Chapter 3 Dynamics of Rigid Body Systems
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Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism
sensors Article Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism Linkang Wang 1 , Jingjing You 1,* , Xiaolong Yang 2, Huaxin Chen 1, Chenggang Li 3 and Hongtao Wu 3 1 College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China; [email protected] (L.W.); [email protected] (H.C.) 2 School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210037, China; [email protected] 3 School of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; [email protected] (C.L.); [email protected] (H.W.) * Correspondence: [email protected] Abstract: The solution of the dynamic equations of the six-axis accelerometer is a prerequisite for sensor calibration, structural optimization, and practical application. However, the forward dynamic equations (FDEs) and inverse dynamic equations (IDEs) of this type of system have not been completely solved due to the strongly nonlinear coupling relationship between the inputs and outputs. This article presents a comprehensive study of the FDEs and IDEs of the six-axis accelerometer based on a parallel mechanism. Firstly, two sets of dynamic equations of the sensor are constructed based on the Newton–Euler method in the configuration space. Secondly, based on the analytical solution of the sensor branch chain length, the coordination equation between the output signals of the branch chain is constructed. The FDEs of the sensor are established by combining the coordination equations and two sets of dynamic equations. Furthermore, by introducing generalized momentum and Hamiltonian function and using Legendre transformation, the vibration differential equations (VDEs) of the sensor are derived. -
Lie Group Formulation of Articulated Rigid Body Dynamics
Lie Group Formulation of Articulated Rigid Body Dynamics Junggon Kim 12/10/2012, Ver 2.01 Abstract It has been usual in most old-style text books for dynamics to treat the formulas describing linear(or translational) and angular(or rotational) motion of a rigid body separately. For example, the famous Newton's 2nd law, f = ma, for the translational motion of a rigid body has its partner, so-called the Euler's equation which describes the rotational motion of the body. Separating translation and rotation, however, causes a huge complexity in deriving the equations of motion of articulated rigid body systems such as robots. In Section1, an elegant single equation of motion of a rigid body moving in 3D space is derived using a Lie group formulation. In Section2, the recursive Newton-Euler algorithm (inverse dynamics), Articulated-Body algorithm (forward dynamics) and a generalized recursive algorithm (hybrid dynamics) for open chains or tree-structured articulated body systems are rewritten with the geometric formulation for rigid body. In Section3, dynamics of constrained systems such as a closed loop mechanism will be described. Finally, in Section4, analytic derivatives of the dynamics algorithms, which would be useful for optimization and sensitivity analysis, are presented.1 1 Dynamics of a Rigid Body This section describes the equations of motion of a single rigid body in a geometric manner. 1.1 Rigid Body Motion To describe the motion of a rigid body, we need to represent both the position and orien- tation of the body. Let fBg be a coordinate frame attached to the rigid body and fAg be an arbitrary coordinate frame, and all coordinate frames will be right-handed Cartesian from now on. -
Rigid Body Dynamics 2
Rigid Body Dynamics 2 CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017 Cross Product & Hat Operator Derivative of a Rotating Vector Let’s say that vector r is rotating around the origin, maintaining a fixed distance At any instant, it has an angular velocity of ω ω dr r ω r dt ω r Product Rule The product rule of differential calculus can be extended to vector and matrix products as well da b da db b a dt dt dt dab da db b a dt dt dt dA B dA dB B A dt dt dt Rigid Bodies We treat a rigid body as a system of particles, where the distance between any two particles is fixed We will assume that internal forces are generated to hold the relative positions fixed. These internal forces are all balanced out with Newton’s third law, so that they all cancel out and have no effect on the total momentum or angular momentum The rigid body can actually have an infinite number of particles, spread out over a finite volume Instead of mass being concentrated at discrete points, we will consider the density as being variable over the volume Rigid Body Mass With a system of particles, we defined the total mass as: n m m i i1 For a rigid body, we will define it as the integral of the density ρ over some volumetric domain Ω m d Angular Momentum The linear momentum of a particle is 퐩 = 푚퐯 We define the moment of momentum (or angular momentum) of a particle at some offset r as the vector 퐋 = 퐫 × 퐩 Like linear momentum, angular momentum is conserved in a mechanical system If the particle is constrained only -
Existence and Uniqueness of Rigid-Body Dynamics With
EXISTENCE AND UNIQUENESS OF RIGIDBODY DYNAMICS WITH COULOMB FRICTION Pierre E Dup ont Aerospace and Mechanical Engineering Boston University Boston MA USA ABSTRACT For motion planning and verication as well as for mo delbased control it is imp ortanttohavean accurate dynamic system mo del In many cases friction is a signicant comp onent of the mo del When considering the solution of the rigidb o dy forward dynamics problem in systems such as rob ots wecan often app eal to the fundamental theorem of dierential equations to prove the existence and uniqueness of the solution It is known however that when Coulomb friction is added to the dynamic equations the forward dynamic solution may not exist and if it exists it is not necessarily unique In these situations the inverse dynamic problem is illp osed as well Thus there is the need to understand the nature of the existence and uniqueness problems and to know under what conditions these problems arise In this pap er we show that even single degree of freedom systems can exhibit existence and uniqueness problems Next weintro duce compliance in the otherwise rigidb o dy mo del This has two eects First the forward and inverse problems b ecome wellp osed Thus we are guaranteed unique solutions Secondlyweshow that the extra dynamic solutions asso ciated with the rigid mo del are unstable INTRODUCTION To plan motions one needs to know what dynamic b ehavior will o ccur as the result of applying given forces or torques to the system In addition the simulation of motions is useful to verify motion -
Newton Euler Equations of Motion Examples
Newton Euler Equations Of Motion Examples Alto and onymous Antonino often interloping some obligations excursively or outstrikes sunward. Pasteboard and Sarmatia Kincaid never flits his redwood! Potatory and larboard Leighton never roller-skating otherwhile when Trip notarizes his counterproofs. Velocity thus resulting in the tumbling motion of rigid bodies. Equations of motion Euler-Lagrange Newton-Euler Equations of motion. Motion of examples of experiments that a random walker uses cookies. Forces by each other two examples of example are second kind, we will refer to specify any parameter in. 213 Translational and Rotational Equations of Motion. Robotics Lecture Dynamics. Independence from a thorough description and angular velocity as expected or tofollowa userdefined behaviour does it only be loaded geometry in an appropriate cuts in. An interface to derive a particular instance: divide and author provides a positive moment is to express to output side can be run at all previous step. The analysis of rotational motions which make necessary to decide whether rotations are. For xddot and whatnot in which a very much easier in which together or arena where to use them in two backwards operation complies with respect to rotations. Which influence of examples are true, is due to independent coordinates. On sameor adjacent joints at each moment equation is also be more specific white ellipses represent rotations are unconditionally stable, for motion break down direction. Unit quaternions or Euler parameters are known to be well suited for the. The angular momentum and time and runnable python code. The example will be run physics examples are models can be symbolic generator runs faster rotation kinetic energy. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
Investigation Into the Utility of the MSC ADAMS Dynamic Software for Simulating
Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms A thesis presented to The faculty of The Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Xiao Xue May 2013 © 2013 Xiao Xue. All Rights Reserved 2 This thesis titled Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms by XIAO XUE has been approved for the Department of Mechanical Engineering and the Russ College of Engineering and Technology by Robert L. Williams II Professor of Mechanical Engineering Dennis Irwin Dean, Russ College of Engineering and Technology 3 ABSTRACT XUE, XIAO, M.S., May 2013, Mechanical Engineering Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating Robots and Mechanisms (pp.143) Director of Thesis: Robert L. Williams II A Slider-Crank mechanism, a 4-bar mechanism, a 2R planar serial robot and a Stewart-Gough parallel manipulator are modeled and simulated in MSC ADAMS/View. Forward dynamics simulation is done on the Stewart-Gough parallel manipulator; Inverse dynamics simulation is done on Slider-Crank mechanism, 2R planar serial robot and 4- bar mechanism. In forward dynamics simulation, the forces are applied in prismatic joints and the Stewart-Gough parallel manipulator is actuated to perform 4 different expected motions. The motion of the platform is measured and compared with the results in MATLAB. An example of inverse dynamics simulation is also performed on it. In the inverse dynamics simulation, the Slider-Crank mechanism and four-bar mechanism both run with constant input angular velocity and the actuating forces are measured and compared with the results in MATLAB. -
Lecture L27 - 3D Rigid Body Dynamics: Kinetic Energy; Instability; Equations of Motion
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L27 - 3D Rigid Body Dynamics: Kinetic Energy; Instability; Equations of Motion 3D Rigid Body Dynamics In Lecture 25 and 26, we laid the foundation for our study of the three-dimensional dynamics of rigid bodies by: 1.) developing the framework for the description of changes in angular velocity due to a general motion of a three-dimensional rotating body; and 2.) developing the framework for the effects of the distribution of mass of a three-dimensional rotating body on its motion, defining the principal axes of a body, the inertia tensor, and how to change from one reference coordinate system to another. We now undertake the description of angular momentum, moments and motion of a general three-dimensional rotating body. We approach this very difficult general problem from two points of view. The first is to prescribe the motion in term of given rotations about fixed axes and solve for the force system required to sustain this motion. The second is to study the ”free” motions of a body in a simple force field such as gravitational force acting through the center of mass or ”free” motion such as occurs in a ”zero-g” environment. The typical problems in this second category involve gyroscopes and spinning tops. The second set of problems is by far the more difficult. Before we begin this general approach, we examine a case where kinetic energy can give us considerable insight into the behavior of a rotating body. This example has considerable practical importance and its neglect has been the cause of several system failures. -
An Inverse Dynamics Model for the Analysis, Reconstruction and Prediction of Bipedal Walking
Pergamon J. Biomechanics, Vol. 28, No. 11, pp. 1369%1376,1995 Copyri& 0 1995 Elsevicr Science Ltd hinted in Great Britain. All rifits reserved 0021-9290/95 $9.50 + .oO 0021-9290(94)00185-J AN INVERSE DYNAMICS MODEL FOR THE ANALYSIS, RECONSTRUCTION AND PREDICTION OF BIPEDAL WALKING Bart Koopman, Henk .I. Grootenboer and Henk J. de Jongh University of Twente, Faculty of Mechanical Engineering, Laboratory of Biomedical Engineering, P.O. Box 217,750O AE Enschede, The Netherlands Abstract-Walkingis a constrainedmovement which may best be observed during the double stance phase when both feet contact the floor. When analyzing a measured movement with an inverse dynamics model, a violation of these constraints will always occur due to measuring errors and deviations of the segments model from reality, leading to inconsistent results. Consistency is obtained by implementing the constraints into the model. This makes it possible to combine the inverse dynamics model with optimization techniques in order to predict walking patterns or to reconstruct non-measured rotations when only a part of the three-dimensional joint rotations is measured. In this paper the outlines of the extended inverse dynamics method are presented, the constraints which detine walking are defined and the optimization procedure is described. The model is applied to analyze a normal walking pattern of which only the hip, knee and ankle flexions/extensions are measured. This input movement is reconstructed to a kinematically and dynam- ically consistent three-dimensional movement, and the joint forces (including the ground reaction forces) and joint moments of force, needed to bring about thts movement are estimated. -
8.09(F14) Chapter 2: Rigid Body Dynamics
Chapter 2 Rigid Body Dynamics 2.1 Coordinates of a Rigid Body A set of N particles forms a rigid body if the distance between any 2 particles is fixed: rij ≡ jri − rjj = cij = constant: (2.1) Given these constraints, how many generalized coordinates are there? If we know 3 non-collinear points in the body, the remaing points are fully determined by triangulation. The first point has 3 coordinates for translation in 3 dimensions. The second point has 2 coordinates for spherical rotation about the first point, as r12 is fixed. The third point has one coordinate for circular rotation about the axis of r12, as r13 and r23 are fixed. Hence, there are 6 independent coordinates, as represented in Fig. 2.1. This result is independent of N, so this also applies to a continuous body (in the limit of N ! 1). Figure 2.1: 3 non-collinear points can be fully determined by using only 6 coordinates. Since the distances between any two other points are fixed in the rigid body, any other point of the body is fully determined by the distance to these 3 points. 29 CHAPTER 2. RIGID BODY DYNAMICS The translations of the body require three spatial coordinates. These translations can be taken from any fixed point in the body. Typically the fixed point is the center of mass (CM), defined as: 1 X R = m r ; (2.2) M i i i where mi is the mass of the i-th particle and ri the position of that particle with respect to a fixed origin and set of axes (which will notationally be unprimed) as in Fig. -
2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 1.0 Lecture L22 - 2D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L11-13 for particle dynamics, and extend it to 2D rigid body dynamics. Kinetic Energy for a 2D Rigid Body We start by recalling the kinetic energy expression for a system of particles derived in lecture L11, n 1 2 X 1 02 T = mvG + mir_i ; 2 2 i=1 0 where n is the total number of particles, mi denotes the mass of particle i, and ri is the position vector of Pn particle i with respect to the center of mass, G. Also, m = i=1 mi is the total mass of the system, and vG is the velocity of the center of mass. The above expression states that the kinetic energy of a system of particles equals the kinetic energy of a particle of mass m moving with the velocity of the center of mass, plus the kinetic energy due to the motion of the particles relative to the center of mass, G. For a 2D rigid body, the velocity of all particles relative to the center of mass is a pure rotation. Thus, we can write 0 0 r_ i = ! × ri: Therefore, we have n n n X 1 02 X 1 0 0 X 1 02 2 mir_i = mi(! × ri) · (! × ri) = miri ! ; 2 2 2 i=1 i=1 i=1 0 Pn 02 where we have used the fact that ! and ri are perpendicular. -
AN ALGORITHM for the INVERSE DYNAMICS of N-AXIS GENERAL MANIPULATORS USING KANE's EQUATIONS
Computers Math. Applic. Vol. 17, No. 12, pp. 1545-1561, 1989 0097-4943/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press plc AN ALGORITHM FOR THE INVERSE DYNAMICS OF n-AXIS GENERAL MANIPULATORS USING KANE'S EQUATIONS J. ANGELES and O. MA Department of Mechanical Engineering, Robotic Mechanical Systems Laboratory, McGill Research Centre for Intelligent Machines, McGill University, Montrral, Qurbec, Canada A. ROJAS National Autonomous University of Mexico, Villa Obregrn, Mexico (Received 4 April 1988) Abstract--Presented in this paper is an algorithm for the numerical solution of the inverse dynamics of robotic manipulators of the serial type, but otherwise arbitrary. The algorithm is applicable to manipulators containing n joints of the rotational or the prismatic type. For a given set of Hartenberg- Denavit and inertial parameters, as well as for a given trajectory in the space of joint coordinates, the algorithm produces the time histories of the n torques or forces required to drive the manipulator through the prescribed trajectory. The algorithm is based on Kane's dynamical equations of mechanical multibody systems. Moreover, the complexityof the algorithm presented here is lower than that of the most ett~cient inverse-dynamics algorithm reported in the literature. Finally, the applicability of the algorithm is illustrated with two fully solved examples. 1. INTRODUCTION The inverse dynamics of robotic manipulators has been approached with a variety of methods, the most widely applied being those based on the Newton-Euler (NE) and the Euler-Lagrange (EL) formulations [1-11]. Although essentially equivalent [12], the two said formulations regard the dynamics problems associated with rigid-link manipulators from very different viewpoints.