Differential Kinematics and Statics Ref: 理论力学，洪嘉振，杨长俊，高 等教育出版社， 2001

Incremental Motion  What small (incremental) motions at the end- effector (  x,  y,  z) result from small motions of the joints (  1,  2, …,  n )?  Alternatively, what velocities at the end- effector (v x, v y, v z ) result from velocities at the joints (  1,  2, …  n )?

Some Definitions  Linear Velocity: The instantaneous rate-of- change in linear position of a point relative to some frame. v=(v x, v y, v z ) T  Angular Velocity: The instantaneous rate-of-change in the orientation of one frame relative to another. –Angular Velocity depends on the way to represent orientation (Euler Angles, Rotation Matrix, etc.) –Angular Velocity Vector and the Angular Velocity Matrix.

Some Definitions  Angular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to another and whose magnitude is the rate of rotation about that axis.

Free Vector  Linear velocity are insensitive to shifts in origin but are sensitive to orientation. {D} x x

Free Vector  Angular velocity are insensitive to shifts in origin but are sensitive to orientation. {A} {B} {D} x x x x

Velocity Frames  frame of reference: this is the frame used to measure the object’s velocity  frame of representation.: this is the frame in which the velocity is expressed.

X0X0 Y0Y0 x0x0 y0y0 0  Y1Y1 X1X1 0 x2x2 a1a1 v v v v  R a2a2 y2y2 Figure 2.13: Two-Link Planar Robot

X0X0 Y0Y0 x0x0 y0y0 0  0 v v v v End-effector velocity for  1 r 0n

X0X0 Y0Y0 x0x0 y0y0 0  0 v v v v End-effector velocity for  2 r 1n

Two-Link Planar Robot  Direct kinematics equation

Incremental Motion  taking derivatives of the position equation w.r.t. time we have  note that

Incremental Motion  written in the more common matrix form,  or in terms of incremental motion,

Differential Kinematics  Find the relationship between the joint velocities and the end-effector linear and angular velocities. Linear velocity Angular velocity for a revolute joint for a prismatic joint

Differential Kinematics  Differential kinematics equation  Geometric Jacobian

Relationship with T(q)  Direct kinematics equation  Linear velocity  Angular velocity?

Vector (Cross) Product  Vector product of x and y  Skew-symmetric matrix

Vector (Cross) Product  Skew-symmetric matrix

Derivative of a Rotation Matrix define S(t) is skew-symmetric

Interpretation of S(t)

Given R(t)

Example 3.1: Rotation about Z