机器人学导论(英) 第五讲
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Chapter5: Velocities and Static Forces
•Objective : we will examine the linear and angular velocity of a rigid body and analyze the motion of a manipulator.t
t Q t t Q Q dt d V B
B t B Q B
∆−∆+==→∆)()(lim
)(0•The derivative of a vector
•It is important to indicate the frame in which the vector is
differentiated.
))
(()(Q dt
d V B
A
Q B
A =•The velocity vector can be represented in terms of any frame (reference frame)
)
()(Q B
A B
Q B
A V R V =differentiated in {B}
differentiated in {B},written in {A}
•We often consider the velocity of the origin of a frame relative to some non-moving universe reference frame.
CORG
U
c V v ={U} is the reference frame Example 5.1
{T}
{C}
{U}
mph
v C 30=mph
v T 100=?)(CORG U U
P dt
d ?
)(TORG U
C V ?
)(CORG T
C V R
U T
R
U C
•While linear velocity vector describes motion of a point, angular velocity vector describes motion of a body.
•Angular velocity can be utilized to describe rotational motion of a frame.{A}
{B}
B
A
Ω
rotation of {B} relative to {A}
C
U
C Ω=ω{U} is some understood universe frame
{A}
{B}
Q
B
BORG
A
P R
A B
•Linear Velocity: we assume that the orientation
is not changing with time , the motion of point P with respect to
{A} is due to or changing in time.R A
B
BORG
A
P Q B )
)((Q B
A B
BORG A
Q A
V R V V +=
{B}
•Rotational Velocity
We consider two frame with : (1) coincident origin; (2) zero linear relative velocity
=Q B
V ))
(()(Q R V R V B
A B
B A
Q B
A B
Q A
×Ω+={A}
B
A
ΩQ
B
Q
V A
B A Q A
×Ω=0
≠Q B
V
•Simultaneous linear and rotational velocity
))
(()(Q R V R V V B
A B
B A
Q B A B
BORG A
Q A
×Ω++=•Time-derivative of rotation matrix R
S
R R =−1 skew-symmetric
matrix
n
T
S S 0=+•Vector cross-product and skew-symmetric matrix
T z y
x
T
z y x q q q Q p p p P ]
[][==Q p p p p p p Q P x
y x z
y z ⎥⎥⎥⎦
⎤⎢⎢⎢
⎣
⎡−−−=×000)
(P S
•At any instant, each link of a manipulator in motion has some linear and angular velocity.
•We can calculate the velocity of each link in order, starting
from the base. The velocity of link i+1 will be that of link i, plus the new velocity components added by joint i+1, this is called as “velocity propagation”.
Axis i
link i
i
Z ˆi
X ˆi
i
ωi i v i
Y ˆ