In this paper we derive a Poisson bracket on the phase space so(3)^*x so(3)^*x SO(3) such that the dynamics of two three- dimensional rigid bodies coupled by a ball and socket joint can be written as a Hamiltonian system

Grossman, R.

Krishnaprasad, P. S.

Marsden, Jerrold E.

R. Grossman

P. S. Krishnaprasad

Jerrold E. Marsden

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The Dynamics of Two Coupled Rigid Bodies,

In this paper we derive a Poisson bracket on the phase space so(3)*x so(3)*x S0(3) such that the dynamics of two three- dimensional rigid bodies coupled by a ball and socket joint can be written as a Hamiltonian system

Krishnaprasad, Perinkulam S.

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The Dynamics of Two Coupled Rigid Bodies.

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The Dynamics of Two Coupled Rigid Bodies 
R. GROSSMAN,* P. S. KRISHK'APRASAD1" AND JERROLD E. M.ARSDEN~: 
Abstract 
In this paper we derive a Poisson bracket on the phase space so(3)* x so(3)* x SO(3) such that 
the dynamics of two three dimensional rigid bodies coupled by a baJl and socket joint can be written 
as a Hamiltonian system. 
§1. Introduction 
In this paper we introduce a Poisson bracket on the phase space 
so(3)* x so(3)* x 80(3), 
where so(3)* is the dual of the Lie algebra of 80(3), so that the dynamics of two rigid bodies 
coupled by a ball and socket joint can be written as the Hamilitonian system if = {F, H}. This 
sets the stage so that the stability and asymptotics of the system can be studied using the energy 
Casimir method as in Holm, Marsden, Ratiu and Weinstein [1985] and Krishnaprasad [1985]; so that 
chaotic solutions can be found using the Melnikov method such as in Holmes and Marsden [1983]; 
so that bifurcations of the system can be described using the techniques in Golubitsky and Stewart 
[1986] and Lewis, Marsden and Ratiu [1986]; and so that control issues can be studied, as in Sanchez 
de Alvarez [1986]. 
The dynamics of planar coupled rigid bodies has been studied using similar ideas in 8reenath, 
Krishnaprasad and Marsden [1986]. 
Research supported in part by AFOSR-URl grant # AFOSR-87-0073. 
• Supported by an NSF postdoctoral fellowship held at the University of California, Berkeley. 
t Department of Electrical Engineering and the Systems Research Center, University of 
Maryland, College Park, Maryland, 20742. Partially supported by NSF grant 
OIR-85-00108, and by the Minta Martin Fund for Aeronautical Research. 
*Department of Mathematics, University of California, Berkeley, Berkeley, CA 
94720. Research partially supported by DOE contract DE-A 1'03-85ER-12097. 
373 
374 GROSSMAN. KRISHNAPRASAD AND MARSDEN 
§2. Kinematics 
In this section we will derive the Lagrangian describing the free motion of two rigid bodies 
coupled by a ball and socket joint. At time 0 we assume that the two coupled rigid bodies are in a 
reference configuration denoted B. Fix a.n inertial frame a.nd let Q denote a point in the reference 
configuration 8. Let 8 1 denote those points Q E B which belong to body 1 and let 8 2 denote 
those points which belong to body 2. 
The configuration at time" is determined by a smooth map 
We can also specify the configuration at time t as follows. First we specify the position of the joint 
with respect to the inertial frame. Denote this as wet). Fix a frame centered at tbe joint and parallel 
to the inertia.! frame. With respect to this frame, the configuration of body 1 is determined as usua.! 
by three Euler angles. The Euler angles determine the orientation of a body fixed frame relative to 
tbe spatial frame centered at the joint. Alternatively these two frames are related by an element 
Al(t) E SO(3). Similarly the configuration of body 2 is determined by an element A2(t) E SO(3). 
We conclude that the configuration space is 
and that 
C = SO(3) x SO(3) x R3 
q(Q, t) = A1(t)Q + wet), for Q E 8 1 
q(Q, t) = A2(t)Q + wet), for Q E 82. (2.1) 
We now proceed to compute the kinetic energy of the system. This requires that we keep track 
of the centers of mass of the two bodies and the center of mass of the system relative to the fixed 
inertia.! frame as well as the frame centered at the joint. Let m1 and m2 denote the masses of the 
two bodies and let m denote the total mass. Let sP denote the center of mass of body 1 in the 
reference configuration relative to the inertia.! frame and let S~ denote the center of mass for body 
2. Let r1(t) denote the center of mass of body 1 at time t relative t.o the inertial frame and let "2(1) 
denote the center of mass for body 2. Let 81 (1) and 82(1) denote the center of mass of bodies 1 and 
2, respectively, measured with respect to the frame centered at the joint. Finally let aCt) denote the 
center of mass of the ensemble measured with respect to the inertia.! frame. Figures 1 and 2 show 
the relationships of these quantites. For example the following equations can be read off from the 
figures 
F1<;1Ure 1 
81(t) = AI (t)S~ 
82(t) = A2(I)S~ 
r1(t) = w(i) + 81(t) 
r2(1) = w(t) + 82(t) 
F1<;1Ure 2 
(2.2) 
THE DYNAMICS OF TWO COUPLED RIGID HODIES 375 
Given two vectors tI = (vi ,11',1111) and w = (wi, w', w ll), denote their inner product relative to 
the standa.rd Euclidea.n structure as 
II 
(II,W) = LviWi , 
i=1 
and the corresponding norm by 1111. Let p(Q) denote the mass measure of the ensemble in the 
reference configuration. The kinetic energy KE of the configuration is 
KE = ~ k 14(t)1 dp(Q) 
= ~ is
l 
IAl(t)Q + w(t)12 dp{Q) + (1 ..... 2) 
= i kl (Al"O" + wi) . (A~,Q' + Wi) dp(Q) + (1 ..... 2) (2.3) 
= {~kl Q"Q' dl'(Q)} AhA{, + {~fal dP(Q)} wiWi 
+ {~k1 0" dl'(Q)} Ai"Wi + {i k1 Q' dp(Q)} Ai, til + (1 -- 2). 
Let It denote the coefficient. of inertia matrix of body 1, defined by 
for i.i = 1.2,3. (2.4) 
The coefficient of inertia i2 of body 2 is defined similarly. Using these definitions and the defintion 
of the center of mass, we can rewrite the expressions above as 
1 . . ml . 
KE = itr (AI/lAD + T(u', w) + ml(AISr.w) + (1 ..... 2) 
= ~tr (A1/IAD + ml(AISf.w) + (1 ..... 2)+ ;(w.w) (2.5) 
The Lagrangian is simply the total kinetic energy. To summarize. the velocity phase space for 
our system is TSO(3) x TSO(3) x TR3 and the Lagrangian is given by (2.5). 
§3. Reduction by the Euclidean Group 
Consider the following action of an element 9 of the Euclidean group E(3) 
where A E SO(3) and b E R 3 , on a point (AI. A 2 • w) in the configuration space C: 
(3.1) 
It is easy to check that the Lagrangian (2.5) is invariant under this action. Since the Lagrangian 
is invariant under E(3). so is the Hamiltonian. The purpose of this section is to perform the 
reduction by this group. This will be done in two steps. First we will reduce by R3; this accounts 
for conservation of total linear momentum. Then we will reduce by SO(3); this accounts for 
conservatic.n of total angular momentum. 
We begin by rewriting the Lagragian in terms of the linear momentum p. Using (2.2), we can 
write the total linear momentum as 
p = rna = ml"'l + m2"'2 
= rnw + rn1A I sr + m2A2~' (3.2) 
376 GROSSMA N, KRISII N APRASAD AND MARSDEN 
It is convenient to introduce the expression 
(3.30) 
and write 
ml ,.,0 m2 nO 
W = Q - -A1,:)i - -A2,:):j 
m m (3.3b) 
W=ir-p 
SubstitutiDl! (3.3a and b) into (2.5) and simplifying gives the following form for the Lagrangian 
(3.4) 
The Legendre transformation F L induces a symplectic structure on 
TC = TSO(3) x TSO(3) x TR3 
and the tangent of the action of R3 on C (see (3.1» is symplectic. To verify this statement and the 
ones that follow, see Abraham and Marsden [1978], chapters 3 and 4. The momentum map for this 
action is given by 
J : 1'80(3) x TSO(3) x TR3 __ RS" 
(A b Ab A2 ,A2 ,w,w) t-+ p. 
The corresponding reduced space at pis: 
(J-l(p») IR3 = TSO(3) x TSO(3). 
From (3.4) we see that the Lagrangian on the reduced space is simply 
Since p is constant, we can drop the last term. This completes the first stage of th~ reduction. 
(3.5) 
We DOW perform the reduction corresponding to conservation of total angular momentum. This 
time we use Poisson reduction; see Krishnaprasad and Marsden [1986] for a summary of Poisson 
reduction. Consider the map 
~ : T* (SO(3) x SO(3» -- so(3)" x so(3)" x SO(3) 
(1I"A" 7I"A.) ..... (n l , n2 , A) = (T" L",1I""" T* LA, 7I"A" All A2) . (3.6) 
We will define a Poisson bracket on the target space so that the map A becomes a Poisson map with 
respect to this bracket and the canonical bracket on the cotangent bundle T' (SO(3)x SO(3». 
Introduce the body angular velocities of each of the bodies 
(3.7) 
where n is the linear map v t-+ n x v on R3. We will also need the moment of inertia J1 of bod~ 1 
pven by 
(Jd j = - f QiQi dp(Q), if if: i 
JBI 
= kl (IQI2_(Qi )2) dll(Q), ifi=j. (3.8) 
The moment of inertia h of body 2 is defined in a similar way. 
TilE DYNAMICS 01' TWO COCPLED RIGlD BOIlIES 377 
In order to derive an explicit exprssion for the angular momenta of the system, it is helpful 
to write the total kinetic energy as a quadratic form. After a certain amount of algebra, the total 
kinetic energy (3.5), can be written 
(OL O~) (1, j~) (010 ( 2), (3.90) 
where 
Let J denote the matrix 
2 
i1 = J1 + ~ (I~12 . 1 - .s1 ® ~) 
m 
2 
i2 = J2 + ~ (I~12 . 1 - ~ ® ~) 
m 
A = (Srf A (S~) 
mlm2 f=--
m 
(3.9b) 
(3.9c) 
From the Legendre transformation applied to Lagrangians quadratic in the velocities, the angular 
momenta are found to be 
The inertial orientation matrices AI, A2 determine the relative orientation matrix 
A = AlIA •. 
Using the definitions (3.7) - (3.10), we can rewrite the Lagragian (3.5) as 
L = !Il. rIll 
2 
(3JJd) 
(3.10) 
(3.11) 
and conclude, using the fact that the Lagragian is quadratic in the momenta, that the Hamiltonian 
H(Ill,Il2.A) is also given by (3.11). 
We next derive the Poisson bracket on 50(3)" x 50(3)" x 50(3). Given a function F on 50(3)" x 
80(3)- x 50(3), define a function FA on T* (50(3) x 50(3» by 
FA = FooX. (3.12) 
The canonical bracket on T" (SO(3) x SO(3» is 
{F,\, G,\} = DA,F,\ . ~ - DA,H,\· ~ 
+ D. F,\.!l!.J.. - DA H,\ .lli. 
""2 li"2 2 6"2 
(3.13) 
Using the chain rule, we can introduce a bracket on 80(3)" x 50(3)* x SO(3), so that oX becomes a 
Poisson map. This is straghtforward but tedious. To organize the computation, it is helpful to note 
the following facts. 
Fact 1. H'lrA E TASO(3), then 
IT := 1; LA' 'irA = A'1I'A E 50(3)* (3.14) 
Fact 2. Let HA(A I , 11'1> A 2 , 11'2) be a function on T" (SO(3) x SO(3) and let ~ denote the func-
tional derivative of H,\ with respect to 11'1' Then 
~-AHL 
'''' - .n,· 
Fact 3. Let FA (All 11'1, A 2 ,'lr2) be a function on T" (SO(3) x SO(3), where we have, by abuse of 
.Dotation, written an element in the cotangent space at Al &5' (All 'lrd. Then 
DA,F,\(Ab lrl,A2,'lr2)(Md = (tk,(M2)''lr2) + <*f. All(M2». 
It is DOW straightforward to combine these fa.cts to see that the canonical bracket (3.13) may 
be written in terms of the fuctions F a.nd G using the chain rule as 
{F. H}(Il l , IT2 ,A) = -(lIlt [ff., 1tf.]) - (Il2 , [M;. Ilf,-) 
- (if. flf,A - AHf;) + (if, ff.A - AM;). (3.15) 
To summarize, we have 
:178 GROSSMAN, KRISHNAPRASAD AND MARSDEN 
Theorem 1. (i) With tbe canonical bracket Oll r(50(3) x 50(3» and the bracket (3.15) on 
80(3)" x 80(3)* x 50(3), tbe map 
l: r(50(3) x 50(3» - 80(31 x 80(3)" x 50(3) 
8iven by (3.6) is a POissoli map. (ii) The dynamics Oll the reduced space is given by 
F= {F,H}. 
54. Further Remarks 
1. We begin with a brief discussion of the Casimirs. Consider the momentum map 
J : r (50(3) x 50(3)) - 80(3)· 
(lI"Al'lI"A,) ..... (T"'RAlll"Al +rRA,:lrA,)· 
The composition of this with the Casimir 
C :80(3)"-R 
II ..... llll2 
(4.1) 
(4.2) 
produces a collective Hamiltonian on ro (50(3) x 50(3)) whose Hamiltonian vector field is tangent 
to the G-orbits (see Guillemin and Sternberg [1980J or Holmes and Marsden [1983]) and therefore 
induces a Casimir (: on the space 8O(3t x 80(3)" x 50(3) via the Poisson map (3.6). 
Tracing through the diagram shows that (: = Illl + AIT212 ; and, hence, any function of the 
form 4> (1111 + AIT2f) is a Casimir for tbe bracket (3.15). 
2. The symplectic leaves in the nine dimensional space 80(3)" x 80(3)" x SO(3) appear to be 
eight dimensional (level sets of the function IITI + AIT2f) and in the case of J = 0, (given by (4.1),) 
the six dimensional space roSO(3); and, finally, if ITI = 0, IT2 = 0, a two dimensional space 52 of 
trival equilbria. We expect to explore the geometry of these leaves and the other topics listed in the 
introduction in a future publication. 
References 
Abraham, R. and Marsden, J. [1978] Foundations of Mechanics, second edition, Addison-
Wesley. 
Guillemin, V. and Sternberg, S. [1980], The Moment Map and Collective Motion, Ann. of 
Phys., vol. 127, pp. 220-253. 
Golubitsky, M. and Stewart [1986],1., Generic Bifurcation of Hamiltonian Systems with Sym-
metry to appear in Physico. D. 
Holm, D., Marsden, J., Ratiu, T., and Weinstein, A. [1985], Nonlinear Stability of Fluid and 
Plasma Equilbria, Physics Reports, vol. 123, nos. 1-2, pp. 1-116. 
Holmes, P. and Marsden [1983], J., Horseshoes and Arnold Diffusion for Hamiltonian Systems 
on Lie Groups, Indiana University Mathematics Journal, vol. 32, no. 2, pp. 273-309. 
Krishnaprasad, P. [1985], Lie-Poisson Structures, Dual-Spin Spacecraft, and Asymptotic Sta-
bility, Nonlinear Analysis, Theory, Methods and Appl., vol. 9, pp. 1011-1035. 
Krishnaprasad, P. and Marsden, J. [1987], Hamiltonian Structures and Stability for Rigid Bodies 
with Flexible Attachments, Arch. Rat. Mech. Anal., vol. 98, no. 1, pp. 71-93. 
Lewis, D., Marsden, J. and Ratiu, T. [1986], Stability and Bifurcation ofa Rotating Planar 
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The Dynamics of Two Coupled Rigid Bodies

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