scholarly journals Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130249 ◽  
Author(s):  
Christopher L. Burnett ◽  
Darryl D. Holm ◽  
David M. Meier
Keyword(s):  
Lie Group ◽  
Trajectory Planning ◽  
Quantum Control ◽  
Higher Order ◽  
Integration Scheme ◽  
Planning Problem ◽  
Lagrange Equations ◽  
Lie Group Action ◽  
Potential Applications ◽  
New Interpretation

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler–Lagrange equations is obtained from a higher-order Hamilton–Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre–Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton–Pontryagin principle and preserves the geometric properties of the continuous-time solution.

scholarly journals MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE

2016 ◽  
Vol 4 ◽  
Author(s):  
TÂNIA M. N. GONÇALVES ◽  
ELIZABETH L. MANSFIELD
Keyword(s):  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Linear Action ◽  
Independent Variables ◽  
Lie Group Action ◽  
Standard Linear

In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

scholarly journals A New Methodology for Solving Trajectory Planning and Dynamic Load-Carrying Capacity of a Robot Manipulator

2016 ◽  
Vol 2016 ◽  
pp. 1-28 ◽  
Author(s):  
Wanjin Guo ◽  
Ruifeng Li ◽  
Chuqing Cao ◽  
Xunwei Tong ◽  
Yunfeng Gao
Keyword(s):  
Trajectory Planning ◽  
Dynamic Load ◽  
Carrying Capacity ◽  
Robot Manipulator ◽  
Time Interval ◽  
Planning Problem ◽  
Load Carrying Capacity ◽  
Objective Functions ◽  
Decision Variables ◽  
Load Carrying

A new methodology using a direct method for obtaining the best found trajectory planning and maximum dynamic load-carrying capacity (DLCC) is presented for a 5-degree of freedom (DOF) hybrid robot manipulator. A nonlinear constrained multiobjective optimization problem is formulated with four objective functions, namely, travel time, total energy involved in the motion, joint jerks, and joint acceleration. The vector of decision variables is defined by the sequence of the time-interval lengths associated with each two consecutive via-points on the desired trajectory of the 5-DOF robot generalized coordinates. Then this vector of decision variables is computed in order to minimize the cost function (which is the weighted sum of these four objective functions) subject to constraints on joint positions, velocities, acceleration, jerks, forces/torques, and payload mass. Two separate approaches are proposed to deal with the trajectory planning problem and the maximum DLCC calculation for the 5-DOF robot manipulator using an evolutionary optimization technique. The adopted evolutionary algorithm is the elitist nondominated sorting genetic algorithm (NSGA-II). A numerical application is performed for obtaining best found solutions of trajectory planning and maximum DLCC calculation for the 5-DOF hybrid robot manipulator.

scholarly journals The stratified spaces of a symplectic lie group action

2006 ◽  
Vol 58 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Stratified Spaces ◽  
Lie Group Action

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Stratifications of equivariant varieties

1977 ◽  
Vol 16 (2) ◽  
pp. 279-295 ◽  
Author(s):  
M.J. Field
Keyword(s):  
Lie Group ◽  
General Position ◽  
Higher Order ◽  
Order Conditions ◽  
Compact Lie Group ◽  
Algebraic Varieties ◽  
Equivariant Maps

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

scholarly journals Higher-order orbifold Euler characteristics for compact Lie group actions

2015 ◽  
Vol 145 (6) ◽  
pp. 1215-1222 ◽  
Author(s):  
S. M. Gusein-Zade ◽  
I. Luengo ◽  
A. Melle-Hernández
Keyword(s):  
Euler Characteristic ◽  
Lie Group ◽  
Group Actions ◽  
Wreath Products ◽  
Higher Order ◽  
Compact Lie Group ◽  
Euler Characteristics ◽  
Finite Group Actions ◽  
Lie Group Actions ◽  
Generating Series

We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.

scholarly journals Trajectory Planning in UAV Emergency Networks with Potential Underlaying D2D Communication Based on K-means

2021 ◽  
Author(s):  
Shuo Zhang ◽  
Shuo Shi ◽  
Tianming Feng ◽  
Xuemai Gu
Keyword(s):  
Trajectory Planning ◽  
Communication Systems ◽  
High Speed ◽  
Large Scale ◽  
High Reliability ◽  
Base Station ◽  
Planning Problem ◽  
Communication Signal ◽  
User Coverage ◽  
Trajectory Length

Abstract Unmanned aerial vehicles (UAVs) have been widely used in communication systems due to excellent maneuverability and mobility. The ultra-high speed, ultra-low latency, and ultra-high reliability of 5th generation wireless systems (5G) have further promoted vigorous development of UAVs. Compared with traditional means of communication, UAV can provide services for ground terminal without time and space constraints, so it is often used as air base station (BS). Especially in emergency communications and rescue, it provides temporary communication signal coverage service for disaster areas. In the face of large-scale and scattered user coverage tasks, UAV's trajectory is an important factor affecting its energy consumption and communication performance. In this paper, we consider a UAV emergency communication network where UAV aims to achieve complete coverage of potential underlying D2D users (DUs). The trajectory planning problem is transformed into the deployment and connection problem of stop points (SPs). Aiming at trajectory length and sum throughput, two trajectory planning algorithms based on K-means are proposed. Due to the non-convexity of sum throughput optimization, we present a sub-optimal solution by using the successive convex approximation (SCA) method. In order to balance the relationship between trajectory length and sum throughput, we propose a joint evaluation index which is used as an objective function to further optimize trajectory. Simulation results show the validity of the proposed algorithms which have advantages over the well-known benchmark scheme in terms of trajectory length and sum throughput.

scholarly journals Certain Types of Complex Lie Group Action

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Ahmed Khalaf Radhi ◽  
Taghreed Hur Majeed
Keyword(s):  
Tensor Product ◽  
Group Action ◽  
Lie Group ◽  
The Other ◽  
Smooth Representation ◽  
Equivalent Relation ◽  
Complex Group ◽  
Lie Group Action ◽  
Group 10 ◽  
Definition Of

     The main aim in this paper is to look for a novel action with new properties on       from the  , the Literature are concerned with studying the action of  of two representations , one is usual and the other is the dual, while our  interest in this work  is focused on some actions on complex Lie group[10] . Let G be a matrix complex  group , and  is representation of   In this study we will present and analytic  the  concepts of action of complex  group on    We recall the definition of  tensor  product of two representations of  group and construct  the definition of action of   group on , then by using the equivalent  relation   between  and  , we get a new action : The two actions are forming smooth  representation of    This  we have new action which called     denoted by    which acting on      This  is smooth representation of   The theoretical Justifications are developed and prove supported by some concluding  remarks and illustrations.

scholarly journals Trajectory optimization of UAV based on Hp-adaptive Radau pseudospectral method

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yi Cui ◽  
Xintong Fang ◽  
Gaoqi Liu ◽  
Bin Li
Keyword(s):  
Trajectory Planning ◽  
Trajectory Optimization ◽  
Optimization Problem ◽  
Tracking Error ◽  
Pseudospectral Method ◽  
Planning Problem ◽  
A Algorithm ◽  
Planning Scheme ◽  
Path Search ◽  
Simulation Results

<p style='text-indent:20px;'>Unmanned Aerial Vehicles (UAVs) have been extensively studied to complete the missions in recent years. The UAV trajectory planning is an important area. Different from the commonly used methods based on path search, which are difficult to consider the UAV state and dynamics constraints, so that the planned trajectory cannot be tracked completely. The UAV trajectory planning problem is considered as an optimization problem for research, considering the dynamics constraints of the UAV and the terrain obstacle constraints during flight. An hp-adaptive Radau pseudospectral method based UAV trajectory planning scheme is proposed by taking the UAV dynamics into account. Numerical experiments are carried out to show the effectiveness and superior of the proposed method. Simulation results show that the proposed method outperform the well-known RRT* and A* algorithm in terms of tracking error.</p>

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