Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical

systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system

Bloch, Anthony M.

Marsden, Jerrold E.

Zenkov, Dmitry V.

English

Caltech Authors - Main

Controlled Lagrangian Methods
and Tracking of Accelerated Motions
Dmitry V. Zenkov*
Department of Mathematics
North Carolina State University
Raleigh, NC 27695
dvzenkov@unity.ncsu.edu
Anthony M. Bloch**
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109
abloch@math.lsa.umich.edu
Jerrold E. Marsden***
Control and Dynamical Systems
California Institute of Technology 107-81
Pasadena, CA 91125
marsden@cds.caltech.edu
Abstract— Matching techniques are applied to the problem
of stabilization of uniformly accelerated motions of mechanical
systems with symmetry. The theory is illustrated with a simple
model—a wheel and pendulum system.
I. INTRODUCTION
In this paper we apply the method of controlled La-
grangians to the problem of stabilization of accelerated
motions of Lagrangian mechanical systems with symme-
try. The method of controlled Lagrangians for stabilization
of relative equilibria (steady state motions) originated in
Bloch, Leonard, and Marsden [4] and was then developed
in Auckly [1], Bloch, Leonard, and Marsden [5], [6], [7],
Bloch, Chang, Leonard, and Marsden [8], and Hamberg
[10], [11]. A similar approach for Hamiltonian controlled
systems was introduced and further studied in the work of
Blankenstein, Ortega, van der Schaft, Maschke and Spong
and their collaborators (see [2], [14], [15], [16]) and the two
methods were shown to be equivalent in [9]. A nonholonomic
version of the method of controlled Lagrangians was studied
in [3], [17], [18].
According to the method of controlled Lagrangians, the
original controlled system is represented as a new, uncon-
trolled Lagrangian system for a suitable controlled Lagran-
gian. The energy associated with this controlled Lagrangian
is designed to be positive or negative definite at the (relative)
equilibrium to be stabilized. The time-invariant feedback con-
trol law is obtained from the equivalence requirement for the
new and old systems of equations of motion. If asymptotic
stabilization is desired, dissipation emulating terms are added
to the control input.
In Bloch, Chang, Leonard, and Marsden [8], the problem
of tracking was briefly discussed and, based on some nu-
merical evidence and the study of some simple cases, it was
proposed that tracking problems could be studied by means
of the method of controlled Lagrangians. The idea is to create
a time-dependent function that has a minimum at the point
one wishes to track. The goal of the present paper is, in
fact, to study the stabilization of a certain class of motions
* Research partially supported by NSF grant DMS-0306017
** Research partially supported by NSF grants DMS 0103895 and
0305837
*** Research partially supported by ONR Contract N00014-02-1-0826
of mechanical systems with symmetry, which one may view
as a special case of the general tracking problem. Assuming
that the symmetry group is commutative, one can represent
motions for which the component of acceleration in the group
direction is constant as equilibria of the reduced system.
Stabilization of such equilibria will thus produce orbitally
stable accelerated trajectories.
In this paper we suggest a stabilization strategy using the
framework of time-dependent Lagrangians. We expect this
method to be applicable to more general tracking problems.
In particular, we anticipate implementing our approach in
problems of simultaneous tracking of a given trajectory in
the symmetry group and stabilizing of an appropriate shape
equilibrium.
The paper is organized as follows: In Section II we intro-
duce a simple mechanical example—a wheel coupled with a
pendulum—that demonstrates unstable accelerated dynamics.
The main results are presented in Sections III and IV. In
Section III we study a class of time-dependent Lagrangians
with uniformly accelerated group dynamics represented by
relative equilibria. We also discuss an energy-based stability
analysis for these relative equilibria. In Section IV we derive
the matching conditions and then illustrate the theory using
the wheel and pendulum system.
In a future publication we intend to treat systems with
noncommutative symmetry as well as systems with nonholo-
nomic constraints.
II. THE MECHANICAL EXAMPLE
Consider a homogeneous vertical disk that is rolling with-
out slipping along a horizontal straight line. A pendulum
is attached to the center of the disk. The configuration
coordinates are the angles (θ, φ); note that φ is measured
from the rod. See Fig. 1 for details. This system is SO(2)-
invariant; the group action is given by φ 7→ φ + α. See [12]
and [13] for details about symmetry in mechanics.
We use the following notation for the parameters of the
system:
M = the mass of the disk,
R = the radius of the disk,
PSfrag replacements
θ
φ
Fig. 1. The disk with inverted pendulum.
A = the principal moment of inertia of the disk,
l = the rod length,
m = the bob mass.
The kinetic energy of this system is
K =
1
2
(
A(θ)θ˙2 + 2B(θ)θ˙φ˙ + Cφ˙2
)
,
where
A = A + MR2 + m(R2 + 2Rl cos θ + l2),
B = A + MR2 + m(R2 + Rl cos θ),
C = A + MR2 + mR2.
The potential energy is
V (θ) = mgl cos θ.
The Lagrangian equals the kinetic minus potential energy,
K−V . To simplify the exposition, we divide the Lagrangian
by C, i.e., we put
l(θ, θ˙, φ˙) =
1
2
(
α(θ)θ˙2 + 2β(θ)θ˙φ˙ + φ˙2
)
− U(θ),
where
α(θ) = A(θ)/C, β(θ) = B(θ)/C, U(θ) = V (θ)/C.
Assume there is a constant torque k applied to the disk.
The equations of motion are
d
dt
∂l
∂θ˙
=
∂l
∂θ
,
d
dt
∂l
∂φ˙
= k.
This dynamics can be rewritten in the form of the Euler–
Lagrange equations
d
dt
∂L
∂θ˙
=
∂L
∂θ
,
d
dt
∂L
∂φ˙
= 0
if a new, time-dependent Lagrangian
L(θ, θ˙, φ˙, t) = l(θ, θ˙, φ˙)− kφ˙t
is used instead of l(θ, θ˙, φ˙).
One can check that the system performs the uniformly
accelerated motion
θ = θ0, φ¨ = φ¨0 (1)
if k and θ0 satisfy the condition
kβ(θ0) + U
′(θ0) = 0.
The value of φ¨ for such a motion is k. Straightforward
computations confirm spectral instability of (1). Below we
discuss how to design a torque actuator that stabilizes (1)
with respect to θ.
III. TIME-DEPENDENT LAGRANGIANS AND
ACCELERATED DYNAMICS
A. Accelerated Motions
Consider the following class of moving system La-
grangians:
L(r, r˙, s˙, t) =
1
2
(
gαβ(r)r˙
α r˙β + 2gαa(r)r˙
α s˙a + δabs˙
as˙b
)
+ (aα(r)r˙
α + ba(r)s˙
a)t− U(rα). (2)
See [13] for details about moving systems.
Here and below α, β, γ, . . . = 1, . . . , m and a, b, c, . . . =
1, . . . , n, and summation over repeated indices is understood.
Remark. The variables s are cyclic. Without loss of gen-
erality, we assume that the quadratic form obtained from the
kinetic energy of the system by setting r˙α = 0 is 1
2
δabs˙
as˙b.
(One can find an r-dependent basis in the commutative Lie
algebra Rn that takes any positive-definite quadratic form
gab(r)s˙
as˙b to its canonical form δabs˙as˙b.)
We intend to study here accelerated motions of the form
r = r0, s¨ = s¨0. (3)
The forces and/or torques that influence such motions are
produced by the time-dependent terms in the Lagrangian.
The dynamics is governed by the Euler–Lagrange equa-
tions
d
dt
∂L
∂r˙α
=
∂L
∂rα
,
d
dt
∂L
∂s˙a
= 0
or
gαβ r¨
β + gαas¨
a =
(
1
2
∂gβγ
∂rα
− ∂gαβ
∂rγ
)
r˙β r˙γ
+
(
∂gβa
∂rα
− ∂gαa
∂rβ
)
r˙β s˙a
+
(
∂aβ
∂rα
− ∂aα
∂rβ
)
tr˙β
+
∂ba
∂rα
ts˙a − aα(r) − ∂U
∂ra
, (4)
gαar˙
α + δabs˙
b + bat = pa. (5)
Equation (5) represents the momentum conservation law.
Substituting (3) in (4) and (5), we obtain
s˙a = δab(pb − bb(r0)t),
s¨a
0
= −δabbb(r0),
−gαa(r0)δabbb(r0) = ∂ba
∂rα
δab(pb − bb(r0)t)t
− aα(r0)− ∂U
∂rα
.
The last equation implies
∂ba
∂rα
(r0) = 0.
The latter can be satisfied by setting ba(r) = −ka = const,
which is assumed in the rest of the paper. The accelerated
motions become
r = r0, s¨
a = ka, (6)
where r0 is determined from
gαa(r0)δ
abkb + aα(r0) +
∂U
∂rα
(r0) = 0.
Assuming that (6) is unstable, we impose a control input
u in the group direction in order to stabilize (6) with respect
to the shape variable r.
B. Reduced Dynamics and Stability Analysis
Recall that ba(r) = −ka. Since sa are cyclic variables, the
reduced dynamics is
d
dt
∂R
∂r˙α
=
∂R
∂rα
, (7)
where the Routhian R is
R(rα, r˙α, pa, t) =
1
2
(
gαβ − δabgαagβb
)
r˙αr˙β
+
(
δabgαa(pb + kbt) + aαt
)
r˙α
− 1
2
∑
a
(pa + kat)
2 − U. (8)
Since pa are the flow-invariant cyclic momenta, the terms
1
2
∑
a(pa + kat)
2 in the Routhian are independent of the
reduced phase variables and thus can be safely omitted. This
is assumed in the rest of the paper. The partial derivatives of
the Routhian are computed below:
∂R
∂r˙α
=
(
gαβ − δabgαagβb
)
r˙β
+ δabgαapb + (aα(r) + δ
abgαabb)t,
∂R
∂rα
=
1
2
∂
∂rα
(
gβγ − δbcgβbgγc
)
r˙β r˙γ
+
∂
∂rα
(
δabgβb(pa + kat) + aβt
)
r˙β − ∂U
∂rα
.
The equilibria of (7) correspond to the accelerated motions
(6). The accelerated motions are orbitally stable if the equi-
libria of the reduced system (7) are stable.
The reduced energy associated with (8) is
E = r˙α
∂R
∂r˙α
−R = 1
2
(
gαβ − δabgαagβb
)
r˙αr˙β + U.
Its flow derivative E˙ equals
−(aα + δabgαakb)r˙α.
Assuming that the one-form
(aα + δ
abgαakb) dr
α
is closed, define the modified energy by
E = E +
∫
(aα + δ
abgαakb) dr
α. (9)
The modified energy is flow-invariant and thus can be used
as a Lyapunov function.
IV. MATCHING AND STABILIZATION OF UNIFORMLY
ACCELERATED MOTIONS
A. Matching Conditions
Given the Lagrangian (2), one writes the controlled dy-
namics as
d
dt
∂L
∂r˙α
=
∂L
∂rα
,
d
dt
∂L
∂s˙a
= ua. (10)
Consider the controlled Lagrangian
L˜(r, r˙, s˙, t) =
1
2
(
g˜αβ(r)r˙
α r˙β + 2g˜αa(r)r˙
α s˙a + δabs˙
as˙b
)
+ (a˜α(r)r˙
α − kas˙a)t− U(rα).
We require that the dynamics determined by L˜,
d
dt
∂L˜
∂r˙α
=
∂L˜
∂rα
,
d
dt
∂L˜
∂s˙a
= 0, (11)
is equivalent to (10); this imposes certain conditions on
the controlled kinetic energy. These are called the matching
conditions. They are specified in Theorem 1 below.
Put
F δα = GαβG˜
βδ,
where
Gαβ = gαβ − δabgαag˜βb, G˜αβ = g˜αβ − δabg˜αag˜βb.
The matrix G˜αβ is invertible as it represents the controlled
Lagrangian’s reduced kinetic energy metric, which is as-
sumed to be non-degenerate.
Theorem 1: Equations (10) and (11) are equivalent if and
only if
F δα
[
1
2
∂g˜βγ
∂rδ
− ∂g˜δβ
∂rγ
+ δabg˜δa
∂g˜βb
∂rγ
]
=
1
2
∂gβγ
∂rα
− ∂gαβ
∂rγ
+ δabgαa
∂g˜βb
∂rγ
,
F δα
[
∂g˜βa
∂rδ
− ∂g˜δa
∂rβ
]
=
∂gβa
∂rα
− ∂gαa
∂rβ
,
F δα
[
∂a˜β
∂rδ
− ∂a˜δ
∂rβ
]
=
∂aβ
∂rα
− ∂aα
∂rβ
,
F δα
[
a˜δ + δ
abkbg˜δa +
∂U
∂rδ
]
= aα + α
abkbgαa +
∂U
∂rα
and the control inputs are
ua = (gαa − g˜αa)r¨α +
(∂gαa
∂rβ
− ∂g˜αa
∂rβ
)
r˙αr˙β . (12)
Using equations (11), one can eliminate the accelerations
r¨α from the control law (12).
The controlled dynamics is s-invariant, and thus one can
use the modified energy (9) for stability analysis of its relative
equilibria. These relative equilibria represent the accelerated
motions (3). Below we demonstrate this approach using our
mechanical example.
B. Stabilization of the Accelerating Wheel-Pendulum Sys-
tem
1) The Stability Condition: Recall that the Lagrangian has
the following structure:
L =
1
2
(
α(θ)θ˙2 + 2β(r)θ˙φ˙ + φ˙2
)
− kφ˙t− U(θ).
The controlled dynamics is governed by the equations
αθ¨ + βφ¨ +
1
2
α′θ˙2 = −U ′,
βθ¨ + φ¨ + β′θ˙2 = k + u. (13)
Consider the controlled Lagrangian
L˜ =
1
2
(
α˜(θ)θ˙2 + 2β˜(θ)θ˙φ˙ + φ˙2
)
+ a˜(θ)θ˙t− kφ˙t− U(θ).
The equations of motion associated with L˜ become
α˜θ¨ + β˜φ¨ +
1
2
α˜′θ˙2 + a˜ = −U ′,
β˜θ¨ + φ¨ + β˜′θ˙2 = k. (14)
We require that these equations are equivalent to (13). This
equivalence implies the following matching conditions (see
Theorem 1):
α˜′(α− ββ˜) + 2β˜′(−αβ˜ + βα˜) = α′(α˜− β˜2), (15)
(a˜ + U ′)(α− ββ˜)− k(−αβ˜ + βα˜) = U ′(α˜ − β˜2). (16)
After β˜ has been chosen, (15) becomes a linear first order
differential equation for α˜. After solving (15), one finds a˜ that
satisfies (16). The equivalence requirement also determines
the feedback control input u.
We now discuss the conditions for stability of the acceler-
ated motion (1). As before, we discuss stability with respect
to θ.
Using the Routh reduction, one finds the reduced dynamics
d
dt
∂R
∂θ˙
=
∂R
∂θ
,
or, explicitly,
(α˜− β˜2)θ¨ + (α˜′/2− β˜β˜′)θ˙2 + a˜ + β˜k = −U ′.
The (time-dependent) Routhian is
R =
1
2
(α˜− β˜2)θ˙2 + (β˜kt + a˜t + β˜p)θ˙ − U.
According to (9),
E(θ, θ˙) = 1
2
(α˜− β˜2)θ˙2 + U +
∫
(a˜ + β˜k) dθ. (17)
The relative equilibrium θ = θ0 is stable if E is definite at
(θ0, 0).
We now discuss how one achieves stability. First, we
obtain a new representation of the second matching condition
(16). We have
(a˜ + U ′)(α − ββ˜)− k(−αβ˜ + βα˜)
= (a˜ + U ′ + β˜k)(α − ββ˜)
− β˜k(α− ββ˜)− k(−αβ˜ + βα˜)
= (a˜ + U ′ + β˜k)(α − ββ˜)− βk(α˜− β˜2)
and thus (16) becomes
(a˜ + U˜ ′ + β˜k)(α − ββ˜) = (U ′ + βk)(α˜ − β˜2). (18)
At θ = θ0, both U ′ + kβ and a˜ + U ′ + kβ˜ vanish, and
therefore
(a˜′ + U ′′ + β˜′k)(α − ββ˜) = (U ′′ + β′k)(α˜− β˜2).
Recall that U ′′ + β′k is negative at θ = θ0 and
the stability condition requires that a˜′ + U ′′ + β˜′k and
α˜ − β˜2 are of the same sign. The stability condition thus
becomes
α(θ0)− β(θ0)β˜(θ0) < 0. (19)
After choosing β˜ that satisfies (19) one can assign a suitable
initial condition α˜(θ0), find α˜(θ) from (15), and find a˜(θ)
from (18). The above procedure determines the controlled
Lagrangian L˜.
2) The Control Input: The equivalence of (13) and (14)
implies that the control input is given by
u =
[(
(α˜− β˜2)(β′ − β˜′)− (β − β˜)(α˜′/2− β˜β˜′))θ˙2
− (β − β˜)(a˜ + β˜k + U ′)]/(α˜− β˜2). (20)
Summarizing, we have:
Theorem 2: If (19) holds for the system (13) and the
control is dened by (20), then the accelerated motion (1)
of the wheel-pendulum system is stable in the orbital sense,
i.e., it is stable with respect to the variables (θ, θ˙, φ¨).
3) The Stability Region: We now demonstrate that the
controller proposed in this paper is capable of producing a
large region of stability. Choose the numerical values of the
parameters of the wheel-pendulum system to be such that
mRl
A + MR2 + mR2
=
1
8
,
ml2
A + MR2 + mR2
=
1
16
,
mgl
A + MR2 + mR2
= 1.
For the motion with the pendulum tilt of θ = pi/4 one
computes
φ¨ =
8
8
√
2 + 1
.
The controlled Lagrangian can be evaluated explicitly for
this problem (details are omitted here and will appear in
a future publication). The stability region for the relative
equilibrium θ = pi/4, which is the region inside the critical
level of the modified energy (17) is shown in Fig. 2.
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PSfrag replacements θ
θ˙
Fig. 2. The stability region for θ = pi/4.
V. CONCLUSIONS
This paper has extended matching techniques to tracking
of relative equilibria of a class of time-dependent Lagrangian
dynamical systems as well as developed an energy-based
procedure for stability analysis of these equilibria. Although
the stability analysis proposed here relies on the time-
independence of the modified energy, we expect our approach
to be applicable to more general tracking problems and we
intend to address this issue in a future publication.
ACKNOWLEDGMENTS
The authors would like to thank the reviewers for helpful
remarks.
VI. REFERENCES
[1] Auckly, D., L. Kapitanski, & W. White, Control of
Nonlinear Underactuated Systems, Commun. Pure Appl.
Math. 53, 2000, 354–369.
[2] Blankenstein, G., R. Ortega & A. van der Schaft,
The Matching Conditions of Controlled Lagrangians
and Interconnection and Damping Assignment Passivity
Based Control, Internat. J. Control 75, 2002, 645–665.
[3] Bloch, A.M., Nonholonomic Mechanics and Con-
trol, Interdisciplinary Appl. Math. 24, Springer-Verlag,
2003.
[4] Bloch, A.M., N. Leonard, & J.E. Marsden, Stabilization
of Mechanical Systems Using Controlled Lagrangians,
Proc. CDC 36, 1997, 2356–2361.
[5] Bloch, A.M., N. Leonard, & J.E. Marsden, Matching
and Stabilization by the Method of Controlled La-
grangians, Proc. CDC 37, 1998, 1446–1451.
[6] Bloch, A.M., N. Leonard, & J.E. Marsden, Potential
Shaping and the Method of Controlled Lagrangians,
Proc. CDC 38, 1999, 1652–1657.
[7] Bloch, A.M., N.E. Leonard, & J.E. Marsden, Controlled
Lagrangians and the Stabilization of Mechanical Sys-
tems I: The First Matching Theorem, IEEE Trans. on
Systems and Control 45, 2000, 2253–2270.
[8] Bloch, A.M., D-E. Chang, N.E. Leonard, & J.E. Mars-
den, Controlled Lagrangians and the Stabilization of
Mechanical Systems II: Potential Shaping, Trans. IEEE
on Autom. Contr. 46, 2001, 1556–1571.
[9] Chang, D-E., A.M. Bloch, N.E. Leonard, J.E. Marsden,
& C. Woolsey, The Equivalence of Controlled Lagran-
gian and Controlled Hamiltonian Systems, Control and
the Calculus of Variations (special issue dedicated to
J.L. Lions) 8, 2002, 393–422.
[10] Hamberg, J., General Matching Conditions in the The-
ory of Controlled Lagrangians, Proc. CDC 38, 1999,
2519–2523.
[11] Hamberg, J., Controlled Lagrangians, Symmetries and
Conditions for Strong Matching. In: Lagrangian and
Hamiltonian Methods for Nonlinear Control, Elsevier,
2000.
[12] Marsden, J.E., Lectures on Mechanics, London Math-
ematical Society Lecture Note Series 174, Cambridge
University Press, 1992.
[13] Marsden, J.E. & T.S. Ratiu, An Introduction to Mechan-
ics and Symmetry. Texts in Appl. Math. 17, Springer-
Verlag, 1999.
[14] Maschke, B., R. Ortega, & A. van der Schaft, Energy-
Based Lyapunov Functions for Forced Hamiltonian
Systems with Dissipation, IEEE Trans. Automat. Con-
trol 45, 2001, 1498–1502.
[15] Ortega, R., A. Loria, P.J. Nicklasson, & H. Sira-
Ramirez, Passivity-based Control of EulerLagrange
Systems, Springer-Verlag, Communication & Control
Engineering Series, 1998.
[16] Ortega, R., M.W. Spong, F. Go´mez-Estern, &
G. Blankenstein, Stabilization of a Class of Under-
actuated Mechanical Systems via Interconnection and
Damping Assignment, IEEE Trans. Aut. Control 47,
2002, 1218–1233.
[17] Zenkov, D.V., A.M. Bloch, N.E. Leonard, & J.E. Mars-
den, Matching and Stabilization of Low-Dimensional
Nonholonomic Systems. Proc. CDC 39, 2000, 1289–
1295.
[18] Zenkov, D.V., A.M. Bloch, & J.E. Marsden, Flat Non-
holonomic Matching. Proc. ACC, 2002, 2812–2817.


Controlled Lagrangian Methods and Tracking of Accelerated Motions

An Introduction to Mechanics and Symmetry.

An Introduction to Mechanics and Symmetry. Texts in

Controlled Lagrangians and the Stabilization of Mechanical Systems I: The First Matching Theorem,

Controlled Lagrangians and the Stabilization of Mechanical Systems II: Potential Shaping, Trans.

Controlled Lagrangians, Symmetries and Conditions for Strong Matching. In: Lagrangian and Hamiltonian Methods for Nonlinear Control,

EnergyBased Lyapunov Functions for Forced Hamiltonian Systems with Dissipation,

Flat Nonholonomic Matching.

General Matching Conditions in the Theory of Controlled Lagrangians,

Lectures on Mechanics, London Mathematical Society Lecture Note Series 174,

Matching and Stabilization by the Method of Controlled Lagrangians,

Matching and Stabilization of Low-Dimensional Nonholonomic Systems.

Nonholonomic Mechanics and Control,

Potential Shaping and the Method of Controlled Lagrangians,

Stabilization of a Class of Underactuated Mechanical Systems via Interconnection and Damping Assignment,

Stabilization of Mechanical Systems Using Controlled Lagrangians,

The Matching Conditions of Controlled Lagrangians and Interconnection and Damping Assignment Passivity Based Control,

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