The physics of positioning systems relate the parameters of motion control components to the overall system performance. While the relationships obey, in general, known kinematics, dynamics, and structural equations, they become rather complex as many parameters affect many variables. Consequently, designers can lose that intuitive feel for the right optimization direction, and frequently turn to computer aided engineering tools, with a model representation of their system to simplify the design and analysis process.
There are several tools known in the market to handle such problems and help engineers to better understand the behavior of their system of interest. For example, Matlab/Simulink from MathWorks (Natick, MA) and Easy5 (a Boeing design tool that is now commercialized for general industrial applications), assist design and application engineers in developing a good intuitive feel to the otherwise complex relationship between motion control component parameters and overall system performance.
For positioning systems, the Bimo transfer function model includes a controller, servo amplifier, motor, actuator, structure, and position feedback. 
This article discusses how Bayside developed its own premodeled analysis tool for positioning systems using Excel spreadsheet software. The company uses the new analysis tool Bimo (Bayside's Integrated Motion Optimization) internally as a customer support tool and service package to help its customers design better machines. However, plans exist for future commercialization.
Bimo uses simplified positioning system models to integrate structural kinematics, dynamics, frequency response simulation, time domain analysis, 3D precision calculations, and a component selection database in a package that doesn't require any sophisticated analytical background or special programming skills to use.
The first model is used for frequency response analysis. It assists in understanding the effects of each positioning system component on the overall system stability and responsiveness. The second model is used for time domain analysis. It assists in analyzing kinematics, dynamics, settling time, smoothness of motion, and precision of positioning systems, as well as sizing linear motors, amplifiers, and structures.
The Bimo model assumes a mass, spring, and damper structure riding over a rigid body mass. Motor force acts directly or through an actuator on the rigid mass. 
The requirements. There are many different configurations that can be used to provide a required motion profile. For each configuration, such as gantry, split axes or compounded axes, various component types can be selected for the controllers, amplifiers, motors, actuators, stages, structures, and feedback devices. Typical performance specifications for a positioning system, as shown above, are:

X (step), Y (scan), Z (focus)

Travel500 mm

Max Acceleration2 g

S Curve50 msec

Maximum velocity2 m/sec

Position Accuracy plus or minus 1 micron (typical)

Repeatability plus or minus 0.1 micron (typical)

Resolution20 nanometers

Encoder Transmission Rate20 MHz

Constant Velocity1 m/sec, plus or minus 0.01 % at 1 kHz sampling rate (equivalent to position precision of ( plus or minus 0.01/100 ) * (1,000,000 micron/sec) / (1000 cycle/sec) = 0.1mu)

Settling Time to submicron accuracy30 msec (typical with linear motors)

Natural Frequency300 Hz

Position Closed Loop Bandwidth50 Hz (to settle to submicron in 12 cycles, taking 2040 msec)

Moving Weight120 kg (typical)

Environment10^{79} Torr vacuum
The objective of Bimo is to analyze high performance positioning systems with motion requirements such as listed above. It uses a simplified servo control model for the linear motor positioning system. The controller provides a low power output signal to the servo amplifier. The servo amplifier in turn amplifies the signal and drives a linear motor, which applies a direct force on the stage. The stage, in turn, accelerates and moves the flexible structure to the desired position. The encoder feedback device mounts to the flexible structure.
Bode plots display the "gain" in units of dB, (20*log(output position/input command)) and "phase angle" in degrees for the positioning system's open loop transfer function. 
Frequency response analysis. The analysis starts by considering the frequency response of the system. It provides the means to study the effects of component parameters on the overall system stability and bandwidth. System bandwidth is defined as the maximum frequency that the system can follow with a minimal error of 3dB.
The purpose of frequency response analysis, is to help understand the motion characteristic of each component in the positioning system, as well as the characteristics of the system as a whole. The frequency in the plots is displayed in logarithmic scale; for example, 1 represents 10^{1} rad/sec, 2 represents 10^{2} = 100 rad/sec, etc. The analysis is important in determining the closed loop bandwidth of the system, and its stability.
Increasing the system gain, K, and the structural natural frequency, then decreasing moving mass increases the closed loop system bandwidth to about 100Hz. 
The closed loop bandwidth of the positioning system is about 2 Hz (12 rad/sec, between 10^{1} and 10^{2} in the chart). The phase margin is about 30 degrees (between 150 and 180 degrees) and the gain margin is a few dB, indicating a low performance, marginally stable system. The characteristics of the PID (proportionalintegral differential), motor/actuator and the structure are noticeable in the overall plot.
In order to study the effect of component parameters on system bandwidth, to yield higher system performance, parameters such as gain, and natural frequency, can quickly be modified. The results indicate an improved performance and stability with 100 Hz bandwidth and a phase angle of about 80 degrees.
Time response analysis. While frequency response analysis considers the effects of component parameters on the closed loop bandwidth and stability of the system in frequency domains, simulations are often used in analyzing complex system performance in time domains. The following example demonstrates Bimo in simulating dynamic settling time and smoothness of motion.
Dynamic analysis alos helps engineers determine the effects of structural damping, systems bandwidth, moving mass, resistance forces, and natural frequency on the overall system performance such as settling time and constant velocity. 
The dynamics analysis assists in finding the required motor forces, needed to drive the positioning system in a motion profile, which was selected in the kinematics phase. It also determines the settling time and settling distance, at the end of the acceleration phase, needed to reach constant velocity, as well as the settling time at the end of the deceleration phase, needed to reach the desired accuracy. Finally, the model provides the structural natural frequency and its stiffness, which are required to meet the desired closedloop bandwidth. These values can be used as a guideline for a finite element analysis design of the machine structure. The results of the dynamic analysis are then carried over to motor/amplifier sizing.
A 3D precision analysis determines pitch, yaw, roll and Abbe offsets on the overall accuracy of the machine. ALthough each stage in this example exhibits 5micron precision, the overall contribution of pitch, yaw, roll, and various Abbe offsets of the stages, yield an order of magnitude lower 3D accuracy of 42 microns for the assembled system. 
After sizing each positioning stage individually, a precision analysis of the assembled machine is conducted.