Download - Motion Kinematics – Lecture Series 3
Motion Kinematics – Lecture Series 3ME 4135 – Fall 2011R. Lindeke
Outline Of Motion KinematicsRigid Body Motions
◦ Includes rotation as well as translationsThe Full blow Homogenous
Transformation Matrix◦Coupling origin movement with reorientation◦Physical Definition◦Making Use of its power◦Building its Inverse
Compound HTM’s is Rigid MotionScrew Coordinates
Rigid Motion
The Body Frame (B) has been coincidently displaced by a vector d and reoriented about ZG, XG and zi
axes
Accounting for this overall change:
where:
P is a point on the body
R is a rotational effect 3x3 Matrix
; ;
G G B GP B P
P P oG G G
P P P P o
P P o
r R r d
X x Xr Y r y d Y
Z z Z
An Example: Find the global position of a body point: [.5, 1.25, 3]T if
the Body frame has been subjected to the following ‘operations’. A Rotation about ZG of 30˚ followed by Rotation XG of 45˚ and a translation of [7,4,-10]T
,45 ,30
0.5 71.25 4
3 10
3 1 02 2 0.5 762 2 1.25 44 4 2
3 1062 24 4 2
6.80802.82096.9364
G G GP X Z
GP
GP
r Q Q
r
r
Like this one with some extras
(As found in MathCad:)
Trying Another – A Rotational /Translational Device
Initially (a) B and G are coincident – in (b) the Device has been rotated and then the upper arm has been extended, and note that B has been translated and rotated in this second image
Accounting for these – Where is Pi in G space for both cases
CASE 1: P1 defined wrt the origin
Case 1.5: After Rotation (45˚) about ZG
1 1
13500
900
G BP Pr r
1.5 1
1.5
(45 ) (45 ) 0 1350 0(45 ) (45 ) 0 0 00 0 1 900 0
954.6954.6900
G G B GP B P
GP
r R r d
Cos SinSin Cos
r
And Finally: After an Elongation of 600 in the xB direction:
2 1.5
2
1.5 ,
954.6 600 (45 ) 1378.9954.6 600 (45 ) 1378.9900 0 900
G G GP P x B
GP
r r d
Cosr Sin
Where Gdx,B1.5 is the motion of the elongation axis of the
“Upper Arm” resolved to the Ground Space
Wouldn’t it be Nice if …Combining Rotational and Translational Effects into a Grand Transformation could be done
This is the role of the Homogenous Transformation Matrix
It includes a “Rotational Submatrix” a “Origin Translational Vector” a “Perspective Vector” and finally a “Spatial Scaling Factor”
11 12 13
21 22 23
31 32 33
0 0 0 1
X
YMN
Z
R R R dR R R d
TR R R d
Lets see how it can be used in the two jointed robot Example
1.5
2 2 1 1.5 2 1
1.5 1.52
1.5 1.5 2
2
1.5 1.5
1.5
2
13500
0 1 0 1 900
Here:(45 ) (45 ) 0 0(45 ) (45 ) 0 & 00 0 1 0
GB BG G B BP B P B B P
B BBGG B B BG
P
G GB B
BB
r T r T T r
R d R dr
Cos SinR Sin Cos d
R
1.5
2
1 0 0 6000 1 0 & 00 0 1 0
BBd
Dropping into MathCad:
And Note: To use the original positional vector we needed to append a scaling factor to it as seen here Thus the position
of P2 in the Ground space is this vector: [1378.9,1378.9,900] just as we found earlier
What’s NextEquipped with the ideas of the HTM and
individual effects “easily” separated we should be able to address multi-linked machines – like robots
But, before we dive in let’s examine some other Motion Kinematic tools before we!◦Axis Angle Rotation and Translation◦Inverse Transformations◦Screw Motions – see the text, they are a
general extension of Axis Angle Rot/trans motion
Turning about a body axis – Developing the Rodriguez Transformation sub-matrix
We’ll consider rotation about and translation along a Vector u
Developing an HTM
1. Develop the unit vector in the direction of u
2. Develop the Rodriguez Rotation Matrix
2 2 2
ˆ,
0.5771,1,1 0.577
1 1 1 0.577
Where: is the magnitude of the vector u
I is an identity matrixvers is versin = 1- cos
is the skew symetr
ˆ ˆ
ic mat
cos si
ˆ
n
r
, ,
i
u u u
TuR I u u vers u
X Y Zu
u
u
u
ˆ,
x of the (unit) vector uand is:
0 .577 .577.577 0 .577.577 .577 0
1 0 00 1 0 cos(45 )0 0 1
0.5770.577 0.577 0.577 0.577 (1 cos(45 )0.577
0 .577 .577.577 0 .577.577 .577 0
uR
sin(45 )
Building Rodrigues Matix (MathCad)
Continuing with HTM3. The
Translational Vector:
4. The Transform:
ˆ ,
111
0.805 .311 0.506 10.506 0.805 0.311 10.311 0.506 0.805 10 1
0 0 0 1
G
GG u
B
d
R dT
The HTM in Use: (MathCad)
What of the Inverse of the HTM?It is somewhat like
the Inverse of the orientation matrix
The Rotational sub-matrix is just the transpose (since we are reversing the point of view when doing an inverse)
The positional vector changes to:
in Foward Sense:
in Inverse sense:
B
B
B
G
G
G
o
Go
o
GO
B GO
GO
x
d y
z
X n dd Y o d
a dZ
Leading to:11 12 13
21 22 23
31 32 33
11 21 31
1 12 22 32
13 23 33
Given:
0 0 0 1The Inverse is then
0
0 0
seen as
10 1
:
B
B
B
O
OGB
O
G G
G T G GBG B
B G G G
R R R x
R R R yT
R R R z
n d R R R n dR o d R R R o d
T Ta d R R R a d
11 12 13
21 22 23
31 32 33
11 21 31
& (or s) &
Similarly for & B B B
GO O O
G G
WhereR R R
n R o R a RR R R
Finally
n d R x R y R z
o d a d
Note these are DOT
Products of 2 vectors – or scalars!
SummaryThe Homogeneous Transformation Matrix
is a general purpose operator that accounts for operations (rotations and translations) taking place between Ground and Remote Frames of reference
As such, they allow us to relate geometries between these spaces and actually perform the operations themselves (mathematically)
Finally, they can be studied to understand the relationships (orientation and position) of two like geometried – SO3 – coordinate frames
SummaryTheir Inverses are simply
constructed since they represent the geometry of the Ground in the geometry defined in the Remote Frames space
Thus they are powerful tools to study the effects of motion in simple situations, complex single spaced twisting /translating motion as well as multi-variable motion as is seen in robotics