Optimal Motion Generation for Groups of Robots: A Geometric Approach

Belta, Călin; Kumar, Vijay

doi:10.1115/1.1641190

Cited by 45 publications

(26 citation statements)

References 22 publications

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“…Ribbons find applications in a number of domains including computer aided geometric design [34], to deform 3D shapes in solid modeling [24], and geometry modeling of structures such as DNA strands [17]. Ribbons have also been used to layout arrangements of roads [39] and for path planning for rigid formations of nonholonomic vehicles [2,21]. Sampling-based planning was used to generate variable width ribbon-like paths between a camera and moving object in the environment [16].…”

Section: Related Workmentioning

confidence: 99%

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Planning Curvature and Torsion Constrained Ribbons in 3D With Application to Intracavitary Brachytherapy

Patil

Pan

Abbeel

et al. 2015

IEEE Trans. Automat. Sci. Eng.

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Abstract. A "ribbon" is a surface traced out by sweeping a constant width line segment along a spatial curve. We consider the problem of planning multiple disjoint and collision-free ribbons of finite thickness along curvature and torsion constrained curves in 3D space. This problem is motivated by the need to route multiple smooth channels through a 3D printed structure for a healthcare application and is relevant to other applications such as defining cooling channels inside turbine blades, routing wires and cables, and planning trajectories for formations of aerial vehicles. We show that this problem is equivalent to planning motions for a rigid body, the cross-section of the ribbon, along a spatial curve such that the rigid body is oriented along the unit normal to the curve defined according to the Frenet-Serret frame. We present a two stage approach. In the first stage, we use sampling-based rapidly exploring random trees (RRTs) to generate feasible curvature and torsion constrained ribbons. In the second stage, we locally optimize the curvature and torsion along each ribbon using sequential quadratic programming (SQP). We evaluate this approach for a clinically motivated application: planning multiple channels inside 3D printed implants to temporarily insert high-dose radioactive sources to reach and cover tumors for intracavitary brachytherapy treatment. Constraints on the curvature and torsion avoid discontinuities (kinks) in the ribbons which would prevent insertion. In our experiments, our approach achieves an improvement of 46% in coverage of tumor volumes as compared to an earlier approach that generates each channel in isolation.

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Section: Related Workmentioning

confidence: 99%

“…Planning smooth motions of rigid bodies in 3D has also been studied [41,2,4,9]. This requires planning in the space of 3D positions and orientations, also commonly referred to the SE(3) group.…”

Section: Related Workmentioning

confidence: 99%

Planning Curvature and Torsion Constrained Ribbons in 3D With Application to Intracavitary Brachytherapy

Patil

Pan

Abbeel

et al. 2015

IEEE Trans. Automat. Sci. Eng.

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“…21 By changing the original constant kinetic energy, the optimal smooth trajectories for a set of mobile robots are generated. 22 Generally speaking, all existing works are on rigid body motion with one degree of freedom.…”

Section: Introductionmentioning

confidence: 99%

Optimal three-dimensional biped walking pattern generation based on geodesics

Zhang

Zhou

2017

International Journal of Advanced Robotic Systems

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The innovative three-dimensional humanoid biped gait planning method using geodesics is introduced in this article. In order to control three-dimensional walking, the three-dimensional linear inverted pendulum model is studied in our energy-optimal gait planning based on geodesics. The kinetic energy of the three-dimensional linear inverted pendulum model is calculated at first. Based on this kinetic energy model, the Riemannian metric is defined and the Riemannian surface is further determined by this Riemannian metric. The geodesic is the shortest line between two points on the Riemannian surface. This geodesic is the optimal kinetic energy gait for the center of gravity because the kinetic energy along the geodesic is invariant according to the geometric property of geodesics and the walking is energy-saving. Finally, a simulation experiment using a 12-degree-of-freedom biped robot model is implemented. The gait sequences of the simulated RoboErectus humanoid robot are obtained in the ROS (Robot Operating System) Gazebo environment. The proposed geodesics approach is compared with the traditional sinusoidal interpolation method by analyzing the torque feedback of each joint of both legs, and our geodesics optimization gait planning method for three-dimensional linear inverted pendulum model walking control is verified by the assessment results.

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“…In other words, the dynamical systems of ODEs need to be integrated for many different initial conditions. Examples include geodesics computation in computational geometry [11], robotics [2] and the theory of general relativity. Our motivation for this comes from high frequency wave propagation problems.…”

Section: Introductionmentioning

confidence: 99%

A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems

Motamed

Runborg

2007

Communications in Mathematical Sciences

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Abstract. We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization. The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a threedimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing. We apply the method to two different problems: the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.Key words. ODEs on a manifold, phase space method, escape equations, high frequency wave propagation, geodesics, creeping rays, seismic waves, travel-time AMS subject classifications. 53C22, 65N06, 65Y20, 78A05, 78A40 IntroductionWe want to compute trajectories on two-dimensional compact manifolds possibly embedded in a higher-dimensional space. The dynamics of the trajectories we consider are given by systems of ODEs in a phase space. In many problems, we need to compute a large number of trajectories. In other words, the dynamical systems of ODEs need to be integrated for many different initial conditions. Examples include geodesics computation in computational geometry [11], robotics [2] and the theory of general relativity. Our motivation for this comes from high frequency wave propagation problems. We consider the problem of scattering of a time-harmonic incident field by a bounded scatterer D. We split the total field into an incident and a scattered field. The scattered field in the region outside D is given by the Helmholtz equation,

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Optimal Motion Generation for Groups of Robots: A Geometric Approach

Cited by 45 publications

References 22 publications

Planning Curvature and Torsion Constrained Ribbons in 3D With Application to Intracavitary Brachytherapy

Planning Curvature and Torsion Constrained Ribbons in 3D With Application to Intracavitary Brachytherapy

Optimal three-dimensional biped walking pattern generation based on geodesics

A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems

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