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Dynamic model and disturbance
observer based nonlinear positiontracking control of a new XYz motion
platform
M.Santhakumar , Ramavatar Meena, Jayant Kr. Mohanta and Sandip patidarCenter for robotics and control ,Indian Institute of Technology ,Indore
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Introduction
What are XYz motion platforms?
Difference between serial and parallel motion
platforms
Need(s) of a proposed platform
Advantages and limitations of the proposed
platform
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Conceptual and frame diagram
3
P(x,y)
O(0,0)
z
r2
r1
r3
Passive joint
Prismatic
joint 2
Prismaticjoint 2
Work
table
Fixed
base
Rotary
joints
Guide ways for
prismatic joints
Guide ways for
prismatic joints
Prismatic joint
1
s
Inertial
frame
endeffector
frame
Number of links = 6
Number of joints = 6
Degrees of freedom = 3 (6-1)2 (6) = 3
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Kinematic model
Forward kinematic model
Inverse kinematic model
4
s
rr
s
rrrry
rx
z231
231
2
1
tan
z
z
xsyr
xyr
xr
tan
tan
3
2
1
where,
r1, r2 and r3are prismatic joint
displacements (joint parameters)
sis the fixed distance (horizontal span
of the platform)
xand yare the endeffector (work table
/ tool) positions
zis the yaw angle of the endeffectorfrom the inertial frame
x, yand zare the task space
parameters
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Dynamic model
Euler-Lagrangian formulation method
It is an energy based approach.
Since, it is planar platform potential energy of the
robotic platform considered to be zero. Total kinetic energy (KE) of the robotic platform is
sum of individual prismatic (slider) joint kinematicenergies and work table kinetic energy.
5
zzzw
w
wT
z
sss y
x
I
m
m
y
x
rmrmrmKE
00
00
00
2
1
2
1
2
1
2
1 211
2
11
2
11
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Dynamic model continued...
Jacobian (velocity mapping) matrix Joint space velocities to task space velocities
Use this relation for deriving kinetic energy of the roboticplatform. Therefore, total kinetic energy is in terms of jointspace variables
6
3
2
1
22
1123
coscos0
1001
r
rr
ss
s
r
s
r
s
rryx
zzz
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Dynamic model continued...
Euler-Lagrangian formulation method continued... Joint space input variables (joint forces) can be derived from the
following relation as given below:
By combining all input variables and formulating the equationsof motion of the platform in state space form as follows:
where,
7
i
iifr
KE
r
KE
dt
d
disCM ,
edisidisdis
T
TTT
fff
rrrrrrrrr
vectoreDisturbanc
,,
321
321321321
matrixlcentripetaandCoriolistheis,Cmatrixinertiatheis
M
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Disturbance observer based nonlinear
position tracking controller
Proposed control law based on disturbance
observer as given below:
Disturbance observer scheme as follows:
8
disPDd CKKM ,~~
0,0i.e.,constants,positiveareand
,,,simplicityfor
matrix,gainobserveranis,ector,arbitary vanis,,,
0,~~~,~
~~
,,,,
,
1
1
1
M
zMdt
d
MMCzz
z
disdisdisdisdis
dis
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Block diagram of the proposed
controller
9
Robotic
platform
Disturbance
observer
M
DK
PK
Trajectory
planner
Inverse
kinematic
model
,Cx(t),
y(t),
z(t)d
d
d
~
~ dis
Disturbances
dis
,
s
zsss yx ,,
+
+-
-
-+
+
Sensor
systems
User input block Proposed controller block
Robotic System
Santhakumar
EKF
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Stability proof of the proposed
controller Applying the proposed nonlinear disturbanceobserver-based control law to the robot platformdescribed by its equations of motion results in thefollowing closed-loop equation:
We can show that all the tracking errors will be
converged to zero asymptotically by the Lyapunovsdirect method. i.e., The proposed controller stability
(closed loop) can be proved using the Lyapunovsdirect method.
10
disPD MKK ~~~~ 1
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Stability proof continued...
Consider the following positive Lyapunovs candidate function as given
below:
Time derivative of the above function along with its state variable
trajectories, and using closed-loop equation and time derivative of
disturbance error dynamics, we get
11
disTdisDPTT
dis IKKV ~~
2
1~~
2
1~~~~
2
1~,~,~2
0i.e.,dynamics,errorestimationn with thecomparisioin
negligibleisplatformroboticon theactingedisturbanctheofchangeofrateThe
thatsatisfyingisand,0,0i.e.,
matrices,definitepositiveandsymmeticconstantareand
dis
DDDP
DP
IKKKK
KK
disT
disP
T
D
T MKIKV ~~~~~~ 1
0i.e.,property,bymatrixdefinitepositiveais 11 MM Santhakumar
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Stability proof continued...
Lyapunovs function is positive definite and its time derivative isnegative definite in the entire state space.
It is assumed that, the estimated state vectors based on extend
Kalman filter (EKF) are converged to their true state vector values.In this research, the EKF convergence is not discussed.
Therefore, based on LaSalles invariance theorem and Lyapunovsdirect method the closed-loop equation is globally asymptotically
stable. i.e., the velocity, position and disturbance tracking errorsconverge to zero.
12
0~and0~,0~ then,if dist
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Numerical Simulations
Robotic platform details
Joint limits
Horizontal span (s) = 20 cm
Controller details
Controller gains
Observer details
Observer constants
13
cm20cm0
cm20cm0
cm20cm0
3
2
1
r
r
r
5,
5,
33
33
ppP
ddD
kIkK
kIkK
2,2
EKF details Slider displacements, work table
positions and turning angle can bemeasure using linear encoders (orpotentiometers) and vision system
No velocity and accelerationfeedback
Slowly varying Gaussian noises andinternal disturbances are considered
Simulation cases Case 1: without any disturbances and no
sliding friction forces
Case 2: with disturbances, parameteruncertainties and sliding friction forces
Task 1: Rectangular trajectory
following Task 2: Circular trajectory
following
Comparisons With disturbance compensation and
without disturbance compensation
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Simulation Case 1: without any disturbances and no sliding frictionforces
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Proposed controller without
disturbance compensation
Proposed controller with
disturbance compensation
XY task space motion of the robotic
platform
Simple rectangle trajectory has
considered for the analysis with
four straight line segments.
Results are in satisfactory level and
the system behaviors are almost
equal for both controllers.
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Trajectories of task space tracking
errors
Trajectories of task space tracking
errors
Trajectories of joint space tracking errorsTrajectories of joint space tracking errors
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Simulation case 2: with disturbances, parameter uncertainties and sliding friction
forces
Simple rectangular trajectory with constant joint disturbances and sliding
friction forces on their prismatic joints.
Desired task space trajectory details:
Santhakumar 15
seconds10050
102.3024.010
cm1028.10096.05
cm13
seconds500
106.1012.0
cm5
cm1028.10096.05
342
342
342
342
t
tt(t)
tty(t)
x(t)
t
tt(t)
y(t)
ttx(t)
z
z
seconds200150
106.1012.010
cm1028.10096.013
cm5
seconds150100
102.3024.010
cm13
cm1028.10096.013
342
342
342
342
t
tt(t)
tty(t)
x(t)
t
tt(t)
y(t)
ttx(t)
z
z
XY task space motion trajectories of the robotic platform for
a given rectangular trajectory in the presence of disturbances
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the joint space position tracking errors
(without disturbance compensation)
Time trajectories of the joint space position tracking errors (with
disturbance compensation)
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the task space position tracking errors
(without disturbance compensation)
Time trajectories of the task space position tracking errors (with
disturbance compensation)
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the estimated joint disturbances for the given rectangular
trajectory
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Simulation Case 2: with disturbances, parameter uncertainties and sliding
friction forces
Simple rectangular trajectory with constant
joint disturbances and sliding friction forces
on their prismatic joints.
Desired task space trajectory details:
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4cos10
cm2sin510
cm2cos510
t(t)
ty(t)
tx(t)
z
XY task space motion trajectories of the robotic platform for a given
circular trajectory in the presence of disturbances
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the joint space position tracking errors
(without disturbance compensation)
Time trajectories of the joint space position tracking errors (with
disturbance compensation)
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the task space position tracking errors
(without disturbance compensation)
Time trajectories of the task space position tracking errors (with
disturbance compensation)
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Simulation Case 2: with disturbances, parameter uncertainties and sliding friction
forces continued
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Time trajectories of the estimated joint disturbances for the given circular
trajectory
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Conclusions and future work
The effectiveness of the proposed scheme wasdemonstrated using extensive numerical simulations withsuitable manipulation tasks.
The results confirmed the effectiveness and robustness ofthe proposed scheme in terms of tracking errors in thepresence of unknown external disturbances and parameteruncertainties.
The proposed controller is simple structure, ease ofcomputation and positional feedback inputs are sufficient.Therefore, it can be implemented easily and extended tospatial platforms as well.
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