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- A& Maths Mechs, Vol. 65, No. 5, pp. 731-739, 0 2002 Elsevier Science Ltd
PII: S0021-8928(01)00079-X
All rights reserved. Printed in Great Britain 0021_8928/01/$-seefront matter
THE LAGRANGE MULTIPLIERS ASSOCIATED
WITH THE ROTATION MATRIX CHARACTERIZING
THE MOTION OF A RIGID BODY ABOUT
ITS CENTRE OF MASS”f
C. VALLBE and D. DUMITRIU
Chasseneuil, France e-mail: [email protected]; [email protected] (Received 17 November 2000) To describe the motion of a rigid body, parametrization based on the use of a rotation matrix consisting of nine components is chosen instead of angular parameters. The equations of motion of mechanical systems consisting of many bodies coupled to one another turn out to be linear. The description of the rotations is provided by six Lagrange multipliers, grouped in a symmetrical 3 x 3 matrix, denoted by A, the components of which are related to the volume averages of the internal couplings in the body. The following properties are proved for a rigid body rotating about its centre of mass: the negative of the Lagrange multiplier matrix is positive, and at each instant of time an orthonormalized basis exists in which new components of the matrix A are constant, which gives six first integrals of the equations of motion [l]. It is proved that three eigenvalues of the matrix A do not change with time and, moreover, they can be found in explicit form. 0 2002 Elsevier Science Ltd. AI1 rights reserved.
To obtain the optimal trajectories of systems of many bodies coupled to one another, parametrization of their configuration is used as the first step. After this, the equations of motion are written out and the problem of optimization is formulated, for example, in the form of Pontryagin’s maximum principle. The non-linear system of differential equations with mixed (initial and terminal) boundary conditions thus obtained is generally solved by the “shooting method”. However, its realization very delicate at the initial stage, since there is no numerical information on the associated variables (multipliers), introduced within the framework of Pontryagin’s principle. These multipliers have to be “guessed” [l]. Moreover, their mechanical interpretation, if, of course, it exists, is not always obvious. In this connection it turns out to be reasonable to eliminate the non-linearity of the equations of dynamics. It was suggested in [2] that one could dispense with parametrization of the rotations using angular variables, which are chosen in a more or less involved way (like, for example, the Euler angles
or the Denavit-Hartenberg parameters). Here all nine components of the rotation matrix R (a non-
singular 3 x 3 matrix) are preserved in the equations of motion. However, these nine components are dependent: they are related by six conditions that can be represented in the form of the matrix equation
RRr = RTR = I, which is to be regarded as the constraint equation (1 is the identity matrix and the
superscript T denotes transposition). This constraint is taken into account by introducing six Lagrange
multipliers, grouped in a symmetrical 3 x 3 matrix A. Using this approach, rotational motion can be modelled as simply as translational one. The classical result consists of the fact that in the equations of translational motion the second derivative with respect to time of the displacement vector of the rigid body considered, multiplied by the mass, already appears in the first term. Within the framework of the proposed approach, a similar property also occurs when describing rotational motion - the second derivative with respect to time of the components of the rotation matrix, multiplied by the inertia matrix, also occurs in the first term of the equations of motion. Here these equations are derived directly both for rotational and for translational motion. The dynamic part of the new equations is linear, which is a considerable advantage from the point of view of numerical investigation. Below we investigate a number of properties of the matrix of Lagrange multipliers A. A simple case of the motion of a rigid body about its centre of mass is investigated, corresponding to the motion of a rigid body about a fixed point, if this point coincides with the centre of mass. Finally, we refine some properties of the matrix A in the case of an axisymmetrical body, i.e. in the Euler-Lagrange problem.
tfikl. Mat. Mekh. Vol. 65, No. 5, pp. 755-764, 2001.
731
732 C. Vallee and D. Dumitriu
1. THE PRELIMINARY PROPERTIES OF THE MATRIX A
The motion of a rigid body about a fixed point is defined by the matrix of its rotations R. In the special case when the fixed point coincides with the centre of mass of the body, the following system was obtained in [l, 21
kK, = RA, RTR = I (^) (1.1)
where K,, is the constant symmetrical positive-definite Poinsot inertia matrix of the rigid body with respect to its centre of mass, which depends on the geometry and distribution of the mass of the body. It is related to the classical inertia matrix Jo as follows:
Jo = tr(&)I - K0 or K0 = tr(&)l/2 -Jo (^) (1.2)
The matrices Jo and Ka are symmetrical. Numerical solution of system (1.1) enabled a number of its interesting properties to be established, including the constancy in time and negativity of the eigenvalues of the matrix A. We have succeeded in obtaining and proving these properties. Finally, the eigenvalues of the matrix of Lagrange multipliers were obtained in explicit form. We have the following property.
Prope@ 1. The negative of the matrix of the Lagrange multipliers (-A) is positive.
Proof. We need to show that the scalar product (-Av, V) is positive for any vector V. It is sufficient to show this solely in the case when Vis an eigenvector of the matrix A. By twice differentiating the second relation of system (1.1) and eliminating R from the expression obtained using the first relation of system (l.l), we obtain
AK,-’ •t-K,-‘A = -2dTK (^) (1.3)
In view of the symmetry of the matrices Kil and A for any vector V the following equality holds
(V, K,-‘AV) = -2(dV, lb’) (^) (1.4)
If Vis the eigenvector of the matrix A, related to the eigenvalue A, the quantity
h=-(kV,liV)l(V,K,$) (^) (1.
is negative, since the matrix Ki’, like the matrix Ks, is positive. Hence, since all the eigenvalues of the matrix A are negative, the negative of the matrix A is positive.
2. THE CONSTANCY OF THE EIGENVALUES
OF THE MATRIX A
A numerical check of the negativity of the eigenvalues of the matrix A of the Lagrange multipliers showed that these eigenvalues do not change with time (here the correctness of this assertion only concerns the Euler-Lagrange problem). In order to prove that the eigenvalues of the matrix A are constant with time, we only need to verify that the derivatives h = 0. By virtue of relations (1.1)
RTiiK, = A (^) (2.1)
and a priori the matrix A is not constant. Differentiation of the characteristic equation of the matrix A
det(h - hl) = det(RTiK 0 - hl) - 0- (^) (2.2)
enables us to obtain certain information on its eigenvalues. As is well known, differentiation of the determinant of the matrix A(t), which depends on time, gives [3]
734 C. VallCe and D. Dumitriu
(Au, Au, Aw) = (detAh u. WI
we obtain
ATj(Au)A = (detA)j(u)
Applying this formula to the matrixA = JO1 and the vector u = (Jso) x w from the second relation of (3.3), we obtain
after which the formula for the double vector product enables us to convert the expression j[(&w) @w]. Using the formula for the double vector product, we have
j(uxw)=w@u-u@w
Assuming u = Jaw and w = w, we reduce expression (3.6) to the form (3.5).
Property 2. The matrix A of Lagrange multipliers can be expressed as a function of o only
A= T(Eo -tlol)+‘To, -(co, +&o)+roJo& -r&o, +~,o)+r,&,lJ,
Proof. Writing the formula for the double vector product in the form
j(u)j(u) = u @Iu - (u, ZJ)/
and assuming u = u = o, we have
Combining this result with the result of Lemma 1 and bearing in mind the second expression of (1.2), the formula for A can be represented in the form
~=~,~50-rl0~~+~0~5,2-521~-50,+7)01-~0~513-522~ (3.8)
whence formula (3.7) follows. The symmetry of the matrix A is not obvious from this expression. But, by virtue of the Hamilton- Cayley theorem, the matrixJo satisfies the characteristic equation
whence we can express Ji as a function of Jo and Ji, which enable us to convert (3.8) to the following symmetrical form
- THE CONSTANCY OF THE EIGENVECTORS OF THE MATRIX A
The previous results hold without any special assumptions regarding the form or distribution of the masses of the body. We will now consider the Euler-Lagrange problem. We will assume that the body is symmetrical about a certain axis both as regards it shape and from the point of view of the inertia properties. We will use the following notation: k is the unit vector of this axis, C is the moment of inertia of the rigid body about the axis of symmetry andA is the moment of inertia about an axis passing through the centre of mass and perpendicular to the vector k. In this notation, the inertia tensor of the rigid body of revolution at a fixed point has the form
J,=A/+(C-A)k@k (^) (4.1)
Since,4 - C/2 is the integral over all points of the body of the product of the density of the body and
The rotation matrix characterizing the motion of a rigid body about its centre 735
the square of the distance through the plane orthogonal to k and passing through the centre of mass, this quantity is positive. Note that JO is the inertia matrix at the initial instant of time, and the vector k is constant. At the instant t the value of the inertia matrix is
J = RJoRT = Al + (C - A)(Rk) @ (Rk)
By virtue of the fact that the matrixJO has a special form, due to the rotational symmetry of the body, not only can Ji be expressed as a function of Jo2and JO, but also 502,and then 3; can also be expressed as a function of JO,which can be shown using the following result.
Lemma 2. The inertia tensor ./a possesses the following properties
- rI = A + C/2; 2) l/z; =AC; 3) J; =(C+A&-ACI; 4)7=A(A+2C)
Proof. The first two properties are obvious. By virtue of the equation (k 8 k)’ = k @ k we have
./o’= A/+[(C-A) +ZA(C-A)ytBk
whence, from expression (4.1) we have property 3. Property 4 can be derived directly from properties 3 and 1.
Remark. Property 3, which expresses Jg in terms of JO, is related to the fact that the matrix JO has two equal eigenvalues.
Since the matrixJo has two different eigenvalues, it is fairly easy to integrate Eqs (3.3). Here, by virtue of results obtained previously [l], the instantaneous angular velocity vector has the form
0=EQ,, (^) E = exp[f(C/ A - l)(k&)j(k)] (^) (4.2)
where Szs is the initial value of the instantaneous angular velocity vector 52. Since E is the rotation around the vector k, we have the following simple properties
EJ,E’ = J,, JoET = ETJ,, EJ, = J,E (^) (4.3)
They follow from the equalities
j(k)J, = J,,j(k) = Aj(k)
which are a consequence of the commutivity of Jo with all integer powers ofj(k) and with the exponent ofE. We will introduce the following notation, similar to notation (3.4),
5; =Q,@Q,, 11; =(Q,,Q,> (^) (4.4)
6; = J&Jd, qi = J~T$J~, i, j = 0,1,2 ,...
To simplify expression (3.8) we will need certain relations which link the quantities (3.4) and (4.6). It is clear from relations (4.3) and (4.4) that
(4.5)
We have the following properties.
Property 3. The symmetrical matrix ETAE is constant with time, and its form is determined by the right-hand side of (3.7), if, in the latter, we replace the expressions without asterisks by the corresponding expressions with asterisks, i.e. we replace w by Qs. Moreover, the six independent components of the matrix ETAE specify the first six integrals of the equations of motion.
Proof. Bearing in mind relations (4.5) and replacing o by the expression EQo in relation (3.6) for A, we obtain
The rotation matrix characterizing the motion of a rigid body about its centre 737
Property 1 is obtained directly by considering the symmetrical part of (5.2) Multiplying relation (5.2) on the right by JO,we obtain the relation
5;, = A% ~(Jon,)+(C-A)~k((Jono)
from which we obtain Property 2.
The following result can be derived directly from (5.1) using relations 1 and 2 of Lemma 3
ErAE= -Cq;,I/2+~;,/2-Cl(2A)(C-A)j.t(~,, +clo)+
+(C-A)[T$,+(C-A)A-‘]~G (^) (5.3)
If the rigid body is a uniform fall, we have A = C, and the matrix A can be evaluated directly. We have
Jo = CI, trJ, =3C, K. =C1/2, E=I; h=O, o=R,
Then
This expression is identical with expression (5.3) if we make the substitution,4 = C. Hence it can be seen that the matrix
-A = cj(no)[j(no)]T / 2
is positive (but not positive-definite).
Property 5. Suppose r = [qi - p2]1’ r-’
is the length of the vector Q. - pk, and n is the unit vector (Qo - @c). Then the matrix ETAE allows of the following tensor representation
ETAE=-Cq;(nxk)@(nxk)12-C~2(n~‘n)/2+
+CA-‘(A-C/2)+(n@k+k@n)+(A-C/2)r2(k63&)
Proof. From the equality
we have sequentially
Ro=rn+~
co, +clo =r(n@k+k@n)+2@@k
5; =r2(n~n)+rCl(n~&++k~)+C12k8k
Substituting the previous diads into (5.3), we obtain
ETAE=-C?$l/2+Cr2(n@n)12+CA-‘(A-C/2)r~o(n@k+k@n)+
+[(A-C/2)~2+(C-A)r9k@k
But the identity matrix I can be represented in the form
I=(nxk)~(nxk)+n~~++k~
whence we finally obtain representation (5.4). We will not obtain the eigenvalues of the matrix A. We will introduce the scalars
cl = -(A - C/2)r2 - Cp2 12, c2 = -C(A - C/2)(1 -C/A)2r2p2 I
A=c; +4C
738 C. VallCe and D. Dumitriu
Three eigenvalues of the matrix A can be represented as
h, = 5+, J-^ h,=- Cl^ +a 2
, +c 2 2 ll”
By virtue of Property 4 the quantity -CQ2 is an eigenvalue of the matrices ETAE and A. By relation (5.4) the other two eigenvalues of the matrix are identical with the eigenvalues of the matrix
I/
-cp= I2 CA-‘(A^ -C/2)+ CA-‘(A -C/2)/y -(A - C/2)r2 (^) /I
w
The result mentioned holds since the characteristic polynomial of this matrix has the form
Remarks 1. Since the quantity (A - C/2) is positive, Property 1 can be verified by confining ourselves to proving that the matrix which is the negative of matrix (5.5) is positive.
- The discriminant A of the characteristic polynomial is positive. We have
A=c~+4c2=(A-C/2)2r4+(C/2)2~4+C(A-CC2)~2~2[1-2(1-C/A)2]
By (4.2) the quantity
1-2(1-C/A)= =-1+4CA-=(A-C/2)
exceeds -1, while the discriminant A exceeds the positive quantity
[(A-C/2)r2 -Cp
3. It can be verified that three eigenvalues of the matrix A are negative. Since the scalar cl is negative, it must be shown that the quantity h2 is negative. We have the sequence of equivalences
h2 cO~&c-c, c3A2<c; ec2 <
although the negativity of c can be derived from the positivity of (A - C/2).
- THE CAUCHY CONSTRAINT TENSOR OVER THE VOLUME
AVERAGED
The equations of motion of a rigid body (1.1) can be derived from the principle of virtual work using the constraint equations
RTR=I
which are taken into account using the matrix of the Lagrange multipliers A. On the other hand, the body can be regarded as a continuous medium S. Then, using the principle of virtual work for a continuous medium, one can obtain [l] an expression for the Cauchy constraint tensor o for the matrix A averaged over the volume
j odv =-RART S
where du is the element of volume. The simple expression for the final rotation
RE = exp[rj(ao + (Cl A - l)M)]
and the constancy of the matrix ETAE enable us to represent relation (6.1) in the form
(6.1)
