CONTROL OF THE NONHOLONOMIC INTEGRATORbanavar/lectures/twente.pdf · Control of the Nonholonomic...

June 6, 2005

CONTROL OF THE NONHOLONOMIC INTEGRATOR

R. N. Banavar

(Work done with V. Sankaranarayanan)

Systems & Control Engg.

Indian Institute of Technology, Bombay

Mumbai -INDIA.

[email protected]

Control of the

Nonholonomic Integrator A Talk at University of Twente

Outline

• Nonholonomic systems - a brief introduction

• Nonholonomic integrator (NI) (Kinematic model of a wheeled mobile

robot)

• Four stabilizing control laws

• The Extended Nonholonomic Double Integrator (ENDI) - Dynamic

model of a wheeled mobile robot

• A Sliding mode controller

• Other problems

1

Control of the


Classification of constraints in mechanical systems

• Holonomic constraints - restrict the allowable configurations of the

system

• Nonholonomic constraints - do not restrict the allowable configurations

of the system but restrict instantaneous velocities/accelerations

– Velocity level constraints - parking of a car, wheeled mobile robots,

rolling contacts in robotic applications

– Acceleration level constraints - fuel slosh in spacecrafts/launch

vehicles, underwater vehicles, underactuated mechanisms (on

purpose or loss of actuator) systems - serial link manipulators

2

Control of the


A coin rolling on a horizontal plane

x

y

θ

φ

Figure 1: Vertical coin on a plane

3

Control of the

Nonholonomic Integrator A Talk at University of TwenteThe Differential geometric view

• Rewriting the constraints in terms of annihilator codistributions

[sin(θ) − cos(θ) 0

] x

y

θ

= 0

• Permissible motions of the coin are such that the vector field is

annihilated by the codistribution

Ω =[

sin(θ) − cos(θ) 0]

• Can Ω be expressed as the gradient of a function

λ : (IR1 × IR1 × S1) → IR1 as

Ω =[

dλ]

4

Control of the


The NI or Brockett Integrator

• Nonholonomic integrator

x1 = u1

x2 = u2

x3 = x1u2 − x2u1

x ∈ IR3

• – Third order driftless system

– Equilibria - the whole of IR3

– Not linearly controllable at any equilibrium point

– No continuous feedback control law that can globally stabilize the

system

– Solution: Time varying feedback OR discontinuous feedback

– Mimics the kinematic model of a mobile robot with a nonholonomic

constraint

5

Control of the


Past work on the NI

• Motion planning - Given x(0) = x0 and xf and a time interval

[0, tf ], find a control history u(·) (from an admissible class of functions)

such that x(tf) = xf .

• Stabilization - Compute a feedback control law that stabilizes the

system in a region (set) near the origin

• Exhaustive survey - A. M. Bloch - Nonholonomic Mechanics and

Control -Springer , 2003

6

Control of the


A wheeled mobile robot on a horizontal plane

θ

τ1

τ 2

X

Y

(x,y)

Figure 2: Schematic of a mobile robot

7

Control of the

Nonholonomic Integrator A Talk at University of TwenteWheeled mobile robot

• Notation

M = Mass of the vehicle

I = Inertia of the vehicle

F =1

R(τ1 + τ2)

τ =L

R(τ1 − τ2)

L = Distance between the center of mass and the wheel

R = Radius of the rear wheel

τ1 = Left wheel motor torque

τ2 = Right wheel motor torque

• Nonholonomic constraint - no lateral (sideways) motion

x sin θ − y cos θ = 0

8

Control of the


Kinematics and Dynamics

• Generalized coordinates - position of the center of mass (x, y) and

orientation θ

• Constraints of motion

x sin θ − y cos θ = 0 No lateral motion

• Equations of motion

θ = ω x = v cos θ y = v sin θ

• Control inputs (at the kinematic level) - drive v and steer ω

• Dynamics - (control inputs are forces and torques)

Mv = F

Iω = τ

9

Control of the


Transforming the variables

• Step 1 - State transformation from (θ, x, y) to (z1, z2, z3)

z1= θ

z2= x cos θ + y sin θ

z3= x sin θ − y cos θ

• Step 2 - State transformation from (z1, z2, z3) to (η1, η2, η3)

η1= z1

η2= z2

η3= −2z3 + z1z2

10

Control of the


The new representation

• Define modified control inputs as

u1= ω u2

= z3ω + v

• The system in the new states and inputs is η1

η2

η3

=

u1

u2

η1u2 − η2u1

11

Control of the


Technique 1 - Exponential Stability in a Dense Set

Definition 1.1 If the system is exponentially stable in an open dense

set O, and the control law is well defined and bounded along the closed

loop trajectories, then the closed system is said to be almost

exponentially stable (AES).

• The open dense set in the system under consideration is

O = x ∈ IR3|x1 = 0.

• Proposition 1.1 The NI is AES with the control law

u =

(−k1x1

−k3x3x1− k1x2

)x1 = 0

where 0 < k1 < k3.

12

Control of the


Proof

• The closed loop dynamics can be written in the form

x = Ax + g(t)

where

A =

−k1 0 0

0 −k1 0

0 0 −k3

; g(t) =

0

−k3x3(t)x1(t)

0

Since g(t) ≤ |k3

x3(0)x1(0)|e(k1−k3)t and the eigen values of A are

negative-real , the system is exponentially stable.

• If x1(0) = 0, then an open loop control can be used to steer the system

to any non zero value of x1 and then the control law can be applied.

13

Control of the


Technique 2 - Bounded Control Law

• In Technique 1, if the initial condition is close to the surface x1 = 0,

the magnitude of u2 is very large.

• Define α= |x3

x1| and a set

M1= x ∈ IR3 : α ≤ c

and three other sets as

M2= x ∈ IR3 : x /∈ M1 and |x2 + c| ≥ 1

M3= x ∈ IR3 : x /∈ M1 and |x2 + c| < 1

M4= x ∈ IR3 : x1 = x3 = 0

14

Control of the


Technique 2

• Proposition 1.2 The NI is exponentially stable in the set M1

containing the origin under the following control law

u =

(−k1x1

−k3x3x1− k1x2

)if x ∈ M1(

ksS1/3

x2+c

0

)if x ∈ M2(

0

d

)if x ∈ M3(

−k1x1

−k1x2

)if x ∈ M4

where S(x)= x3 − cx1 and 0 < k1 < k3, 0 < ks and d is any non

zero constant.

15

Control of the


Proof of Technique 2

• Proof: Case 2

If x ∈ M2, then the closed loop system becomes

x1 = ksS1/3

x2 + cx2 = 0

x3 = −x2ksS1/3

x2 + cThe dynamics of S becomes

S = −ksS1/3

This is called a power rate reaching law and it is finite-time stable. So

there exists a finite-time T1 such that S(t) = 0 ∀ t ∈ [T1,∞). So all

the trajectories will reach the boundary of the set M1 in some

finite-time after which case 1 follows.

16

Control of the


Finite time stabilization to the surface S(x) = 0

• The dynamics of the surface is given by

S = −ksS1/3

• Consider a Lyapunov candidate function and its rate of change

V = S2/2

V = −ksS4/3 < 0

Hence S = 0 is attractive globally.

• For finite time convergence we show

V + kV α ≤ 0 ([Haimo] condition for finite-time stability)

where α ∈ (0, 1) and k > 0

17

Control of the


• For α = 2/3 and k < 223ks we have

V + kV23 = −S4/3(ks −

k

223

) < 0

• Further

S = 0

which implies that S = 0 is positively invariant

18

Control of the


Proof

Case 3

If x ∈ M3 , then the dynamics of x2 is

x2 = d

So there exists a finite-time T2 such that |x2(T2) + c| = 1. Then Case 2

follows.

Case 4

If x ∈ M4, then closed loop system becomes

x1 = 0

x2 = −k1x2

x3 = 0

and we have exponential stability

19

Control of the


Techniques 1 and 2

• Methods 1 and 2 are related though the AES property of 1 does not

hold in 2. Method 2 however guarantees boundedness of the control law

in a local domain around the origin.

20

Control of the


Technique 3 - Sliding Mode Control

• The next two techniques rest on the variable structure (sliding mode)

philosophy. The methodologies rest on bringing the system to a surface

(defined by the dependent state in the dynamics) in finite time and

then exponentially making the other two states reach the origin.

• Proposition 1.3 The NI is exponentially stable in the set

M = x ∈ IR3 : x3 = 0 with the following control law

u =

−k1x1

−k3x

1/33x1

− k1x2

if x1 = 0

where k1, k3 > 0.

21

Control of the



• If x1(0) = 0 then the closed loop system is

x1 = −k1x1

x2 =−k3x

1/33

x1− k1x2

x3 = −k3x1/33

Since k1, k3 > 0, the dynamics of x3 implies finite time stability while

that of x1 implies exponential stability. So there exists a finite time

T1 > 0 such that x3(t) ≡ 0 ∀t ≥ T1.

• The system on M is then governed by the set of equations

x1 = −k1x1 x2 = −k1x2 x3 ≡ 0

22

Control of the


Technique 4

• Define S(x)= x2

1 − 3k1x2/33 where k1 > 0 is a predefined constant,

• Proposition 1.4 The NI is exponentially stable in the set

M = x ∈ IR3 : x3 = 0 with the following time-varying control

law

u =

(−k1x1

−x1x1/33 − k1x2

)∀ s ∈ [t,∞) if S(x(t)) > 0(

x1

x2

)if S(x(t)) ≤ 0(

−k1x1

−k1x2

)if x3 = 0

23

Control of the



Case 1 Assume S(x(0)) > 0

• We have

x1 = −k1x1

x2 = −x1x1/33 − k1x2

x3 = −x21x

1/33

• The evolution of x1(t) is given by

x1(t) = x1(0)e−k1t

• The evolution of x3 is

x3(t) = ±[x1(0)2e−2k1t − S(x(0))

3k1]3/2 ∀t ∈ [0, T1)

x3(t) = 0 ∀t ≥ T1

24

Control of the



• The positive (negative) sign corresponds to the solution for an initial

condition x3(0) > 0 (x3(0) < 0).

• The closed loop system after time T1 is

x1 = −k1x1

x2 = −k1x2

x3 ≡ 0

and is exponentially stable.

25

Control of the



• Case 2 If x1(0) = 0 and S(x(0)) ≤ 0 , then we have

x1 = x1

x2 = x2

x3 = 0

The above dynamics leads to an increase in the value of S(x(t)) as time

increases. Note that x3 remains a constant and x1 has an exponential

growth. So there exists a finite-time such that S(x(t)) > 0. Then Case

1 follows.

26

Control of the


• The third case is straightforward. If x1(0) = 0 then any open loop

control can push x1 to some non-zero value, then either case 1 or case 2

follows.

27

Control of the


Techniques 3 and 4

• Methods 3 and 4 bring in a variable structure (or sliding mode) feature;

the latter is also time-varying.

• Both techniques use a power rate reaching law to enter a set in which

the system is exponentially stable.

• The idea in Method 4 is to utilize the state x1 as a gain for the

convergence of x3 to zero in finite-time. So we have the term x1x1/33

instead ofx

1/33x1

. But since this control law valid in a local domain, we

have introduced a swtiching strategy to move from anywhere to this

domain.

• All the four control laws are not defined for x1(0) = 0.

28

Control of the


• Closest parallels - notions of sigma process, paper by Bloch and

Drakunov

29

Control of the


Dynamic model of the mobile robot

• θ

x

y

v

ω

=

ω

v cos θ

v sin θ

0

0

+

0

0

01M

0

F +

0

0

0

01I

τ

• Fifth order system with drift

• Discontinuous controllers (velocity level) designed from the kinematic

model cannot be applied to obtain controllers at the acceleration level

for the dynamic model.

30

Control of the



• Step 1 - State transformationz1

z2

z3

z4

z5

=

θ

x cos θ + y sin θ

x sin θ − y cos θ

ω

v − (x sin θ − y cos θ)ω

• The dynamic model in the z variables

z1

z2

z3

z4

z5

=

z4

z5

z2z4

τI

FM − τ

I z3 − z2z24

31

Control of the



• Alternate inputs

u1=

τ

Iu2

= −z2

4z2 −τ

Iz3 +

F

M

• Step 2 - State transformation from (z1, z2, z3, z4, z5) to

(x1, x2, x3, y1, y2)

x1= z1 x2

= z2 x3

= −2z3 + z1z2 y1

= z4 y2

= z5

• The system equations are now in the form

x1 = u1

x2 = u2

x3 = x1x2 − x2x1

32

Control of the


The Extended Nonholonomic Double Integrator (ENDI)

x = f (x) +

2∑i=1

gi(x)ui

where x= [x1, x2, x3, y1, y2]

T ∈ IR5 is the state vector and

u= [u1, u2] ∈ IR2 is the control

f (x)=

y1

y2

x1y2 − x2y1

0

0

; g1(x)=

0

0

0

1

0

; g2(x)=

0

0

0

0

1

33

Control of the


Features of the ENDI

• The equlibria of the ENDI system are of the form

xe = x ∈ IR5 : y1 = y2 = 0 and satisfy the following properties

1. xe is not stabilized by any smooth feedback control laws.

2. The ENDI is locally strongly accessible from any x ∈ IR5.

3. The ENDI is small time locally controllable (STLC) from any xe.

34

Control of the


A Sliding Mode Approach

• Define S1(x)= kx1 + y1 and S2(x)

= x1y2 − x2y1 + k3x3 where

k3 > k > 0

• Define the sets

M1= x ∈ IR5 : S1(x) = 0 M2

= x ∈ M1 : S2(x) = 0

• Proposition 1.5 The origin of the ENDI is exponentially stable in

the set M2 with the following control law

u1 = −ky1 − S1/31 if (x1, y1) = 0

u2 =

−S1/32

x1− k3y2 − (k3 − k)kx2 x ∈ M1 (x = 0)

−x2 − y2 otherwise

35

Control of the


Proof

• The closed loop dynamics for (x1, y1) = 0 and x /∈ M1 becomes

x1 = y1

x2 = y2

x3 = x1y2 − x2y1

y1 = −ky1 − S1(x)1/3

y2 = −x2 − y2

• The dynamics of S1(x) is

S1 = −S1/31

and the surface S1(x) = 0 is finite-time stable

36

Control of the


• So there exists a time T1 ≥ 0 such that the closed loop trajectory for

any permissible initial condition reaches the set M1 and stays there for

all future time. The control law u1 allows both x1 and y1 to converge

to zero as time gets large.

• At the same time as the system is being driven towards M1, the PD

control law u2 with unity gain makes states x2 and y2 converge to zero.

37

Control of the


The dynamics on the set M1

• The closed loop system on M1 is

x1 = −kx1

x2 = y2

x3 = x1y2 − x2y1

y1 = −ky1

y2 =−S

1/32

x1− k3y2 − (k3 − k)kx2

• The dynamics of S2 is

S2 = −S1/32

So there exists a time T2 ≥ T1 ≥ 0 such that the closed loop

trajectories reach the set M2 and stay there for all future time.

38

Control of the


The dynamics on the set M2

• The closed loop system on M2

x1 = −kx1

x2 = y2

x3 = −x3

y1 = −ky1

y2 = −k3y2 − (k3 − k)kx2

• All trajectories converge to the origin

39

Control of the


Simulation

• Initial conditions are x(0) = −1.5m, y(0) = 4m, θ(0) =

−2.3rad, ω(0) = 1rad/sec, v(0) = −1m/sec.

• Controller parameters are k = 0.5, k3 = 1.

• Vehicle parameters are

M = 10Kg, I = 2Kgm2, L = 6cm, R = 3cm.

40

Control of the


−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

intial condition

x in meters

y in

met

ers

Figure 3: Stabilization to the origin

41

Control of the


0 10 20 30 40 50−3

−2

−1

0

1

2

3

4

5

time in sec

x,y

in m

eter

s, θ

in r

ad

xyθ

Figure 4: Position and orientation of the vehicle

42

Control of the


0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

time in sec

v in

m/s

ec, ω

in r

ad/s

ec

ωv

Figure 5: Velocities of the vehicle

43

Control of the


0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

time in sec

τ 1,τ2 in

Nm

τ1τ2

Figure 6: Input to the motors

44

Control of the


Thank You

45

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