Transcript 下載/瀏覽Download

Fundamentals of Robotics
Linking perception to action
2. Motion of Rigid Bodies
南台科技大學
電機工程系
謝銘原
1
Chapter 2. Motion of Rigid Bodies

2.2 Cartesian Coordinate Systems

2.3 Projective Coordinate Systems

2.4 Translational Motions

2.5 Rotational Motions

2.6 Composite Motions






Homogeneous Transformation
Differential Homogeneous Transformation
Successive Elementary Transformation
Successive Elementary Rotations
Euler Angles
Equivalent Axis and Angle of Rotation
2
2.2 Cartesian Coordinate Systems

Two references for describing the motions of a rigid
body –



The time reference (for velocity and acceleration, t )
The spatial reference (for position and orientation, X, Y, Z )
Cartesian Coordinate System –
O0 – X0Y0Z0
rx i y jz k R r
1 0 0  x 
  0 1 0   y 
0 0 1 z 

 
3
2.3 Projective Coordinate Systems

Geometric operations


Typical projection –



Translation, Rotation, Scaling, Projection
Display of 3D objects onto a 2D screen
Visual perception (onto an image plane)
Projective coordinates  X , Y , Z , k 
t
(k – homogeneous coordinate)
P   x, y , z 
t
Q   kx, ky, kz 
t
4
2.4 Translational Motions

2.4.1 Linear Displacement
O1 (t2 )  O1 (t1 )  T
 O1 (t1 )  (t x , t y , tz )t
O1  1O1  0T1
0

The relative distance (displacement)
 Between the origin of frame 0 and
the origin of frame 1
0
P (t )  P (t )  T1
1
0
0
T1
( x0 , y0 , z0 )t  ( x1 , y1 , z1 ) t  (t x , t y , t z )
5
Translational transformation

The Cartesian coordinates
0
P (t )  1P (t )  0T1
( x0 , y0 , z0 )t  ( x1 , y1 , z1 ) t  (t x , t y , t z )

The equivalent projective coordinates
 x0   1
 y  0
 0 
 z0   0
  
 1  0
0
0 0 t x   x1 
1 0 t y   y1 
 
0 1 t z   z1 
 
0 0 1  1 
0
T1
P   0 M 1 1P
Homogeneous motion transformation matrix
6
2.4.2 Linear Velocity and Acceleration

The displacement vector

The corresponding linear velocity and acceleration
7
2.5 Rotational Motion

Rotational motion –



The rotation of a rigid body about any straight line (a rotation axis)
Force, or torque
2.5.1 Circular displacement
8
Rotation matrix

Rotation matrix, R

Initial configuration

After rotation
– inverse
9
Example 2.4

Rotational transformation

from frame1 to frame0

Inverse (from frame0 to frame1)
10
The homogeneous motion transformation matrix

Rotational motion

Translational motion
 x0   1
 y  0
 0 
 z0   0
  
 1  0
0
0 0 t x   x1 
1 0 t y   y1 
 
0 1 t z   z1 
 
0 0 1  1 
P   0 M 1 1P
11
2.5.2 Circular Velocity and Acceleration

Circular velocity


caused by a rigid body’s angular velocity
Circular acceleration

Vector
is parallel to the rotation axis and its norm is equal to
at
tangential acceleration vector
an
centrifugal acceleration vector
12
2.6 Composite Motions

Any complex motion can be treated as the combination of
translational and rotational motions.
13
2.6.1 Homogeneous Transformation

Homogeneous Transformation
with
14
2.6.2 Differential Homogeneous Transformation

Homogeneous Transformation

By differentiating,
15
In a matrix form

Homogeneous Transformation

Differential Homogeneous Transformation
16
2.6.3 Successive Elementary Translations

Three successive translations
17
2.6.4 Successive Elementary Rotations

Three successive rotations of frame 1 to frame 0
18
The equivalent projective coordinates

In robotics, the above equation describes the forward kinematics of a
spherical joint having three degrees of freedom.
19
Useful expression

Imagine now that qx, qy, and qz can undergo instantaneous variations
with respect to time.

The property of skew-symmetric matrix

We can derive the following equalities:
20
2.6.5 Euler Angles

The three successive rotations

1st elementary rotation can choose X, Y, or Z axes as its rotation
axis.
2nd, 3rd, both have two axes to choose from.
Total, 3*2*2 = 12 possible combinations

These sets are commonly called Euler Angles.


21
Example 2.5 ZYZ Euler Angles




1st – about Z axis,
rotation angle a
2nd – about Y axis,
rotation angle b
3rd – about Z axis,
rotation angle f
a, b, f are called
ZYZ Euler Angles
22
2.6.6 Equivalent Axis and Angle of rotation

Euler angles



The set of minimum angles which fully determine a frame’s
rotation matrix with respect to another frame.
Each set has three angles which make three successive elementary
rotations.
Thus,
the orientation of a frame, with respect to another frame,
depends on three independent motion parameters
even through the rotation matrix is a 3*3 matrix with 9 elements
inside.
23
In robotics

It is necessary
to interpolate the orientation of a frame from its initial orientation to
an actual orientation
so that
a physical rigid body (e.g. the end-effector)
can smoothly execute the rotational motion in real space.
24
Equivalent axis of rotation

Now, imagine that there is am intermediate frame i which has the Z
axis that coincides with the equivalent rotation axis r.
25
Rotation

If r is the equivalent axis of rotation for the rotational motion between
frame 1 and frame 0:

The orientation of frame i with respect to frame 0,
26
The solutions for r and q

To derive,

If q = 0, Rz = I 3*3
given
27