1. Modeling and control of a Stewart Platform (Hexapod Mount) 1 Frank Janse van Vuuren Supervisor: Dr Y. Kim

2. Presentation Overview 1. Objective 2. Hexapod Mount 3. Design Challenges 4. Kinematics 5. Dynamics 6. Configurations 7. Deliverables to Date 8. Results 9. Future Work 2

3. 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 3

4. 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 Example3.avi 4 Figure 1: Stewart Platform [1]

5. Alternative Alt-Azimuth Mount is currently widely used in telescopic positioning. Advantages of the Hexapod Mount:  High Load Carrying capacity  High Stiffness  Precise positioning Accuracy Disadvantages  Small Workspace  Complex Control  Singularities 5 Figure 2: Altitude- Azimuth Mount [2] (top) Figure 3: Hexapod Mount [3] (bottom)

6. Design Challenges 1. 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. 6

7. Kinematics Forward 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. 7 Figure 4: Alternative Hexapod Mount Configurations

8. Different Configurations 8  From the above analysis it was decided to use the 6-3 design as it avoids the main difficulties. 6-66-33-3 Construction Difficulty LowMediumHigh Forward Kinematics Difficulty HighMediumLow Leg Actuation to adjustment ratio HighMediumLow StiffnessLowMediumHigh

9. 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 D’Alembert’s Principle:

10. Singularities Two Types of Singularities 1. Gain DOF 2. Lose DOF Calculated by Jacobian 10 No Possible Movement Joints Track Sliders Multiple Possible Positions Figure 7: Different Types of Singularities

11. 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. 11 Figure 8: Effect of Leg Length Resolution on Pointing Direction

12. 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. 12 Figure 9: Cumulative Pointing Error Simulation 150% Decrease in Error

13. Results (Work Complete)  Forward Kinematics  Inverse Kinematics  Dynamics  Graphical User Interface (GUI) An algorithm has been developed to decrease the pointing error of the system caused by discretization. 13

14. Figure 6: Graphical User Interface Graphical User Interface 1. Set the size and layout of the Platform 2. Calculate the inverse Kinematics 3. Calculate the forward Kinematics 4. Run Simulations in the workspace. 5. Make a video file of the simulation. 14

15. Future Work of Project  Construct 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) 15

16. References 16 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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