Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics

2007 ◽  
Author(s):  
Sigrid Leyendecker ◽  
Sina Ober-Blo¨baum ◽  
Jerrold E. Marsden ◽  
Michael Ortiz
Keyword(s):  
Optimal Control ◽  
Rigid Body ◽  
Null Space ◽  
Numerical Dissipation ◽  
Discrete Version ◽  
Dynamical Equations ◽  
Time Stepping ◽  
Minimal Dimension ◽  
Energy And Momentum ◽  
External Forces

This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.

scholarly journals Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

scholarly journals Propagation of Elastic Waves in Prestressed Media

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Inder Singh ◽  
Dinesh Kumar Madan ◽  
Manish Gupta
Keyword(s):  
Elastic Waves ◽  
Plane Waves ◽  
Dynamical Equations ◽  
P Waves ◽  
General Solutions ◽  
External Forces ◽  
Propagation Of Elastic Waves

3D solutions of the dynamical equations in the presence of external forces are derived for a homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves. Expressions for slowness surfaces and apparent velocities for these waves are derived analytically as well as numerically and represented graphically.

scholarly journals Direct Optimal Control Approach to Laser-Driven Quantum Particle Dynamics

2021 ◽  
Vol 9 ◽  
Author(s):  
A. R. Ramos Ramos ◽  
O. Kühn
Keyword(s):  
Optimal Control ◽  
Gaussian Approximation ◽  
Gaussian Function ◽  
Dynamical Equations ◽  
Control Approach ◽  
Wide Range ◽  
Bistable Potential ◽  
Wavepacket Dynamics ◽  
Direct Optimal Control ◽  
Backward Propagation

Optimal control theory is usually formulated as an indirect method requiring the solution of a two-point boundary value problem. Practically, the solution is obtained by iterative forward and backward propagation of quantum wavepackets. Here, we propose direct optimal control as a robust and flexible alternative. It is based on a discretization of the dynamical equations resulting in a nonlinear optimization problem. The method is illustrated for the case of laser-driven wavepacket dynamics in a bistable potential. The wavepacket is parameterized in terms of a single Gaussian function and field optimization is performed for a wide range of particle masses and lengths of the control interval. Using the optimized field in a full quantum propagation still yields reasonable control yields for most of the considered cases. Analysis of the deviations leads to conditions which have to be fulfilled to make the semiclassical single Gaussian approximation meaningful for field optimization.

Gradient Optimization of Inverse Dynamics for Robotic Manipulator Motion Planning Using Combined Optimal Control

2017 ◽  
Author(s):  
Shangdong Gong ◽  
Redwan Alqasemi ◽  
Rajiv Dubey
Keyword(s):  
Optimal Control ◽  
Motion Planning ◽  
Inverse Dynamics ◽  
Null Space ◽  
Optimal Solution ◽  
Human Motion ◽  
Optimization Techniques ◽  
Redundant Manipulators ◽  
Robot Arm ◽  
Gradient Optimization

Motion planning of redundant manipulators is an active and widely studied area of research. The inverse kinematics problem can be solved using various optimization methods within the null space to avoid joint limits, obstacle constraints, as well as minimize the velocity or maximize the manipulability measure. However, the relation between the torques of the joints and their respective positions can complicate inverse dynamics of redundant systems. It also makes it challenging to optimize cost functions, such as total torque or kinematic energy. In addition, the functional gradient optimization techniques do not achieve an optimal solution for the goal configuration. We present a study on motion planning using optimal control as a pre-process to find optimal pose at the goal position based on the external forces and gravity compensation, and generate a trajectory with optimized torques using the gradient information of the torque function. As a result, we reach an optimal trajectory that can minimize the torque and takes dynamics into consideration. We demonstrate the motion planning for a planar 3-DOF redundant robotic arm and show the results of the optimized trajectory motion. In the simulation, the torque generated by an external force on the end-effector as well as by the motion of every link is made into an integral over the squared torque norm. This technique is expected to take the torque of every joint into consideration and generate better motion that maintains the torques or kinematic energy of the arm in the safe zone. In future work, the trajectories of the redundant manipulators will be optimized to generate more natural motion as in humanoid arm motion. Similar to the human motion strategy, the robot arm is expected to be able to lift weights held by hands, the configuration of the arm is changed along from the initial configuration to a goal configuration. Furthermore, along with weighted least norm (WLN) solutions, the optimization framework will be more adaptive to the dynamic environment. In this paper, we present the development of our methodology, a simulated test and discussion of the results.

Sign in / Sign up

Export Citation Format

Share Document