spie 6422-32 risley prisms torales

High precision prism scanning system

G. García-Torales *, J. L. Flores, Roberto X. Muñoz.Dpto. of Electronic, Univ. of Guadalajara/CUCEI, Av. Revolución 1500,

Guadalajara Jal., MX CP 44840;

ABSTRACT

Risley prisms are commonly used in continuous scanning manner. Each prism is capable of rotating separately about a common axis at different speeds. Scanning patterns are determined by the ratios of the wedge angles, the speed and direction of rotation of both prisms. The use of this system is conceptually simple. However, mechanical action in most applications becomes a challenge often solved by the design of complex control algorithms. We propose an electronic servomotor system that controls incremental and continuous rotations of the prisms wedges by means of an auto-tuning PID control using a Adaline Neural Network Algorithm, NNA.

Keywords: Risley prism, scnning system, Adaline, automatic control

1. INTRODUCTION The Risley prism scanning system consists in a pair of sequential wedge prisms, which have wedge angles used to deviate a light beam has been used in several applications. Each prism of the system is capable of rotating about the optical scan axis at angular speeds. A focussed laser beam propagated through the prisms, along the optical scan axis is deviated in a direction according to the relative orientation of the prisms with respect to each other.

When the individual prisms are rotated clockwise or counterclockwise, the combined deviation angle and the orientation phase change with time, such that the image spot traces out a vector pattern1. These set of wedge prisms has been used in many scanning and laser applications where accurate wave front positioning is required. Such is the case also in many interferometric systems, like the shearing types, where the shear of the wavefront under test has to be known. Advantages and limitations of the Risley prism system in specific applications have been presented and solved for many authors2-7.

Figure 1 shows the propagation of a ray, incident in the object space along the optical axis, as it passes through the Risley prisms to the detection plane. Consider that both prisms have the same refraction index, a fixed refracting angle and rotates at a fixed selectable angular speed 1 and 2, respectively. A set of linear and circular scan patterns can be generated regarding two cases: when the two wedge prisms rotate in the same direction with selected angular velocities, and when the two wedge prisms rotate in opposite directions, always whit the possibility of selecting the initial relative orientation of the prisms.

Due to the wide use of Risley prisms, special-purpose mathematical ray-trace models have been developed. In scanning techniques, for example, the ratio of velocity, the deviation angles and the relative angle between prisms rotations is of interest. However, in the alignment and positioning applications we are interested solely in the image or pupil. We develop a simple ray trace formulation to demonstrate the performance of the laser pointer director, by the use of generalized matrix equations.

*[email protected]; phone (052)33 394-259-20; fax (052)33 394-259-20 ext. 7744

Sixth Symposium Optics in Industry, edited by Julio C. Gutiérrez-Vega, Josué Dávila-Rodríguez, Carlos López-Mariscal,Proc. of SPIE Vol. 6422, 64220X, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.742275

Proc. of SPIE Vol. 6422 64220X-1

(a) (b)

Fig. 1 Risley system: The relative angle between prisms determines the deviation angle and also the position of the ray on the plane. (a) View of a circular pattern generated when one is prism is rotated 360º respect to stationary prism. (b)Lateral view of the prism system.

2. RAY TRACE EQUATIONS

We use the vector ray-trace methodology, incorporating the vector transfer and the refraction equations at the prism surfaces.We developed a simple ray-trace program to model the ray propagation through the Risley prisms. With Pi isthe point at the corresponding surface Ui the direction of propagation of each point. We can find the refraction Ri andtranslation Ti equations for the hole system that can be written as:

I

PINFIR i

Tii

i0

(1)

and

IUAIdn

tI

T T

i

ii 1*'

0

,1 . (2)

Here the termidn

t

' is constant and depending on the physical parameter of the prisms and their relative position, and A

is the unit vector along the optical axis, Fi is defined as the deviation power of the prisms given by

iiiii nnF coscos 11 . (3)

Then for a six surfaces system we can apply iteratively equations 1 and 2 in order to obtain the position and direction for

any ray that was propagated through the wedge prisms by


0

00111,222,3323,444,55

5

55 ''''

P

UnRTRTRTRTR

P

Un. (4)

According with eq.4, it is necessary to know the initial conditions; the starting point coordinates and its direction, theprisms parameters; refraction index, wedge angle and the relative position between prisms, besides, the systemparameters; distances between prism and the detection plane. All these parameters feed the data requirements of servocontroller which drives the servomotor in order to achieve the desire position of the laser beam. Next, we show somegeneral characteristic of servo controller system.

3. OPTO-MECHANICAL DESING

In practice, accuracy in the determination of laser point displacement is limited by the error in the deflection angle thatdepends on three essential parameters: (a) the apex angle , which depends on the fabrication process, (b) the index ofrefraction n that depends on the quality of the glass, and (c) the relative angle between prisms which depends on theaccuracy of the rotary holders. All surfaces of the wedges must be of high quality, e. g. figure error less than /20, andpex angles with tolerances on the order of arc seconds. Nowadays, transparent materials with high degree ofhomogeneity are commercially available, e.g., ultra pure synthetic fused silica.

Therefore, the deciding parameter over the net deviation is the relative angle. Hence, a precise opto mechanical designfor mutual orientation and rotation of the wedges is necessary. The precision and accuracy of the mechanical systemdetermine the degree of control upon the wedge rotation. Commercial rotary holders perform rotations in the order ofminutes or even seconds of arc, however, most of them do not permit change parameters freely. Next, we show ourproposed to control the wedges by servo motors controlled with Adaline Neural Network8.

Figure 2 shows a schematic drawing of the servo motion system that controls the angular position of the prisms. Eachprism is mounted in a plastic base attached to the tension pulley that is driven by the servo motor. Speed motion andposition are controlled by the servo motor is controlled by a servo amplifier. Some of the main specifications of theservo motor used are its size down to 40 mm square, fast response with rapid acceleration and mechanical time constants down to 2 msecs, maintenance-free 3 phase brushless construction, peak torque ratings up to 7.2 Nm, ratings up to 250Nm. Encoder commutation facilitates sinewave drive technology to provide smooth operation and high resolution up to131,072 counts/rev.

Servo motor 1

Encoder 1 Prism system

Tension pulley Servo motor 2

(a) (b)

Fig. 2 Opto-mechanical device for the prism system. (a) View of the hole system, (b) Cross-section showing the prismsposition.


4. PID CONTROLLER

Servomotors are extensively used to perform highly precise positioning systems. Servo control in general can be dividedinto two fundamental classes of problems. The first class deals with command tracking. It addresses the question of howwell does the actual motion follow what is being commanded. The typical commands in rotary motion control are position, velocity, acceleration and torque. A second problem deals with the disturbance rejection characteristics of thesystem. Disturbances can be anything from torque disturbances on the motor shaft to incorrect motor parameterestimations used in the feed-forward control. Here, “P.I .D.” (Proportional Integral and Derivative position loop) controlis used to combat these types of problems. In contrast to feed-forward control, which predicts the needed internalcommands for zero following error, disturbance rejection control reacts to unknown disturbances and modelling errors. Complete servo control systems combine both these types of servo control to provide the best overall performance. Eventhough the fundamental concepts of servo motion control have not changed significantly in the last years, newtechnologies using servo systems have improved the accuracy on control positioning. Improve transient response times,reduce the steady state errors and reduce the sensitivity to load parameters9 .

Controllers are designed to eliminate the need for continuous operator attention. Controllers are used to automaticallyadjust some variable to hold the measurement (or process variable) at the set-point. The set-point is where you wouldlike the measurement to be. Error is defined as the difference between set-point and measurement. The variable being adjusted is called the manipulated variable which usually is equal to the output of the controller. The output of PIDcontrollers will change in response to a change in measurement or set-point . Manufacturers of PID controllers usedifferent names to identify the three modes. Depending on the manufacturer, integral or reset action is set in eithertime/repeat or repeat/time. One is just the reciprocal of the other. Note that manufacturers are not consistent and oftenuse reset in units of time/repeat or integral in units of repeats/time. Derivative and rate are the same.

Figure 3 shows, using Laplace notation, the basic components of a servo motor system. The servo drive, modeled asG(s), receive a voltage command. It can be approximated as the unity for the relatively lower motion frequencies. Theservomotor is modelled as a lump inertia, J, a viscous damping term, b, and a torque constant, Kt. The lump inertia term is comprised of both the servomotor and load inertia. I t is also assumed that the load is rigidly coupled such that the torsional rigidity moves the natural mechanical resonance point well out beyond the servo controller’s bandwidth. Thisassumption allows us to model the total system inertia as the sum of the motor and load inertia for the frequencies we can control. Somewhat more complicated models are needed if coupler dynamics are incorporated. The output of the P.I .D. controller is a torque signal. I ts mathematical expression in the time domain isgiven in eq. (5).

dt

terrordKdtterrorKirerrorKtDIP dp

))(()())(().(.. (5)

pK

s

iK

sK d

tK

1 sG

ServoDrive

tKJs

1

s

1

b

dT RisleyPrismSystem

P.I.D. Control

SpeedandPosition

ServoMotor

Fig. 3. Basic P.I.D. servo control configuration


5. TUNING THE P.I.D.

One of the problems in servo control is to accomplish the tuning of controllers PID. There are logic programmable controls on the market that permit the use of mathematical tools using digital controls for servo-drivers. A PID tuning controller with a linear adaptive element is compounded by a servo-driver of type multi-axle programmed on its own programming language, it makes use of learning algorithms with NNA, it learns to make the tuning simpler and automatic in its function, no only of constants. Nets with a RNA’s process discreet signals, in order to use adaptive filters, the analog signal converts by analogical to digital to digital values, and processed by a microprocessor.

Tuning methods are divided in open loop and feedback loop methods, where the Ziegler and Nichol method (feedback loop) is widely used. Theoretical PID control structures become very useful since Minorsky publications in 1922. In spite of the plenty of work published about PID control systems are still used practically in most of the industry controlling more than the 95 % of the feedback control process.

Tuning a PID controller involves adjustment of three parameters: the proportionality constant KI, the integral time constant TI and the anticipative time constant Td. First to all, the dynamic parameters of the process must be identified in order to obtain a reliable and robust performance of those parameters of the system for. The variables of in loop control are the reference value wish r(t), the output c(t) from a transmitter that generates the wish value, the feedback signal y(t)the output of the controller that modifies the final control element, ands the error xW(t) defined as the difference between the real value and the expected value.

The dynamics of the process is often identifying analytically and experimentally. The analytical method must solve the mathematical expression that describes the system as a function of time. This method is very complex in most of the real systems applications. In most cases mathematical functions are included o digital controllers and the distributed control due to their calculus power.

The experimental method, the static and dynamical features are obtained from direct measurements, where the Ziegler and Nichols is commonly used in two ways limit gain and the reaction curve10. Ziegler and Nichols technique is one of the first tuning methods applied to the feedback loop systems. The tuning process can be achieved following a two step procedure. Step 1: Set Ki and Kd to zero. Excite the system with a step command. Slowly increase KP until the shaft position begins to oscillate. At this point, record the value of Kp and set Ko equal to this value. Record the oscillation frequency, fo.Step 2: Set the parameters of the PID controller according with the table 1.

Table 1. Parameter of the controller: using the limit gain method.

Action of control Kp Ki Kd

Proportional-Integral-Derivative 0.6 KO 2 fO Kp/8Kd

6. NEURAL NETWORK ADALINE

The first meaning for Adaline was ADAptive LInear NEuron but its meaning change to o Adaptive LInear Element and only has one neural element; it has been used to solve linearly separable systems 6. This one element adds the product of the input vectors and its weights applying an output function in order to obtain one value response. Using a NNA, it is possible to establish a proceeding to modify the weight function and get a correct value to a given input. Signal to be proceeded by a NNA must be digitalized with sample times and the variables are the set point the reference signal w(t), a feedback signal x(t) (the output of system) then is possible to asses the error e(t), that correspond to the input GC(s). The PID controller generates an output gC(k), where Kp is the proportional gain, Ti is the integration time, and Td is the derivative time, where the discrete representation of the output of the PID controller in the time domain is given by


Ts

eeKTseKkeKkg nn

n

jDjIpC

1

1

)( (8)

The tuning process for discrete signals using the Ziegler-Nichols method is similar to those before explained and as a result only preserve the error and proportional constant KP as is shown next.

)()()( kwkxKkxKkg PWpC (9)

At the output of the ADALINE controller we obtain the signal that is represented by vector u(k) that becomes in thesequential input u(k-1), generating the knowledge matrix W(Kp). This matrix information is useful to calculate the newAdaline control outputs. Then, the constant for the conventional system are obtained applying the limit gain methodregarding the transfer function of he servo-driver. The learning rule of Adaline needs aprori information about the inputparameters. Once the NNA have learned, the error is reduced by an factor, at the same time, the weights of the systemchange according with the input values. The weights are renewed every time until the system converges.

)()(

2ke

kp

kpkpkeke

T

(10)

The best selection of , helps to manage the training process stability and velocity of convergence. Stability condition inachieve if 0.1 < 2. The selection of does not depends on the input magnitude, every weight actualized is collinear

respect the inputs parameters and its magnitude is inversely proportional to2kp . Figure

pK

s

iK

ffK

sK d

TransferFunctionofServoSystem

Adaline auto-tuning

SpeedandPosition

dd

ii

p

TK

TK

KControlSignal

P.I.D. Control Encoderposition

Fig. 4. Basic P.I.D. with Adaline servo control configuration


'I d

I lIO

IOA

k p p

• • .

IjClA

flm.

I

7. RESULTS

In order to implement the rotary holder control, it was regarded a security factor ranging from 40% to 100% in somecases, e.g. load effects. We used two servomotors with resolution 131,072 pulses per revolution without considering thereducer factor of 100 due to use a pulley system.

Typically, servo systems are first tuned with a step input in order to get a feel for the system response. Once this is done,the user now is often interested in how their actual motion will behave. At this point, the user must decide on the natureof the velocity profile. By far the most common velocity profile is the trapezoid. This is due to the relative ease of calculating all the state variables needed for motion: position, velocity and acceleration. As the need for smootheraccelerations and decelerations becomes greater, either "S" profiles or cubic splines are often employed. For the purposes of our investigation, we will focus on the use of a simple trapezoidal velocity profile. The matrix of weight is formed applying the conditions before exposed tuning with the Adaline. Figure 5 shows the response to the step functionapplying the auto tuner ADALINE. Note that the velocity does not change, but the position is corrected smoothly.Figure 6 shows two pictures of the rotary system. At left, show the lateral view and a perspective view on the right.

Fig. 5. Response to the step function applying the auto tuner Adaline.

Fig. 6. Pictures of the rotary prism system.


8. CONCLUSIONS

Risley prism system has been designed, assembled and tested using a P.I.D. control with an Adaline neural network. P.D.I. tuning process was implemented in a rotary holder servo-controller using the neural network Adaline. Algorithms were programmed in Motion Arquitect Program (c) in order to facilitate the introduction of constants. PID tuning can be used in SISO process automatically. The rotary holder can be programmed using exact ray trace equations until find the proper position of the prism system in order of perform scanning applications.

REFERENCES

1. G. F. Marshall, “Risley prism scan patterns”, Optical Scanning: Design and Application, Proc. SPIE 3787, 74-86 (1999).

2. M. Ostaszewski, et. al, “Risley prism beam pointer”, Free-Space laser communications VI, edited by Arun K. Majumar, Chritopher C. Davis, Proc. Of SPIE Vol. 6304, 630406, (2066).

3. D. C. Weber, J. D. Trolinger, R. G. Nichols, A.K. Lal, “Diffractively corrected Risley prism for infrared imaging”, Acquisition, Tracking, and Pointing XIV, Proc. SPIE 4025, 79-86 (2000).

4. J. Lacoursiere, et. al., “Large-deviation achromatic Risley prisms pointing systems”, Optical Scanning 2002, Proc. SPIE 4773, 123-131 (2002).

5. Michael Sánchez, David Gutow, “Control laws for a 3-element Risley prism optical beam pointer” Free-Space laser communications VI, edited by Arun K. Majumar, Chritopher C. Davis, Proc. Of SPIE Vol. 6304, 630403, (2066).

6. B. D. Duncan, P. J. Bos, V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications”, Opt. Eng 42(4), 1038-1047 (2003).

7. G. Garcia-Torales, M. Stronjnik, G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer”, Applied Optics 41(7), 1380-1384 (2002).

8. James, A. Freeman-David M. Skapura, Artifcial Neural Networks, Addison Wesley, (1991). 9. K.J. Åström and T. Hägglund, “PID Controllers: Theory, Design, and Tuning.Research” Triangle Park, NC:

Instrum. Soc. Amer. (1995). 10. Ziegler, J.G., and Nichols, N.B., Optimum Settings for Automatic Controllers, Transactions of the American

Society of Mechanical Engineers (ASME). v. 64, 1942, pgs. 759-768.


spie 6422-32 risley prisms torales

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