Differential Kinematics and Statics Ref: 理理理理 理理理理 理理理理 理理理理理理理理 ,,,, 2001
Differential Kinematics and Statics
Ref: 理论力学,洪嘉振,杨长俊,高等教育出版社, 2001
Incremental MotionIncremental 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 (vx, vy, vz) result from velocities at the joints (1, 2, … n)?
ii
t
vx
tv
y
tv
z
txn
yn
zn, , ,
Some DefinitionsSome Definitions
Linear Velocity: The instantaneous rate-of-change in linear position of a point relative to some frame.
v=(vx, vy, vz)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 orientat
ion (Euler Angles, Rotation Matrix, etc.)– Angular Velocity Vector and the Angular Velocity Matrix.
Some DefinitionsSome 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.
Tzx y )( =
x
y
z
Free VectorFree Vector
Linear velocity are insensitive to shifts in origin but are sensitive to orientation.
{D}
DBA vvv
x
x
Free VectorFree Vector
Angular velocity are insensitive to shifts in origin but are sensitive to orientation.
DBA
{A}
{B}
{D}x
x
xx
Velocity FramesVelocity 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.
X0
Y0
x0
y0
0
Y1X1
0
x2
a1
v
vv
v
R
a2
y2
Figure 2.13: Two-Link Planar Robot
X0
Y0
x0
y0
0
0 v
vv
v
End-effector velocity for 1
r0n
n01 r
X0
Y0
x0
y0
0
0 v
vv
v
End-effector velocity for 2
r1n
n12 r
Two-Link Planar RobotTwo-Link Planar Robot
Direct kinematics equation Direct kinematics equation
x x c y s a c a c
y x s y c a s a s
2
0 2
0 2 12 12 2 12 1 1
2 12 12 2 12 1 1
Incremental MotionIncremental Motion taking derivatives of the position taking derivatives of the position
equation w.r.t. time we have equation w.r.t. time we have
note thatnote that
v x a y c a s
v x a ) c y s a c
x 2
y 2
( )( ) ( )
( ( ) ( )
2 2 1 2 12 1 2 12 1 1 1
2 2 1 2 12 1 2 12 1 1 1
s
d(c
dt)s
d(s
dt)c
12
1 2 1212
1 2 12
)( ,
)(
Incremental MotionIncremental Motion
written in the more common matrix written in the more common matrix form, form,
or in terms of incremental motion,or in terms of incremental motion,
v
v =
x a y c a s x a y c
x a )c y s a c x a )c y sx
y
2 2
2 2
( ) ( )
( (2 2 12 12 1 1 2 2 12 12
2 2 12 12 1 2 2 12 12
1
2
s s
x
y =
x a y c a s x a y c
x a )c y s a c x a )c y s2 2
2 2
( ) ( )
( (2 2 12 12 1 1 2 2 12 12
2 2 12 12 1 2 2 12 12
1
2
s s
Differential KinematicsDifferential Kinematics
Find the relationship between the joint velocities and the end-effector linear and angular velocities.
Linear velocity
Angular velocity
i
ii d
q for a revolute joint
for a prismatic joint
Differential KinematicsDifferential Kinematics
Differential kinematics equation
Geometric Jacobian 6Rv
Relationship with T(q)Relationship with T(q)
Direct kinematics equation
Linear velocity
Angular velocity?
)(qpp
?)(qRq
qpJ P
)(
Vector (Cross) ProductVector (Cross) Product
Vector product of x and y
Skew-symmetric matrix
321
321
yyy
xxx
kji
yx
0SS T
Vector (Cross) ProductVector (Cross) Product
Skew-symmetric matrix
Derivative of a Rotation MatrixDerivative of a Rotation Matrix
define
S(t) is skew-symmetric
Interpretation of S(t)Interpretation of S(t)
Interpretation of S(t)Interpretation of S(t)
Given R(t)
Example 3.1: Rotation about ZExample 3.1: Rotation about Z