velocity propagation between robot links...
TRANSCRIPT
Velocity Propagation Between Robot Links 3/4
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Introduction – Velocity Propagation
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Jacobian Matrix - Introduction
• In the field of robotics the Jacobian matrix
describe the relationship between the joint
angle rates ( ) and the translation and
rotation velocities of the end effector ( ).
This relationship is given by:
• In addition to the velocity relationship, we are
also interested in developing a relationship
between the robot joint torques ( ) and the
forces and moments ( ) at the robot end
effector (Static Conditions). This
relationship is given by:
N
x
Jx
F
F
FJT
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Jacobian Matrix - Calculation Methods
Jacobian Matrix
Differentiation the
Forward Kinematics Eqs. Iterative Propagation
(Velocities or Forces / Torques)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Summary – Changing Frame of Representation
• Linear and Rotational Velocity
– Vector Form
– Matrix Form
• Angular Velocity
– Vector Form
– Matrix Form
Q
BA
BB
A
Q
BA
BBORG
A
Q
A PRVRVV
B
A
Q
BA
B
A
BQ
BA
BBORG
A
Q
A PRRVRVV
Q
BP
TA
B
B
C
A
B
A
B
A
C RRRRR
C
BA
BB
A
C
A R
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Frame - Velocity
• As with any vector, a velocity vector may be described in terms of any frame,
and this frame of reference is noted with a leading superscript.
• A velocity vector computed in frame {B} and represented in frame {A} would be
written
Q
BA
Q
BA Pdt
dV )(
Q
Computed
(Measured)
Represented
(Reference Frame)
0BORG
AV
0RA
B
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Position Propagation
• The homogeneous transform matrix provides a complete description of the
linear and angular position relationship between adjacent links.
• These descriptions may be combined together to describe the position of a link
relative to the robot base frame {0}.
• A similar description of the linear and angular velocities between adjacent links
as well as the base frame would also be useful.
TTTT i
i
oo
i
11
21
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Position Propagation
TTTTTTTT TT
65
6
4
5
3
4
2
3
1
2
0
1
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Motion of the Link of a Robot
• In considering the motion of a robot link we will always use link frame {0} as the
reference frame (Computed AND Represented). However any frame can be used
as the reference (represented) frame including the link’s own frame (i)
Where: - is the linear velocity of the origin of link frame (i) with respect to
frame {0} (Computed AND Represented)
- is the angular velocity of the origin of link frame (i) with respect to
frame {0} (Computed AND Represented)
• Expressing the velocity of a frame {i} (associated with link i ) relative to the robot
base (frame {0}) using our previous notation is defined as follows:
iv
i
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
iii 000
iii VVv 000
Velocities - Frame & Notation
• The velocities differentiate (computed) relative to the base frame {0} are often
represented relative to other frames {k}. The following notation is used for this
conditions
i
k
i
k
i
k
i
k vRVRVv 0
0
0
0
i
k
i
k
i
k
i
k RR 0
0
0
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity Propagation
• Given: A manipulator - A chain of
rigid bodies each one capable of
moving relative to its neighbor
• Problem: Calculate the linear and
angular velocities of the link of a
robot
• Solution (Concept): Due to the
robot structure (serial mechanism)
we can compute the velocities of
each link in order starting from the
base.
The velocity of link i+1 will be that of
link i , plus whatever new velocity
components were added by joint i+1
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 0/5
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 1/5
• From the relationship developed previously
• we can re-assign link names to calculate the velocity of any link i relative to the
base frame {0}
• By pre-multiplying both sides of the equation by ,we can convert the frame
of reference for the base {0} to frame {i+1}
C
BA
BB
A
C
A R
1
0
iC
iB
A
1
00
1
0
i
i
iii R
Ri 1
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 2/5
• Using the recently defined notation, we have
- Angular velocity of frame {i+1} measured relative to the robot base, and
expressed in frame {i+1} - Recall the car example
- Angular velocity of frame {i} measured relative to the robot base, and
expressed in frame {i+1}
- Angular velocity of frame {i+1} measured relative to frame {i} and
expressed in frame {i+1}
1
01
0
01
01
01
0
i
i
i
i
i
i
i
i RRRR
1
11
1
1
i
ii
ii
i
i
i R
1
1
i
i
i
i 1
1
1
i
ii
i R
c
c
c
wcvV
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 3/5
• Angular velocity of frame {i} measured relative to the robot base, expressed in
frame {i+1}
1
11
1
1
i
ii
ii
i
i
i R
i
ii
ii
i R 11
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 4/5
• Angular velocity of frame {i+1} measured (differentiate) in frame {i} and
represented (expressed) in frame {i+1}
• Assuming that a joint has only 1 DOF. The joint configuration can be either
revolute joint (angular velocity) or prismatic joint (Linear velocity).
• Based on the frame attachment convention in which we assign the Z axis
pointing along the i+1 joint axis such that the two are coincide (rotations of a link
is preformed only along its Z- axis) we can rewrite this term as follows:
1
1
1 0
0
i
i
ii
i R
1
2
3
i+1
i
1
11
1
1
i
ii
ii
i
i
i R
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Angular Velocity 5/5
• The result is a recursive equation that shows the angular velocity of one link in
terms of the angular velocity of the previous link plus the relative motion of the
two links.
• Since the term depends on all previous links through this recursion, the
angular velocity is said to propagate from the base to subsequent links.
1
1
1
1 0
0
i
i
ii
ii
i R
1
1
i
i
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 0/6
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 1/6
• Simultaneous Linear and Rotational Velocity
• The derivative of a vector in a moving frame (linear and rotation velocities) as
seen from a stationary frame
• Vector Form
• Matrix Form
Q
BA
BB
A
Q
BA
BBORG
A
Q
A PRVRVV
B
A
Q
BA
B
A
BQ
BA
BBORG
A
Q
A PRRVRVV
Q
BP
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 2/6
• From the relationship developed previously (matrix form)
• we re-assign link frames for adjacent links (i and i +1) with the velocity
computed relative to the robot base frame {0}
• By pre-multiplying both sides of the equation by ,we can convert the
frame of reference for the left side to frame {i+1}
Q
BA
B
A
BQ
BA
BBORG
A
Q
A PRRVRVV
1
0
iC
iB
A
1
00
1
00
1
0
i
i
iii
i
iii VRVPRRV
Ri 1
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 3/6
• Which simplifies to
• Factoring out from the left side of the first two terms
1
01
0
01
01
001
01
01
0
i
i
i
i
i
i
i
i
ii
i
i
i VRRVRPRRRVR
1
101
01
001
01
01
0
i
ii
ii
i
i
i
ii
i
i
i VRVRPRRRVR
Ri
i
1
1
10
01
00
0
1
1
01
0
i
ii
ii
i
i
i
ii
ii
ii
i VRVRPRRRRVR
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 4/6
- Linear velocity of frame {i+1} measured relative to frame {i} and
expressed in frame {i+1}
• Assuming that a joint has only 1 DOF. The joint configuration can be either
revolute joint (angular velocity) or prismatic joint (Linear velocity).
• Based on the frame attachment convention in which we assign the Z axis
pointing along the i+1 joint axis such that the two are coincide (translation of a
link is preformed only along its Z- axis) we can rewrite this term as follows:
1
10
01
00
0
1
1
01
0
i
ii
ii
i
i
i
ii
ii
ii
i VRVRPRRRRVR
1
1
1 0
0
i
i
ii
i
d
VR
1
1
i
ii
i VR
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 5/6
1
0
01
00
0
1
1
01
0 0
0
i
i
i
i
i
ii
ii
ii
i
d
VRPRRRRVR
i
i
i
i
i
iTi
i
i
ii
i RRRRRRRR 0
0
00
0
0
00
0
Multiply by Matrix
Definition
i
i
i
i vVR 0
0
Definition
1
1
1
01
0
i
i
i
i vVR
Definition
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Linear Velocity 6/6
• The result is a recursive equation that shows the linear velocity of one link in
terms of the previous link plus the relative motion of the two links.
• Since the term depends on all previous links through this recursion, the
angular velocity is said to propagate from the base to subsequent links.
1
1
1
1
1 0
0
i
i
i
i
i
i
ii
ii
i
d
vPRv
1
1
i
i v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Velocity of Adjacent Links - Summary
• Angular Velocity
• Linear Velocity
1
1
1
1
1 0
0
i
i
i
i
iii
ii
i
d
vPRv
1
1
1
1 0
0
i
i
ii
ii
i R
0 - Prismatic Joint
0 - Revolute Joint
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• For the manipulator shown in the figure, compute the angular and linear velocity
of the “tool” frame relative to the base frame expressed in the “tool” frame (that
is, calculate and ).
4
44
4v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• Frame attachment
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• DH Parameters
i 1i 1ia id i
1 0 0 0 1
2 90 L1 0 2
3 0 L2 0 3
4 0 L3 0 0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• From the DH parameter table, we can specify the homogeneous transform
matrix for each adjacent link pair:
1000
0
1111
1111
1
1
dccscss
dsscccs
asc
Tiiiiii
iiiiiii
iii
i
i
1000
0100
0011
0011
0
1
cs
sc
T
1000
0022
0100
1022
1
2cs
Lsc
T
1000
0100
0033
2033
2
3
cs
Lsc
T
1000
0100
0010
3001
3
4
L
T
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• Compute the angular velocity of the end effector frame relative to the base
frame expressed at the end effector frame.
• For i=0
1
1
1
1 0
0
i
i
ii
ii
i R
111
0
01
01
1 0
0
0
0
0
0
0
100
011
011
0
0
cs
sc
R
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• For i=1
• For i=2
• For i=3
• Note
2
1
1
212
1
12
12
2 2
2
0
0
0
0
010
202
202
0
0
c
s
cs
sc
R
32
1
1
32
1
1
3
2
23
23
3 23
23
0
0
2
2
100
033
033
0
0
c
s
c
s
cs
sc
R
32
1
1
32
1
1
3
34
34
4 23
23
0
0
0
23
23
100
010
001
0
0
0
c
s
c
s
R
4
4
3
3
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• Compute the linear velocity of the end effector frame relative to the base frame
expressed at the end effector frame.
•
• Note that the term involving the prismatic joint has been dropped from the
equation (it is equal to zero).
1
1
1
1
1 0
0
i
i
i
i
iii
ii
i
d
vPRv
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• For i=0
• For i=1
0
0
0
0
0
0
0
0
0
0
0
0
100
011
011
0
0
1
0
0
01
01
1 cs
sc
vPRv
111
1
1
2
1
1
12
12
2 0
0
0
0
0
0
0
1
0
0
010
202
202
L
L
cs
sc
vPRv
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• For i=3
1
2
2
11
1
12
1
1
2
2
3
2
2
23
23
3
)221(
32
32
122
2
0
100
033
033
1
0
0
0
0
2
2
2
100
0303
033
cLL
cL
sL
LcL
Lcs
sc
L
L
c
s
cs
sc
vPRv
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• For i=4
1
32
2
1
2
2
32
1
1
3
3
4
3
3
34
34
4
)233221(
3)332(
32
)221(
32
32
0
0
3
23
23
100
010
001
cLcLL
LLcL
sL
cLL
cL
sLL
c
s
vPRv
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• Note that the linear and angular velocities ( ) of the end effector where
differentiate (measured) in frame {0} however represented (expressed) in frame
{4}
• In the car example: Observer sitting in the “Car”
Observer sitting in the “World”
Solve for and by multiply both side of the questions from the left by
i
k
i
k
i
k
i
k vRVRVv 0
0
0
0
i
k
i
k
i
k
i
k RR 0
0
0
0
4
4
04
4 vRv
4
4
04
4 R
4
4
4
4 , v
C
WCV
C
WWV
4v4
14
0
R
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
• Multiply both sides of the equation by the inverse transformation matrix, we
finally get the linear and angular velocities expressed and measured in the
stationary frame {0}
4
40
44
44
04
414
04 vRvRvRv T
4
40
44
44
04
414
04 RRR T
TTTTT 3
4
2
3
1
2
0
1
0
4
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
1
1
1 0
0
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
2
1
1
2
2 2
2
c
s
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
32
1
1
3
3 23
23
c
s
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
4
4
3
3
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
0
0
0
1
1v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
11
2
2 0
0
L
v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
1
2
2
3
3
)221(
32
32
cLL
cL
sL
v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Angular and Linear Velocities - 3R Robot - Example
1
32
2
4
4
)233221(
3)332(
32
cLcLL
LLcL
sL
v
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA