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High Precision Position Control using an Adaptive Friction Compensation Approach A. Amthor, St. Zschaeck, Ch. Ament
Abstract— The presented work concerns the development of a trajectory tracking controller which is able to improve clearly the dynamical performance of a high precision positioning stage. Experiments in the pre-rolling and rolling friction regimes are conducted and a hybrid parameter estimation algorithm is used to fit the parameters of a simple dynamic friction model based on experimental data. Further experiments show that the identified model does not represent the system behaviour over the whole operating range of 200 mm. To solve this problem the linear model parameters are adjusted online to ensure precise dynamic friction compensation. Finally, the extended friction model is utilized in a feed-forward controller in combination with a standard feedback controller to compensate for the effects of the friction force and other disturbances while moving. Index Terms—Nanopositioning, precision motion control
I
Friction
models,
High
I. INTRODUCTION
N micro-technology and nano-technology positioning with an extremely high resolution (< 1 nm) it is necessary to measure or manipulate objects on an atomic level. To realize this task, high-precision positioning stages are commonly used. Due to developments in the electronic components industry and in the production of optical components the need arises for an enlarged operating range of such stages. To realize motions up to several hundred millimeters in a vacuum, ball bearing guides are commonly utilized to support the motion axes. The position of the stage has to be controlled in all axes in order to minimize external disturbances such as sound waves, thermal expansion of the mechanical components, ground motion, et cetera. The state of the art in nanopositioning utilizes modified standard controllers, which are designed to compensate for external disturbances at a stationary set point [9]. With increasing measurement volume a dynamic positioning system is absolutely essential and thus a trajectory tracking controller has to be designed. In order to Manuscript received July 14, 2008. This work is supported by the Deutsche Forschungsgemeinschaft (German Research Council) under grant SFB 622. A. Amthor is with the System Analysis Group at the Ilmenau University of Technology, Ilmeanu, CO 98693 Germany (phone: +49-3677-691467; fax: +49-3677-691434; e-mail:
[email protected]). S. Zschaeck is with the System Analysis Group at the Ilmenau University of Technology, Ilmeanu, CO 98693 Germany (e-mail:
[email protected]). Ch. Ament is with the System Analysis Group at the Ilmenau University of Technology, Ilmeanu, CO 98693 Germany (e-mail:
[email protected]).
achieve high-precision dynamic positioning over wide velocity ranges, adaptive compensation of nonlinear effects is necessary, for example to overcome frictional effects [3]. Over the last decades, dynamic friction modeling and compensation have made great advances. Accurate friction modeling based on physical principles, e.g. in [1], and based on material/surface properties are not yet possible. Hence physically motivated models in combination with efficient identification methods based on experimental data are often used in the control community [17]. Several dynamic friction models have been proposed like the Karnopp [11], LundGrenoble [6], Elasto-plastic [7], Leuven [12] or Generalized Maxwell Slip (GMS) model [1]. In this work, we consider a 200x200 mm2 high precision positioning stage which is supported by ball bearing guides. First, the dynamical behaviour is modeled with an identification method based on the Basic Maxwell Slip model (DNLRX model) [13]. After explaining the structure of the model and the parameter identification algorithm, the experimental setup is introduced. Then, results of the system identification process are presented and a control scheme is proposed using an adaptive version of the DNLRX model as a feed-forward friction compensator. It is shown by experimental results that this approach is able to improve the dynamic performance of the system significantly. II. SYSTEM MODELING A. The Dynamic NonLinear Regression with direct application of eXcitation (DNLRX) Identification Method In order to estimate the model parameters all the physicallymotivated friction models mentioned above utilize the actual velocity as well as the measured friction force. In the application considered here none of these signals are directly measurable or computable (e.g. velocity by differentiation) with the precision needed. Thus the DNLRX identification algorithm which was proposed by Rizos et al. is used [13]. This identification method is based upon the measurable signals which are the applied force (via the applied current) and measured displacement. The extended model reflects the dynamics of a simple mechanical system with friction, which consists of a mass m , a linear spring with the spring constant c and a damper with the damping coefficient d (see fig. 1a). The system is stimulated by a force u ( t ) while the (immeasurable) friction force f ( t ) resists the excited motion.
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x1 ( k ) x( t )
m
0, ..., nx ) and the spring deformation vector δ k ( k ) =
m1
⎡⎣δ 1 ( k )… δ M ( k ) ⎤⎦ driven through a M -dimensional FIR filter of order nδ (with vector coefficients Θ j for j = 0, ... ,
ΔM mM
nδ ). Placing the expressions for velocity, acceleration (Eqn. (2)) and friction force (Eqn. (4)) into Eqn.(1) leads to a timediscrete system model:
x( k )
c u( t )
response (FIR) filter of order nx (with coefficients rj for j =
δ1 ( k )
d f (t )
xM ( k )
δM ( k )
T
na
Fig. 1: Simple mechanical system with friction (a) and an illustration of the basic Maxwell-slip model structure (b)
j =0
The system dynamics can be described as follows m ⋅ a( t ) + d ⋅ v( t ) + c ⋅ x( t ) = u( t ) − f ( t )
Conversion of Eqn. (5) yields to an inverse model of considered system
moving average representations of orders nv and na and the
j =0
(2)
with t = k ⋅ T
Herein, T is the sample time. The friction force is modeled using the Basic Maxwell Slip model structure [15], which is shown in fig. 1b. This approach approximates friction with M parallel elasto-plastic elements, all of which have the displacement x ( t ) as common input. A single element is characterized by a certain stiffness ki , a slipping threshold Δi and a state variable δ i , which represents the appropriate spring deflection. Since it is assumed that the elasto-plastic elements have no mass, there is a static relationship between the generated counterforce Fi and the spring deflection δ i . From this it follows that the spring deflection depends on the displacement while sticking (while there is zero relative motion) and remains constant at the slipping threshold limit Δi after the considered Maxwell Slip element relative motion begins [14]. The explained sticking/slipping behaviour of each element could be expressed in discrete time with one nonlinear state expression: min { x( k + 1 ) − x( k ) + δ i ( k ) , Δ i } with i = 1,...,M
nδ
j =0
j =0
f ( k ) = ∑ rj ⋅ x( k − j ) + ∑ ΘTj ⋅ δ( k − j )
j =0
j =0
(6)
This representation facilitates the possibility to model the inverse system dynamics with two FIR filters based on the displacement history. For further information the reader is referred to [2], [14]. B. Parameter Estimation Algorithm The parameter identification algorithm proposed in [13] uses pairs of displacement-applied force signals to determine the model parameters via a quadratic cost function based on the output error: N
J
∑λ e ( k )
(3)
(4)
Eqn. (4) suggests that the friction force could be calculated by having the displacement driven through a finite impulse
2
with e 2 ( k )
ˆ k )) ( u( k ) − u(
2
(7)
k=
where N is the number of signal samples, u ( k ) the applied force and uˆ ( k ) the model provided force. With the objective of identifying the model parameters Eqn. (6) can be rewritten [13]: DNLRX ( M , nδ , nx ,Δ , Θ ) : u( k ) = T
ΦT ⋅ ⎡⎣ x( k )...x( k − nx ) δT ( k )...δT ( k − nδ ) 1⎤⎦ + e( k )
In Eqn. (8)
Now the spring forces from all operators are summed up and a displacement history is added to account for unmodeled dynamics: nx
nδ
with n = max {na , nv , nx } and e.g. g0 = m ⋅ q0 + d ⋅ p0 + c + r0 .
v( k ) ≈ ∑ p j ⋅ x( k − j ), a( k ) ≈ ∑ q j ⋅ x( k − j )
δ i ( k + 1 ) = sgn [ x( k + 1 ) − x( k ) + δ i ( k )] ⋅
n
u( k ) = ∑ g j ⋅ x( k − j ) + ∑ ΘTj ⋅ δ( k − j )
coefficients p j and q j :
j =0
(5)
nδ ⎛ ⎞ u( k ) − ⎜ ∑ rj ⋅ x( k − j ) + ∑ ΘTj ⋅ δ( k − j ) ⎟ ⎜ j =0 ⎟ j =0 ⎝ ⎠
(1)
velocity v ( t ) can be approximated in discrete time with
na
j =0
nx
with x ( t ) as the displacement. The acceleration a ( t ) and
nv
nv
m ⋅ ∑ q j ⋅ x( k − j ) + d ⋅ ∑ p j ⋅ x( k − j ) + c ⋅ x( k ) =
e(k )
(8)
designates the model error and T
⎡ g ...g ΘT ...ΘT b ⎤ contains all model parameters to 0 nδ ⎢⎣ 0 n ⎥⎦ be identified, including an additional offset b . The offset has to be introduced due to the displacement dataset containing a certain offset, which is caused by small levelling errors of the experimental set up, dust, electrical wiring, and so on. The model is linear with respect to Φ and nonlinear with respect Φ
to the threshold vector Δ
[ Δ1 … Δ M ]
T
. Thus, a sequential
two-stage optimization problem is formulated:
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(9)
In the outer loop iteration, a genetic algorithm in combination with a Nelder-Mead simplex algorithm is used to determine the nonlinear threshold vector [10]. In the inner loop iteration the linear parameters are identified using a least squares algorithm. In order calculate the cost function J , the initial spring deformation vector δ0 is required. Following the algorithm described in [13], the system has to operate in both friction regimes while performing the identification experiment. If the displacement reaches a “dominant” extreme ( x ( tsl ) max { x ( t )} ), the breakaway displacement is already exceeded and the system is clearly in sliding regime. At this moment, by definition, all Maxwell Slip elements should slip in the direction indicated by x ( tsl ) . Now the assumption
δ0 = sgn ⎡⎣ x ( tsl ) ⎤⎦ ⋅ Δ is satisfied and data pairs with t ≥ tsl are selected for identification. To evaluate the quality of the identified model, the Normalized Root Mean Square Error (NRMSE) is used. The NRMSE is defined as
1 ⎛ N ⎞ 2 ˆ k )) N ⎟ × 100% where Θu is the ⋅ u( k ) − u( 2 ⎜ ∑( Θ u ⎝ k =1 ⎠ variance of the applied force. The model order is determined during the identification process as described in [13, 14, 15]. III. EXPERIMENTAL SET-UP The experimental set-up is a two dimensional fine positioning stage (see fig. 2). It was developed at the Collaborative Research Centre 622 ‘Nanopositioning and Nanomeasuring Machines’ at the Ilmenau University of Technology. Every axis is driven by two linear voice coil actuators. The engines are powered by proprietary developed analogue amplifiers, which provide the needed current with the required precision. Commutation of the engines is achieved by the control system using magnetic field intensity measurements, provided online by integrated Hall sensors. The operating range of this positioning stage is 200x200 mm2.
Each axis is supported by two linear V-grooved guideways. The position is measured by stabilized HeNe-laser interferometers with a resolution of less than 0.1 nm [16]. For data acquisition and control, a modular dSpace® real-time system in combination with Matlab/Simulink® is utilized. The control algorithm works with a sample rate of 10 kHz and operates on the amplifiers with a 16-bit resolution. For the present study, only the outer axis of the demonstrator is used. The inner axis is mechanically fixed at the position shown in fig. 2. Extensive identification experiments show that the ball bearings induce rolling friction effects like those described by De Moerlooze et al., i.e. with predominantly (pre)rolling hysteresis. Additional in-depth analysis is beyond the scope of this paper and so the reader is referred to [5]. IV. EXPERIMENTAL RESULTS
A. Parameter Identification Results For identification, the system was excited by a certain force Fmeas (see fig. 3b) and the resulting displacement was captured (see fig. 3a). It should be mentioned that tsl is the first sample of the plots in fig. 3. Afterwards, the model parameters were identified using these displacement-applied force signals. Regarding the model order, a DNLRX model with M = 7 , n = 6 , nδ = 0 generates the best NRMSE of 4.2 %. 4
1.5 position [nm]
}
x 10
a)
1 0.5 0 -0.5 -1 0
5
10
15
20
25
30 b)
1 force [N]
{
ˆ ˆ⎤ ⎡Δ min J ( Φ ,Δ ) ⎣ Φ ⎦ = arg min Δ Φ
0.5 0 -0.5
Fmeas
-1 0
Fmodel 5
10
15 20 time [s]
25
30
Fig. 3: Identification data-set: (a) displacement, (b) applied Fmeas and model provided force Fmodel
Fig. 2: xy- high precision positioning stage
Figure 4 shows the validation data-set in another position of the slider (more than 3 cm away). The identification experiment was repeated and the exciting force was reconstructed with the already identified DNLRX model. Figure 4 shows the ability of the DNLRX model to reflect the global characteristics of the validation data-set. But the model produces an offset error at another position and the NRMSE rises to 23.2 %. This observation leads to the finding that position dependent forces caused by dust, electrical wiring, mechanical tolerances in the ball bearing guides, bearing variabilities, et cetera inhibit the motion. Hence, the parameters identified partially lose their validity at another
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position. Extensive validation runs lead to the finding that a recursive estimation of the linear parameter vector Φ enables the capability that the already identified model reflects the system behaviour at every position. As an alternative method a position dependent parameter estimation was implemented but failed because the system behaviour is not only positiondependent but also time-dependent. This is caused by slow changes in the surrounding conditions such as temperature, humidity and so on. Also the dust in the linear ball bearings is moved by motions of the slider. It should be noted that the nonlinear threshold vector Δ is not adapted. However by using the proposed adaptation mechanism a precise inverse system modeling at every point of the operating range at any time was possible. 4
x 10
adapt the linear parameter vector Φ ( k + 1) by minimizing eADNLRX ( k ) . In the following time step the new linear
parameter vector Φ ( k + 1) is used by both DNLRX models and the adaptation process starts again. Notice that both models are initialized with the same initial values and they use the same internal states at work. Hence it can be assured that both models have the identical behaviour and the parameter adaptation mechanism is stable. u ff ( k )
x ff ( k ) u fb ( k ) + +
0 -1 0
5
10
15
20
25
Fmodel 0 -1 0
b) 5
10
15 20 time [s]
25
x( k )
+ −
eADNLRX ( k )
uˆ ( k )
30 Fmeas
1
u( k )
+
−
Φ( k + 1 )
position [nm]
model output uˆ ( k ) and the commanded force u ( k ) is used to
a)
1
force [N]
nonlinear system. The difference eADNLRX ( k ) between the
30
Fig. 4: Validation data-set: (a) displacement, (b) applied Fmeas and model provided force Fmodel
B. Model based control scheme The maximum acceleration in the identification experiments which is performed using a sinusoidal motion as the initial excitation is 0.003 m/s2. Figure 4 shows that the engines have to generate more than 1.15 N to perform this movement and the force which is needed to accelerate the slider (24.5 kg) consumes only 0.0735 N, that is, only 6 % of the total force. These data clearly indicate the dominant impact of the forces caused by friction, dust, electrical wiring, et cetera. Hence a pure acceleration feed-forward of the initial forces does not work at all. Only a model based feed-forward of the inverse system dynamics can improve the dynamical behaviour of the controlled system and therefore an adaptive version of the DNLRX (ADNLRX) model is utilized as an adaptive feedforward friction compensator in a trajectory-tracking control scheme (see fig. 5). In the feedback path of the proposed scheme there is a well-tuned PID controller which accounts for external disturbances such as sound waves, ground motion and so on. To realize the online estimation of the linear model parameters, the identical model is implemented twice. While “DNLRX model 1” operates as feed-forward friction compensator, the “DNLRX model 2” works in parallel to the
Fig. 5: Schematic diagram of the adaptive trajectory tracking controller
For the adaption algorithm first a conventional recursive least squares algorithm was implemented. Using this algorithm we found that it tends to become unstable during runtime. As a solution, a recursive least squares algorithm with U-Dfactorization was used to realize a stable and reliable parameter estimation online. For further information on the algorithm the reader is referred to [4].
Statements on the stability of the adaptive feed-forward friction compensator Theorem 1: The adaptive feed-forward compensator is stable if the output u ( t ) is bounded. Proof:
By
definition:
Φ (t )
determination of uˆ ( t ) and u ff ( t )
friction T
⎡⎣ϕ1 , ..., ϕ n + nδ + 1 ⎤⎦ . The (see Eqn. (6)) fulfills the
conditions defined by the ring theory described in [8]. If ϕi ∈ ∀i ∈ {1, … , n + nδ + 1} , the result according the ring theory is that u ( t ) ∈ Theorem 2: ϕi ∈
. ∀i ∈ {1, … , n + nδ + 1}
Proof: ϕi ∀i ∈ {1, … , n + nδ + 1} are determined by a least
squares algorithm. The least square algorithm also fulfills the conditions defined by the ring theory [8]. Thus ϕi ∈ ∀i ∈ {1, … , n + nδ + 1} if ( u ( t ) − uˆ ( t ) ) ∈ . Theorem 3: ( u ( t ) − uˆ ( t ) ) ∈ Proof: Under the assumption that model uncertainties as well as external disturbances are bounded, we can define
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u ff ( t ) = u( t ) . Furthermore we can describe the inverse system model with f ( x ff ( t ) ) = u( t ) , the real system with
g ( u ( t ) ) = x ( t ) = x ff ( t ) + ξ ( t ) and the adaption mechanism
( (
f g f ( x ff ( t ) )
with
)) =
f ( x ff ( t ) + ξ ( t ) ) = uˆ ( t ) .
The
nonlinear system model can be decomposed into a linear
(( x f (( x f
ff
(t ) + ξ (t )))
and
ff
(t ) + ξ (t ))) =
f
rewritten
as
(( x
an ff
affine
part
( t ) + ξ ( t ) ) ) + af
. This can be
f ( x ff ( t ) ) + f (ξ ( t ) ) + af = uˆ ( t ) .
f ( x ff ( t ) ) + af = u ( t ) , we obtain
where
af
Because
f (ξ ( t ) ) = ( u ( t ) − uˆ ( t ) )
and this leads to the result d f ( x ( t ) ) ⋅ ( x ( t ) − x ff ( t ) ) = ( uˆ ( t ) − u ( t ) ) . The tracking dx ( t ) error ( x ( t ) − x ff ( t ) ) is bounded because x ( t ) as well x ff ( t ) are bounded (max. 200 mm). Since
d f ( x ( t0 ) ) is dx ( t0 )
bounded due to the fact, that the parameter identification it follows that described in section IIIA assures ϕ1 g0 ∈ the error, which is minimized by the least squares algorithm, is also bounded ∀t ≥ t0 . By way of example, fig. 6 shows the tracking error with and without the proposed adaptive feed-forward compensator, in the first 5 seconds and in the last 5 seconds, respectively. The reference trajectory is a sinusoidal motion with a frequency of 1 Hz and amplitude of 10,000 nm (see fig. 6a)). 4
x 10 position [nm]
1
xref x
0.5 0 -0.5
a)
-1 0
2
4
6
8
tracking error [nm]
1000
10 b)
500 0 -500 -1000 0
2
4
time [s]
6
8
10
Fig. 6: Tracking performance with and without feed-forward control
In the first 5 s the position is only controlled by a well tuned PID controller (see fig. 6b)) and it can be clearly seen that the feedback controller is not able to follow the reference trajectory satisfactorily. Since the NRMSE of the tracking error is large at 9.01 %. After 5.2 s the ADNLRX model structure is utilized in the feed-forward path and the hybrid
control scheme is able to follow the trajectory significantly better (see fig. 6b)). With the hybrid control scheme the NRMSE is reduced to 0.36 % and this corresponds to a reduction in the tracking error by a factor of more than 25. Long-duration experiments also indicate that the ADNLRX model structure is quite robust against disturbances (e.g. variations in temperature, dust, sound waves) and works over the whole operating range with a uniform quality. In order to confirm this statement fig. 7 shows the evolution of the adapted linear model parameters over 500 s, while performing the experiment shown in fig. 6. It can be clearly seen that the model parameters converge quite rapidly and remain afterwards nearly constant if the parameter adaption algorithm is enabled. A large number of similar long-duration experiments also indicate the stability of the proposed online parameter adaption algorithm. -5
x 10 10 5 0 -5
1
10
2
log time [s]
10
Fig. 7: Developing of the adapted linear model parameters over 500 s
V. CONCLUSION Modeling and dynamic control of a system with rolling friction has been addressed. The so-called extended DNLRX model structure is utilized to represent the dynamics of a highprecision positioning stage. After identification and validation the usability of the DNLRX model to reflect the system behaviour was examined. The results show that the system behaviour varies with the position of the stage due to stochastic disturbances such as dust, mechanical tolerances of the ball bearing guides and so on. Thus, the parameters identified offline are not universally valid. As problemsolving technique a recursive least-squares algorithm was utilized to adapt the linear model parameters online. At the end of the paper an adaptive version of the DNLRX model structure is used as feed-forward part in the trajectory-tracking controller in combination with a standard PID controller in the feedback loop. Exploiting this hybrid control scheme, the tracking error is reduced by a factor of more than 25 over the whole operating range. We conclude that these findings show the effectiveness of an adaptive friction model as a friction compensator controlling motions on a nanometer scale. In future we will analyse the performance of more advanced friction models, for example, an adaptive version of the GMS model.
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[16]
SIOS Meßtechnik GmbH, Available: http://www.SIOS.de
This work was done within the framework of the Collaborative Research Centre 622 ‘Nanopositioning and Nanomeasuring Machines’ at the Ilmenau University of Technology, which is supported by the German Research Council (Deutsche Forschungsgemeinschaft) and the Thuringian Ministry of Science. The authors would also like to thank all colleagues who offered assistance in the work presented here.
[17]
Worden, K.; Wong, C. X.; Parlitz, U.; Hornstein, A.; Engster, D.; Tjahjowidodo, T.; Al-Bender, F.; Rizos, D. D.; Fassois, S. D., “Identification of Pre-Sliding and Sliding Friction Dynamics: Grey Box and Black Box Models” – In: Mechanical Systems and Signal Processing, vol. 21(1), pp. 514-534, 2006
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