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# Controller Synthesis for Nonholonomic Robots

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### Controller Synthesis for Nonholonomic Robots

2. 2. Pioneer 2 DX Lexus Parking Assist LAAS-CNRS h2 Beam Rattleback
3. 3. Pure rolling, no slip Nonholonomic constraint: Any configuration can be reached Three Degrees of Freedom Refs. [1] and [2]
4. 4. Generalized coordinates: Generalized velocities: Geometric constraint: Kinematic constraints: Pfaffian kinematic constraints: ‘s are independent. Ref. [1]
5. 5. Kinematic constraints: These constraints can be integrated to geometric constraints: One Degree of Freedom Ref. [2]
6. 6. Given a set of kinematic constraints, how can we tell if they are nonholomic? Nonholonomic constraints allow accessing all configurations (do not decrease Degrees of Freedom). It is a controllability question. Ref. [1]
7. 7. For a system with the following constraints, the kinematic model is: Ref. [1]
8. 8. Nonholonomic constraint: Kinematic model: is the driving input and is the steering input. Ref. [1]
9. 9. It can be shown that a system with the following kinematic model, is nonholonomic if the set is not closed with respect to Lie bracket: Ref. [1]
10. 10. Let . A filtration generated by is the sequence where: The involutive closure of is where is the smallest number such that: If , there are geometric constraints and nonholonomic constraints. If , the system is controllable and fully nonholonomic. Ref. [1]
11. 11. Therefore unicycle is a nonholonomic system. Ref. [1]
12. 12. Brockett Theorem (1983): If the system is locally asymptotically stabilizable at with smooth state feedback , then the image of the map contains some neighborhood of (a necessary condition) Nonholonomic systems cannot be stabilized at a point by smooth feedback. There is no time invariant linear controller for stabilizing a nonholonomic system! Ref. [1]
13. 13. Dynamic model: Using the kinematic model, we have: Ref. [1]
14. 14. Choose: Then, we have: Ref. [1]
15. 15. Reference model: where the timing law is :
16. 16. Dynamic Model: Therefore:
17. 17. Let us: To find consider: Control law:
18. 18. Vector Control: Control law:
19. 19. 1. G. Oriolo, Control of Nonholonomic Systems, Lecture Notes, http://www.dis.uniroma1.it/~oriolo/cns/cns_slides.pdf 2. M. Manson, Nonholonomic Constraint, Lecture Notes, http://www.cs.rpi.edu/~trink/Courses/RobotManipulation/lectures/lecture 5.pdf 3. G. Oriolo, Wheeled Mobile Robots: Modeling, Planning and Control, 2010 SIDRA Doctoral School on Robotics, http://bertinoro2010.dii.unisi.it
20. 20. ありがとうございます Thank you