| 949 |
|
method using quaternion representation was developed by Omelyan. |
| 950 |
|
However, both of these methods are iterative and inefficient. In |
| 951 |
|
this section, we will present a symplectic Lie-Poisson integrator |
| 952 |
< |
for rigid body developed by Dullweber and his coworkers\cite{}. |
| 952 |
> |
for rigid body developed by Dullweber and his |
| 953 |
> |
coworkers\cite{Dullweber1997}. |
| 954 |
|
|
| 955 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 956 |
|
|
| 976 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 977 |
|
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 978 |
|
\begin{equation} |
| 979 |
< |
Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ |
| 979 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 980 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 981 |
|
\end{equation} |
| 982 |
|
|
| 988 |
|
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 989 |
|
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 990 |
|
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 991 |
< |
\frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 991 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 992 |
|
\end{array} |
| 993 |
|
\] |
| 994 |
|
|
| 995 |
+ |
In general, there are two ways to satisfy the holonomic constraints. |
| 996 |
+ |
We can use constraint force provided by lagrange multiplier on the |
| 997 |
+ |
normal manifold to keep the motion on constraint space. Or we can |
| 998 |
+ |
simply evolve the system in constraint manifold. The two method are |
| 999 |
+ |
proved to be equivalent. The holonomic constraint and equations of |
| 1000 |
+ |
motions define a constraint manifold for rigid body |
| 1001 |
+ |
\[ |
| 1002 |
+ |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1003 |
+ |
\right\}. |
| 1004 |
+ |
\] |
| 1005 |
|
|
| 1006 |
+ |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1007 |
+ |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1008 |
+ |
transformation, the cotangent space and the phase space are |
| 1009 |
+ |
diffeomorphic. Introducing |
| 1010 |
|
\[ |
| 1011 |
< |
M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} |
| 997 |
< |
\right\} . |
| 1011 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1012 |
|
\] |
| 1013 |
+ |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1014 |
+ |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1015 |
+ |
\[ |
| 1016 |
+ |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1017 |
+ |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1018 |
+ |
\] |
| 1019 |
|
|
| 1020 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 1021 |
< |
|
| 1022 |
< |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} |
| 1020 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1021 |
> |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1022 |
> |
given as |
| 1023 |
> |
\begin{equation} |
| 1024 |
> |
X_i^{lab} = Q X_i + q. |
| 1025 |
> |
\end{equation} |
| 1026 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1027 |
> |
\[ |
| 1028 |
> |
V(q,Q) = V(Q X_0 + q). |
| 1029 |
> |
\] |
| 1030 |
> |
Hence, |
| 1031 |
> |
\[ |
| 1032 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
| 1033 |
> |
\] |
| 1034 |
|
|
| 1035 |
+ |
\[ |
| 1036 |
+ |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1037 |
+ |
\] |
| 1038 |
|
|
| 1039 |
+ |
As a common choice to describe the rotation dynamics of the rigid |
| 1040 |
+ |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1041 |
+ |
rewrite the equations of motion, |
| 1042 |
+ |
\begin{equation} |
| 1043 |
+ |
\begin{array}{l} |
| 1044 |
+ |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1045 |
+ |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1046 |
+ |
\end{array} |
| 1047 |
+ |
\label{introEqaution:RBMotionPI} |
| 1048 |
+ |
\end{equation} |
| 1049 |
+ |
, as well as holonomic constraints, |
| 1050 |
+ |
\[ |
| 1051 |
+ |
\begin{array}{l} |
| 1052 |
+ |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1053 |
+ |
Q^T Q = 1 \\ |
| 1054 |
+ |
\end{array} |
| 1055 |
+ |
\] |
| 1056 |
+ |
|
| 1057 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1058 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
| 1059 |
+ |
\begin{equation} |
| 1060 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1061 |
+ |
{\begin{array}{*{20}c} |
| 1062 |
+ |
0 & { - v_3 } & {v_2 } \\ |
| 1063 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
| 1064 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
| 1065 |
+ |
\end{array}} \right), |
| 1066 |
+ |
\label{introEquation:hatmapIsomorphism} |
| 1067 |
+ |
\end{equation} |
| 1068 |
+ |
will let us associate the matrix products with traditional vector |
| 1069 |
+ |
operations |
| 1070 |
+ |
\[ |
| 1071 |
+ |
\hat vu = v \times u |
| 1072 |
+ |
\] |
| 1073 |
+ |
|
| 1074 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1075 |
+ |
matrix, |
| 1076 |
+ |
\begin{equation} |
| 1077 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1078 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1079 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1080 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1081 |
+ |
\end{equation} |
| 1082 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1083 |
+ |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
| 1084 |
+ |
multiplier $\Lambda$ is ignored in the integration. |
| 1085 |
+ |
|
| 1086 |
+ |
Hence, applying hat-map isomorphism, we obtain the equation of |
| 1087 |
+ |
motion for angular momentum on body frame |
| 1088 |
+ |
\[ |
| 1089 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
| 1090 |
+ |
\times X_i } |
| 1091 |
+ |
\] |
| 1092 |
+ |
In the same manner, the equation of motion for rotation matrix is |
| 1093 |
+ |
given by |
| 1094 |
+ |
\[ |
| 1095 |
+ |
\dot Q = Qskew(M^{ - 1} \pi ) |
| 1096 |
+ |
\] |
| 1097 |
+ |
|
| 1098 |
+ |
The free rigid body equation is an example of a non-canonical |
| 1099 |
+ |
Hamiltonian system. |
| 1100 |
+ |
|
| 1101 |
+ |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
| 1102 |
+ |
|
| 1103 |
+ |
\[ |
| 1104 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1105 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
| 1106 |
+ |
\] |
| 1107 |
+ |
|
| 1108 |
+ |
\[ |
| 1109 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
| 1110 |
+ |
_{\Delta t,\pi } |
| 1111 |
+ |
\] |
| 1112 |
+ |
|
| 1113 |
+ |
\[ |
| 1114 |
+ |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1115 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1116 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1117 |
+ |
_1 } |
| 1118 |
+ |
\] |
| 1119 |
+ |
|
| 1120 |
+ |
\[ |
| 1121 |
+ |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1122 |
+ |
_{\Delta t/2,\tau } |
| 1123 |
+ |
\] |
| 1124 |
+ |
|
| 1125 |
+ |
|
| 1126 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1127 |
|
|
| 1128 |
|
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
