Model Abstraction in Dynamical Systems: Application to Mobile Robot Control

126 Pages·2008·2.35 MB·English

by  Patricia Mellodge

Lecture Notes
in Control and Information Sciences 379
Editors:M.Thoma,M.Morari
Patricia Mellodge, Pushkin Kachroo
Model Abstraction in
Dynamical Systems:
Application to Mobile
Robot Control
ABC
SeriesAdvisoryBoard
F.Allgöwer,P.Fleming,P.Kokotovic,
A.B.Kurzhanski,H.Kwakernaak,
A.Rantzer,J.N.Tsitsiklis
Authors
Dr.PatriciaMellodge
DepartmentofElectricalandComputerEngineering
UniversityofHartford
200BloomfieldAvenue
WestHartford,CT06117
USA
E-Mail:[email protected]
Dr.PushkinKachroo
UniversityTransportationCenter
HowardR.HughesCollegeofEngineering
4505MarylandParkway
LasVegas,NV89154-4007
USA
&
DepartmentofElectricalandComputerEngineering
UniversityofNevada,LasVegas
4505S.MarylandPkwy
LasVegas,NV89154-4026
USA
E-Mail:[email protected]
ISBN978-3-540-70792-9 e-ISBN978-3-540-70799-8
DOI 10.1007/978-3-540-70799-8
LectureNotesinControlandInformationSciences ISSN0170-8643
LibraryofCongressControlNumber:2008930993
(cid:2)c 2008Springer-VerlagBerlinHeidelberg
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Contents
1 Introduction................................................ 1
1.1 Motivation .............................................. 1
1.2 Previous Research ........................................ 2
1.2.1 Modeling.......................................... 2
1.2.2 Nonholonomic Motion Planning and Control........... 3
1.2.3 Other Control Approaches........................... 4
1.3 Contributions of This Work................................ 4
1.4 Organizationof This Work ................................ 4
2 Mathematical Preliminaries ................................ 7
2.1 Differential Geometric Description of Systems ................ 7
2.2 Control System Properties ................................ 17
2.3 Nonholonomic Systems ................................... 20
2.4 Chained Form ........................................... 24
3 Kinematic Modeling and Control .......................... 27
3.1 Modeling Approach....................................... 27
3.2 Kinematic Modeling ...................................... 28
3.2.1 Nonholonomic Contraints ........................... 29
3.2.2 Frames of Reference ................................ 30
3.2.3 Unicycle Model .................................... 30
3.2.4 Global Coordinate Model ........................... 31
3.2.5 Path Coordinate Model ............................. 32
3.3 Control Design .......................................... 34
3.3.1 Chained Form ..................................... 34
3.3.2 Derivation......................................... 35
3.4 Curvature Estimation..................................... 36
3.5 Estimation Methods ...................................... 38
3.5.1 Estimation Based on the Steering Angle φ ............ 38
3.5.2 Estimation Based on the Vehicle Kinematics .......... 38
3.6 Simulation Results........................................ 39
3.6.1 Control Using the Actual Curvature .................. 39
VI Contents
3.6.2 Control Using the φ Estimator ....................... 41
3.6.3 Control Using the Model Estimator................... 46
4 Vision Based Modeling and Control ........................ 49
4.1 Image Dynamic Modeling ................................. 49
4.1.1 Ground Curves..................................... 49
4.1.2 Image Dynamics ................................... 50
4.2 Unicycle Controller ....................................... 55
4.3 Simulation Results........................................ 56
5 Abstraction ................................................ 61
5.1 Preliminary Definitions.................................... 61
5.2 Consistent Control System Abstractions .................... 67
5.3 Application to the Robotic Car and Unicycle................. 68
5.4 Traceable Control Systems................................. 70
5.5 Traceability and Differential Flatness ....................... 76
6 Control Design ............................................. 81
6.1 Problem Statement ....................................... 81
6.2 Existence and Uniqueness of the Input Transformation ........ 81
6.2.1 Special Case ....................................... 82
6.3 Application to the Car/Unicycle System..................... 84
6.4 Connection with Traceability .............................. 85
7 Open-Loop Control Design ................................. 87
7.1 Input Transformation ..................................... 87
7.2 Open-Loop Optimal Control .............................. 88
7.3 Simulation .............................................. 94
8 Uncertainty Propagation in Abstracted Systems ........... 97
8.1 The Liouville Equation and Its Solution .................... 99
8.1.1 Derivation of the Liouville Equation .................. 99
8.1.2 Solution to the Liouville Equation .................... 100
8.2 Definition of the Φpdf Mapping ............................ 102
8.3 Example of Φ-Related Linear Systems ...................... 103
8.4 Autonomous Φ-Related Systems ........................... 105
8.5 The Liouville Equation for Control Systems ................. 106
8.6 Φ-Related Control Systems ............................... 108
9 Conclusion ................................................. 111
Bibliography................................................ 113
List of Figures
1.1 A vector traveling along a curve on a Mo¨bius strip............. 8
1.2 The local representation of an open set on a sphere by a
plane .................................................... 9
1.3 C∞-compatible coordinate charts (U,φ) and (V,ψ) ............ 10
1.4 The manifold for Example 2.8............................... 11
1.5 The geometric interpretation of a tangent space ............... 12
1.6 Vector fields on a manifold ................................. 13
1.7 An integral curve on manifold M ............................ 14
1.8 The interpretation of the Lie derivative ...................... 16
1.9 An example of a controllable linear system ................... 18
1.10 The geometric and kinematic constraints of a robotic arm ...... 21
1.11 The unicycle moving in a plane ............................. 22
1.12 The unicycle can move from position A to position B even
though its velocity is restricted.............................. 23
3.1 The control hierarchy of a mobile robot ...................... 28
3.2 The velocity constraints on a rolling wheel with no slippage..... 29
3.3 Top view of the global and mobile robot frames ............... 30
3.4 The model for the unicycle. (a) Top view. (b) Side view. ....... 31
3.5 The global coordinate system for the car ..................... 31
3.6 The path coordinates for the car ............................ 33
3.7 A sample path showing the constraints....................... 36
3.8 The curvature of the path in Figure 3.7 with respect to the
path length s ............................................. 37
3.9 The path used for simulation. Adapted from [24]............... 40
3.10 The curvature profile of the path shown in Fig. 3.9. Adapted
from [24].................................................. 40
3.11 The states, x2, x3, and x4, resulting from using the actual
errors and curvature. Adapted from [24]. ..................... 41
3.12 The control inputs, v1 and v2, resulting from using the actual
errors and curvature. Adapted from [24]. ..................... 41
VIII List of Figures
3.13 The heading angle, θp, and steering angle, φ, resulting from
using the actual errors and curvature. Adapted from [24]........ 42
3.14 The states, x2, x3, and x4, resulting from using the discretized
errors. Adapted from [24]. .................................. 42
3.15 The control inputs, v1 and v2, resulting from using the
discretized errors.Adapted from [24]. ........................ 43
3.16 The heading angle, θp, and the steering angle, φ, resulting
from using the discretized errors.Adapted from [24]............ 43
3.17 The curvature estimated using only the steering angle, φ, with
θp initially nonzero. Adapted from [24]. ...................... 44
3.18 The car’s states resulting from the using the φ estimator.
Adapted from [24]. ........................................ 44
3.19 The control inputs resulting from the using the φ estimator.
Adapted from [24]. ........................................ 45
3.20 The heading angle, θp, and steering angle, φ, resulting from
using the φ estimator. Adapted from [24]. .................... 45
3.21 The curvature determined by thresholding aˆ with θp initially
nonzero. Adapted from [24]. ................................ 46
3.22 The car’s states resulting from the using the model estimator.
Adapted from [24]. ........................................ 46
3.23 The control resulting from the using the model estimator.
Adapted from [24]. ........................................ 47
3.24 The heading angle, θp, and steering angle, φ, resulting from
using the model estimator. Adapted from [24]. ................ 47
4.1 The camera frame of reference and its relation to the robot
frame .................................................... 50
4.2 Representation of point P on the ground curve in the camera
coordinates ............................................... 51
4.3 The circular path to be followed by the unicycle............... 56
4.4 The spiral path to be followed by the unicycle ................ 57
4.5 The inputs for the circular path ............................. 57
4.6 The states for the circular path ............................. 58
4.7 The unicycle’s path following the circular path ................ 58
4.8 The inputs for the spiral path............................... 58
4.9 The states for the spiral path ............................... 59
4.10 The unicycle’s path following the spiral path.................. 59
5.1 The relationshipbetweenvectorfields inM andN.The vector
fields X and Y are Φ-related if TΦ(X(p))=Y(Φ(p)). .......... 62
5.2 Local description of a control system. Adapted from [44]........ 63
5.3 Global description of a control system. Adapted from [44]....... 63
5.4 Fiber bundle. Adapted from [44]............................. 64
5.5 The mapping of vector fields in SM to SN .................... 66
5.6 A surface in the car’s manifold .............................. 71
5.7 A unicycle path that the car cannot follow.................... 72
List of Figures IX
5.8 Γ1 and Γ2 are possible paths that the car can follow ........... 72
5.9 The car with n trailers. Adapted from [49]. ................... 79
7.1 The unicycle trajectory for the car to follow .................. 89
7.2 The car follows an inscribed circle based on its maximum
steering angle............................................. 91
7.3 The car follows two circles based on its maximum steering
angle .................................................... 92
7.4 The intersection of the inscribed circle with the unicycle’s
trajectory ................................................ 93
7.5 The method of sequential circles for the car to follow the
unicycle’s trajectory ....................................... 94
7.6 The trajectory resulting from the initial inscribed circle with
w1 =1 and w2 =0 ........................................ 96
7.7 The car’s inputs resulting from the initial inscribed circle with
w1 =1 and w2 =0 ........................................ 96
8.1 The relationship between Φ-related systems with known initial
conditions ................................................ 98
8.2 The relationship between Φ-related systems with uncertain
initial conditions .......................................... 99
8.3 The area under ρ(x,t) must be conserved. The change in
shaded area is equal to the difference between the flow into
and out of the area Δx. ................................... 100
8.4 The time evolution of the pdf ρY(y,t) with initial Gaussian
distribution............................................... 105
8.5 The relationship between Φ-related control systems with
uncertain initial conditions ................................. 107
8.6 The relationship between the endpoints of trajectories of SX
and SY .................................................. 109
List of Tables
7.1 The value of the cost function J for various weights and
maximum steering angles................................... 95
7.2 The value of the cost function J for the different initial
inputs ................................................... 95
7.3 The number of iterations for the different initial inputs......... 95