Thermal displacement error compensation in temperature domain

Precision Engineering 42 (2015) 66–72

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Precision Engineering journal homepage: www.elsevier.com/locate/precision

Thermal displacement error compensation in temperature domain Takeshi Morishima a,∗ , Ron van Ostayen a , Jan van Eijk b , Robert-H. Munnig Schmidt a a b

Research Group Mechatronic System Design, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands MICE BV, Schweitzerlaan 18, Eindhoven 5644 DL, The Netherlands

a r t i c l e

i n f o

Article history: Received 9 September 2014 Received in revised form 23 February 2015 Accepted 31 March 2015 Available online 8 April 2015 Keywords: Thermal error reduction Thermal displacement error Thermal modal analysis Temperature control

a b s t r a c t Thermal displacement errors are becoming more and more important in the precision engineering field where the specifications are ever increasing. This paper proposes a novel technique to compensate inplane thermal displacement errors in a thin plate under a moving disturbance heat load. The displacement error is evaluated at the point of the moving heat load. The technique utilizes Thermal Modal Analysis (TMA) as a means to analyze the transient temperature distribution in the plate and estimate the thermal displacement field resulting from that temperature field. The temperature field is controlled by applying additional heat loads to the plate to control and eliminate some of the modes in the temperature domain which have the largest influence to the thermal displacement error. The theory of thermal modal analysis and the developed technique of controlling modes in the temperature domain are explained. The experimental setup and results are shown to validate the control of the thermal modes. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Thermal displacement errors have been one of the major sources of errors in precision engineering and will become even more dominant in the near future with the trend of increasing accuracies [1–4]. The compensation of thermal errors usually takes place with a thermal error model and a positioning system of the machine to compensate the predicted thermal displacement error at the point of interest by an additional displacement of the positioning device. Therefore the efforts in this field are mainly oriented on how to obtain an accurate thermal error model of the target machine. These approaches mainly can be divided into two categories. One is the experimental approach in which the temperature is measured at several locations in the machine structure and the thermal displacement at the point of interest is correlated to the temperature data with a multivariable regression, or an artificial neural network, etc. [5–10]. The main limitation of this approach is that the performance of the prediction within the test operating condition is good but the performance outside of the testing condition is not guaranteed. The other is an analytical or numerical approach which tries to predict the thermal displacement error from first principles. The benefit of the analytical approach is that it gives insight into the thermal behavior of the target structure and hence gives insights

∗ Corresponding author. Tel.: +31 15 27 81945; fax: +31 15 27 82150. E-mail address: [email protected] (T. Morishima). http://dx.doi.org/10.1016/j.precisioneng.2015.03.012 0141-6359/© 2015 Elsevier Inc. All rights reserved.

on how to improve the machine design as well. The drawback of the analytical approach is that it is usually restricted to a simple geometry with simple boundary conditions. Therefore a numerical approach is required such as finite element analysis (FEM) or finite difference element analysis (FDEM) which is a combination of finite difference method (FDM) for temperature field calculation and FEM for the corresponding thermal displacement field calculation to speed up the computation [11–13]. Also there is an approach which lies in between these experimental and analytical/numerical approaches: Optimal sensor location identification based on thermal modal analysis or proper orthogonal decompositions [14–17]. In these studies, some mutually orthogonal shapes of temperature distributions are used to obtain the best reduced order reconstruction of the original temperature field from the limited numbers of sensor measurements. In this paper, a novel thermal error compensation technique which is conducted in the temperature domain is introduced. Thermal error compensation in the temperature domain means that control actuation effort is applied not by a mechanical force but by external heat loads to control and reduce the thermal displacement errors by controlling the temperature distribution. Thermal Modal Analysis is used to calculate what kind of temperature distribution is required to achieve the thermal error reduction. As an example, we tested the proposed technique on a problem of in-plane thermal displacement error reduction of a thin aluminum plate under a moving disturbance heat load (Section 3.2). The thermal displacement error at the same location as the moving

T. Morishima et al. / Precision Engineering 42 (2015) 66–72

67

Similar to the displacement in vibration modal analysis, the temperature distribution of a structure can be expressed by the sum of the mode shapes multiplied with their amplitudes  i (t). T (t) =

N 

˚i i (t) = (t)

(5)

i=1

where  is a matrix in which each column consists of a mode shape i . Substitute Eq. (5) into the discretized heat conduction Eq. (1), ˙ C(t) + K(t) = q(t)

(6)

Multiply T onto the both side of the equation, T

T

T

˙  C(t) +  K(t) =  q(t)

(7)

Due to the orthogonality among the mode shapes with respect to the capacity and conductivity matrices [18].

Fig. 1. Positions of extra heat loads and trace of moving heat load.

heat load is minimized by the control of external heat loads at each corner of the plate. This example resembles applications such as inkjet printer with hot inks, photolithography process, etc. which have a moving heat load within the system as a given. In this paper, first the theory of thermal modal analysis is introduced in Section 2 and then our novel technique of controlling the temperature in terms of modes to reduce thermal displacement error is explained in Section 3. Then the experimental setup and its results are shown in Section 4. And finally the discussion based on the obtained results and conclusion is given in Section 5. 2. Thermal modal analysis

T

(8)

T

(9)

 C = [Cr ]  K = [Kr ]

where [Cr ] and [Kr ] are diagonal modal capacitance and conductivity matrices and the following relation holds:

1 r

= [Cr ]−1 [Kr ]



(10)



where 1/r is a diagonal matrix consisting of each time constant of the modes. Then from Eqs. (7) to (10), T

˙ [Cr ](t) + [Kr ](t) =  q(t) −1

(11)

2.1. Introduction into thermal modes

˙ (t) + [1/r ](t) = [Cr ]

From Fourier’s law of heat conduction, the discretized equation of heat conduction in a structure can be described as follows:

This equation determines the change of the modal amplitudes over time. Assuming a quasi-static linear thermo-mechanical relation between temperature and thermal deformation, the thermal deformation of the structure can be decomposed into the thermal deformation shapes that correspond to the thermal mode shapes.

CT˙ (t) + KT (t) = q(t)

(1)

C is a capacity matrix of size N*N where N is the number of degree of freedom being considered. K is a conductivity matrix and is a symmetric matrix of size N*N. T (t) is a vector of size N consisting of temperature values at each node in the discretized model over time. q(t) is a vector of size N containing all heat loads affecting the system. An analysis similar to that used for the determination of the vibration modes of a structure can be applied to this equation. Consider the following homogeneous equation of the discretized heat conduction equation, CT˙ (t) + KT (t) = 0

(2) t

Assuming the form of temperature vector T (t) as T (t) = T˜ e−  and substitute it into Eq. (2).

 1



t

− C + K T˜ e−  = 0 

(3)

For the above equation to have non-trivial solution, the following equation needs to hold.





U(t) =

N 

T

 q(t)

 i i (t) = (t)

(12)

(13)

i=1

where  i is the thermal displacement field of the structure when only mode shape ˚i is present in the temperature field and  is a matrix which has  i in each column. Each vector  i can be calculated using, for example, multi-physics finite element software. Generally, the thermal boundary conditions and mechanical boundary conditions are not the similar in a structure. For example, if a 2D plate is supported with thermally insulating material at one end as shown in Fig. 1, then the thermal boundary condition for this structure is free-free, whereas the mechanical boundary condition is fixed-free. The thermal deformation shapes in this configuration are shown in Fig. 2. 2.2. Reduced order thermal model based on thermal modes

 1  det − C + K = 0 

(4)

This equation has N solutions,  i (for i = 1 . . . N), which are the  eigen values of the matrix −1  C + K . By substituting each of the i

eigen values  i into Eq. (3), the corresponding mode shape i of the structure can be derived. The eigen value  i (for i = 1 . . . N) is the time constant of mode i. For our example problem of a thermally insulated 2D plate in Fig. 1, the first 9 modes (modes with longer time constants) are presented in Fig. 2.

As shown in the previous sections, the temperature field and the corresponding thermal displacement field in a machine structure can be reconstructed from the weighted linear combination of thermal amplitudes and thermal displacement shapes, respectively. Since each mode shows first order system behavior, its response is mainly influenced by its time constant  i . This  i is monotonically decreasing towards larger indices which means that modes with higher indices have faster response speed to inputs but smaller amplitudes while modes with lower indices have slower response

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Fig. 2. (Left): First 9 thermal modes of an insulated 2D plate. (Right): The corresponding thermal displacement fields with the left edge mechanically clamped (color scaling is different for each displacement shape.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with larger amplitudes. Fig. 3 shows the convergence of the temperature and the thermal displacement at a point under a stationary heat load versus the number of modes included in the reconstruction. Generally, the temperature value requires a large number of modes to obtain an accurate reconstruction under a heat load because of the need to represent high spatial frequency components in the temperature field. But the thermal displacement at the same location requires a smaller number of corresponding displacement shapes to reach a similar error. Therefore in the next chapter, our novel technique to control the thermal displacement error is introduced which aims to control mainly the thermal modes with lower indices. Hence the error at the point of interest will be reduced.

3. Mode cancellation 3.1. Basic idea This chapter introduces our novel idea of reducing a thermal displacement error by applying and controlling additional heat loads, which we call “mode cancellation”. The idea of mode cancellation is to control dominant thermal modes, minimize their amplitudes and therefore the influences of those modes in the temperature distribution and the thermal deformation. To control the dominant modes in the temperature field, external control heat loads q1 (t), . . ., qm (t) are applied together. Most practical methods to insert heat into the system do not allow for a similar controlled

Thermal displacement at t=11.67[s], x=46[mm], y=26.7[mm] 10.4

Temperature value at t=11.67[s], x=46[mm], y=26.7[mm] 35

10.2 10 25 Temperature [K]

Thermal displacement [um]

30

9.8 9.6 9.4

20

15 9.2

8.8

10

Thermal Modal Analysis model FEM software

9

0

20

40 60 80 mode numbers included [−]

100

5

Thermal Modal Analysis model FEM software 0

Fig. 3. Accuracy of reconstruction.

20

40 60 80 mode numbers included [−]

100

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69

10 no control mode cancellation 8

modal amplitudes [K]

6

4

2

0 Fig. 4. Control block diagram.

−2

cooling of the system. Therefore a separate, uncontrolled cooling heat flow qcool (t) is assumed to extract the heat from the system. Then selected number of modes (i = 1, . . ., n) are canceled out by applying m extra point heat loads provided where m ≥ n. Let (t) be a vector of amplitudes of only the selected modes which you would like to control and eliminate. From this point on, other matrices such as [1/ r ], [Cr ], [] are also reduced matrices which include only the elements corresponding to the selected modes. When a moving disturbance heat load qmov (t) is applied to the target system together with the cooling qcool (t) and the external control heat loads qext (t) = [q1 (t). . .qm (t)], Eq. (12) becomes ˙ (t) +

1 r

(t) = [Cr ]

−1



[mov (t)] qmov (t) + [cool ] qcool (t)

where [mov (t)] is a time-dependent extract from the mode shape matrix ˚ for the selected numbers of modes at the location of the T T moving load qmov (t), and [cool ] and [nm ] are also similarly extracted matrix for qcool (t) and qext (t). Let T

T

[mov (t)] qmov (t) + [cool ] qcool (t)

(15)

i

which describes the excitation of mode i due to the moving disturbance load and the cooling. So Eq. (14) can be rewritten as follows:

⎛

˙ (t) +

1 r

⎜ ⎝

(t) = [Cr ]−1 ⎜e(t) + [nm ]

⎡

T

⎢ ⎢ ⎣

q1 (t) .. .

⎤⎞ ⎥⎟ ⎥⎟ ⎦⎠

⎢ ⎢ ⎣

.. .

⎤

⎥ ⎥ = −([nm ]T )−1 e(t) ⎦

20

25

Fig. 5. Amplitude of the mode 1 to 4 eith and without control.

Regulator method from the state space model of the system below (Eq. (18)) with the objective function J (Eq. (19)): ˙ (t) = −

1 r



(t) + [Cr ]−1 [mov (t)] qmov (t) + [cool ] qcool (t) T

T

J=

N  

(18)



(tk )T I(tk ) + qext (tk )T Rqext (tk )

(19)

k=0

where I is a unit matrix which weights all the selected modes equally and R is a cost matrix for the control effort. In this paper R is chosen to be negligibly small compared to the unit matrix. Then the optimal feed-back gain K with respect to the objective function J can be obtained which minimize the sum of the selected modal amplitudes. Fig. 4 shows the control block diagram of the combination of feed-forward and feed-back control of target modes. 3.2. Simulation and result

where  e(t) is a vector with ei (t) in each element. To avoid a set of target modes to be excited, the external loads need to excite those same modes by − e(t). Therefore the amplitudes of the external heat loads to cancel out the excitation of the target four modes are given by the following equation. q1 (t)

15

(16)

qm (t)

⎡

10 time [s]

(14)

T



5

+[nm ] qext (t)

T

+[nm ] qext (t)

ei (t) =

0

T

T

T

−4

(17)

qm (t) Selection of the extra heat load locations should be made so that the inverse matrix has only small values and does not amplify the required extra heat loads’ amplitudes. Also techniques used to determine the optimal sensor placement [15] can be applied to the selection of the extra heat load positions. Using the Eq. (17), target modes can be controlled in a feedforward manner. The remaining error can be further controlled by feed-back. In this paper, a proportional gain feed-back control is used which gain K (Fig. 4) is calculated using the Linear Quadratic

A simulation study is conducted to study how the control of a particular set of modes results in the reduction of the thermal displacement error. The example simulation model used for this simulation study is presented in Fig. 1. It consists of a 2D aluminum plate (76 mm * 44.5 mm * 0.1 mm) with thermally insulated boundary condition with mechanical clamped boundary condition at its left edge. A moving disturbance heat load of 1 W is applied as depicted in the same figure for 25 s. This movement mimics a moving heat load from an inkjet printhead with hotink or a photolithography exposure process over a wafer. Convective cooling with a natural convection coefficient of h = 16.5 W/(m2 K) and different ambient temperatures are applied to the plate. The thermal displacement at the location of the moving heat load is evaluated over the simulation time, thus this becomes a moving load/ moving point of interest problem. Four external control heat loads are placed at each corner of the plate as depicted in Fig. 1. Each element −1 in ˚44 in Eq. (17) is small and this thus results in a smaller control power q(t). These four external loads’ amplitudes are controlled using Eq. (17) and the thermal displacement error at the moving point of interest is simulated over the simulation time. The simulation result of mode cancellation in this example is summarized below. Fig. 5 shows the amplitudes of the four target

70

T. Morishima et al. / Precision Engineering 42 (2015) 66–72 q1(t)

External heat load amplitude [W]

q2(t) 0.9

q3(t)

0.8

q4(t)

0.7 0.6 0.5 0.4 0.3 0.2 0.1

temperature values at the moving point of interest at an arbitrary chosen point in time. The resultant thermal displacement error under the moving heat load after applying our novel technique, mode cancellation, is smaller than that obtained using “best cooling”. One thing to note is that this technique reduces the contribution from the target four modes in the temperature field and the corresponding thermal displacement error while the contribution from the remaining modes actually increases. However as already stated in Section 2.2, higher order modes have a smaller contribution in the thermal displacement field. Therefore it is possible to reduce the thermal displacement error by controlling the four lower modes even considering the increasing contribution or the remainder of the modes.

0 −0.1

4. Experimental verification 0

5

10

15

20

25

time [s]

Fig. 6. The amplitudes of four extra heat loads.

modes (mode 1 to mode 4) with and without mode cancellation. The solid lines show the amplitudes of the target modes without any control and the dotted lines are the amplitudes of the same modes with mode cancellation. The result shows that the mode cancellation eliminates the target modes over the entire simulation time. And the calculated amplitudes of the four external control heat loads are shown in Fig. 6. Each of the amplitudes is of the same order of magnitude as the disturbance heat load of 1 W. The resultant deformation at the point under the moving heat load is shown in Fig. 7 with two other results for comparison. Blue line (with legend “moving disturbance load only”) is the error caused by the moving disturbance load plus a convection cooling (h = 16.5 W/(m2 K), Tref = Troom [K]). The green line (with legend “best cooling”) is the error in case uniform and constant strength cooling is applied together with the moving disturbance load. The convection coefficient in this simulation is kept constant (h = 16.5 W/(m2 K)) and the reference temperature Tref is chosen to the value which is below the ambient temperature and minimizes the average error over the simulation (Tref = Troom − 10 K). Then the red line (with legend “mode cancellation”) is the error with mode cancellation (with h = 16.5 W/(m2 K), Tref = Troom − 29 K). Fig. 3 shows the contribution of each mode in the thermal deformation and the

50 moving disturbance load only best cooling mode cancellation

45

thermal displacement error [um]

40 35 30 25 20 15 10 5 0

The previous chapter introduced the idea of mode cancellation and showed simulation results for the thermal displacement error in a thin plate under a moving heat load with additional control heat loads with uniform strength convective cooling. In this chapter an experiment of mode cancellation in the temperature domain shows that the selected number of modes can be controlled. The thermal displacement error is not measured in this setup. This measurement is both difficult to realize at the required accuracy, and also not really necessary, as the link between the temperature field and the thermal deformation has been well established in thermo-elastic theory. 4.1. Experimental setup The experimental setup is shown in Fig. 8. The setup consists of a large video projector, lens, mirror, a thin aluminum plate, and infrared (IR) camera. The video projector provides a means to easily produce a changing heat load both in time and space. Lens and mirror focus and reflect the image pattern from the beamer onto a thin aluminum plate. The aluminum plate is mounted by strings from the four corners to the supporting structure to reduce the thermal conduction through the mechanical support as much as possible. The aluminum plate is illuminated by the beamer, absorbs the light energy and converts it into heat. As a result, the temperature of the aluminum plate rises and thermal deformation occurs. This temperature rise is measured by the IR camera placed above the aluminum plate. Using this setup, the moving heat load to the plate and also the compensation external heat loads at the corners of the same plate are applied to control the amplitudes of the target modes and eliminate them. There are some differences compared with the simulation: cooling is not applied in this setup and this results in the control of only three modes: mode 2, 3, and 4 but not mode 1 which represents the average temperature of the plate for this thermal boundary condition. The IR camera measures the temperature of the aluminum plate at 4 Hz frequency. The computer connected to the beamer and IR camera controls the image projected from the beamer and records the data from the IR camera. The beamer projects four printing lines as shown in Fig. 1 during 25 s while amplitudes of the four external loads are controlled at the same frequency as the measurement frequency. The amplitudes of the external loads are controlled in feed-forward, feed-back or the combination of both using control scheme in Fig. 4. 4.2. Results

0

5

10

15

20

time [s] Fig. 7. The deformation at the moving point of interest.

25

Fig. 9 shows the amplitudes of the target modes mode 2, 3, and 4 over 25 s. The average values of those three modes controlled by three different controls are summarized in Fig. 10. Fig. 9 shows

T. Morishima et al. / Precision Engineering 42 (2015) 66–72

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Fig. 8. Experimental setup.

Modal Amplitude [K] mode 2 [K]

0.03

mov. load FF FB FF+FB

0.02 0.01 0 −0.01 0

5

10

15

20

25

time [s]

mode 3 [K]

0.04

mov. load FF FB FF+FB

0.02 0 −0.02 −0.04 0

5

10

15

20

25

time [s]

mode 4 [K]

0.02

mov. load FF FB FF+FB

0

−0.02 0

5

10

15

20

25

time [s] Fig. 9. Experimental result.

that the moving load (‘mov. load’, blue) excites the target three modes by about 0.02 K. The amplitudes of the external loads at the corners are calculated based on Eq. (17) and controlled in feed forward, feed back and combination of both. Feed forward control (‘FF’, red) reduces the excitation of the target modes. Feed back control (‘FB’, green) reduces the error caused by the moving load and also

Modal Amplitude [K]

Average Amplitude

5. Discussion

0.02

mov. Load

0.015

FF

0.01

FB

0.005

FF+FB

0 mode2

mode3

the remaining error in feed forward control. Due to relatively slow control frequency of the setup, feed back control has large errors when the moving heat load jumps from one line to another. The combination of feed forward and feed back control (‘FF + FB’, cyan) of the external loads gives the best results by reducing errors by both feed forward and feed back controls.

mode4

Fig. 10. Mean amplitudes of controlled modes.

Fig. 9 shows the experimental results of thermal modes control. Feed forward control (red) manages to control the amplitudes of the target three modes for the initial time period but the errors will grow as time goes by. There remain some errors, especially in mode 2 due to the imperfection of the model correspond to the reality. Therefore, feedback control is applied in addition to feed forward control (cyan) to avoid this growth of the error over time. The experimental results show good control of the target modes. As the simulation result in chapter 3 shows, the thermal displacement error along the moving point of interest under the

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T. Morishima et al. / Precision Engineering 42 (2015) 66–72

moving point load can be well reduced by controlling and eliminating the modes with large influences. 6. Conclusion The thermal modal analysis technique has been introduced as an analogy to modal analysis of structural vibration but for temperature and thermal displacement fields. The theoretical derivation of thermal modes, time constants and decomposition and reconstruction of the temperature field and thermal displacement field is explained. A novel thermal displacement error reduction technique has been introduced. In this method, the temperature is controlled by applying external heat loads to a thin plate to eliminate the dominant modes in the temperature and thermal displacement fields. In simulation, it is shown that the control of four dominant modes reduces the thermal error in a moving point of interest. The experiment shows the control of three modes in the temperature domain and shows the validity of the novel idea. Acknowledgment This HiPrins project is financed by the Dutch Ministry of Economic Affairs, Province of Limburg, Province of Overijssel, Province of Noord-Brabant and Eindhoven Regional Government. We would like to express our appreciation for Oce and NTS Mechatronics for providing technical expertise in the application field of industrial inkjet printing. References [1] Bryan J. International status of thermal error research (1990). Ann CIRP 1990;39/2/1990:645–56.

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