trajectory planning for robot manipulatorsrohan/teaching/en3562/lecturenotes/lec 8...trajectory...
TRANSCRIPT
TRAJECTORY PLANNING FOR
ROBOT MANIPULATORS
Prof. Rohan Munasinghe
Based on MSc Research by Chinthaka Porawagama
� Pick-and-place operations
� Assembling operations
� Loading and stacking
� Automated welding, etc.
Industrial Robotics Involves in
Proper motion planning is needed in these applications
Trajectory Planning
Manipulators with multi degree of
freedom for accomplishing various
complex manipulation in the work
space
Path: only geometric description
Trajectory: timing included
Trajectory Planning
Joint Space Vs Operational Space
Planning in Operational Space
Planning in Operational Space
Planning in Joint Space
“Expressing the desired positions of a manipulator
in the space, as a parametric function of time”
Path Definition
Task to Trajectory
� End effector moves from a start point to start
1. Point to point motion:
Types of Motion
end point in work space
� All joints’ movements are coordinated for
the point-to-point motion
� End effector travels in an arbitrary path
2. Motion with Via Points
end
start
� End effector moves through an
intermediate point between start and end
� End effector moves through a via point
without stoppingend via
Point to point motion:“Describing of joints’ motions from start to end by smooth functions
θ3B
End point ‘B’θ1B θ2B θ3B
Joint Space Planning
Basic stages of solving of joint space trajectory planning problem ?
θ3B
θ2A
θ3A
θ2B
θ1
θ2 θ3
Start point ‘A’
θ1B θ2B θ3B
θ2(t)
θ3(t)
θ1B
θ1A
Start point ‘A’θ1A θ2A θ3A
ATravel time T
θ1(t)
Arbitrary path in the task space
Task Space Parametric Representation
θ3B
End point ‘B’θ1B θ2B θ3B
Point to point motion:
“Describing of joints’ motions from start to end by smooth functions
Joint Space Planning
θ3B
θ2A
θ3A
θ2B
θ1
θ2 θ3
Start point ‘A’
θ1B θ2B θ3B
1. Inverse kinematics of start and end points (A & B)
θ2(t)
θ3(t)
2. Joint angles for start and end points
θ1B
θ1A
Start point ‘A’θ1A θ2A θ3A
A BTravel time T
θ1(t)
Arbitrary path in the task space
θ1A θ1B
θ2A θ2B
θ3A θ3B
Task Space Parametric Representation
θ3B
End point ‘B’θ1B θ2B θ3B
Joint Space Planning
θ3B
θ2A
θ3A
θ2B
θ1
θ2 θ3
Start point ‘A’
θ1B θ2B θ3B
3. Interpolation of start and end joint angles by smooth functions
θ1(t): θ1A → θ1B
θ2(t): θ2A → θ2B
θ3(t): θ3A → θ3B
θ2(t)
θ3(t)
θ1B
θ1A
Start point ‘A’θ1A θ2A θ3A
A BTravel time T
θ1(t)
Arbitrary path in the task space
4. Joint space trajectories for each joint
Task Space Parametric Representation
Smooth Motion Quality of Work
� Non smooth trajectories lead to low quality in
production.production.
Vibration
Error in path tracking
Manipulator wear
Poor quality in task
Linear Trajectory
−+=
sgs
tqq
qtq )(
<<
=∞=
−=
+=
g
g
g
sg
g
tt
tttq
t
qqtq
tt
qtq
00
,0)(
)(
)(
&&
&
� Infinite accelerations at endpoints
� Discontinuous velocity when two trajectory
segments are connected (at via points)
Triangular Velocity Trajectory
min22
ggsgt
qt
=− &&
2min
min
)(4
22
g
sg
t
qqq
−=
&&
≤≤−+
<≤+=
gs
g
s
ttttqqq
tttqqtq
5.05.0)(5.0
5.005.0)(
2
2
min
&&
&&
≤≤−+=
gg
gsttttqqq
tq5.05.0)(5.0
)(2
min&&
� Acceleration discontinuity at endpoints and at
the midpoint of the trajectory
� Zero acceleration in middle
segments.
Linear Trajectory with Parabolic Blends
Acceleration segments.
� Constant acceleration at end
segments.
Acceleration discontinuity at blend points
Linear Trajectory with Parabolic Blends
)(2 LinearparabolicS +=
� Total angular motion
)(2
12
)(2
2
bg
b
b
bsgttqtqqq
LinearparabolicS
−+×=−
+=
&&&
0)(2
=−+−sg
bg
b
b
bqqttqtq &&&&
sg
g
g
b
qqt
tt
)(41 2 −−±=
bgbq
tt&&22
−±=
� For a linear part to exist
2
2 )(4 0
)(4
g
sgb
b
sg
gt
qqq
q
qqt
−>>
−− &&
&&
Minimum joint
acceleration
Linear Trajectory with Parabolic Blends
� Via points (knot points) can be introduced between start and goal (qs, qg) positions with
Inclusion of Via Points into a Linear Trajectory with Parabolic Blends
between start and goal (qs, qg) positions with
constant acceleration at via point.
Multi-stage linear parabolic blend spline
0 0.2 0.4 0.6 0.8 10
20
40
60
time
posit
ion
Cubic Polynomial (Bring in Smoothness)
CubicJoint Position
(1)
time
0 0.2 0.4 0.6 0.8 10
50
100
time
Velo
city
0
200
400
Acc
ele
rati
on
Parabolic
Linear
Joint Velocity
Joint Acceleration
(2)
(3)
0 0.2 0.4 0.6 0.8 1-400
-200
time
Acc
ele
rati
on
0 0.2 0.4 0.6 0.8 1-721
-720.5
-720
-719.5
-719
time
Jerk
Joint Jerk
(3)
(4)
Cubic Polynomial (zero sped at end-
points)
� Satisfies position and velocity at end-points
� Eg: zero speed at end-points� Eg: zero speed at end-points
0)(
0)0(
)(
)0(
=
=
=
=
g
g
g
s
tq
q
qtq
&
&
2
221
1
3
3
2
210
0
32 ,for (2)
,0for (2)
,for (1)
,0for (1)
gg
g
g
sggg
g
g
s
tataaqtt
aqt
tatataaqtt
aqt
++==
==
+++==
==
&
&
23
� Acceleration is linear and uncontrollable
33
22
1
0
)(2
)(3
)0(
g
sg
g
sg
s
s
t
qqa
t
qqa
qa
qa
−−=
−=
==
=
&
g
sg
g
sg
g
s
tt
tqqt
tqqt
qtq
≤≤
−−−+=
0
)(2
)(3
)(3
3
2
2
gtttaatq ≤≤+= 0 62)( 22&&
Cubic Polynomial (zero sped at end-
points)
Cubic Polynomial (nonzero speeds at
end- points)
� Satisfies position and velocity at end-points
� Eg: non-zero speeds at end-points� Eg: non-zero speeds at end-points
g
g
s
g
g
s
qtq
qtq
&&
&&
=
=
=
=
)(
)0(
)(
)0(
2
221
1
3
3
2
210
0
32 ,for (2)
,0for (2)
,for (1)
,0for (1)
gg
g
g
sggg
g
g
s
tataaqtt
aqt
tatataaqtt
aqt
++==
==
+++==
==
&
&
0
sqa =tatataatq +++=)(
32
� Acceleration is linear and
uncontrollable )(1
)(2
)(1
)(3
233
22
1
0
sg
g
sg
g
sg
g
sg
g
s
qqt
qqt
a
qqt
qqt
a
qa
qa
&&
&&
&
−+−−=
−−−=
=
=
gtttatataatq
≤≤
+++=
0)(
3
3
2
210
gtttaatq ≤≤+= 0 62)( 22&&
Cubic Spline Trajectory
� Stitching cubic polynomials together
� Acceleration is not continuous at via (stitching points)
0 0.2 0.4 0.6 0.8 10
20
40
60
80
time
posit
ion
5th Order Polynomial
(more oscillatory)
0 0.2 0.4 0.6 0.8 10
50
100
150
time
Velo
city
0
200
400
Acc
ele
rati
on
0 0.2 0.4 0.6 0.8 1-400
-200
time
Acc
ele
rati
on
0 0.2 0.4 0.6 0.8 1-2000
0
2000
4000
time
Jerk