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contact control of a robotic manipulator
Computers Elect. Engng Vol. 18, No. i, pp. 99-108, 1992 Printed in Great Britain. All rights reserved
0045-7906/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press plc
IMPACT/CONTACT CONTROL OF A ROBOTIC MANIPULATOR YASUMASA SHOJI I, MAKOTO INABAt and TOSHIOFUKUDA2 'Applied Technology Department, Toyo Engineering Corporation 2-8-1 Akanehama, Narashino, Chiba 275, 2Department of Mechanical Engineering, Nagoya University l Furo-cho, Chikusa-Ku, Nagoya 464-01, Japan (Received I0 October 1990; accepted in final revisedform 8 July 1991)
Abstract--In this paper, a methodologyusing the Lyapunov direct method is proposed to analyze the stability of a manipulator system which is positioned or forced against a deformablewall with collision phenomena. The stability and response of the system are examined by parameter studies of numerical simulations. As collision is a phenomenon with energy dissipation, it is effectivelyused in the field of industry to suppress vibration. However, in the field of robotics little research has been undertaken to the collisionof robotics with their environmentat the viewpointof active control. Up until now systems have been controlled to avoid rapid contact. We have adopted a Hertz-type model with an energy loss parameter to expressthe impact force betweenthe manipulator and the wall. Using this model, we proved that stable positioning of the manipulator against the deformablewall is assured to be possible and that the stabilization effect of collision using a nonlinear Lyapunov function. The effect is confirmedby the simulation. 1. I N T R O D U C T I O N When manipulators are acted on the environment, an interaction must occur between them. This interaction becomes a collision if the motion of the manipulator is fast enough at contact [1,6,8]. As the collision has the effect o f causing an energy dissipation, it is effectively being utilized in suppressing vibration in the field of industry, such as for the support of pipes or towers. On the contrary, the phenomenon of collision often results in worse controllability in the field of robotics. This is because the ordinary control theory of the robotic manipulator is unable to properly deal with the strong nonlinearity of the collision. Moreover, the motion during the collision is much faster than the motion during normal controlled phase and the ordinary active control is inefficient in catching up with the increased speed in such cases. For these reasons, manipulator systems are controlled to move slowly on contact to avoid collisions. On the other hand, as factory automation has been developed, recently the demands for faster performance of robotic manipulators has become essential, especially in the field of industry [4,5]. In the fields o f application varieties, such as faster assembling and insertion, it is increasing gradually. As well as the faster tasks, more precise and variable control is demanded. A master-slave type teleoperation is an example of this kind of task. However, even by the latest control methodology, the slave side is controlled so that the relative velocity becomes very low on contact to avoid collision with precise position control [7]. At this stage, though little research has been undertaken, the collision problem has become one of the major problems to be solved. More research is needed to satisfy these control demands. In this paper, a positioning and forcing of manipulators is discussed with consideration of collision phenomena. Its stability and transient response are studied. On the stability of manipulators contact, Hogan analyzed by the Lyapunov method in the Ref. [2], but transient state with relative velocity during contact was not discussed and the control law did not include contact interaction. To focus the collision effect to the manipulator-wall system, for simplicity, the system consists o f one l-d.f, manipulator and a deformable wall. Their motion is confined to a straight line. The manipulator is controlled with nonlinear feedback and the wall vibrates according to its dynamics. Basically, they move independently of each other. The manipulator collides and rebounds from the wall without any limit upon repetition. The stability o f this nonlinear system is analyzed by the Lyapunov direct method and is validated by numerical simulation. The transient responses are examined by the simulation at the same time. 99
100
YASUMASASHOJI et
2. M A T H E M A T I C A L
al.
MODELING
AND
DYNAMICS
Because the major interest in this paper is the stability of the total manipulator object system including collision, we considered the manipulator dynamics as simplest and sufficient and also consider the modeling of the wall and collision phenomena as sufficient and detail. On this basis. a mathematical model is defined as shown in Fig. 1 for examination of stability and for numerical simulation.
2. I. Assumptions The following assumptions are considered to establish the model shown in Fig. 1.
Assumption 1: the manipulator is expressed as a mass-spring-damper system. Assumption 2: motion of the manipulator and the wall is confined to a straight line. Their motions are independent of each other, when they are not in contact. Assumption 3: the wall is expressed as an equivalent mass-spring-damper system to express its dynamics. Assumption 4: impact force is considered to occur due to the local deformation of the manipulator pointer and the wall. It is numerically modeled as the production of Hertz type interaction force and damping term. The damping term is the primary function of the relative velocity multiplied by a damping parameter. Assumption 5: impact force must be positive (only the condition of repulsion is considered). 2.2. Modeling and dynamics The equations of motion of the two masses in Fig. 1 are as follows: Manipulator: + f --f~.
(1)
c2-~2 -~-.]#~v
(2)
mlS~l = - - k l x l - - c l k l Wall:
m2£2 = - k2x2
-
where x, m, k and c are the displacements, masses, spring constants and damping coefficients, respectively, f is the input force applied to the manipulator and fw is the impact force between the manipulator and the wall. The subscripts 1 and 2 denote the manipulator and the wall, respectively. For the control law, nonlinear feedback is adopted and is expressed by the following formula: f
=
-
Kpx,
-
KvA,
-
Krf,~ :~'~
(3)
where Kp and Kv denote the position and the velocity feedback gains, respectively. Kf denote the feedback gain of the production of impact force and manipulator velocity. Because this nonlinear term includes impact force, interaction (i.e. collision) is to be considered in the manipulator control. Impact force is modeled and expressed by the following equation (see Ref. [3]):
L=
~'(1 +pu)Hu3"2(ifu >>,0 and u ~> - l / ' p ) (otherwise)
(4)
where, u = x l - x2
actuator arm
I
'
'n
--- - ~ L def°rmable pointer [~ wall
Fig. 1. Mathematical model.
(5)
Impact/contact control of a robotic manipulator
101
In these equations, p is an energy loss parameter, and H is a Hertz interaction force coefficient which is normally positive with a high value. We discuss p in more detail. If this value is large, energy dissipation is large or damping effect is large, and vice versa. In Appendix 1 the relationship between p and the coefficient of restitution e is discussed more mathematically: p ~>0, H > 0 .
(6)
In this paper, in order to analyze the system stability both theoretically and by numerical simulation and to study transient behavior by numerical simulation, collision is assumed to occur in a finite measurable duration, not to occur instantaneously. Because the impact force model basically consists of a Hertz force model which is widely approved and it describes the local relative deformation of the manipulator and the wall, it is considered to be most suitable for the purpose of this study. Due to this model's capability to treat the local relative deformation during collision, the acceleration/deceleration by the impact force can be properly treated. This is the reason why the above impact force model is adopted in this paper. Based on the impact force model above, the contact/noncontact condition is examined by the relative displacement and relative velocity of the manipulator pointer and the wall. In other words, the condition when the impact force occurs is considered to be contact condition, and, if not, it is considered to be noncontact. It is expressed by the following forms: contact:
u ~>0 and t~ >/ -
1/p
(7)
noncontact: otherwise. 3. S Y S T E M
STABILITY
WITH
COLLISION
The stability of the system is analyzed by the Lyapunov direct method. First a nonlinear scalar function is defined in the quadratic form as: V = xTpx
(8)
where,
p =
I kl + Kp + Ak 0 0 ml
- Ak 0
0 0
- Ak
0
k 2 + Ak
0
0
0
0
m2
(9)
x = {x, :e,
~aHul/2(ifu >10 and Ak = [0 (otherwise)
(lO)
ti i> -
l/p) (11)
where a is defined as an arbitrary uncontrollable constant. 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 the position feedback gain (control force) and Hertz's coefficient with nonlinear relative deformation (impact force) also contribute to the function. Here we discuss differentiability of the function. According to equation (1 I), condition of contact/noncontact is defined by both relative displacement and relative velocity. However, because in Ref. [3] p is normally a nonnegative and small value, - lip becomes negative and a large value. This means that, in most cases, the rebounding velocity does not contribute to the judgment of contact/noncontact, and Ak is continuous and differentiable at contact/contact state change i.e. at u = 0, which indicates that the function in the quadratic form is differentiable (see Appendix 2 for more details). In this system, the conditions of contact and that of noncontact are discussed separately, as follows.
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YASUMASASHOJ1e t al.
3.1. Condition o f noncontact state
In this condition, u<0
or t i < - l / p
Ak = 0 , f w = 0 by equations (8)-(11) V = (k, + Kp)x~ + m, yc2t+ k2 x2 + m2Ycz~
(12)
Equation (12) means that, if kl + Kp > 0, x=0,
V=0
x¢0,
V>0.
(13)
The derivative of equation (12) is I2 = 2~,{(k~ + Kp)x~ + rnlS~~} + 2~2(k2x2 + m2.~2) • Substitute equations (1) and (2) into this equation and after being arranged, the following equation is obtained: 12 = - 2(cl + Kv).~ - 2c2 ~2. (14) In equation (14), if
c, +Kv > 0 ,
(15)
12-..<0
(16)
from the result of equations (13) and (16) in the condition of noncontact, the system is stable in the sense of Lyapunov theory. 3.2. Condition o f contact state
In this condition, u / > 0 and t i > 1 - 1 / p A k = aHu 1/2, fw = (1 + p f t ) H u 3/2
by equations (8)-(11) V = (kl + Kp)x~ + ml Yc2 + k2 x2 + rn2Jc2 + all, u~/2
(17)
from this, if k~ + Kp > 0, a > 0, x=0,
V=0
x#0,
V>0.
(18)
The derivative of equation (17) is 12 = 2-xt{(k~ + Kp)x) + mt xz } + 2-~2(k2x2 + m2-~2) + ~aHu3/2f~"
(19)
Substitute equations (1) and (2) into this I2 = -- 2(c, + Kv)Yc~ -- 2pHu3/2ft 2 - 2Krfw2~ - 2c22~ -- (2 - ~a)Hu3/2fi. This results in, if cL+K~>O, K f > O , 12 ~<0.
a=~,
(20) (21) (22)
From the result of equations (18) and (22) in the condition that the manipulator and the wall are in contact, the system is stable in the sense of Lyapunov theory. Here we should pay attention to the value of a. In equation (21), from the above derivation a must be fixed to 4/5. As described before, a is an uncontrollable constant, and the condition of a = 4/5 is the condition so that the function V is a Lyapunov function rather than the condition for the stable control.
Impact/contact control of a robotic manipulator
103
As shown by the above results, the system is stable even with collision phenomena. Furthermore, in equations (14) and (20), 2pHu3/2f42 is the term expressing the collision effect on the Lyapunov function. The derivative of the Lyapunov function I;"is smaller by this value than that without the collision term (2pHu3/2ft2). This implies that the collision has a stabilizing effect. For instance, in the case of marginal stability, such as the case where there is no damper term, I7 is zero in the condition of noncontact, but it becomes negative in the contact condition. As the characteristic of the Lyapunov function is the total energy of the system, the negative derivative stands for energy loss. This means that the system is stable, because the system loses energy during the collision and V goes to zero as time passes and as the system experiences collision a number of times. 4. S I M U L A T I O N
METHOD
AND
RESULTS
The system stability and transient response are studied by numerical simulation using the same model in Fig. 1 as follows.
4. I. Simulation method The following state equation is defined using the equations of motion, equations (1) and (2) = A x + b f + fw
(23)
where x denotes a state vector, b f is an input vector and fw is a collision force vector: (24)
X ~---{X 1"~1X2 "~2)T
A =
0
1
0
0
kl ml
el m~
0
0
0
0
0
1
0
0
k2 --m2
I ,w--{0 b=
0
(25)
--1 m]
0
:.
ml
0
o
c2 m2
t
(26)
m2 J
The equations of motion are solved by the Runge-Kutta method. As shown by matrix A in equation (25) both manipulator and the wall are considered individual systems. The difference between contact and noncontact states is only whether collision force is applied or not, which is determined by the relative deformation and relative velocity of the manipulator and the wall. It means that these systems are not connected even in the contact condition. This method allows the calculation with a nonlimited number of collision.
4.2. Boundary condition Contact/noncontact condition is judged by the relativedeformation u and the relativevelocity [equation (7)]both at the beginning and the end of every time step of the Rungc--Kutta method. On thisjudgment, the time steps, which include contact/noncontact condition changes, are critical and discussed in more detail below. 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 tc by the linear interpolation of u and ti, and calculate all state values such as xt and x2 by the linear interpolation using step time and tc. For the rest of this time step (from to to the end of the time step), the response is calculated under contact condition. After this time step the
YASUMASASHOJI et al.
104
Y, 1L
EO ×'
E E
2o
1
~E0 x '
~_
z
0
5
~
20
:ii 20 Time
2O
Time
s
Fig. 2. Marginally stable, perfect elastic collision.
s
Fig. 3. Marginally stable, collision with energy dissipation.
collision force fw is considered to include the effect of contact. Naturally, before the contact time step, collision force is zero, expressing the noncontact condition. The time step for the state changes from contact to noncontact which is just the contrary of the previous process. That is, by monitoring every time step, the exact state change time is obtained and the condition of simulation is switched from contact to noncontact. 4.3. Simulation results
Simulation results are shown in Figs 2-6. The conditions of parameters and others are shown in Table 1 and as follows. Case numbers are also indicated. No. 1. The response of the marginally stable condition with all damping factors are zero (c, K,. and p = 0) (Fig. 2). This response continues periodically without divergence or convergence. In this simulation, the desired condition is that the manipulator is touching the wall with zero contact force. The following three simulations (i.e. Nos 2-4) have the same desired condition. In Fig. 2 the part where x~ is larger than x2 means the manipulator plunges into the wall. This is applicable in the following cases. No. 2. The response of the marginally stable manipulators with the damping effect of collision (p = 1) (Fig. 3). These two simulations (Nos 1 and 2) show that marginally stable systems can be stabilized by the collision with energy dissipation (Figs 2 and 3). This fact agrees with the result of the analysis using Lyapunov method. No. 3. The response of stably controlled case (p = 0.1) (Fig. 4). By this simulation, it is shown that even a simple nonlinear feedback control rather than complex control enables stable contact control.
~E
°
,
- ~
,
zo
~:~
~
0
._ C3
_1 ~
~E
2~0
o
-5
;s °-°
1 .........
2b Time
s
Fig. 4. S t a b l y c o n t r o l l e d .
20
Time
s
Fig. 5. Stably controlled with larger energy loss parameter.
Impact/contact control o f a robotic manipulator
iE n
-
' -
105
2'0
-1
2'0
~-~0tilI~-
¢D
Time
/0
s
Fig. 6. Stably controlled with pushing force.
Table 1. Simulation parameters No. m 1 k I 1 2 3 4 5
1 1 1 1 1
10 10 10 10 10
cI rn2 k 2 0 0 0.2 0.2 0.2
10 10 10 10 10
300 300 300 300 300
c2
Ko
Kv
p
H
xl0
Figure
0 0 0.5 0.5 0.5
15 15 15 15 15
0 0 3 3 3
0 1 0.1 1 0.1
10000 10000 10000 10000 10000
0 0 0 0 0.3
2 3 4 5 6
No. 4. The response of No. 3 with larger value of p (p = 1) for larger energy dissipation in collision (Fig. 5). These two simulations show the same result as Nos 1 and 2, and stabilizing effect is observed even in the stably controlled system. From this result, we can estimate that we can positively utilize collision in impact/contact control. No. 5. The response of No. 3 with pushing force (Fig. 6). Oscillation of contact force is observed while the contact condition is kept. The time for continuous contact condition is obviously shortened relative to the zero force control cases. This means that, if the object and the manipulator has adequate strength, we should use impact/contact phenomenon to shorten the time to grasp or to stub. This means it is possible to shorten the total operation time of manipulators including contact tasks.
5. C O N C L U S I O N In this paper, positioning of manipulators against the wall has been analyzed by the Lyapunov direct method and its transient behavior has been studied by extensive numerical simulations. As a result, the following conclusions have been obtained: (1) The Lyapunov direct method is effective even if the controlled system includes collision. It is proved theoretically that positioning with collision can be stable. (2) Collision is a phenomenon which stabilizes the system under some conditions such as marginal stability. REFERENCES 1. E. Colgate and N. H o g a n , A n analysis o f contact instability in terms of passive physical equivalent. Proc. IEEE Conf. Robotics and Automation '89, pp. 4 0 4 - 4 0 9 (1989). 2. N. H o g a n , O n the stability of manipulators performing contact tasks. IEEE J. Robot. Automat. 4, 6 7 7 - 6 8 6 (1988). 3. T. Fujita and S. H a t t o r i , Periodic vibration and impact characteristics o f a nonlinear system with collision. Bull. JSME 23, 4 0 9 - 4 1 8 (1980). 4. T. F u k u d a , N. Kitamura and K. Tanie, Flexible handling by gripper with consideration of characteristics of objects. Proc. IEEE Conf. Robot. Automat., pp. 703-708 (1986). 5. T. F u k u d a , N. Kitamura and K. Tanie, Adaptive force control of manipulators with consideration of object dynamics. Proc. IEEE Conf. Robot. Automat., pp. 1543-1548 (1987). 6. J. Kahng and F. M. L. A m i r o u c h e , Impact force analysis in mechanical hand design--part I. Int. J. Robot. Automat. 3, 158-164 (1988).
YASUMASA SHOJI el al.
106
7. K. Tanie, K. Komoriya, T. Kaneko, T. Ohno and T. Fukuda, Bilateral remote control with dynamics reflection. Proc. Sixth C I S M - I F T o M M Symp. on Theory and Practice of. Robots and Manipulators, pp. 296--308 (1988). 8. K. Youcef-Toumi and D. Gutz, Impact and force control. Proc. Conf. 1EEE Robot. Automat. '89, pp. 410-416 (198%
APPENDIX
1
To obtain the relation between the energy loss parameter p and the coefficient of restitution e. we will consider the contact of two masses, m and M, with velocities of v0 and O, respectively. Equations of motion/during contact: m.%, = --/[,
(A 1)
"15,:M= 1,, I,, - (1 + pu)Hu v2
(A2) (A3)
u = x m - x M.
(A4)
From equations (A1) (A4),
D1 m(&n - -~:~) = - . L - M,L
(m)
mii= -
( l + p i O H u 3'2.
I+M
(A5)
If we consider that u = z, from equation (A5): dz dz du M + m ........... (l + pz)Hu 3'2. dt du dt Mm Since du/dt = z, equation (A6) is rewritten into equation (A7):
du
Mm
+ p Nu3'e du.
Equation (A7) is integrated using the initial condition of z, :., £[
-
(A6)
(A7)
= v 0 at u,--0 = 0
Mm dz £7 M+ m [:z+p = Hu 32du.
(AS)
The left hand of equation (A8) is calculated as follows:
f,[
Mm -M-+m
dz Mm 1/::+p =-]~+m
£:
~
,,pz
dz
Mm l { ln(PZ+ 1~ M+mp ~ pvo-pz+ \ p v o ~ l l ),
(A9)
The right hand of equation (AS) is as follows:
f ' Hu 32 du = 2sHu52.
(A10)
Mm , { , _ ,n(PZ+l~f M + n i p S ~ p , o p_~ + \ p r ~ , + i ] j = e s H u 52.
{All)
From equations (A9) and (AI0), if p > 0,
If we substitute
pt' o =
Vo
and pz
= Z,
( A I 1) is written into equation (A12) Mm 1 / . Z+I\ M + m p 2 ~ [ % - Z + I n V 0 + i ) =~HuS"2" equation
(A12)
Here, if we consider e as the coefficient of restitution, we obtain the following two equations at the exact time when two masses, m and M, detach u=0,
Z=
- e V 0.
(AI3)
As the condition of equation (A12) is applicable at the time of detach, using equation (AI3),
Mm I { V+tnP ~ (l+e)V0+
ln(-el/f'+l~ 0 \ VI;+ i } 3 = .
(AI4)
Since
Mm - > O, physically, M+m (l+e)l~)+ln\
( eVo±! V0+l /=0.
(AI5)
From equation (AI5), equation (AI6) is derived, - - - 1 ( V1 l+0 ) exp{-(l+e)V,}.'.e . . . . .1. ( P1~+0 ) e - (>o ' pro This is the equation which expresses the relation between e and p.
exp{-(l+e)pvo}.
(A16)
107
Impact/contact control of a robotic manipulator VM=0 Before Contact r
~xM
Xm
CONTACT
After Contact
i VM
V In
Fig. A1.
d
O
U
- -
U
'B
Co l ck-wsi e
Fig. A2. (a) Contour of the Lyapunov function with small p. (b) Trajectory of the Lyapunov function. vc: contour of the Lyapunov function in contact state; v.e: contour of Lyapunov function in noncontact state.
APPENDIX 2 On the Differentiability o f the Lyapunov Function
In this paper, in a sense, two scalar functions are used for contact and noncontact states. We consider this as follows: Among the impact/contact parameters, H and p are very important. As mentioned in the paper, H represents a kind of spring constant, or stiffness, for the local deformation between a manipulator and an object or a wall. p denotes energy loss parameter, or a kind of damping parameter, which has come from the motion difference in the acceleration during collision. On the judgment of contact or noncontact, the parameter p is more important. We will discuss the Lyapunov function focusing on p. From physical consideration, p is assumed nonnegative and it is true in the following discussion. According to our definition of the Lyapunov function, the contour of the function can be drawn as in Figs A2 and A3. In the case wherep is sufficiently small, the contour is shown in Fig. A2(a). The line is connected smoothly at the two points, A and B where the contact/noncontact condition changes. In this case, a trajectory of the Lyapunov function is shown in Fig. A2(b). Here the inner closed line means that the contour value on this line is smaller than that on the outer line. At the contact, the trajectory must be clockwise. If the system has stable dynamics, the trajectory converges to the origin, which means the system goes to the equilibrium point and stop moving. It is not questionable that the Lyapunov function is differentiable in this case. In the case, however, where p is not so small, we must be more careful. It will occur that the contour line does not have the closed loop, because the line jumps to somewhere else when it meets the line of d u / d t = - lip. The contour of this case is shown in Fig. A3. According to this plot the contour jumps from point C to point D. However, we can guarantee that the jumping direction is always inward, because the Lyapunov function loses collision terms with positive value drastically, and becomes smaller than before the jump. We have another reason for the guarantee. That is the direction of the trajectory. As described above, the direction is always clockwise. Thus the changing rate of the Lyapunov function should be considered only in one direction, which means we have no need to guarantee the motion with counter-clockwise trajectory and no need to consider the jump from point D to point C in Fig. A3. In this case, at the point C and D of Fig. A3 the Lyapunov function is not differentiable but it is obvious that the value of the function reduces on the plane at any time, which means the system is stable.
/ vc2
Vnclf
Vn~
Vol
U
Fig. A3. Contour of the Lyapunov function with larger p.
108
YAS~:MASASHOJ1et al. AUTHORS'
BIOGRAPHIES
Yasumasa Shoji--Yasumasa Shoji graduated from Keio University, Tokyo, Japan in 1982 and received a M.Eng. degree
from the University in 1984. Since 1990 he has been working for TOYO Engineering Corporation, mainly engaging in the field of engineering analyses. From 1988 to 1989 he joined the Science University of Tokyo, and has researched stable control of manipulators including collision.
Makoto Inaba--Makoto Inaba graduated from Keio University, Tokyo, Japan in 1969. Since this year he has been working
for TOYO Engineering Corporation, mainly engaging in the field of aseismic design and intelligence of chemical and other plants. He is the leader of the Advanced Technology Group, Technology R&D Center.
Toshio Fukuda--Professor Toshio Fukuda received the D.Ing. degree from the University of Tokyo in 1977. After he joined
the National Mechanical Engineering Laboratory in 1977 and the Science University of Tokyo in 1981, he joined Nagoya University, Japan in 1989 as Professor of Mechanical Engineering Department. He is the Vice President of IEEE Industrial Electronics Society (1991-).
0045-7906/92 $5.00 + 0.00 Copyright © 1992 Pergamon Press plc
IMPACT/CONTACT CONTROL OF A ROBOTIC MANIPULATOR YASUMASA SHOJI I, MAKOTO INABAt and TOSHIOFUKUDA2 'Applied Technology Department, Toyo Engineering Corporation 2-8-1 Akanehama, Narashino, Chiba 275, 2Department of Mechanical Engineering, Nagoya University l Furo-cho, Chikusa-Ku, Nagoya 464-01, Japan (Received I0 October 1990; accepted in final revisedform 8 July 1991)
Abstract--In this paper, a methodologyusing the Lyapunov direct method is proposed to analyze the stability of a manipulator system which is positioned or forced against a deformablewall with collision phenomena. The stability and response of the system are examined by parameter studies of numerical simulations. As collision is a phenomenon with energy dissipation, it is effectivelyused in the field of industry to suppress vibration. However, in the field of robotics little research has been undertaken to the collisionof robotics with their environmentat the viewpointof active control. Up until now systems have been controlled to avoid rapid contact. We have adopted a Hertz-type model with an energy loss parameter to expressthe impact force betweenthe manipulator and the wall. Using this model, we proved that stable positioning of the manipulator against the deformablewall is assured to be possible and that the stabilization effect of collision using a nonlinear Lyapunov function. The effect is confirmedby the simulation. 1. I N T R O D U C T I O N When manipulators are acted on the environment, an interaction must occur between them. This interaction becomes a collision if the motion of the manipulator is fast enough at contact [1,6,8]. As the collision has the effect o f causing an energy dissipation, it is effectively being utilized in suppressing vibration in the field of industry, such as for the support of pipes or towers. On the contrary, the phenomenon of collision often results in worse controllability in the field of robotics. This is because the ordinary control theory of the robotic manipulator is unable to properly deal with the strong nonlinearity of the collision. Moreover, the motion during the collision is much faster than the motion during normal controlled phase and the ordinary active control is inefficient in catching up with the increased speed in such cases. For these reasons, manipulator systems are controlled to move slowly on contact to avoid collisions. On the other hand, as factory automation has been developed, recently the demands for faster performance of robotic manipulators has become essential, especially in the field of industry [4,5]. In the fields o f application varieties, such as faster assembling and insertion, it is increasing gradually. As well as the faster tasks, more precise and variable control is demanded. A master-slave type teleoperation is an example of this kind of task. However, even by the latest control methodology, the slave side is controlled so that the relative velocity becomes very low on contact to avoid collision with precise position control [7]. At this stage, though little research has been undertaken, the collision problem has become one of the major problems to be solved. More research is needed to satisfy these control demands. In this paper, a positioning and forcing of manipulators is discussed with consideration of collision phenomena. Its stability and transient response are studied. On the stability of manipulators contact, Hogan analyzed by the Lyapunov method in the Ref. [2], but transient state with relative velocity during contact was not discussed and the control law did not include contact interaction. To focus the collision effect to the manipulator-wall system, for simplicity, the system consists o f one l-d.f, manipulator and a deformable wall. Their motion is confined to a straight line. The manipulator is controlled with nonlinear feedback and the wall vibrates according to its dynamics. Basically, they move independently of each other. The manipulator collides and rebounds from the wall without any limit upon repetition. The stability o f this nonlinear system is analyzed by the Lyapunov direct method and is validated by numerical simulation. The transient responses are examined by the simulation at the same time. 99
100
YASUMASASHOJI et
2. M A T H E M A T I C A L
al.
MODELING
AND
DYNAMICS
Because the major interest in this paper is the stability of the total manipulator object system including collision, we considered the manipulator dynamics as simplest and sufficient and also consider the modeling of the wall and collision phenomena as sufficient and detail. On this basis. a mathematical model is defined as shown in Fig. 1 for examination of stability and for numerical simulation.
2. I. Assumptions The following assumptions are considered to establish the model shown in Fig. 1.
Assumption 1: the manipulator is expressed as a mass-spring-damper system. Assumption 2: motion of the manipulator and the wall is confined to a straight line. Their motions are independent of each other, when they are not in contact. Assumption 3: the wall is expressed as an equivalent mass-spring-damper system to express its dynamics. Assumption 4: impact force is considered to occur due to the local deformation of the manipulator pointer and the wall. It is numerically modeled as the production of Hertz type interaction force and damping term. The damping term is the primary function of the relative velocity multiplied by a damping parameter. Assumption 5: impact force must be positive (only the condition of repulsion is considered). 2.2. Modeling and dynamics The equations of motion of the two masses in Fig. 1 are as follows: Manipulator: + f --f~.
(1)
c2-~2 -~-.]#~v
(2)
mlS~l = - - k l x l - - c l k l Wall:
m2£2 = - k2x2
-
where x, m, k and c are the displacements, masses, spring constants and damping coefficients, respectively, f is the input force applied to the manipulator and fw is the impact force between the manipulator and the wall. The subscripts 1 and 2 denote the manipulator and the wall, respectively. For the control law, nonlinear feedback is adopted and is expressed by the following formula: f
=
-
Kpx,
-
KvA,
-
Krf,~ :~'~
(3)
where Kp and Kv denote the position and the velocity feedback gains, respectively. Kf denote the feedback gain of the production of impact force and manipulator velocity. Because this nonlinear term includes impact force, interaction (i.e. collision) is to be considered in the manipulator control. Impact force is modeled and expressed by the following equation (see Ref. [3]):
L=
~'(1 +pu)Hu3"2(ifu >>,0 and u ~> - l / ' p ) (otherwise)
(4)
where, u = x l - x2
actuator arm
I
'
'n
--- - ~ L def°rmable pointer [~ wall
Fig. 1. Mathematical model.
(5)
Impact/contact control of a robotic manipulator
101
In these equations, p is an energy loss parameter, and H is a Hertz interaction force coefficient which is normally positive with a high value. We discuss p in more detail. If this value is large, energy dissipation is large or damping effect is large, and vice versa. In Appendix 1 the relationship between p and the coefficient of restitution e is discussed more mathematically: p ~>0, H > 0 .
(6)
In this paper, in order to analyze the system stability both theoretically and by numerical simulation and to study transient behavior by numerical simulation, collision is assumed to occur in a finite measurable duration, not to occur instantaneously. Because the impact force model basically consists of a Hertz force model which is widely approved and it describes the local relative deformation of the manipulator and the wall, it is considered to be most suitable for the purpose of this study. Due to this model's capability to treat the local relative deformation during collision, the acceleration/deceleration by the impact force can be properly treated. This is the reason why the above impact force model is adopted in this paper. Based on the impact force model above, the contact/noncontact condition is examined by the relative displacement and relative velocity of the manipulator pointer and the wall. In other words, the condition when the impact force occurs is considered to be contact condition, and, if not, it is considered to be noncontact. It is expressed by the following forms: contact:
u ~>0 and t~ >/ -
1/p
(7)
noncontact: otherwise. 3. S Y S T E M
STABILITY
WITH
COLLISION
The stability of the system is analyzed by the Lyapunov direct method. First a nonlinear scalar function is defined in the quadratic form as: V = xTpx
(8)
where,
p =
I kl + Kp + Ak 0 0 ml
- Ak 0
0 0
- Ak
0
k 2 + Ak
0
0
0
0
m2
(9)
x = {x, :e,
~aHul/2(ifu >10 and Ak = [0 (otherwise)
(lO)
ti i> -
l/p) (11)
where a is defined as an arbitrary uncontrollable constant. 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 the position feedback gain (control force) and Hertz's coefficient with nonlinear relative deformation (impact force) also contribute to the function. Here we discuss differentiability of the function. According to equation (1 I), condition of contact/noncontact is defined by both relative displacement and relative velocity. However, because in Ref. [3] p is normally a nonnegative and small value, - lip becomes negative and a large value. This means that, in most cases, the rebounding velocity does not contribute to the judgment of contact/noncontact, and Ak is continuous and differentiable at contact/contact state change i.e. at u = 0, which indicates that the function in the quadratic form is differentiable (see Appendix 2 for more details). In this system, the conditions of contact and that of noncontact are discussed separately, as follows.
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YASUMASASHOJ1e t al.
3.1. Condition o f noncontact state
In this condition, u<0
or t i < - l / p
Ak = 0 , f w = 0 by equations (8)-(11) V = (k, + Kp)x~ + m, yc2t+ k2 x2 + m2Ycz~
(12)
Equation (12) means that, if kl + Kp > 0, x=0,
V=0
x¢0,
V>0.
(13)
The derivative of equation (12) is I2 = 2~,{(k~ + Kp)x~ + rnlS~~} + 2~2(k2x2 + m2.~2) • Substitute equations (1) and (2) into this equation and after being arranged, the following equation is obtained: 12 = - 2(cl + Kv).~ - 2c2 ~2. (14) In equation (14), if
c, +Kv > 0 ,
(15)
12-..<0
(16)
from the result of equations (13) and (16) in the condition of noncontact, the system is stable in the sense of Lyapunov theory. 3.2. Condition o f contact state
In this condition, u / > 0 and t i > 1 - 1 / p A k = aHu 1/2, fw = (1 + p f t ) H u 3/2
by equations (8)-(11) V = (kl + Kp)x~ + ml Yc2 + k2 x2 + rn2Jc2 + all, u~/2
(17)
from this, if k~ + Kp > 0, a > 0, x=0,
V=0
x#0,
V>0.
(18)
The derivative of equation (17) is 12 = 2-xt{(k~ + Kp)x) + mt xz } + 2-~2(k2x2 + m2-~2) + ~aHu3/2f~"
(19)
Substitute equations (1) and (2) into this I2 = -- 2(c, + Kv)Yc~ -- 2pHu3/2ft 2 - 2Krfw2~ - 2c22~ -- (2 - ~a)Hu3/2fi. This results in, if cL+K~>O, K f > O , 12 ~<0.
a=~,
(20) (21) (22)
From the result of equations (18) and (22) in the condition that the manipulator and the wall are in contact, the system is stable in the sense of Lyapunov theory. Here we should pay attention to the value of a. In equation (21), from the above derivation a must be fixed to 4/5. As described before, a is an uncontrollable constant, and the condition of a = 4/5 is the condition so that the function V is a Lyapunov function rather than the condition for the stable control.
Impact/contact control of a robotic manipulator
103
As shown by the above results, the system is stable even with collision phenomena. Furthermore, in equations (14) and (20), 2pHu3/2f42 is the term expressing the collision effect on the Lyapunov function. The derivative of the Lyapunov function I;"is smaller by this value than that without the collision term (2pHu3/2ft2). This implies that the collision has a stabilizing effect. For instance, in the case of marginal stability, such as the case where there is no damper term, I7 is zero in the condition of noncontact, but it becomes negative in the contact condition. As the characteristic of the Lyapunov function is the total energy of the system, the negative derivative stands for energy loss. This means that the system is stable, because the system loses energy during the collision and V goes to zero as time passes and as the system experiences collision a number of times. 4. S I M U L A T I O N
METHOD
AND
RESULTS
The system stability and transient response are studied by numerical simulation using the same model in Fig. 1 as follows.
4. I. Simulation method The following state equation is defined using the equations of motion, equations (1) and (2) = A x + b f + fw
(23)
where x denotes a state vector, b f is an input vector and fw is a collision force vector: (24)
X ~---{X 1"~1X2 "~2)T
A =
0
1
0
0
kl ml
el m~
0
0
0
0
0
1
0
0
k2 --m2
I ,w--{0 b=
0
(25)
--1 m]
0
:.
ml
0
o
c2 m2
t
(26)
m2 J
The equations of motion are solved by the Runge-Kutta method. As shown by matrix A in equation (25) both manipulator and the wall are considered individual systems. The difference between contact and noncontact states is only whether collision force is applied or not, which is determined by the relative deformation and relative velocity of the manipulator and the wall. It means that these systems are not connected even in the contact condition. This method allows the calculation with a nonlimited number of collision.
4.2. Boundary condition Contact/noncontact condition is judged by the relativedeformation u and the relativevelocity [equation (7)]both at the beginning and the end of every time step of the Rungc--Kutta method. On thisjudgment, the time steps, which include contact/noncontact condition changes, are critical and discussed in more detail below. 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 tc by the linear interpolation of u and ti, and calculate all state values such as xt and x2 by the linear interpolation using step time and tc. For the rest of this time step (from to to the end of the time step), the response is calculated under contact condition. After this time step the
YASUMASASHOJI et al.
104
Y, 1L
EO ×'
E E
2o
1
~E0 x '
~_
z
0
5
~
20
:ii 20 Time
2O
Time
s
Fig. 2. Marginally stable, perfect elastic collision.
s
Fig. 3. Marginally stable, collision with energy dissipation.
collision force fw is considered to include the effect of contact. Naturally, before the contact time step, collision force is zero, expressing the noncontact condition. The time step for the state changes from contact to noncontact which is just the contrary of the previous process. That is, by monitoring every time step, the exact state change time is obtained and the condition of simulation is switched from contact to noncontact. 4.3. Simulation results
Simulation results are shown in Figs 2-6. The conditions of parameters and others are shown in Table 1 and as follows. Case numbers are also indicated. No. 1. The response of the marginally stable condition with all damping factors are zero (c, K,. and p = 0) (Fig. 2). This response continues periodically without divergence or convergence. In this simulation, the desired condition is that the manipulator is touching the wall with zero contact force. The following three simulations (i.e. Nos 2-4) have the same desired condition. In Fig. 2 the part where x~ is larger than x2 means the manipulator plunges into the wall. This is applicable in the following cases. No. 2. The response of the marginally stable manipulators with the damping effect of collision (p = 1) (Fig. 3). These two simulations (Nos 1 and 2) show that marginally stable systems can be stabilized by the collision with energy dissipation (Figs 2 and 3). This fact agrees with the result of the analysis using Lyapunov method. No. 3. The response of stably controlled case (p = 0.1) (Fig. 4). By this simulation, it is shown that even a simple nonlinear feedback control rather than complex control enables stable contact control.
~E
°
,
- ~
,
zo
~:~
~
0
._ C3
_1 ~
~E
2~0
o
-5
;s °-°
1 .........
2b Time
s
Fig. 4. S t a b l y c o n t r o l l e d .
20
Time
s
Fig. 5. Stably controlled with larger energy loss parameter.
Impact/contact control o f a robotic manipulator
iE n
-
' -
105
2'0
-1
2'0
~-~0tilI~-
¢D
Time
/0
s
Fig. 6. Stably controlled with pushing force.
Table 1. Simulation parameters No. m 1 k I 1 2 3 4 5
1 1 1 1 1
10 10 10 10 10
cI rn2 k 2 0 0 0.2 0.2 0.2
10 10 10 10 10
300 300 300 300 300
c2
Ko
Kv
p
H
xl0
Figure
0 0 0.5 0.5 0.5
15 15 15 15 15
0 0 3 3 3
0 1 0.1 1 0.1
10000 10000 10000 10000 10000
0 0 0 0 0.3
2 3 4 5 6
No. 4. The response of No. 3 with larger value of p (p = 1) for larger energy dissipation in collision (Fig. 5). These two simulations show the same result as Nos 1 and 2, and stabilizing effect is observed even in the stably controlled system. From this result, we can estimate that we can positively utilize collision in impact/contact control. No. 5. The response of No. 3 with pushing force (Fig. 6). Oscillation of contact force is observed while the contact condition is kept. The time for continuous contact condition is obviously shortened relative to the zero force control cases. This means that, if the object and the manipulator has adequate strength, we should use impact/contact phenomenon to shorten the time to grasp or to stub. This means it is possible to shorten the total operation time of manipulators including contact tasks.
5. C O N C L U S I O N In this paper, positioning of manipulators against the wall has been analyzed by the Lyapunov direct method and its transient behavior has been studied by extensive numerical simulations. As a result, the following conclusions have been obtained: (1) The Lyapunov direct method is effective even if the controlled system includes collision. It is proved theoretically that positioning with collision can be stable. (2) Collision is a phenomenon which stabilizes the system under some conditions such as marginal stability. REFERENCES 1. E. Colgate and N. H o g a n , A n analysis o f contact instability in terms of passive physical equivalent. Proc. IEEE Conf. Robotics and Automation '89, pp. 4 0 4 - 4 0 9 (1989). 2. N. H o g a n , O n the stability of manipulators performing contact tasks. IEEE J. Robot. Automat. 4, 6 7 7 - 6 8 6 (1988). 3. T. Fujita and S. H a t t o r i , Periodic vibration and impact characteristics o f a nonlinear system with collision. Bull. JSME 23, 4 0 9 - 4 1 8 (1980). 4. T. F u k u d a , N. Kitamura and K. Tanie, Flexible handling by gripper with consideration of characteristics of objects. Proc. IEEE Conf. Robot. Automat., pp. 703-708 (1986). 5. T. F u k u d a , N. Kitamura and K. Tanie, Adaptive force control of manipulators with consideration of object dynamics. Proc. IEEE Conf. Robot. Automat., pp. 1543-1548 (1987). 6. J. Kahng and F. M. L. A m i r o u c h e , Impact force analysis in mechanical hand design--part I. Int. J. Robot. Automat. 3, 158-164 (1988).
YASUMASA SHOJI el al.
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7. K. Tanie, K. Komoriya, T. Kaneko, T. Ohno and T. Fukuda, Bilateral remote control with dynamics reflection. Proc. Sixth C I S M - I F T o M M Symp. on Theory and Practice of. Robots and Manipulators, pp. 296--308 (1988). 8. K. Youcef-Toumi and D. Gutz, Impact and force control. Proc. Conf. 1EEE Robot. Automat. '89, pp. 410-416 (198%
APPENDIX
1
To obtain the relation between the energy loss parameter p and the coefficient of restitution e. we will consider the contact of two masses, m and M, with velocities of v0 and O, respectively. Equations of motion/during contact: m.%, = --/[,
(A 1)
"15,:M= 1,, I,, - (1 + pu)Hu v2
(A2) (A3)
u = x m - x M.
(A4)
From equations (A1) (A4),
D1 m(&n - -~:~) = - . L - M,L
(m)
mii= -
( l + p i O H u 3'2.
I+M
(A5)
If we consider that u = z, from equation (A5): dz dz du M + m ........... (l + pz)Hu 3'2. dt du dt Mm Since du/dt = z, equation (A6) is rewritten into equation (A7):
du
Mm
+ p Nu3'e du.
Equation (A7) is integrated using the initial condition of z, :., £[
-
(A6)
(A7)
= v 0 at u,--0 = 0
Mm dz £7 M+ m [:z+p = Hu 32du.
(AS)
The left hand of equation (A8) is calculated as follows:
f,[
Mm -M-+m
dz Mm 1/::+p =-]~+m
£:
~
,,pz
dz
Mm l { ln(PZ+ 1~ M+mp ~ pvo-pz+ \ p v o ~ l l ),
(A9)
The right hand of equation (AS) is as follows:
f ' Hu 32 du = 2sHu52.
(A10)
Mm , { , _ ,n(PZ+l~f M + n i p S ~ p , o p_~ + \ p r ~ , + i ] j = e s H u 52.
{All)
From equations (A9) and (AI0), if p > 0,
If we substitute
pt' o =
Vo
and pz
= Z,
( A I 1) is written into equation (A12) Mm 1 / . Z+I\ M + m p 2 ~ [ % - Z + I n V 0 + i ) =~HuS"2" equation
(A12)
Here, if we consider e as the coefficient of restitution, we obtain the following two equations at the exact time when two masses, m and M, detach u=0,
Z=
- e V 0.
(AI3)
As the condition of equation (A12) is applicable at the time of detach, using equation (AI3),
Mm I { V+tnP ~ (l+e)V0+
ln(-el/f'+l~ 0 \ VI;+ i } 3 = .
(AI4)
Since
Mm - > O, physically, M+m (l+e)l~)+ln\
( eVo±! V0+l /=0.
(AI5)
From equation (AI5), equation (AI6) is derived, - - - 1 ( V1 l+0 ) exp{-(l+e)V,}.'.e . . . . .1. ( P1~+0 ) e - (>o ' pro This is the equation which expresses the relation between e and p.
exp{-(l+e)pvo}.
(A16)
107
Impact/contact control of a robotic manipulator VM=0 Before Contact r
~xM
Xm
CONTACT
After Contact
i VM
V In
Fig. A1.
d
O
U
- -
U
'B
Co l ck-wsi e
Fig. A2. (a) Contour of the Lyapunov function with small p. (b) Trajectory of the Lyapunov function. vc: contour of the Lyapunov function in contact state; v.e: contour of Lyapunov function in noncontact state.
APPENDIX 2 On the Differentiability o f the Lyapunov Function
In this paper, in a sense, two scalar functions are used for contact and noncontact states. We consider this as follows: Among the impact/contact parameters, H and p are very important. As mentioned in the paper, H represents a kind of spring constant, or stiffness, for the local deformation between a manipulator and an object or a wall. p denotes energy loss parameter, or a kind of damping parameter, which has come from the motion difference in the acceleration during collision. On the judgment of contact or noncontact, the parameter p is more important. We will discuss the Lyapunov function focusing on p. From physical consideration, p is assumed nonnegative and it is true in the following discussion. According to our definition of the Lyapunov function, the contour of the function can be drawn as in Figs A2 and A3. In the case wherep is sufficiently small, the contour is shown in Fig. A2(a). The line is connected smoothly at the two points, A and B where the contact/noncontact condition changes. In this case, a trajectory of the Lyapunov function is shown in Fig. A2(b). Here the inner closed line means that the contour value on this line is smaller than that on the outer line. At the contact, the trajectory must be clockwise. If the system has stable dynamics, the trajectory converges to the origin, which means the system goes to the equilibrium point and stop moving. It is not questionable that the Lyapunov function is differentiable in this case. In the case, however, where p is not so small, we must be more careful. It will occur that the contour line does not have the closed loop, because the line jumps to somewhere else when it meets the line of d u / d t = - lip. The contour of this case is shown in Fig. A3. According to this plot the contour jumps from point C to point D. However, we can guarantee that the jumping direction is always inward, because the Lyapunov function loses collision terms with positive value drastically, and becomes smaller than before the jump. We have another reason for the guarantee. That is the direction of the trajectory. As described above, the direction is always clockwise. Thus the changing rate of the Lyapunov function should be considered only in one direction, which means we have no need to guarantee the motion with counter-clockwise trajectory and no need to consider the jump from point D to point C in Fig. A3. In this case, at the point C and D of Fig. A3 the Lyapunov function is not differentiable but it is obvious that the value of the function reduces on the plane at any time, which means the system is stable.
/ vc2
Vnclf
Vn~
Vol
U
Fig. A3. Contour of the Lyapunov function with larger p.
108
YAS~:MASASHOJ1et al. AUTHORS'
BIOGRAPHIES
Yasumasa Shoji--Yasumasa Shoji graduated from Keio University, Tokyo, Japan in 1982 and received a M.Eng. degree
from the University in 1984. Since 1990 he has been working for TOYO Engineering Corporation, mainly engaging in the field of engineering analyses. From 1988 to 1989 he joined the Science University of Tokyo, and has researched stable control of manipulators including collision.
Makoto Inaba--Makoto Inaba graduated from Keio University, Tokyo, Japan in 1969. Since this year he has been working
for TOYO Engineering Corporation, mainly engaging in the field of aseismic design and intelligence of chemical and other plants. He is the leader of the Advanced Technology Group, Technology R&D Center.
Toshio Fukuda--Professor Toshio Fukuda received the D.Ing. degree from the University of Tokyo in 1977. After he joined
the National Mechanical Engineering Laboratory in 1977 and the Science University of Tokyo in 1981, he joined Nagoya University, Japan in 1989 as Professor of Mechanical Engineering Department. He is the Vice President of IEEE Industrial Electronics Society (1991-).