עקיבה אחר מטרה נעהעקיבה אחר מטרה נעהStable tracking control method for a mobile robotStable tracking control method for a mobile robot

מנחה : ולדיסלב זסלבסקימנחה : ולדיסלב זסלבסקי

: :מציגיםמציגיםרונן ניסיםרונן ניסים

מרק גרינברגמרק גרינברג

1.1 Project objective1.1 Project objective Kinematical analysis of the system in order to determine it's Kinematical analysis of the system in order to determine it's

stability and controllability properties.stability and controllability properties.

Design of a non-linear stabilizing controller by applying direct Lyaponov method. Design of a non-linear stabilizing controller by applying direct Lyaponov method.

Modification of the controller to suit the robot’s dynamic constraints.Modification of the controller to suit the robot’s dynamic constraints.

1.2 Project outline.1.2 Project outline. This Project is based on the work of This Project is based on the work of YY.Kanayama, .Kanayama,

Y.Kimura, F.Miyazaki and T.NoguchiY.Kimura, F.Miyazaki and T.Noguchi "A stable tracking "A stable tracking control method for an autonomous mobile robot" control method for an autonomous mobile robot"

This project deals with the control law of a vehicle that’s This project deals with the control law of a vehicle that’s tracking a moving target.tracking a moving target.

The controller manufactures the linear and angular speed The controller manufactures the linear and angular speed outputs in order to place the robot near the targetoutputs in order to place the robot near the target

The nature of this control system is nonlinear. As in Many The nature of this control system is nonlinear. As in Many control problems of this kind, the controller will be designed control problems of this kind, the controller will be designed using a Lyaponov function.using a Lyaponov function.

The controller parameters, efficiency and error The controller parameters, efficiency and error characteristics will be determined through simulations.characteristics will be determined through simulations.

1.2 Project outline. – Cont’1.2 Project outline. – Cont’

A linearized version of the controlled system will A linearized version of the controlled system will be studied in order to obtain the relation between be studied in order to obtain the relation between the controller parameters and characteristics like the controller parameters and characteristics like overshoot and settling timeovershoot and settling time

In order to examine our controller model in more In order to examine our controller model in more realistic environment, velocity limiters and realistic environment, velocity limiters and dynamics will be applied do the model. dynamics will be applied do the model. Appropriate simulations will demonstrate the Appropriate simulations will demonstrate the updated system performance updated system performance

1.3 Theoretical background1.3 Theoretical background

Lyapunov Stability Theorem Lyapunov Function

1.3 Theoretical background1.3 Theoretical background

Non-holonomic systems

A non-holonomic system is a system whose motion is restricted by non-holonomic constraints.

Definition: A non-holonomic constraint is a limitation on the allowable velocities of an object. For example, two wheeled robots: the robot can move in some directions (forwards and backwards), but not others (side to side).

We write a constraint equation for this kinematical system: cos sin 0y x

What does this equation tell us? It tells us the direction we can’t move inSo if θ=0, then the velocity in y = 0 ; if θ =90, then the velocity in x = 0.

2.1 2.1 Problem definition Problem definition

Consider the following 2D representation of the target and controlled vehicle:Consider the following 2D representation of the target and controlled vehicle:

2.1 2.1 Problem definition – Cont’ Problem definition – Cont’ In the robot’s frame of reference, the error coordinates Pe are :In the robot’s frame of reference, the error coordinates Pe are :

2.2 2.2 Nonlinear controller design Nonlinear controller design..

Our aim is to minimize Pe. For that purpose we shall Our aim is to minimize Pe. For that purpose we shall examine the following proposed Liapunov equation:examine the following proposed Liapunov equation:

In order for this equation to be definite semi-negative, we can In order for this equation to be definite semi-negative, we can

simply determine:simply determine:

1cosc r e eV V x K

2sinc r e r y ey V K K

Which is our control ruleWhich is our control rule

y

eee KyxV

cos1

2

1 22

2.2 2.2 Nonlinear controller design Nonlinear controller design..

Block diagram of the basic layout of the tracking system:Block diagram of the basic layout of the tracking system:

2.2 2.2 Nonlinear controller design Nonlinear controller design..

A simulation of this controller :A simulation of this controller :

2.3 Controller parameter analysis2.3 Controller parameter analysis

For Larger Values of K, the tracking For Larger Values of K, the tracking vehicle performs better.vehicle performs better.

1cosc r e eV V x K

2sinc r e r y ey V K K

Each K value has different impact on the controller’s Each K value has different impact on the controller’s velocity accommodation to the reference trajectory. velocity accommodation to the reference trajectory.

Larger values of K will close the gap more rapidly, enabling Larger values of K will close the gap more rapidly, enabling the system to cope with higher frequency inputs.the system to cope with higher frequency inputs.

1cosc r e e

Horizontal gaplinear velocityclosingconservation

V V x K

2sinc r e r y e

anglar velocity Angular gapVertical gapconservation closingclosing

y V K K

2.3 Controller parameter analysis2.3 Controller parameter analysis

To illustrate this, let us now view the vehicle's reaction to high reference speed withTo illustrate this, let us now view the vehicle's reaction to high reference speed with

varying gain parameters: varying gain parameters:

2.32.3 LinearizationLinearizationMotivation

Linearizing the system will benefit us in a number of ways:

We can use the linearization method of Liapunov in order to prove that the controlled system is indeed stable : :

We would like to examine the performance of the linearized system in comparison to that of the original system to show how accurately the linearized system represents the original one

Linearization enables us to use simple tools from the theory of linear system in order to characterize the system by overshoot, settling time and bandwidth, which of course isn't possible in the nonlinear system as these concepts lose their meaning.

2.32.3 LinearizationLinearizationSystem linearizationthe state-space representation of the controlled closed loop system is given by:

Linearizing the system around the origin :

1 1 1

1

2 2 2

2

3 3 30

0

0

0

0

0

e e er

linlin lin line r re

e e er y

Xee e e Ye

e

f f f

x yK

f f fP A P A V

x yV K K

f f f

x y

1

2

3

, ,

, ,

, ,

e e c c

e e e c c

e e c c

x f P V

P y f P V

f P V

2.32.3 LinearizationLinearization

2coslinc x e r e r y e t e e

linc r r y e t e

V K x V V K y K y

V K y K

System linearization

By comparing the linear and nonlinear error vector around the equilibrium, we extract the new: ,c cV

These changes perform the linearization in practice.

(Using Routh-Horowitz algorithm, we get that, for all K values, all the eigenvalues of the matrix A are negative i.e.

A is negative definite ,)thus, According to Lyaponov linearization method, the origin is a globally asymptotic equilibrium point) .

2.32.3 LinearizationLinearizationA simulation illustrating a comparison between the linear and nonlinear controller performance:

2.32.3 LinearizationLinearizationA simulation illustrating the limited convergence region of the linearized system:

2.32.3 LinearizationLinearizationA simulation illustrating a comparison between the linear and nonlinear controller performance:

The simulation demonstrates the limited convergence region of the linearized system.

2.32.3 LinearizationLinearizationWe can determine the location of each pole by Applying the following:

By approximating the linearized system to a second order LTI system, we can link the controller parameters with bandwidth, overshoot and settling time requirements.

1 2 0

2 2 21 2 0 0

2 2 22 1 0

: 2

: 2

:

r r y

r y r

A K K

B K K V K

C K K K V

We can determine the location of each pole by Applying the following :

1 2 0

2 2 21 2 0 0

2 2 22 1 0

2

2r r y

r y r

K K

K K V K

K K K V

By approximating the linearized system to a second order LTI system, we can link the controller parameters with bandwidth, overshoot and settling time requirements.

2.32.3 LinearizationLinearization

2.4 Additional features intended for 2.4 Additional features intended for the robot application the robot application

Distance between wheels

rV lV

The considered robotic vehicle has two wheels (left and right) each of which is spinning in a different velocity. Creating the new state variables: , .

The dynamics of such vehicle are given by:

11

t

Command CommandV t V e V VV

Where V is the current velocity and Vcommand is the requested (by the controller) velocity

2.4 Additional features intended for 2.4 Additional features intended for the robot aplication the robot aplication

Applying the Dynamics to each of the wheels:

We get the Following dependency on Tao

2.4 Additional features intended for 2.4 Additional features intended for the robot aplication the robot aplication

Now, adding limitors to each wheel’s velocity :

2.4 Additional features intended for 2.4 Additional features intended for the robot application the robot application

This dynamic response means that there are two additional state variables. Therefore an adjustment of the Lyapunov function and of the controller is needed to stabilize the new system:

2222

2

1cos1

2

1ee

y

eee V

KyxV

The above is one of the possible Lyapunov functions leading to a possible controller.

2.5 Concludion 2.5 Concludion

The controller was developed using classic The controller was developed using classic design methods.design methods.

Our controller is simple and stabilizes the system Our controller is simple and stabilizes the system globally-asymptotically once the target is in globally-asymptotically once the target is in motion.motion.

We have explored the limits of our controller in We have explored the limits of our controller in face of more realistic constraints such as face of more realistic constraints such as velocity limiters and dynamic inhibition.velocity limiters and dynamic inhibition.

Possible direction for future projects:Possible direction for future projects:

Implementation on a real lab robot.Implementation on a real lab robot.

2.6 Thank you! 2.6 Thank you!

We would like to thank our instructor Vladislav for his help and guidance and also We would like to thank our instructor Vladislav for his help and guidance and also to all the control lab staff. to all the control lab staff.