- Email: [email protected]
Yet Another Fractal in Pendulum System
Copyright © IFAC Mechatronic Systems, Sydney, Australia, 2004
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocatelifac
YET ANOTHER FRACTAL IN PENDULUM SYSTEM Masaki Yamakita • Toshiyasu Yonemura ••
• Tokyo Tech.!RlKEN BMC, Japan •• Tokyo Tech, Japan
Abstract: In this paper we investigate a sensitivity of initial state for swing up control of Acrobot by a non-linear feedback control inspired by a paper by Furuta. The nonlinear control is an output zeroing controller where the output function is composed of angular momentum of the system and the zero dynamics is stable due to the dynamic nature. Initial states are a~signed each color depending on a time in which the acrobot is swung up to the upright position. It will be shown that the colored region of the initial state exhibits an another fractal structure. Copyright © 2004 [FAC Keywords: Acrobot, Swing-up Control, Output Zeroing, Fractal
input so that the divergence of the control input is avoided. If we use the modified control input, global stability can not be assured any more, however, it is shown that the control can swing up the acrobot in a wide range of the initial state of the system and it is also shown that the time period in which the system is swung up is very sensitive to the initial state by numerical simulations. Inspired by a paper (K.Furuta, 2003) the initial state is colored according to the period of swing up and it is shown that the resultant figure shows an another fractal structure.
1. INTRODUCTION
Under-actuated mechanical systems are systems whose number of actuators is less than that of the freedom. (R.M.Murray and A.S.Sastry, 1993) A two-link serial manipulator in a plain whose 1st joint is passive is called acrobot (M.W.Spong, 1995) and considered as a simple model of gymnast with a horizontal bar. Several swing up control methods were investigated for the system and Nam proposed a swing up control method based on a special coordinate transformation and introduction of an output function which is composed of the angular momentum and which attains a stable zero dynamics. (L. Cambrini, 2(00)(M. Yamakita and T. Yonemura, 2002)(T.Nam and M.Yamakita, 2002) If we consider that the output function is one of the state coordinates, state transformation between the original physical coordinates and the new ones is not defined globally and it becomes singular in a set of the state space. If we consider the swing up control, every trajectory of the solution should traverse the region and the control input is not defined on the set. In order to avoid the problem. a sinusoidal open loop control is used to pomp up the energy for the system to go across the singular regiolL
Active joint I st link Passive joint
-.I
x
Fig. 1. 2 links acrobot model 2. MODEL
In this paper we modify the control method to introduce a regularization term to determine the control
As in Fig! we consider a two-link acrobot where the first joint is passive and control torque is only applied
343
to the second one. We assume that mass, length, position of the center of mass, inertia moment are denoted by 1nl , 1n2, 11 , 12, ai, a2 and J 1 , h, respectively. Angles are defined and denoted e and 1jJ as in the figure.
(-7r/ 2, 0) to (e, 1/J ) = (~(2n+1),2k7r),(n,k = 0, 1", . ), which means that the control objective is to derive a stabilizing controller of q defined as eq. (8).
The dynamic equation can be describe as (T.Nam and M.Yamakita, 2002):
+ ')' + 2(3 cos 1jJ)0 + (r + (3 cos 1jJ);j; (1) - (3J) (2B + ~) sin 1/J + k1 cose + k2 cos(e + 1/J) = 0
(0:
As in eq.(8). if we can consider that the angular velocity,1jJ, is a control input and it is easily determined so that (p , L) is stabilized, then u = ;j; can be easily determined by a backstepping approach as follows .
(2) ('Y + (J cos 1/J) + ,),;j; + (30 2 sin 1jJ + k2 cos(e + 1/J)
=T
2.1 ContrallAw
(3)
In this section a fundamental control is derived and the modification is explained. In order to derive the control input u. tentatively we assume that v = ~ is a control input and the following system is considered :
where T is a control torque for 1jJ and 0:,(3 ,')', kl,k2 are defined as
0: = h
+ mIai + 1n21i, (3 = 1n21Ia2,
+m2a~ kl = (mlal + 1n2h)g, k2
')' = h
= m2 a 2g ·
First, we consider a state transformation so that a nonlinear control is easily derived. Using nonlinear feedback we can partially linearize the dynamics so that it can be represented as ~; = u where u is a new input. The angular momentum,L, around the 1st joint can be represented as
= (0: + ')' + 2{3cos'1jJ)B + (r + (3 COS '1jJ) ~
L
For this system we define an output function y as
y
kl cose - k2 cos(1jJ
+ e) .
(13)
(5)
where a2 > 0, a3 > 0 are designed parameters. Since from eq.(8) we have L = h(1/J )p. y = stands for
°
Using the angular momentum expressed in eq(4), a new function p satisfying
L
= (0: + ')' + 2(3 cos 1/J )p.
y
(6)
:=
r'" ')' + (3 cos 1jJ
10
0:
+ ')' + 2(3 cos 1/J
The determined control input can be also interpreted as follows . Let assume a Lyapunov function candidate
(7)
where C is an integral constant and it is determined as P = 0 when the system is at the upright position. If we choose a set of coordinate functions as q = (p , L , 1/), J)). a new system representation can given as
[
~P1 ~
g(r
[L / h(1/J)1
(15)
Then the control input above is the one that satisfies
+ [01 ~
(8)
u,
The control input v can be determined from eq(ll) and by substituting
where h(1/!)
= 0: + ')' + 2(3 cos 1jJ > 0, = -kl cos (p - w(1/J )) -
g(p,1/J)
(14)
°
d1jJ _ C
e + w(,V) )
= h(1/J )p + al P = O.
Due to the dynamic nature of the system it is shown that h( 1/J) > and the dynamics of p is stable and it is assured that p converges to zero.
is defined. From this equation P can be determined as P := e +
(12)
where al > 0 is a design parameter. Since the output function y has a relative degree 2 from the input ~. the control input is determined so that the dynamics of the output function satisfies
(4)
and it can be easily shown that the time derivative of L just contains a gravity term and it is calculated as
t =-
= L + alP ,
(9)
k2 cos(p - w(1jJ)
.. ay L ag L=aph + a1/J v
+ 1/J) .
(10)
.. t
P=
The control becomes a regulation problem so that the state of the above system is transfered from (e , 1jJ) =
to (13) as
344
h-
al
(ahla1/J )L h2
(17)
(18)
D '= -( f!JL
.
a~
8gL N := (8p h
_ CLI (ah/a~)L) h2
'
9
. + al h + a2Y + a3Y).
Then we consider the following criterion function to determine the control : "Inlll)
J = iiDv Fig. 2. Singularity of control law v
= F1(p, L,~)
8g
= -(81/.] - al
8gL
x (8p
)
9
v=DN/(D*D+f:).
h + al h + a2Y + a3y).(19)
Please notice here that v becomes zero when D Based on the obtained v, u is detennined using a backstepping. Let assume that v - Fl i= 0 and eq. (l3) is not satisfied, and define a Lyapunov function candidate, Vi! , of the state q = (p, L , 7/), .,j;) as
V2
= Vi +
(25)
=
O.
Furthermore, we introduce a truncation of signals as follows
0- { 0 - 27f (0:2: 3/27f)
0 + 27f (0 ~ -5/27f)
-
(v - F 1 )2 2 > O.
(24)
where the second term is a regularization term. The v is determined to minimize the criterion function and it is given by
(8h/8~)L - 1
h2
NW + fiivW
(20)
~=
For this Lyanplmov function candidate u is determined so that V2 is negative semi-definite. From v = u, eq.(l7), eq.(18) and the definition of FI in eq.(l9), the control input u is determined to satisfy
~ - 27f (~
{
:2: 27f)
~ + 27f (~ ~
-27f)
(26)
(27)
since even though the state of the system is same physically, the value of the output function is different. 3. NUMERICAL SIMULATION
(21)
In order to test the validity of the control method, we measured a time in which the swing up control is finished where it is determined when the norm of the state becomes less than 0.1 for each initial state and if the the state did not converge to the specified region within 20 [sec] and the swing up time was set to 20 [sec]. In Fig. 3 the initial state is colored depending on the time where the bright color correspond~ to short period and the dark color does to long one and the vertical and horizontal lines show e and ~ , respectively where the min. and max. values of each axes are -7f ~ e ~ 7f, - 7f ~ V) ~ 7r. From this figure it can be observed that a fractal structure exists in the initial state space.
and it can be expressed as
.
ag
IJ.=Fl - a4(v - Fd-(a~-CLl
where CL4
(ah/a~)L
h2
)f:?2)
°
> is a design parameter.
2.2 Singularity of control We consider a set of the state in which the control input obtained in the last section is not defined. From eq. (19) the set is given the state for which the denominator of the control input becomes zero and it expressed as the state which satisfies
ag (ah/a~)L if> := (j ;J; - al -----~. = 0,
4. CONCLUSIONS
(23)
As inspired by a paper by Furuta, we investigated a sensitivity of time in which swing up control is completed by a nonlinear feedback control w.r.t. the initial state of the acrobot. The nonlinear feedback control considered in the paper is practically global controller though the convergence is not guaranteed theoretically but the numerical simulation showed that region of the attractor is very large. It was also shown that some fractal structure exists in the initial state space when we consider a swing up time as a parameter.
and the set is actually a plain in a space of (0, ~, L). In Fig. 2 a projected line of the singularity plain onto (O,~) plain is plotted when aj = 1 and L = 1. From this figure it can be seen that any trajectory cormecting the hanging position,(O,~) = (-7f /2, ~ = 0), to the upright position, (0, ~) = (7f /2 + 7f X n, 27f x k)(n , k = 0,1 ", , ), does cross the plain. Therefore the control of eq. (22) can not be used globally. Then we consider a simple modification of the control law. Let assume that the control law in eq (19) be expressed as v=N/ D where
345
20 [sec]
O[sec]
-J[ - J[
lj; [rad ]
Fig. 3. Sensitivety of the control time w.r.t. the initial state. S. REFERENCES
K.Furuta (2003) . Control of pendulum: From super mechano-system to human adaptive mechatronics. Proc. of CDC 20030, 1498-1507. L. Cambrini, et. a1. (2000). Stable trajectory tracking for biped robots. Proc. of CDC 2000. M.W.Spong (1995). The swing up control problem for the acrobat. IEEE control systems pp. 49-55. M. Yamakita and T.Yonemura (2002). Stabilization of acrobat robot in upright position on a horizontal bar. Proc. of1EEE lnt. ConI on Robotics and Automation. M.Yarnakita and TYonemura (2004). Swing up control of acrobat based on switched output functions. Proc. of SICE Annual Con! R.M.Murray, Z.Li and A.S.Sastry (1993). A Mathematical Introduction to Robotic Manipulation. CRC Press. TNarn, TMita and M.Yarnakita (2002). Swing-up control and singular problem of a acrobat sytem. journal of Robotics Society, Japan (in Japanese).
346
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocatelifac
YET ANOTHER FRACTAL IN PENDULUM SYSTEM Masaki Yamakita • Toshiyasu Yonemura ••
• Tokyo Tech.!RlKEN BMC, Japan •• Tokyo Tech, Japan
Abstract: In this paper we investigate a sensitivity of initial state for swing up control of Acrobot by a non-linear feedback control inspired by a paper by Furuta. The nonlinear control is an output zeroing controller where the output function is composed of angular momentum of the system and the zero dynamics is stable due to the dynamic nature. Initial states are a~signed each color depending on a time in which the acrobot is swung up to the upright position. It will be shown that the colored region of the initial state exhibits an another fractal structure. Copyright © 2004 [FAC Keywords: Acrobot, Swing-up Control, Output Zeroing, Fractal
input so that the divergence of the control input is avoided. If we use the modified control input, global stability can not be assured any more, however, it is shown that the control can swing up the acrobot in a wide range of the initial state of the system and it is also shown that the time period in which the system is swung up is very sensitive to the initial state by numerical simulations. Inspired by a paper (K.Furuta, 2003) the initial state is colored according to the period of swing up and it is shown that the resultant figure shows an another fractal structure.
1. INTRODUCTION
Under-actuated mechanical systems are systems whose number of actuators is less than that of the freedom. (R.M.Murray and A.S.Sastry, 1993) A two-link serial manipulator in a plain whose 1st joint is passive is called acrobot (M.W.Spong, 1995) and considered as a simple model of gymnast with a horizontal bar. Several swing up control methods were investigated for the system and Nam proposed a swing up control method based on a special coordinate transformation and introduction of an output function which is composed of the angular momentum and which attains a stable zero dynamics. (L. Cambrini, 2(00)(M. Yamakita and T. Yonemura, 2002)(T.Nam and M.Yamakita, 2002) If we consider that the output function is one of the state coordinates, state transformation between the original physical coordinates and the new ones is not defined globally and it becomes singular in a set of the state space. If we consider the swing up control, every trajectory of the solution should traverse the region and the control input is not defined on the set. In order to avoid the problem. a sinusoidal open loop control is used to pomp up the energy for the system to go across the singular regiolL
Active joint I st link Passive joint
-.I
x
Fig. 1. 2 links acrobot model 2. MODEL
In this paper we modify the control method to introduce a regularization term to determine the control
As in Fig! we consider a two-link acrobot where the first joint is passive and control torque is only applied
343
to the second one. We assume that mass, length, position of the center of mass, inertia moment are denoted by 1nl , 1n2, 11 , 12, ai, a2 and J 1 , h, respectively. Angles are defined and denoted e and 1jJ as in the figure.
(-7r/ 2, 0) to (e, 1/J ) = (~(2n+1),2k7r),(n,k = 0, 1", . ), which means that the control objective is to derive a stabilizing controller of q defined as eq. (8).
The dynamic equation can be describe as (T.Nam and M.Yamakita, 2002):
+ ')' + 2(3 cos 1jJ)0 + (r + (3 cos 1jJ);j; (1) - (3J) (2B + ~) sin 1/J + k1 cose + k2 cos(e + 1/J) = 0
(0:
As in eq.(8). if we can consider that the angular velocity,1jJ, is a control input and it is easily determined so that (p , L) is stabilized, then u = ;j; can be easily determined by a backstepping approach as follows .
(2) ('Y + (J cos 1/J) + ,),;j; + (30 2 sin 1jJ + k2 cos(e + 1/J)
=T
2.1 ContrallAw
(3)
In this section a fundamental control is derived and the modification is explained. In order to derive the control input u. tentatively we assume that v = ~ is a control input and the following system is considered :
where T is a control torque for 1jJ and 0:,(3 ,')', kl,k2 are defined as
0: = h
+ mIai + 1n21i, (3 = 1n21Ia2,
+m2a~ kl = (mlal + 1n2h)g, k2
')' = h
= m2 a 2g ·
First, we consider a state transformation so that a nonlinear control is easily derived. Using nonlinear feedback we can partially linearize the dynamics so that it can be represented as ~; = u where u is a new input. The angular momentum,L, around the 1st joint can be represented as
= (0: + ')' + 2{3cos'1jJ)B + (r + (3 COS '1jJ) ~
L
For this system we define an output function y as
y
kl cose - k2 cos(1jJ
+ e) .
(13)
(5)
where a2 > 0, a3 > 0 are designed parameters. Since from eq.(8) we have L = h(1/J )p. y = stands for
°
Using the angular momentum expressed in eq(4), a new function p satisfying
L
= (0: + ')' + 2(3 cos 1/J )p.
y
(6)
:=
r'" ')' + (3 cos 1jJ
10
0:
+ ')' + 2(3 cos 1/J
The determined control input can be also interpreted as follows . Let assume a Lyapunov function candidate
(7)
where C is an integral constant and it is determined as P = 0 when the system is at the upright position. If we choose a set of coordinate functions as q = (p , L , 1/), J)). a new system representation can given as
[
~P1 ~
g(r
[L / h(1/J)1
(15)
Then the control input above is the one that satisfies
+ [01 ~
(8)
u,
The control input v can be determined from eq(ll) and by substituting
where h(1/!)
= 0: + ')' + 2(3 cos 1jJ > 0, = -kl cos (p - w(1/J )) -
g(p,1/J)
(14)
°
d1jJ _ C
e + w(,V) )
= h(1/J )p + al P = O.
Due to the dynamic nature of the system it is shown that h( 1/J) > and the dynamics of p is stable and it is assured that p converges to zero.
is defined. From this equation P can be determined as P := e +
(12)
where al > 0 is a design parameter. Since the output function y has a relative degree 2 from the input ~. the control input is determined so that the dynamics of the output function satisfies
(4)
and it can be easily shown that the time derivative of L just contains a gravity term and it is calculated as
t =-
= L + alP ,
(9)
k2 cos(p - w(1jJ)
.. ay L ag L=aph + a1/J v
+ 1/J) .
(10)
.. t
P=
The control becomes a regulation problem so that the state of the above system is transfered from (e , 1jJ) =
to (13) as
344
h-
al
(ahla1/J )L h2
(17)
(18)
D '= -( f!JL
.
a~
8gL N := (8p h
_ CLI (ah/a~)L) h2
'
9
. + al h + a2Y + a3Y).
Then we consider the following criterion function to determine the control : "Inlll)
J = iiDv Fig. 2. Singularity of control law v
= F1(p, L,~)
8g
= -(81/.] - al
8gL
x (8p
)
9
v=DN/(D*D+f:).
h + al h + a2Y + a3y).(19)
Please notice here that v becomes zero when D Based on the obtained v, u is detennined using a backstepping. Let assume that v - Fl i= 0 and eq. (l3) is not satisfied, and define a Lyapunov function candidate, Vi! , of the state q = (p, L , 7/), .,j;) as
V2
= Vi +
(25)
=
O.
Furthermore, we introduce a truncation of signals as follows
0- { 0 - 27f (0:2: 3/27f)
0 + 27f (0 ~ -5/27f)
-
(v - F 1 )2 2 > O.
(24)
where the second term is a regularization term. The v is determined to minimize the criterion function and it is given by
(8h/8~)L - 1
h2
NW + fiivW
(20)
~=
For this Lyanplmov function candidate u is determined so that V2 is negative semi-definite. From v = u, eq.(l7), eq.(18) and the definition of FI in eq.(l9), the control input u is determined to satisfy
~ - 27f (~
{
:2: 27f)
~ + 27f (~ ~
-27f)
(26)
(27)
since even though the state of the system is same physically, the value of the output function is different. 3. NUMERICAL SIMULATION
(21)
In order to test the validity of the control method, we measured a time in which the swing up control is finished where it is determined when the norm of the state becomes less than 0.1 for each initial state and if the the state did not converge to the specified region within 20 [sec] and the swing up time was set to 20 [sec]. In Fig. 3 the initial state is colored depending on the time where the bright color correspond~ to short period and the dark color does to long one and the vertical and horizontal lines show e and ~ , respectively where the min. and max. values of each axes are -7f ~ e ~ 7f, - 7f ~ V) ~ 7r. From this figure it can be observed that a fractal structure exists in the initial state space.
and it can be expressed as
.
ag
IJ.=Fl - a4(v - Fd-(a~-CLl
where CL4
(ah/a~)L
h2
)f:?2)
°
> is a design parameter.
2.2 Singularity of control We consider a set of the state in which the control input obtained in the last section is not defined. From eq. (19) the set is given the state for which the denominator of the control input becomes zero and it expressed as the state which satisfies
ag (ah/a~)L if> := (j ;J; - al -----~. = 0,
4. CONCLUSIONS
(23)
As inspired by a paper by Furuta, we investigated a sensitivity of time in which swing up control is completed by a nonlinear feedback control w.r.t. the initial state of the acrobot. The nonlinear feedback control considered in the paper is practically global controller though the convergence is not guaranteed theoretically but the numerical simulation showed that region of the attractor is very large. It was also shown that some fractal structure exists in the initial state space when we consider a swing up time as a parameter.
and the set is actually a plain in a space of (0, ~, L). In Fig. 2 a projected line of the singularity plain onto (O,~) plain is plotted when aj = 1 and L = 1. From this figure it can be seen that any trajectory cormecting the hanging position,(O,~) = (-7f /2, ~ = 0), to the upright position, (0, ~) = (7f /2 + 7f X n, 27f x k)(n , k = 0,1 ", , ), does cross the plain. Therefore the control of eq. (22) can not be used globally. Then we consider a simple modification of the control law. Let assume that the control law in eq (19) be expressed as v=N/ D where
345
20 [sec]
O[sec]
-J[ - J[
lj; [rad ]
Fig. 3. Sensitivety of the control time w.r.t. the initial state. S. REFERENCES
K.Furuta (2003) . Control of pendulum: From super mechano-system to human adaptive mechatronics. Proc. of CDC 20030, 1498-1507. L. Cambrini, et. a1. (2000). Stable trajectory tracking for biped robots. Proc. of CDC 2000. M.W.Spong (1995). The swing up control problem for the acrobat. IEEE control systems pp. 49-55. M. Yamakita and T.Yonemura (2002). Stabilization of acrobat robot in upright position on a horizontal bar. Proc. of1EEE lnt. ConI on Robotics and Automation. M.Yarnakita and TYonemura (2004). Swing up control of acrobat based on switched output functions. Proc. of SICE Annual Con! R.M.Murray, Z.Li and A.S.Sastry (1993). A Mathematical Introduction to Robotic Manipulation. CRC Press. TNarn, TMita and M.Yarnakita (2002). Swing-up control and singular problem of a acrobat sytem. journal of Robotics Society, Japan (in Japanese).
346