Day 05
Forward Kinematics
1
1/15/2017
Links and Joints
joint 3
joint 4
link 3
joint n
.................
link 2
joint 1
joint n-1
link n-1
link n
link 1
link 0



n joints, n + 1 links
link 0 is fixed (the base)
joint i connects link i – 1 to link i

2
joint 2
i
qi  
di
revolute
prismatic
link i moves when joint i is actuated
1/15/2017
Forward Kinematics

given the joint variables and dimensions of the links what is
the position and orientation of the end effector?
p0 ?
a2
2
y0
a1
1
x0
3
Forward Kinematics
1/15/2017
Forward Kinematics

because the base frame and frame 1 have the same
orientation, we can sum the coordinates to find the position
of the end effector in the base frame
(a1 cos 1 + a2 cos (1 + 2),
a1 sin 1 + a2 sin (1 + 2) )
y1
( a2 cos (1 + 2),
a2 sin (1 + 2) )
2
y0
1
a1
1
x0
4
a2
Forward Kinematics
x1
( a1 cos 1 , a1 sin 1 )
1/15/2017
Forward Kinematics

from Day 02
p0 = (a1 cos 1 + a2 cos (1 + 2),
a1 sin 1 + a2 sin (1 + 2) )
y2
x2
x2 = (cos (1 + 2),
sin (1 + 2) )
a2
y2 = (-sin (1 + 2),
cos (1 + 2) )
2
y0
1
a1
1
x0
5
Forward Kinematics
1/15/2017
Frames
y2
y1
x2
a2
2
y0
x1
a1
1
x0
6
Forward Kinematics
1/15/2017
Forward Kinematics

using transformation matrices
T  Rz ,1 Dx, a1
0
1
T  Rz ,2 Dx, a2
1
2
T 02  T 10 T 12
7
1/15/2017
Links and Joints
joint 3
joint 4
link 3
joint n
.................
link 2
joint 1
joint n-1
link n-1
link n
link 1
link 0



n joints, n + 1 links
link 0 is fixed (the base)
joint i connects link i – 1 to link i

8
joint 2
i
qi  
di
revolute
prismatic
link i moves when joint i is actuated
1/15/2017
Forward Kinematics

attach a frame {i} to link i


all points on link i are constant when expressed in {i}
if joint i is actuated then frame {i} moves relative to frame {i - 1}

motion is described by the rigid transformation
T ii 1

the state of joint i is a function of its joint variable qi (i.e., is a function of qi)
T ii 1 T ii1(qi )

this makes it easy to find the last frame with respect to the
base frame
T  T T T T
0
n
9
0
1
1
2
2
3
n1
n
1/15/2017
Forward Kinematics

more generally
T ii 1 T ij12  T jj 1 if i  j
i 
T j 
I
if i  j
i 1

Tj
if i  j

 

10
the forward kinematics problem has been reduced to matrix
multiplication
1/15/2017
Forward Kinematics

Denavit J and Hartenberg RS, “A kinematic notation for lowerpair mechanisms based on matrices.” Trans ASME J. Appl. Mech,
23:215–221, 1955

described a convention for standardizing the attachment of frames
on links of a serial linkage

common convention for attaching reference frames on links of
a serial manipulator and computing the transformations
between frames
11
1/15/2017
Denavit-Hartenberg
T ii 1 Rz ,i Tz ,di Tx,ai Rx,i
ci  si ci si si ai ci 
s

c
c

c
s
a
s
i  i
i  i
i i 
  i
0
si
ci
di 


0
0
1 
0
ai
i
di
i
12
link length
link twist
link offset
joint angle
1/15/2017
Denavit-Hartenberg
13
1/15/2017
Denavit-Hartenberg

notice the form of the rotation component
ci  si ci si si 


si ci ci  ci si 
0

s
c


i
i




14
this does not look like it can represent arbitrary rotations
can the DH convention actually describe every physically
possible link configuration?
1/15/2017
Denavit-Hartenberg


yes, but we must choose the orientation and position of the
frames in a certain way

(DH1)
xˆ1  zˆ0

(DH2)
xˆ1 intersects zˆ0
claim: if DH1 and DH2 are true then there exists unique
numbers
a, d ,  ,  such that T10  R z , Dz ,d Dx,a R x,
15
1/15/2017
Denavit-Hartenberg

16
proof: on blackboard in class
1/15/2017