Modeling and control of a Stewart Platform (Hexapod Mount)

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Frank Janse van VuurenSupervisor: Dr Y. Kim

Presentation Overview1. Objective 2. Hexapod Mount3. Design Challenges4. Kinematics5. Dynamics6. Configurations7. Deliverables to Date8. Results9. Future Work

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Objective Design a Hexapod mount to position a 3.4 m antenna

at the PED (Phased Experimental Demonstrator).

The PED is used as a testbed for the KAT project and also used to educate and train students.

Project Activities : Model Simulation Controller Design Construct a scale model Programme a Graphical User Interface Verify the performance capabilities

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Hexapod Mount (Stewart Platform)Positioning mechanism

consisting of base and platform.

Linear actuator legs. Leg lengths change length to

alter the orientation of the platform.

Example:Example3.avi

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Figure 1: Stewart Platform [1]

AlternativeAlt-Azimuth Mount is currently widely used in telescopic positioning.

Advantages of the Hexapod Mount:High Load Carrying capacityHigh StiffnessPrecise positioning Accuracy

DisadvantagesSmall WorkspaceComplex ControlSingularities

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Figure 2: Altitude- Azimuth Mount [2] (top) Figure 3: Hexapod Mount [3] (bottom)

Design Challenges1. Calculate the best route between two

positions while avoiding singularities (unstable positions).

2. Calculating the direct Kinematics.3. Optimal layout of the Hexapod.4. Calculating a singularity free workspace.5. Taking discrete space into account.

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KinematicsForward Kinematics - Calculate the position of the platform given the leg lengths. (Difficult – up to 40 solutions for 6-6 configuration)

Inverse Kinematics - Calculate the leg lengths given the position of the platform.

Solving for the Kinematics is an Iterative Process.

There are alternative layouts that will make the forward kinematics easier.

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Figure 4: Alternative Hexapod Mount Configurations

Different Configurations

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From the above analysis it was decided to use the 6-3 design as it avoids the main difficulties.

6-6 6-3 3-3

Construction Difficulty

Low Medium High

Forward Kinematics Difficulty

High Medium Low

Leg Actuation to adjustment ratio

High Medium Low

Stiffness Low Medium High

Dynamics

Dynamics verified using results from literature.

Important with high speed mechanisms. Since dynamics do not play a significant role

there is no need for a complex control system.9

Figure 5: Forces in each Leg for an acceleration of 5 m/s in x-direction

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-50

0

50

100

150

200

time (s)

For

ce (

N)

Total Force

F1F2

F3

F4

F5F6

D’Alembert’s Principle:

SingularitiesTwo Types of

Singularities1. Gain DOF2. Lose DOF

Calculated by Jacobian

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No Possible Movement

Joints

Track

Sliders

Multiple Possible Positions

Figure 7: Different Types of Singularities

Discrete Space Leg lengths of all hexapod

mounts have a discrete resolution.

The conversion from ideal leg lengths to real leg lengths causes pointing errors.

Problem: the rounding off of leg lengths creates errors in the pointing direction that are greater than the rounding off error.

Given a path that must be tracked, develop an algorithm to minimize the pointing error.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 183

84

85

86

87

88

89

90

Time

altitude a

ngle

ideal vs real altitude angle

Figure 8: Effect of Leg Length Resolution on Pointing Direction

Approach to Discrete Space 1. All the discrete leg

lengths were converted to elevation-azimuth angles (direct kinematics).

2. Calculate altitude and azimuth errors (difference between the ideal and the real path).

3. These errors were each assigned a weighting.

4. Switching was done when the current error was equal to the next error.

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Figure 9: Cumulative Pointing Error Simulation

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

3

3.5

4

Time

Cum

ulat

ive

Err

or

Error Comparison

Altitude

AzimuthTotal

Previous

150% Decrease in Error

Results (Work Complete)Forward KinematicsInverse KinematicsDynamicsGraphical User Interface

(GUI)

An algorithm has been developed to decrease the pointing error of the system caused by discretization.

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Figure 6: Graphical User Interface

Graphical User Interface1. Set the size and layout of the Platform2. Calculate the inverse Kinematics3. Calculate the forward Kinematics4. Run Simulations in the workspace.5. Make a video file of the simulation.

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Future Work of ProjectConstruct a model and controller.

(March)Run Simulations. (April)Use Results to make design

decisions for the final model.(June)Final Design of a hexapod mount

for the 3.4 m Dish of the Phased Experimental Demonstrator (PED). (August)

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References

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1. Tsai, Lung-Wen. Robot Analysis: the mechanics of serial and parallel manipulators. NY : John Wiley & Sons, 1999. 0-471-32593-7.

2. Jangan, Manisha. Giant Metrewave Radio Telescope. [Online] January 20, 2005. [Cited: June 18, 2008.] http://www.gmrt.ncra.tifr.res.in/.

3. Kingsley, Jeffrey S., Martin, Robert N. and Gasho, Victor L. A Hexapod 12m Antenna Design Concept for the MMA. Taipei : s.n., 1997.

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