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Trajectory Planning for a Rigid Body Based on Voronoi Tessellation and Linear-Quadratic Feedback Control
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2012 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

When a

rigid body moves in 3-dimensional space, it is of interest to nd a trajectory

such that it avoids obstacles. With this report, we create an algorithm that nds such a

trajectory for a 6-

DOF rigid body. For this trajectory, both the rotation and translation

of the rigid body are included.

This is a trajectory planning problem in the space of Euclidean transformations


Because of the complexity of this problem, we divide it into two consecutive parts where

the rst part comprises translational path planning in the 3-dimensional Euclidean space

and the rotational path planning. The second part comprises the design of control laws

in order to follow the designed trajectory.

In the rst part we create

virtual spheres surrounding each obstacle in order to obtain

approximate central points for each obstacle, which are then used as input for our


Voronoi tessellation method. We then create a graph containing all feasible

paths along the faces of the convex polytopes containing the central points of the virtual

spheres. A simple global graph-search algorithm is used to nd the shortest path between

the nodes of the polytopes, which is then further improved by approximately 3%. Where

random obstacles are uniformly distributed in a conned space. During the work process,

we discovered that our translational path planning method easily could be generalized to

be used in

n-dimensional space.

In the second part, the control law for the rigid body such that it follows the rotation

and translation trajectory is designed to minimize the cost, which is done by creating a


feedback loop for a linear system. The translational control is designed

in an inertial reference frame and the rotational control is designed in a body frame,

rigidly attached to the rigid body's center of mass. This allows us to separate the control

laws for the translational and rotational systems. The rotational system is a nonlinear

system and has been linearized to be able to use the

Linear-Quadratic feedback controller.

All this resulted in a well performing algorithm that would nd and track a feasible

trajectory as long as a feasible path exists.

Place, publisher, year, edition, pages
2012. , 59 p.
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-118351OAI: diva2:605820
Available from: 2013-02-15 Created: 2013-02-15 Last updated: 2013-02-15Bibliographically approved

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