In this paper we extend the previous work on the

asymptotic stabilization of pure Euler-Poincaré mechanical

systems using controlled Lagrangians to the

study of asymptotic stabilization of Euler-Poincaré mechanical

systems such as the heavy top

Chang, Dong Eui

Marsden, Jerrold E.

English

Caltech Authors - Main

Asymptotic Stabilization of the Heavy Top
Using Controlled Lagrangians 1
Dong Eui Chang Jerrold E. Marsden
Control and Dynamical Systems 107-81
California Institute of Technology
Pasadena, CA 91125
{dchang@cds.caltech.edu, marsden@cds.caltech.edu
Proc. CDC , 39, (2000), 269–273
Abstract
In this paper we extend the previous work on the
asymptotic stabilization of pure Euler-Poincare´ me-
chanical systems using controlled Lagrangians to the
study of asymptotic stabilization of Euler-Poincare´ me-
chanical systems such as the heavy top.
1 Introduction
This paper considers asymptotic stabilization of the
Euler-Poincare´ mechanical systems using controlled La-
grangians. Control of Euler-Poincare´ mechanical sys-
tems involves both generic control theory and knowl-
edge of the mechanical structure of the Euler-Poincare´
equations (see Marsden and Ratiu [1999]). This knowl-
edge helps us to construct Lyapunov functions by using
the energy-Casimir method. We illustrate this in the
example of control of the upright spinning top.
The guiding principle behind our method of controlled
Lagrangians is to consider a class of control laws that
yield closed-loop systems which remain in Lagrangian
form. The method thus provides a natural class of
energy-based Lyapunov functions whose shape can be
changed by control gains. This type of energy shap-
ing technique, combined with dissipation, allows us
to achieve asymptotic stabilization. Initial work on
the controlled Lagrangian method was done in a se-
ries of papers by Bloch, Leonard and Marsden [1997,
1998, 1999a,b]. Complete discussion of the basic the-
ory is given in Bloch, Leonard and Marsden [2000a]
and Bloch, Chang, Leonard and Marsden [2000a,b], il-
lustrating such examples as an inverted pendulum on
a cart and a spherical pendulum on a puck. The con-
trolled Lagrangian method for pure Euler-Poincare´ me-
chanical systems is addressed in detail in Bloch, Chang,
Leonard, Marsden and Woolsey [2000] and Bloch,
1Research partially supported by AFOSR contract F49620-99-
1-0190.
Leonard and Marsden [2000b], examples of which are a
satellite with an internal rotor and an underwater vehi-
cle with internal rotors and coincident centers of buoy-
ancy and gravity. An alternative control method for
the underwater vehicle on the Hamiltonian side is stud-
ied in Woolsey and Leonard [1999a,b]. Related work on
controlled Lagrangians is Andreev, Auckly, Kapitanski,
Kelkar and White [2000], and Hamberg [2000].
This paper is organized as follows. In §2 we discuss the
Euler-Poincare´ matching. In §3 we apply the theory to
the asymptotic stabilization of the upright spinning of
the heavy top.
2 Euler-Poincare´ Matching
In this section we address the controlled Lagrangians
method for the (general) Euler-Poincare´ equations. The
Euler-Poincare´ matching conditions are found in Bloch,
Leonard and Marsden [1998] for pure Euler-Poincare´
equations. Here we apply the same conditions to the
general Euler-Poincare´ equations. See Holm, Marsden
and Ratiu [1998] for more detail about Euler-Poincare´
equations.
Assume that there is a left representation of Lie group
H on a vector space V . Let G be an abelian Lie group
on which H acts trivially. Let L : TH × V ∗ × TG→ R
be a H-invariant function. We consider the class of
mechanical systems whose kinetic energy depends on
TH × TG and potential energy on V ∗. The left H-
invariance of L allows us to deﬁne the reduced La-
grangian l : h × V ∗ × TG → R by l(h−1vh, h−1x,w) =
L(vh, x, w) for (vh, x, w) ∈ TH × V ∗ × TG and h ∈ H
where h is the Lie algebra of H. In coordinates
l(ηα, xα, θ˙a) =
1
2
gαβη
αηβ + gαaηαθ˙a +
1
2
gabθ˙
aθ˙b
− U(xα) (1)
for (ηα, xα, θ˙a) ∈ h×V ∗×TG. We assume that gαβ , gαa
and gab are constant and that the controls ua act in the
θa directions. The controlled Euler-Poincare´ equations
are given by
d
dt
∂l
∂η
= ad∗η
∂l
∂η
+
∂l
∂x
 x (2)
d
dt
∂l
∂θ˙
= u (3)
with
x˙(t) = −η(t)x(t), (4)
where the bilinear map  : V × V ∗ → h is deﬁned by
〈ηx, v〉 = −〈v  x, η〉
for v ∈ V, x ∈ V ∗ and η ∈ h. Choose the following form
of the controlled Lagrangian lτ,σ,ρ:
lτ,σ,ρ(ηα, xα, θ˙a) =
1
2
(gαβ − gαagabgβb + σabτaατ bβ)ηαηβ
+
1
2
ρab(θ˙a + (gacgαc + τaα)η
α)(θ˙b + (gbdgβd + τ bβ)η
β)
− U(xα). (5)
Deﬁne the controlled momentum J˜a conjugate to θa by
J˜a =
∂lτ,σ,ρ
∂θ˙a
= ρab(θ˙b + (gbdgβd + τ bβ)η
β). (6)
We wish to transform the equations in (2)-(4), by an ap-
propriate feedback u, to the following controlled Euler-
Poincare´ equations of lτ,σ,ρ:
d
dt
∂lτ,σ,ρ
∂η
= ad∗η
∂lτ,σ,ρ
∂η
+
∂lτ,σ,ρ
∂x
 x (7)
d
dt
∂lτ,σ,ρ
∂θ˙
= udiss (8)
with
x˙(t) = −η(t)x(t). (9)
The following is the the Euler-Poincare´ matching con-
ditions from Bloch, Leonard and Marsden [1999a]:
EP-1. τaα = −σabgbα,
EP-2. σab + ρab = gab.
We can then prove the following theorem along the
same lines as the proof in Bloch, Leonard and Mars-
den [2000b].
Theorem 2.1 Under the assumptions EP-1 and EP-
2, the Euler-Poincare´ equations in (7)-(9) coincide with
the Euler-Poincare´ equations in (2)-(4) with the follow-
ing choice of the control u:
ua = uconsa + (gab − kαa gαb)ρbcudissc
where
uconsa = k
α
a
(
cψαδη
δ(gψβηβ + gψbθ˙b)− dψαδ
∂U
∂xδ
xψ
)
,
kαa = Dabσ
bcgcβB
αβ ,
Bαβ = gαβ − gαbgabgaβ ,
Dba = gba + σbcgcβBαβgαegae,
where cψαδ are the structure constants of the Lie algebra
h and dψαδ are the coordinate expression of the bilinear
map  : V × V ∗ → R.
O
x y
z
CM
Figure 1: Heavy top with two rotors, each consisting of
two rigidly coupled disks. The center of mass is
at CM.
3 Asymptotic Stabilization of the Heavy Top
It is well-known in mechanics that the upright spinning
top is unstable if the angular velocity is small. The
motion of the heavy top and the stability of the La-
grange top are well studied in Marsden and Ratiu [1999]
and Holm, Marsden and Ratiu [1998]. In this section
we use the controlled Lagrangian method to asymptoti-
cally stabilize the upright spinning motion of the heavy
top with small angular velocity, including zero velocity.
Related work on the stabilization of the heavy top is
Egorov and Posbergh [2000].
We ﬁrst describe the heavy top with two rotors. We
mount two rotors within the top so that each rotor’s
rotation axis is parallel to the ﬁrst and the second prin-
cipal axes of the top; see Figure 1. Let I1, I2, I3 be
the moments of inertia of the top in the body ﬁxed
frame. Let J1, J2 be the moments of inertia of the rotors
around their rotation axes. Let Ji1, Ji2, Ji3 be the mo-
ments of inertia of the ith rotor with i = 1, 2 around the
ﬁrst, the second and the third principal axes, respec-
tively. Let I¯1 = I1 +J11 +J21, I¯2 = I2 +J12 +J22, and
I¯3 = I3 + J13 + J23. Let λ1 = I¯1 + J1 and λ2 = I¯2 + J2.
Let M be the total mass of the system, g the magni-
tude of the gravitational acceleration, and h the dis-
tance from the origin O to the center of mass of the
system.
In this example we have H = SO(3), V ∗ = R3 and G =
T 2 = S1 × S1. Let Ω = (Ω1,Ω2,Ω3) ∈ so(3) ∼= R3 be
the angular velocity of the top in the body ﬁxed frame.
The vector Γ = (Γ1,Γ2,Γ3) represents the motion of the
unit vector with the opposite direction of the gravity
as seen from the body. θ = (θ1, θ2) is the rotation
angle s of rotors around their axes. Then the reduced
Lagrangian l : so(3)× R3 × TT 2 → R is given by
l(Ω,Γ, θ˙) =
1
2


Ω1
Ω2
Ω3
θ˙1
θ2


T 

λ1 0 0 J1 0
0 λ2 0 0 J2
0 0 I¯3 0 0
J1 0 0 J1 0
0 J2 0 0 J2




Ω1
Ω2
Ω3
θ˙1
θ2


−MghΓ3. (10)
The angular momentum Π = (Π1,Π2,Π3) is
Π =
∂l
∂Ω
= (λ1Ω1 + J1θ˙1, λ2Ω2 + J2θ˙2, I¯3Ω3). (11)
The equations of motion are derived from (2)-(4) as
follows:
Π˙ = Π× Ω +MghΓ× χ (12)
Γ˙ = Γ× Ω (13)
d
dt
∂l
∂θ˙i
= ui (14)
for i = 1, 2 where χ = (0, 0, 1) and ui’s are the control
torques acting on the rotors.
Choose the following diagonal matrix form of σab and
ρab
[σab] =
[
J1σ1 0
0 J2σ2
]
, [ρab] =
[
J1ρ1 0
0 J2ρ2
]
.
In this case the two matching conditions EP-1 and EP-2
become
[τaθ ] =
[− 1σ1 0 0
0 − 1σ2 0
]
,
and
1 =
1
σi
+
1
ρi
for i = 1, 2. The controlled Lagrangian lτ,σ,ρ in (5) is
computed as
lτ,σ,ρ(Ω,Γ, θ˙) =
1
2
(
λ1 − J1
ρ1
)
Ω21 +
1
2
(
λ2 − J2
ρ2
)
Ω22
+
1
2
I¯3Ω23 +
1
2
J1ρ1
(
Ω1
ρ1
+ θ˙1
)2
+
1
2
J2ρ2
(
Ω2
ρ2
+ θ˙2
)2
−MghΓ3. (15)
The controlled momentum J˜ = (J˜1, J˜2) conjugate to
θ = (θ, θ2) is
J˜i =
∂lτ,σ,ρ
∂θ˙i
= JiΩi + Jiρiθ˙i (16)
with i = 1, 2. By Theorem 2.1 we have only to ﬁnd an
asymptotically stabilizing controller udissi for the follow-
ing controlled Euler-Poincare´ equations:
Π˙ = Π× Ω +MghΓ× χ (17)
Γ˙ = Γ× Ω (18)
˙˜J1 = udiss1 (19)
˙˜J2 = udiss2 (20)
where Π is the same as that in (11) by EP-1 and EP-2.
We have two constants of motion; Π · Γ and ||Γ||2.
Let Ω(0),Γ(0) and θ˙(0) with ||Γ(0)||2 = 1 be an initial
condition with
Ω◦3 := Π(0) · Γ(0)/I¯3 <
√
Mgh/I¯3. (21)
We are interested in the equilibrium e = (Ωe,Γe, θ˙e):
Ωe = (0, 0,Ω◦3), Γe = (0, 0, 1), θ˙e = (0, 0) (22)
or
Ωe = (0, 0,Ω◦3), Γe = (0, 0, 1),
˙˜Je = (0, 0)
which corresponds to the upright spinning top with the
rotors at rest. Notice that this equilibrium lies in the
same level set of (Π · Γ, ||Γ||2) as the initial condition.
We construct a Lyapunov function using the energy-
Casimir method (see Bloch, Chang, Leonard, Marsden,
and Woolsey [2000] for more detail of this method). Set
EΦ˜ = Kτ,σ,ρ + U + Φ(Π · Γ, ||Γ||2) + Ψ(J˜1, J˜2) (23)
where U(Γ) = MghΓ3 and Kτ,σ,ρ is given by
Kτ,σ,ρ =
1
2
(
λ1 − J1
ρ1
)
Ω21 +
1
2
(
λ2 − J2
ρ2
)
Ω22 +
1
2
I¯3Ω23
+
J˜21
2J1ρ1
+
J˜22
2J2ρ2
which is the controlled kinetic energy, consisting of
the quadratic terms in (15), in the new coordinates
(Ω,Γ, J˜). Choose the function Ψ as follows
Ψ(J˜1, J˜2) =
J˜21
2)1J1
+
J˜22
2)2J2
(24)
where coeﬃcients )i will be determined later. Choose
the function Φ of the form
Φ(x, y) = −Ω◦3(x− I¯3Ω◦3) +
1
2
(
I¯3(Ω◦3)
2 −Mgh) (y − 1)
+
1
2
a1(x− I¯3Ω◦3)2 +
1
2
a2(y − 1)2
where the constants a1 and a2 are chosen such that
a1 < −1/I¯3
and
4a2 + a1(I¯3Ω◦3)
2 + I¯3(Ω◦3)
2 −Mgh < I¯3(a1I¯3Ω
◦
3 − Ω◦3)2
1 + a1I¯3
.
One can check that the equilibrium e is a critical point
of EΦ˜. We now ﬁnd conditions under which this critical
point is a local maximum. First choose ρi satisfying
I¯3(Ω◦3)
2 −Mgh
(Ω◦3)2
< λi − Ji
ρi
< 0 (25)
for i = 1, 2, and then we can choose )1 and )2 such that
the second derivative of EΦ˜ becomes negative deﬁnite
at e, which implies that EΦ˜ has a local maximum at e.
For later use we impose an additional condition on ρi
and )i as follows:
Ji(Ω◦3)
2 + ()i + ρi)
(
(Ω◦3)
2(I¯3 − λi)−Mgh
) = 0 (26)
for i = 1, 2. With (26), it is still possible to ﬁnd ρi and
)i to ensure negative deﬁniteness of the second deriva-
tive of EΦ˜ at e.
The following choice of udiss
udissi = ci
(
θ˙i +
J˜i
)iJi
)
(27)
with ci > 0 for i = 1, 2, implies
d
dt
EΦ˜ =
2∑
i=1
ci
(
θ˙i +
J˜i
)iJi
)2
≥ 0 (28)
which proves the Lyapunov stability of the equilibrium
e in the closed-loop system. The complete control law
u can be obtained from Theorem 2.1.
Asymptotic stabilization will now be shown by using
the LaSalle invariance principle. Since EΦ˜ has a local
maximum at e, it is nondecreasing in time, and Π·Γ and
||Γ||2 are conserved, there is a number c such that S =
{x ∈ so(3)×R3×TT 2|EΦ˜ ≥ c, Π ·Γ = Πe ·Γe, ||Γ||2 =
1} is compact and positively invariant. Deﬁne E = {x ∈
S|E˙Φ˜ = 0} = {x ∈ S|udiss = 0}. Let M be the largest
invariant subset of E . One can show M = {e} by (26)
after shrinking the set S if necessary. Thus, by the
LaSalle invariance principle, e is asymptotically stable.
Remark. The above procedure shows that the choice
of control gains depends on the initial condition. This
is unavoidable because we need to know the value of
the constant of motion Π · Γ which the internal actua-
tion cannot change; however, our suggested controller
is robust to small errors in the measurement of the ini-
tial condition. Let e˜ be the equilibrium of the form
(22) with Ω˜◦3 instead of Ω
◦
3. Suppose the Ω
◦
3 used in
constructing the control law is very close to the value
Ω˜◦3. Let E˜Φ˜ be the function of the form (23) with Ω
◦
3
replaced by Ω˜◦3. Then e˜ is a critical point of E˜Φ˜. By
continuity, the second derivative of E˜Φ˜ at e˜ will remain
negative deﬁnite, proving Lyapunov stability of e˜. The
condition (26) also holds by continuity, ensuring asymp-
totic stability of e˜.
Remark. The same form of controller works for the
asymptotic stabilization of the upright spinning top
with Ω◦3 >
√
Mgh/I¯3, which is the opposite of (21).
All that needs to be done is to choose ρi and )i to make
EΦ˜ have a local minimum at the equilibrium and to
choose negative ci such that EΦ˜ decreases in time. The
same LaSalle invariance principle argument guarantees
asymptotic stability.
Acknowledgments. We thank Tony Bloch, Naomi
Leonard, Tudor Ratiu, Troy Smith and Craig Woolsey
for helpful comments.
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The Euler-Poincare equations and semidirect products with applications to continuum theories,

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Asymptotic stabilization of the heavy top using controlled Lagrangians

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