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Variable Structure Control of a Manipulator with Four Degrees of Freedom
671
VARIABLE STRUCTURE CONTROL OF A MANIPULATOR WITH FOUR DEGREES OF FREEDOM
Figen Ozen, Ahmet Denker Department of Electrical and Electronic Engineering, Bo~azi~i University Bebek 80815, istanbul, TURKEY
Keywords : manipulator control, variable structure control, decoupling theory, PVA command generator
Abstract : A VSS control which switches between PO (Proportional and Derivative) and CEA (Constant Error Acceleration) schemes is presented in this article. It is applied to steer robot manipulators in order to satisfy "no overshoot" and "high response speed" requirements . Simulation results verify that with this approach, maximum actuator capacity can be exploited while avoiding danger of overshoots .
INTRODUCTION
Close tracking of the reference without overshoots is an essential concern in many robotic applications, such as collision avoidance . While the "no overshoot" requirement can be met by applying critically damped PO controller to joints with Iinearized dynamics, it must be assured that limits of the actuators must not be exceeded . This consideration leads to overcautious designs which refrain from using the full capacity of the actuators, thereby leading to sluggish behaviour. In order to meet the "no overshoot" requirement while maintaining high response speed, an alternative approach is presented here which implements the strategy proposed in (Denizhan, 1988). This strategy is based upon switching between PO (proportional and Derivative) control and CEA (Constant Error Acceleration) schemes, in order to exploit the maximum actuator capacity, while avoiding danger at overshoots .
PO CONTROL SCHEME With the task description y solved in the joint space, the robot manipulator dynamics can be represented (Koivo, 1989) as
x
f[x(t),U(t)]
y
h( x)
(1 )
where x(t) E Rn and u(t) E Rm. The objective of the VSS control strategy is to steer the states of the system in a subspace where the requirements of the "high response speed" and "no overshoot" can be satisfied . Using the nonlinear feedback of the form (Isidori, 1985)
672
u(t)
=
(2)
A(x, t) Cl(t) + 8(x, t)
the input output behaviour of the system in Eq .1 becomes the same as the one of a linear system
y . (t)
Q .(t , x ) +
1
1
I=,
i
0
(3)
jh .(t-t)U .(t)dt 0
1
1
When PD control is implemented, task equations for each joint can be written as y .(t)
=
1
(4)
t.(t) + Cl, . [f .(t)- y .(t)]+ Clo .[r.(t)- y .(t)] 1
1
1
1
1
1
1
Since both "high response speed" and "no overshoot" requirements can be expressed in terms of error and its first derivative, it is quite convenient to design the control law with respect to error phase plane behaviour (Denizhan, 1988). Noting that
=
e.(t) 1
r .(t)-y .(t) 1
(5)
I
and choosing for critical damping 2
Cl .
Cl
=
_ 11
(6)
4
Oi
Using unsubscribed variables for the sake of clarity, error equations for the "no overshoot" condition can be obtained as
a 1
{ e (0) + [ e (0) + :
e (t)
e(t)
=
e( 0 )]t} e -
-+
e(O)+2e(O)
Cl, --
l
1
Cl1
+ --2 e(O i +[e(O)+
:1
(7)
1
(8)
e(t)
e (O)}
The design goal can now be formulated as bringing any initial point on the error phase plane to the origin as fast as possible without overcrossing the e -axis. Taking the limit as t tends to infinity, the following relation between
e
and e can be obtained
ex
lim e(t) t--+oo
= - _J.
(9)
lim e (t) 2 1--+ 00
Hence the state trajectories in
e -e approach the
origin along the asymptote
673
(10)
Besides, the choice of parameters as in Eq.S leads to a critically damped behaviour, giving rise to convergent parabolic phase trajectories. The PO phase plane trajectories for a specific 01 value and for different initial conditions are shown in Fig .1. In Fig.1 one of the trajectories overcrosses the 9 -axis, thereby indicating an overshoot situation.
e
.1"I,'
e
f.jg.1; PO phase trajectories
It can be shown that for a particular 0 1 value the overshoot zone can be identified as 0 9(t) e(t)
0 <
1
""2
1
9(t) <- --- e(t) 2
if
e( t) >0 (11 )
~
0 9(t»-ye(t)
if
e( t) < 0
That is, any phase trajectory which starts from or passes through the overshoot zone is bound to overcross the e -axis, hence causing an overshoot . By increasing 0 1 the overshoot zone can be made arbitrarily small, however this corresponds to higher torque or force demands from actuators. If the demanded values exceed the actuator limits, this phenomenon is observed in the e -e plane as a phase trajectory deviating from the normal PO behaviour. It eventually overcrosses the asymptote described by Eq.10, enters the "overshoot zone" and finally overcrosses ttle 9 -axis which means overshoot. This consideration leads to too precautious designs which in order to meet the "no overshoot" requirement almost never make use of the actual capacity of the actuators . The VSS control scheme which is implemented here releases the constraints by switching automatically between the PO controller and an algorithm imposing constant error acceleration upon the system.
674
Application of the "Constant Error Acceleration" of the largest possible magnitude in the appropriate direction leads to the fastest way of reducing the error w~hout causing overshoot. In the following section this approach will be formulated as the "Constant Error Acceleration" (CEA) control scheme.
CEA CONTROL SCHEME The idea of CEA control is established upon first making the system accelerate very fast towards the reference trajectory, and then forcing ~ to decelerate as fast as possible so as to make e and ~ to converge to zero before an overshoot occurs . Obviously, the later the deceleration starts the better it is with respect to the response time. The fastest way of doing this is to apply the maximum available "Constant Error Acceleration" in the appropriate direction. Consider that maximum reference trajectory r( t) is bounded as follows (12)
where +
f
~ + and ~ - are the "maximum always available" accelerations in respective directions, and -
and f
are the factors between 0 and 1.
The "maximum always available error acceleration" is defined in both directions as follows : +
r- -
~ maa
~-
(13)
+
~~aa = r - ~+ Let us now consider the error dynamics for CEA control scheme . The analytical description of the phase trajectories can be found as follows =
de
(14)
~
Noting that ~ (t)
constant, and integrating in the time interval [to' t+to), the following expression
which represents a parabola as e(t) versus
e(t +t)
o
[~(t 0 + t) ] =
is obtained
2
~--2~(t)
~ (t)
(15)
+ e(t ) 0
As long as e and ~ have the same sign, the system is going to accelerate towards the reference trajectory. A danger of overshoot exists only when e and ~ have opposite signs, which corresponds to phase trajectories starting from the 2nd and 4th quadrants of the phase plane . From Eq.2 the e-axis intercept of the phase trajectory of a system starting with e(tO) and experiencing a constant error acceleration
~
can be found as:
~
(t 0) and
675
2 [ e( t )] 0
e (t + t) = e (t ) _ o
(16)
2~(t)
0
11 the e-axis intercept 01 a phase trajectory has the opposite sign as the initial position error e(tO)' this indicates an overshoot. Now, by applying time shift in Eq.16 overshoot conditions can be expressed as:
e (t)
[e(t)]
--
2
<
+
0
if
e(t) > 0
2 ~ maa
(17) e (t)
[e(t)]
2
- --
<
2~~aa
if
0
e (t) < 0
where +
r- - p-
~ maa
(18) +
+
= r -p =
++
1
+
++
P -P = -
(1- 1
)P
Hence the forbidden zone can be described as
e (t)
-
e (t)
-
2
[e(t)]
-2(1-f)~-
[ e(t)] - 2( 1- f
<
0
if
e( t) > 0
0
if
e( t) < 0
2
+)p +
>
This zone is shown as the shaded region in Fig .2. 11
• •••••••••• •••• • •• • e
...
•••••• • •••••
,-----.-...;~
e
=
•
e III
IV ~
Forbidden zone
(19)
676
In order to avoid overshoots the system should be prevented from entering this "forbidden zone" .
THE SWITCHING ALGORITHM If a PO controller is used along with a high gain u1 value, phase trajectories will approach a steep asymptote and may eventually enter the forbidden zone as shown in Fig .3. PO controller scheme is used until the phase trajectory hits the border line at the forbidden zone. From there onwards controller switches to CEA scheme , steering the system along the border line. When the border line intersects the PO asymptote the controller switches back to the PO scheme and the system converges to the origin along the PO asymptote . The switching points are depicted in Fig.3 . Consequently, the following control strategy which switches between PO and CEA controls can be outlined: 1. Calculate the error acceleration
~ pd
which can be realized through PO control and call it
2. Check if trajectory enters forbidden zone if 3. If YES, go to 5. 4. ELSE, apply ~ .
In
~ . In
were applied.
and go to 1.
5. Assign ~ . = ~ maa , apply CEA control. In
6. Check if trajectory leaves forbidden zone. 8. If YES, go to 1. 9. ELSE, go to 5.
e
for b idden zone
~:
Switching between PO and CEA controls
~ in .
677
SIMULATION
RESULTS
The approach which is described above has been applied to a SCARA type manipulator. The non linear and coupled equations of motion of the manipulator have been converted into second order decoupled linear differential equations to enable the application of this control scheme. For this particular application , critical value of a has been determined as 1.54. That is, a had to be greater than or equal to 1.54. In this work a has been chosen as 20 . Inverse kinematics problem has been solved by using a PVA (Position + Velocity + Acceleration) command generator, Vaccaro, Hill, 1988. The simulation results are presented in the Figs 4 through 7 .. Rewriting Eq.7 as
e (t)
k e
a 1t
alt
-2
-2
1
+ k2 t e
(20)
where
k
:;:
1
e(O)
and
k
2
Ta
(21 )
e(O) + 9(0)
Different choices of k1 and k2 have been considered in (Czen, 1989), to observe their effects on system behaviour. Results verify that the proposed strategy provides close tracking of the reference without overshoots.
0, 34
E
0 ,3 2
x
"
<1>
L
-0-
-+-
0, 30
-a-
tfl
<1>
"
desired x (m) PVA o ut x Cm) V SC out x (m )
0,28 time (sec)
0,26
iO
°
20
.EiQ.....4.; Trajectory tracking in the x component, k1 =2.0, k2=8 .0
2,OOe - l
E >-
~
-0-
1,00e - 1
-+-
L
-a-
tfl
d esired y ('11) PVA ou t y (m) VSC out y (m)
Cl.>
" I ,3 6e - 20 - t - - - --- , - - - - - - - - r - - - - - . - - - - - - , o ~
10
time (sec)
20
Trajectory tracking in the y component, k1 =2.0, k2=8 .0
678
E x "U I!>
L
If)
I!>
"U
0,33 0,32 0, 3 1 0,30 0,29
.........
-G-
0,28 C,2 7
desired x (rn) PVA out x (m) VSC out x (rn)
time (sec)
0, 26
0
10
20
.Eig,.,..6.; Trajectory tracking in the x component, k1 =0.5, k2=2.0
2,OOe-l
E >-
-g
deSlred y (rn) .... PV Aouty(rn) ou t ( rn )
-G-
1,OO e - 1
L
..... vsc
(f)
I!>
y
"U
! ,36e - 20
+------.-----r----.....-----.. o
10
time (sec)
20
.E.i.Q...L Trajectory tracking in the y component, k1 =0 .5, k2=2 .0
REFERENCES Denizhan , Y., 1988. "A Variable Structure Control Algorithm for Robotic Systems and a Learning Scheme for Load Adaptation", Ph,D Dissertation. SoOazici University. Isidori, A., 1985, "Nonlinear Control Systems : An Introduction",Lecture Notes in Control and Information Sciences , Springer-Verlag. Koivo, A. J., 1989, "Fundamentals for Control of Robotic Manipulators", John Wiley & Soos. Inc . Ozeo, F., 1989, "Variable Structure Control of a Manipulator with Four Degrees of Freedom", M,S. Thesis, BoQazici University. Vaccaro, R. J. and Hill, S.D., 1968, "A Joint Space Command Generator for Cartesian Control of Robotics Manipulators", IEEE Journal of Robotics and Automation, Vol. 4, No. 1, pp. 70-76.
VARIABLE STRUCTURE CONTROL OF A MANIPULATOR WITH FOUR DEGREES OF FREEDOM
Figen Ozen, Ahmet Denker Department of Electrical and Electronic Engineering, Bo~azi~i University Bebek 80815, istanbul, TURKEY
Keywords : manipulator control, variable structure control, decoupling theory, PVA command generator
Abstract : A VSS control which switches between PO (Proportional and Derivative) and CEA (Constant Error Acceleration) schemes is presented in this article. It is applied to steer robot manipulators in order to satisfy "no overshoot" and "high response speed" requirements . Simulation results verify that with this approach, maximum actuator capacity can be exploited while avoiding danger of overshoots .
INTRODUCTION
Close tracking of the reference without overshoots is an essential concern in many robotic applications, such as collision avoidance . While the "no overshoot" requirement can be met by applying critically damped PO controller to joints with Iinearized dynamics, it must be assured that limits of the actuators must not be exceeded . This consideration leads to overcautious designs which refrain from using the full capacity of the actuators, thereby leading to sluggish behaviour. In order to meet the "no overshoot" requirement while maintaining high response speed, an alternative approach is presented here which implements the strategy proposed in (Denizhan, 1988). This strategy is based upon switching between PO (proportional and Derivative) control and CEA (Constant Error Acceleration) schemes, in order to exploit the maximum actuator capacity, while avoiding danger at overshoots .
PO CONTROL SCHEME With the task description y solved in the joint space, the robot manipulator dynamics can be represented (Koivo, 1989) as
x
f[x(t),U(t)]
y
h( x)
(1 )
where x(t) E Rn and u(t) E Rm. The objective of the VSS control strategy is to steer the states of the system in a subspace where the requirements of the "high response speed" and "no overshoot" can be satisfied . Using the nonlinear feedback of the form (Isidori, 1985)
672
u(t)
=
(2)
A(x, t) Cl(t) + 8(x, t)
the input output behaviour of the system in Eq .1 becomes the same as the one of a linear system
y . (t)
Q .(t , x ) +
1
1
I=,
i
0
(3)
jh .(t-t)U .(t)dt 0
1
1
When PD control is implemented, task equations for each joint can be written as y .(t)
=
1
(4)
t.(t) + Cl, . [f .(t)- y .(t)]+ Clo .[r.(t)- y .(t)] 1
1
1
1
1
1
1
Since both "high response speed" and "no overshoot" requirements can be expressed in terms of error and its first derivative, it is quite convenient to design the control law with respect to error phase plane behaviour (Denizhan, 1988). Noting that
=
e.(t) 1
r .(t)-y .(t) 1
(5)
I
and choosing for critical damping 2
Cl .
Cl
=
_ 11
(6)
4
Oi
Using unsubscribed variables for the sake of clarity, error equations for the "no overshoot" condition can be obtained as
a 1
{ e (0) + [ e (0) + :
e (t)
e(t)
=
e( 0 )]t} e -
-+
e(O)+2e(O)
Cl, --
l
1
Cl1
+ --2 e(O i +[e(O)+
:1
(7)
1
(8)
e(t)
e (O)}
The design goal can now be formulated as bringing any initial point on the error phase plane to the origin as fast as possible without overcrossing the e -axis. Taking the limit as t tends to infinity, the following relation between
e
and e can be obtained
ex
lim e(t) t--+oo
= - _J.
(9)
lim e (t) 2 1--+ 00
Hence the state trajectories in
e -e approach the
origin along the asymptote
673
(10)
Besides, the choice of parameters as in Eq.S leads to a critically damped behaviour, giving rise to convergent parabolic phase trajectories. The PO phase plane trajectories for a specific 01 value and for different initial conditions are shown in Fig .1. In Fig.1 one of the trajectories overcrosses the 9 -axis, thereby indicating an overshoot situation.
e
.1"I,'
e
f.jg.1; PO phase trajectories
It can be shown that for a particular 0 1 value the overshoot zone can be identified as 0 9(t) e(t)
0 <
1
""2
1
9(t) <- --- e(t) 2
if
e( t) >0 (11 )
~
0 9(t»-ye(t)
if
e( t) < 0
That is, any phase trajectory which starts from or passes through the overshoot zone is bound to overcross the e -axis, hence causing an overshoot . By increasing 0 1 the overshoot zone can be made arbitrarily small, however this corresponds to higher torque or force demands from actuators. If the demanded values exceed the actuator limits, this phenomenon is observed in the e -e plane as a phase trajectory deviating from the normal PO behaviour. It eventually overcrosses the asymptote described by Eq.10, enters the "overshoot zone" and finally overcrosses ttle 9 -axis which means overshoot. This consideration leads to too precautious designs which in order to meet the "no overshoot" requirement almost never make use of the actual capacity of the actuators . The VSS control scheme which is implemented here releases the constraints by switching automatically between the PO controller and an algorithm imposing constant error acceleration upon the system.
674
Application of the "Constant Error Acceleration" of the largest possible magnitude in the appropriate direction leads to the fastest way of reducing the error w~hout causing overshoot. In the following section this approach will be formulated as the "Constant Error Acceleration" (CEA) control scheme.
CEA CONTROL SCHEME The idea of CEA control is established upon first making the system accelerate very fast towards the reference trajectory, and then forcing ~ to decelerate as fast as possible so as to make e and ~ to converge to zero before an overshoot occurs . Obviously, the later the deceleration starts the better it is with respect to the response time. The fastest way of doing this is to apply the maximum available "Constant Error Acceleration" in the appropriate direction. Consider that maximum reference trajectory r( t) is bounded as follows (12)
where +
f
~ + and ~ - are the "maximum always available" accelerations in respective directions, and -
and f
are the factors between 0 and 1.
The "maximum always available error acceleration" is defined in both directions as follows : +
r- -
~ maa
~-
(13)
+
~~aa = r - ~+ Let us now consider the error dynamics for CEA control scheme . The analytical description of the phase trajectories can be found as follows =
de
(14)
~
Noting that ~ (t)
constant, and integrating in the time interval [to' t+to), the following expression
which represents a parabola as e(t) versus
e(t +t)
o
[~(t 0 + t) ] =
is obtained
2
~--2~(t)
~ (t)
(15)
+ e(t ) 0
As long as e and ~ have the same sign, the system is going to accelerate towards the reference trajectory. A danger of overshoot exists only when e and ~ have opposite signs, which corresponds to phase trajectories starting from the 2nd and 4th quadrants of the phase plane . From Eq.2 the e-axis intercept of the phase trajectory of a system starting with e(tO) and experiencing a constant error acceleration
~
can be found as:
~
(t 0) and
675
2 [ e( t )] 0
e (t + t) = e (t ) _ o
(16)
2~(t)
0
11 the e-axis intercept 01 a phase trajectory has the opposite sign as the initial position error e(tO)' this indicates an overshoot. Now, by applying time shift in Eq.16 overshoot conditions can be expressed as:
e (t)
[e(t)]
--
2
<
+
0
if
e(t) > 0
2 ~ maa
(17) e (t)
[e(t)]
2
- --
<
2~~aa
if
0
e (t) < 0
where +
r- - p-
~ maa
(18) +
+
= r -p =
++
1
+
++
P -P = -
(1- 1
)P
Hence the forbidden zone can be described as
e (t)
-
e (t)
-
2
[e(t)]
-2(1-f)~-
[ e(t)] - 2( 1- f
<
0
if
e( t) > 0
0
if
e( t) < 0
2
+)p +
>
This zone is shown as the shaded region in Fig .2. 11
• •••••••••• •••• • •• • e
...
•••••• • •••••
,-----.-...;~
e
=
•
e III
IV ~
Forbidden zone
(19)
676
In order to avoid overshoots the system should be prevented from entering this "forbidden zone" .
THE SWITCHING ALGORITHM If a PO controller is used along with a high gain u1 value, phase trajectories will approach a steep asymptote and may eventually enter the forbidden zone as shown in Fig .3. PO controller scheme is used until the phase trajectory hits the border line at the forbidden zone. From there onwards controller switches to CEA scheme , steering the system along the border line. When the border line intersects the PO asymptote the controller switches back to the PO scheme and the system converges to the origin along the PO asymptote . The switching points are depicted in Fig.3 . Consequently, the following control strategy which switches between PO and CEA controls can be outlined: 1. Calculate the error acceleration
~ pd
which can be realized through PO control and call it
2. Check if trajectory enters forbidden zone if 3. If YES, go to 5. 4. ELSE, apply ~ .
In
~ . In
were applied.
and go to 1.
5. Assign ~ . = ~ maa , apply CEA control. In
6. Check if trajectory leaves forbidden zone. 8. If YES, go to 1. 9. ELSE, go to 5.
e
for b idden zone
~:
Switching between PO and CEA controls
~ in .
677
SIMULATION
RESULTS
The approach which is described above has been applied to a SCARA type manipulator. The non linear and coupled equations of motion of the manipulator have been converted into second order decoupled linear differential equations to enable the application of this control scheme. For this particular application , critical value of a has been determined as 1.54. That is, a had to be greater than or equal to 1.54. In this work a has been chosen as 20 . Inverse kinematics problem has been solved by using a PVA (Position + Velocity + Acceleration) command generator, Vaccaro, Hill, 1988. The simulation results are presented in the Figs 4 through 7 .. Rewriting Eq.7 as
e (t)
k e
a 1t
alt
-2
-2
1
+ k2 t e
(20)
where
k
:;:
1
e(O)
and
k
2
Ta
(21 )
e(O) + 9(0)
Different choices of k1 and k2 have been considered in (Czen, 1989), to observe their effects on system behaviour. Results verify that the proposed strategy provides close tracking of the reference without overshoots.
0, 34
E
0 ,3 2
x
"
<1>
L
-0-
-+-
0, 30
-a-
tfl
<1>
"
desired x (m) PVA o ut x Cm) V SC out x (m )
0,28 time (sec)
0,26
iO
°
20
.EiQ.....4.; Trajectory tracking in the x component, k1 =2.0, k2=8 .0
2,OOe - l
E >-
~
-0-
1,00e - 1
-+-
L
-a-
tfl
d esired y ('11) PVA ou t y (m) VSC out y (m)
Cl.>
" I ,3 6e - 20 - t - - - --- , - - - - - - - - r - - - - - . - - - - - - , o ~
10
time (sec)
20
Trajectory tracking in the y component, k1 =2.0, k2=8 .0
678
E x "U I!>
L
If)
I!>
"U
0,33 0,32 0, 3 1 0,30 0,29
.........
-G-
0,28 C,2 7
desired x (rn) PVA out x (m) VSC out x (rn)
time (sec)
0, 26
0
10
20
.Eig,.,..6.; Trajectory tracking in the x component, k1 =0.5, k2=2.0
2,OOe-l
E >-
-g
deSlred y (rn) .... PV Aouty(rn) ou t ( rn )
-G-
1,OO e - 1
L
..... vsc
(f)
I!>
y
"U
! ,36e - 20
+------.-----r----.....-----.. o
10
time (sec)
20
.E.i.Q...L Trajectory tracking in the y component, k1 =0 .5, k2=2 .0
REFERENCES Denizhan , Y., 1988. "A Variable Structure Control Algorithm for Robotic Systems and a Learning Scheme for Load Adaptation", Ph,D Dissertation. SoOazici University. Isidori, A., 1985, "Nonlinear Control Systems : An Introduction",Lecture Notes in Control and Information Sciences , Springer-Verlag. Koivo, A. J., 1989, "Fundamentals for Control of Robotic Manipulators", John Wiley & Soos. Inc . Ozeo, F., 1989, "Variable Structure Control of a Manipulator with Four Degrees of Freedom", M,S. Thesis, BoQazici University. Vaccaro, R. J. and Hill, S.D., 1968, "A Joint Space Command Generator for Cartesian Control of Robotics Manipulators", IEEE Journal of Robotics and Automation, Vol. 4, No. 1, pp. 70-76.