**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30982

##### Planning Rigid Body Motions and Optimal Control Problem on Lie Group SO(2, 1)

**Authors:**
Nemat Abazari,
Ilgin Sager

**Abstract:**

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.

**Keywords:**
Optimal Control,
Rigid Body Motion,
maximum principle,
Hamiltonian vector field,
Darboux vector,
lie group,
Lorentz metric

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1079938

**References:**

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[5] A.Yucesan, A.C. Coken, N.Ayyildiz,(2004), On the darboux rotation axis of Lorentz space curve, Applied Mathematics and Computation, 155:345-351.

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