A new solution to the measurement process planning for machine tool assembly based on Kalman filter

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A new solution to the measurement process planning for machine tool assembly based on Kalman filter Junkang Guo, Baotong Li ∗ , Zhigang Liu, Jun Hong, Qiang Zhou State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China

a r t i c l e

i n f o

Article history: Received 23 May 2015 Received in revised form 19 August 2015 Accepted 29 August 2015 Available online xxx Keywords: Machine tool assembly Volumetric error Measurement process planning State space model Kalman filter

a b s t r a c t This paper introduces a novel approach for measurement process planning in machine tool assembly by applying the observability principle of state space model which is widely used in control engineering. Initially, state space modeling was carried out for describing the variation propagation and accumulation of assembly process. Then, a mathematical explanation of measurement uncertainty accumulation was presented by means of optimal estimation using Kalman filter. Based on this, an analysis algorithm is developed to find an optimal solution with small estimation errors and money and working time costs, so as to provide designers with feasible measurement plans in assembly. When such method is used, a quantitative approach could be offered to the evaluation of uncertainties of measurement process, which is totally different from traditional methods dependent on experiences. The suggested approach was finally applied to the assembly of a horizontal machining center, on which numerical analyses was conducted to validate the proposed method, and some guidance for measurement planning is summarized from this example analysis. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Machining accuracy has been improved for modern machine tools in order to meet the requirements of every walk of industry, such as aerospace, automobile, ship building, energy equipment, and so on. The factors affecting the machining accuracy involve the volumetric error of machine tool [1,2], the stiffness error related to the exciting of cutting forces and the heating from various thermal sources [3,4], the wear and backlash effects in the transmission system [5], etc. Among these factors, the volumetric error has the fiercest impact to the machining quality compared with other factors [6,7]. This is because it directly reflects the position and orientation errors of tool path with respect to the ideal path. For this reason, volumetric error is no doubt a criterion with priority for the design of modern machine tools with high precision, speed and productivity. The volumetric error of precision machine tools mainly comes from two sources: one is the kinematic errors of a single axis and the other is the relative angular errors between two axes. Analytical model of a single axis has been developed by many researchers to reveal the relationship between geometric errors of individual components and kinematic errors [8–10]. The orientation of

∗ Corresponding author. Tel.: +86 18729973331. E-mail address: [email protected] (B. Li).

each axis is determined by the mounting surfaces of supporting component. Due to the accumulation and propagation of the geometric errors of supporting components in step-by-step assembly process, the orientation deviation of each axis will generate and change, leading to relative angular errors between two axes. Thus, a series of adjustment processes should be performed to keep the angular errors at an acceptable level. All the adjustment operations are based on the accurate measurement of variation accumulation. Therefore, it is important to develop a systematic variation reduction strategy through optimal measurement process planning to achieve high volumetric accuracy. The goal of measurement is to know the exact value of variables concerned. However, due to the noises existing in almost all measurements, it is really hard to obtain the true values in reality. Estimation theory was developed to filter noises to reach an utmost exactness. The estimation error is generally influenced by the measurement uncertainties and data processing algorithm. State space model has been used to describe the geometrical variation propagation in multistage assembly process, especially for the compliant parts such as aircraft and automotive body [11–14]. Based on this model, process variation estimators are developed for in-process dimensional measurement and control [15]. The estimation methods also find applications in industry such as variation source identification [16], optimal sensor allocation [17] and design of fixture layout [18]. However, the assembly process is quite different between machine tools and compliant parts.

http://dx.doi.org/10.1016/j.precisioneng.2015.08.016 0141-6359/© 2015 Elsevier Inc. All rights reserved.

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Fig. 1. Modeling of the kinematic errors of a horizontal machining center.

The variation accumulation of compliant parts is controlled by fixtures at each assembly stage [12]. Thus, the control term (B(k)U(k)) in state space equation becomes the key issue in the study of quality improvement. The geometrical accuracy of machine tools is assured by the key product characteristics (KPCs) of the components [19]. It is the state variables (X(k)) that should be optimally estimated to reduce the variation accumulation at each assembly stage. Kalman filter is a popularly used and effective estimation algorithm for time series analysis, which is based on the recursive process of state space model [20,21]. The variance of estimation error of Kalman filter is mainly determined by measurement uncertainty and observation matrix. In machine tool assembly, they are influenced by the selection of measurement instrument and KPCs. It is critical to select proper measurement plan with multiple objectives, such as minimizing the cost of measurement and instruments use as well as the estimation errors. The satisfactory measurement strategy has to select the most appropriate process plan among candidate plans and make reasonable trade-offs in the above mentioned three objectives. Research efforts on process plan evaluation and selection for minimal product variability and cost has been conducted over the past four decades and they are still going on today [22]. Ding et al. [15] studied the interrelationships and properties of the available variance estimators and compared their performance for in-process dimensional measurements and control. Jiao and Djurdjanovic [23] proposed a reactive tabu search algorithm to joint optimal allocation of measurement points and controllable tooling machines. Liu and Shi [24] established a definition of sensor allocation in a Bayesian network and developed an algorithm to obtain the optimal sensor allocation design at minimum cost under different specified requirements. By using new fixture technologies, Abellan-Nebot [25] proposed a methodology to facilitate the implementation of sensor-based fixture in multi-station machining process. They all focused on the fixture positioning multistage assembly process. New estimation methodology and inspection strategy should be developed for the specific requirements of machine tools assembly.

state space model is introduced to mathematically describe the variation propagation in assembly process. In Section 3, an optimal estimation method of variation accumulation using Kalman filter is presented. The searching strategy is proposed to obtain the optimal measurement process plan. In Section 4, a horizontal machining center is studied to exemplify the proposed optimal planning method and some findings are discussed. The research work is concluded in the last section. 2. Modeling of machine tool assembly process 2.1. Geometric errors of machine tools In precision machine tools, the machining accuracy is influenced by the combined effect of supporting parts, feeding systems, spindle systems, etc. All these structured components and the control system and other auxiliary systems are assembled to enable the machine tool to realize required functions. The geometric accuracy is generally defined as the relative motion error between tool and workpiece in a three-dimensional coordinate system. They are derived from the kinematic chains including all axes, the workpiece, the tool and the supporting components. With the assumption of rigid body, the translation and rotation of each supporting surface of components can be described by six degrees of freedom: three translations and three rotations. Viewing a machine tool as a multi-body system (MBS), Datum Flow Chain (DFC) and homogeneous transfer matrix (HTM) are widely used in error modeling [26,27]. Fan et al. [28] developed a universal kinematics modeling of a machine tool based on MBS. To date, it is the best method for geometric error modeling of machine tools. Based on MBS and HTM, Zhu et al. presented an integrated method for geometric error modeling, identification and compensation of CNC machine tools [29]. A horizontal machining center is taken as an example as shown in Fig. 1. By using 4 × 4 HTM to represent the transformation relationship between a pair of adjacent bodies, the position errors in each direction of the tool tip can be represented in Eq. (1).

x = xx + xy − xz − xB + (yB + B ) · y + (ˇz + ˇB ) · z + (yB + B − y ) · pty − (ˇxz + ˇz + ˇB − ˇy − ˇxy − ˇx ) · ptz y = yx + yy − yz − yB − (B + yB + x + xy ) · x − (˛x + ˛xy + ˛yB − ˛xz ) · z − (yB + B − y ) · ptx + (˛B + ˛yB − ˛y ) · ptz

(1)

z = zx + zy − zz − zB + (ˇB + ˇz + ˇxz ) · x − (˛B + ˛yB ) · y + (ˇxz + ˇz + ˇB − ˇy − ˇxy − ˇx ) · ptx − (˛B + ˛yB − ˛y ) · pty

The remainder of this paper is organized as follows. In Section 2, geometric error of machine tool is modeled and analyzed and

According to this result, it can be seen that there are three components in each position error: (1) the kinematic errors of motion systems; (2) the position error coming from angular errors between

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Fig. 2. Key product characteristics (KPCs) in assembly model.

two axes; (3) the position error coming from mounting errors of cutting tool and workpiece. The first and the third components are fixed when the motion system, cutting tool and workpiece are mounted on the supporting components. The angular errors between two axes are generated in the step-by-step processes in machine tool assembly. Therefore, in the discussion of geometric accuracy assurance for machine tool assembly, we focus on these angular errors, also called link geometric error [30].

defined as the reference surface. O2 is the derived surface, which is perpendicular to O1 . O2  is the actual surface of O2 . Defining T i,j as a 4 × 4 homogeneous transformation matrix from coordinate Oi to Oj , so the actual transformation T ci,j between two KPCs (Oi to Oj  ) is T ci,j = T i,j + dT i,j or



2.2. State space modeling of machine tool assembly process 2.2.1. Representation of part deviations The angular errors (link geometric errors) generally depend on the machined surfaces on which the motion systems (rails, carriages, bearings) are mounted. These surfaces of the supporting components are defined as the key product characteristics (KPCs) in machine tool assembly model. As shown in Fig. 2, an assembly contains three supporting components and the X-axis. All the KPCs are represented with purple lines and explained in Table 1. The KPCs can be denoted by coordinates to indicate the position and orientation. The relative position or orientation of KPCs in one component is specified by tolerances in design, where one KPC is usually defined as a reference surface. However, geometric errors inevitably exist in any component with respect to the ideal geometry. These errors accumulate and propagate in assembly process. As shown in Fig. 3, taking the column as an example, O1 is the bottom surface of the column, also

Table 1 Description of key product characteristics. Coordinate

Description

O1

Mating surface of bed and column

O2 O3

Mounting surface of X-axis (rails) Mounting surface of X-axis (carriages) Mounting surface of Y-axis (rails) Mounting surface of Z-axis (rails)

O4 O5

Reference surface of bed and column Reference surface of X-axis Reference surface of saddle Reference surface of Y-axis Reference surface of Z-axis

(2)



T ci,j = I + ıT i,j T i,j

(3)

Based on the small displacements hypotheses, the differential error notations (d) can be replaced by the actual (small) errors () during linear or angular transformations [31]. For machine tool assembly, only the angular errors of the KPCs affect the machining accuracy. The angular errors of the incoming component bringing to the assembly induced by machining errors can be expressed as



 = x

y

z

T

The angular errors are defined based on the reference surface (coordinate) in one component. For example, the angular errors  column is the 3 × 1 differential rotation vector of T1,2 with respect to coordinate O1 as shown in Fig. 3. They are considered small according to the tolerance specification, such as perpendicularity and parallelism [32]. 2.2.2. State space model of machine tool assembly process As shown in Fig. 3, O1 can be defined as the datum coordinate in this assembly, since the bed is usually first mounted. All the variations of KPCs will accumulate and propagate along the DFCs of the assembly [26]: (1) O1 → O2 → O3 → O4 ; (2) O1 → O5 . In DFCs, assuming that the deviation coordinate of (n-1)th frame is also the reference coordinate of nth frame. For example, O2  is the deviation coordinate of column, which is also the reference coordinate of X-axis. O3  is the deviation coordinate of X-axis, which is also the reference coordinate of the saddle. By using differential motion vector (DMV), the total deviation in the orientation of a coordinate

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Fig. 3. Deviation geometric model.

on the nth KPC along a DFC, expressed in the datum coordinate at the base of the chain is given as follows:



ın = ıxn

y

T

ızn

ın

⎡

where ın is the first order differential rotation vector of the nth KPC with respect to datum coordinate. ın can be represented in the following matrix form [31]:

⎡

ıxn

⎤

⎡

W(5,1),1

W(5,1),2

W(5,3),1

W(5,3),2

⎢ y⎥ ⎢ ın = ⎣ ın ⎦ = ⎣ W(5,2),1 W(5,2),2 ızn

⎤ ⎡ ⎤ x1 · · · W(5,1),n ⎢ ⎥ x2 ⎥ ⎥⎢ ⎥ ⎢ · · · W(5,2),n ⎦ ⎢ .. ⎥ ⎣ . ⎦ ···

···

W(7,1),n

W(7,3),1

W(7,3),2

···

W(7,3),n

⎢ + ⎣ W(7,2),1 W(7,2),2

⎤ ⎤ z1 · · · W(9,1),n ⎢ ⎥ ⎥ ⎢ z2 ⎥ ⎥ · · · W(9,2),n ⎦ ⎢ ⎢ . ⎥ ⎣ .. ⎦

W(9,1),1

W(9,1),2

W(9,3),1

W(9,3),2

···

(4)

W(5,1),1

W(7,1),1

W(9,1),1

W(5,3),1

W(7,3),1

W(9,3),1

x1

⎤

⎡

⎤⎡

⎤

W(5,1),n

W(7,1),n

W(9,1),n

+ · · · + ⎣ W(5,2),n

W(7,2),n

W(9,2),n ⎦ ⎣ yn ⎦

W(5,3),n

W(7,3),n

W(9,3),n

⎢

⎥⎢

xn

zn

W2

···

 1

⎤

⎥ ⎢  ⎥ ⎥⎢ 2 ⎥ ⎥⎢ ⎥ .. ⎥ ⎢ .. ⎥ ⎣ ⎦ . ⎦ .

O

WN

⎥

(6)

 N

T

y

ı1 (k + 1)

⎤

ıN (k + 1)

⎢ ⎥⎢ ⎥ ın = ⎣ W(5,2),1 W(7,2),1 W(9,2),1 ⎦ ⎣ y1 ⎦ z1

W1

⎤⎡

O

⎡

⎤ ⎡

ı1 (k)

W1

O

···

⎢ ı (k + 1) ⎥ ⎢ ı (k) ⎥ ⎢ W W · · · 2 ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 1 ⎢ ⎥=⎢ ⎥+⎢ ⎢ ⎥ ⎢ ⎥ ⎢ .. .. .. ⎣ ⎦ ⎣ ... ⎦ ⎣ ... . . .

Eq. (4) can be rewritten as

⎤⎡

···

Machine tool assembly is a step-by-step process. In each assembly step, one component is mounted on the preceding assembly, or two components are assembled to be a subassembly. Machining errors of the components are introduced into the assembly in these processes. Therefore, the total accumulated variation in the (k + 1)th step and the total variation accumulated in the kth step can be represented in the following matrix form:

⎡

W(9,3),n

zn

⎡

O

˛yB = ıxB-axis − ıxY -axis and ˛yz = ıxY -axis − ıxZ-axis

⎡

⎢ + ⎣ W(9,2),1 W(9,2),2

W1

of each KPC is given, the relative angular errors ıxn ın ızn (geometric error parameters in Eq. (1)) can be calculated as:

yn

⎡

⎡

⎢ ı ⎥ ⎢W W ··· 2 ⎢ 2⎥ ⎢ 1 ⎢ ⎥=⎢ ⎢ . ⎥ ⎢ . .. .. ⎣ .. ⎦ ⎣ .. . .



⎤ ⎡ ⎤ y1 ⎢  ⎥ y2 ⎥ ⎥⎢ ⎥ · · · W(7,2),n ⎦ ⎢ ⎢ . ⎥ ⎣ .. ⎦

W(7,1),2

⎤

where O is a zero matrix,  n = [ xn ,  yn ,  zn ]T is the deviation vector of the nth KPC along DFC, which can be obtained from measurement or designer specified tolerances. As shown in Fig. 4, if the differential rotation vector ın =

W(5,3),n

W(7,1),1

ı1

ıN

xn

⎡

Based on Eq. (5), the differential rotation vector of all KPCs can be expressed as

(5)

ıN (k)

W1

W2

···

O

⎤

⎥ ⎥ ⎥ (k) .. ⎥ . ⎦

O

(7)

WN

where (k) contains one or several element vectors of [ 1 ,  2 ,. . .,  N ]T corresponding to the newly assembled component in the kth assembly step. For example, part 1 and part 3 are assembled in the kth step, (k) can be written as:



(k) =  1

O3×1

 3

Om×1

T

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Fig. 4. Relationship between differential rotation vector and relative angular error.

Each element vector is calculated only once in the assembly process. When the assembly is finished, all the element vectors should be introduced. The above variation accumulation modeling is a special case where only one DFC (from KPC 1 to KPC N) exists in the assembly process. For more complex case with more DFCs, the differential rotation vector of KPCs can be represented in the following form:

⎡

ıDFC1 (k + 1)

⎤

⎢ı ⎥ ⎢ DFC2 (k + 1) ⎥ ⎢ ⎥= ⎢ ⎥ .. ⎣ ⎦ . ıDFCm (k + 1)

⎡

ıDFC1 (k)

⎤

⎢ı ⎥ ⎢ DFC2 (k) ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . ıDFCm (k)

⎡ ⎢ ⎢ +⎢ ⎢ ⎣

W DFC1

O

···

O

O

W DFC2

···

O

.. .

.. .

..

.. .

O

O

···

.

⎤ ⎥ ⎥ ⎥ (k) (8) ⎥ ⎦

W DFCm

The recursive form of Eq. (8) illustrates the variation propagation and accumulation induced by the machining error in assembly process. Considering the measurement and adjustment process, the machine tool assembly process can be expressed in the state space equation [33] x(k + 1) = A(k)x(k) + B(k)u(k) + F(k)w(k) y(k) = C(k)x(k) + v(k)

(9)

where A(k) u(k) B(k)

w(k)

F(k)

C(k)

v(k)

Identity matrix; Re-machining or scraping adjustment vector; Transforming u(k) from the local part coordinates to the base coordinates of the DFC. If u(k) is defined in the same coordinate as w(k), B(k) = F(k); w(k) = (k), describing the variation associated with the part being assembled at the kth assembly step, expressed in local part coordinates; Transforming the variation of the incoming part at the kth assembly step from local part coordinate to the base coordinate of ⎡ ⎤ W DFC1 O ··· O W DFC2 · · · O O ⎢ ⎥; the DFC, F(k) = ⎣ . . . .. ⎦ . . . . . . . O · · · W DFCm O Observation matrix of 1s, −1s and 0s, defining values that we are interested in, for a particular KPC; and The potential measurement noise.

2.2.3. Example study A horizontal machining center is taken as an example to illustrate the modeling of variation propagation in assembly process as shown in Fig. 5. For the linear or rotation motion systems are usually assembled on their supporting component before general assembly. For example, the rails and carriages of X-axis are mounted on the column before the column is assembled on the bed. Therefore, KPCs are defined as the mating surfaces between supporting components or mounting surfaces of motion systems. The bed is firstly mounted on the ground. The mating surface of bed and column is selected as the datum surface in DFCs. In this machine tool structure, all these defined KPCs are parallel or perpendicular to each other. For simplification, the nominal coordinates associated on the KPCs are defined in the same orientation as shown in Fig. 5. The state space equation of angular errors can be expressed as

⎡

ı1 (k + 1)

⎤

⎡

ı1 (k)

⎤

⎡ I ⎢ ı (k + 1) ⎥ ⎢ ı (k) ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢I ⎢ ⎥ ⎢ ⎥ ⎢ ı3 (k + 1) ⎥ ⎢ ı3 (k) ⎥ ⎢ I ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥=⎢ ⎥+⎢ O ⎢ ı4 (k + 1) ⎥ ⎢ ı4 (k) ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎣O ı ı (k + 1) (k) ⎣ 5 ⎦ ⎣ 5 ⎦ ı6 (k + 1)

O

O O O O

⎤

I

O O O O⎥

I

I

O O O⎥

O O

O

I

⎥ ⎥ (k) O O I O O⎥ ⎥ O O I I O⎦ I

(10)

I

ı6 (k)

where I is a 3 × 3 identity matrix, O is a 3 × 3 zero matrix and (k) is the incoming error of one or several assembled components in the kth assembly step. In the first assembly step, the bed is mounted on the ground. The mounting surface of Z-axis is adjusted (usually by scraping) to keep horizontal. The column is assembled on the bed in the second step. The perpendicularity and parallelism of the mounting surfaces of X-axis and Z-axis must be checked in this assembly step. In Fig. 6(a), the perpendicularity between X-axis and Z-axis is measured by a dial gauge and a calibrated rule. The parallelism between X-axis and Z-axis plane can be measured either by a dial gauge and a calibrated rule (Fig. 6(b)) or by a level and a calibrated rule (Fig. 6(c)). Observation matrix C(k) indicates which KPCs are measured in the kth step. In machine tool assembly, the perpendicularity or parallelism of two KPCs is most frequently measured. The relative angular error of a KPC pair can be obtained by setting the elements in the corresponding column of the observation matrix to 1 or −1. The measurement error is decided by the measurement method. Corresponding to Eq. (10), the observation equation in the

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Fig. 5. KPCs and DFCs of a horizontal machining center.

second step can be written as y(2)

=

C(2)x(2) + v(2)

=

0

1

0

0

0

1

+

=

vdial

⎡

O2×6

gauge (2)

vlevel (2)

0

−1

0

0

y



z

z

+

ı1 (2) − ı4 (2)

⎤

 ⎢ ı (2) ⎥ ⎢ 2 ⎥ ⎥ O2×6 ⎢ ⎢ . ⎥ −1 ⎣ .. ⎦ 0

ı6 (2)

y

ı1 (2) − ı4 (2)

ı1 (2)

vdial

gauge (2)

vlevel (2)

The step-by-step assembly of machine tools including measurement process is mathematically modeled in the state space form. Optimal estimation of KPC variation based on state space model is further discussed in the following section. 3. Measurement process planning of machine tools assembly

KPC variation. However, in each assembly step, only the output vector y(k) can be directly obtained from measurements. The accumulated variation of KPCs x(k) cannot be directly obtained due to the limitation of measurement methods and assembled structure. Some KPCs cannot be measured in specific assembled structures, such as the mating surface of two supporting parts. Therefore, effective and accurate estimation of the accumulated variation of KPCs is important and necessary in accuracy assurance of machine tools.

3.1.1. Assumptions (1) Input error induced by newly assembled component, w(k) w(k) is the deviation associated with the incoming part during assembly process. As discussed in Section 2.1, the elements of this vector are defined as the angular error of two KPCs in one part. The deviation induced by machining error is constrained by tolerance specification. Because of the inherent sources of variability, the actual deviations may randomly lie anywhere in the range of permissible variation. However, in most cases the distribution of variation has been found to be a normal distribution [34]. In this paper, the variation is defined as a sequence of mutually uncorrelated, zero-mean, stochastic vectors with the following properties: E [w(k)] = 0





(12)

3.1. Optimal estimation of KPC variation based on Kalman filter

E w(k)w T (j) = Q k ıkj

In machine tool assembly process, adjustment process is carried out by scraping or re-machining to control the variation accumulation. The adjustment value depends on the estimation of

where ıkj is the Kroneker delta function, 0 is a zero vector. (2) Measurement noise dependent on the measurement method, v(k)

Fig. 6. Measurements for the subassembly of bed and column.

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v(k) is assumed to be mutually uncorrelated, zero-mean and determined by the measurement equipment and methods. In this paper, in order to evaluate the measurement process, the variance of measurement noise is replaced by measurement uncertainty. The machining errors of components and measurement uncertainties are uncorrelated. The properties can be expressed as E [(k)] = 0









E (k)T (j) = R k ıkj E

w(k)T (j)

(13)

=O

where O is a zero matrix. (3) Initial values of state vector x(0) and variance matrix of estimated error P(0) In the initial state of machine tool assembly, no part is introduced, the assembly is theoretical an ideal model without geometric errors. The state vector x(0) and variance matrix of estimated error P(0) are given as

P(0) = E



xˆ (0) = E [x(0)] = 0



x(0) − xˆ (0)

T 

x(0) − xˆ (0)

(14)

=O

C(k) and measurement method Rk , yielding to assembly sequence) lead to different variation propagation scenarios, and result in different product qualities. Cost-effectiveness is another critical concern in process planning. It is desirable that the optimal measurement process plan is the one that satisfies the estimation accuracy requirements using relatively imprecise instruments and less KPCs observation to minimize the money and time costs [18]. In assembly process, each step has a set of candidate measurement KPCs and measurement methods. The state space model and Kalman filter provide the capability to assess estimation accuracy for each intermediate setup. This modeling technique can be effectively incorporated into the decision making process for optimal measurement plan. An appropriate measurement process can reduce the uncertainty of final geometric errors, and also lead to a reduction in manufacturing cost. This two-objective optimization problem can be formulated and solved by investigating possible tradeoffs of various factors as follows: objective: min

3.1.2. Optimal estimation of KPC variation based on Kalman filter Based on the state space model and assumptions, the optimal estimation of state vectors in the assembly process without control process can be recursively calculated based on Kalman filter as follows: xˆ (k + 1/k) = A(k + 1, k)ˆx(k/k) T

T

P(k + 1/k) = A(k + 1, k)P(k/k)A (k + 1, k) + F(k + 1, k)Q k F (k + 1, k)





xˆ (k + 1/k + 1) = xˆ (k + 1/k) + K (k + 1) y(k + 1) − C(k + 1)ˆx(k + 1/k) P(k + 1/k + 1) = [I − K (k + 1)C(k + 1)] P(k + 1/k)



K (k + 1) = P(k + 1/k)C T (k + 1) C(k + 1)P(k + 1/k)C T (k + 1) + R k+1

(15)

−1

where k is the kth assembly step, xˆ (k + 1/k) is the one-step optimal prediction of the state vector from the kth step, xˆ (k + 1/k + 1) is an optimal estimation of state vector in the (k + 1)th step, P(k + 1/k) and P(k + 1/k + 1) are the variance matrices of estimation errors, K(k) is the optimal gain matrix. For machine tools, the variation accumulation should be controlled and reduced in assembly process through adjustments. The adjustment process is based on the optimal estimation of state vector xˆ (k). The optimal adjustment process planning has been discussed in our past studies [33]. The variances matrix of estimation errors P(k) is also important. It can ensure the estimation error of state vector xˆ (k) within a determined probability, where x˜ (k) = x(k) − xˆ (k). Therefore, the variance of estimation error is also expected to be in a very small range in assembly process so that it can ensure the reliable estimation of variation accumulation. From Eq. (15), the variance of estimation error is mainly influenced by the variance matrix of input error Qk , the variance matrix of measurement noise Rk and the observation matrix C(k). The variance matrix of input error Qk depends on the probability density function of the deviation of incoming component, which is decided by tolerance design and machining process. The variance matrix of measurement noise Rk mainly depends on the measurement instrument and methods. The observation matrix C(k) indicates which KPCs are measured in the kth step. 3.2. Problem formulation of measurement process planning Final volumetric accuracy of machine tool assembly is affected by the outcome of measurement process planning since the geometric deviation estimation directly influences the adjustment decision to control variation accumulation. Different measurement process planning schemes (selection of observation KPCs

7

 N 

s,C,m



(c(s, C, m))

(16)

k=1

subject to: (s, C, m) ≤ s

(17)

where c is the cost; s, C, m are design variables (the assembly sequence s, the selection of measurement KPCs C and measurement method m);  is the variance of estimation error of state variables;  s is the design requirement. Assembly sequence s should be firstly determined so as to provide candidate measurement KPCs (KPC pairs for machine tools) in each assembly step. One or more KPC pairs C(k) can be selected to be measured in one assembly step. For each measurement KPC pair, one measurement method m is selected to evaluate the angular errors from the possible instruments and methods. Once the instrument and method is selected, the measurement uncertainty for the specific KPC pair (Rk ) can be determined. The costs of instrument and measurement methods c can be seen as the cost of instrument used for each measurement, which is usually related to the precision of the instrument. 3.3. Optimization strategy All the possible assembly sequences can be obtained by using the search method in graph theory. Each assembly sequence is a sequence of part combination in a particular order (Table 2). In each assembly step, an incoming part is assembled to the preceding assembly or two components are combined to a sub-assembly. All the possible combination of parts or components can be collected in a set, called set of component combinations. In machine tool assembly, the relative angular error is measured by a KPC pair. As shown in Fig. 7, all the possible KPC pairs constitute the set of measurement KPC pairs. Giving a component combination, the possible measurement KPC pairs must be the elements of this set. Based on the machine tool structure, three sets (set of assembly sequences, set of component combinations and set of measurement KPC pairs) should be constructed before optimization. Possible measurement methods (instruments) to evaluate the angular error of a KPC pair can be listed based on the practical experiences. Each method has the attributes of measurement uncertainty and cost of use. The resulting optimization process is shown in Fig. 8. There are three searching loops in this measurement process planning. The

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8 Table 2 Assembly sequences of horizontal machining center.

outer loop is the selection of assembly sequences. The middle loop is the possible measurement KPC pairs yielding to the given sequence. The inner loop is the possible measurement methods for KPC pairs. In each circle, giving s, C and m, the cost of measurement c and the variance of estimation error of the state variables  can be calculated. If the uncertainty  goes beyond the requirement  s , this measurement plan should be abandoned. If it meets the requirement, the costs of all the feasible plans should be compared to get the lowest cost. Then the optimal measurement process plan can be obtained. For machine tools, the variance of estimation error is more important than process cost. Therefore, in practice, it is necessary to search the plans of small estimation error under the condition

that the cost is in an acceptable range. Assuming the state vector is  in the form of a normal distribution, x(k)∼N (x(k), P(k)), by using the concept of information entropy, the uncertainty in the state estimation can be measured by the determinant of variance matrix [35]. The judgments in the plan searching can be revised to 1) Uncertainty requirements of specific KPC pairs:  ≤  s 2) Cost requirements: c ≤ cs 3) Search the optimal plan which minimizes the conditional entropy in final assembly: C(k)∗ =

arg min

det(P(k))

(18)

c≤c s ,≤s C(k)

Fig. 7. Measurement KPC pairs and component combinations.

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9

Table 3 Variances of component angular errors.

Variance (× 10−10 rad2 )

A

B

C

D

1.5

1

2

2.5

process planning, depending on quantitative comparison instead of experiences. 4. Example analysis The horizontal machining center shown in Fig. 1 in Section 2.2.3 is also used to illustrate the application of proposed optimization methods. Three linear motion axes (X, Y, Z) and one rotation axis (B) are contained in this machine. KPCs and DFCs are defined as shown in Fig. 5. A state space model is established in the expression of Eq. (10). Each KPC has three rotary degrees of freedom around X, Y and Z respectively. To simply and clearly illustrate the modeling and optimization processes, without loss of generality, five KPCs are selected in the example analysis as shown in Fig. 10. Four basic supporting components are contained in this model: bed (A), column (B), saddle (C) and table (D). The datum flow chain is also illustrated in Fig. 10. Five KPCs which represent the joint surfaces are contained. The angular error between two KPCs in one component is the input error of the assembly. In this example, it is assumed that all the KPCs only have the rotation freedom around X-axis. 4.1. Optimal estimation based on Kalman filter Fig. 8. Searching for optimal measurement process plan of minimum cost.

where, N is the final assembly step. Specific KPC pairs are the sensitive KPC pairs, whose angular errors significantly affect the volumetric error of machine tools. The variation accumulation of specific KPC pairs must be strictly measured and controlled in assembly process. The determinant of variance matrix of estimation error det(P(N)) is an overall evaluation of the uncertainty in final assembly. Two measurement plans can be directly compared by the values of determinant. The flow chart is illustrated in Fig. 9. The proposed methods enable designers to optimize the measurement

The variances of input errors of the supporting components ( A ,  B ,  C ,  D ) are listed in Table 3. The mating surface of A and B (KPC0) is taken as the base coordinate in modeling. The final deviations of the other four KPCs can be expressed as:

⎡

⎤ ⎡ ⎤  B ⎢ ⎥ ⎥ ⎢ ⎢ ı2 (N) ⎥ ⎢ 1 1 0 0 ⎥ ⎢  C ⎥ ⎢ ⎥=⎢ ⎥ ⎥⎢ ⎢ ı (N) ⎥ ⎣ 0 0 1 0 ⎦ ⎢  ⎥ ⎣ 3 ⎦ ⎣ A⎦ ı1 (N)

⎤

⎡

0

ı4 (N)

⎡

1

0

0

0

0

1 1

 D

The state space equation can be written as:

⎤

⎡

⎤

⎡ ⎤ 1 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ı2 (k + 1) ⎥ ⎢ ı2 (k) ⎥ ⎢ 1 1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ı (k + 1) ⎥ = ⎢ ı (k) ⎥ + ⎣ 0 0 1 0 ⎦ w(k) ⎣ 3 ⎦ ⎣ 3 ⎦ ı1 (k + 1)

ı4 (k + 1)

Fig. 9. Search for optimal measurement process plan of minimum uncertainty.

(19)

ı1 (k)

ı4 (k)

0

0

(20)

1 1

Assembly sequence, variance matrix of input error and measurement KPC pairs (observation matrix and measurement uncertainty) in each assembly step are illustrated in Table 4. where ujk is the measurement uncertainty of the KPC pairs j and k, determined by the measurement instrument and method. The measurement uncertainties are all set to 1 × 10−5 rad in this example. Based on Eq. (15), the variances of estimation errors pm of ım (k) in each assembly step can be calculated and illustrated in Fig. 11(b). The variance of estimation error of ım (k) calculation based on the directly sum of variances without measurement is shown in Fig. 11(a), where p1 =  B , p2 =  B +  C , p3 =  A , p4 =  A +  D , assuming that all the angular errors are mutually independent. The variances of the angular errors of each component are listed in Table 3. The variance of the component  is taken into account when it is assembled in the kth step.

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10

Fig. 10. KPCs and DFCs of the horizontal machining center.

Table 4 Assembly sequence of the horizontal machining center. Assembly step k

Step 1

Step 2

Step 3

Step 4

[0, 0, 0,  D ]T Mounting D on part ⎡ ⎤ A 0 0 0 0 ⎣0 0 0 0 ⎦ 0 0 0 0 0 0 0 D

Measurement KPC pairs

Input error w(k) (= (k))

[0, 0,  A , 0]T

[ B ,  C , 0, 0]T

[0, 0, 0, 0]T

Mounting A on the ⎡ ⎤ ground

Assembling B and⎤C ⎡

No newly assembled ⎡ ⎤ part

⎣

⎣0

0

Variance matrix of input error Q(k)

0 0 0 0 0 0 A 0 0 0

⎣0

0 0⎦ 0 0

Measurements  3 to 0

Observation matrices C(k)

Measurement uncertainty R(k)



0 0 1 0

u203



B 0 0 0 0 C 0 0 ⎦ 0 0 0 0 0 0 0 0 ⊥ 1 to 0 ⊥ 2 to 0  2 to 1 

1 0 0 0 0 1 0 0

1u2 −10 0 00  01 0 u202 0 0 0 u212

0 0 0 0 0 0 0

0 0 0 0

0 0⎦ 0 0

⊥ 2 to 4  3 to 4

⊥ 2 to 3



0 1 −1

0





 u223

0 1 0 −1 0 0 1 −1 u224 0

0 u234





Fig. 11. Variances of estimation errors of KPCs in assembly process.

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J. Guo et al. / Precision Engineering xxx (2015) xxx–xxx Table 5 Set of assembly sequences. No.

1 2 3 4 5 6

11

Table 7 Uncertainty and cost of use of measurement methods.

Assembly sequences (combination of components) Step 1

Step 2

Step 3

AB AB BC BC AD AD

ABC ABD AD ABC BC ABD

ABCD ABCD ABCD ABCD ABCD ABCD

Measurement method

Measurement uncertainty (× 10−5 rad)

Cost of use

a b c

1 2 3

3 2 1

From the comparison of the results, it can be seen that the variances of estimation errors are significantly reduced by using Kalman filter. It is an effective approach to evaluate the variation accumulation in assembly process.

4.2. Measurement process planning Possible sequences can be explored based on the adjacent matrix by using graph theory. The assembly sequences of the four components in Fig. 10 are listed in Table 5. The assembly sequence determines the component combination in each assembly step. In each assembly step, according to the assembled components, one or more KPC pairs are selected to be measured. The set of measurement KPC pairs is listed in Table 6. In machine tool assembly, to measure the relative angular errors between two KPCs (generally the perpendicularity and parallelism), several instruments are commonly used. The instruments include: level (bubble or electric), dial gauge with calibrated rules, collimator, laser interferometer, etc. They are different in measurement uncertainty and cost of use. Small measurement uncertainty usually means high cost of use. Geometric measurement uncertainty is specified in the standards (ISO 14253-1, -2, -3) of new generation Geometrical Product Specifications (GPS). The measurement uncertainty of each instrument can be evaluated according to these standards. In this example, the uncertainty and cost of use of the measurement methods are listed in Table 7. Variances of component angular errors are the same as shown in Table 3. Final assembly accuracy is evaluated by 6 relative angular errors between any two KPCs (1, 2, 3 and 4 as shown in Fig. 10). According to the approach proposed in Section 3.3, the variances of estimation errors, costs and determinant det(P(N)) are calculated for each assembly plan (assembly sequence and measurement process). Optimal assembly sequences and measurement plans can be selected according to the judgment of least costs and estimation uncertainty. For example, giving the constraint ( ≤ [0.5, 0.5, 0.8, 0.5, 0.8, 0.9] × 10−10 and cost ≤ 11), four assembly plans are selected in this searching approach as shown in Tables 8 and 9. Exhaustive searching provides us a possible method to compare all the candidate assembly and measurement plans. However, considering more components and possible measurement plans, the workload of calculation will grow exponentially. Principles

Fig. 12. Determinant det( P(3)) of different measurement KPC pairs.

should be summarized to give the designers some guidance in assembly measurement planning. The assembly sequence shown in Table 10 is selected for a detail discussion. Part A is firstly mounted on the ground. The angular error 3–0 is measured. Part B is assembled on A and KPC pairs 1–0 and 3–0 are measured. Then component D is assembled and all the possible measurement plans are evaluated. Two questions are expected to be answered: How many KPC pairs need to be measured in one assembly step? Which KPC pair should be measured firstly? With the increase of measurement KPC pairs, the determinant det(P(3)) (indicates the uncertainty of estimation in step 3) varies as shown in Fig. 12. From Fig. 12, it can be seen that the determinant det(P(3)) decreases with the increase of measurement KPC pairs. The estimation error covariances of relative angular error of KPCs 1, 3, 4 (Cov(1,3), Cov(1,4), Cov(3,4)) in different measurement plans are shown in Fig. 13. The estimation error covariance of angular error Cov(1,3) is mostly smaller than Cov(3,4) and Cov(1,4). This is because KPC1 and KPC3 are respectively measured in the first and second step (3–0 and 1–0) and 3–4 is only affected by the deviation of component D while 1–4 has the longest DFC through part A, B and D. In the subfigure, only one KPC pair is selected to be measured in the third assembly step. The estimation error covariances of KPC pairs 4–0 and 1–4 are much smaller than the other plans. This is because the deviation information of the incoming component D in step 3 is not obtained from the measurement of KPC pairs 1–0, 3–0 and 1–3. However, the measurement plan of KPC pair 1–3 is

Table 6 Set of measurement KPC pairs. Component combination

A

B

C

D

AB

BC

AD

ABD

ABC

ABCD

Measurement pairs

0–3

0–1

1–2

3–4

1–3

0–1 0–2 1–2

0–3 0–4 3–4

1–3 1–4 3–4

1–2 1–3 2–3

1–2, 1–3 1–4, 2–3 2–4, 3–4

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12 Table 8 Assembly and measurement plans satisfy given constraints. No.

Step 1

Step 2

Step 3

Step 4

0–3 (c) 0–3 (b)

1–3 (a) 1–3 (b)

1–2 (a), 1–3 (c), 2–3 (c) 1–2 (a), 1–3 (c), 2–3 (c)

1–4 (c), 2–4 (c) 1–4 (c), 2–4 (c)

0–1 (c) 0–1 (c)

0–1(c), 1–2 (c) 0–2(c), 1–2 (c)

1–3 (a), 2–3 (a) 1–3 (a), 2–3 (a)

1–4 (c), 2–4 (c) 1–4 (c), 2–4 (c)

Assembly sequence

I II

Measurement plan Measurement plan

Assembly sequence

III IV

Measurement plan Measurement plan

Table 9 Estimation errors of satisfy plans. Assembly and measurement plan

Cost

Determinant det(P(N))

I II III IV

11 11 11 11

0.038 0.042 0.037 0.034

Covariance of estimation error (1 × 10−10 rad2 )  1–2

 1–3

 1–4

 2–3

 2–4

 3–4

0.410 0.424 0.420 0.388

0.381 0.463 0.441 0.447

0.776 0.795 0.790 0.779

0.470 0.497 0.490 0.466

0.639 0.639 0.639 0.634

0.796 0.819 0.813 0.807

Table 10 One assembly process of horizontal machining center. Step 1

Step 2

Step 3

Assembly sequence

Measurement KPC pairs

...

...

3–0

3–0, 1–0

Candidate KPC pairs: 1–0, 3–0, 4–0, 1–3, 1–4, 3–4

...

Fig. 13. Estimation error covariances of relative angular error.

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better than plans 1–0 and 3–0, since KPC pairs 1–0 and 3–0 have been measured in the first and second steps, no “new information” is introduced. Plans 4–0, 1–4 and 3–4 include the evaluation of the incoming component D. Among these three plans, plan 1–4 has the longest DFC through components A, B and D, leading to the least uncertainty and can be considered as the best measurement plan. Therefore, in the measurement process planning, KPC pair having the new assembled part and the longest DFC should be selected to measure. It can be inferred from Fig. 12 that with the increase of measurement pairs, the decreasing rate of uncertainty becomes even and smooth. Two or three KPC pairs are recommended in consideration of time and working cost in measurement. The same result is showed in Table 8, where the optimization measurement plans usually contain 1–3 KPC pairs in one assembly step. 5. Conclusion Combining the observability concept in control theory and the engineering knowledge about machine tool assembly, a new and practical measurement process planning is developed by using Kalman filter, which is based on the state space modeling of variation propagation in assembly process. According to the discussion of modeling and searching methods and the numerical analysis results presented in this paper, the following conclusions can be drawn: (1) State space model is a compliance effective modeling method to describe the variation propagation in machine tool assembly. The observability and controllability characteristics in state space model are of interest for an innovative measurement and adjustment process planning, which are the key issues for the assurance of machine tool precision. (2) Based on state space model, the optimal estimation method by using Kalman filter is developed to evaluate the variation accumulation in assembly process, which can also be used to identify variation sources and determine adjustment strategy. As an initial attempt to extend the application of optimal estimation toward measurement process planning, this paper suggests an optimization strategy which is based on Kalman filter and evaluated by using the concept of information entropy. The cost of use and measurement uncertainties of instruments are considered and all the candidate KCs are searched to obtain an optimal measurement process plan with least cost, estimation error and uncertainty. (3) The measurement process planning of an actual horizontal machining center is conducted to validate the proposed method. Some guidance for the measurement process planning is proposed based on the numerical analysis results. The paper assumes that the components are rigid. Part deformation, form error and other factors are not considered in the state space modeling. More accurate estimation models and evaluation of measurement cost and uncertainty will further reduce the estimation error and make the process planning more reliable. The improvement of variation propagation model in theory and the accumulation of measurement data in practice will be involved in our future work. Acknowledgments The work reported in this paper is supported by the National Natural Science Foundation of China [51421004, 51135004] and the National High-Tech Research and Development Program of China [2012AA040701].

13

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