A Jacobian-based algorithm for planning the motion of an underactuated rigid body undergoing forward and reverse rotations

Robotica ◽  
2009 ◽  
Vol 28 (5) ◽  
pp. 747-757 ◽  
Author(s):  
Sung K. Koh
Keyword(s):  
Rigid Body ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Practical Importance ◽  
The Body ◽  
Coordinate Frame ◽  
Torque Generation ◽  
Generation Mechanisms ◽  
Jacobian Algorithm ◽  
Null Matrix

SUMMARYA Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.

Jacobian-Based Algorithm for Motion Planning for Underactuated Systems Undergoing Forward and Reverse Rotations

2008 ◽  
Author(s):  
Sung Koh ◽  
Wankyun Chung
Keyword(s):  
Rigid Body ◽  
Motion Planning ◽  
Degrees Of Freedom ◽  
Practical Importance ◽  
Underwater Vehicles ◽  
Coordinate Frame ◽  
Underactuated System ◽  
Torque Generation ◽  
Generation Mechanisms ◽  
Jacobian Algorithm

A Jacobian-based algorithm for motion planning for an underactuated system that is a rigid-body operated by two input-rotations is discussed in this paper. The rigid body undergoes a four-rotation fully-reversed (FR) sequence of rotations which consists of a series of initial two rotations about the axes of a coordinate frame attached to the rigid body and subsequent two rotations that undo the proceeding rotations. Due to the insufficient degrees of freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve some orientations. In order to best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input-rotations, the switching-Jacobian algorithm proposed in this paper has significant practical importance for motion planning for aerospace and underwater vehicles maneuvered using only two input-rotations due to the failure of one of torque-generation mechanisms.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Performance-Oriented Design of Inverse Kinematics Algorithms: Extended Jacobian Approximation of the Jacobian Pseudo-Inverse

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Joanna Karpińska ◽  
Krzysztof Tchoń
Keyword(s):  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Ritz Method ◽  
Approximation Error ◽  
Robotic Manipulators ◽  
Error Functional ◽  
Free Representation ◽  
Pseudo Inverse ◽  
Jacobian Inverse ◽  
Jacobian Algorithm

For redundant robotic manipulators, we study the design problem of Jacobian inverse kinematics algorithms of desired performance. A specific instance of the problem is addressed, namely the optimal approximation of the Jacobian pseudo-inverse algorithm by the extended Jacobian algorithm. The approximation error functional is derived for the coordinate-free representation of the manipulator’s kinematics. A variational formulation of the problem is employed, and the approximation error is minimized by means of the Ritz method. The optimal extended Jacobian algorithm is designed for the 7 degrees of freedom (dof) POLYCRANK manipulator. It is concluded that the coordinate-free kinematics representation results in more accurate approximation than the coordinate expression of the kinematics.

scholarly journals The Motion of a Rigid Body in the Presence of a Gyrostatic : الحركات المغزلية السريعة لجسم متماسك في مجال قوة نيوتوني في حالة وجود العزم

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Ghadir Ahmed Sahli
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
First Integral ◽  
Geometric Interpretation ◽  
The Body ◽  
Parameter Method ◽  
Principal Axes ◽  
Linear Autonomous System ◽  
Newtonian Force Field

In this study، the rotational motion of a rigid body about a fixed point in the Newtonian force field with a gyrostatic momentum  about the z-axis is considered. The equations of motion and their first integrals are obtained and reduced to a quasi-linear autonomous system with two degrees of freedom with one first integral. Poincare's small parameter method is applied to investigate the analytical peri­odic solutions of the equations of motion of the body with one point fixed، rapidly spinning about one of the principal axes of the ellipsoid of inertia. A geometric interpretation of motion is given by using Euler's angles to describe the orientation of the body at any instant of time.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

1994 ◽  
Vol 446 (1926) ◽  
pp. 169-193 ◽  
Keyword(s):  
Flow Field ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Surface Deformation ◽  
Nonlinear Differential Equations ◽  
Deformable Body ◽  
Equations Of Motion ◽  
First Integral ◽  
Uniform Flow ◽  
The Body

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.

Series Solution for Finite Displacement of Single-Loop Spatial Linkages

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Coordinate Frame ◽  
Recursive Procedure ◽  
Single Loop ◽  
Analytical Technique ◽  
Finite Displacement ◽  
Instantaneous Screw

The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

Study on Kinematics and Simulation of Robotic Arm

2011 ◽  
Vol 335-336 ◽  
pp. 619-624 ◽  
Author(s):  
Xiao Long Lei ◽  
Rong Chao Ma ◽  
Wen Gou
Keyword(s):  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Robotic Arm ◽  
Coordinate Frame ◽  
Matlab Software ◽  
Improve Efficiency ◽  
Frame Model ◽  
Inverse Transform ◽  
Motion Parameters

In order to improve efficiency and automation of spraying, a automatic spray lance, which was regard as an robotic arm, was studied. It could reach the desired position automatically. A study was carried out using Denavit and Hartenberg (D-H) approach to automate the four Degrees of Freedom (DOF) robotic arm. The kinematics of robotic arm was constructed us D-H coordinate frame model. And the inverse kinematics has been solved with inverse transform technique. Some motion parameters could be acquired. Simulation of the robotic arm was studied using the Robotics Toolbox which is MATLAB software. Simulations’ content included homogeneous, construction of robot object, kinematics and trajectory. The results of simulation indicated that the movement of robotic arm could satisfy the demands of four DOF. The robotic was founded able to move to the desired position by simulating. The links’ motion parameters were reasonable and it provided a new spray lance for automatic sprayer.

An Investigation on the Issues in Modelling of Free Flight in Animal Locomotion

2011 ◽  
Author(s):  
Masateru Maeda ◽  
Toshiyuki Nakata ◽  
Hao Liu
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Aerodynamic Force ◽  
Three Dimensional ◽  
Micro Air Vehicles ◽  
Rigid Body Dynamics ◽  
Free Flight ◽  
Comprehensive Investigation ◽  
Body Dynamics ◽  
Generation Mechanisms

Aiming at establishing an effective computational framework to accurately predict free-flying dynamics and aerodynamics we here present a comprehensive investigation on some issues associated with the modelling of free flight. Free flight modelling/simulation is essential for some types of flights e.g. falling leaves or auto-rotating seeds for plants; unsteady manoeuvres such as take-off, turning, or landing for animals. In addition to acquiring the deeper understanding of the flight biomechanics of those natural organisms, revealing the sophisticated aerodynamic force generation mechanisms employed by them may be useful in designing man-made flying-machines such as rotary or flapping micro air vehicles (MAVs). The simulations have been conducted using the coupling of computational fluid dynamics (CFD) and rigid body dynamics, thus achieving the free flight. The flow field is computed with a three-dimensional unsteady incompressible Navier-Stokes solver using pseudo-compressibility and overset gird technique. The aerodynamic forces acting on the flyer are calculated by integrating the forces on the surfaces. Similarly, the aerodynamic torque around the flyer’s centre of mass is obtained. The forces and moments are then introduced into a six degrees-of-freedom rigid body dynamics solver which utilises unit quaternions for attitude description in order to avoid singular attitude. Results are presented of a single body model and some insect-like multi-body models with flapping wings, which point to the importance of free-flight modelling in systematic analyses of flying aerodynamics and manoeuvrability. Furthermore, a comprehensive investigation indicates that the framework is capable to predict the aerodynamic performance of free-flying or even free-swimming animals in an intermediate range of Reynolds numbers (< 105).

Sign in / Sign up

Export Citation Format

Share Document