Transcript An Introduction to Robot Kinematics Renata Melamud Kinematics studies the motion of bodies.

An Introduction to
Robot Kinematics
Renata Melamud
Kinematics studies the motion of bodies
An Example - The PUMA 560
2
3
4
1
There are two more
joints on the end
effector (the gripper)
The PUMA 560 has SIX revolute joints
A revolute joint has ONE degree of freedom ( 1 DOF) that is
defined by its angle
Other basic joints
Revolute Joint
1 DOF ( Variable - )
Prismatic Joint
1 DOF (linear) (Variables - d)
Spherical Joint
3 DOF ( Variables - 1, 2, 3)
We are interested in two kinematics topics
Forward Kinematics (angles to position)
What you are given:
The length of each link
The angle of each joint
What you can find:
The position of any point
(i.e. it’s (x, y, z) coordinates
Inverse Kinematics (position to angles)
What you are given:
The length of each link
The position of some point on the robot
What you can find:
The angles of each joint needed to obtain
that position
Quick Math Review
Dot Product:
a x 
a 
 y
Geometric Representation:
A  B  A B cosθ
A
b x 
b 
 y
θ
B
Matrix Representation:
a x  b x 
A  B        a xb x  a y b y
a y  b y 
Unit Vector
Vector in the direction of a chosen vector but whose magnitude is 1.
B
uB 
B
B
uB
Quick Matrix Review
Matrix Multiplication:
An (m x n) matrix A and an (n x p) matrix B, can be multiplied since
the number of columns of A is equal to the number of rows of B.
Non-Commutative Multiplication
AB is NOT equal to BA
a b   e
c d    g

 
f  ae  bg


h  ce  dg
af  bh
cf  dh
Matrix Addition:
a b   e
c d    g

 
f   a  e


h  c  g 
b  f 
d  h
Basic Transformations
Moving Between Coordinate Frames
Translation Along the X-Axis
Y
O
(VN,VO)
VO
P
VN
X
Px
Px = distance between the XY and NO coordinate planes
Notation:
V
XY
V X 
  Y
V 
V
NO
V N 
  O
V 
Px 
P 
0
N
Writing V XY in terms of V NO
Y
O
VO
P
X
V XY

VN
PX  V N 
NO


P

V

O
V


N
Translation along the X-Axis and Y-Axis
O
Y
VO
VN
X
V XY  P V NO
P
PX  V N 

O
PY  V 
XY
 Px 
 
 PY 
N
Using Basis Vectors
Basis vectors are unit vectors that point along a coordinate axis
O
n
o
Unit vector along the N-Axis
Unit vector along the N-Axis
V NO
Magnitude of the VNO vector
VO
o
n
V NO
VN
N
NO
V NO cosθ  V NO  n 
V N   V cosθ 
  O    NO
   NO
   NO 
V   V sinθ   V cos(90 θ)  V  o 
Y
Rotation (around the Z-Axis)
Y
X
Z
V
VY

VX
X
 = Angle of rotation between the XY and NO coordinate axis
V XY
V X 
  Y
V 
V NO
V N 
  O
V 
x
Y
Unit vector along X-Axis
V
Can be considered with respect to
the XY coordinates or NO coordinates
V
V XY
 V NO
VY


X
VX
V X  V XY cos α  V NO cos α  V NO  x
V  (V  n  V  o)  x
X
N
O
(Substituting for VNO using the N and O
components of the vector)
V X  V N ( x  n )  V O ( x  o)
 V N (cosθ)  V O (cos(θ  90))
 V N (cosθ)  V O (si nθ)
Similarly….
V Y  V NO sinα  V NO cos(90  α)  V NO  y
V Y  (VN  n  VO  o)  y
V Y  V N (y  n)  VO (y  o)
 V N (cos(90 θ))  V O (cosθ)
 V N (si nθ)  V O (cosθ)
So….
V  V (cosθ)  V (sinθ)
V Y  V N (sinθ)  VO (cosθ)
X
N
O
V XY
V X 
  Y
V 
Written in Matrix Form
V
XY
V X  cosθ  sinθ V N 
  Y  
 O 
sinθ
cosθ
V
 V 
  
Rotation Matrix about the z-axis
Y1
(VN,VO)
Y0
VNO
VXY

P
X1
Translation along P followed by rotation by 
X0
V
XY
V X  Px  cosθ
  Y     
V  Py   si nθ
 si nθ V N 


cosθ  V O 
(Note : Px, Py are relative to the original coordinate frame. Translation followed by
rotation is different than rotation followed by translation.)
In other words, knowing the coordinates of a point (VN,VO) in some coordinate
frame (NO) you can find the position of that point relative to your original
coordinate frame (X0Y0).
HOMOGENEOUS REPRESENTATION
Putting it all into a Matrix
V
XY
V X  Px  cosθ
  Y     
V  Py   si nθ
 si nθ V N 


cosθ  V O 
V X   Px  cosθ


 V Y   Py    si nθ
 1   1   0


 si nθ
V X  cosθ


 V Y    si nθ
 1   0


cosθ
H   si nθ
 0
 si nθ
cosθ
0
cosθ
0
 si nθ
cosθ
0
Px 
Py 
1 
What we found by doing a
translation and a rotation
0  V N 


0 V O 
1  1 
Px  V N 


Py  V O 
1   1 
Padding with 0’s and 1’s
Simplifying into a matrix form
Homogenous Matrix for a Translation in
XY plane, followed by a Rotation around
the z-axis
Rotation Matrices in 3D – OK,lets return from
homogenous repn
 si nθ
cosθ
R z   si nθ
 0
cosθ
0
 cosθ
R y   0
 si nθ
1
R z  0
0
0
cosθ
si nθ
0
1
0
0
0
1 
si nθ 
0 
cosθ 

 si nθ 
cosθ 
Rotation around the Z-Axis
Rotation around the Y-Axis
0
Rotation around the X-Axis
Homogeneous Matrices in 3D
H is a 4x4 matrix that can describe a translation, rotation, or both in one matrix
O
Y
N
P
X
A
Translation without rotation
1
0
H
0

0
0
1
0
0
n x
n
H y
n z

0
ox
oy
oz
0
0 Px 
0 Py 
1 Pz 

0 1
Z
Y
O
N
X
Z
Rotation without translation
A
ax
ay
az
0
0
0
0

1
Rotation part:
Could be rotation around z-axis,
x-axis, y-axis or a combination of
the three.
Homogeneous Continued….
V
V XY
XY
V N 
 O
V 

H A
V 
 
 1 
n x
n
 y
n z

0
The (n,o,a) position of a point relative to the
current coordinate frame you are in.
ox
ax
oy
oz
ay
az
0
0
Px   V N 
 O

Py  V 
Pz  V A 
 
1   1 
V X  nx V N  ox VO  ax V A  Px
The rotation and translation part can be combined into a single homogeneous
matrix IF and ONLY IF both are relative to the same coordinate frame.
Finding the Homogeneous Matrix
EX.
W N 
 O
W 
W A 


Y
T
P
X
Z
W X 
 Y  Point relative to the
 W  X-Y-Z frame
W Z 


 W   Pi   n i
 J   
 W    Pj    n j
 W K  Pk  n k


I
oi
oj
ok
 WI 
 J  Point relative to the
 W  I-J-K frame
W K 


ai  W 


a j   W O 
a k   W A 
N
 WI   ni
 J 
 W   n j
 W K  n k

 
 1   0
W N 
 O  Point relative to the
 W  N-O-A frame
W A 


oi
ai
oj
ok
aj
ak
0
0
Pi   W N 


Pj   W O 
Pk   W A 


1   1 
W N 
 O
W 
W A 


Y
T
P
X
Z
 W  Tx  i x
 Y   
 W   Ty   i y
 W Z  Tz  i z


X
W 


Substituting for  W J 
W K 


I
jx
jy
jz
 W X  i x
 Y 
 W   i y
 W Z  i z

 
1

  0
k x  W 


k y   W J 
k z   W k 
I
 W X  i x
 Y 
 W   i y
 W Z  i z

 
 1   0
jx
kx
jy
ky
jz
kz
0
0
Tx   n i
Ty   n j
Tz  n k

1  0
Tx   W I 


Ty   W J 
Tz   W K 


1   1 
jx
kx
jy
ky
jz
kz
0
0
oi
ai
oj
aj
ok
ak
0
0
Pi   W N 


Pj   W O 
Pk   W A 


1   1 
W X 
W N 
 Y
 O
W

  H W 
W Z 
W A 




 1 
 1 
i x
i
H y
i z

0
jx
jy
jz
0
kx
ky
kz
0
Tx   n i
Ty   n j
Tz  n k

1  0
oi
oj
ok
0
ai
aj
ak
0
Pi 
Pj 
Pk 

1
Product of the two matrices
Notice that H can also be written as:
1
0
H
0

0
0
1
0
0
0 Tx  i x
0 Ty  i y
1 Tz  i z

0 1  0
jx
jy
jz
0
kx
ky
kz
0
0 1
0 0
0  0

1  0
0
1
0
0
0 Pi   n i
0 Pj   n j
1 Pk  n k

0 1  0
oi
oj
ok
0
ai
aj
ak
0
0
0
0

1
H = (Translation relative to the XYZ frame) * (Rotation relative to the XYZ frame)
* (Translation relative to the IJK frame) * (Rotation relative to the IJK frame)
The Homogeneous Matrix is a concatenation of numerous
translations and rotations
W N 
 O
W 
W A 


Y
T
X
P
Z
One more variation on finding H:
H=
(Rotate so that the X-axis is aligned with T)
* ( Translate along the new t-axis by || T || (magnitude of T))
* ( Rotate so that the t-axis is aligned with P)
* ( Translate along the p-axis by || P || )
* ( Rotate so that the p-axis is aligned with the O-axis)
This method might seem a bit confusing, but it’s actually an easier way to
solve our problem given the information we have. Here is an example…
Forward Kinematics
The Situation:
You have a robotic arm that
starts out aligned with the xo-axis.
You tell the first link to move by 1
and the second link to move by 2.
The Quest:
What is the position of the
end of the robotic arm?
Solution:
1. Geometric Approach
This might be the easiest solution for the simple situation. However,
notice that the angles are measured relative to the direction of the previous
link. (The first link is the exception. The angle is measured relative to it’s
initial position.) For robots with more links and whose arm extends into 3
dimensions the geometry gets much more tedious.
2. Algebraic Approach
Involves coordinate transformations.
Example Problem:
You are have a three link arm that starts out aligned in the x-axis.
Each link has lengths l1, l2, l3, respectively. You tell the first one to move by 1
, and so on as the diagram suggests. Find the Homogeneous matrix to get the
position of the yellow dot in the X0Y0 frame.
Y3
3
Y2
2
2
X3
3
X2
H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 )
Y0
1
1
X0
i.e. Rotating by 1 will put you in the X1Y1 frame.
Translate in the along the X1 axis by l1.
Rotating by 2 will put you in the X2Y2 frame.
and so on until you are in the X3Y3 frame.
The position of the yellow dot relative to the X3Y3 frame is
(l1, 0). Multiplying H by that position vector will give you the
coordinates of the yellow point relative the the X0Y0 frame.
Slight variation on the last solution:
Make the yellow dot the origin of a new coordinate X4Y4 frame
Y3
Y4
3
Y2
2
2
X3
3
X2
X4
H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) * Tx3(l3)
Y0
1
This takes you from the X0Y0 frame to the X4Y4 frame.
1
X0
X
0 
Y 
0 
   H 
Z
0 
 
 
1
 
1 
The position of the yellow dot relative to the X4Y4 frame
is (0,0).
Notice that multiplying by the (0,0,0,1) vector will
equal the last column of the H matrix.
More on Forward Kinematics…
Denavit - Hartenberg Parameters
Denavit-Hartenberg Notation
Z(i - 1)
Y(i -1)
Yi
Zi
Xi
a(i - 1 )
X(i -1)
ai
di
i
( i - 1)
IDEA: Each joint is assigned a coordinate frame. Using the DenavitHartenberg notation, you need 4 parameters to describe how a frame (i)
relates to a previous frame ( i -1 ).
THE PARAMETERS/VARIABLES:
, a , d, 
The Parameters
Z(i - 1)
Y(i -1)
X(i -1)
( i - 1)
Yi
a(i - 1 )
Zi
Xi
di
ai
You can
align the
two axis
just using
the 4
parameters
i
1) a(i-1)
Technical Definition: a(i-1) is the length of the perpendicular between the joint
axes. The joint axes is the axes around which revolution takes place which are the
Z(i-1) and Z(i) axes. These two axes can be viewed as lines in space. The common
perpendicular is the shortest line between the two axis-lines and is perpendicular
to both axis-lines.
a(i-1) cont...
Visual Approach - “A way to visualize the link parameter a(i-1) is to imagine an
expanding cylinder whose axis is the Z(i-1) axis - when the cylinder just touches the
joint axis i the radius of the cylinder is equal to a(i-1).” (Manipulator Kinematics)
It’s Usually on the Diagram Approach - If the diagram already specifies the
various coordinate frames, then the common perpendicular is usually the X(i-1)
axis. So a(i-1) is just the displacement along the X(i-1) to move from the (i-1) frame
to the i frame.
If the link is prismatic, then a(i-1)
is a variable, not a parameter.
Z(i - 1)
Y(i -1)
X(i -1)
( i - 1)
Yi
a(i - 1 )
Zi
Xi
di
ai
i
2) (i-1)
Technical Definition: Amount of rotation around the common perpendicular so that
the joint axes are parallel.
i.e. How much you have to rotate around the X(i-1) axis so that the Z(i-1) is pointing
in the same direction as the Zi axis. Positive rotation follows the right hand rule.
3) d(i-1)
Technical Definition: The displacement
along the Zi axis needed to align the a(i-1)
common perpendicular to the ai common
perpendicular.
In other words, displacement along the
Zi to align the X(i-1) and Xi axes.
Z(i - 1)
Y(i -1)
X(i -1)
Yi Z
i
a(i - 1 )
Xi
di
( i 1)
4) i
Amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi
axis.
ai
i
The Denavit-Hartenberg Matrix
cosθi

sinθ cosα
i
(i 1)

 sinθi sinα (i 1)

0

 sinθi
cosθi cosα (i 1)
cosθi sinα (i 1)
0
a(i 1)

 sinα (i 1) d i 
cosα (i 1) d i 

1

0
 sinα (i 1)
cosα (i 1)
0
Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a
transformation matrix from one coordinate frame to the next. Using a series of
D-H Matrix multiplications and the D-H Parameter table, the final result is a
transformation matrix from some frame to your initial frame.
Put the transformation here
Z(i Y(i -
1)
1)
X(i ( i
- 1)
1)
Y Z
i
a(i -
d
1)
i
i
X a
i
i

i
3 Revolute Joints
Y2
Z1
Z0
X2
X0
d2
X1
Y0
Y1
a0
Denavit-Hartenberg Link
Parameter Table
a1
i
(i-1)
a(i-1)
di
i
1) To describe the robot with its
variables and parameters.
0
0
0
0
0
2) To describe some state of the
robot by having a numerical values
for the variables.
1
0
a0
0
1
2
-90
a1
d2
2
Notice that the table has two uses:
Y2
Z1
Z0
X2
X0
X1
Y0
Y1
a0
V
X 0 Y0 Z 0
i
(i-1)
a(i-1)
di
i
0
0
0
0
0
1
0
a0
0
1
2
-90
a1
d2
2
d2
a1
V X 2 
 Y2 
V 

T Z
V 2 


 1 
T ( 0T)( 01T)( 12T)
Note: T is the D-H matrix with (i-1) = 0 and i = 1.
i
(i-1)
a(i-1)
di
i
0
0
0
0
0
1
0
a0
0
1
2
cosθ1
 sinθ
1
0

T

1
 0

 0
-90
a1
 sinθ1
cosθ1
0
0
d2
2
0 a0 
0 0 
0 0

0 1
This is a translation by a0 followed by a
rotation around the Z1 axis
cosθ 0  sinθ 0
 sinθ
cosθ 0
0

0T 
 0
0

0
 0
0
0
1
0
0
0
0

1
This is just a rotation around the Z0 axis
 cosθ 2
 0
1

2T 
 sinθ 2

 0
 sinθ 2
0
 cosθ 2
0
0 a1 
1 d 2 
0 0

0 1
This is a translation by a1 and then d2
followed by a rotation around the X2 and
Z2 axis
T ( 0T)( 01T)( 12T)
Inverse Kinematics
From Position to Angles
A Simple Example
Revolute and
Prismatic Joints
Combined
Finding :
y
θ  arctan( )
x
More Specifically:
(x , y)
y
θ  arctan 2( )
x
Y
S
1
X
Finding S:
S  (x 2  y 2 )
arctan2() specifies that it’s in the
first quadrant
Inverse Kinematics of a Two Link Manipulator
(x , y)
Find:
2
l2
1
l1
Given: l1, l2 , x , y
 1,  2
Redundancy:
A unique solution to this problem
does not exist. Notice, that using the
“givens” two solutions are possible.
Sometimes no solution is possible.
(x , y)
The Geometric Solution
2
l2
(x , y)
Using the Law of Cosines:
c 2  a 2  b 2  2ab cosC
( x 2  y 2 )  l1  l2  2l1l2 cos(180 θ 2 )
2
2
cos(180 θ 2 )   cos(θ 2 )
x 2  y 2  l1  l2
cos(θ 2 ) 
2l1l2
2
l1

1
Using the Law of Cosines:
sin B sin C

b
c
sin θ1 sin(180 θ 2 )
sin(θ 2 )


l2
x 2  y2
x 2  y2
θ1  θ1  α
y
α  arctan2 
x
2
 x 2  y 2  l12  l2 2 

θ 2  arccos

2l1l2


Redundant since 2 could be in the
first or fourth quadrant.
Redundancy caused since 2 has two possible
values
 l sin(θ ) 
 y
2 
θ1  arcsin 2
 arctan2 
 x 2  y2 
x


The Algebraic Solution
2
l2

(x , y)
c1  cosθ1
c1 2  cos(θ 2  θ1 )
(1) x  l1 c1  l2 c1 2
l1
(2) y  l1 s1  l2 sin 1 2
1
(3)θ  θ1  θ 2
(1)2  (2) 2  x 2  y 2 

 
 l1 c1  l2 (c1 2 ) 2  2l1l2 c1 (c1 2 )  l1 s1  l2 (sin1 2 ) 2  2l1l2 s1 (sin1 2 )
2
2
2
2
2
2

 l1  l2  2l1l2 c1 (c1 2 )  s1 (sin1 2 ) 
2
2
 l1  l2  2l1l2 c 2
2
2
Only Unknown
 x 2  y 2  l12  l2 2 

 θ 2  arccos

2
l
l
1 2


Note :
cos(ab)  (cosa)(cosb)  (sin a)(sin b)
sin(ab)  (cosa)(sin b)  (cosb)(sin a)
x  l1 c1  l2 c1 2
 l1 c1  l2 c1c 2  l2 s1s2
 c1 (l1  l2 c 2 )  s1 (l2 s2 )
y  l1 s1  l2 sin 1 2
 l1 s1  l2 s1c 2  l2 s 2 c1
 c1 (l2 s 2 )  s1 (l1  l2 c 2 )
c1 
s1 
cos(ab)  (cosa)(cosb)  (sin a)(sin b)
sin(ab)  (cosa)(sin b)  (cosb)(sin a)
We know what 2 is from the previous
slide. We need to solve for 1 . Now
we have two equations and two
unknowns (sin 1 and cos 1 )
x  s1 (l 2 s 2 )
(l1  l 2 c 2 )
x  s1 (l 2 s 2 )
y
(l 2 s 2 )  s1 (l1  l 2 c 2 )
(l1  l 2 c 2 )

Note :

Substituting for c1 and simplifying
many times
1
x l 2 s 2  s1 (l12  l 2 2  2l1l 2 c 2 )
(l1  l 2 c 2 )
y(l1  l 2 c 2 )  x l 2 s 2
x 2  y2

Notice this is the law of cosines
and can be replaced by x2+ y2
 y(l1  l 2 c 2 )  x l 2 s2
θ1  arcsin
2
2
x

y




