Force Control of Robotic Manipulator with Consideration of Collision Phenomena

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IFAC 11th Tril'lIl1i,d \I'"rld

Congress. Tallillll. FSlollia. l·SSR.

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STABLE POSITION/FORCE CONTROL OF ROBOTIC MANIPULATOR WITH CONSIDERATION OF COLLISION PHENOMENA T. Fukuda*, Y. Shoji** and M. Inaba** *Department of Mechanical Engineering, Nagoya University, 1 Furo-cho, Chigusa-ku, Nagoya 464-01, Japan **Applied Technology Group, Toyo Engineering Corporation, 12-10, 6-chome, HigashifwUlbashi, Funabashi-shi, Chiba 273, Japan

Abstruct. This paper presents the stabilization effect of collision in the position/force controlled manipulator-object systems. The stability of the system behavior is analyzed by Lyapunov's direct method. Some simulation results are also shown to confirm the effect. Collision is one of the nonlinear problems with structure changing. In the industrial fields, the phenomenon is utilized effectively to suppress vibrations, while collision is a difficult problem to treat in the fields of control, as the methodology is mostly based on the linear theory. In this paper, the nonlinear impact force is modelled as a lierz type force with damping. A nonlinear matrix and a quadratic form is defined to examine the system stability. As a result, stable control with collision phenomena is obtained and is expected be applied to the practical robot control in a wide variety of applications such fast assembling or fast insertion tasks. Keywords.

Stabili ty; collision; simulation

Lyapunov method;

INTRODUCTION

Recently the demands for faster performance of robotic manipulators has become essential, especially in the industrial fields, as factory automation has been developed. (Fukuda, Kitamura and Tanie, 1987a, 1987b )This results in higher efficiency in faster assembling or other tasks. Furthermore, the demand for more variable and more precise performances are also increasing . These tasks must include the interaction of manipulators and environment or objects to be treated. Particulary when the motion of the manipulator is fast, collision occurs between them. (Colgate and Hogan, 1989; Kahng and Amirouche, 1988; YousefToumi and Gutz, 1989 ) The collision phenomenon often causes poor controllability for the manipulators . This is because even the latest control theory hardly treats the strong nonlinearity of the phenomenon, also because little research has been and undertaken to solve its poor controllability. Thus, even now manipulators are controlled to avoid the collision crudely. For instance, in a masterslave type teleoperation, the slave side is controlled so that the relative velocity becomes very small on contact to avoid the collision. ( Tanie and co-workers, 1988 ) However, it is obvious that collision is a problem to be solved if manipulators are needed to be used for fast tasks in practical applications, such as hammering or insertion. In this paper, a position / force controlled grasping of objects is discussed with consideration of COllision phenomena, and its stability and transient response are studied. For the simplicity, the system consists of two I - D.D.F. manipulators With an object. Their motions are confined to a

nonlinear

systems;

to as

robots;

straight line. The two manipulators are individually controlled with linear feedback and the object moves freely along the line. The manipulators collide and repulse against the object without any limitation of repetation in number. The stability of this nonlinear system is analyzed by the Lyapunov's direct method and is validated by numerical simulation. The transient responses are examined by the simulation at the same time. As a result, it is shown thnt stahle grasping is possible even if the manipulators motion includes collision with the environment. Moreover, under some conditions such as marginally stable conditions, the collision phcnomenon has a stabilizing effect on the system. MATHFJlATICAL MODELLING AND DYNAMICS

For examination of stability and for numerical simulation, a mathematical model is defined as shown in Fig . 1. Assu.ptions

The following assumptions are establish the model in Fig . 1. actuator arm ..... S=D • I .{). pOinter kl

~x 1

4TIJ

~l· -

considered

e

to

object

f-I'

f-I'

k.

~fWl~fw'~ ~

ffio

-I

Fig. I Mathemalical model

~

4ijJ

~

Assu.ption 1: The manipulators are expressed as mass-spring-damper systems. Control forces are input independently to the parameters of system dynamics. Properties of left- and right-arms are individually defined and need not be equal to eac h other. Assu.ption 2: Motion of the 2 manipulators and the object is confined to a straight line. The object moves freely along the line . Assu.ption 3: Impact force is considered to occur due to the local deformation of the manipulator pointers and the object. It is numerically modelled as the production of Herz type interaction force and damping parameter. The damping parameter depends on the primary function of the relative velocity. Assu.ption 4: Impact force must be positive. ( Only the condition of repulsion is considered . )

SYSTEM STABILITY 11TH COLLISION

The stability of the system is analyzed by Lypaunov's direct method. First a nonlinear scalar function is defined in the quadratic form as Eq. (11). Physically, the function expresses the energy stored in the potential form of the springs and in the kinematic form of the masses . The spring effects of control for ces ( position fe edback gains ) and impact forc es ("erz's coefficients with nolinear r elative de formation) also co ntribute to the function. (11) V=xTpx

p=

kl+Kpl+t:.kl

0

0

ml

- t:.kl

0

0

0

0 0

0 0

Equations of Motion The equations of motion of the 3 masses in Fig. are as follows; Left Manipulator : m IXI=-kIXI-CIXI+fl-fwl Right Manipulator : m rXr= -krxr-crxr+fr+fwr

wr

0

mo

0

- t:.k,

0

0

0

0

0

k,+Kpr+t:.kr 0 0

mr

(14 )

t:.k = {b Hru/!2 (ur ~ 0 and ur ~ - I /Pr)

(15)

r

0

(

others

)

In this system the following four contact/noncontact conditions are possible: (1) All ( the 1- and r-manipulators and the object ) are in contact. (2) Only the left-manipulator and the object are in contact . (3) Only the right - manipulator and th e object are in contact. (4) None of them are in contact. Each condition is discussed separately below.

(5 )

Condtion of All in contact In this condition, UI ~ 0 and tll ~ -l /pl

(6)

ur ~ 0 and u, ~ -I/Pr

(others)

0

llr=xo - xr

by Eqs. (11) to (15) V = ( kl + Kpl ) Xf + mlxr + mo"~ + ( kr + Kpr ) Xt + mrxt + a Hlur!2 + b Hru(!2 Eq . (16) mea ns that if kl + Kpl > 0, kr + Kpr > 0, a > 0, b> 0

(8)

In th ses equations PI and Pr ar e energy loss parameters, and HI and Hr are He rz ' s repulsion force coefficients which are normally positive with a high value.

x=O,V=O X;O Th e derivative of Eq. (16) is

PI , Pr ~ 0, HI, Hr ~ 0 (9) In thi s paper, in order to analyze the system stability and to inspect transient behavior by numeri cal simulation, collision is assu med to occur in finite meas urable duration. The above impact force mod e l is adopted beca use it is co nsid ered to be optimal for this purpo se.

(16)

(17)

Y = 2"1 ( kl + Kpl ) XI + mlxl ) + 2mo xoxo + 2x r ( kr + Kpr ) Xr + mrxr )

t

+ ~ a HIUfi2 UI + b H,ut i2 ur Substitute Eqs. (1) , (2) and (3) to this equation and after being arranged, th e following equations is obtained :

Based on the impact forc e model in the above, the contact/noncontact condition is examined by the relative displa cement and relative velocity of the manipulator point e rs and the obj ec t. In other words, the condition when the impact force occurrs is considered to be contact co ndition, and, if not, it is considered to be noncontact. It is expressed by th e following forms : Contact Ui ~ 0 and tli ~ - l /pi (i = I, r) Son co ntact : others

fwr = ( I+Prur ) H rui /2

fwl = ( I+PIUI ) HIUf /2

(7)

(others)

UI=XI - Xo , UI=xl - Xo llr=Xo - Xr ,

0

t:.k = { a Hlu/!2 ( UI ~ 0 and UI ~ - I /pl ) 1 0 ( others )

(4 )

= {( I +Prur ) Hrui!2 (u r ~ 0 and tlr ~ -l/pr )

f where.

0

0

- t:.k r

0

0

t:.kl+t:.kr 0

(2 )

where Kp and Kv denote the position and the velocity feedback gain respectively. Impact forces are modelled and expressed by the following equations. ( Fujita and Hattori, 1979a, 1979b ) 1

0

(13)

where XI. X, and Xo are the di splacement of the masses respectively, fl and fr are the input forces of left and right manipulators respectively, and fwl and fwr are impact forces between each manipulator and the object. For the control law. linear state feedback is adopted and is expressed by the following formula.

fw = {( I+PIUI) HIUf!2 (UI ~ 0 and UI ~ -l/pl)

0

x = (XI XI Xc Xo x, xr)T where a and b are de fined as constants,

(3 )

fr = - Kprxr - Kvrxr

(12)

0

(1)

Object

fl = - KplXI - KvlXI

- t:.kl

Y = - 2 ( Cl + Kvl ) xr - 2 PIH,Uf /2ur - ( 2 - 2 ( c,. + Kv,) xt - 2 PrH,ut /2ut - ( 2 In Eq. (18) if cl + Kvl > 0

(10)

c,. + Kvr > 0, a = Y::;O

~-I-I

t

tal HI Uri2U I

tt) Hrut i2u,

4 b = "5

(18)

(19) (20)

from the result of Eqs . (17) and (20) in the condition that all are in contact, the system is stable in the sense of Lyapunov theory.

Xl = XI = 0, VI = 0 (30) others ,VI> 0 When VI is differentiated and is substituted by Eq. (1) ,

CondtIon of L-.anipulator and Object In Contact

= - 2 ( Cl + K yl ) xr

In this condition, UI 2: 0 and III 2: -l/pI • Ur < 0 or ur < -I/Pr t>.kl = a HIU/ 12 , t>.kr = 0 fwl = ( I +PIUI ) HIUf l2 , fwr = 0 by Eqs. (11) to (15) V =( kl + Kpl) Xf + mlxr + lIlox~ + ( kr + Kpr ) xt + mrx; + a Hluri2 from this, if kl + Kpl > 0, kr + Kpr > 0, a > 0 x =0 , V=O x;oO,V>O The derivative of Eq. (21) is V = 2xI (( kl + Kpl ) XI + mlXI }+ 2lIloXoXo + 2x r (( kr + Kpr ) Xr + mrx r } + ~ a H Iuf /21l1

(2) Right manipulator In the same way as the left manipulator (32) (3) (21)

(3) ,

(34) Strictly speaking, if a system is stable according to Lyapunov theory, V is zero only when the state vector x equals a zero vector. In the manipulator/object system discussed here, Vo can be zero even when Xo is not zero. Thus, it is not allowed to determine whether the system is stable or not by Lyapunov's direct method . However, when Vo is zero at the time Xo is not zero and Xo is zero, physically the object is stopped at a point deviated from its origin. For this kind of system its state does not change unless external forces are applied. Here in this discussion, we only concentrate on the condition of noncontact .

(22)

(~3)

This results ,if Cl + KYI > 0 , Cr + KYr > 0 , a = %

(24) For convenience, we will focus only on the left manipulator and the object . Phisically, relative deformation of them is limited. The condition of noncontact satisfying III < - I/pI must be

V~O

(25) From the result of Eqs. (22) and (25) in the condition that the left-manipulator and the object are in contact, the system is stable in the sense of Lyapunov theory.

~ > ~

Er >

This condition is equivalent to the previous condition if the right-manipulator is substituted to the left manipulators. Thus, it is obvious that the following equations are satisfied.

(26)

x;oO,V>O

V~O

As shown by the above results, the system is stable even with collision phenomena . (27)

Futhermore, in Eqs. (18) and (23), 2 pH u3/2u2 is the term expressing the collision effect on the Lyapunov's function. The derivative of the Lyapunov's function V is smaller by this value than that without the collision term ( 2pH u3i2( 2 ) . This implies that the collision has a stabilizing effect . For instance, in the case of marginal stability, such that there is no damper term, V is zero in the condition of noncontact, but it becomes negative in th e contact condition. It means that the system is stable if considered as a whole motion depending on time .

As a result, in this condition the system is stable in the sense of Lyapunov theory. ConditIon of all in noncontact In this condition, UI < 0 or UI < -l/pI t>.kl = 0 , t>.kr = 0

(36)

XI - El < Xo < Xr + Er (37) According to Eq. (37), Xo can be converged, because Xl and xr converge to ze ro in any condition. This means that the system is stable even in the condition of noncontact .

Cl + KYI > 0 , Cr + KYr > 0

x=O,V=O

Ur

From Eqs . (35), (36) and (8)

kl + Kpl > 0, kr + Kpr > 0

b=1 5

(35)

where El (>0) denote the maximum magnitude of local relative deformation of the manipulators . In the same way, for the right side

CondItion of R-.anipulator and Object in Contact

If

Object :

xo=O, Vo=O and xo;OO, Vo>O (33) When Vo is differentiated and is substituted by Eq .

Substitute Eqs. (1), (2) and (3) to this V = - 2 ( Cl + Kyl ) xr - 2 PIHluf12llr - 2 ( Cr + KYr ) x; - ( 2 - ~a) Hluf12ul

(31)

Ur < 0 or ur < -l/pr fwl = 0 ,fwr = 0

by Eqs . (11) to (15) V = ( kl + Kpl ) Xf + mlxr + lIloX~ (28) + ( kr + Kpr ) xt + mrxt In this condition the two manipulators and the object move independently of one another. So we try to treat these separately and define the subset of the scalar function respectively as follows: VI = ( kl + Kpl ) xf + mlxr Vo = lIlox~ (29) Vr = ( kr + Kpr ) xt + mrxt The following is the discussion of each portion of the total system. (1) Left manipulator :

SIJIIULATlON JlllETHOD AND RESULTS The system stability and transient response are studied by numerical simulation using the same model in Fig. 1 as follows: Si.ulation Ncthod The equations of motion are solved by the Rungekutta method. Both manipulators and the object are considered individual systems. The difference

245

between contact and non contact is only whether collision force is applied or not, which is determined by the relative deformation and relative velocity of manipulators and the object. It means that these systems are not connected even in the contact condition . This method allows the calculation with nonlimited number of collision times.

the object in the same time step, the process is just the mixture of the previous ones. The first collision ( either left or right side) is adopted to calculate with one manipulator - contact condition until the second collision ( the other side), and the response is calculated in the rest of the time step.

Si.ulation Results Boundary condition Simulation results are shown from Fig. 2 to Fig. 15 . The conditions of parameters and others are as follows. Case numbers are also indicated. No.l The standard case of the following study cases. Parameters are shown in Table 1. The two manipulators have the same properties. Their initial conditions are different. ( Fig. 2 ) In Fig. 1 the part that "I is larger than Xo means the left manipulator plunges into the object. For the right manipulator, "0 is larger than X r • These conditions are applied in the following cases. No.2 The response of the marginally stable condition with all damping factors are zero. ( q, Cr. KvI. Kvr , Ph Pr = 0 ) Fig. 3 ) This response continues periodically without divergence or convergence. No.3 The response of the marginally stable manipulators with damping effect of collision ( Fig. 4 ). No.4 The response of No . l with larger value of PI and Pr for larger energy loss in collision.

Contact/noncontact condition is judged by the relative deformation UI. Ur and the relative velocity Or. ~ ( Eq . (10) ) both at the beginning and the end of every time step of Runge-kutta method. On this judgement, the time steps which include contact/noncontact condition changes are critical and discussed in more detail below . For convenience, focus is placed upon the left side of the system. If the condition is noncontact at the beginning of one time step and it is contact at the end, the manipulator must exactly contact at a certain time in this time step. We obtain this exact contact time tic by the linear interpolation of UI and 01 , and calculate all state values such as "0 and "r by linear interpolation using step time and ~c . For the rest of this time step( from ~c to the end of the time step ), the response is caluculated under contact condition. After this time step the collision force ~c is considered to include the effect of contact. Naturally, before the contact time step, collision force is zero, expressing the noncontact condition.

"I.

c: Q)

The release time step is just the contrary of the previous process. That is, by monitoring every time step, the exact release time is obtained and the condition is switched.

X,

E Q)

u ctl

20

c. Cfl

The right side is the same way as the left side. If both left and right manipulators collide or release

0

T ABLE I Standard Parameters for Simulation

1 10 0 .2 1 10 0.2

ml kl Cl mr kr C r

mo

-Kpl Kvl Kpr Kvr

5

-5 0.5 5 0.5

PI Pr HIHr Xlo V 10 X ro V ro

Time

-1 10

0.1 10000 0.3 0

f,

20

0 .5 0

Time

-10

Fig. 2 Standard parameters

-

C

E

E

c:

Q)

Q)

Q)

X,

Q)

u

U

ctl

tU

c.

Cfl

0

20

x,

c.

x,

Cfl

0

Time

-1

Time

-1

10

10 f,

f,

"'3

c.

20

c:

-10

"'30 c. c:

Time

-1 0

Fig. 3 Marginally stable, perfect elastic collision

20

f,

Time

Fig. 4 Marginally stable, collision with energy loss

246

( Fig. 5 ) From No . 5 to No.S the initial conditions are symmetrical to emphasize the effect of asymmetrical manipulator properties. No.5 The response of No.l with 10 times the masses for both manipulators (ml= m r = 10). ( Fig. 6 ) No.6 The response of No.l with 10 times the mass for left manipulator (ml= 10 ). ( Fig. 7 ) No . 7 The response of No.l with 10 times the stiffness for both manipulators. ( kl = kr = 100 ) ( Fig. S ) No.S The response of No.l with 10 times the number of damping coefficients for both manipulators. (Cl=C r =2 ) ( Fig. 9 ) No.9 The response of No.1 with 10 times the position feedback gains. ( Kpl =Kpr =50) ( Fig. 10 No.lO The response of No.l with 10 times the velocity feedback gains. ( Kvl = Kvr = 5 ) ( Fig. 11 ) No.11 The response of No.l with 10 times the mass for the object. ( lIlo= 50) ( Fig . 12 ) As shown by the results of these simulations the following aspects are observed. (1) Marginally stable systems can be stabilized by the collision with energy loss by Figs.3 and 4 . This fact agrees with the result of Lyapunov's method. (2) Energy loss in collision causes the manipulators good convergence with the same property and the same control by Figs. 2 and 5. (3) Linear feedback control enables stable

grasping. (4) Stablity is maintained after such parameters as mass. elasticity and damping coefficients change. (5) The increase of the manipulator mass causes the higher vibrating frequency and lower controllability. ( Figs. 2 and 6 ) (6) The increase of manipulator stiffness causes better convergence to a degree but demands longer time for convergence. ( Figs. 2 and S ) (7) The increase of damping coefficients causes better convergence . ( Figs. 2 and 9 ) Manipulators with asymmetric properties (S) converge asymmetrically but have lower controllability in total. (9) The behavior is affected by the change of feedback gains . For example, (10) The increase of position feedback gains causes larger control force demand and worse convergence due to the relative decrease of damping effect. Figs. 2 and 10 ) (11) The increase of velocity feedback gains causes better convergence due to rclatively larger damping effect. ( Figs. 2 and 11 ) (12) The increase of the object mass causes worse controllability and the demand of longer time for convergence. ( Figs. 2 and 12 ) CONCLUSION In

this paper, grasping of object by

c

-

E

E

Q)

Q)

C,)

co

-

c

manipulators

X,

Q)

Q)

C,)

co

20

x,

0.

en

0.

en

x,

o

0

Time

-1

Time

-1 10

10

f, f,

'5 0 0. c

20

f,

Time - 10 Fig. S Standard with larger energy loss in collision

Time

- 10

Fig. 6 10 times the masses. both manip.

-

c

c

Q)

Q)

E

E

Q)

Q)

C,)

C,)

co

co

0.

0.

20

en

en

0

0

-1

Time

Time

-1 10

10

I, '5

0.

-10

20

.s

20 Time

-10

Fig. 7 10 times the mass. left manip.

Time

Fig. 8 10 times the stiffness. both manip. 247

has been analyzed by Lyapunov's direct method its transient behavior has been studied extensive numerical simulation.

and by

collision. Trans. ,JSME, 45-397, 985-992. ( in Japanese ) Fukuda, T., Kitamura, N. and Tanie, K. (1987a). Adaptive force control in robot manipulator with consideration of characteristics of objects ( 1st report, Grasping control of gripper by the adaptivehybrid force control ). Trans. ,JSME, 53-487, 726-730. ( in Japanese) Fukuda, T., Kitamura, N. and Tanie, K. (1987b). Adaptive force control in robot manipulator with consideration of characteristics of objects (2nd report, Control of one degreeof-freedom manipulator by adaptive hybrid force control ) . Trans. ,JSME, 53-496, 25772583. ( in Japanese) Kahng, J. and Ami rouche, F. M. L. (1988) . Impact force analysis in mechanical hand design part 1. Int'l ~ Robotics! Automation, 3-3, 158-164 Youcef-Toumi, K. and Gutz, D. (1989). Impact and force control. Proc. IEEE R&A '89, 410-416

As a result, the following conclusions were obtained: (1) Lyapunov's direct method is effective even if the controlled system includes collision. It is proved theoretically that grasping with collision can be stable. (2) Collision is a phenomenon which stabilizes the system under some conditions such as marginal stabili ty. (3) If the characteristics of two manipulators are different and are controlled in the same way, system controllability is lower than that of symmetrical system. REFERENCES

Colgate, E. and Hogan, n. (1989). An anlysis of contact instability in t erms of passive physical eq uivalents . Proc. IEEE R&A ~ 404409 Fujita, T. and Hattori, s. (1979a). Impulse behavior in periodically vibration with collision. Trans. ,JSME, 45-395, 737-746. ( in Japanese ) Fujita, T. and Hattori, s . (1979b) . Static irregular vibration of a system with

c

x.

Q)

c

E

Q)

x.

E Q)

U

ell

Q)

u

~

0

20

X, Xo

C.

20

c. C/l

0

C/l

0

-1

Time

-1

10 10

fI

r

fI

:; 0 c. c

20

f.

Time -10 Fig. ID 10 times the position feedback gains

Time -10 Fig. 9 ID times the damping coef. , both manip.

c

c

X.

Q)

E

Q)

E

Q)

Q)

u

U

ell

c.

X,

C/l

ell

20

xo

20

c.

C/l

0

0

Time

-1

-1

10

Time

10 f,

:;

:; 0 c.

c. .E

20

c

f.

Time Fig. 11 10 times the velocity feedback gains

-10

-10

248

20 Time Fig. 12 ID times the mass, object