A Mathematical Introduction to Robotic Manipulation 輪講 七回と八回.pdf

A mathematical introduction to robotic manipulation の輪講を行った際の第六・七回発表資料です（最終回）

1. A mathematical introduction
to robotic manipulation
Chapter 7&8
Harada Lab, Osakauniversity
D2 Zhenting Wang
2021.11.19

2. Overview
Chapter 7 -Nonholonomic Behavior in Robotic System
7.1 Controllability and Frobenius’ Theorem
7.2 Examples ofNonholonomic Systems
7.3 Structure of Nonholonomic Systems
Chapter 8 -Nonholonomic Motion Planning
8.1 Steering Model Control Systems Using Sinusoids
8.2 General Methods for Steering
•Fourier techniques
•Conversion to chained form
•Optimal steering of nonholonomic systems
•Steering with piecewise constant inputs
8.3 Dynamic Finger Repositioning

3. Overview
Chapter 7 -Nonholonomic Behavior in Robotic System
7.1 Controllability and Frobenius’ Theorem
7.2 Examples ofNonholonomic Systems
7.3 Structure of Nonholonomic Systems
Chapter 8 -Nonholonomic Motion Planning
8.1 Steering Model Control Systems Using Sinusoids
8.2 General Methods for Steering
•Fourier techniques
•Conversion to chained form
•Optimal steering of nonholonomic systems
•Steering with piecewise constant inputs
8.3 Dynamic Finger Repositioning
11/19
12/03

4. Nonholonomic System
Pfaffian constraints:
holonomic: it restricts the motion of a system to a smooth hypersurface of the configuration space.
Integrable: An integrable Pfaffian constraint is called a holonomic constraint.
A set of Pfaffian constraints is said to be nonholonomic if it is not equivalent to a set of holonomic
constraints.
Chapter 7 -Nonholonomic Behavior in Robotic System
Introduction

5. Nonholonomic System
In this chapter:
Understand when we can exploit the nonholonomyof the constraints to achieve motion between
configurations.
Examples:
Chapter 7 -Nonholonomic Behavior in Robotic System
Introduction

6. Lie derivative and Lie bracketChapter 7 -Nonholonomic Behavior in Robotic System

7. Lie derivative and Lie bracketChapter 7 -Nonholonomic Behavior in Robotic System

8. Two kinds of situations:
1.Bodies in contact with each other which roll without slipping
•Dextrousmanipulation with a multi-fingered robot hand
•Path planning for mobile robots or automobiles
2.Conservation of angular momentum in a multibody system
•Motion of a satellite with robotic appendages moving in space
Chapter 7 -Nonholonomic Behavior in Robotic System
Examples of Nonholonomic Systems

9. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 1: Kinematic car
Configuration:
Pfaffian constraints:
Control system:

10. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 2: Hopping robot in flight
Configuration:
Pfaffian constraints: (total angular momentum is initially 0)
Control system:

11. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 3: Fingertip rolling on an object
Configuration:
Kinematic equations of contact:
The rolling constraint is obtained by setting the sliding velocity and the velocity of
rotation about the contact normal to 0:

12. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 3: Fingertip rolling on an object
Control system:

13. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 3: Fingertip rolling on an object
Specialize the example to the case that the object is flat and the fingertip is a sphere
of radius one. The curvature forms, metric tensors and torsions for the fingertip and
the object are (chapter 5):

14. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 3: Fingertip rolling on an object
With the input being the rates of the rolling about the two tangential directions:

15. Distribution:
Chapter 7 -Nonholonomic Behavior in Robotic System
Structure of Nonholonomic Systems

16. Chapter 7 -Nonholonomic Behavior in Robotic System
Structure of Nonholonomic Systems

17. Degree of nonholonomy
Chapter 7 -Nonholonomic Behavior in Robotic System
Structure of Nonholonomic Systems
Growth vector, relative growth vector

18. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 1: Kinematic car
Control system:
To calculate the growth vector, we build the filtration:

19. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 2: Hopping robot in flight
Control system:
The controllability Lie algebra:

20. Chapter 7 -Nonholonomic Behavior in Robotic System
Example 3: Fingertip rolling on an object
Input:
Constructing the filtration:

21. Optimal control
By attaching a cost functional to each trajectory, we can limit our search of trajectories
which minimize a cost function.
Typical cost functions might be the length of the path, the control cost, or the time
required to execute the trajectory.
Piecewise constant inputs
The most naive way of using constant inputs is to pick a time interval and generate a
graph by applying all possible sequences of input. Each node on the graph
corresponds to a configuration, and branches indicate the choice of a fixed control
over the time interval.
Chapter 8 –Nonholonomic Motion Planning
Introduction

22. Canonical paths
This approach solves the nonholonomic path planning problem by choosing a family
of paths which can be used to produce desired motions.
The set of canonical paths used for a given problem is usually specific to that problem.
In some cases the paths may derived from some unifying principle.
Chapter 8 –Nonholonomic Motion Planning
Introduction

23. First-order controllable systems: Brockett’s system
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

24. First-order controllable systems: Brockett’s system
Case:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

25. First-order controllable systems: Brockett’s system
Case:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

26. First-order controllable systems: Brockett’s system
Case:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

27. First-order controllable systems: Brockett’s system
Case:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

28. First-order controllable systems: Brockett’s system
Case:
Total cost:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

29. First-order controllable systems: Brockett’s system
Case: input m
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

30. First-order controllable systems: Brockett’s system
Control Algorithm:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

31. Second-order controllable systems
By computation:
The maximum number of controllable is :
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

32. Second-order controllable systems
Control Algorithm:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

33. Second-order controllable systems
Control Algorithm:
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
m=2

34. Higher-order systems: chained form systems (12/03)
Generalization: conversion to chained form
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids

35. Fourier techniques
Example 1: Hopping robot in flight
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Control equations:

36. Fourier techniques
Example 1: Hopping robot in flight
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Control equations:

37. Fourier techniques
Example 2: Kinematic car
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

38. Fourier techniques
Example 2: Kinematic car
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Step C:

39. Overview
Chapter 7 -Nonholonomic Behavior in Robotic System
7.1 Controllability and Frobenius’ Theorem
7.2 Examples ofNonholonomic Systems
7.3 Structure of Nonholonomic Systems
Chapter 8 -Nonholonomic Motion Planning
8.1 Steering Model Control Systems Using Sinusoids
8.2 General Methods for Steering
•Fourier techniques
•Conversion to chained form
•Optimal steering of nonholonomic systems
•Steering with piecewise constant inputs
8.3 Dynamic Finger Repositioning
11/19
12/03

40. Higher-order systems: chained form systems
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
2 input system:
One-chain system

41. Higher-order systems: chained form systems
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
One-chain system

42. Higher-order systems: chained form systems
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
One-chain system

43. Higher-order systems: chained form systems
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
One-chain system

44. Higher-order systems: chained form systems
Chapter 8 –Nonholonomic Motion Planning
Steering Model Control System Using Sinusoids
One-chain system

45. Conversion to chained system
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Control system:
The system is in chained form in the z
coordinates with input v.

46. Conversion to chained system
The system can be converted into one-chain form if these distributions are all constant
rank and involutive, and there exists a function such that:
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Sufficient conditions for the 2 input case, in which case the system is to be converted
into the one-chain form:

47. Conversion to chained system
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Chosen independent of :

48. Conversion to chained system
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example 2: Kinematic car
One-chain system

49. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

50. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

51. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

52. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

53. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

54. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

55. Optimal steering of nonholonomic systems
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

56. Optimal steering of nonholonomic systems
The solution of this equation is of the form:
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Optimal inputs of Engel’s system

57. Optimal steering of nonholonomic systems
The solution of this equation is of the form:
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Optimal inputs of Engel’s system

58. Optimal steering of nonholonomic systems
The optimal inputs for the Engel’s system come from elliptic function.
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Optimal inputs of Engel’s system
Elliptic integral

59. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
We consider a nilpotent Lie algebra of order k generated by the vector fields
The Philip Hall basis of the controllability Lie algebra generated by are
defined as

60. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
All flows of the nonlinear control system:
are of the form
for some suitably chosen functions known as the Philip Hall coordinates.
The mean of (8.22) is:
All flows that could possibly be generated by the control system may be obtained by
composing flows along the Philip Hall basis elements

61. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
All flows of the nonlinear control system:
are of the form
for some suitably chosen functions known as the Philip Hall coordinates.
The mean of (8.22) is:
All flows that could possibly be generated by the control system may be obtained by
composing flows along the Philip Hall basis elements

62. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
satisfies a differential equation involving the basis elements:

63. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Differentiating equation:
We introduced the notation:

64. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

65. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

66. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Nilpotent system of degree three with two inputs

67. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Nilpotent system of degree three with two inputs

68. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering
Example: Nilpotent system of degree three with two inputs

69. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

70. Steering with piecewise constant inputs
Chapter 8 –Nonholonomic Motion Planning
General Methods for Steering

71. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning

72. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Problem description
The overall constraints on the system have the form:

73. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Problem description

74. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Steering using sinusoids

75. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Steering using sinusoids

76. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Steering using sinusoids
The location of the contact on
the finger is unchanged
The location of the contact on
the face of the object

77. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Geometric phase algorithm

78. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Geometric phase algorithm

79. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Geometric phase algorithm

80. Chapter 8 –Nonholonomic Motion Planning
Dynamic Finger Repositioning
Geometric phase algorithm

81. Thank you!