Trajectory representation and landmark projection for continuous-time structure from motion

This paper revisits the problem of continuous-time structure from motion, and introduces a number of extensions that improve convergence and efficiency. The formulation with a [Formula: see text]-continuous spline for the trajectory naturally incorporates inertial measurements, as derivatives of the sought trajectory. We analyze the behavior of split spline interpolation on [Formula: see text] and on [Formula: see text], and a joint spline on [Formula: see text], and show that the latter implicitly couples the direction of translation and rotation. Such an assumption can make good sense for a camera mounted on a robot arm, but not for hand-held or body-mounted cameras. Our experiments in the Spline Fusion framework show that a split spline on [Formula: see text] is preferable over an [Formula: see text] spline in all tested cases. Finally, we investigate the problem of landmark reprojection on rolling shutter cameras, and show that the tested reprojection methods give similar quality, whereas their computational load varies by a factor of two.

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Open Higher Order Continuous-Time Dynamic Model with Mixed Stock and Flow Data and Derivatives of Exogenous Variables

Econometric Theory ◽

10.1017/s0266466600004540 ◽

1991 ◽

Vol 7 (3) ◽

pp. 404-408 ◽

Cited By ~ 12

Author(s):

K. Ben Nowman

Keyword(s):

Continuous Time ◽

Higher Order ◽

Flow Data ◽

Continuous Time Model ◽

Exogenous Variables ◽

Time Model ◽

Time Dynamic ◽

Mixed Stock ◽

Gaussian Estimation ◽

Derivatives Of

This paper is concerned with deriving formulae for higher order derivatives of exogenous variables for use in estimating the parameters of an open secondorder continuous time model with mixed stock and flow data and first and second order derivatives of exogenous variables which are not observable. This should provide the basis for the future estimation of continuous time models in a range of applied areas using the new Gaussian estimation computer program developed by Nowman [4].

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Spatiotemporal Neural Integration for Bistable Perception in a Response-Time Structure-From-Motion Task

IEEE Transactions on Biomedical Engineering ◽

10.1109/tbme.2009.2027332 ◽

2009 ◽

Vol 56 (12) ◽

pp. 2937-2948 ◽

Cited By ~ 1

Author(s):

Zhisong Wang ◽

Nikos K. Logothetis ◽

Hualou Liang

Keyword(s):

Response Time ◽

Structure From Motion ◽

Time Structure ◽

Bistable Perception ◽

Neural Integration

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Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.2478/v10175-012-0037-2 ◽

2012 ◽

Vol 60 (2) ◽

pp. 279-284 ◽

Cited By ~ 8

Author(s):

M. Busłowicz

Keyword(s):

Linear Systems ◽

Continuous Time ◽

State Equation ◽

The State ◽

Natural Order ◽

State Variables ◽

The Stability ◽

Derivatives Of ◽

Fractional Orders ◽

Time Linear

Abstract. The stability problem of continuous-time linear systems described by the state equation consisting of n subsystems with different fractional orders of derivatives of the state variables has been considered. The methods for asymptotic stability checking have been given. The method proposed in the general case is based on the Argument Principle and it is similar to the modified Mikhailov stability criterion known from the stability theory of natural order systems. The considerations are illustrated by numerical examples.

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Non-Linear Macroeconomic Models of Growth with Memory

Mathematics ◽

10.3390/math8112078 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2078 ◽

Cited By ~ 1

Author(s):

Vasily E. Tarasov

Keyword(s):

Economic Growth ◽

Differential Equations ◽

Exact Solutions ◽

Fractional Differential Equations ◽

Continuous Time ◽

Non Linear ◽

Standard Models ◽

Fractional Differential ◽

Derivatives Of ◽

With Memory

In this article, two well-known standard models with continuous time, which are proposed by two Nobel laureates in economics, Robert M. Solow and Robert E. Lucas, are generalized. The continuous time standard models of economic growth do not account for memory effects. Mathematically, this is due to the fact that these models describe equations with derivatives of integer orders. These derivatives are determined by the properties of the function in an infinitely small neighborhood of the considered time. In this article, we proposed two non-linear models of economic growth with memory, for which equations are derived and solutions of these equations are obtained. In the differential equations of these models, instead of the derivative of integer order, fractional derivatives of non-integer order are used, which allow describing long memory with power-law fading. Exact solutions for these non-linear fractional differential equations are obtained. The purpose of this article is to study the influence of memory effects on the rate of economic growth using the proposed simple models with memory as examples. As the methods of this study, exact solutions of fractional differential equations of the proposed models are used. We prove that the effects of memory can significantly (several times) change the growth rate, when other parameters of the model are unchanged.

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A robust real-time structure from motion for situational awareness and RSTA

10.1117/12.778074 ◽

2008 ◽

Cited By ~ 2

Author(s):

M. Shim ◽

S. Yilma ◽

K. Bonner

Keyword(s):

Real Time ◽

Structure From Motion ◽

Situational Awareness ◽

Time Structure

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Development of a Dynamics-Based Continuous Time Motion Planning Algorithm for a 6 Degrees of Freedom Robot arm under constraints

The Proceedings of JSME annual Conference on Robotics and Mechatronics (Robomec) ◽

10.1299/jsmermd.2016.2p2-04a4 ◽

2016 ◽

Vol 2016 (0) ◽

pp. 2P2-04a4

Author(s):

Keisuke UTO ◽

Gaku TAKANO ◽

Makoto OBAYASHI

Keyword(s):

Motion Planning ◽

Continuous Time ◽

Degrees Of Freedom ◽

Robot Arm ◽

Time Motion ◽

Planning Algorithm ◽

6 Degrees Of Freedom

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Computer Aided Synthesis of Piecewise Rational Motions for Spherical 2R and 3R Robot Arms

Journal of Mechanical Design ◽

10.1115/1.2976444 ◽

2008 ◽

Vol 130 (11) ◽

Cited By ~ 6

Author(s):

Anurag Purwar ◽

Zhe Jin ◽

Q. J. Ge

Keyword(s):

Motion Planning ◽

Spline Interpolation ◽

Robot Arm ◽

Spherical Robot ◽

Kinematic Constraints ◽

Robot Arms ◽

Revolute Joints ◽

Computer Aided ◽

Planar Robot ◽

Circular Interpolation

This paper deals with the problem of synthesizing smooth piecewise rational spherical motions of an object that satisfies the kinematic constraints imposed by a spherical robot arm with revolute joints. This paper brings together the kinematics of spherical robot arms and recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. The kinematic constraints under consideration are workspace related constraints that limit the orientation of the end link of robot arms. This paper extends our previous work on synthesis of rational motions under the kinematic constraints of planar robot arms. Using quaternion kinematics of spherical arms, it is shown that the problem of synthesizing the Cartesian rational motion of a 2R arm can be reduced to that of circular interpolation in two separate planes. Furthermore, the problem of synthesizing the Cartesian rational motion of a spherical 3R arm can be reduced to that of constrained spline interpolation in two separate planes. We present algorithms for the generation of C1 and C2 continuous rational motion of spherical 2R and 3R robot arms.

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Solvability of cardinal spline interpolation problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001578x ◽

1983 ◽

Vol 95 (1-2) ◽

pp. 39-57

Author(s):

T. N. T. Goodman

Keyword(s):

Incidence Matrix ◽

Bounded Solution ◽

Spline Interpolation ◽

Piecewise Polynomial ◽

Piecewise Polynomials ◽

Cardinal Spline ◽

Interpolation Problems ◽

Bounded Data ◽

Derivatives Of ◽

Cardinal Spline Interpolation

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

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Computer Aided Synthesis of Piecewise Rational Motions for Planar 2R and 3R Robot Arms

Journal of Mechanical Design ◽

10.1115/1.2756082 ◽

2006 ◽

Vol 129 (10) ◽

pp. 1031-1036 ◽

Cited By ~ 4

Author(s):

Zhe Jin ◽

Q. J. Ge

Keyword(s):

Motion Planning ◽

Spline Interpolation ◽

Circular Ring ◽

Robot Arm ◽

Kinematic Constraints ◽

Robot Arms ◽

Revolute Joints ◽

Computer Aided ◽

Planar Robot ◽

Circular Interpolation

This paper deals with the problem of synthesizing piecewise rational motions of an object that satisfies kinematic constraints imposed by a planar robot arm with revolute joints. This paper brings together the kinematics of planar robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Through the use of planar quaternions, it is shown that for the case of a planar 2R arm, the problem of rational motion synthesis can be reduced to that of circular interpolations in two separate planes and that for the case of a planar 3R arm, the problem can be reduced to a combination of circular interpolation in one plane and a constrained spline interpolation in a circular ring on another plane. Due to the limitation of circular interpolation, only C1 continuous rational motions are generated that satisfy the kinematic constraints exactly. For applications that require C2 continuous motions, this paper presents a method for generating C2 continuous motions that approximate the kinematic constraints for planar 2R and 3R robot arms.

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