Proceedings of ICAM2005: 2005 International Conference on Advanced Manufacture Nov 28 - Dec 2, Taipei, Taiwan
3D Model-Based Design for Control of a Mechatronic Machine Tools System Chin-Yin Chen and Chi-Cheng Cheng Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University 70, Lien-hai road. Kaohsiung, 804 Taiwan, R.O.C. Email:
[email protected];
[email protected]
Keywords: machine tools; integrated structure and control design; design for control; multilevel decomposition Abstract. The goal of this research is to develop a design and optimization methodology for the mechatronic modules of machine tools by treating all important characteristics from all involved engineering domains in one single process. In this study, a mechatronic system of machine tools was broken into a structure and control two-level systems. In the first stage for structure design process, the Pro/E was used to build up the 3D models and the AnSys was employed to design the mechanical structure and select the optimal components for the machine tools. Next, in the control design process, a common controller type was designed by MATLAB in this stage. Then, three important parameters were established for the machine tools design to achieve the overall system performance. Introduction The traditional mechatronic systems design process can be formulated as Fig. 1(a) shown. Let given design group and passed to the next group, who uses the previous group’s output variables as input variables. This “over the wall” design approach is intuitive to implement, but lacks the sophistication, ingenuity, and efficiency required by the high-technology designs in this new century. Because of this reason, the serial design approach is slowly becoming obsolete in favor of a methodology known as integrated design. The integrated design concept and method were applied to the aerospace industry in early years. In 1985, Hale [1], and Bodden and Junkins [2] separately proposed rest-to-rest maneuver algorithm and eigenvalue optimization algorithms in view of the structure and controller design for flexible spacecrafts. In 1994 and 1996, Asada [3,4] focused on the structure and controller design problem. An optimization scheme with a recursive experiment analysis method was presented to tune the structure and controller parameters in this study. Integrated design is based on a multidisciplinary team organized on a project basis. The same group is in charge of the complete design process, from the definition of the limits of the system under study and from its objectives to the implementation of control laws on prototypes. This process includes modeling, analysis, architectural choice of the sensors and actuators and safety consideration. To achieve success, a crucial factor is the choice of an applicable model during all the steps of this design procedure. The model will need to be at the same time a knowledge model
allowing real physical insight and a representation model for controller design and validation. In this study, the detailed and quickly integrated mechatronic design method was formulated as Fig. 1(b) shown, and used to design feed drive system of machine tools. The process and results will be presented in the following sections. Project objective
Project objective
Mechanical design
Mechanical Design (Pro/E, AnSys)
Controller design
Not ok
Performance test
Not OK
System parameters
Controller Design (Matlab)
Controller paprmeters
Performance test
Integrated design methodology
OK
Ok
Finish
Finish
(a) Traditional design process (b) Concurrent design process Fig. 1 Design process for a mechatronic system. Integrated Design for Multidisciplinary System With a multilevel decomposition approach [5], considering the structure and control two-level problem for a mechatronic system, the multilevel decomposition procedure can be written as below. In the structure level, min . Y Ni ( X N ), i = 1,...., n N s.t. g Nk ( X N ) ≤ 0, k = 1,...., nc N NDV N
∂Y Rj
i =1
∂X Ni
∑
∆X Ni ≤ ε 2 j , j = 1,......, n R
(1)
X NiL ≤ X Ni ≤ X UNi , i = 1,......., NDV N * X RjL ≤ X Rj +
NDV N
* ∂X Rj
i =1
∂X Ni
∑
∆X Ni ≤ X URj , j = 1,..., NDV R
.
where YN and YR are the objective function vectors at the structure level and the control level, respectively; g N is the corresponding constraint vectors; X N and X R are the corresponding design variable vectors, ε 2 j is a tolerance on the change in the jth objective of control level during optimization at the structure level; L and U are lower and upper bounds of design vectors, ∂YRj* ∂X Ni and ∂X Rj* ∂X Ni represent the optimal sensitivity parameters of the control level objective function and design variable vectors, respectively, with respect to the structure level design variables. n N and n R denote numbers of objective functions for each level; nc N is the number of constraint for structure level; NDVN and NDV R are numbers of design variables for the structure and the control levels.Similarly, the process of control level becomes *
min YRj ( X N , X R ), j = 1,..., nR *
s.t. g Rk ( X N , X R ) ≤ 0, k = 1,..., ncR X
L Ri
≤ X Ri ≤ X , i = 1,..., NDVR U Ri
.
(2)
where g R is the corresponding constraint vectors; X N * is the optimum design variable vector from the structure level and must be fixed during optimization at the control level. In 2000, Li [6] proposed the “Design for Control” to design a mechatronic system. In that study, she not only broke whole mechatronic parameters into control parameters and structure parameters, but also defined the performance error vectors and control power into the optimal design method. So the mechatronic integrated design problem from Eq. (1) and Eq. (2) can be described by the following models. Let E = Y − Yd be the performance error vector representing the discrepancy between the desired behavior variable vector Yd and the actual behavior variable vector Y . min s.t. Y
I (E a ,W ) I R_L
< FRI ( X R , X N ) < Y RI _ U
Y NI _ L < FNI ( X N ) < Y NI _ U
.
(3)
Y RE = FRE ( X R , X N ) Y RE = FRE ( X N )
where E a = min ∑ α i E N i + β i E R i
i
and W = min ∑ γ iτ i . E a is total absolute behavior error for both i
YN and YR ; W is total power generated by actuators in the system; α i , β i , γ i is weighting factors
Design of a Mechatronic Feed Drive System A typical feed drive system in machine tools is shown in Fig. 2, and the design specifications: max speed of the table; micro-motion sensitivity; position lose motion (under cutting); bandwidth of position loop and max cutting force are must achieve 40 m/min, 1 µm, 0.01 mm, 15 Hz and 100 Kgf, respectively. Where the position lose motion is defined as the position deviation caused by material deformation due to cutting force. 1 S
Velocity loop
Position Loop
Command
Fig. 2 Typical mechatronic feed drive system. Mechanical structure design. The structure of machine tools can be broken into the base, saddle, and table. The structure of a machine tool should be able to support the heavy weight and withstand the cutting action. Therefore, a precision machine has to be carefully designed to avoid the undesirable deflection and vibration. Based on the design process described before, the mechanical structure of an experimental feed drive system is designed as shown in Fig. 3(a). The masses of the base, the saddle, and the table are 570 Kg, 140 Kg and 60 Kg, respectively. And the result of first resonance frequency for mechanical structure is 378.8 Hz as shown in Fig. 3(b). Mechanism components design. In order to analyze dynamic performance of the feed drive system, a multi-body model of the feed drive system is established as illustrated in Fig. 4. In this figure, parameters are defined as below: T : motor torque (N-m), J m : rotor inertia (Kg-m2), J s : ball screw inertia (Kg-m2), J : total inertia J m + J s , M t : table mass (Kg), θ m : motor shaft angle
(rad), X t : table position (m), K s : longitudinal stiffness of ball screw (N/m), K b : longitudinal stiffness of support bearing (N/m), K n : longitudinal stiffness of nut (N/m), K all : overall stiffness of the feed drive system (N/m), (1 / K s + 1 / K b + 1 / K n ) −1 , Ct : damping coefficient of the guideway (N-s/m), p : lead of ballscrew (m), R : transformation ratio p / 2π (m/rad).
(a) Major mechanical structure (b) The first resonance mode of the mechanical structure Fig. 3 Characteristics of major mechanical structures for a feed drive system. Xt
Rθ m Mt K all
J m , θm
Ct
T
Fig. 4 Dynamic model of the feed drive system. In the mechanism components design process, the nut and support bearing is dependent on the screw diameter. Therefore, considering the screw diameter is the same as considering the system stiffness. And the range of the diameter for the ball screw d can be determined by the critical speed, buckling load and DN value. From those conditions, the objective of the mechanical design parameters I N for feed drive system can be written as below and the results are listed in Table 1. min I N = E N = (1− 5 − s.t.
Fc 2 ) K all
−3
.
(4)
−3
8 × 10 m < d < 35 × 10 m
where FC is cutting force.
Controller design. A two-inertia block diagram of the feed drive system is depicted in Fig. 5(a). The corresponding transfer function from the motor torque to table position can be calculated as: Xt ω 2 mech R . = 2 2 2 T Js (s + ω a )
(5)
Where ω mech = K all / M t and ω a = K all (1 / M t + R 2 / J ) represent the anti-resonance and resonance frequencies, respectively. As shown in Fig. 5(b), the closed loop transfer function from the table reference speed to actual table speed can be expressed by Eq. 6. where K p is the proportional feedback gain and K i is the integral feedforward gain. 2 X& t K i ω mech . = 2 2 2 2 Js ( s + K p / J s + ω a2 ) + K p ω mech s + K i ( s 2 + ω mech ) X& tr
(6)
According to the standard 4th order system, the closed loop controller gain can be arranged as Eq. 7, and the design results are listed in Table 1. In addition, the constraints can be formulated 2 2 2 2 as ω a2 (ω12 + ω 22 + 4ξ1ω1ξ 2ω 2 ) − ω12ω 22 = ω mech ω all and ω1ξ1 (ω 22 − ω mech ) = ω 2ξ 2 (ω mech − ω12 ) . ω12 ω 22 J. 2 ω mech
K p = 2(ξ 1ω1 + ξ 2 ω 2 ) J ; K i =
(7)
2 inertia model
Servo controller
X tr R X tr 1 θ mr R + -
K pp
+
Ki S
-
T +
-
1 S
1 J mS
θm
1/R
θ mr
K pp
+
ω mr +
R
+
-
K all
1 Mt S
1 S
-
Ki s
T
+
-
s 2 + ω 2 mech 2 Js ( s 2 + ω a )
ωm
1 s
θm
R ω 2 mech s 2 + ω 2 mech
Xt
Xt
Kp Kp
Speed loop
Position loop
(a) Two inertia system (b) Control loop for feed drive system Fig. 5 Block diagram of the feed drive system. After the speed controller design is accomplished, design for the position loop gain can be preceded. As shown in Fig. 5(b), the only proportional gain for the position loop is K pp . Consequently, the design objective I Rθ can be expressed as an ITAE (Integral Time Absolute Error) criterion, and constraints can be summarized as below and the resulting design parameters for X axis and Y axis are shown in Table 1. I Rθ = min ∫ t eθ (t ) dt s.t T (t ) ≤ 14.4 eθ ≥ 0
.
(8)
where eθ = θ mr − θ m and T (t ) = K i ∫ (eθ K pp − ω m )dt − K p ω m , which are the tracking error of motor angle and the control torque, respectively. Table 1 Optimal design results Mechanical parameters M t (Kg) 2 J m (Kg-m )
First integrated design step
Second integrated design step Variation (%)
X axis
Y axis
X axis
Y axis
160 13e-4
300 13e-4
160 13e-4
275 13e-4
-
-8.3 -
J s (Kg-m )
6.4e-4
6.4e-4
6.4e-4
6.4e-4
-
-
K s (N/m)
2.81e8 4.27e8 1e9 20/π
2.81e8 4.27e8 1e9 20/π
2.81e8 4.27e8 1e9 20/π
2.81e8 4.27e8 1e9 20/π
-
-
2
K b (N/m) K n (N/m) R (mm/rad)
X axis Y axis
Controller parameters Kp
1.72
1.61
1.72
1.55
-
-3.7
Ki
370
207.65
370
215.40
-
+3.7
K pp
65 16
48 12.5
65 16
55 15.3
-
+14.5 +22
Bandwidth(Hz)
Design modification for the Y axis by Design for Control. From Table 1, it appears that the position loop bandwidth of the Y axis is unable to achieve the design specification. Therefore, an integrated design steps needs to be performed. According to Eq. 6 and “Design for Control” methodology, we will aim at the parameter M t and to seek for other better solutions. In the Y axis of the feed drive system, the parameter M t represents the overall mass of the moving parts, which include table mass and saddle mass, and the shape of the saddle for the first design result is shown in Fig. 6(a). The value of M t directly affects the physical properties of the saddle. The saddle is therefore modified to a less weight model as depicted in Fig. 7(b) without reducing the structural natural frequency. As a result, the mass of saddle can be decreased to 25 Kg and the first resonance frequency of the saddle is increased to 513 Hz from 490 Hz. After the saddle modification, a set of new controller parameters can be found by employing (7) and (8). The final design parameters are also shown in Table 1. That bandwidth of the position loop is increased from 12.5 Hz up to 15.3 Hz and satisfies design requirement.
(a) First integrated design step (b) Second integrated design step Fig. 6 Structural model and dynamic performance of the saddle. Conclusions The structure and controller integrated design of feed drive system for machine tools, which based on 3D model design for control is presented in this paper. In this approach, the 3D model and FEM method was applied to assist structure performance in detail and the embedded controller also help to improvement the system performance by structure parameters modification. During the integrated design process, the aim is not only to complete a final design, but also identify the performance-limiting factors of the design proposal and to choose satisfactory specifications for these factors. References [1] [2] [3] [4] [5] [6]
A.L. Hale, R.J. Lisowski, and W.E. Dahl: J. of Guidance Controller and Dynamics. Vol. 8 (1985), p.86. D.S. Bodden and J.L. Junkins: J. of Guidance Controller and Dynamics. Vol. 8 (1985), p. 697. J.H. Park and H. Asada: ASME J. dyn. syst. meas. control. Vol. 116 (1994), p. 344. A.C. Pil and H. Asada: IEEE/ASME Transactions on Mechatronics. Vol. 1 (1996), p. 191. A. Chattopadhyay and N. Pagaldipti: Computers Math. Applic. Vol. 29 (1995), p.55. Q. Li, W.J. Zhang, and L. Chen: IEEE/ASME Transactions on Mechatronics. Vol. 6 (2001), p.161.
