Abstract. In the past decades, there were considerable number. The backlash, which is caused by a gap between an actuator-side gear tooth and a link-side ...
FP04 5:lC
Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
Nonlinear Adaptive Control for Manipulator System with Gear Backlash Jung-Hua Yang’ and Li-Chen Fu2 1. Dept. of Electrical Engineering Yungta Jr. College of Technology and Commerce, Taiwan, R.O.C. 2. Dept. of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C. Abstract
Backlash is a common phenomenon in servomechanism, especially in gear-driven mechanical systems. A certain amount of backlash is necessary when two gears are in mesh. In other words, if the space between teeth were not made greater than the tooth width measured on the operating pitch circle, the gears could mesh without jamming. However, any amount of backlash greater than the maximum allowed backlash which is to ensure satisfactory meshing of gears can lead to system instability and cause position errors in gear chains. In a majority of past researches about backlash, the effect of backlash is usually taken to be static. That is, it was thought of only as a gap between two mashing gears, and once their contact is established and no change is made in the direction of rotation, the gear chain may work as if the backlash does not exist. However, from a practical point of view, this is far from being reasonable, When two meshing gears, between which backlash exists, begin to rotate, it is expected that a series of collisions take place between them if they keep the original direction of rotation all the time. Therefore, it is more
In the past decades, there were considerable number of researches concerning the control issues of robotic manipulators or their similar systems. However, few of them have taken the effect of backlash into account. Coleman el al. [l]approximated the backlash and Coulomb friction by well-behaved functions to ease the restrictions due to hard nonlinearities inherent in the gun-turret system, and then synthesized a nonlinear controller combining ”feedback stabilization” part and ”feedforward steadystate compensation” part. In [a], a robust digital design method was presented to solve this problem by first constructing a quadratic Lyapunov function and then iteratively modifying the feedback gain matrix. On the other hand, Taylor and Lu [3] decomposed the whole system into the drive subsystem and the wheel/barrel subsystem, and utilized a two-stage control method. The control system incorporated a scheme based on sinusoidal-input describing function model of the drive system to reduce the effect of backlash and nonlinear friction, and a dissipative control scheme to make the system insensitive to the parameter uncertainty and unmodeled dynamics. In addition, an adaptive control scheme based on the anti-backlash model was also developed in [4]. To accommodate all factors that influence the performance severely due to backlash, we first develop a more detailed model which can describe all the characteristics of backlash. They are similar to that of rigid-link flexiblejoint robots by inspection. Then, based on the derived model, a nonlinear adaptive controller, in which only the position and velocity signals are needed, is further devised on the basis of [7]. Via Lyapunov analysis, a SOcalled semi-global tracking is achieved, and simulation studies are also provided to demonstrate the validity of the proposed strategy. The remaining part of this paper is organized as follows: Section 2 presents the dynamic modeling of backlash existing in the gear chain of the manipulators. In section 3, based on the mathematical model derived in section 2, an adaptive scheme is developed. Simulation
realistic to consider backlash dynamically instead of stat-
results are shown to demonstrate the validity of our pro-
ically merely as some kind hysteresis or deadzone.
posed strategy in section 4. Finally, some conclusions are
The backlash, which is caused by a gap between an actuator-side gear tooth and a link-side gear tooth, is a common phenomenon in manipulator systems with gear in their joints. However, if the amount of backlash is greater than the maximum allowed amount which is to ensure satisfactory meshing of gear, the system instability may appear in dynamic situations and cause position errors in the gear chains. Hence, in this paper, a nonlinear adaptive controller is devised to cope with the effects due to backlash as well as parameter variation. A socalled semi-global tracking is achieved, and simulation studies are also provided to demonstrate the validity of the proposed strategy.
1 Introduction
0-7803-3590-2/96 $5.00 0 1996 IEEE
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given in section 5 .
2
i={
Dynamic Modeling
Considering the planar mating (or conjugate) gears as shown in Fig.1, the angle of the driving gear designated as gear 1 is q1 and the one of the driven gear designated as gear 2 is q 2 . Their operating pitch radii are rpl and rp2, respectively. Without loss of generality, we assume both angles, when the driving and driven gears are just in contact without deformation the profile surface, are viewed as the reference angles. If the driving gear moves with some slight displacement of rotation such that two contacting gear teeth are breaking away, then the two gears become out of contact. Thereby the motion switches to ”backlash mode”. The following equation shows the condition under which the operation is in ”backlash” mode: r p 2 ~ 2-
B
rplql
~ p 2 ~ 2
Tp2Q2
+ Crl + Crz
,”positive impact” mode , ”backlash” mode ,”negative impact” mode
(4)
+
where r1 = rplql - rp2q2 and 1-2 = r l B.Then, we can easily multiply the above equation by individual operation pitch radii to get the impact torques ?I and ?2 imposed on gear 1 and gear 2, respectively. Assume n = PP2 is the gear ratio and, hence, then we obtain
*
?j =
I?rs + C r s 0
i
zr4
T2
+
CTq
,”positive impact” mode , ”backlash” mode (5) ,”negative impact” mode
= n-lT1
(6)
c
where I? = K r P ~ r p 2 , = CrP1rp2,rg = nql - 9 2 , and B 7-4 = 7-3 $- -. PP2
3
Adaptive Controller Design
(1)
where B is the amount of backlash. If the driving gear rotates across the backlash and the contact with the driven gear is established again, then the mode switches over again such that the impact force builds up immediately. Hence, there are totally two conditions for the ”impact” mode. For convenience, we refer to the condition of (2) as the ”positive” mode:
rp1q1 2
Iirl 0 ICr2
(2)
On the other hand, the second condition described below is referred to as ”negative” mode.
In order to facilitate the controller design, we have to mathematically formulate the motion of the three modes, namely, the backlash mode, the positive mode, and the negative mode. In the operation of impact mode, although the gear teeth are mechanically rigid, the contact model of the two gears is formulated as a mechanical device composed of a spring and a damper, as shown in Fig.1. In this model, the stiffness coefficient I< describes the material rigidity of the gear teeth, whereas the damping action with damping coefficient C proportional to the relative velocity is aimed at dissipating the energy due to impact. Hence, the impact collision will only result in one-half collision of a simple lumped parameter spring-mass-damper system since one gear tooth will break away with the other once the contact is removed. Thus, we can write down the impact force f determined by the involved impact model in a more compact form:
Consider an np-link robot manipulator with gear backlash in its joints. Based on the Lagrangian-Euler formulation, the dynamic equations of the robot system can be written as:
where the definitions of M I , C1, and GI can be found in the literatures of robot dynamics, N is a diagonal matrix with diagonal elements nil i = l , 2 , . . . , np being the gear ratios of each actuator i, respectively, J,,,, C, are the inertia constant and viscous coefficient of the motor, shaft and gear assemblies, r~ is the Coulomb friction torques on the actuator part, and the control vector r, represents the torque input at each actuator. Furthermore, the term T is the impact torque where the driving and driven gears establish contact, and can be written as r = [q, 7 2 , . . . , rnJT where
ri =
{
+C+5 0 I(r6 + Cf6
Er5
,”positive impact” mode , ”backlash” mode ,”negative impact” mode
+ 3,
(9)
where rg = nq,; - q1i and 1-6 = rg and i denotes the ith link of the manipulator system. The control objective of this paper is to force the link position to smoothly track the desired path. To fulfill such control objective, the control system must be kept in the ”impact” mode as long as po2sible and then be applied appropriate control laws. Therefore, two phases of control are developed. One is the ”impact” mode control which is based on the whole dynamics, and the other is the ”backlash” mode control which is, since the link part
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is uncontrollable, based only on the actuator dynamics to force the motion into "impact" mode as quickly as possible. Before we proceed with the adaptive controller design, certain properties and additional assumptions about the whole control system needed in the process of design will be given below.
In the following, we will present an adaptive control synthesis which contains two stages of design. In the first stage, we consider the virtual input z d as: z d
= Y d ( q l d , i l d r iild)el + N - ' I ( ~ r + N - ' C q i d - P 2 k - ~ 2 e ~ (17)
and the adaptive laws as following:
Property 1: (Linear-in-Parameter) The dynamic equations of the link part is linear in parameters and, hence, can be expressed in the following form:
M(qr)i'r + Cr(qr,ir)ir + Gr(qr) = Y(qr,Qr,qr)Or
(10) where Y(qr,qr, il) is called the regressor which is a matrix of known functions of the joint variables, and 81 is the vector of constant parameters. Property 2: iGfr(qr) - 2Cr(~l, Q r ) is skew-symmetric. Property 3: The time-derivative of inertia matrix is bounded by the following equation:
d
I Qilleill+
l,M(nr)ll
(11)
a2
where a l p h a l , a2 are some known positive constants, and er = qr - q l d , where q l d is the link desired trajectory, is the position tracking error.
Property 4: Given two vectors x and y, the equation Ci(qr,z)y = C/(ql,y ) will ~ hold. Property 5: The norm of the centrifugal-Coriolis matrix Cr(ql,Qi) can be represented by the inequality IlCi(ql,41)11 5 rllilll, where y is a known positive constant. As stated previously, if the control objective is achieved, each axis will be supposed to be in the impact mode. Hence, we first present the design of an adaptive controller for the operation in impact mode. Consider the robot dynamics which is operated in impact mode, as sho:vn in (7) and (8), and then we define a new variable called the virtual input to the link part:
z= Rq, + cqm
(12) By virtue of the definition of 2, we can reformulate (7) and (8) as
+
Mr(qi)&'r(qr,qr)Qi Gr(qi) JmC-lZ (Cm - C-'R)Qm
+
where qr)
f1
+
= Z
fi
+ f2
=
~m
(13) (14)
where the notation 'overhat' represents an estimate of that variable, rl is a positive definite, diagonal adaptation gain matrix, p is a sufficiently large positive number, and the conditions "A" and "B" denote "when all axes are in impact mode" and "otherwise", respectively.
Remark: The auxiliary variable k is introduced to serve as a surrogate for the link velocity tracking error. The scalar gain /3 is a positive designed constant which will be devised to obtain the desired stability result. Remark: The use of dead-zone in the adaptive law 8, and the auxiliary variable k is suggested since then both of them will depend on the vector term of tracking error el. Turning off the updating parameters and the filtering states when any of the joints switches to the "backlash" mode will prevent incorrect operations of adaptation or filtering. As soon as all axes switch to the "impact" mode, the updating or filtering will be resumed. Remark: It is noteworthy that only link position measurements are needed in implementing the actuator desired manifold in spite of the appearance of i[ explicitly in (18) and (19). T h a t is, although k and B r depend on ir, k and f$ can be computed using el only by way of integration by part. In the second stage, based on the results of the firststage design, we propose an adaptive control law for the actual input as:
- kfsgn(ez) - [er
1 + -er P
1
- p k ] (20)
with the adaptation law being:
f 2 = rr + [E(Nqm- Q i ) ] . Nonsurprisingly, through such defini-
= N - ' R q l + N-lCqr and
+ C(N q ,
tion, the dynamics of actuator is reduced to be first-order. Moreover, we define the virtual input error as ez = z - z d where z d will be determined in the sequel, then
+
bJr(qr)qr Cr(qr,h)h4-Gr(qr) f i J,C-'Z (C, C-'IT')im fz
+
-
+
=
z d
=
T,
-I-ez(l5)
(16)
+ (c,
- C-'R)q,, r2 is where W ( i d r q m )= J , , , C - ' Z d a positive definite, diagonal adaptation gain matrix and ku can be chosen arbitrarily as a positive scalar number. By applying the adaptive control laws presented above, we can obtain the following theorem.
437 1
Theorem 1 Consider an np-link rigid manipulator with gear backlash in its joints as formulated in (7) and (8). If the control inputs are designed as (17), (18), (19), (20), (21), and the designed constant /3 is chosen sufficiently large, then all signals in the closed-loop system are bounded and the tracking error el will converge to zero asymptotically, i.e., limt,, er(t) = 0.
Outline of proof: In the stability proof, we similarly make a two-stage analysis. For the sake of limited space, we only give the sketch of stability proof in this paper and the readers can see [7] for more details. In the first stage, we define a non-negative function:
Vi
+
= (1/2),B2eTei+ ( 1 / 2 ) 4 ~ l e , ( 1 / 2 ) k T k (l/2)xpc1
(22)
(23)
=
(132
= 423 = P
433
413
and a i , y;,vZ,i=l, 2 are some positive constants. It can be shown [7] that Q is a positive definite matrix as long as P is chosen large enough. In the second stage, similarly, we define another nonnegative function as
+ (1/2)J;rzJz
VZ = (l/2)e;J,C-1Nez
VZI -e;kvez
- e z [ h + (1/P)er - (I/@]
v = VI + v2 = [e?, eT, k T , @] Mr/P P21 MI M / P P =
[
0 0
-M/P 0
As a result of
0
-M/P I 0
Vi I -xTQxz +e;[& where
with (111
= p-
711 7
+ (l/P)er - ( l / P ) k ]
and
V
0 0 rOl :
and P is chosen to be sufficiently large. Apparently, P can be made positive-definite if ,B is large enough. Then, differentiating VI with respect to time t along the link part dynamics of the closed-loop system we have
(24)
(26)
To obtain the complete analysis of stability, we simply combine the non-negative functions VI and V2 to form the Lyapunov function candidate
where 21
(25)
and take its first-order derivative along the actuator part dynamics of the closed-loop system to obtain
- t ( ~ / p ) ~ T-~( , i e/ ~, ~ ) k ~ + ~ (, iep ,) i T r l i j T
=
431
V2
(27)
mentioned above, we get
+
=
VI
5
-xTQx2 - esk,eZ
v2
and, hence, the asymptotical stability of the overall system is concluded. Q.E.D. Now, we turn our attention to the control for the operations in "backlash" mode. As in the preceding statements, if the mode of operation switches to the backlash mode, there is no connection between the actuator part and the link part, so that the links become free and uncontrollable. The urgent matter of this moment is to establish contact rapidly but preventing excessive control force to result in unacceptable collision. Conceivably, a simple regulation scheme for the actuator dynamics will work well here. But entering into "positive impact" or "negative impact" mode depends on the value of the position error signal at that time. If el; is less than zero, which means the driven gear must continue to rotate clockwise, then we set the regulation point of the driving gear at (nflqri +6), where the tiny value 6 is to guarantee the achievement of establishment of contact. On the contrary, if eli is greater than zero, we set 6). the regulation point at (nf'qri -
e+
421
=
Y12
4
Simulation Study
In this section, some computer simulation results are
given to illustrate the performance of the proposed con-
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trollcr. A two-link rigid manipulator is utilized for simulation and a sinusoidal function @ d = sin(2f) is used for both links as the desired trajectory. Fig.(2)-Fig.(7) show the results of simulation. Fig.(2)Fig.(3) depict the tracking responses of both links while Fig.(4)-Fig.(5) present the associated tracking errors. In Fig.(G)-Fig.(7), the contact patterns of both links during operation mode are given.
5
[7] Yeh, C-S, L. C. Fu, and J. H. Yang,”Nonlinear Adap. tive Control of a two-axis gun-turret system with backlash”, Proc. of IFAC, Vol. I(, p p . 91-96, 1996.
Conclusion
In this paper, we have briefly discussed the dynamic behavior of backlash. Instead of viewing it as a static hysteresis phenomenon, a more realistic mathematical formulation describing the dynamics of backlash is given. Based on that, we take a n,-link rigid manipulator as a target system for study, and an adaptive nonlinear controller is then devised via two-stage design approach. The so-called semi-global tracking is achieved. The term semi-global means that the region of convergence can be extended as large as possible by properly designing the control gains. Simulation results are given and demonstrate the good performance of the proposed strategy.
References
Driving Gear
(GEAR I)
Fig.( 1) Impact Model
-
h
Huang, J., C. F. Lin, N. Coleman, M. Mattice and S. Banks,” A Nonlinear Controller for the Gun-Turret System” Proc. of the American Control Conference, p p . 424-128, 1992.
-0.5 -1
0
Chia, W., N. I
