MT411 Robotic Engineering Chapter 10 Velocity Kinematics – The Jacobian Narong Aphiratsakun, D.Eng Asian Institution of Technology (AIT)

Velocity Kinematic To follow any prescript velocity, it is important to know the relationship between the velocity of the tool and the joint velocity. So, we are going to find velocity relationship relating the linear and angular velocities of the end effector to the joint velocities. The velocity relationships are determined by the Jacobian of the function. The jacobian is a matrix that generalizes the notions of the ordinary derivative of the scalar function.

Velocity Kinematics: 2 links manipulator Differentiate both side, (prove is on next page)

Velocity Kinematics: 2 links manipulator

Velocity Kinematics: 2 links manipulator Put in vector form Rewrite as Tool velocity Joints velocity

Velocity Kinematics: 2 links manipulator **J is called Jacobian of the manipulator

Velocity Kinematics: Tool and Joint Velocity Joints velocity Tool velocity In this chapter, we are going to find Jacobian matrix of the manipulator. Tool velocity Joints velocity

Derivation of the Jacobian As robot moves about, both joint variables and end effector position, d, and orientation, R, will be function of time. The objective of this section is to relate the linear and angular velocity of the end effector to the vector of joint velocities . Rewrite d position as o position in function of time as:

Derivation of the Jacobian Linear velocity and Angular velocity Joint velocity Body velocity Manipulator Jacobian is a 6xn matrix where n is the number of link

Angular and Linear Velocity Angular velocity: k is a unit vector in the direction of the axis of rotation. is a time derivative of the . Linear velocity: r is a vector from the origin to the point.

Combining Linear and Angular Velocity Linear velocity Jacobian : Angular velocity Jacobian :

Angular Velocity Angular velocity for different frames: denotes the angular velocity of frame 2 that corresponds to the changing , express relative to the coordinate system frame 1. Thus the product of expresses this angular velocity relative to the coordinate system frame 0. Angular velocity can be obtained from:

Linear Velocity Linear velocity: This is linear velocity of the end effector that would result if were equal to one and the other were zero.

Jacobian Analysis : 2 links manipulator Tool velocity joint velocity 3 linear + 3 Revolute velocity (x,y,z)

Jacobian Analysis : 2 links manipulator

Jacobian Analysis : 2 links manipulator 2 links, n =2:

Jacobian Analysis : 2 links manipulator

Jacobian Analysis : 2 links manipulator

Jacobian Analysis : RRP (SCARA) Tool velocity joint velocity 3 linear + 3 Revolute velocity (x,y,z)

Jacobian Analysis : RRP (SCARA) 3 links, n =3:

Jacobian Analysis : RRP (SCARA)

Jacobian Analysis : RRP (SCARA)

Velocity Kinematics: Further Work Joints velocity Tool velocity The joint velocities are found from the end-effector velocities via inverse Jacobian. Our next problem is to determine what the joint velocity needed for the desired end effector velocity. Tool velocity Joints velocity

Tool Velocity It is necessary to relate the velocity of the tool frame to the velocity of the end-effector frame. Assume frame 6 is end-effector, therefore Prove of this equation can be read from Robot Modeling and Control, M. W. Spong, S. Hutchinson, M. Vidyasagar pg. 138-139