### Abstract

Mathematical models of complicated mechanical systems such as multibody systems with kinematic constraints, whether holonomic or nonholonomic, are generally represented by implicit nonlinear Differential-Algebraic Equations (DAEs). For the numerical integration of the constrained multibody dynamics represented by the DAEs, we eventually need to calculate inversion of a large scale sparse Jacobian matrix during Newton iteration at each time step. This directly causes in taking much CPU time, since the Jacobian matrix of such mathematical models has generally the characteristic of random sparseness and hence it is very difficult to solve the matrix inversion fast unless employing some efficient sparse matrix technique based on topological structure of the DAEs. In this paper, we propose a graph theoretic approach to symbolic generation of sparse matrix inversion for large scale multibody systems, by which one can " symbolically " calculate matrix inversions of random sparse Jacobian matrices associated to nonlinear implicit differential equations for the sake of fast numerical integrations. It is noteworthy that one can easily calculate the sparse matrix inversion without numerical Gaussian elimination at each time step. To do this, we first show how kinematical and dynamical relations can be effectively set up by introducing connection matrices by mechanical analogy with KCL and KVL constraints in circuit theory. Second, we show how to assign input-output relations among variables in all the kinematical and dynamical relations that appear in the mathematical models of DAEs by introducing bipartite graphs. Then, we demonstrate how one can choose pivots of the Jacobian matrix associated to the kinematical and dynamical relations by using input-output relations in the context of bipartite graphs. Furthermore, we explain solvability for the sparse Jacobian matrix inversion associated to the DAEs by using the bipartite graph. Finally, we propose symbolic generation of the sparse matrix inversion for the Jacobian matrix, which is to be explicitly done by symbolic manipulation and we demonstrate the validity of the proposed approach in numerical efficiency by an example of the Stanford manipulator.

Original language | English |
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Title of host publication | 5th Asian Conference on Multibody Dynamics 2010, ACMD 2010 |

Publisher | International Federation for the Promotion of Mechanism and Machine Science (IFToMM) |

Pages | 242-250 |

Number of pages | 9 |

Volume | 1 |

ISBN (Print) | 9781618390882 |

Publication status | Published - 2010 |

Event | 5th Asian Conference on Multibody Dynamics 2010, ACMD 2010 - Kyoto Duration: 2010 Aug 23 → 2010 Aug 27 |

### Other

Other | 5th Asian Conference on Multibody Dynamics 2010, ACMD 2010 |
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City | Kyoto |

Period | 10/8/23 → 10/8/27 |

### Fingerprint

### Keywords

- Bipartite graphs
- DAEs
- Multibody systems
- Sparse tableau approach
- Symbolic generation

### ASJC Scopus subject areas

- Mechanical Engineering
- Automotive Engineering
- Control and Systems Engineering

### Cite this

*5th Asian Conference on Multibody Dynamics 2010, ACMD 2010*(Vol. 1, pp. 242-250). International Federation for the Promotion of Mechanism and Machine Science (IFToMM).

**A graph-theoretic approach to large scale multibody systems.** / Noguchi, Takashi; Yoshimura, Hiroaki.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*5th Asian Conference on Multibody Dynamics 2010, ACMD 2010.*vol. 1, International Federation for the Promotion of Mechanism and Machine Science (IFToMM), pp. 242-250, 5th Asian Conference on Multibody Dynamics 2010, ACMD 2010, Kyoto, 10/8/23.

}

TY - GEN

T1 - A graph-theoretic approach to large scale multibody systems

AU - Noguchi, Takashi

AU - Yoshimura, Hiroaki

PY - 2010

Y1 - 2010

N2 - Mathematical models of complicated mechanical systems such as multibody systems with kinematic constraints, whether holonomic or nonholonomic, are generally represented by implicit nonlinear Differential-Algebraic Equations (DAEs). For the numerical integration of the constrained multibody dynamics represented by the DAEs, we eventually need to calculate inversion of a large scale sparse Jacobian matrix during Newton iteration at each time step. This directly causes in taking much CPU time, since the Jacobian matrix of such mathematical models has generally the characteristic of random sparseness and hence it is very difficult to solve the matrix inversion fast unless employing some efficient sparse matrix technique based on topological structure of the DAEs. In this paper, we propose a graph theoretic approach to symbolic generation of sparse matrix inversion for large scale multibody systems, by which one can " symbolically " calculate matrix inversions of random sparse Jacobian matrices associated to nonlinear implicit differential equations for the sake of fast numerical integrations. It is noteworthy that one can easily calculate the sparse matrix inversion without numerical Gaussian elimination at each time step. To do this, we first show how kinematical and dynamical relations can be effectively set up by introducing connection matrices by mechanical analogy with KCL and KVL constraints in circuit theory. Second, we show how to assign input-output relations among variables in all the kinematical and dynamical relations that appear in the mathematical models of DAEs by introducing bipartite graphs. Then, we demonstrate how one can choose pivots of the Jacobian matrix associated to the kinematical and dynamical relations by using input-output relations in the context of bipartite graphs. Furthermore, we explain solvability for the sparse Jacobian matrix inversion associated to the DAEs by using the bipartite graph. Finally, we propose symbolic generation of the sparse matrix inversion for the Jacobian matrix, which is to be explicitly done by symbolic manipulation and we demonstrate the validity of the proposed approach in numerical efficiency by an example of the Stanford manipulator.

AB - Mathematical models of complicated mechanical systems such as multibody systems with kinematic constraints, whether holonomic or nonholonomic, are generally represented by implicit nonlinear Differential-Algebraic Equations (DAEs). For the numerical integration of the constrained multibody dynamics represented by the DAEs, we eventually need to calculate inversion of a large scale sparse Jacobian matrix during Newton iteration at each time step. This directly causes in taking much CPU time, since the Jacobian matrix of such mathematical models has generally the characteristic of random sparseness and hence it is very difficult to solve the matrix inversion fast unless employing some efficient sparse matrix technique based on topological structure of the DAEs. In this paper, we propose a graph theoretic approach to symbolic generation of sparse matrix inversion for large scale multibody systems, by which one can " symbolically " calculate matrix inversions of random sparse Jacobian matrices associated to nonlinear implicit differential equations for the sake of fast numerical integrations. It is noteworthy that one can easily calculate the sparse matrix inversion without numerical Gaussian elimination at each time step. To do this, we first show how kinematical and dynamical relations can be effectively set up by introducing connection matrices by mechanical analogy with KCL and KVL constraints in circuit theory. Second, we show how to assign input-output relations among variables in all the kinematical and dynamical relations that appear in the mathematical models of DAEs by introducing bipartite graphs. Then, we demonstrate how one can choose pivots of the Jacobian matrix associated to the kinematical and dynamical relations by using input-output relations in the context of bipartite graphs. Furthermore, we explain solvability for the sparse Jacobian matrix inversion associated to the DAEs by using the bipartite graph. Finally, we propose symbolic generation of the sparse matrix inversion for the Jacobian matrix, which is to be explicitly done by symbolic manipulation and we demonstrate the validity of the proposed approach in numerical efficiency by an example of the Stanford manipulator.

KW - Bipartite graphs

KW - DAEs

KW - Multibody systems

KW - Sparse tableau approach

KW - Symbolic generation

UR - http://www.scopus.com/inward/record.url?scp=84912081938&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912081938&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84912081938

SN - 9781618390882

VL - 1

SP - 242

EP - 250

BT - 5th Asian Conference on Multibody Dynamics 2010, ACMD 2010

PB - International Federation for the Promotion of Mechanism and Machine Science (IFToMM)

ER -