Open access peer-reviewed chapter

FPGA-Realization of a Motion Control IC for X-Y Table

By Ying-Shieh Kung and Ting-Yu Tai

Published: January 1st 2010

DOI: 10.5772/6977

Downloaded: 3542

1. Introduction

The development of a compact and high performance motion controller for precision X-Y table, CNC machine etc. has been a popular field in literature (Goto et al., 1996; Wang & Lee, 1999; Hanafi et al., 2003). In position control of X-Y table, there are two approaches to be considered. One is semi closed-loop control and the other is full closed-loop control. The full closed-loop control with feed-backed by a linear encoder as the table position signal has a better positioning performance than the semi closed-loop control that a rotary encoder attached to PMSM is feed-backed as the position signal. However, to develop a motion control IC for X-Y table, the fixed-point digital signal processor (DSP) and FPGA provide two possible solutions in this issue. Compared with FPGA, DSP suffers from a long period of development and exhausts many resources of the CPU (Zhou et al., 2004).

For the progress of VLSI technology, the FPGA has been widely investigated due to its programmable hard-wired feature, fast time-to-market, shorter design cycle, embedding processor, low power consumption and higher density for implementing digital control system (Monmasson & Cirstea, 2007; Naouar et al., 2007; Jung & Kim, 2007). FPGA provides a compromise between the special-purpose ASIC (application specified integrated circuit) hardware and general-purpose processors (Wei et al., 2005). Therefore, using an FPGA to form a compact, low-cast and high performance servo system for precision machine has become an important issue. However, in many researches, the FPGA is merely used to realize the hardware part of the overall control system. Recently, fuzzy control has been successfully demonstrated in industrial control field (Sanchez-Solano et al., 2007; Kung & Tsai, 2007).Compared with other nonlinear approaches, FC has two main advantages, as follows: (1) FC has a special non-linear structure that is universal for various or uncertainty plants. (2) the formulation of fuzzy control rule can be easily achieved by control engineering knowledge, such as dynamic response characteristics, and it doesn’t require a mathematical model of controlled plant. In literature, Li et al. (2003) utilized an FPGA to implement autonomous fuzzy behavior control on mobile robot. Lin et al. (2005) presented a fuzzy sliding-mode control for a linear induction motor drive based on FPGA. But, due to the fuzzy inference mechanism module adopts parallel processing circuits, it consumes much more FPGA resources; therefore limited fuzzy rules are used in their proposed method. To solve this problem, a FSM joined by a multiplier, an adder, a LUT (Look-up table), some comparators and registers are proposed to model the FC algorithm of the PMSM drive system. Then a VHDL is adopted to describe the circuit of the FSM (Hsu et al., 1996). Due to the FSM belongs to the sequential processing method; the FPGA resources usage can be greatly reduced. Further, in recent years, an embedded processor IP and an application IP can now be developed and downloaded into FPGA to construct a SoPC environment (Altera, 2004), allowing the users to design a SoPC module by mixing hardware and software in one FPGA chip (Hall & Hamblen, 2004). The circuits required fast processing but fixed computation are suitable to be implemented by hardware in FPGA, and the heavy computation or complicated processing can be realized by software in FPGA (Kung et al., 2004; Kung & Shu, 2005). The results of the software/hardware co-design increase the programmability, flexibility of the designed digital system, enhance the system performance by parallel processing and reduce the development time.

To exploit the advantages, a motion control IC for X-Y table based on the new-generation FPGA technology is developed in this study and shown in Fig.1 (Kung et al., 2006), which the scheme of position/speed/current vector control of two PMSMs can be realized by hardware in FPGA, and the motion trajectory for X-Y table can be realized by software using Nios II embedded processor. Hence, all functionalities, which are based on software/hardware co-design, required to construct a full closed-loop control for X-Y table can be integrated and implemented in one FPGA chip. In addition, the FPGA resources usage can be greatly reduced by using the FSM in the control algorithm design. Herein, the Altera Stratix II EP2S60F672C5ES (Altera, 2008), which has 48,352 ALUTs (Adaptive Look-UP Tables), maximum 718 user I/O pins, total 2,544,192 RAM bits, and a Nios II embedded processor which has a 32-bit configurable CPU core, 16 M byte Flash memory, 1 M byte SRAM and 16 M byte SDRAM, are used. Finally, an experimental system included by an FPGA experimental board, two inverters, two sets of A/D converter and an X-Y table, is set up to verify the correctness and effectiveness of the proposed FPGA-based motion control IC.

Figure 1.

The architecture of the FPGA-based motion control system for X-Y table.


2. System description and controller design of X-Y table

The X-Y table is driven by two PMSMs which the current, speed and position loop in each PMSM drive adopts vector control, P control and fuzzy control, respectively The architecture of the proposed FPGA-based motion control IC for X-Y table is shown in Fig. 1. The modeling of PMSM, the fuzzy control algorithm and the motion trajectory planning are introduced as follows:

2.1. Mathematical model of PMSM and current vector controller

The typical mathematical model of a PMSM is described, in two-axis d-q synchronous rotating reference frame, as follows

diqdt=ωeLdLqidRsLqiqωe λf Lq+1LqvqE2

where v d, v qare the d and q axis voltages; id, iq, are the d and q axis currents, Rs is the phase winding resistance; Ld, Lq are the d and q axis inductance;ωeis the rotating speed of magnet flux;λfis the permanent magnet flux linkage.

The current loop control of PMSM drive in Fig.1 is based on a vector control approach. That is, if the id is controlled to 0 in Fig.1, the PMSM will be decoupled and controlling a PMSM like to control a DC motor. Therefore, after decoupling, the torque of PMSM can be written as the following equation,

Te=3P4λfiqΔ__ KtiqE3



Finally, considering the mechanical load with linear table, the overall dynamic equation of linear table system is obtained by


where Teis the motor torque, Ktis force constant, Jmis the inertial value, Bmis damping ratio, TLis the external torque, sprepresents the displacement of X-axis or Y-axis table and ris the lead of the ball screw.

The current loop of the PMSM drive for X- or Y-table in Fig.1 includes two PI controllers, coordinate transformations of Clark, Modified inverse Clark, Park, inverse Park, SVPWM (Space Vector Pulse Width Muldulation), pulse signal detection of the encoder etc. The coordination transformation of the PMSM in Fig. 1 can be described in synchronous rotating reference frame. Figure 2 is the coordination system in rotating motor which includes stationary a-b-cframe, stationary - frame and synchronously rotating d-qframe. Further, the formulations among three coordination systems are presented as follows.

  1. 1.Clarke: stationary a-b-cframe to stationary - frame.

  1. 2. ModifiedClarke1: stationary - frame to stationary a-b-cframe.

  1. Parkd-q
  1. 4.Park1: rotating d-qframe to stationary - frame.


whereθeis the electrical angle.

In Fig. 1, two digital PIcontrollers are presented in the current loop of PMSM. For the example in dframe, the formulation is shown as follows.

vp_d(k)=kp_d ed(k)E11
vi_d(k)=vi_d(k1)+ki_d ed(k1)E12
theedis the error between current command and measured current. Thekp_d,ki_dare Pcontroller gain and Icontroller gain, respectively. Thevp_d(k),vi_d(k),vd(k)are the output of Pcontroller only, Icontroller only and the PIcontroller, respectively. Similarity, the formulation of PIcontroller in qframe is the same.

2.2. Fuzzy controller (FC) for position control loop

The position controllers in X-axis and Y-axis table of Fig. 1 adopt fuzzy controller, which includes fuzzification, fuzzy rules, inference mechanism and defuzzification. Herein, an FC design method for X-axis and Y-axis table is presented. At first, position error and its error change,e,deare defined by

Figure 2.

Transformation between stationary axes and rotating axes.


The Kerand Kderare the gains of the input variableseandde, respectively, as well as ufis the output variables of the FC. The design procedure of the FC is as follows:

  1. a. Take the Eand dEas the input linguist variables, which are defined by {A0, A1, A2, A3, A4, A5, A6} and {B0, B1, B2, B3, B4, B5, B6}, respectively. Each linguist value of Eand dEare based on the symmetrical triangular membership function which is shown in Fig.3. The symmetrical triangular membership function are determined uniquely by three real numbersξ1ξ2ξ3, if one fixesf(ξ1)=f(ξ3)=0andf(ξ2)=1. With respect to the universe of discourse of [-6.6], the numbers for these linguistic values are selected as follows:

A0=B0: {6,6,4}, A1=B1: {6,4,2},A2=B2: {4,2,0},A3=B3: {2,0,2},A4=B4: {0,2,4},A5=B5: {2,4,6},A6=B6: {4,6,6}E16
  1. b. Compute the membership degree of eand de. Figure 3 shows that the only two linguistic values are excited (resulting in a non-zero membership) in any input value, and the membership degreeμAi(e)can be derived, in which the error eis located between eiand ei+1, two linguist values of Aiand Ai+1are excited, and the membership degree is obtained by

μAi(e)=ei+1e2   and  μAi+1(e)=1μAi(e)E17

whereei+1Δ__6+2*(i+1). Similar results can be obtained in computing the membership degreeμBj(de).

Figure 3.

Fuzzification, fuzzy rule table, fuzzy inference and defuzzication.

  1. c. Select the initial fuzzy control rules by referring to the dynamic response characteristics (Liaw et al., 1999), such as,

IF  e  is Ai  and  Δe is Bj   THEN  uf  is  cj,iE18

where iand j =0~6, Aiand Bjare fuzzy number, and cj,iis real number. The graph of fuzzification and fuzzy rule table is shown in Fig. 3.

  1. d. Construct the fuzzy system uf(e,de) by using the singleton fuzzifier, product-inference rule, and central average defuzzifier method. Although there are total 49 fuzzy rules in Fig. 3 will be inferred, actually only 4 fuzzy rules can be effectively excited to generate a non-zero output. Therefore, if the error eis located between eiand ei+1, and the error change deis located between dejand dej+1, only four linguistic values Ai, Ai+1, Bj, Bj+1and corresponding consequent values cj,i, cj+1,i, cj,i+1, cj+1,i+1can be excited, and the (18) can be replaced by the following expression:

uf(e,de)=n=ii+1m=jj+1cm,n[μAn(e)*μBm(de)]n=ii+1m=jj+1μAn(e)*μBm(de) Δ__ n=ii+1m=jj+1cm,n*dn,mE19

wheredn,mΔ__ μAn(e)*μBm(de). And thosecm,ndenote the consequent parameters of the fuzzy system.

2.3. Motion trajectory planning of X-Y table

The point-to-point, circular and window motion trajectories are usually considered to evaluate the motion performance for X-Y table.

  1. a. In point-to-point motion trajectory, for smoothly running of the table, it is designed with the trapezoidal velocity profile and its formulation is shown as follows.

s(t)={12At2+s0                                 0ttavm(t-ta)+s(ta)                           tattd -12A(t-td)2+vm(t-td)+s(td)     tdtts    E20

Where 0<t<tais at the acceleration region, ta<t<tdis at the constant velocity region, and td<t<tsis at the deceleration region. The srepresents the position command in X-axis or Y-axis table; Ais the acceleration/deceleration value; s0is the initial position; vmis the maximum velocity; ta, tdand tsrepresents the end time of the acceleration region, the start time of the deceleration region and the end time of the trapezoidal motion, respectively.

  1. b. In circular motion trajectory, it is computed by

xi=r sin(θi)E21
yi=r cos(θi)E22
withθi=θi1+Δθ. WhereΔθ,r,xi,yiare angle increment, radius, X-axis trajectory command and Y-axis trajectory command, respectively.
  1. c. The window motion trajectory is shown in Fig.4. The formulation is derived as follows:

           atrajectory :xi=xi1,yi=S+yi1btrajectory :(θi:64π2π, and θi=θi1+Δθ)E23
        ctrajectory :xi=S+xi1,yi=yi1dtrajectory :(θi:π64π, and θi=θi1+Δθ)E25
         etrajectory :xi=xi1,yi=S+yi1ftrajectory :(θi:12ππ, and θi=θi1+Δθ)E27
         gtrajectory :xi=S+xi1,yi=yi1htrajectory :(θi:012π, and θi=θi1+Δθ)E29
itrajectory :xi=xi1,yi=S+yi1E31

whereS,Δθ,xi,yiare position increment, angle increment, X-axis trajectory command and Y-axis trajectory command, respectively. In addition, the(Ox1,Oy1),(Ox2,Oy2),(Ox3,Oy3),(Ox4,Oy4)are arc center of b-, d-, f-, and h-trajectory in the Fig. 4 and r is the radius. The motion speed of the table is determined byΔθ.

Figure 4.

Window motion trajectory.

3. Design of an FPGA-based motion control IC for X-Y table

The architecture of the proposed FPGA-based motion control IC for X-Y table is shown in Fig. 1, in which the motion trajectory is implemented by software using Nios II embedded processor and the current vector controller, the position and speed controller for two PMSMs are implemented by hardware in FPGA chip. However, in this section, we firstly introduce the concept of finite state machine (FSM). Then use FSM to design the complicated control algorithm, such as the FC and the vector controller in PMSM drive.

3.1. Finite state machine (FSM)

To reduce the use of the FPGA resource, FSM is adopted to describe the complicated control algorithm. Herein, the computation of a sum of product (SOP) shown below is taken as a case study to present the advantage of FSM.


Two kinds of design method that one is parallel processing method and the other is FSM method are introduced to realize the the computation of SOP. In the former method, the designed SOP circuit is shown in Fig. 5(a), and it will operate continuously and simultaneously. The circuit needs 2 adders and 3 multipliers, but only one clock time can complete the overall computation. Although the parallel processing method has fast computation ability, it consumes much more FPGA resources. To reduce the resource usage in FPGA, the designed SOP circuit adopted by using the FSM method is proposed and shown in Fig. 5(b), which uses one adder, one multiplier and manipulates 5 steps (or 5 clocks time) machine to carry out the overall computation of SOP. Although the FSM method needs more operation time (if one clock time is 40ns, the 5 clocks needs 0.2μs) than the parallel processing method in executing SOP circuit, it doesn’t loss any computation power. Therefore, the more complicated computation in algorithm, the more FPGA resources can be economized if the FSM is applied. Further, VHDL code to implement the computation of SOP is shown in Fig.6

Figure 5.

Computation of SOP by using (a) parallel operation (b) FSM operation.

Figure 6.

VHDL code to implement the computation of SOP.

3.2. Design of an FPGA-based motion control IC for X-Y table

The internal architecture of the proposed FPGA-based motion control IC for X-Y table is shown in Fig. 7. The FPGA is used by Altera Stratix II EP2S60 and a Nios II embedded processor can be downloaded into FPGA to construct an SoPC environment. The Altera Stratix II EP2S60 has 48,352 ALUTs (Adaptive Look-UP Tables), maximum 718 user I/O pins, total 2,544,192 RAM bits, and Nios II embedded processor is a 32-bit configurable CPU core, 16 M byte Flash memory, 1 M byte SRAM and 16 M byte SDRAM. A custom software development kit (SDK) consists of a compiled library of software routines for the SoPC design, a Make-file for rebuilding the library, and C header files containing structures for each peripheral. The motion control IC, which is designed in this SoPC environment, comprises a Nios II embedded processor IP and an application IP. The application IP implemented by hardware is adopted to realize two position/speed/current vector controllers of PMSMs and two QEP circuits of linear encoder. The circuit of each current vector controller includes a current controller and coordinate transformation (CCCT), SVPWM generation, QEP detection and transformation, ADC interface, etc. The speed loop uses P controller and the position loop adopts FC. The sampling frequency of the position control loop is designed with 2kHz. The frequency divider generates 50 Mhz (Clk), 25 Mhz (Clk-sp), 16 kHz (Clk-cur), and 2 kHz (Clk-po) clock to supply all circuits in Fig. 7.

Figure 7.

Internal circuit of the proposed FPGA-based motion control IC.

The internal circuit of CCCT performs the function of two PI controllers, table look-up for sin/cosfunction and the coordinate transformation for Clark, Park, inverse Park, modified inverse Clarke. The CCCT circuit designed by FSM is shown in Fig. 8, which uses one adder, one multiplier, an one-bit left shifter, a look-up-table and manipulates 24 steps machine to carry out the overall computation. The data type is 12-bit length with Q11 format and 2’s complement operation. In Fig. 8, steps s0~s1 is for the look-up sin/costable; steps s2~s5 and s5~s8 are for the transformation of Clarke and Park, respectively; steps s9~s14 is for the computation of d-axis and q-axis PI controller; and steps s15~s19 and s20~s23 represent the transformation of the inverse Park and the modified inverse Clarke, respectively. The operation of each step in FPGA can be completed within 40ns (25 MHz clock); therefore total 24 steps need 0.96 s operation time. Although the FSM method needs more operation time than the parallel processing method in executing CCCT circuit, it doesn’t loss any control performance in overall system because the 0.96 s operation time is much less than the designed sampling interval, 62.5 s (16 kHz) of current control loop in Fig. 1. To prevent numerical overflow and alleviate windup phenomenon, the output values of I controller and PI controller are both limited within a specific range.

Figure 8.

Designed CCCT circuit inFig. 7.

An FSM is employed to model the FC of the position loop and P controller of the speed loop in PMLSM and shown in Fig. 9, which uses one adder, one multiplier, a look-up table, comparators, registers, etc. and manipulates 23 steps machine to carry out the overall computation. With exception of the data type in reference model are 24-bits, others data type are designed with 12-bits length, 2’s complement and Q11 format. Although the algorithm of FC is highly complexity, the FSM can give a very adequate modeling and easily be described by VHDL. Furthermore, steps s0~s2 are for the computation of speed, position error and error change; steps s3~s6 execute the function of the fuzzification; s7 describes the look-up table and s8~s16 defuzzification; and steps s17~s22 execute the computation of speed and current command output. The SD is the section determination of eand de, and its flow chart of circuit design is shown in Fig.10. And the RS,1 represents the right shift function with one bit. The operation of each step in Fig.9 can be completed within 40ns (25 MHz clock) in FPGA; therefore total 23 steps need 0.92s operation time. It doesn’t loss any control performance in the overall system because the operation time with 0.92s is much less than the sampling interval, 500 s (2 kHz), of the position control loop in Fig.1.

In Figure 7, with exception of the CCCT circuit, others circuit design, like SVPWM and QEP, are presented in Fig. 11(a) and 11(b), respectively. The SVPWM circuit is designed to be 12 kHz frequency and 1s dead-band. The circuit of the QEPmodule is shown in Fig.11(b), which consists of two digital filters, a decoder and an up-down counter. The filter is used for reducing the noise effect of the input signals PAand PB. The pulse count signal PLSand the rotating direction signal DIRare obtained using the filtered signals through the decoder circuit. The PLSsignal is a four times frequency pulses of the input signals PAor PB. The QEPvalue can be obtained using PLSand DIRsignals through a directional up-down counter.

Figure 9.

State diagram of an FSM for describing the FC in position loop and P controller in speed loop.

Figure 10.

Section determination inFig. 9.

Figure 11.

Block diagram of (a) SVPWM circuit (b) QEP circuit.

The Nios II embedded processor IP is depicted to perform the function of the motion trajectory for X-Y table in software. Figure 12 illustrates the flow charts of the main program and the interrupt service routine (ISR), where the interrupt interval is designed with 2ms. All programs are coded in the C programming language in Fig.10. Then, through the complier and linker operation in the Nios II IDE (Integrated Development Environment), the execution code is produced and can be downloaded to the external Flash or SDRAM via JTAG interface. Using the C language to develop the control algorithm has the portable merit and is easier to transfer the mature code from the other processor to the Nios II embedded processor. Finally, Table 1 shows the FPGA utility of the proposed motion control IC and the overall circuits included a Nios II embedded processor IP (5,059 ALUTs and 78,592 RAM bits) and an application IP (10,196 ALUTs and 102,400 RAM bits), use 31.5% ALUTs resource and 7.1% RAM resource of Stratix II EP2S60.

Figure 12.

Flow chart of the main and ISR program in Nios II embedded processor.

Table 1.

Utility evaluation of a motion contron IC for X-Y table in FPGA.

4. Experiments and results

The overall experimental system is depicted in Fig. 1. This system includes an FPGA experimental board, two sets of voltage source IGBT inverter and an X-Y table which is driven by two PMSMs and two ball-screws. The power, rating, voltage, current and rating speed of PMSM are 200W, 92V, 1.6A and 3000rpm, respectively. A 2500 ppr rotary encoder attached to PMSM is used to measure the motor’s electrical angle. Two linear encoders with 5m resolution are mounted on the X-axis and Y-axis table as a position sensor. Each ball-screw has 5mm lead. The inverter has 6 sets of IGBT type power transistors. The collector-emitter voltage of the IGBT is rating 600V, the gate-emitter voltage is rating 12V, and the collector current in DC is rating 25A and in short time (1ms) is 50A. The photo-IC, Toshiba TLP250, is used for gate driving circuit of IGBT. Input signals of the inverter are PWM signals from FPGA chip. The FPGA-Altera Stratix II EP2S60 in Fig.1 is used to develop a full digital motion controller for X-Y table. The motion trajectory are implemented by software using Nios II embedded processor, and the two axis position/speed/current vector controller are implemental by hardware in FPGA. In the experimental system, the PWM switching frequency of inverter is designed with 12k Hz, dead-band is 1s, and the sampling frequency in current loop and position loop of the PMSM are designed with 16kHz and 500Hz, respectively. The motion control algorithms are coded by C language.

In experiment, the position step response and the motion trajectory control are used to evaluate the dynamic performance of the proposed system. In the experiment of the step response, the results of X-axis and Y-axis table under 10 mm amplitude and 0.5Hz square wave command are shown in Fig. 13. The rising time, overshoot and steady-state value in Fig. 13(a) are 110ms, 14% and near 0mm, and in Fig. 13(b) are 90ms, 15% and near 0mm. It reveals that the mass carried in X-axis table is heavier than those in Y-axis table. In the experiment of the motion trajectory tracking, one-dimensional trapezoidal motion trajectory, two-dimensional circular and window motion trajectory are tested and its experimental tracking results are shown in Figs. 14 ~ 16. In one-dimensional motion trajectory, the trapezoidal velocity profile is considered which the acceleration and deceleration is designed with 500mm/s2, maximum speed is 125mm/s, and the overall displacement is designed with moving from 0 mm to 100 mm position. The trajectory tracking results in each axis corresponding with the aforementioned input commands is shown in Fig. 14. It can be seen that the motion of X-axis and Y-axis table can give a perfect tracking with command target both in position or speed trajectory. Further, in two-dimensional motion trajectory, the circular motion trajectory control with center (60, 60) mm and radius 50mm is evaluated and the tracking errors are the maximum±0.55 mm in X-axis, and±0.75 mm in Y-axis in Fig. 15. The window motion trajectory designed as Fig.4 and its experimental result is shown in Fig. 16, which also shows the tracking errors maximum±0.5 mm in X-axis, and±0.9 mm in Y-axis. Therefore, from the experimental results of Figs. 13~16, it demonstrates that the proposed FPGA-based motion controller IC for X-Y table is effective and correct.

Figure 13.

Step response for (a) X-axis table (b) Y-axis table.

Figure 14.

a) Position and speed tracking response in X-axis and in (b) Y-axis table.

Figure 15.

Response of the circular trajectory (a) circular trajectory response (b) response for X- and Y- axis (c) control effort (d) tracking error.

Figure 16.

Response of the window trajectory (a) window trajectory response (b) response for X- and Y- axis (c) control effort (d) tracking error.

5. Conclusion

This study successfully presents a motion control IC for X-Y table based on novel FPGA technology. The works herein are summarized as follows.

  1. The functionalities required to build a fully digital motion controller of X-Y table, such as the two current vector controllers, two speed P controllers, and two position fuzzy controllers and one motion trajectory planning, have been integrated in one FPGA chip.

  2. An FSM joined by one multiplier, one adder, one LUT, or some comparators and registers has been employed to model the overall FC algorithm and the CCCT in vector control of the PMSM, such that it not only is easily implemented by VHDL but also can reduce the FPGA resources usage.

  3. The software/hardware co-design technology under SoPC environment has been successfully applied to the motion controller of X-Y table.

However, the experimental results by step response, point-to-point, window and circular motion trajectory tracking, has been revealed that the software/hardware co-design technology with the parallel processing well in the motion control system of X-Y table.

© 2010 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Ying-Shieh Kung and Ting-Yu Tai (January 1st 2010). FPGA-Realization of a Motion Control IC for X-Y Table, Motion Control, Federico Casolo, IntechOpen, DOI: 10.5772/6977. Available from:

chapter statistics

3542total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

A Long-Stroke Planar Actuator with Multiple Degrees of Freedom by Minimum Number of Polyphase Currents

By Yasuhito Ueda and Hiroyuki Ohsaki

Related Book

First chapter

Introduction to Infrared Spectroscopy

By Theophile Theophanides

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us