Nonlinear Systems. Orbital Stabilization of Underactuatedweb.eecs.umich.edu/~grizzle/papers/CDC2003Canudas_Orbital... · Orbital Stabilization of Underactuated ... Slide # 1 CONTENTS

Orbital Stabilization of Underactuated

Nonlinear Systems.∗

Anton Shiriaev† and Carlos Canudas-de-Wit‡ †‡

November 24, 2003

∗This presentation is based on [10] and [7]†† Department of Applied Physics and Electronics, University of Umea, SE-901 87

Umea, SWEDEN. Email: [email protected]‡‡ Laboratoire d’Automatique de Grenoble, UMR CNRS 5528, ENSIEG-INPG,

B.P. 46, 38 402, ST. Martin d´Heres, FRANCE. Email: carlos.canudas-de-wit@

inpg.fr.

0-0

Virtual Constraints a Constructive Tool For Orbital Stabilization. Slide # 1

CONTENTS

1. Virtual constraints.

2. Virtual Limit System

• Definition, examples• Distinction to physically constrained systems• Main properties

3. Control design

• Partial feedback linearization• Choice of the orbit• Controllability test• Stability

4. Examples

c©C. Canudas-de-Wit Nov. 2003.


Previous Works

1. Transversal Lyapunov Functions. Hauser and Choo’94,[6].

2. Zero dynamic matching, Grognard and Canudas’02, [5];Canudas-Espiau-Urrea’02, [7]; Marconi-Isidory-Sarrani’02 [8].

3. Hamiltonian Formalism. Aracil-Gordillo-Acosta’02 [2],Vivas-Rubio’03 [4]

4. In: Shiriaev and Canudas-de-Wit’03 [10]

• Notion of Virtual Limit system• Linear time-varying controllability• Explicit use of integral forms for stabilization



Balancing: the problem

Zero dynamics

q2

q1

q1 = φ(q2, θ)

Create a sustained oscillation in the full robot coordinates at thesingle leg support configuration (high-dimensional invertedpendulum).

• Learn how to create stable orbits without impacts,• A tool for transition strategy between different motions phases(i.e. run/walk),



Key steps

We consider underactuated Lagrangian systems

d

dt

(∂L∂q

)− ∂L

∂q= B(q)u

with a degree of underactuation equal to one.

• n − 1 virtual constraint ϕ(q, p) = 0, with p, being parametervector.

• The virtually constrained system has the form

α(θ, p)θ + β(θ, p)θ2 + γ(θ, p) = 0

which posses a conserved quantity I = I(θ, θ, θ(0), θ(0), p)

• Characterize the LTV controllability.• Construct a (local) stabilizable control law.



Balancing vs Walking:What changes ?

Conceptual differences:

• Motion without impacts, then• Cycles should be stabilized dynamically.• “Passive” stable cycles do not exist.

Two possible methods:

• Matching a desired limit cycle exo-system,• Use the trajectory controllability of the target cycle.



The matching solution

The parameter p(t) may be time-varying i.e.

y = h0(q)− hd(θ(q), p(t))

Find hd(θ(q), p(t)) and an adaptation law (dynamic feedback) for p,such that:

the zero dynamics

x1 = x2

x2 = β(x, hd, p(t), p(t), p(t))

exhibits stable periodic behaviour, with stable solutions for p(t).



• Target orbit (exosystem) defines a target orbit Ωd(x)defines a closed path in the plane.

x1 = x2

x2 = βd(x)

• Dynamic Matching condition. Solve for p, to satisfy

β(x, hd, p, p, p) = βd(x)

while ensuring boundedness of the obtained solutions for p(t).



Internal Stability Issues

• Is the internal dynamics of p(t) stable ? average equation for

p(t) = 1T

∫p(τ)dτ

¨p+ ρ1 ˙p+ ρ2 ˙p2 + ρ3g(p) = 0

• Integral form

I = f((p, ˙p, p(0), ˙p(0)) = 0

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x1

x2

Evolution of (x, x)

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5−5

−4

−3

−2

−1

0

1

2

3

4

5

Evolution of (p, ˙p)



II. Virtual constrains

1-DOF Naturally

constrained system

y ≡ 0 3-DOF Free system

Feedback-constrained

1-DOF system

y → 0

Implicit form

y1 =

y2 =

L1 cos(θ1) + L2 cos(θ1 + θ2) = 0

θ1 + θ2 + θ3 − π = 0.

Explicit form

y1 =

y2 =

θ2 − (π − θ1 − arccos(L1L2

cos(θ1))) = 0

θ3 − arccos(L1L2

cos(θ1)) = 0.



Virtual constraints: more examples

q4q2

q3q1

virtual camshaft

θl1

l2

q1

q2virtual

link

(x1, y1)

virtualtoolbar

(x2, y2)

ψ2

φ2

l2

x

y

v2

q1

q3

virtual joints

q4

q5q2



Time-invariance of periodic motions

Reference trajectory

Robot delayed trajectory

Reference trajectory

angle

TimeRobot delayed trajectory

Ψ = f(θ)

0 2π

Ψ(θ)

θθ

A A

B

C B

C

Time invariance implies Ψ = f(θ)



Multi-dimensional robots manipulatorswith open chain structure

Assuming that the link in contact with the ground is clamped, the

equations of motion if a n-degree manipulator, with (n− 1) inputs is:

H(q)q + C(q, q)q + g(q) = Bu; rank(B) = (n− 1)

Let θ = qk for some k ∈ 1, 2, . . . , N. Let the constraints be of the form

ϕi = qi −N∑j=0

p(i,j)θj = 0, ∀i = 1, . . . , n, i = k.

The zero dynamic

α(θ, p)θ + β(θ, p)θ2 + γ(θ, p) = 0

For the particular case where θ is chosen to be cyclic, then the above



equation has the equivalent form

θ =

σ =

1I(θ)

σ

f(θ)(1)

Else, In general (i.e. Pendubot) f = f(θ, σ) depends on σ as well.

Implication of selection a cyclic coordinate is that the above equation

can be integrated by eliminating the time;

dσ

dθσ = f(θ) · I(θ) =⇒ σ2

1

2− σ2

0

2= P1 − P0

where P =∫ θ1θ0f(s)I(s)ds.

The function

LV S = K − P =σ2

2−

∫f(s)I(s)ds

that can be interpreted as the Lagrangian for the virtual system, is

different to Lagrangian of the full mechanisms projected into the

constraints.



Distinction between physically andvirtually constrained systems

Consider a system of the general from:

x = f(x) +B(x)u,

y = h(x).

with x ∈ Rn, y ∈ Rm, and u ∈ Rm, m < n.

Lagrange reduction procedure:

x = f(x) +B(x)u+ J(x)λ︸︷︷︸extra inputs

where J(x)T = ∂h∂x

(x). The reduced system has the form

x =(I − J(x)J(x)†

)[f(x) +B(x)u] , J† = (JTJ)−1JT

Remark: the motion on this invariant manifold (n−m) is still forced

by the input u.



Systems subject to virtual constrains.

Use u in

x = f(x) +B(x)u,

y = h(x).

to zeroing the outputs y.

The resulting constrained dynamics is, in this case, given by

x =(I −BBTJ(JTBBTJ)−1JT

)f(x)

Remark: inputs are not present in this autonomous equation which

contains the zero dynamics.



Example

x1 = x2 + u,

x2 = bu,

y = x1 + x2.

The two reduced systems are, with x1 = −x2:

x2 = −1

2(x2 + (1 − b)u)︸︷︷︸

Lagrange red. syst.

x2 = − b

1 + bx2︸︷︷︸

Virtual limit system

• Conclusion: the limit system obtained from the use of the virtual

constrains, will lead to different equations than the one obtained by

standard reduction Lagrange method.

• The equation of the zero dynamics resulting from the virtual

constrains is named the “virtual limit system”.



Properties of Virtual Limit SystemTheorem 1 (Perram-Shriaev-Canudas’03)

Given initial conditions[θ0, θ0

], if the solution

[θ(t, θ0), θ(t, θ0)

], of the

limit system

α(θ, p)θ + β(θ, p)θ2 + γ(θ, p) = 0

exists for these initial conditions, then the function

I(θ, θ, θ0, θ0

)= θ2 − ψ(θ0, θ)θ

20 + ψ(θ0, θ) ·

θ∫θ0

ψ(s, θ0)2γ(s)

α(s)ds

ψ(θ0, θ1) = exp

−2

θ1∫θ0

β(τ)

α(τ)dτ

preserves its value along this solution. This holds irrespective of the

boundedness of the solution[θ(t, θ0), θ(t, θ0)

].



Proof (outline)Introducing the new variable Y = θ2(t) One can then rewrite the virtual

limit system in the equivalent form

d

dθY +

2β(θ)

α(θ)Y +

2γ(θ)

α(θ)= 0

This is a linear equation with respect to function Y and with θ (instead

of t) as independent variable. Its general solution has the following form:

Y (θ) = ψ(θ0, θ) · Y (θ0) − ψ(θ0, θ) ·θ∫

θ0

ψ(s, θ0, s)2γ(s)

α(s)ds

Introducing function I as

I(θ, θ, θ0, θ0) = Y (θ) − ψ(θ0, θ) · Y (θ0) + ψ(θ0, θ) ·θ∫

θ0

ψ(s, θ0, s)2γ(s)

α(s)ds

results in the identity I(θ(t), θ(t)) = 0.



Other useful relations of I• I the virtual energy of the limit virtual system (conserved quantity).

• Consider now the forced virtual limit system

α(θ)θ + β(θ)θ2 + γ(θ) = u

then the following relation holds:

d

dtI = θ

2

α(θ)u− 2β(θ)

α(θ)I

• I(θ, θ, θ0, θ0) = 0 is the only invariant under u = 0. Thus it is not the

first integral of system, but in number of examples this function

could be readily rewritten in the form of the first integral U(θ, θ) of

the system, that is

f(θ, θ, θ0, θ0) · I(θ, θ, θ0, θ0) = U(θ, θ) − U(θ0, θ0),

where f(θ, θ) is some function. The ‘modified’ energy functionddtU = θ · u



Controller Design

Problem: Derive a family of feedback laws and conditions, that ensure

exponential orbital stabilization of a particular class of periodic solution

of the virtual limit system.

Main control design steps:

1. Partial feedback linearization,

2. Choice of a periodic solution for the limit virtual system,

3. Controllability test of the auxiliary LPTV system,

4. Construction of the feedback law.



1. Partial Feedback LinearizationThe original nonlinear system is first transformed, via partial feedback

linearization, to a form

α(θ)θ + β(θ)θ2 + γ(θ)︸︷︷︸virtual limit system

= g1(θ, θ, y, y) + g2(θ, θ, y, y) · v

y = v

Here y ∈ Rn−1, θ ∈ R1; v ∈ Rn−1. The gi, are smooth functions

g1(θ, θ, 0, 0) + g2(θ, θ, 0, 0) · v = 0, v = 0(n−1)×1.

In turn, due to the smoothness of g1 = gy(θ, θ, y, y) · y + gy(θ, θ, y, y) · y.This yields the Auxiliary System:

I = θ

2

α(θ)[gy · y + gy · y + g2 · v] − 2β(θ)

α(θ)I

y = v



2. Choice of Periodic SolutionSelect a set of parameter p such that the limit virtual system has a

periodic solution

[θγ(t), θγ(t)

]=

[θγ(t+ T ), θγ(t+ T )

], ∀ t (2)

of a period T . The problem is then: to determine a feedback controller

that orbitally stabilize the following periodic solution, ∀ t[θγ(t), θγ(t), y(t), y(t)

]=

[θγ(t+ T ), θγ(t+ T ), 0, 0

](3)

of the nonlinear system

α(θ)θ + β(θ)θ2 + γ(θ)︸︷︷︸virtual limit system

= g1(θ, θ, y, y) + g2(θ, θ, y, y) · v

y = v



3. Controllability of Auxiliary LTVPsystem

Evaluation of input functions of the Auxiliary system along the solutions[θγ(t), θγ(t), 0, 0

]gives

I =2θγ(t)

α(θγ(t)

) gy(t) · y + gy(t) · y + g2(t) · v − β

(θγ(t)

)· I

y = v

where

gy(t) = gy(θγ(t), θγ(t), 0, 0

)

gy(t) = gy(θγ(t), θγ(t), 0, 0

)

g2(t) = g2(θγ(t), θγ(t), 0, 0

)with periodic coefficients.



Its state representation with ζ = [ I, y, y ]T is

ζ = A(t)ζ + b(t)v

with A(t) = A(t+ T ), and b(t) = b(t+ T ).

This system is controllable iff:

K =

T∫0

[X0(t)

]−1

b(t)b(t)T[X0(t)T

]−1

dt > 0

where the matrix function X0(t) is defined as

d

dtX0 = A(t)X0, X0(0) = I2n+1.



Examples of possible feedbacks for theAuxiliary LTVP system

Let Γ = ΓT > 0 G = GT > 0, suppose that ζ = A(t)ζ + b(t)v is

completely controllable, then ∃ R(t) = R(t)T = R(t+ T ) that satisfies to

the Riccati equation,

R(t) +A(t)TR(t) +R(t)A(t) +G = R(t) b(t)Γ−1b(t)TR(t),

and, the feedback controller

v = −Γ−1 b(t)TR(t)ζ

renders the linear periodic system exponentially stable. Moreover, along

any solution[I(t), y(t), y(t)

]of the closed loop system, the following

holdsd

dtV (t) = −ζ(t)TGζ(t) − v(t)T Γv(t)

with V (t) = ζ(t)TR(t)ζ(t).



4. Constructive Procedure for ControlDesign

The previous LTV feedback controller

v = −Γ−1 b(t)TR(t)ζ

suggest the following control structure

v = −Γ−1 b(θ, θ, y, y)T︸︷︷︸=b(t)

R(t)ζ

with

b(θ, θ, y, y)T =

2 θ g2

(θ, θ, y, y

)α(θ) , 0(n−1)×(n−1), I(n−1)

T

Question: Does this controller leads to any type of orbitally stability ?



Stability

Consider any underactuated system of the form ddt

(∂L∂q

)− ∂L

∂q= B(q)u,

with B(q) of full rank. Assume that:

(i) A set of constraints (or outputs)

yi = φi(q, p) = 0 ∀i = 1, 2, . . . , (n− 1), are defined, so that the

resulting virtual limit system exhibes cycles;

(ii) Given the periodic solution [θγ(t), θγ(t)] of the virtual limit system

with a period T , the corresponding auxiliary linear system

ζ = A(t)ζ + b(t)v is completely controllable on [0, T ].

Then the control law v = −Γ−1 b(θ, θ, y, y)TR(t)[I, y, y]T with

I = θ2 − ψ(θγ(0), θ)θ2γ(0) + ψ(θγ(0), θ) ·θ∫

θγ(0)

ψ(s, θγ(0), s)2γ(s)

α(s)ds

makes the chosen solution of the closed loop system orbitally

exponentially stable in a local sense.



Proof: outlineTaking V = ζTR(t)ζ, it can be shown that

V = ζT

− G︸︷︷︸

<0

−R(t)b(t)Γ−1b(t)TR(t)︸︷︷︸<0

+ ∆(t)︸︷︷︸perturbation

ζ

Where the perturbation term ∆

∆(t) =(A − A(t)

)TR(t) + R(t)

(A − A(t)

)R(t)

(b − b(t)

)Γ−1

(b + b(t)

)TR(t)

where A = A(θ, θ, y, y), and b = b(θ, θ, y, y).

• limζ→0 |∆(t)| = 0

• |∆(t)| can be made arbitrarily small in [0, T ]

• It is possible to shown that

V (nT ) ≤ V (0) −12

minλ(G)

maxt∈[0,T ]

∥∥∥R(t)∥∥∥

nT∫0

V(τ)dτ.



Examples of systems controlled by thismethod



Generating Oscillations in 3-LinksPendulum

Virtual Constraints

y1 = qa1 − a1 (qp − ε1)

y2 = qa2 − a2 (qp − ε2)

(y = 0) where a1, a2 and ε1, ε2 are the constant parameters to be

determined later.



Partial Feedback LinearizationAfter partial linearization of the outputs y, we get

α(qp)qp + β(qp)q2p + γ(qp) = −c12 y1 − c13 y2 −m12 · v1 −m13 · v2

y1 = v1

y2 = v2

with,

α(qp) = m11 + a1 ·m12 + a2 ·m13

β(qp) = c11 + a1 · c12 + a2 · c13γ(qp) = g1



Choice of Parameters of Virtual limitsystem

Conditions need to be fulfilled by the choice of a1, a2, ε1 and ε2

1. The limit system

α(qp)qp + β(qp)q2p + γ(qp) = 0 (4)

has a stable equilibrium around qp = π/2; i.e. search zero of γ(qp) in

some vicinity of π2

;

2. Compute the linearization of (4) at the found equilibrium qe

z + ωez = 0, (5)

where ωe = ddqp

(γ(qp)

α(qp)

)∣∣∣qp=qe

. If ωe is positive, then the linear

system (5) is a focus, and its solutions are periodic;

3. To verify that the nonlinear system (4) has periodic solutions in some

neighborhood of the equilibrium qe, check level sets of its integral I.

If these sets are closed curves, then there exist periodic solutions.



Selected range of possible values:

0 ≤ a1 ≤ 10, 0 ≤ a2 ≤ 10, ε1 = −π2− 0.05, ε1 =

π

2+ 0.1

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

a1

a 2

Figure 1: The areas in black correspond to those values of a1 and a2,

where ωe is positive and the equilibrium qe of the virtual limit system (4)

deviates from π2

no more than 0.1 rad.



Choose any point (a1, a2) belonging to areas in black depicted at

Figure 1 and verify that limit system has a periodic solutions.

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

qp

d q p /

dt

Orbit chosen to be stabilized

Figure 2: Level sets of the integral I when the constraints’ parameters a1,

a2, ε1, ε2 are chosen as in (1).

We have chosen the values as

a1 = 8.85, a2 = 8.9, ε1 = −π2− 0.05, ε1 =

π

2+ 0.1



Evolution of the Rabbit restricted to 3 links

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Equilibrium position (center)−0.6 −0.4 −0.2 0 0.2 0.4

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Motion over a period



0 0.5 1 1.5 2.215 2.5 31.4

1.45

1.5

1.55

1.6

1.643

1.7

1.75

1.8

time

q p(t)

Periodic Solution Chosen to Be Stabilized

Verify the controllability of the auxiliar system along the chosen solution

d

dt

I

y

y

=

κ1(t) κ2(t) κ3(t)

0 0 I

0 0 0

︸︷︷︸A(t)

I

y

y

+

ρ(t)

0

I

︸︷︷︸b(t)

v,



Simulated closed-loop responses

0 5 10 15 20 25 30 35 40−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

time (sec)

I(t)

Behaviour of I

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

qp

d q p /

dt

Motion of theunder-actuated angle



Simulated closed-loop responses

0 5 10 15 20 25 30 35 40−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (sec)

y 1(t)

Behaviour of y1

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time (sec)

y 2(t)

Behaviour of y2



References

[1] Aracil, J. Gomez-Estern, F. and Gordillo, F. (2003)“A Family of

oscillating generalized Hamiltonian Systems”. IFAC workshop on

Hamiltonian Systems, Sevilla, April, 2003.

[2] Aracil, Gordillo, F. and Acosta, J.(2002) “Stabilization of oscillations in

the inverted pendulum”. IFAC word Congress. , Barcelona, Spain, Jully,

2002

[3] C. Chevallereau, G. Abba, Y. Aoustin, F. Plestan, E.R. Westervelt, C.

Canudas-de Wit, and J.W. Grizzle.“A Testbed for Advanced Control

Theory”. to appear in: IEEE Control Systems Magazine, Dec. 2003.

[4] Vivas-Venegas, C. and Rubio, F.(2003) “Improving the performance of

orbitally-stabilizing controls for the furuta pendulum ”. IFAC workshop

on Hamiltonian Systems, Sevilla, April, 2003.

[5] Grognard, F. and C. Canudas-de-Wit (2002) “Design of orbitally stable

zero-dynamics for a class of nonlinear systems,”. In Proceeding of the

IEEE Decision and Control Conference, Dec. 2002 Las Vegas, Nevada,

USA. Also to apprear in: Systems & Control Letters.

[6] Hauser, J. and C.C. Chung (1994). Converse Lyapunov functions for



exponentially stable periodic orbits. Systems & Control Letters

(23), 27–34.

[7] Canudas de Wit C., Espiau B., Urrea C. (2002), “Orbital stabilization of

underactuated mechanical systems,” IFAC word Congress, 2002,

Barcelona, Spain.

[8] Marconi L., Isidori A., Sarrani A., “Autonomous vertical landing on an

oscillating platform: an internal-model based approach,” Automatica 38

(2002), no. 1, 21-32.

[9] Perram J., Shiriaev A., Canudas-de-Wit C., Grognard F. “Explicit

Formula for A General Integral Of Motion For A Class Of Mechanical

Systems Subject To Holonomic Constraint”. IFAC workshop on

Hamiltonian Systems, Sevilla, April, 2003.

[10] Shiriaev A. and Canudas-de-Wit “Virtual Constraints a constructive tool

for orbital stabilisation of underacutaed nonlinear systems: theory and

examples”. Paper in two parts. Submitted to IEEE Trans. on Aut.

Control, Nov. 2003.


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Nonlinear Systems. Orbital Stabilization of Underactuatedweb.eecs.umich.edu/~grizzle/papers/CDC2003Canudas_Orbital... · Orbital Stabilization of Underactuated ... Slide # 1 CONTENTS