Cubic Splines on Curved Spaces
01 Jan 1989-Ima Journal of Mathematical Control and Information (Oxford University Press)-Vol. 6, Iss: 4, pp 465-473
About: This article is published in Ima Journal of Mathematical Control and Information.The article was published on 1989-01-01. It has received 271 citations till now.
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01 Jan 2005
TL;DR: In this article, a comprehensive set of modeling, analysis and design techniques for a class of simple mechanical control systems is presented, that is, systems whose Lagrangian is kinetic energy minus potential energy.
Abstract: This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.
848 citations
TL;DR: A method for computing weighted averages on spheres based on least squares minimization that respects spherical distance is introduced, and existence and uniqueness properties of the weighted averages are proved, and fast iterative algorithms with linear and quadratic convergence rates are given.
Abstract: This article introduces a method for computing weighted averages on spheres based on least squares minimization that respects spherical distance. We prove existence and uniqueness properties of the weighted averages, and give fast iterative algorithms with linear and quadratic convergence rates. Our methods are appropriate to problems involving averages of spherical data in meteorological, geophysical, and astronomical applications. One simple application is a method for smooth averaging of quaternions, which generalizes Shoemake's spherical linear interpolation.The weighted averages methods allow a novel method of defining Bezier and spline curves on spheres, which provides direct generalization of Bezier and B-spline curves to spherical spline curves. We present a fast algorithm for spline interpolation on spheres. Our spherical splines allow the use of arbitrary knot positions; potential applications of spherical splines include smooth quaternion curves for applications in graphics, animation, robotics, and motion planning.
312 citations
Additional excerpts
...Gabriel-Kajiya [12], Jupp-Kent [16], Noakes-Heinzinger-Paden [ 23 ],...
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TL;DR: A general framework for the control of Lagrangian systems with as many inputs as degrees of freedom is presented, and particular insight is provided into both aerospace and underwater applications where the configuration manifold is a Lie group.
284 citations
Cites background from "Cubic Splines on Curved Spaces"
...Other recent papers on modeling (Bloch & Crouch 1995), controllability (Lewis & Murray 1997), interpolation ( Noakes, Heinzinger & Paden 1989 ) and dynamic feedback linearization (Rathinam & Murray 1998) share the same theoretical tools....
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TL;DR: In this paper, the authors consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifold, and they consider the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory.
Abstract: We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.
225 citations
Cites background or methods from "Cubic Splines on Curved Spaces"
...A necessary condition for a curve x to be a critical point for problem (P1) was given in [14, Theorem 1]. The same condition for problem (7~1) without the interpolating conditions (2) was first derived independently in Noakes et al. [ 26 ]....
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...respectively, where w = (wl, w2, wa) T and v = (vl, v2, v3) T. Equation (37), the equation of a cubic spline in SO(3), appeared in Noakes et al. [ 26 ] and Jackson [21] for the first time....
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...d~(O) d~(T) -_ vN." (5) �9 (0) =x0, d-'~ =v~ ~(T) = x~v, T Theorem 2.2 (Noakes et al. [ 26 ])....
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...d3v d2v dt a +vx~-~=0, where (Noakes et al. [ 26 ], and Jackson [21])....
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...In Sec. 1 we derive the second variation formulas for a variational problem whose extremals turn out to be generalizations of cubic splines to Pdemannian manifolds and which appeared before in Noakes, Heinzinger, and Paden [ 26 ] and Crouch and Silva Leite [14]....
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TL;DR: A new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements is proposed, to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates.
Abstract: We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.
196 citations