Leng-Feng Lee Dec 3, 2004 Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo Leng-Feng Lee ([email protected]) Advisor : Dr

Leng-Feng Lee Dec 3, 2004Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo

Leng-Feng Lee ([email protected])

Advisor : Dr. Venkat N. Krovi

Mechanical and Aerospace Engineering Dept.

State University of New York at Buffalo

Decentralized Motion Planning within an Artificial Potential Framework (APF) for Cooperative

Payload Transport by Multi-Robot Collectives


Agenda

Motivation & System Modeling

Literature Survey & Research Issues

Part I

Part II

Local APF & limitations

Global APF-Navigation Function

Case Studies-Single robot with APF

Dynamic Formulation-Group of Robots

Motion Planning-Three Approaches

Case Studies-Multi Robots with APF

Performance Evaluation of Three Approaches

Conclusion & Future Work


MotivationExamples of Multi-robot groups:

– Tasks are too complex;– Gain in overall performance;– Several simple, small-sized robot are easier, cheaper to built,

than a single large powerful robot system;– Overall system can be more robust and reliable.

Group Cooperation in Nature:

Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

Schools of Fish Armies of Ants Flocks of Birds

How do we incorporate similar cooperation in artificial multi robot group?


Motivation

Example of Multi robot groups:

Cooperative payload transport


Robots in formation

Robots in group


Why Motion Planning?

– To realize all the functionalities for mobile robots, the fundamental problem is getting a robot to move from one location to another without colliding with obstacles.

Motion Planning (MP) for Robot Collectives

Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion

Definition:

– The process of selecting a motion and the associated set of input forces and torques from the set of all possible motions and inputs while ensuring that all constraints are satisfied.

MP for Robot Collective -

– MP exist for individual robots such as manipulator, wheeled mobile robot (WMR), car-like robot, etc.

– We want to examine extension of MP techniques to

Robot Collectives


Explicit Motion Planning:

– Decompose MP problem into 3 tasks:– Path Planning, Trajectory Planning, & Robot Control;

– Example: Road Map Method, Cell Decomposition, etc.

Motion Planning Algorithm Classification


Implicit Motion Planning:

– Trajectory and actuators input are not explicitly compute before the motion occur.– Artificial Potential Field (APF) Approach belongs to this category.– Combine Path Planning, Trajectory Planning, and Robot Control in a single framework.


Motion Planning (cont’)


Artificial Potential Field (APF) Approach:

– Obstacles generated a artificial Repulsive potential and goal generate an Attractive potential.

– Motion plan generated when attractive potential drives the robot to the goal and repulsive potential repels the robot away from obstacles.

– Combine Path Planning, Trajectory Planning, and Robot Control in one framework.

Subclass of Implicit Motion Planning Algorithm


Research Issues

Specific Research Questions:

– Which type of potential function is more suitable for MP for multi robot groups?

– How can we use the APF framework to help maintain formation? and

– How this framework be extended to realize the tight formation requirement for cooperative payload transport?


Broad Challenges:

– Extending APF approach for Multi-robot collectives.

– Ensuring tight formations required for Cooperative Payload Transport application.


Research Issues (cont’)

Part I:

– Study various APF & their limitations;

– Determined a suitable APF as our test bed;

– Create a GUI to design and visualize the potential field;

– Case studies: MP for single robot using APF approach.


To answer these research questions:

Part II:– E.O.M. for group of robots with formation constraints;– Solved the MP planning problem using three approaches; – Performance evaluation using various case studies.


Research Issues (cont’)


Hierarchical difficulties in MP:

Our results:

– Multiple point-mass robots;

– Sphere World;

– Stationary Obstacles & Target.

(Dynamic Model)


System ModelingIndividual level system models include:

– Point Mass Robot;– Differentially Driven Nonholonomic Wheel Mobile Robot (NH-WMR).


,T

x yq , ,T

x y q


System Modeling (cont’)

1 1 1 2 2 2 3 3 3, , , , , , , ,T

x y x y x y q


Group level system model is formed using:

– Point Mass Robot;– Differentially Driven Nonholonomic Wheel Mobile Robot.

1 1 2 2 3 3, , , , ,T

x y x y x yq


PART I: Artificial Potential Approach

Examine:

– Variants of APF & their limitations;

– Navigation function ;

– Single module formulations;

– Simulation studies.


Local APF -background

Artificial Potential Field Approach– Proposed by Khatib in early 80’s.– FIRAS Function. [Khatib, 1986]

Later, various kind of Potential Functions were proposed:

– GPF Function. [Krogh, 1984]

– Harmonic Potential Function. [Kim, 1991]

– Superquadric Potential Function. [Khosla, 1988]

– Navigation Function. [Koditschek, 1988]

– Ge New Potential. [Ge, 2000]



Local APF Approach-Formulation

Idea:

– Goal generate an attractive potential well;

– Obstacle generate repulsive potential hill;

– Superimpose these two type of potentials give us the total potential of the workspace.


) ( )( )( AtTot l pta ReUU U q q q

( )TotalU q denote the total artificial potential field;

( )AttU q denote the attractive potential field; and

( )RepU q is the repulsive potential field.

,T

i ix yq

Where:

is the position of the robot.


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Target

Attractive Potential Field Gradient Plot

x-Position

y-P

ositio

n

Local APF -Attractive potential

Characteristics:

– Affect every point on the configuration space;

– Minimum at the goal.

– The gradient must be continuous.




Example 1:


Where:

1( )

2

m

Att Tar RobU p q q

Tar Robq q = Euclidean distance between the robot and the target

Tarq

Robq

= Position of the target.

= Position of the robot.

= Positive scaling factor

2m is commonly used.



Example 2:


Where:

For distance smaller than s, conical well. For distance larger than s, constant attractive force.

k = Positive scaling factor

2

2

, ( )

2 , Tar Rob Tar Rob

Att

Tar Rob Tar Rob

sU p

ks ks s

q q q q

q q q q


-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Obstacle

Repulsive Potential Field Gradient Plot

x-Position

y-P

ositi

on

Local APF -Repulsive potential

Characteristics:– The potential should have

spherical symmetry for large distance;

– The potential contours near the surface should follow the surface contour;

– The potential of an obstacle should have a limited range of influence;

– The potential and the gradient of the potential must be continuous.




Example 1 - FIRAS Function:


Where:

= Positive scaling factor

2

00

0

1 1 1, if

2

0 , if

RORep RO

RO

U

q

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x position

y po

sitio

n

Contour plot of FIRAS Function of a Square Obstacle

Obstacle

1

2

3

4

5

6

7

8

9

RO Obs Rob q q

= the shortest Euclidean distance between the robot from the obstacle surface

RO



Example 2 - Superquadric Potential Function:– Approach Potential;– Avoidance Potential.– Avoid creation of local minima result from flat surface by creating

a symmetry contour around the obstacle.




Example 3 - Harmonic Potential Function:


20 20

– Superimpose of another harmonic potential is also a harmonic potential. – More complicated shape can be modeled using ‘panel method’.

Repulsive Potential Attractive Potential

log2

r

Detail



Example 4 - Ge New Function:


Where:

–Modified from FIRAS function to solve the ‘Goal NonReachable for Obstacle Nearby’ -GNRON problem. – Ensures that the total potential will reach its global minimum, if and only if the robot reaches the target where

2

00

0

1 1 1, if

2

0 , if

nRT RO

Rep RO

RO

U

q

RT = Minimal Euclidean distance from robot to the target.

0RT 0 2 4 6 8 10 12 14 16 18 20

0

2

4

6

8

10

12

14

16

18

20

TargetObstacle

x position

y po

sitio

n

5

10

15

20

25

30

Obstacle



Potential Function with Velocities Information:

– Some potential function include the velocities information of the robots, obstacles and target.

– Example: Ge & Cui Potential [Dynamic obstacle & Target]. – Provide a APF for dynamic workspace.– Example: GPF Function. [Dynamic obstacles only].

– Can be used with our formulation for group of robots for motion planning in dynamic workspace.



Local APF –Total Potential

Total Potential of Workspace:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Tar

Target

Obstacle

Total Potential Field Gradient Plot

x-Position

y-P

ositi

on


( ) ( ) ( )Total Att Repi jU U U q q q

– Superimpose different repulsive potential from obstacles and different attractive potential from the goal, we get the total potential for the workspace.

– At any point of the workspace, the robot will reach the target by

following the negative gradient flow of the total potential.

( ) , , .Rep jU FIRAS Function Harmonic Function etcq

1( ) , .

2

m

Att Tar RobiU etc

q q q


Local APF –Total Potential

Example: FIRAS Function


-3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

4

x positiony

posi

tion

Contour plot of Potential Field Generated Using FIRAS Function

Obs

Obs

Target 5

10

15

20

0, 0 2, 2

1.5, -1.5

Rectangular Obstacle:

2 unit in height, 1 unit in width.

Circular Obstacle: Radius

Target :

0.5

More


Local APF –Limitations

Local Minimum - result from single obstacle


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Target

Contour Plot of Attractive Potential Field

x position

y p

ositio

n

2

4

6

8

10

12

14

16

18

20

22

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x position

y po

sitio

n

Contour plot of Repulsive Potential

Obstacle

1

2

3

4

5

6

7

8

9

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Local Minima

ObstacleTarget

x position

Contour plot Total Potential

y po

sitio

n

2

4

6

8

10

12

3D View



Local Minimum - result from multiple obstacles


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Target

Obstacle

Local Minima

x position

y po

sitio

n

2

4

6

8

10

12

14

16


-1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

Target(0.0, 0.0)

Influence Range ofRepulsive Potential

Obstacle

Goal Nonreachable with Obstacle Nearby, GNRON Problem

x position

y po

sitio

n


Limitation - Target close to obstacle:


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x position

y po

sitio

n

Contour Plot Showing GNRON Problem

Target

Local Minima(-0.4, 0.0)

Obstacle

1

2

3

4

5

6

7

8

9

10

11


Local APF -Limitations

Some other limitations include:

– No passage between closely spaced obstacle.– Non optimal path.– Implementation related limitations.

• Oscillation in the presence of obstacle;• Oscillation in narrow passages;• Infinite torque is not possible.



Global APF – Navigation Function

Properties:– Guarantee to provide a global minimum

at target.– Bounded maximum potential.


Let be a robot free configuration space, and let

Tarq be a goal point in the interior of , A map : [0,1] is a Navigation Function if it is:

1. Smooth on , that is, at least a 2 function.

2. Polar at Tarq ,i.e., has a unique minimum at Tarq on the path-connected

containing Tarq

.

component of

3. Admissible on , i.e., uniformly maximal on the boundary of

.

4. A Morse Function

[ Proposed by: Rimon & Koditschek]


Navigation Function

Navigation Function of a sphere world :

2 2i i i q q q


Where:

2

1 1/2 for 0

( ) ( ) ( )

1 for 0

Tar

Total TarU

q q

q q q q q

2 20 0 0 q q q

0

M

ii

1,2i M Number of obstacles

is the implicit form of bounding sphere.

is the implicit form of obstacle geometric Eq.

Feature: Tunable by a single parameter :

Detail


Navigation Function

Example - Navigation Function of a sphere world :

0.4 4.0 Introduction APF Approaches Group Robot Formulation Simulation Results Conclusion

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

BoundedWorkspace

ConfigurationSpace

Obstacles

Target

A Euclidean Sphere World with 4 Obstacles

x Position

y P

ositi

on

Where:


Navigation Function -Constructions

At low value of , local minima may exist:


-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

x position

y po

sitio

n

Contour Plot of Navigation Function with = 2.6

Target

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

-3

-2

-1

0

1

2

3

4

5

x position

y po

sitio

n

Contour Plot of Navigation Function with = 3.6

Target

3.6 2.6


Navigation Function – MATLAB GUI

A GUI to properly select a value:



APF Approach – Formulation & Simulation

Idea:– We want the robot to follow the

negative gradient flow of the workspace potential field;

– Analogy to a ball rolling down to the lowest point in a given potential.

– Thus the gradient information will serve as the input to the robot system.



APF Approach – Formulation

Formulation – Single point-mass robot:


Kinematic Model:

f Uqq u K

f UqMq u Kq K Kq

Dynamic Model:

Uq is the gradient of the potential field

2 2f fkK I is a positive diagonal scaling matrix

Kq is dissipative term added to stabilize the system


APF Approach – FormulationFormulation – Nonholonomic Wheeled Mobile Robot (NH-WMR):


Kinematic Model:

desired y-direction velocity. the desired x-direction velocity.

1 2

1

2

cos 0

sin 0

0 1

cos sin

,

p d d

d d

x

y u u

u k x y

u k atan2 x y

q

d

Ux

x

d

Uy

y

is the projected gradient onto the direction of forward velocity.

is the proportional to the angular error between the gradient and robot direction.

1u

2u


APF Approach – Formulation

Formulation – Group robot without formation constraints:


Generalize position:

1 1 2 2, , , , ,T

n nx y x y x yq

Kinematic Model:

Dyanamic Model:

n -number of point-mass robot

f U qq u K

Mq u Kq f Uqu K


APF Approach – Simulations

Simulation 1 – Single robot with single obstacle:


f UqMq u Kq K Kq

2

00

0

1 1 1, if

2

0 , if

RORep RO

RO

U

qDetail


APF Approach – SimulationsSimulation 2 – Single robot with two obstacles:


f UqMq u Kq K Kq

2

00

0

1 1 1, if

2

0 , if

nRT RO

Rep RO

RO

U

qDetail



Simulation 3 – Single NH-WMR with four obstacles:


2

1 1/2 for 0

( ) ( )

1 for 0

Tar

Total TarU

q q

q q q q

Detail

1 2

cos 0

sin 0

0 1

x

y u u

q

More



Simulation 4 – Group robots without formation constraint:

1( )

2

m

Att Tar RobU p q q


f UqMq u Kq K Kq

Detail



Simulation 5 – Group robots without formation constraint:


1 1

q 0 I q 0

uq 0 M K q M

2

1/2 for 0

( ) ( )

1 for 0

Tar

Total TarU

q q

q q q qDetail


PART II: Group Robots Dynamic Formulation

Include:– Dynamic Formulation for Group of Robots with Formation;– Solved the E.O.M using three Methods;

– Simulation Studies;

– Performance evaluation of each Methods.


Group Robots Dynamic Formulation

Approaches for formation maintenance:

– Formation Paradigm• Leader-follower [Desai et. al., 2001]

• Virtual structures [Lewis and Tan, 1997]

• Virtual leaders [Leonard and Fiorelli, 2001], [Lawton, Beard et al., 2003]

Our Approaches:

– View as a constrained mechanical system.

– Formation constraints – holonomic constraints added to a unconstrained dynamic system.

– Motion planning now can be treated as a forward dynamic simulation of a constrained mechanical system.



The dynamic of group of robot can be formulated using Lagrange Equation by:


q v

, ,T

t M q v f q, v u J q λ , t C q 0

q is the n-dimensional vector of generalized coordinates

v is the n-dimensional vector of generalized velocities

is the n-dimensional vector of generalized velocities M q

, , ,tf q v u is the n-dimensional vector of external forces

u is the vector of input forces, which is f U qk

C q

J q =q

is the Jacobian matrix


(1)



The Lagrange Equation can be solved using following three methods:

– Method I: Direct Lagrange Multiplier Elimination Approach.• Explicitly computing the Lagrange multiplier by a projection into

the constrained force space.

– Method II: Penalty Formulation Approach.• Approximating the Lagrange multiplier using artificial compliance

elements such as virtual springs and dampers.

– Method III: Constraints Manifold Projection Based Approach• By projecting the equations of motion onto the tangent space of

the constraint manifold in a variety of ways to obtain constraint-reaction free equations of motions.




2 1

12 1 , ,n

Tn t

vq

q M f q, v u J λ

Method I: Direct Lagrange Multiplier Elimination Approach:

– The direct Lagrange multiplier elimination is a centralized approach where the Lagrange multiplier is explicitly calculated to ensure formation constraints are not violated.


(2)

(3)

The resulting Dynamic Equation can be expressed as:

2

1

2, ,t

t

-1 T -1 C

λ J q M J q J q M f q, v u J q q

Detail



Method II: Penalty Formulation Approach:

– The holonomic constraints are relaxed and replaced by linear/non-linear spring with dampers.

– Here, the Lagrange multipliers are explicitly approximated as the force of a virtual spring or damper based on the extent of the constraint violation and assumed spring stiffness and damping constant.

Resulting Dynamic Equation:


S D λ K C q K C qThis can be expressed as:

2 1

12 1 , , ,n

Tn S Dt

vq

q M f q v u J K C q K C q

(4)



1, , ,t

T T Tυ S MS S f q v u S Mγ

2 1

1

2 1 , , ,

n

n m t

T T T

Sυ+ ηq

υ S MS S f q v u S Mγ

Method III: Constraints Manifold Projection Based Approach:

– In this approach, the dynamic equation with constraint-reactions is projected into the tangent space (feasible motion subspace) to obtain the constraint free projected dynamics equations.


(5)

Thus, the resulting Dynamic Equation become:

Is the independent velocities. υ

Detail



Baumgarte Stabilization:– To prevent numerical drift in the simulation, we adopted

Baumgarte stabilization method. – Baumgarte stabilization method involves the creation of an artificial

first or second order dynamical system which has the algebraic position-level constraint as its attractive equilibrium configuration.

For example, the holonomic constraint of Eq.(1) is replaced with:


, 0 C q C q J q q C q 0

0tt e C C

Where the solution of the above equation is :


Case Study - Formulation:

Three point-mass robots forming a triangular shape:


where:

The governing Equation can be written as:

T

q = v

M q q +V q,q +G q E q u J λ

J q q C q = 0

2 2

2 2

2 2

0 0

0 0

0 0

A

B

C

M

M q M

M

A

B

B

q

q q

q

V q,q Kq Tf U qu K

6 6E q I

G q = 0

, , , , ,T

A A B B C Cx y x y x yq

We will use this model to perform various case studies.


Simulation & Results

i j

Performance Evaluation – Formation Error:


Formation Error:

2 2 2

Error AB AB BC BC CA CAc c c c c c

Error is the total formation error;

ijc is the actual Euclidean distance between robot i and robot j

ijc is the desired Euclidean distance between robot and robot


Simulations & Results

Case Study 1 – Three robots in formation, without obstacle:


1

T

vq

M u Kq J λq

Method I:

11 1

Tλ JM J JM u Kq J q q C q

Tf U qu K





Method II:

2 1

12 1

nT

n S

vq

q M u Kq J K C q

1

1

1

, ,4 1

, ,4 1

, ,4 1

AAT

A A A A A A S A D A AA

BBT

B B B B B B S B D B BB

CCT

C C C C C C C S C D C C

vq

M E u K q J K C K Cq

vq

M E u K q J K C K Cq

vq

q M E u K q J K C K C

Decentralized Formulation:





Method III:

1

T T T T

Sυ+ ηq

υ S MS S u S Mγ S Kq

1

2 2 2 2

1

2 2 2 2

1

2 2 2 2

A AA

TA A A A A A A

B BB

TB B B B B B B

B BC

TC C C C C C C

k

k

k

T

T

T

S υ ηq

υ S MS S I u M γ I q

S υ ηq


S υ ηq


Partial Decentralized Formulation:



0 1 2 3 4 5 6 7 8 9 100

1

2

3

4x 10

-14 Total Formation Error Using Method I

Tota

l F

orm

ation E

rror

Time, t0 1 2 3 4 5 6 7 8 9 10

0

0.005

0.01

0.015

0.02

0.025

0.03Total Error Using Method II

Time

Tota

l E

rror

Case Study 1 – Formation Error from three methods:


Method I

Method II

Method III

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7x 10

-5 Total Formation Error Using Method III

Tota

l F

orm

ation E

rror

Time, t



Case Study 1 – Formation Error & Effect of Ks


Method II:

0 1 2 3 4 5 6 7 8 9 1010

-7

10-6

10-5

10-4

10-3

10-2

10-1

Total Formation Error for Different Values of Ks using Method II

Time, t

Tota

l F

orm

ation E

rror

Ks=10

Ks=50

Ks=100

Ks=500

Ks=1000

Ks=5000

Ks=10000

Ks=10

Ks=50

Ks=100

Ks=500

Ks=1000

Ks=5000

Ks=10000



Case Study 1 – Formation Error & Effect of


Method II:

0 1 2 3 4 5 6 7 8 9 10

10-10

10-9

10-8

10-7

Total Formation Error for Different Values using Method III

Time, t

Tota

l F

orm

ation E

rror

=10=50=100=500=1000

=10

=50

=100



-5 -4 -3 -2 -1 0 1 2 3 4 5-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

Robots

RobotsFormations

Obstacle

Target

3 Robots carried a common payload - An 2D Workspace

x Position

y P

ositio

n

Case Study 2 – Three robots in Formation, one obstacle




Case Study 2 – Three robots in formation, one obstacle:


Method II Method III Method I



Case Study 2 – Formation Error from three methods:


Method I

Method II

Method III



0 1 2 3 4 5 6 7 810

-5

10-4

10-3

10-2

10-1

100

Time, t

Tot

al E

rror

Total Formation Error For Different value of

=10=100=300=500=700=1000

=100

=300

=500 =700

=1000

=10

Case Study 2 – Formation Error & Effect of Ks &


Method II Method III

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6x 10

-7

Time, t

Tot

al E

rror

Total Formation Error For Different value of

=10=20=30=40=50=60=70

=70

=60

=50

=40

=30



Case Study 2 – Three robots in Formation with Expansion.


2 0.5 , 0 4

4, 4ij

t tc

t

Each sides change from 2 units to 4 units in 4 seconds:



Case Study 3 – Three robots in Formation with Expansion.


Method II Method III Method I



Case Study 3 – Formation Error & Effect of Ks &


Method II

Method III


Simulation & ResultsCase Study 4 – Three robots in Formation with Shape Change.


Constraint between robot A & B change from 2 units to 4 units in 4 seconds:

2 0.5 , 0 4

4, 4AB

t tc

t

Note: Method I cannot perform this task because when three robots in a straight line, the inverse of the Jacobian matrix become singular.



Case Study 4 – Three robots in Formation with Shape Change.


Method II Method III


Simulation & ResultsCase Study 4 – Formation Error & Effect of Ks &


Method II: Method III:


Conclusion

General Characteristics – Formation Accuracy


The average total formation error for each method :

Method III Method I Method IIError Error Error


Conclusion

Method I Method II Method IIITime Time Time

General Characteristics – Computational Time


The average total Computational Time (sec) for each method :


Conclusion

Method I Method III

General Characteristics – Decentralize formulation capability


The decentralize formulation capability for each method :

Centralized Decentralized

Method II


Conclusion


General Characteristics – Formation related concerns:

– The Jacobian matrix in Method I and Method III can become singular in some specific position.

– Method II has no such limitations.

In Summary:


Conclusion


– Evaluation of various potential functions.

– Development of a GUI to generate navigation function.

– Develop the group motion planning problem as a forward dynamic simulation problem;

– Evaluation of three different method in solving motion planning problem for a group of robots in formation.

– Critical evaluation of the performance by the three approaches.


Future Work


– Provide a way to avoid Jacobian matrix become singular.

– Incorporate nonholonomic constraints in the formulation.

– Implement a more efficient gradient finding method by utilizing the available information from each robot.

– Implement the algorithm in a decentralized computation manner.


Thank You!Acknowledgments:

Dr. V. Krovi, Dr. T. Singh & Dr. J. L. Crassidis

Documents

Leng-Feng Lee Dec 3, 2004 Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo Leng-Feng Lee ([email protected]) Advisor : Dr