3.2 Time Integration Overview
3.2.3 Constrained Discrete Variational Integrators
Extensive use will be made in this chapter of constrained mechanical systems, in which then−dimensional trajectory is required to evolve on the submanifold C ⊂ Q with C described by mholonomic con- straints,g(q), i.e.,
C={q∈ Q|g(q) =0} ⊂ Q, (3.5)
with tangent bundle
L(q,q,˙ λ) =L(q,q)˙ −gT(q)λ.
Discretizing the additional terms involving the Lagrange multipliers and constraints according to
h
2gT(qk)λk+h
2gT(qk+1)λk+1≈ Z tk+1
tk
gT(q)λdt
gives rise to the augmented discrete Lagrangian
Ld(qk−1,qk,λk,λk+1) =Ld(qk,qk+1)−h
2gT(qk)λk−h
2gT(qk+1)λk+1 (3.7) in which a set of Lagrange multipliers,λd={λk}Kk=0, restrict the trajectory to the submanifold on whichg(q) =0. The corresponding action sum is given by
Id=
K−1
X
k=0
Ld(qk,qk+1,λk,λk+1).
Extremizing the action with respect to all variables–now includingλd–yields theconstraineddiscrete Euler-Lagrange equations
D1Ld(qk,qk+1) +D2Ld(qk−1,qk)−hGT(qk)λk =0, g(qk+1) =0,
(3.8)
in whichG is the Jacobian of the constraint vectorg. In this form, the constrained discrete Euler- Lagrange equations are an (n+m)−dimensional system to be solved for qk+2 and λk+1, given (qk,qk+1). The term −hGT(qk)λk represents the constraint forces which prevent the system from deviating from the constraint manifold,C.
Discrete Null Space Approach. In the analog to the continuous null space approach (c.f. [9, 58]), Equations 3.8 can be reduced via premultiplication by a discrete null space matrix to eliminate the Lagrange multipliers (c.f. [50]). In the discrete case, the null space matrixP should satisfy
range(P(qk)) = null(G(qk)) =TqkC.
One such matrix, which will be calledQ, can be found from the natural orthogonal complement of G(qk). Another option is to use the projection given by
Q=In×n−GT
GM−1GT−1
GM−1, (3.9)
withGevaluated at the appropriate discrete configuration,qk. The null space matrixQis a mapping Q(qk) :Tq∗kQ →η(Tq∗kC), in which is an embedding of the lower dimensional sub-manifold into the redundant cotangent bundle, i.e. η : T∗C →T∗Q(see [58] for more details). By premultiplication withQ(qk), Equation 3.8 becomes
Q(qk) [D1Ld(qk,qk+1, h) +D2Ld(qk−1,qk, h)] =0, (3.10a)
g(qk+1) =0. (3.10b)
In 3.10, the Lagrange multipliers have been eliminated from the system, but the equations of motion remain a redundant (n+m)−dimensional system to be solved fornconfiguration variables.
A second null space matrix may be found directly from the definition of a null space matrix as the linear map from generalized velocities to admissible redundant velocitiesP(q) :Rn−m→TqC. For example, the twist (t) of a free rigid body may be described by its translational velocity, ˙ϕ, and its angular velocity,ω, whereupon the redundant velocities, ˙q, of the body may be expressed as
˙
q=P ν,with:
ν=
˙ ϕ ω
.
(3.11)
The resulting integration scheme is given by
In fact,PT(q) can be thought of as a mapPT(q) :Tq∗Q →Tu∗U, in whichu∈ U are a minimal set of local coordinates in the generalized manifoldU, to be introduced in the following section.
Reparametrization in Local Coordinates. In addition to eliminating the Lagrange multipli- ers, the kinematic constraintsg(qk) can be explicitly eliminated from (3.8) via a mapping from the minimal set of local coordinates u ∈ Rm on the constraint manifold to the fully redundant coor- dinatesq ∈Rn, Fd :U 7→ Q, i.e., a local nodal reparametrization for the redundant configuration variables in terms of a minimal set of generalized variables. The local mappingFd, can be defined such thatqk =Fd(uk,qk−1) withg(Fd(uk,qk−1)) =0. It is readily shown that the Jacobian ofFd
w.r.t. usatisfies
range
dFd(uk+1,qk) du
= null(G(qk)),
however it is most convenient to use this reparametrization in conjunction with the null space matrix introduced in Equation (3.11) to arrive at an integration scheme which eliminates the need to solve for the Lagrange multipliers and explicitly enforces the constraints as
PT(qk) [D1Ld(qk,Fd(uk+1,qk), h) +D2Ld(qk−1,qk, h)] =0. (3.13)
Constrained Discrete Legendre Transform. An alternative time-stepping scheme may be formulated using discrete momenta obtained from the constrained discrete Legendre transforms, Fc−Ld:Q × Q →Tq∗kQand Fc
+Ld:Q × Q →Tq∗kQ, which are defined as
Fc− : (qk,qk+1)7→(qk,p−k) p−k =−D1Ld(qk,qk+1, h) +h
2GT(qk)·λk
Fc
+: (qk−1,qk)7→(qk,p+k) p+k =D2Ld(qk−1,qk, h)−h
2GT(qk)·λk.
(3.14)
The resulting scheme to advance the state of the system from (qk,pk) to (qk+1,pk+1) (along with λk−1 andλk, respectively) is then
p+k −p−k =0 g(qk+1) =0,
(3.15)
with pk+1 then updated according to (3.14). The integrator resulting from the discrete Leg- endre transformation will be referred to as the (q,p) scheme, as opposed to the original (q,q) scheme.
With the definition of the discrete Legendre transform in hand, it is clear that Equations (3.8) and (3.12) may be interpreted as enforcing the balance of discrete momenta at each time node, p+k −p−k =0. Likewise, two additional mappings which eliminate the Lagrange multipliers from (3.14),QFcLd:Q × Q →η(Tq∗kC) andPFcLd:Q × Q →T∗U may be defined as follows
Qp−k =Q(qk) [−D1Ld(qk,qk+1)]
Qp+k =Q(qk) [D2Ld(qk−1,qk)],
(3.16)
and
Pp−k =PT(qk) [−D1Ld(qk,qk+1)]
Pp+k =PT(qk) [D2Ld(qk−1,qk)],
(3.17)
which will be referred to as the projected scheme (3.16) and the reduced scheme (3.17), respec- tively.
Thus, all that is required to start the (q,p) time-stepping scheme is the pair, (q0,ν0) from which (q0,p0) may be calculated according to (3.11) and by letting p0 = Mq˙0. For example, using Equation (3.17), q1 may be solved from P p−0 =PTp0, P p+1 set from (q0,q1), and so on. It is important to note that the Lagrange multipliers are not an immediate output of a (q,p) scheme which uses either the projected or reduced discrete Legendre transforms (3.12,3.17), however the term involving the gradient of the constraint forces isa part of the total, unprojected momentum.
Furthermore, when using either the reduced or constrained schemes, the Lagrange multipliers, λn,
h · ·
Hidden Constraints. A full explanation of this topic may be found in [50]. The aforementioned time-stepping schemes do not enforce a set of hidden constraints on the redundant momenta which come from the temporary differentiated form of the constraint vector g. On the momentum level, these constraints constraints read
G(qk)M−1pk =0. (3.19)
As can be seen in [50], the discrete momenta, including the Lagrange multiplier terms, resulting from (3.15) solved with the projected or reduced momenta (3.17,3.16) do not precisely fulfill these constraints in the presence of a potential. That is, they do not lie in the null space of the internal constraint Jacobian. However, the projected momentaQpk, and the redundant moment recovered from the reduced scheme,
pk =M P(qk)Mred−1)(qk)P pk,with Mred(qk) =PT(qk)M P(qk),
(3.20)
fulfill the discrete hidden momentum constraints exactly.