ROBOTICS
01PEEQW
Basilio Bona
DAUIN – Politecnico di Torino
Planar 2 DOF manipulator
Reference planner: a numerical example
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
2
Planar two-arm manipulator – 1
R 2 = R TCP
This simple manipulator
cannot provide 3 DOF,
since it has only two joints
We will compute below
the kinematics functions
y
This is the third
Euler angle
l2
q2
l1
R1
q1
R0
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
3
Planar two-arm manipulator – 2
 Direct position KF, assuming
pT = й
клx 1 x 2
T
йq q щ
fщ
,
q
=
ъ
кл 1 2 ъ
ы
ы
x є x 1 = l 1 cos(q1 ) + l 2 cos(q1 + q2 ) = l 1c1 + l 2c12
y є x 2 = l 1 sin(q1 ) + l 2 sin(q1 + q2 ) = l 1s1 + l 2s12
y = (q1 + q2 )
 Direct velocity KF
x&= - l 1s1q&
- l 2s12 (q&
+ q&
)
1
1
2
y&= l 1c1q&
+ l 2c12 (q&
+ q&
)
1
1
2
y&= (q&
+ q&
)
1
2
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
4
Planar two-arm manipulator – 3
 Analytical Jacobian
й- l s - l s
щ
l
s
2 12
2 12 ъ
к 11
J a = кк l 1c1 + l 2c12
l 2c12 ъ
ъ
к
ъ
1
1
кл
ъ
ы
&щ
 Geometric Jacobian, assuming w T = й
0
0
y
кл
ъ
ы
йJ
щ
J
L2 ъ
J g = кк L 1
ъ
J
J
A2 ы
кл A 1
ъ
All joints are revolute, hence:
Basilio Bona - DAUIN - PoliTo
J L 1 = k 0 ґ r 0, TCP
J L 2 = k1 ґ r1, TCP
J A1 = k0
J A 2 = k1
ROBOTICS 01PEEQW - 2015/2016
5
Reference planner – sampled time
 Use the 2-1-2 approach
 Use the s-function planner
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
6
Accelerations
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
7
Velocities
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
8
Example: angles
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
9
Cartesian space velocities
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
10
Cartesian space positions
y
x
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
11
Cartesian space velocities
Vmax = 2.0
Basilio Bona - DAUIN - PoliTo
Vmax = 1.5
ROBOTICS 01PEEQW - 2015/2016
12
Joint space
q2
q1
Basilio Bona - DAUIN - PoliTo
ROBOTICS 01PEEQW - 2015/2016
13