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An Alternative Architecture and Control Strategy for Hexapod Positioning Systems to Simplify
Structural Design and Improve Accuracy
Ground Based and Airborne Telescopes III ConferenceJuly 2, 2010
Dr. Joe BenoUniversity of Texas Center for Electromechanics
j.beno@cem.utexas.edu; (512)232-1619
Co-authors: John Booth, UT-MDO & Jason Mock, UT-CEM
University of TexasCenter for Electromechanics
CEM
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Tracker Test Stand being Erected in CEM Lab
CEM Machine Shop
CEM High-Bay Lab Facility
Historical Context and Motivation
Hexapods are common in modern telescopes – advantages of parallel vs. series positioning systems
Modern telescopes are optically faster so primary-secondary mirror alignment is more important than older telescopes
Hexapod payloads for modern telescopes are rapidly increasing --- HET Wide Field Upgrade increases hexapod payload by 7x
Typical result:
• Actuator: Drive screw and nut assembly; geared motor
• Displacement Sensor: indirect measurement with rotary encoder on screw or motor shaft, influenced by actuator and mount compliance
• Heavy, bulky actuator-sensor unit
Goal: Enable simple integration of direct linear sensors for better accuracy and reduce stiffness requirement (and mass) of actuators
ForceActuator
DisplacementSensor
Lower Frame
Upper Frame
Overview
Conventional Design Approach• Optimize design & placement
of 6 actuator-sensor units• High emphasis on actuator &
mount stiffness
Alternative Design Approach• Decouple sensors from
actuators• Actuator mission: apply force• Actuator length not needed
for controls• Sensor defines hexapod• Sensor must be stiff, but
carries no load• Added control sophistication
Primary Application: Large, high precision, high accuracy hexapods with large payloads
Enabling Controls Approach
Controls Approach Overview
Conventional Hexapod
– Determine desired hexapod leg lengths from desired pose of upper frame WRT to lower frame (simple geometry problem: “inverse kinematics solution”)
– Use actuator’s imbedded sensor to estimate actual leg lengths
– Use PID feedback controls determine actuator forces necessary to drive leg length error toward 0
(Why not do position control to directly drive error to zero?)
But for any set of leg-lengths there are up to 40 different mathematically possible upper frame orientations (not all physically possible).
. . . PID controller “selects” closest possible solution.
Controls Approach Overview
Decoupled Sensor Hexapod
– Think of sensors as Virtual Actuators
– Determine desired virtual leg lengths from desired pose of upper frame WRT to lower frame (simple inverse kinematics problem)
– Use sensors to determine length of Virtual Actuators
– Use PID feedback controls to determine Virtual Actuator forces necessary to drive virtual leg length error toward 0
– Determine actual pose of upper frame from sensor values (“forward kinematics” problem – more to follow)
– Use actual pose of upper frame to determine line of action of actual actuators (length of actual actuators not needed)
– Determine force required from real actuators to apply same net force and moment on upper frame as Virtual Actuator would apply
Forward Kinematics Solution
Totally general hexapods:
• 6 unknown degrees of freedom; 6 known leg lengths
• Analytical solution not yet known, but there are 40 solutions (not all physically possible) for any given set of leg lengths.
• Numerical solutions not accurate enough
Approach:
• Identify hexapod leg configuration that can be solved analytically (An = Bn)
• Deploy sensors (virtual actuators) according to identified configuration
• Deploy real actuators in any other convenient, stable nonsingular configuration
Forward Kinematics Solution
(An = Bn)
ForceActuator
DisplacementSensor
Lower Frame
Upper Frame
Forward Kinematics Solution
Set #1 of 3 equations in 3 unknowns
Ji & Wu 2001
Set #2 of 3 equations in 3 unknowns
Forward Kinematics Solution
Solve for rotation matrix first; most foolproof approach:
Choose Type I Euler Angle Sequence (example rotate about X axis; about new Z axis; about new Y axis)
R = [ cc -csc + ss css + scs cc -cs
-sc ssc + cs -sss + cc ]
where s=sin, c=cos
Forward Kinematics Solution
Solve for rotation matrix first; most foolproof approach:
Choose Type I Euler Angle Sequence (example rotate about X axis; about new Z axis; about new Y axis)
R = [ cc -csc + ss css + scs cc -cs
-sc ssc + cs -sss + cc ]
where s=sin, c=cos
Start with Equation Set #2
Terms picked outby Eqn Set #2
Forward Kinematics Solution
Solve for rotation matrix first; most foolproof approach:
Choose Type I Euler Angle Sequence (example rotate about X axis; about new Z axis; about new Y axis)
R = [ cc -csc + ss css + scs cc -cs -
sc ssc + cs -sss + cc ]
where s=sin, c=cos
Start with Equation Set #2. Use sin = sqrt (1- cos2) to get three equations with cos(), cos (), and cos () as three unknowns. Turn the crank with an algebraic solver (e.g., MATLAB or Maple).
Move to Equation set #1, use substitution and turn the crank to solve for P(x,y,z).
Terms picked outby Eqn Set #2
Forward Kinematics Solution
Result:
Solved with one-time 6x6 matrix inversion; one 6x6 matrix-vector multiplication and ~ 20 analytical expressions
8 solutions, typically ~ half are real: Pick solution closest to desired– assumes hexapod in good control
(same assumption made with conventional hexapod when PID controller gravitates toward one solution)
Determine Real Actuator Force
Sum of Moments on Upper Frame from Virtual Actuators = Sum of Moments on Upper Frame from Real Actuators
Sum of Forces on Upper Frame from Virtual Actuators = Sum of Forces on Upper Frame from Real Actuators
Result: 6 linear equations, solved with one 6x6 matrix inversion
Sensor Considerations
Precision external linear sensors and custom mounting system
• Absolute position feedback
• Micron accuracy and sub-micron resolution
• High-stiffness sensor mount
• Appropriate degrees of freedom in the integration and mounting scheme
• Housed in telescoping tube with linear bearings to ensure alignment of read head
• Attached to hexapod frames with degrees of freedom typical of hexapod actuators, but much smaller because of negligible loads
Sensor example: Heidenhain LC 183
Benefits
Weight Savings: Preliminary assessment:
• 50% actuator weight reduction:
– Base actuator design on material survival limits (no yield, fatigue life, etc.)
– Stiffness just adequate to allow actuator to act as two-force member
• 25% actuator weight reduction:
– Material limits same as above
– Stiffness requirement half that of conventional stiff actuator-sensor design approach
Additional Design Freedom: actuators and actuator configuration
Retrofits: May allow inexpensive upgrade to existing hexapods that do not meet desired performance needs – add sensor set