 home  |  products  |  how to buy  |  support & downloads  |  literature         Engineering GuidesLinear Mechanics Guide                 ## Linear and Rotary Positioning Stages Engineering Reference

1. Linear Positioner Terms

2. Rotary Motion Terms

3. Motion Control Components (motors, drives, controls)

4. Positioning System Analysis

## 1. Linear Positioning Stages

### LINEAR

Accuracy - The difference between a commanded position and an actual position of a positioning stage. Accuracy is typically specified in microns that represent specified number of standard deviation "Sigma" (see definition below), per given travel, at a specified height above the stage mounting plate. For example: a +3 micron accuracy, 3 Sigma, per 500 mm travel means that if the controller commands the positioning stage to move to a location 500mm away from a known "home" position in space, then, in 99.8% of the times that this move will be made, the actual position of the stage, at 25mm above the mounting surface, will end up being between 499.997 and 500.003mm.

Repeatability - Repeatability represents the maximum deviation between actual position values, obtained in repetitive moves of a positioning stage, to a desired position. Repeatability, like accuracy, corresponds to a specified number of "Sigma", per specified travel, at a specified height above the mounting surface of the stage. Resolution (Motion) - The smallest positioning movement that can be achieved by a positioning stage. This is typically limited by the stiction or static friction of the system mechanics, and the encoder resolutions.

Resolution (Encoder) - The smallest increment of the position feedback signal that can be measured by a feedback device (e.g., encoder).

Standard Deviation ("sigma") - The average deviation of a Random Variable (a variable such as position error, whose outcome is of a statistical nature) from its average value ("mean"). The chart below represents a Standard Normal distribution of a random variable with zero mean and sigma of 1. The X Axis represents the random variable in units of "sigma" , and the Y Axis represents the Probability Density function of the random variable. The density function is used to calculate the probability that the random variable will occur between two values on the X Axis. More specifically, the probability of a random variable occurring between two values on the X Axis equals to the area under the Probability Density Function between these two values. The total area under the curve equals 1. Some important areas are as follows: the area between +1 sigma is 0.84, between +2 sigma it is 0.977 and between +3 sigma it is 0.998. This means, for example, that the probability of a random variable occurring between +3 Sigma is 99.8%. (see image below) Flatness - The maximum boundaries of positioning path of motion projected on the vertical plane. (see image below)

Straightness - The maximum boundaries of positioning path of motion projected on a horizontal plane. (see image below) ### ANGULAR

Pitch - An angular deviation possible in positioning systems, in which the table leading edge rises or falls as the table translates along the direction of travel. This represents rotation around a horizontal axis, perpendicular to the axis of travel. (see image below)

Yaw - An angular deviation from ideal straight line motion, in which the positioning table rotates around the Z (vertical) Axis as it translates along its travel axis. (see image below)

Roll - An angular deviation from ideal straight line motion, in which the positioning table rotates around its axis of travel as it translates along that axis. (see image below) Abbe Error - A linear positioning error caused by a combination of an angular error in the bearing of the positioning stage, and an offset between the bearing and the actual point of interest. (See also 3-D Precision Analysis in Section 4.3.) (see image below)

### DYNAMIC

Constant Velocity - A measure of smoothness of motion of a positioning stage. Typically measured in percent variation from a nominal value at a given sampling interval. High smoothness of motion can be achieved by using crossed roller or air bearing stages with ironless linear motors. Settling time - The time required for a step response of a system parameter to stop oscillating or ringing and reach its final value. For example, the time it takes for a velocity profile to settle to a specified value of constant velocity after the acceleration ramp phase. Also, the time it takes for a displacement profile to settle to specified accuracy after the deceleration phase at the end of a positioning move. Settling time is greatly affected by the shock, jerk, structural damping and resonance frequencies. Improved settling time in positioning systems can be achieved by high structural stiffness, low moving mass, high natural frequency of the structure, structural damping, high closed loop band width at the overall positioning system and good servo tuning.

#### Here are sample graphs of the Parker MX80L with a 1 micron encoder resolution. The sample stage was moving 1 mm with a maximum velocity of 150 mm/s and an acceleration of 0.3 G. From the chart below we see that settling time is dependant upon the distance traveled as well as velocity, orientation (vertical or horizontal), payload, encoder resolution, and most importantly tuning parameters.  Mx - When the load is cantilevered off the sides of an axis, perpendicular to the direction of travel

My - When a load is cantilevered off the end of an axis, parallel to the direction of travel

Mz - When a force causes a rotational moment about the center of an axis. Maximum Axial Force - The maximum thrust force that the stage can generate in the direction of travel. This force is used to overcome friction, damping, tool resistance and acceleration.

## 1.3 Assembly

### CONFIGURATIONS

Single Axis - The simplest form of positioning stage. Sometimes referred to as "Table", "Slide", "Actuator" or "Stage". It typically consists of slide, base, bearing, motor, encoder, limits, home, cable carrier and hard stops. The base can be mounted to a rigid structure or to the slides of other stages in various configurations as shown below. The slide, which is the moving part, can be used to move another stage, or any object such as a tool, work, test and measuring devices. Compound XY - This configuration provides the simplest form of 2 linear degrees of freedom of a positioning system where the base of the top axis is bolted to the slide of the lower axis. For a highperformance positioning application, a "monolithic" design can be used where the base of the top axis and the slide of the bottom axis are rigidly made as a single part. In a compound XY configuration care should be given in consideration to the Abbe Error of the top axis due to cantilever "diving board" effect. (see sections 1.1 for the definition of Abbe Error, and 4.3 for 3D accuracy analysis.) Compound XYZ - This configuration provides the simplest form of 3 linear degrees of freedom of a positioning system with the smallest footprint. In using this configuration care must be given to calculate the three dimensional accuracy. In particular the Abbe error. (Due to large offset between the bearing of the lowest stage and the point of interest at the top of the vertical stage.) (see sections 1.1 for the definition of Abbe Error, and 4.3 for 3D accuracy analysis.) Split XYZ Axes - A split axes positioning stage typically provides higher precision and higher stiffness than a compound configuration of the same number of axes. The reason is that at least 2 axes are mounted to a flat, rigid, stationary base with a fewer number of stages that ride on other stages. The result is smaller Abbe Errors and less cantilever effects at the expense of a larger footprint. Note that although this structure looks similar to a Gantry
configuration, as shown below, the Z Axis is rigidly mounted to a stationary bridge, and the X Axis is mounted to a stationary Base. Gantry - This configuration has the best accessibility to the space around it per footprint of the machine. It is commonly used as single cell or in process application where several machines are operating over a conveyor. Gantry configuration, driven by linear motors and designed for high natural frequency (typically 150 Hz), can provide an excellent solution that combines high precision, high speed and low settling time. Gantry can further be classified according to the following options:

• Single-sided motor drive typically used for smallsize applications
• Double-sided motor, driven together by a single amplifier with 1 sided encoder typically used in large system, with low accuracy requirements
• Double-sided motor, driven as two independent axes X1, X2 operating as master slave with two sided encoder typically used for large machines that require high precision. Flexure slides may be needed on the X Axes to prevent cleavage (motion resistance at the bearing of the X Axis due to skewed movement of the Y Axis.) ### STAGE COMPONENTS

#### Bearings

Recirculation Bearing - Typically used for highest stiffness and high speed (Pitch, Yaw and Roll on the order of 10 arc sec).

Crossed Roller Bearing - Typically used for a combination of high stiffness and high smoothness of motion (Pitch, Yaw, Roll on the order of 5 arc sec).

Air Bearing - Typically used for highest precision (sub micron) and highest smoothness of motion. (Pitch, Yaw Roll on the order of 1 arc sec).

Motors - (See section 3.1 for more details.)

Rotary Motor / Gear Box / Ball Screw - Typically used for high acceleration, high force.

Rotary Motor / Gear Box / Lead Screw - Typically used for high smoothness of motion.

Linear Motor ( Ironless ) - Typically used for very high smoothness of motion at low or high velocity.

Linear Motor ( Iron Core ) - Typically used for achieving a combined high force (up to 20,000 N), long travel (unlimited) and high speed (up to 10 m / sec).

Piezo Ceramic Motor - Typically used for submicron positioning applications.

Lead Screw: A device for translating rotary motion into linear motion, consisting of an externally threaded screw and an internally threaded carriage (nut).

Ball Screw: A lead screw which has its threads formed as a ball bearing race; the carriage contains a circulating supply of balls for increased efficiency.

Encoders - (see section 3.2 for more details)

Rotary Encoder - Typically mounted to the back of a rotary motor and used for lower precision at lower cost.

Linear Encoder - Typically used for higher precision at higher cost.

# 2. Rotary Positioning Stages

## 2.1 Precision

Axial Runout Error - The total indicated reading (TIR) of axis movement along the axis of rotation Radial Runout Error - The total indicated reading of the horizontal movement of the rotary table. Backlash Error - The error in rotational position due to clearance between a worm and a gear as a result of changing direction of motion. Backlash has an effect on two directional repeatability since the motion of worm is lost while reversing direction and traveling through the gap it has with the gear. Wobble Error - The angular error between the actual axis of rotation and the theoretical axis of rotation.

Axial Load Capacity - The maximum allowable force acting along the axis of rotation of the rotary stage. Perpendicular Load Capacity - The maximum load perpendicular to the positioning stage top surface, applied at a specified radius from the axis of rotation of the table.

# 3. Motion Control Components

## 3.1 Motors

### Various Types of Motors Used in Positioning Systems

Brushless Rotary Motor & Brushless Direct Drive Servomotor - A device that converts electrical current to mechanical energy where the current is varied by a servo
amplifier in a closed loop control system.

DC Motor - A device that converts electrical direct current into mechanical energy. It requires a commutating device, either brushes or electronic. Usually requires source of DC power.

AC Motor - A device that converts electrical alternating current into mechanical energy. Requires no commutation devices such as brushes. Normally operated off commercial AC power. Can be single or multiple phase.

Synchronous Motor - Another term for a Brushless DC motor. Permanent Magnet Motor - A motor utilizing permanent magnets to produce a magnetic field. Has linear torque / speed or force / speed characteristic.

Brushless Motor - A type of direct current motor that utilizes electronic commutation rather than brushless to transfer current.

Linear Motor Iron Core Linear Motor - A permanent magnet motor consisting of laminated ferrous coil assembly and a single-sided secondary magnet assembly.

Ironless Linear Motor - A permanent magnet motor consisting of a non laminated coil assembly and a u-channel secondary magnet assembly.

Piezo - Ceramic Motor Piezo Ceramic Motor - A motor made of a small ceramic plate, oscillating at high frequency (e.g. 40Khz), causing its tip to form circular motion. As the tip comes in contact with a longer ceramic plate, attached to the slide of a positioning stage, it applies friction forces on the plate and causes it to move in the direction of the tip circular rotation.

## 3.2 Encoders

The encoder motion component as shown below is a position feedback device, which converts mechanical motion into electrical signals to indicate actuator actual position. The basic configuration of an encoder can be linear or rotary, incremental or absolute. A rotary encoder is typically attached to the rotary motor and measures the motor shaft rotation. Therefore, any windage effect at the ball screw or lost motion due to backlash and friction will not be seen at the encoder. The linear encoder, on the other hand, reads the actual position closer to the point it takes place and therefore the resulting precision is higher.

### Linear Encoder

Absolute Encoder - A digital position transducer in which the output is representative of the absolute position of the input shaft within one (or more) revolutions. Output is usually a parallel digital word. Incremental Encoder - A position transducer in which the output represents incremental changes in position.

Linear Encoder - A digital position transducer that directly measures linear position.

Quadrature Encoder - This is a special incremental encoder with two channels A and B, sometimes referred to as A Quad B. The two channels are 90 degrees out of phase. This configuration allows detection of direction as well as increasing the resolution by a factor of four .

## 3.3 Controller/Amplifier

### Motion Controllers .

Motion Controller is an electronic device that communicates with a host computer and has the capability to store a desired motion profile as a function of time or any other reference signal, read the actual position feedback, calculate the error, and send out a command signal to the servo amplifier as a complex function of the error and its derivatives. It can also monitor various I / O signals and control several axes in a coordinated moves.

PID controller block diagram with Feed Forward and ZOH ZOH - Zero Order Hold represents the controller time delay in processing the input signals before the output to the amplifier is updated.

DAC- Digital to Analog Convertor component that receives a digital signal from the controller filter and outputs an Analog signal to the Amplifier.

Compensation: The corrective or control action in a feedback loop system that is used to improve system performance characteristics such as accuracy and response time.

Compensation, Feedforward: A control action that depends on the command only and not the error to improve system response time.

Compensation, Integral: A control action that is proportional to the integral or accumulative time error value product of the feedback loop error signal. It is usually used to reduce static error.

Compensation, Lag: A control action that causes the lag at low frequencies and tends to increase the delay between the input and output of a system while decreasing static error.

Compensation, Lead: A control action that causes the phase to lead at high frequencies and tends to decrease the delay between the input and output of a system.

Compensation, Lead Lag: A control action that combines the characteristics of lead and lag compensations.

Compensation, Proportional: A control action that is directly proportional to the error signal of a feedback loop. It is used to improve system accuracy and response time.

Compensation, Derivative: A control action that is directly proportional to the rate of change of the error signal of the feedback loop. It is used to improve system damping to provide smooth motion and reduce settling time. Servo Amplifier - An Amplifier that utilizes internal servo feedback loops for accurate control of motor current and or velocity.

Block Diagram of a Typical Servo Amplifier Analog Amplifier - An Amplifier that has an analog signal as an input.

Digital Amplifier - An Amplifier in which tuning and parameter setting is done digitally. Input can be an analog or digital signal.

Linear Amplifier - An Amplifier that has output directly proportional to either voltage or current input. Normally both input and output signals are analog.

PWM Amplifier - An Amplifier utilizing Pulse Width Modulation techniques to control power to the motor. Typically a high-efficiency drive that can be used for high response applications.

# 4. Positioning System Analysis

## 4.1 System Modeling

#### Physical Model

System modeling is important for developing a better understanding of the effects that various design variables, operating conditions and selected motion control components have on the overall positioning system performance. Modeling starts with a physical system to be modeled. For example, the picture shows a positioning system in a compound X,Y,Z configuration. I n the following sections we will model and analyze a typical axis of similar machines.

#### Schematic Diagram

Once the physical model is defined, a schematic diagram shows the main mechanical components, which are included in the theoretical model, and the way they interact. The diagram shows for example a model of a positioning stage with mass M, driven by a motor force and carrying a flexible structure with mass m, stiffness K and Damping B. The schematic diagram is then used for writing the equations of motion of the theoretical model.

#### Block Diagram & Transfer Functions (See section 4.2 for Parameter definitions.)

The block diagram represents the motion control process within the system with all of its modeled components. The arrows represent the flow of signals within the system from one component to another. The block themselves contain expressions that are called Transfer Functions. Transfer Functions include operators (e.g., "S" designating differentiation and "1 / S" designating Integration) and parameters that together describe the equations of motion of each block, which relate the output variable of a block to its input variable. Transfer functions are used to determine the ratio between the magnitude of the output variable to the magnitude of the input variable. This ratio is called "gain" and it is measured in units of dB, where dB is defined as 20* Log (output / Input). Furthermore, Transfer Functions are used to calculate the "phase angle" which is the lag or lead of the output signal vs. the input signal measured in degrees. The plot that shows the gain and the phase angle as a function of input frequency is called "Bode Plot". #### System Variables and Parameters

The following diagram represents a product tree of a modeled positioning system. The upper section represents various System Variables, which describe the STATIC, SERVO and DYNAMIC specifications of the machine. These variables are modeled as a function of system parameters as shown below. The bottom section of the diagram represents system parameters that characterize the various motion control components of the positioning system. These parameters are needed to be selected for various reasons including structural design, component sizing, and servo tuning. The model relates these parameters to the performance variables as shown above. It can therefore be used to assist in the selection of these parameters to result in a cost-effective solution.

## 4.2 Frequency Response

The purpose of Frequency Response Analysis, as shown below, is to help in understanding the motion characteristic of each component in the positioning system, as well as the characteristics of the system as a whole. The plots display the "gain" in units of db, (20* log (output / input) and "phase angle" in degrees for each block in the Block Diagram (see section 4.1). Both plots are shown as a function of the frequency of the input variable and referred to as Bode Plots. The frequency in the plots is displayed in logarithmic scale. For example 1 represents 101 rad / sec, 2 represents 102 = 100 rad / sec etc. The analysis is important in determining the Closed Loop Bandwidth of the system, as well as its stability.

#### Controller - PID

The PID transfer function, as shown in section 4.1, has the "positioning error" signal as an input and the "Controller command" signal to the amplifier as an output. It shows high gain (ratio of output signal to input signal) in low frequencies, acting as a low pass filter. It also has high gain at high frequencies, acting as a high pass filter. And finally it has lower gain in some intermediate frequencies, reducing the effects of various vibration causes such as structural resonance, bearing jitter, cogging, and tool vibrations. The low pass filter, caused by the integrator term, Ki, amplifies small errors, such as those caused by friction, and reduces them over time. The highpass filter, caused by the derivative gain, Kd, allows the system to lead its reaction to high frequency errors. The phase angle of the output signal versus the input signal starts at -90 degrees Lag and ends up at 90 degrees lead. The purpose of the PID transfer function is to shape the overall transfer function of the positioning system, by choosing the right set of PID parameters, Kp, Ki, Kd, to obtain a fast responding, stable, system with high closed-loop bandwidth.

#### Servo Amplifier

The amplifier transfer function, as shown in section 4.1, has "controller command" signal as an input and "motor voltage" as an output. As shown, the output signal follows the input signal at low frequencies with a constant gain, as determined by the parameter, Ka, of the amplifier. At a certain frequency, called the cutoff frequency, the gain starts to attenuate as frequency increases. The phase angle shows zero lag until the frequency reached the cutoff value, then the output starts to lag to a maximum of -90 degrees at very high frequencies. The cutoff frequency is the inverse of the amplifier time constant Ta, as shown in the transfer function. A time constant is the time it takes for the output signal to reach the level of 63% of a step in the input signal.

#### Motor/Stage

The combined Motor / Stage transfer function, as shown in section 4.1, has "motor voltage" as an input and "stage position" as an output. The gain shows a characteristic of reducing magnitude at a rate of 20 db/decade (decade is a multiple of 10 in frequency change) until a resonant frequency is reached. Then the gain attenuation becomes steeper and reduces at a rate of 60 db/decade. The phase angle starts out at a -90 degrees until the resonance frequency and then it drops an additional 180 degrees to a total of -270. The transfer function of this block has two time constants. One is the electrical time constant of the motor (L/R) and the other is the mechanical time constant of the stage (M•R /Kf•KE). Where,
L = Motor Coil Inductance
R = Motor Coil Resistance
Kf= Motor Force Constant
KE = Motor Back EMF
M= Stage Moving Weight

#### Structure

The structure transfer function, as shown in section 4.1 has the "stage position" as an input and the actual "structure position" of a point of interest on the structure (e.g. Encoder location) as the output. This is a classical transfer function of a mass, spring, damper system with a positive position excitation of the base. The gain starts at 1 (zero dB) with low frequencies and gradually increases and reaches a peak at the natural frequency of the structure. Then the gain drops at a rate of 40 dB / decade at higher frequencies. The phase angle starts out as zero, at low frequency, and drops 180 degrees around the natural frequency. Finally it gains additional 90 degrees to a total of -90 degrees at very high frequencies. The parameters that characterize this system are as follows:
m- Structural Mass
K- Structural Stiffness
B- Structural Damping.
Where the natural frequency of the structure Wn = sqrt (K / m) #### Complete System Overall Positioning System Bode Plot

The overall transfer function of the positioning system model, as shown in the Bode Plot, is made as the superposition of all transfer functions of the individual components. The most important features of this plot are the closed loop bandwidth of the system and the two stability criteria: Phase Margin and Gain Margin. The closed loop bandwidth is determined by the frequency where the gain of the overall transfer function (known as open loop transfer function) crosses the 0 dB line, also referred to as a cross over frequency. The difference between the phase angle at the cross over frequency and -180 degrees is called Phase Margin. For a stable system the Phase margin must be greater than zero. The difference between the gain of zero db and the gain at -180 degrees is called the Gain Margin. For a stable system the gain margin must be greater than zero. The closed loop bandwidth in the example at the chart is about 48 Hz (300 rad / sec, between 102 and 103 in the chart). The phase margin is about 30 degrees and the gain margin is a few dB, indicating a marginally stable system. The signatures of the PID, Motor / Amplifier and structure are clearly noticeable in the overall plot. ## 4.3 Simulation

While Frequency Response Analysis, as shown in section 4.2, is used to study the effects of system parameters on the closed loop bandwidth and stability, in the frequency domain, Simulations are frequently used in analyzing system performance in real-time domains. The following analysis will demonstrate the usage of simulation for analyzing motion profiles, settling time, smoothness of motion, motor sizing, dynamic braking and 3D accuracy. These tools are useful in understanding the overall system performance as function of component parameters, operating conditions and design constraints. The Positioning stage model used in the simulation analysis is a PID model with a Motor / Stage Structure. For calculation of the required natural frequency, the gain is assumed to drop at a rate of 60 dB per decade at the resonant frequency. Furthermore the Gain Margin at the structural resonance is assumed to be zero. (see section 4.2)

#### Kinematics

Kinematics Analysis assists in selecting the best desired motion profile for a positioning application. There are infinite possibilities to achieve a desired cycle time with a given travel requirement and various constraints on maximum velocity, maximum acceleration or Jerk. The following is an example of such an analysis. Several iterative runs may be needed to achieve optimal results.

#### Dynamics & Settling

The Dynamics analysis assists in finding the required motor forces needed to drive the stage in a motion profile, which was determined at the Kinematics Analysis phase. It also determines the Settling time and the Settling distance, at the end of the acceleration phase, to reach the desired constant velocity, as well as the Settling time to reach the desired accuracy at the end of the deceleration phase, during dwell (see section 1.1 dynamics). The dynamic analysis shown, also determines effects of structural damping, system bandwidth, and system damping on the performance. Finally, recommended structural natural frequency and stiffness, which are required to meet the desired settling time and smoothness of motion, are provided by the model. These values can be used as a basis for a Finite Element Analysis design of the machine structure. #### Linear Motor and Amplifier Sizing

Linear Motor and Amplifier sizing is illustrated at the chart. Force requirements are taken from the results obtained by the dynamic analysis. A motor vendor is then selected. The lowest force motors, which meet the force requirements, are listed automatically for the selected vendor. A choice of motor is made and the specifications are automatically listed. Results display motor temperature, safety margins on forces and required amplifier current and voltage. The graph shows the decaying velocity at the end of constant velocity phase under dynamic braking conditions. Dynamic braking is typically used when the amplifier fails and the stage continues to travel under inertia forces. During dynamic braking motor coils are put in short circuit. The result is that the back emf voltage of the motor generates current in the coil that develops a force opposite to the direction of velocity. The graph shows the residual velocity that is needed to be absorbed by the hard stops of the stage in this crash conditions, when the state reaches end of travel and hits the hard stops. #### 3D Precision Analysis

3-Dimensional Precision Analysis is needed to determine effects of various stage parameters, assembly configuration and Abbe offsets on the overall accuracy of the machine. As shown in the example, although each stage has considerably high precision (5 microns) the overall contribution of pitch, yaw and roll and various Abbe offsets of the various stages, this results in an order of magnitude lower 3D accuracy for the assembled system (42 micron). The analysis further shows that most of the System error is in the global Y direction, and that Stage A has the highest error contribution. It can be therefore concluded that an effective way to improve the error of the entire system, in this case, is by reducing the pitch of Stage A by using, for example, different bearing. (See section 1.3.)          Parker Hannifin Corporation, Electromechanical and Drives Division, 9225 Forsyth Park Drive, Charlotte, NC 28273 704-588-3246 or 800-358-9068 Email:Technical Support Privacy Policy  © Copyright Parker Hannifin Corporation 2005