5,675 research outputs found
A Discrete Geometric Optimal Control Framework for Systems with Symmetries
This paper studies the optimal motion control of
mechanical systems through a discrete geometric approach. At
the core of our formulation is a discrete Lagrange-d’Alembert-
Pontryagin variational principle, from which are derived discrete
equations of motion that serve as constraints in our optimization
framework. We apply this discrete mechanical approach to
holonomic systems with symmetries and, as a result, geometric
structure and motion invariants are preserved. We illustrate our
method by computing optimal trajectories for a simple model of
an air vehicle flying through a digital terrain elevation map, and
point out some of the numerical benefits that ensue
Discrete Variational Optimal Control
This paper develops numerical methods for optimal control of mechanical
systems in the Lagrangian setting. It extends the theory of discrete mechanics
to enable the solutions of optimal control problems through the discretization
of variational principles. The key point is to solve the optimal control
problem as a variational integrator of a specially constructed
higher-dimensional system. The developed framework applies to systems on
tangent bundles, Lie groups, underactuated and nonholonomic systems with
symmetries, and can approximate either smooth or discontinuous control inputs.
The resulting methods inherit the preservation properties of variational
integrators and result in numerically robust and easily implementable
algorithms. Several theoretical and a practical examples, e.g. the control of
an underwater vehicle, will illustrate the application of the proposed
approach.Comment: 30 pages, 6 figure
Rigid Body Motion Estimation based on the Lagrange-d'Alembert Principle
Stable estimation of rigid body pose and velocities from noisy measurements,
without any knowledge of the dynamics model, is treated using the
Lagrange-d'Alembert principle from variational mechanics. With body-fixed
optical and inertial sensor measurements, a Lagrangian is obtained as the
difference between a kinetic energy-like term that is quadratic in velocity
estimation error and the sum of two artificial potential functions; one
obtained from a generalization of Wahba's function for attitude estimation and
another which is quadratic in the position estimate error. An additional
dissipation term that is linear in the velocity estimation error is introduced,
and the Lagrange-d'Alembert principle is applied to the Lagrangian with this
dissipation. This estimation scheme is discretized using discrete variational
mechanics. The presented pose estimator requires optical measurements of at
least three inertially fixed landmarks or beacons in order to estimate
instantaneous pose. The discrete estimation scheme can also estimate velocities
from such optical measurements. In the presence of bounded measurement noise in
the vector measurements, numerical simulations show that the estimated states
converge to a bounded neighborhood of the actual states.Comment: My earlier submitted manuscript (arXiv:1508.07671), is an extended
version of this work, containing detailed proofs and more elaborated
numerical simulations, currently under review in Automatica. This paper will
be cited in the extended journal version (arXiv:1508.07671) upon publicatio
Path planning for simple wheeled robots : sub-Riemannian and elastic curves on SE(2)
This paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryagin’s maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals and the expression for the position reduced to quadratures. Reachable sets are defined in the arbitrary speed case and a numerical plot of the time-limited reachable sets presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment
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