subtracting multiplying the rest of diagonal elements. So, minus 0.2 into minus 0.16 minus 0.49 into minus 0.5. So, that comes out to be 0.92. Now, adjoint of J v can be calculated as just exchanging the position of these 2 terms and exchanging the signs of these 2 multiplying these 2 terms by minus signs. So, we got the adjoint matrix as 0.12, 0.49 minus 0.15 and minus 0.6. So, J v inverse can be calculated as this 1 upon adjoint of 1 upon determinant of A into adjoint of A. So, that comes out to be this one. Afterwards we can calculate the value of theta 1 dot and theta 2 dot is equal to J v inverse into x dot and y dot. So, this we know J v inverse that we have calculated and x dot and y dot value are 0.1 and 0.1. So, we calculate we multiply the first row of this matrix into first the column of this matrix that means, 0.13 into 0.1 plus 0.53 into 0.1. This is the first element that we have got that is 0.066. Second element minus 0.54 into 0.1 minus 0.65 into 0.1. This is the second element that we have got minus 0.119. The unit is degree per second. So, theta 1 dot is 0.066 degree per second and theta 2 dot is 0.1 minus 0.119 degrees per second. So, this is the solution that we got the angular velocity of joint 1 is 0.066 degree per second and angular velocity of joint 2 is minus 0.119 degrees per second. That is the solution we have got from this problem. These are the references that are referred. So, thank you.