Jacobians Read Chapter 5What is Jacobian?• Study the notions of linear and angular velocities of a rigid body• Analyze the motion (linear and angular motions) of a robot manipulator hand (leg foot)• We will use a matrix called Jacobian to relate the joint velocities to the linear and angular velocities of the hand or foot• The same matrix can also be used to study the relationship between the static force acting on the hand (foot) and the joint torquesVelocity of a position vector• The positon of a point can be expressed as a vector in a frame {B}: • The velocity of the position vector can be expressed as  lim∆→∆   ∆• When the velocity is expressed in reference frame {A} • The velocity of the origin of frame {B} in frame {A}{A}{B}Angular velocity vector{B}{A}• The angular velocity vector describes the rotation of frame {B}relative to frame {A}:• A rigid body such as a robotic link could have both linear and rotational motions in the space; we should have ways to describe them• Consider the linear velocity of the point vector due to linear motions of the frame and the point (function of joint angles):• Linear velocity due to rotation of the frame:(also function of joint angles)Linear and rotational velocity of a rigid body {B}{A}{B}   When the frame has both linear and rotational motions• When frame {B} has both linear and rotational motions with respect to frame {A}, the velocity of the position vector can be expressed as• The above is the general result of the velocity of a position vector in the moving frame with respect to a stationary frame; clearly it is a function of - joint angles (offsets).     Velocity propagation from link to link in a robot• The motions of link n of a robot manipulator is affected by all the previous links: j = 1, 2, …n-1. The question is how the link motions are propagated through to link n, or any link? • Rotational motion:orNote:󰇛󰇗󰇜  󰇛󰇗󰇜󰆹 󰇛󰇗󰇜󰆹00󰇗 Inertia tensor examples (1) • Linear motionwhere is the vector from the origin of frame  to that of frame 1expressed in frame • Finally two most important equations:  󰇛  󰇜󰇛󰇗󰇜󰆹 - Rotational propagation- Linear propagationJacobian Matrix• In general if we have:where Y and X are vectors: • Take the derivative of both sides, one may have:• Or we may have:where could be called Cartesian velocity and the vector of joint velocities (X is the vector of joint angles).󰇛󰇜 󰇛󰇜:   1  :   1󰇛󰇜 󰇛󰇜is called Jacobian matrix. What is the dimension of J?󰇗 󰇛󰇜󰇗󰇗󰇗Force and torque relation• Work performed by the joint torques (in the joint coordinates):• Work performed by the hand (in the Cartesian coordinates)• The two should be equal• As a result: ∙    ∙    =  = - Static force and torque