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

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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