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Modeling and robust control of worm-gear driven systems
Simulation Modelling Practice and Theory 17 (2009) 767–777
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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Modeling and robust control of worm-gear driven systems T.-J. Yeh *, Feng-Kung Wu Department of Power Mechanical Engineering, National Tsing Hua University, No. 101, Sec. 2 Kuang-fu Rd., Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 13 August 2008 Received in revised form 30 October 2008 Accepted 2 January 2009 Available online 21 January 2009
Keywords: Modeling Worm-gear driven systems Nonlinear systems Uncertainty Robust control Sliding control
a b s t r a c t This paper investigates modeling and control issues associated with worm-gear driven systems. In the modeling part, static and dynamic analyses are conducted to investigate the characteristics of the worm-gear. The static analysis reveals not only the non-backdrivability but also the dependency of break-in torques on the loading torque, direction of motion as well as crucial system parameters. The dynamic analysis generates four linear equations of motion, which, at any particular instant, only one applies. The applicable equation at any given instant depends on the direction of motion and the relative magnitude between the input torque and the loading torque. In the control part, a sliding controller is designed based on the modeling results. The controller can provide robustness against the uncertain loading torque and variations of system parameters caused by the speed-dependent nature of the coefficient of friction. Because the dependency of the dynamic equation on the operating condition may render the controller ill-defined in some scenarios, a lemma is proved and can be used to select proper control parameters to guarantee the well-definedness of the controller. The proposed control scheme is applied to a worm-gear driven positioning platform. Experimental results indicate the proposed control system exhibits more than 70% improvement in tracking error over PID control. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Worm-gears are frequently used in electro-mechanical systems where high transmission ratios are required. Compared to the other transmission elements such as spur or bevel gears, in which a large number of gears and consequently a large space is required to achieve the desired gear ratio, the worm-gear provides a large transmission ratio in a small space with a single gear set. Therefore, worm-gears allow one to adopt small, high velocity motors in systems involving slow motion without the need to redirect the torque through a complex gear train. In addition to the compactness, that worm-gears can provide the orthogonal redirectioning of the torque is also desirable in many designs. Finally, another potential advantage associated with worm-gears is the unique property of non-backdrivability (or self-locking) [7]. This property allows the gearset to be driven on the input side, but not back driven on the side of the load. Consequently, the servos do not have to be actuated all the time to hold the loads in a specific position. These attractive features make worm-gears desirable in robotic applications in which special demands on the transmission element are needed. The examples include stair-climbing robots[11], pruning robots[2], modular self-reconfiguring robots[10], and spherical robotic manipulators[5]. In conventional gears, the motion of one tooth relative to the mating tooth is mainly a rolling motion. However, the relative motion between worm and gear teeth is pure sliding, and so it should be expected that friction plays a dominant role in the performance of worm-gear driven systems [8]. Friction induces nonlinearities to the system dynamics and are usually the major source of motion disturbances and instability in feedback control. For instance, it is pointed out in [5] that a * Corresponding author. Tel.: +886 3 574 2922; fax: +886 3 572 2840. E-mail address: [email protected] (T.-J. Yeh). 1569-190X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2009.01.002
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vibratory behavior has been observed to be associated with worm gears in positioning systems under gravity loading in the case of spherical robot. Regardless that the frictional effect can diminish the usefulness of worm-gears in servo-control applications, the studies in the literature on the impact of friction on dynamics and control of worm-gear driven systems are limited. [1] developed a load-dependent transmission friction model for the potential use of constructing feed-forward compensation for worm-gear driven servo systems. The model presented therein is based on a wedge-like planar mechanism. Ref. [6] showed that the dynamics of a cantilevered manipulator driven through a worm-gear are switched between two sets of nonlinear differential equations. These two sets of nonlinear equations are linearized around a nominal trajectory and are then used as the plant model for designing a classical lead–lag controller [3] and a QFT controller [4]. The approaches adopted in the aforementioned references contain certain limitations which will prevent one from further improving the system performance. For instance, the model in [1] contains implicit variables in the dynamic equations and thus cannot be directly used for controller design. The classical and the QFT controllers respectively proposed in [3] and [4] are linear controllers. They may not have sufficient robustness to deal with uncertainties in the system. In this paper, we develop a worm-gear model which can be used directly for designing a nonlinear controller. The nonlinear controller is sliding-mode-based and can provide robustness against the uncertain loading torque and variations of system parameters caused by the speed-dependent nature of the coefficient of friction. The paper is organized as follows: To begin with, the static and the dynamic behaviors of the worm-gear are examined. The governing equations resulted from the analyses are then used in Section 3 to design a sliding controller. A lemma concerning the well-definedness of the controller is also given therein. To illustrate the usefulness of the model and the controller, Section 4 presents simulations and experimental results of a worm-gear driven system. Finally, conclusions are given in Section 5. 2. Modeling of the worm-gear In the worm-gear considered, a motor torque sm drives the worm and a loading torque sl exerts on the gear. During the power transmission, the worm-gear engagement occurs either at the left surface (as shown in Fig. 1a) or the right surface ( as shown in Fig. 2a) of the gear tooth. In the subsequent analysis, these two types of engagement are respectively referred to as left engagement and right engagement. In the case of left engagement, the force exerted by the gear on the worm consists of a normal force W n , and a frictional force W f . As shown in Fig. 1b, while W f is tangent to the pitch helix on the pitch cylinder of the worm, W n is normal to the pitch helix with a pressure angle /n . Notice that in this figure, the gear axis is parallel to the x axis, the worm axis is parallel to the y axis, and a right-handed coordinate system is adopted. According to the coordinate
Gear
θg τl
z
τm
θw
y x
Worm
(a) Schematics of left engagement
z
Wn
y
φn λl
x
Wf λl Pitch helix
τm Worm
(b) Force diagram of left engagement Fig. 1. Left engagement for the worm-gear.
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769
Gear
θg ττl out
τm
θw
z
yy x
Worm Worm
(a) Schematics of right engagement
Wn
λ τm
φn
Wf
λ
pitch helix
z Worm
y x
(b) Force diagram of right engagement Fig. 2. Right engagement for the worm-gear.
definition, the lead angle kl of worm is equal to the angle between W f and the x axis. The total force acts on the worm can be decomposed into three orthogonal force components F x , F y , and F z :
F x ¼ W n cos /n sin kl W f cos kl ; F y ¼ W n cos /n cos kl W f sin kl ;
ð1Þ
F z ¼ W n sin /n : Only the force component F x can generate a reaction torque on the worm and the worm dynamics is given by
J w €hw ¼ sm þ F x ‘w ¼ sm þ ðW n cos /n sin kl W f cos kl Þ‘w
ð2Þ
in which hw ; J w ; ‘w are respectively the rotation angle, the inertia, and the pitch radius of the worm. For the gear dynamics, considering that F y has its reactive counterpart (F y ) generating a driving torque on the gear, we have
J g €hg ¼ sl þ F y ‘g ¼ sl ðW n cos /n cos kl þ W f sin kl Þ‘g
ð3Þ
in which hg ; J g ; ‘g are respectively the rotation angle, the inertia, and the pitch radius of the gear. In the case of right engagement1, one can perform similar force analysis using Fig. 2. The resultant dynamics are
J w €hw ¼ sm þ ðW n cos /n sin kl W f cos kl Þ‘w
ð4Þ
J g €hg ¼ sl þ ðW n cos /n cos kl W f sin kl Þ‘g :
ð5Þ
and
(1) Static analysis: If h_ g ¼ € hg ¼ h_ w ¼ € hw ¼ 0 for the left engagement case, then
sm þ ðW n cos /n sin kl W f cos kl Þlw ¼ 0; sl ðW n cos /n cos kl þ W f sin kl Þlg ¼ 0:
ð6Þ ð7Þ
The magnitude W f is not large enough to break the maximum frictional force ls W n , where ls is the coefficient of static friction. Therefore ls W n 6 W f 6 ls W n , or using (6) and (7), the inequalities can be rewritten as:
ls 6 l ¼
1
Wf cos /n ðsm lg cos kl þ sl lw sin kl Þ 6 ls : ¼ sm lg sin kl sl lw cos kl Wn
ð8Þ
The conditions on when left engagement or right engagement should apply depend on the direction of rotation and the relative magnitude between sm and
sl . This issue will be investigated in detail in dynamic analysis given subsequently.
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Next we investigate how sm and sl can be related so that the inequalities in (8) are satisfied and worm-gear system stays motionless. C2 ¼
To
facilitate
lw ðcos /n sin kl þls cos kl Þ ;H lg ðcos /n cos kl ls sin kl Þ
the
discussion,
we
define
¼ llwg cot kl ; lg ¼ cos /n cot kl , and
the
following
parameters:
ðcos /n sin kl ls cos kl Þ ; C 1 ¼ llwg ðcos /n cos k þl sin k Þ l
s
l
lw ¼ cos /n tan kl . It is assumed that 0 6 kl 6 p4 so H > 0 and
lg P lw . The investigation is divided into three cases:
lw 6 ls 6 lg : In this case, C 1 6 0 and C 2 P 0. After algebraic manipulations, it can be found that (8) is satisfied if sm and sl are constrained either by
sm > Hsl and C 1 sl 6 sm 6 C 2 sl
ð9Þ
sm < Hsl and C 2 sl 6 sm 6 C 1 sl :
ð10Þ
or
The constraints indicate that to put the worm-gear system in motion, in case of sm < Hsl ðsm > Hsl Þ, the break-in torque sm is C 1 sl ðC 2 sl Þ if positive rotation is desired and is C 2 sl ðC 1 sl Þ if negative rotation is desired. Moreover, to better visualize the above two constraints, a sm sl plane is plotted in Fig. 3 and (9) and (10) appear as the two shaded regions in the figure. We can see that if the drive torque sm ¼ 0, no matter how large the loading torque s‘ on gear is, the worm-gear set remains motionless. This is referred to as the non-backdrivability [7] or the self-locking of the gear. lg < ls : In this case, C 1 < 0 and C 2 < 0. Again by algebraic manipulations, one can find that (8) is satisfied if sm and sl are constrained in either of the shaded regions in Fig. 4. According to this figure, if sl ¼ 0, no matter how large sm is, the wormgear set remains motionless. This is referred to as the self-locking of the worm. Such a design does not happen in practice because no input torque can overcome the static friction to put the worm-gear set to motion. ls < lw : In this case, C 1 > 0 and C 2 > 0. Eq. (8) is satisfied if sm and sl are constrained in either of the shaded regions in Fig. 5. Neither the worm nor the gear is self-locked in this case. If the investigation is conducted in the right engagement case (using (4) and (5)), same conclusions can be reached. It should be noted that in practical applications, making the gear non-backdrivable is usually desirable because it allows the worm-gear to statically sustain any loading torque without motor efforts. In the experimental system investigated subsequently, ls falls between lw and lg , so in the following only the case corresponding to the non-backdrivability in the gear will be discussed. (2) Dynamic analysis: First we consider the case of left engagement. Assume that the worm-gear has a transmission ratio of iwg . Thus the worm angle hw and the gear angle hg are related by hw ¼ iwg hg . Moreover, assuming that the coefficient of kinetic friction is equal to its static counterpart, the friction force W f equals to ls sgnðh_ g ÞW n . Using these two relationships, (2) and (3) can be rewritten as
J w iwg €hg ¼ sm þ W n ðcos /n sin kl ls sgnðh_ g Þ cos kl Þlw ; J h€g ¼ sl W n ðcos / cos kl þ l sgnðh_ g Þ sin kl Þlg g
n
ð11Þ
s
The above two equations can be combined to solve for the normal force W n . In the case of h_ g > 0, we have
Wn ¼
J w iwg sl J g sm : ðJ w iwg þ C 1 J g Þðcos /n cos kl þ ls sin kl Þlg
ð12Þ
On the other hand, in the right engagement case we have
τl
τ m = −C1τ l τ m = Hτ l τm
τ m = −C2τ l
(μw ≤ μs ≤ μ g ) Fig. 3. Relation between
sm and sl for the system to remain static when lw 6 ls 6 lg .
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τl
771
τ m = −C1τ l τ m = Hτ l τm
τ m = −C2τ l
(μ g < μ s ) Fig. 4. Relation between
sm and sl for the system to remain static when lg < ls .
τ m = −C1τ l
τl
τ m = Hτ l τm τ m = −C2τ l
(μ s < μ w ) Fig. 5. Relation between
sm and sl for the system to remain static when ls < lw .
J w iwg €hg ¼ sm þ W n ð cos /n sin kl ls sgnðh_ g Þ cos kl Þlw ; J h€g ¼ sl þ W n ðcos / cos kl l sgnðh_ g Þ sin kl Þlg n
g
ð13Þ
s
so for h_ g > 0
Wn ¼
J g sm J w iwg sl : ðJ w iwg þ C 2 J g Þðcos /n cos kl ls sin kl Þlg
ð14Þ
Since W n is an internal force, it has to be eliminated from (11) and (13) in order to derive the explicit dynamic equation. Before doing so, we investigate a particular scenario to understand the limitation in (11) and (13). The limitation comes from the fact that when deriving these two sets of dynamic equations, it is assumed that backlash does not exist so that the worm can freely engage with the gear either from the right or the left and still obey the kinematic constraint hw ¼ iwg hg . This will lead to some illed-situations. Consider the case that sl ¼ 0 and sm > 0, if the left engagement occurs, then (11) determines the dynamics. Since the normal force in (12) has to be positive, we must have J w iwg þ C 1 J g < 0. If we eliminate W n from (11), then the equation
ðJ w iwg þ C 1 J g Þ€hg ¼ sm
ð15Þ
is achieved. This indicates that the resultant acceleration is in the opposite direction of the input torque which is completely absurd. Contrarily, if the right engagement occurs, then (13) determines the dynamics. In this case, the associated normal force in (14) is positive because J w iwg þ C 2 J g > 0 and cos /n cos kl ls sin kl > 0 from the assumption of lg P ls P lw . Again eliminating W n from (13) we have
ðJ w iwg þ C 2 J g Þ€hg ¼ sm
ð16Þ
which makes physical sense due to the fact that € hg has the same sign as sm . From the analysis, when h_ g > 0; sl ¼ 0 and sm > 0, only the right engagement occurs.
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In order to avoid the erroneous engagement, we add an assumption that J w iwg þ C 1 J g > 0, or
C1 >
J w iwg Jg
ð17Þ
so that W n automatically becomes negative in the illed-situation. Indeed, with (17), positiveness of W n alone determines whether left engagement (11) or right engagement (13) should be used to derive the explicit dynamic equation and the dynamic equation is given by
ðJ w iwg þ C 1 J g Þ€hg ¼ sm þ C 1 sl if
sm 6
Jw iwg sl ðleft engagementÞ; Jg
ðJ w iwg þ C 2 J g Þ€hg ¼ sm þ C 2 sl if
sm >
J w iwg sl ðright engagementÞ Jg
ðJ w iwg þ C 2 J g Þ€hg ¼ sm þ C 2 sl if
sm 6
Jw iwg sl ðleft engagementÞ; Jg
ðJ w iwg þ C 1 J g Þ€hg ¼ sm þ C 1 sl if
sm >
J w iwg sl ðright engagementÞ Jg
ð18Þ
for h_ g > 0, and is given by
ð19Þ
for h_ g < 0. Therefore, the assumption in (17) actually eliminates the ambiguity in the worm-gear engagement. 3. Controller design hg ¼ sm þ C sl in which the parameter C, According to (18) and (19), the dynamic equation is in the form of ðJ w iwg þ CJg Þ€ depending on the sign of the velocity and the relative magnitudes between sm and
Jw iwg Jg
sl , equals either C 1 or C 2 . The param-
eters C 1 and C 2 are functions of the coefficient of friction. In practice, the coefficient of friction varies with the speed [8], so C 1 or C 2 can be viewed as uncertain parameters in the dynamic equation. Moreover, the loading torque encountered can also be uncertain. Assuming that the control task is to make hg track a desired trajectory hd ðtÞ, in order to achieve high tracking performance, the control system needs to provide robustness again such uncertainties. J i ^l being the esti^l with s For the purpose of robust controller design, we first introduce a new input u, where u ¼ sm wJ wg s g _ mated loading torque. For hg > 0; in terms of u, the dynamic equation of (18) becomes the form
J i €hg ¼ b1 u þ d1 s ^l þ e1 if u < w wg s ~l ; Jg
ð20Þ
J i €hg ¼ b2 u þ d2 s ^l þ e2 if u > w wg s ~l ; Jg J iwg
C 1 þ wJ
J iwg
C 2 þ wJ
sl sl ~l ¼ sl s ^l and b1 ¼ J i 1þC J ; d1 ¼ J i þCg J ; e1 ¼ J i C 1þC where s ; b2 ¼ Jw iwg1þC 2 Jg , and d2 ¼ Jw iwg þCg2 Jg ; e2 ¼ Jw iwgC 2þC . If the upper and 1 g 1 g 1 Jg 2 Jg w wg w wg w wg ~
~
~l are known, one can derive the upper bounds and lower bounds for bðÞ ; dðÞ ; eðÞ . Denote the lower bounds for C 1 ; C 2 and s ^2 and b nominal estimates for these parameters as b b1; b d1; b e1 ; b b2; d e 2 , the gain margins for b1 ; b2 as b1 ; b2 2, and b b jd1 d 1 j 6 D1 ; jd2 d 2 j 6 D2 ; je1 b e 1 j 6 E1 ; je2 b e 2 j 6 E2 . Then the problem is transformed into designing a sliding controller
^l þ eðÞ . The development in [9] sugthat offers robustness against the uncertain control gains bðÞ and uncertain disturbances dðÞ s gests that the sliding control law for h_ g > 0 can be chosen as
h s i J iwg b 1 k1 sat ~l ; if u < w s u ¼ u1 ¼ b b 1 u 1 U Jg h s i J iwg b 2 k2 sat ~l u if u > w s b 1 u ¼ u2 ¼ b 2 U Jg
ð21Þ
_ _ b1 ¼ b b2 ¼ b ^l j þ E1 þ gÞ þ ðb1 1Þj u b 1 j; k2 P ^l ^ ^l ^ hg ; u h g and k1 P b1 ðD1 js with u d1 s e1 þ € hd 2k~ hg k2 e d2 s e2 þ € hd 2k~ hg k2 e ^l j þ E2 þ gÞ þ ðb2 1Þj u b 2 j. Notice that in the control Eq. (21), satðÞ is the saturation function, s is a sliding variable satb2 ðD2 js R _ h g with ~hg ¼ hg hd and k being a strictly positive constant, U is the boundary layer thickness, and g hg þ 2ke h g þ k2 e isfying3 s ¼ ~ is another positive constant dictating how fast the state trajectory reaches the sliding surface. It is worth mentioning that k is equivalent to the bandwidth of the closed-loop system. ~l are uncerThe control law in (21) has potential issues for practical implementation. First of all, because sl , consequently s ~l is known, there might exist a tain, whether u1 or u2 should be used as control cannot be determined. Moreover, even if s 2 3
b
The gain margin bðÞ satisfies b1 6 ^ðÞ 6 bðÞ . ðÞ bðÞ Here an integral term is included in the definition of s to reject possible steady-state errors.
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conflicting scenario that u1 >
Jw iwg Jg
s~l and u2 < JwJigwg s~l . In this case, the control law becomes ill-defined and no control action
can be produced. To circumvent these two problems, the following lemma provides a sufficient condition for selecting appropriate controller parameters so that at least outside the boundary layer (jsj P U), the control law is well-defined and can be ~l . explicitly determined without the exact knowledge of s Lemma 1. Assume that j
J w iwg Jg
^
^
s~l j 6 M. If the gain margins b1 ; b2 P 2 and g P maxð0; M2b1 D1 js^l j E1 ; M2b2 D2 js^l j E2 Þ, then for
jsj P U, the control equation in (21) can be written as
b u ¼ u1 ¼ b b 1 1 ½ u 1 k1 if s P U;ð22Þ 1 b b u ¼ u2 ¼ b ½ u 2 þ k2 if s 6 U 2
moreover, the control equation can track the desired trajectory hd ðtÞ to within a guaranteed precision. Proof. First, in first equation of (21), if s P U,
^1 ½ u ^1 ½ u ^1 ½2ðD1 js ^l j þ E1 þ gÞ ðb1 1Þj u ^l j þ E1 þ gÞ j u ^l j þ E1 þ gÞ b 1 b1 ðD1 js b 1 j 6 b b 1 2ðD1 js b 1 j 6 b u1 6 b 1 1 1 6 M
ð23Þ J w iwg Jg
using the lower-bounds for b1 and g. By the definition of M, the last inequality in (23) indicates that u1 < s~l , so according to (21) u ¼ u1 for s P U. Using a similar procedure, one can also prove that u ¼ u2 for s 6 U. The control action is therefore well-defined outside the boundary layer, which implies that the state trajectory will enter the boundary layer, or jsj < U in a hg is finite time. Once jsj < U, the bounds on jsj can be directly translated into bounds on the tracking error. Specifically, ~ bounded by a guaranteed precision of 2kU. h The lemma provides a nonambiguous way of computing the control action outside the boundary layer. However, it should be noted that when jsj < U, the control action still can not be appropriately defined. To maintain the control continuity, the control action inside the boundary layer can be obtained by interpolating between the two control signals respectively for s 6 U and s P U. Specifically,
u¼
u1 u2 u1 þ u2 sþ if jsj < U: 2U 2
ð24Þ
It should be noted that the control laws in (21) and (24) are applied only for h_ g > 0. For h_ g < 0, similar sliding control and interpolation scheme can be derived using (19). 4. Numerical and experimental validations The hardware setup for numerical and experimental validations is depicted schematically in Fig. 6. In the setup, the worm is driven by a DC motor. A metal block with mass m and moment arm d is attached to the gear axis to simulate a loading torque sl ¼ mgd cos hg . There is also an encoder to measure the rotation angle of the gear. The parameters of the experimental system are given in Table 1. First of all, the static behavior of the system is examined. The metal block is set initially at hg ¼ 0. At this position, the gear is self-locked with sm ¼ 0. Then sm is increased slowly in positive (negative) direction, and the break-in motor torque which just barely puts the gear in motion is recorded. Since the gear can be self-locked and the motor can drive the worm, the characteristics of the system fall into the lw 6 ls 6 lg category of the static analysis. At the instant when the gear is about to rotate, sm < Hsl where H ¼ 3:4957 is computed by the parameters of the worm-gear given in Table 1. By (10) the positive and negative break-in torques are respectively C 1 sl and C 2 sl . The same experiment is repeated for different d (7 and 14 cm), and the corresponding loading torque and motor break-in torque are plotted in Fig. 7. From the experimental data, the parameters C 1 and C 2 are respectively estimated as 0:0471 and 0:0837. Upon obtaining the numerical values for C 1 ; C 2 ,
Fig. 6. Schematics of the experiment set up.
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Table 1 Kinematic parameters of the system. jw ¼ 0:0000253 kg m2
/n ¼ 20
Jg ¼ 0:000739 kg m2 ‘w ¼ 11 mm ‘g ¼ 45 mm iwg ¼ 60
kl ¼ 4 lw ¼ 0:0657 lg ¼ 13:4382 m ¼ 1:0 kg
2 X : experim ental data
1.8 1.6
τ =-C τ
l
loading torque ( τ )
1.4 m
1.2
2 l
1 τ =-C τ
0.8
m
1 l
0.6 0.4 0.2 0 -0.1
-0.05
0 0.05 input torque (τ )
0.1
0.15
m
Fig. 7. Experimental results for the break-in torque.
5 4
Experiment Simulation
3
d=7cm
gear angle (rad)
2 1 d=14cm d=14cm
0 -1 -2 -3
d=7cm
-4 -5 -1.5
-1
-0.5 0 0.5 input torque (N-m)
1
1.5
Fig. 8. Experimental and simulation results for the open-loop operation.
the friction coefficient ls ¼ 0:25 is estimated using the parameters of the worm-gear system and the expressions for the C 1 ; C 2 given in the static analysis. Notice that this estimated ls actually falls between lw and lg . To further validate the estimations, Fig. 8 shows the simulated and experimental sm hg plots when sm is varied linearly with time. It is obvious that the simulated model can reproduce the influence of static friction and the initial acceleration of the worm-gear system. Next the dynamic behavior is examined. According to (18) and (19) when either the gear speed h_ g changes its sign or sm J i goes above or below wJgwg sl , the worm-gear system experiences an engagement switching. In the experiment, a sinusoidal torque sm ¼ 0:05 cosð3pt þ p2 ÞN m is applied to the motor, so the gear oscillates sinusoidally which also leads to a sinuJ i soidal loading torque. In Fig. 9a, both sm and the torque wJgwg sl are plotted. From this figure, one can readily identify the time
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T.-J. Yeh, F.-K. Wu / Simulation Modelling Practice and Theory 17 (2009) 767–777
0.1
0.05
0
-0.05
-0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.6
1.8
2
2
1
0
engagement switching
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fig. 9. Experimental and simulation results for switching engagement.
J i
instants when sm equals wJ wg sl . In Fig. 9b where the experimental gear speed is plotted, it can be verified that the experimeng tal h_ g indeed experiences slope change (which corresponds to the instantaneous change of acceleration) at these time instants as well as the instants when h_ g passes through zero. In Fig. 9b the response of gear speed simulated by the dynamic model of (18) and (19) is also plotted. The simulated response does not exhibit as drastic dynamics change at the switching instants as the experimental response. This is possibly attributed to the following reasons. One is that practically the coefficient of friction is not a constant value but varies with the gear speed [8]. Another reason is that, although the mating between the worm and the gear is carefully adjusted that the backlash is virtually eliminated, impact may still occur at engagement switching instants which subsequently causes more dramatic dynamic change. It should be noted that the dynamic model presented is not intended to accurately describe the system behavior but is to be used as a nominal model for the robust control design. The sliding controller is designed based on the nominal model in (18) and (19) and is intended to provide robustness against 13% and 8% variations in C 1 ; C 2 (which corresponds to 10% variations in ls ) and 10% variation in sl (whose nominal value is 0:0196 cos hg N m). From the nominal values for C 1 ; C 2 ; sl , and the assumed variations, the nominal estimates for bðÞ ; dðÞ ; eðÞ as well as the maximum errors of estimation can be computed. Particularly, it is found that the gain margins b1 ; b2 should respectively be lower bounded by 1:12 and 1:04. In order to meet the lower bounds and satisfy the gain-margin conditions in Lemma 1, both b1 and b2 are chosen to , U ¼ 0:3, and g ¼ 30. With these control parameters the sufficient condition in be 2. Moreover, we also choose k ¼ 30p rad s Lemma 1 is satisfied, so the control system can track the desired trajectory within the precision of 2kU ¼ 0:00628 rad. For the purpose of comparison, a PID controller whose control parameters are obtained from applying LQR design with integral control on the nominal model is also implemented. It should be noted that because the nominal model in (18) and (19) does not account for the system behavior at h_ g ¼ 0, the static analysis motivates us to add both the sliding controller and the PID controller the following control effort
sm ¼ C 1 s^l if h_ d P 0 and h_ g ¼ 0; sm ¼ C 2 s^l if h_ d < 0 and h_ g ¼ 0
ð25Þ
to compensate for the break-in torque at zero velocity. For the parameters C 1 and C 2 in (25), the numerical values identified from Fig. 8 are adopted. Moreover, in order to make the compensation less sensitive to the noise in the measurement of h_ g 4, in the experiments the activating condition is changed from h_ g ¼ 0 to jh_ g j 6 e, where e is a small number. Fig. 10 shows the experimental output responses of the two controllers for hd ¼ 0:14 sin 4pt rad. In the experiments, the actual loading torque is 0:0176 sin hg N m, which deviates from the nominal one by 10%. According to the tracking error
4
The measured gear speed is obtained by first taking the backward difference on the encoder output and then performing low-pass filtering.
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x 10
-3
(a) reference
rad
2 0 -2 7.5
rad
1
x 10
8
8.5
-4
9
9.5
10
(b) tracking error sliding LQR
0 -1 7.5
8
8.5 9 (c) control effort of the LQR controller
9.5
10
8
8.5 9 (d) control effort of the proposed sliding controller
9.5
10
9.5
10
N-m
0.02 0 -0.02 7.5
N-m
0.02 0 -0.02 7.5
8
8.5
9 time (sec)
Fig. 10. Experimental results of proposed sliding and LQR controllers in pendulum system.
plot in Fig. 10b, the sliding controller can track the reference trajectory within the precision of 8:7 104 rad peak-to-peak. Such a tracking error is about 30% of that achieved by the PID controller. Observing that the gear speed h_ g changes its sign J i periodically (from the tracking responses in Fig. 10a and b), and sm also frequently goes above or below wJgwg sl (from the control effort responses in Fig. 10c and d) during the control history, so the worm-gear system experiences frequent engagement switchings. The superior performance of the sliding controller indicates that it indeed can better cope with uncertainties in the loading torque, system parameters, as well as the dynamics change associated with engagement switchings. Regardless that the sliding controller shows better tracking performance than the PID controller, from Fig. 10c and d the control efforts of the two controllers are about the same. 5. Conclusions In this paper, a model and a robust control scheme for worm-gear driven systems are presented. The model accounts for not only the static but also the dynamic characteristics of worm-gears. Since the speed-dependent nature of the coefficient of friction induces uncertain system parameters in the model and the loading torque can be uncertain, to achieve consistent performance under different operating conditions, a robust sliding controller is designed. With the help of a lemma concerning the well-definedness of the control law, appropriate control parameters can be selected to make the sliding control effort well-defined and continuous in the entire space. Experiments on a hardware setup indicate that the derived models can reasonably explain the static and dynamic behaviors of the worm-gear system, and the proposed control system exhibits superior performance to a conventional PID controller. Acknowledgements The authors gratefully acknowledge the support provided by Farley Tech. Co. and National Science Council in Taiwan. References [1] M.E. Dohring, E. Lee, W.S. Newman, A load-dependent transmission friction model: theory and experiments, Proceedings of IEEE International Conference on Robotics and Automation 3 (1993) 430–436. [2] H. Kawasaki, et al., Novel Climbing Method of Pruning Robot, SICE Annual Conference, August, 2008, Japan, pp. 160–163. [3] D.C. May, S. Jayasuriya, Velocity control of a manipulator joint driven through a worm gear transmission, in: Proceedings of the 1990 American Control Conference, San Diego, CA, May 1990, pp. 1272–1278. [4] D.C. May, S. Jayasuriya, A QFT controller design implementation for a worm gear driven manipulator Joint, in: Proceedings of the 1994 American Control Conference, Baltimore, MD, June 1994, pp. 595–595. [5] D.C. May, S. Jayasuriya, B.W. Mooring, Modeling and control of a manipulator joint driven through a worm gear transmission, Journal of Vibration and Control 6 (2000) 85–111.
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[6] B.W. Mooring, D.C. May, M.S. Sutle, Modeling and analysis of a manipulator joint driven through a worm gear transmission, in: Proceedings of ASME Conference on Advances in Design Automation, Sep. Canada, 1989, pp. 67–73. [7] R.M. Phelan, Fundamentals of Mechanical Design, McGraw Hill, New York, 1970. [8] J.E. Shigley, C.R. Mischke, Mechanical Engineering Design, McGraw-Hill, 1989. [9] J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, NJ, 1991. [10] C. Ünsal, P.K. Khosla, Mechatronic design of a modular self -reconfiguring robotic system, in: Proceedings of IEEE International Conference on Robotics & Automation, San Francisco, USA, April 2000, pp. 1742–1747. [11] M.-S. Wang, Y.-M. Tu, Design and implementation of a stair-climbing robot, in: Proceedings of IEEE International Conference on Advanced Robotics and its Social Impacts, Taipei, Taiwan, August, 2008.
Contents lists available at ScienceDirect
Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Modeling and robust control of worm-gear driven systems T.-J. Yeh *, Feng-Kung Wu Department of Power Mechanical Engineering, National Tsing Hua University, No. 101, Sec. 2 Kuang-fu Rd., Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 13 August 2008 Received in revised form 30 October 2008 Accepted 2 January 2009 Available online 21 January 2009
Keywords: Modeling Worm-gear driven systems Nonlinear systems Uncertainty Robust control Sliding control
a b s t r a c t This paper investigates modeling and control issues associated with worm-gear driven systems. In the modeling part, static and dynamic analyses are conducted to investigate the characteristics of the worm-gear. The static analysis reveals not only the non-backdrivability but also the dependency of break-in torques on the loading torque, direction of motion as well as crucial system parameters. The dynamic analysis generates four linear equations of motion, which, at any particular instant, only one applies. The applicable equation at any given instant depends on the direction of motion and the relative magnitude between the input torque and the loading torque. In the control part, a sliding controller is designed based on the modeling results. The controller can provide robustness against the uncertain loading torque and variations of system parameters caused by the speed-dependent nature of the coefficient of friction. Because the dependency of the dynamic equation on the operating condition may render the controller ill-defined in some scenarios, a lemma is proved and can be used to select proper control parameters to guarantee the well-definedness of the controller. The proposed control scheme is applied to a worm-gear driven positioning platform. Experimental results indicate the proposed control system exhibits more than 70% improvement in tracking error over PID control. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Worm-gears are frequently used in electro-mechanical systems where high transmission ratios are required. Compared to the other transmission elements such as spur or bevel gears, in which a large number of gears and consequently a large space is required to achieve the desired gear ratio, the worm-gear provides a large transmission ratio in a small space with a single gear set. Therefore, worm-gears allow one to adopt small, high velocity motors in systems involving slow motion without the need to redirect the torque through a complex gear train. In addition to the compactness, that worm-gears can provide the orthogonal redirectioning of the torque is also desirable in many designs. Finally, another potential advantage associated with worm-gears is the unique property of non-backdrivability (or self-locking) [7]. This property allows the gearset to be driven on the input side, but not back driven on the side of the load. Consequently, the servos do not have to be actuated all the time to hold the loads in a specific position. These attractive features make worm-gears desirable in robotic applications in which special demands on the transmission element are needed. The examples include stair-climbing robots[11], pruning robots[2], modular self-reconfiguring robots[10], and spherical robotic manipulators[5]. In conventional gears, the motion of one tooth relative to the mating tooth is mainly a rolling motion. However, the relative motion between worm and gear teeth is pure sliding, and so it should be expected that friction plays a dominant role in the performance of worm-gear driven systems [8]. Friction induces nonlinearities to the system dynamics and are usually the major source of motion disturbances and instability in feedback control. For instance, it is pointed out in [5] that a * Corresponding author. Tel.: +886 3 574 2922; fax: +886 3 572 2840. E-mail address: [email protected] (T.-J. Yeh). 1569-190X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2009.01.002
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vibratory behavior has been observed to be associated with worm gears in positioning systems under gravity loading in the case of spherical robot. Regardless that the frictional effect can diminish the usefulness of worm-gears in servo-control applications, the studies in the literature on the impact of friction on dynamics and control of worm-gear driven systems are limited. [1] developed a load-dependent transmission friction model for the potential use of constructing feed-forward compensation for worm-gear driven servo systems. The model presented therein is based on a wedge-like planar mechanism. Ref. [6] showed that the dynamics of a cantilevered manipulator driven through a worm-gear are switched between two sets of nonlinear differential equations. These two sets of nonlinear equations are linearized around a nominal trajectory and are then used as the plant model for designing a classical lead–lag controller [3] and a QFT controller [4]. The approaches adopted in the aforementioned references contain certain limitations which will prevent one from further improving the system performance. For instance, the model in [1] contains implicit variables in the dynamic equations and thus cannot be directly used for controller design. The classical and the QFT controllers respectively proposed in [3] and [4] are linear controllers. They may not have sufficient robustness to deal with uncertainties in the system. In this paper, we develop a worm-gear model which can be used directly for designing a nonlinear controller. The nonlinear controller is sliding-mode-based and can provide robustness against the uncertain loading torque and variations of system parameters caused by the speed-dependent nature of the coefficient of friction. The paper is organized as follows: To begin with, the static and the dynamic behaviors of the worm-gear are examined. The governing equations resulted from the analyses are then used in Section 3 to design a sliding controller. A lemma concerning the well-definedness of the controller is also given therein. To illustrate the usefulness of the model and the controller, Section 4 presents simulations and experimental results of a worm-gear driven system. Finally, conclusions are given in Section 5. 2. Modeling of the worm-gear In the worm-gear considered, a motor torque sm drives the worm and a loading torque sl exerts on the gear. During the power transmission, the worm-gear engagement occurs either at the left surface (as shown in Fig. 1a) or the right surface ( as shown in Fig. 2a) of the gear tooth. In the subsequent analysis, these two types of engagement are respectively referred to as left engagement and right engagement. In the case of left engagement, the force exerted by the gear on the worm consists of a normal force W n , and a frictional force W f . As shown in Fig. 1b, while W f is tangent to the pitch helix on the pitch cylinder of the worm, W n is normal to the pitch helix with a pressure angle /n . Notice that in this figure, the gear axis is parallel to the x axis, the worm axis is parallel to the y axis, and a right-handed coordinate system is adopted. According to the coordinate
Gear
θg τl
z
τm
θw
y x
Worm
(a) Schematics of left engagement
z
Wn
y
φn λl
x
Wf λl Pitch helix
τm Worm
(b) Force diagram of left engagement Fig. 1. Left engagement for the worm-gear.
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769
Gear
θg ττl out
τm
θw
z
yy x
Worm Worm
(a) Schematics of right engagement
Wn
λ τm
φn
Wf
λ
pitch helix
z Worm
y x
(b) Force diagram of right engagement Fig. 2. Right engagement for the worm-gear.
definition, the lead angle kl of worm is equal to the angle between W f and the x axis. The total force acts on the worm can be decomposed into three orthogonal force components F x , F y , and F z :
F x ¼ W n cos /n sin kl W f cos kl ; F y ¼ W n cos /n cos kl W f sin kl ;
ð1Þ
F z ¼ W n sin /n : Only the force component F x can generate a reaction torque on the worm and the worm dynamics is given by
J w €hw ¼ sm þ F x ‘w ¼ sm þ ðW n cos /n sin kl W f cos kl Þ‘w
ð2Þ
in which hw ; J w ; ‘w are respectively the rotation angle, the inertia, and the pitch radius of the worm. For the gear dynamics, considering that F y has its reactive counterpart (F y ) generating a driving torque on the gear, we have
J g €hg ¼ sl þ F y ‘g ¼ sl ðW n cos /n cos kl þ W f sin kl Þ‘g
ð3Þ
in which hg ; J g ; ‘g are respectively the rotation angle, the inertia, and the pitch radius of the gear. In the case of right engagement1, one can perform similar force analysis using Fig. 2. The resultant dynamics are
J w €hw ¼ sm þ ðW n cos /n sin kl W f cos kl Þ‘w
ð4Þ
J g €hg ¼ sl þ ðW n cos /n cos kl W f sin kl Þ‘g :
ð5Þ
and
(1) Static analysis: If h_ g ¼ € hg ¼ h_ w ¼ € hw ¼ 0 for the left engagement case, then
sm þ ðW n cos /n sin kl W f cos kl Þlw ¼ 0; sl ðW n cos /n cos kl þ W f sin kl Þlg ¼ 0:
ð6Þ ð7Þ
The magnitude W f is not large enough to break the maximum frictional force ls W n , where ls is the coefficient of static friction. Therefore ls W n 6 W f 6 ls W n , or using (6) and (7), the inequalities can be rewritten as:
ls 6 l ¼
1
Wf cos /n ðsm lg cos kl þ sl lw sin kl Þ 6 ls : ¼ sm lg sin kl sl lw cos kl Wn
ð8Þ
The conditions on when left engagement or right engagement should apply depend on the direction of rotation and the relative magnitude between sm and
sl . This issue will be investigated in detail in dynamic analysis given subsequently.
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Next we investigate how sm and sl can be related so that the inequalities in (8) are satisfied and worm-gear system stays motionless. C2 ¼
To
facilitate
lw ðcos /n sin kl þls cos kl Þ ;H lg ðcos /n cos kl ls sin kl Þ
the
discussion,
we
define
¼ llwg cot kl ; lg ¼ cos /n cot kl , and
the
following
parameters:
ðcos /n sin kl ls cos kl Þ ; C 1 ¼ llwg ðcos /n cos k þl sin k Þ l
s
l
lw ¼ cos /n tan kl . It is assumed that 0 6 kl 6 p4 so H > 0 and
lg P lw . The investigation is divided into three cases:
lw 6 ls 6 lg : In this case, C 1 6 0 and C 2 P 0. After algebraic manipulations, it can be found that (8) is satisfied if sm and sl are constrained either by
sm > Hsl and C 1 sl 6 sm 6 C 2 sl
ð9Þ
sm < Hsl and C 2 sl 6 sm 6 C 1 sl :
ð10Þ
or
The constraints indicate that to put the worm-gear system in motion, in case of sm < Hsl ðsm > Hsl Þ, the break-in torque sm is C 1 sl ðC 2 sl Þ if positive rotation is desired and is C 2 sl ðC 1 sl Þ if negative rotation is desired. Moreover, to better visualize the above two constraints, a sm sl plane is plotted in Fig. 3 and (9) and (10) appear as the two shaded regions in the figure. We can see that if the drive torque sm ¼ 0, no matter how large the loading torque s‘ on gear is, the worm-gear set remains motionless. This is referred to as the non-backdrivability [7] or the self-locking of the gear. lg < ls : In this case, C 1 < 0 and C 2 < 0. Again by algebraic manipulations, one can find that (8) is satisfied if sm and sl are constrained in either of the shaded regions in Fig. 4. According to this figure, if sl ¼ 0, no matter how large sm is, the wormgear set remains motionless. This is referred to as the self-locking of the worm. Such a design does not happen in practice because no input torque can overcome the static friction to put the worm-gear set to motion. ls < lw : In this case, C 1 > 0 and C 2 > 0. Eq. (8) is satisfied if sm and sl are constrained in either of the shaded regions in Fig. 5. Neither the worm nor the gear is self-locked in this case. If the investigation is conducted in the right engagement case (using (4) and (5)), same conclusions can be reached. It should be noted that in practical applications, making the gear non-backdrivable is usually desirable because it allows the worm-gear to statically sustain any loading torque without motor efforts. In the experimental system investigated subsequently, ls falls between lw and lg , so in the following only the case corresponding to the non-backdrivability in the gear will be discussed. (2) Dynamic analysis: First we consider the case of left engagement. Assume that the worm-gear has a transmission ratio of iwg . Thus the worm angle hw and the gear angle hg are related by hw ¼ iwg hg . Moreover, assuming that the coefficient of kinetic friction is equal to its static counterpart, the friction force W f equals to ls sgnðh_ g ÞW n . Using these two relationships, (2) and (3) can be rewritten as
J w iwg €hg ¼ sm þ W n ðcos /n sin kl ls sgnðh_ g Þ cos kl Þlw ; J h€g ¼ sl W n ðcos / cos kl þ l sgnðh_ g Þ sin kl Þlg g
n
ð11Þ
s
The above two equations can be combined to solve for the normal force W n . In the case of h_ g > 0, we have
Wn ¼
J w iwg sl J g sm : ðJ w iwg þ C 1 J g Þðcos /n cos kl þ ls sin kl Þlg
ð12Þ
On the other hand, in the right engagement case we have
τl
τ m = −C1τ l τ m = Hτ l τm
τ m = −C2τ l
(μw ≤ μs ≤ μ g ) Fig. 3. Relation between
sm and sl for the system to remain static when lw 6 ls 6 lg .
T.-J. Yeh, F.-K. Wu / Simulation Modelling Practice and Theory 17 (2009) 767–777
τl
771
τ m = −C1τ l τ m = Hτ l τm
τ m = −C2τ l
(μ g < μ s ) Fig. 4. Relation between
sm and sl for the system to remain static when lg < ls .
τ m = −C1τ l
τl
τ m = Hτ l τm τ m = −C2τ l
(μ s < μ w ) Fig. 5. Relation between
sm and sl for the system to remain static when ls < lw .
J w iwg €hg ¼ sm þ W n ð cos /n sin kl ls sgnðh_ g Þ cos kl Þlw ; J h€g ¼ sl þ W n ðcos / cos kl l sgnðh_ g Þ sin kl Þlg n
g
ð13Þ
s
so for h_ g > 0
Wn ¼
J g sm J w iwg sl : ðJ w iwg þ C 2 J g Þðcos /n cos kl ls sin kl Þlg
ð14Þ
Since W n is an internal force, it has to be eliminated from (11) and (13) in order to derive the explicit dynamic equation. Before doing so, we investigate a particular scenario to understand the limitation in (11) and (13). The limitation comes from the fact that when deriving these two sets of dynamic equations, it is assumed that backlash does not exist so that the worm can freely engage with the gear either from the right or the left and still obey the kinematic constraint hw ¼ iwg hg . This will lead to some illed-situations. Consider the case that sl ¼ 0 and sm > 0, if the left engagement occurs, then (11) determines the dynamics. Since the normal force in (12) has to be positive, we must have J w iwg þ C 1 J g < 0. If we eliminate W n from (11), then the equation
ðJ w iwg þ C 1 J g Þ€hg ¼ sm
ð15Þ
is achieved. This indicates that the resultant acceleration is in the opposite direction of the input torque which is completely absurd. Contrarily, if the right engagement occurs, then (13) determines the dynamics. In this case, the associated normal force in (14) is positive because J w iwg þ C 2 J g > 0 and cos /n cos kl ls sin kl > 0 from the assumption of lg P ls P lw . Again eliminating W n from (13) we have
ðJ w iwg þ C 2 J g Þ€hg ¼ sm
ð16Þ
which makes physical sense due to the fact that € hg has the same sign as sm . From the analysis, when h_ g > 0; sl ¼ 0 and sm > 0, only the right engagement occurs.
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In order to avoid the erroneous engagement, we add an assumption that J w iwg þ C 1 J g > 0, or
C1 >
J w iwg Jg
ð17Þ
so that W n automatically becomes negative in the illed-situation. Indeed, with (17), positiveness of W n alone determines whether left engagement (11) or right engagement (13) should be used to derive the explicit dynamic equation and the dynamic equation is given by
ðJ w iwg þ C 1 J g Þ€hg ¼ sm þ C 1 sl if
sm 6
Jw iwg sl ðleft engagementÞ; Jg
ðJ w iwg þ C 2 J g Þ€hg ¼ sm þ C 2 sl if
sm >
J w iwg sl ðright engagementÞ Jg
ðJ w iwg þ C 2 J g Þ€hg ¼ sm þ C 2 sl if
sm 6
Jw iwg sl ðleft engagementÞ; Jg
ðJ w iwg þ C 1 J g Þ€hg ¼ sm þ C 1 sl if
sm >
J w iwg sl ðright engagementÞ Jg
ð18Þ
for h_ g > 0, and is given by
ð19Þ
for h_ g < 0. Therefore, the assumption in (17) actually eliminates the ambiguity in the worm-gear engagement. 3. Controller design hg ¼ sm þ C sl in which the parameter C, According to (18) and (19), the dynamic equation is in the form of ðJ w iwg þ CJg Þ€ depending on the sign of the velocity and the relative magnitudes between sm and
Jw iwg Jg
sl , equals either C 1 or C 2 . The param-
eters C 1 and C 2 are functions of the coefficient of friction. In practice, the coefficient of friction varies with the speed [8], so C 1 or C 2 can be viewed as uncertain parameters in the dynamic equation. Moreover, the loading torque encountered can also be uncertain. Assuming that the control task is to make hg track a desired trajectory hd ðtÞ, in order to achieve high tracking performance, the control system needs to provide robustness again such uncertainties. J i ^l being the esti^l with s For the purpose of robust controller design, we first introduce a new input u, where u ¼ sm wJ wg s g _ mated loading torque. For hg > 0; in terms of u, the dynamic equation of (18) becomes the form
J i €hg ¼ b1 u þ d1 s ^l þ e1 if u < w wg s ~l ; Jg
ð20Þ
J i €hg ¼ b2 u þ d2 s ^l þ e2 if u > w wg s ~l ; Jg J iwg
C 1 þ wJ
J iwg
C 2 þ wJ
sl sl ~l ¼ sl s ^l and b1 ¼ J i 1þC J ; d1 ¼ J i þCg J ; e1 ¼ J i C 1þC where s ; b2 ¼ Jw iwg1þC 2 Jg , and d2 ¼ Jw iwg þCg2 Jg ; e2 ¼ Jw iwgC 2þC . If the upper and 1 g 1 g 1 Jg 2 Jg w wg w wg w wg ~
~
~l are known, one can derive the upper bounds and lower bounds for bðÞ ; dðÞ ; eðÞ . Denote the lower bounds for C 1 ; C 2 and s ^2 and b nominal estimates for these parameters as b b1; b d1; b e1 ; b b2; d e 2 , the gain margins for b1 ; b2 as b1 ; b2 2, and b b jd1 d 1 j 6 D1 ; jd2 d 2 j 6 D2 ; je1 b e 1 j 6 E1 ; je2 b e 2 j 6 E2 . Then the problem is transformed into designing a sliding controller
^l þ eðÞ . The development in [9] sugthat offers robustness against the uncertain control gains bðÞ and uncertain disturbances dðÞ s gests that the sliding control law for h_ g > 0 can be chosen as
h s i J iwg b 1 k1 sat ~l ; if u < w s u ¼ u1 ¼ b b 1 u 1 U Jg h s i J iwg b 2 k2 sat ~l u if u > w s b 1 u ¼ u2 ¼ b 2 U Jg
ð21Þ
_ _ b1 ¼ b b2 ¼ b ^l j þ E1 þ gÞ þ ðb1 1Þj u b 1 j; k2 P ^l ^ ^l ^ hg ; u h g and k1 P b1 ðD1 js with u d1 s e1 þ € hd 2k~ hg k2 e d2 s e2 þ € hd 2k~ hg k2 e ^l j þ E2 þ gÞ þ ðb2 1Þj u b 2 j. Notice that in the control Eq. (21), satðÞ is the saturation function, s is a sliding variable satb2 ðD2 js R _ h g with ~hg ¼ hg hd and k being a strictly positive constant, U is the boundary layer thickness, and g hg þ 2ke h g þ k2 e isfying3 s ¼ ~ is another positive constant dictating how fast the state trajectory reaches the sliding surface. It is worth mentioning that k is equivalent to the bandwidth of the closed-loop system. ~l are uncerThe control law in (21) has potential issues for practical implementation. First of all, because sl , consequently s ~l is known, there might exist a tain, whether u1 or u2 should be used as control cannot be determined. Moreover, even if s 2 3
b
The gain margin bðÞ satisfies b1 6 ^ðÞ 6 bðÞ . ðÞ bðÞ Here an integral term is included in the definition of s to reject possible steady-state errors.
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T.-J. Yeh, F.-K. Wu / Simulation Modelling Practice and Theory 17 (2009) 767–777
conflicting scenario that u1 >
Jw iwg Jg
s~l and u2 < JwJigwg s~l . In this case, the control law becomes ill-defined and no control action
can be produced. To circumvent these two problems, the following lemma provides a sufficient condition for selecting appropriate controller parameters so that at least outside the boundary layer (jsj P U), the control law is well-defined and can be ~l . explicitly determined without the exact knowledge of s Lemma 1. Assume that j
J w iwg Jg
^
^
s~l j 6 M. If the gain margins b1 ; b2 P 2 and g P maxð0; M2b1 D1 js^l j E1 ; M2b2 D2 js^l j E2 Þ, then for
jsj P U, the control equation in (21) can be written as
b u ¼ u1 ¼ b b 1 1 ½ u 1 k1 if s P U;ð22Þ 1 b b u ¼ u2 ¼ b ½ u 2 þ k2 if s 6 U 2
moreover, the control equation can track the desired trajectory hd ðtÞ to within a guaranteed precision. Proof. First, in first equation of (21), if s P U,
^1 ½ u ^1 ½ u ^1 ½2ðD1 js ^l j þ E1 þ gÞ ðb1 1Þj u ^l j þ E1 þ gÞ j u ^l j þ E1 þ gÞ b 1 b1 ðD1 js b 1 j 6 b b 1 2ðD1 js b 1 j 6 b u1 6 b 1 1 1 6 M
ð23Þ J w iwg Jg
using the lower-bounds for b1 and g. By the definition of M, the last inequality in (23) indicates that u1 < s~l , so according to (21) u ¼ u1 for s P U. Using a similar procedure, one can also prove that u ¼ u2 for s 6 U. The control action is therefore well-defined outside the boundary layer, which implies that the state trajectory will enter the boundary layer, or jsj < U in a hg is finite time. Once jsj < U, the bounds on jsj can be directly translated into bounds on the tracking error. Specifically, ~ bounded by a guaranteed precision of 2kU. h The lemma provides a nonambiguous way of computing the control action outside the boundary layer. However, it should be noted that when jsj < U, the control action still can not be appropriately defined. To maintain the control continuity, the control action inside the boundary layer can be obtained by interpolating between the two control signals respectively for s 6 U and s P U. Specifically,
u¼
u1 u2 u1 þ u2 sþ if jsj < U: 2U 2
ð24Þ
It should be noted that the control laws in (21) and (24) are applied only for h_ g > 0. For h_ g < 0, similar sliding control and interpolation scheme can be derived using (19). 4. Numerical and experimental validations The hardware setup for numerical and experimental validations is depicted schematically in Fig. 6. In the setup, the worm is driven by a DC motor. A metal block with mass m and moment arm d is attached to the gear axis to simulate a loading torque sl ¼ mgd cos hg . There is also an encoder to measure the rotation angle of the gear. The parameters of the experimental system are given in Table 1. First of all, the static behavior of the system is examined. The metal block is set initially at hg ¼ 0. At this position, the gear is self-locked with sm ¼ 0. Then sm is increased slowly in positive (negative) direction, and the break-in motor torque which just barely puts the gear in motion is recorded. Since the gear can be self-locked and the motor can drive the worm, the characteristics of the system fall into the lw 6 ls 6 lg category of the static analysis. At the instant when the gear is about to rotate, sm < Hsl where H ¼ 3:4957 is computed by the parameters of the worm-gear given in Table 1. By (10) the positive and negative break-in torques are respectively C 1 sl and C 2 sl . The same experiment is repeated for different d (7 and 14 cm), and the corresponding loading torque and motor break-in torque are plotted in Fig. 7. From the experimental data, the parameters C 1 and C 2 are respectively estimated as 0:0471 and 0:0837. Upon obtaining the numerical values for C 1 ; C 2 ,
Fig. 6. Schematics of the experiment set up.
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Table 1 Kinematic parameters of the system. jw ¼ 0:0000253 kg m2
/n ¼ 20
Jg ¼ 0:000739 kg m2 ‘w ¼ 11 mm ‘g ¼ 45 mm iwg ¼ 60
kl ¼ 4 lw ¼ 0:0657 lg ¼ 13:4382 m ¼ 1:0 kg
2 X : experim ental data
1.8 1.6
τ =-C τ
l
loading torque ( τ )
1.4 m
1.2
2 l
1 τ =-C τ
0.8
m
1 l
0.6 0.4 0.2 0 -0.1
-0.05
0 0.05 input torque (τ )
0.1
0.15
m
Fig. 7. Experimental results for the break-in torque.
5 4
Experiment Simulation
3
d=7cm
gear angle (rad)
2 1 d=14cm d=14cm
0 -1 -2 -3
d=7cm
-4 -5 -1.5
-1
-0.5 0 0.5 input torque (N-m)
1
1.5
Fig. 8. Experimental and simulation results for the open-loop operation.
the friction coefficient ls ¼ 0:25 is estimated using the parameters of the worm-gear system and the expressions for the C 1 ; C 2 given in the static analysis. Notice that this estimated ls actually falls between lw and lg . To further validate the estimations, Fig. 8 shows the simulated and experimental sm hg plots when sm is varied linearly with time. It is obvious that the simulated model can reproduce the influence of static friction and the initial acceleration of the worm-gear system. Next the dynamic behavior is examined. According to (18) and (19) when either the gear speed h_ g changes its sign or sm J i goes above or below wJgwg sl , the worm-gear system experiences an engagement switching. In the experiment, a sinusoidal torque sm ¼ 0:05 cosð3pt þ p2 ÞN m is applied to the motor, so the gear oscillates sinusoidally which also leads to a sinuJ i soidal loading torque. In Fig. 9a, both sm and the torque wJgwg sl are plotted. From this figure, one can readily identify the time
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0.1
0.05
0
-0.05
-0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.6
1.8
2
2
1
0
engagement switching
-1
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fig. 9. Experimental and simulation results for switching engagement.
J i
instants when sm equals wJ wg sl . In Fig. 9b where the experimental gear speed is plotted, it can be verified that the experimeng tal h_ g indeed experiences slope change (which corresponds to the instantaneous change of acceleration) at these time instants as well as the instants when h_ g passes through zero. In Fig. 9b the response of gear speed simulated by the dynamic model of (18) and (19) is also plotted. The simulated response does not exhibit as drastic dynamics change at the switching instants as the experimental response. This is possibly attributed to the following reasons. One is that practically the coefficient of friction is not a constant value but varies with the gear speed [8]. Another reason is that, although the mating between the worm and the gear is carefully adjusted that the backlash is virtually eliminated, impact may still occur at engagement switching instants which subsequently causes more dramatic dynamic change. It should be noted that the dynamic model presented is not intended to accurately describe the system behavior but is to be used as a nominal model for the robust control design. The sliding controller is designed based on the nominal model in (18) and (19) and is intended to provide robustness against 13% and 8% variations in C 1 ; C 2 (which corresponds to 10% variations in ls ) and 10% variation in sl (whose nominal value is 0:0196 cos hg N m). From the nominal values for C 1 ; C 2 ; sl , and the assumed variations, the nominal estimates for bðÞ ; dðÞ ; eðÞ as well as the maximum errors of estimation can be computed. Particularly, it is found that the gain margins b1 ; b2 should respectively be lower bounded by 1:12 and 1:04. In order to meet the lower bounds and satisfy the gain-margin conditions in Lemma 1, both b1 and b2 are chosen to , U ¼ 0:3, and g ¼ 30. With these control parameters the sufficient condition in be 2. Moreover, we also choose k ¼ 30p rad s Lemma 1 is satisfied, so the control system can track the desired trajectory within the precision of 2kU ¼ 0:00628 rad. For the purpose of comparison, a PID controller whose control parameters are obtained from applying LQR design with integral control on the nominal model is also implemented. It should be noted that because the nominal model in (18) and (19) does not account for the system behavior at h_ g ¼ 0, the static analysis motivates us to add both the sliding controller and the PID controller the following control effort
sm ¼ C 1 s^l if h_ d P 0 and h_ g ¼ 0; sm ¼ C 2 s^l if h_ d < 0 and h_ g ¼ 0
ð25Þ
to compensate for the break-in torque at zero velocity. For the parameters C 1 and C 2 in (25), the numerical values identified from Fig. 8 are adopted. Moreover, in order to make the compensation less sensitive to the noise in the measurement of h_ g 4, in the experiments the activating condition is changed from h_ g ¼ 0 to jh_ g j 6 e, where e is a small number. Fig. 10 shows the experimental output responses of the two controllers for hd ¼ 0:14 sin 4pt rad. In the experiments, the actual loading torque is 0:0176 sin hg N m, which deviates from the nominal one by 10%. According to the tracking error
4
The measured gear speed is obtained by first taking the backward difference on the encoder output and then performing low-pass filtering.
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T.-J. Yeh, F.-K. Wu / Simulation Modelling Practice and Theory 17 (2009) 767–777
x 10
-3
(a) reference
rad
2 0 -2 7.5
rad
1
x 10
8
8.5
-4
9
9.5
10
(b) tracking error sliding LQR
0 -1 7.5
8
8.5 9 (c) control effort of the LQR controller
9.5
10
8
8.5 9 (d) control effort of the proposed sliding controller
9.5
10
9.5
10
N-m
0.02 0 -0.02 7.5
N-m
0.02 0 -0.02 7.5
8
8.5
9 time (sec)
Fig. 10. Experimental results of proposed sliding and LQR controllers in pendulum system.
plot in Fig. 10b, the sliding controller can track the reference trajectory within the precision of 8:7 104 rad peak-to-peak. Such a tracking error is about 30% of that achieved by the PID controller. Observing that the gear speed h_ g changes its sign J i periodically (from the tracking responses in Fig. 10a and b), and sm also frequently goes above or below wJgwg sl (from the control effort responses in Fig. 10c and d) during the control history, so the worm-gear system experiences frequent engagement switchings. The superior performance of the sliding controller indicates that it indeed can better cope with uncertainties in the loading torque, system parameters, as well as the dynamics change associated with engagement switchings. Regardless that the sliding controller shows better tracking performance than the PID controller, from Fig. 10c and d the control efforts of the two controllers are about the same. 5. Conclusions In this paper, a model and a robust control scheme for worm-gear driven systems are presented. The model accounts for not only the static but also the dynamic characteristics of worm-gears. Since the speed-dependent nature of the coefficient of friction induces uncertain system parameters in the model and the loading torque can be uncertain, to achieve consistent performance under different operating conditions, a robust sliding controller is designed. With the help of a lemma concerning the well-definedness of the control law, appropriate control parameters can be selected to make the sliding control effort well-defined and continuous in the entire space. Experiments on a hardware setup indicate that the derived models can reasonably explain the static and dynamic behaviors of the worm-gear system, and the proposed control system exhibits superior performance to a conventional PID controller. Acknowledgements The authors gratefully acknowledge the support provided by Farley Tech. Co. and National Science Council in Taiwan. References [1] M.E. Dohring, E. Lee, W.S. Newman, A load-dependent transmission friction model: theory and experiments, Proceedings of IEEE International Conference on Robotics and Automation 3 (1993) 430–436. [2] H. Kawasaki, et al., Novel Climbing Method of Pruning Robot, SICE Annual Conference, August, 2008, Japan, pp. 160–163. [3] D.C. May, S. Jayasuriya, Velocity control of a manipulator joint driven through a worm gear transmission, in: Proceedings of the 1990 American Control Conference, San Diego, CA, May 1990, pp. 1272–1278. [4] D.C. May, S. Jayasuriya, A QFT controller design implementation for a worm gear driven manipulator Joint, in: Proceedings of the 1994 American Control Conference, Baltimore, MD, June 1994, pp. 595–595. [5] D.C. May, S. Jayasuriya, B.W. Mooring, Modeling and control of a manipulator joint driven through a worm gear transmission, Journal of Vibration and Control 6 (2000) 85–111.
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[6] B.W. Mooring, D.C. May, M.S. Sutle, Modeling and analysis of a manipulator joint driven through a worm gear transmission, in: Proceedings of ASME Conference on Advances in Design Automation, Sep. Canada, 1989, pp. 67–73. [7] R.M. Phelan, Fundamentals of Mechanical Design, McGraw Hill, New York, 1970. [8] J.E. Shigley, C.R. Mischke, Mechanical Engineering Design, McGraw-Hill, 1989. [9] J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, NJ, 1991. [10] C. Ünsal, P.K. Khosla, Mechatronic design of a modular self -reconfiguring robotic system, in: Proceedings of IEEE International Conference on Robotics & Automation, San Francisco, USA, April 2000, pp. 1742–1747. [11] M.-S. Wang, Y.-M. Tu, Design and implementation of a stair-climbing robot, in: Proceedings of IEEE International Conference on Advanced Robotics and its Social Impacts, Taipei, Taiwan, August, 2008.