Post on 12-Jan-2016
MAE505: RoboticsFinal Project – Papers Review.
Presented By:Tao Gan
Advisor: Dr. Venkat Krovi.
Main Topic: Nonholonomic-Wheel Mobile Robot (WMR).Sub-Topics: Modeling, Redundancies, & Motion Control.
Focus on two papers:1. “Nonholonomic Mobile Manipulators: Kinematics,
Velocities and Redundancies,” - Journal of Intelligent and Robotic Systems Volume 36 , Issue 1 (January 2003) Pages: 45 – 63.- Authors: B. Bayle, J-Y. Fourquet and M. Renaud.- Present by: Leng-Feng Lee.
2. “Coordinating Locomotion and Manipulation of a Mobile Manipulator” -IEEE Transactions on Automatic Control, Vol. 39, No. 6, June 1994, pp. 1326-1332.- Authors: Yoshio Yamamoto and Xiaoping Yun.- Present by: Tao Gan.
Organization of Presentation:
In each paper, we will present the following:
- Introduction of the paper.- Formulation of the problem using an example.- Simulation setting used.- Simulation Result.- Discussion of the Simulation Result.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Introduction:
-When a person writes across a board, he/she positions his/her arm in a comfortable writing configuration by moving his/her body rather than reaching out the arm.
-The same situation happens in many case such as people transporting a heavy object cooperatively. -Therefore, when a mobile manipulator performs a manipulation task, it is desirable to bring the manipulator into certain preferred configurations by appropriately planning the motion of the mobile platform.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling: Using an Example
Arial View
Modeling
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling:
Three variables describe the position and orientation of the platform
State Variables:
Two variables specify the angular positions for the driving wheels
),,,( , lrcc yxq
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling:Adding Nonholonomic Constraints
stop
No. 1C
No.2,3
C
No.1 constraint is that the platform must move in the direction of the axis of symmetry (holonomic).
No.2,3 constraints are the rolling constraints, i.e. the driving wheels do not slip(nonholonomic).
0sincos dxy cc
rcc rbyx sincos
lcc rbyx sincos
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling: Three Constraints
According to the q, the three constraints can be written in the form of
where
0)( qqA
rb
rb
d
qA
0sincos
0sincos
00cossin
)(
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling: State Space Representation
The Mobile plateform’s equation of motion are described by
N x N inertia matrix
N x 1 vector of position and velocity dependent forces
N x r input transformation matrixr-dimensional input vector
)()(),()( qAqEqqVqqM T
)(qM
),( qqV
)(qE
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling:Dynamic Equations
N x N inertia matrix
)()(),()( qAqEqqVqqM T
w
w
cc
c
c
I
I
Idmdm
dmm
dmm
qM
0000
0000
00cossin
00cos0
00sin0
)(
Modeling:Dynamic Equations
N x 1 vector of position and velocity dependent forces
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
N x r input transformation matrix
r-dimensional input vector
)()(),()( qAqEqqVqqM T
0
0
0
sin
cos
),(
2
2
dm
dm
qqVc
c
10
01
00
00
00
)(qE
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Modeling:Dynamic Equations
Constraints matrix
S(q) are in the null space of A(q).
)()(),()( qAqEqqVqqM T
0)()( qSqA
10
01
)cossin()cossin(
)sincos()sincos(
)()()( 21 cc
dbcdbc
dbcdbc
qsqsqS
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
0)()( qSqA)(qSq
0)( qqA Differentiate and replace the one in the
))()(( VtSMtMSST
)()(),()( qAqEqqVqqM T
Premultiply byTS
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
))()(( VtSMtMSST Using state-space vector
Tlrlrcc
TTT yxqx ],,,,[ ,
1
2 )(
0
MSSf
Sx T
Suppose there is input u
uI
Sx
0
0
lineariz
e
What is u?
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Trr yxqhy )(
Differentiate it.
)()()/)(( SJSJqqqhy hh
Decoupling Matrix
Differentiate it again.
y
)(1
vu
uv
This is u.
)()( yyKyyKyvy dp
dd
d
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
)(1 vu uI
Sx
0
0
Since we know u
Tlrlrcc yxx ],,,,[ ,
We can know every state variables in the system.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation: Simulation Setting
1. Time Span: 0 – 60 sec
2. Path Trace: x = 6, y = t
3. Velocity: x_dot = 0,
y_dot = 1
4. Length of Link 1 = 5/sqrt(2),
Link 2= 5/sqrt(2),
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation: Simulation Setting
1. Time Span: 0 – 60 sec
2. Path Trace: x = t, y = t+t
3. Velocity: x_dot = 1,
y_dot = 1
4. Length of Link 1 = 5/sqrt(2),
Link 2= 5/sqrt(2),
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case i)
Notice:
-Control of WMR with Nonholonomic Constraints with Error Correction.
-The velocity of point Po are shown.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case i)
Notice:
-It indicates that the mobile platform moved backward for a short period of time at the very beginning to achieve the needed heading angle.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case ii)
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case ii)
Notice:
-Control of WMR with Nonholonomic Constraints with Error Correction.
-The velocity of point Po are shown.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case ii)
Notice:
-It indicates that the mobile platform moved backward for a short period of time at the very beginning to achieve the needed heading angle.
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Simulation:Simulation Result (case ii)
2nd paper: “Coordinating Locomotion and Manipulation of a Mobile Manipulator”
Conclusion:
-The second paper introduce a method of using Manipulability Measure of the Mobile Manipulator as the potential function included in the motion control of the WMR.
-We verified the effectiveness of the method by simulations on two representative trajectories