The Force Propagation Method

Recursive dynamic equations fw a single c n chain are derived by Featherstone in [9] and later presented again by Brandi, et al. in [3]. The formulation of these equations is based on the concept of treating a chain of rigid bodies connected by joints as a single articulated body . The recursive equations allow the dynamics of the entire chain to be resolved to a single link called the handle of the articulated body. All interactions with the articulated body are assumed to 

Feathmtone defines the articulated-body inertia of a link as the 6 x 6 matrix which relates a spatial force applied to the link and the spatial acceleration of the link, taking into account the dynamics of the rest of the articulated body. This relationship is linear, and for link t, it may be written as follows [9]  

The development of the recursive articulated-body dynamic equations begins with the simple free-body dynamic equations of the two individual links. In the notation of this book, we may write the free-body equation for link 1 as follows  

The component of a spatial vector which lies on or along a specific axis of a joint may be determined by performing a simple dot product between that spatial vector and the vector which represents the joint axis. This may be refen to as projecting the spatial vector onto the joint axis . Thus, we may project the terms of Equation 4.32 onto the motion space of joint 2 via the following spatial dot product  

We may combine Equations 4.33 through 4.35 to solve for the vector of relative joint accelerations, 2. in terms of the spatial acceleration of link 1, ai, as foUows  

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The scalar ( rations (multiplications, additions) required to compute A and A using the Force Propagation Method are shown in Table 4.6. These scalar operations are given for ap AT degree-of-freedom manipulator with simple Involute and/or prismatic joints. Note that 1)1, K, and L)), may all be computed off-line, and that the initial condition, (Ag) = 0, allows some simplification in the first iteration of the Forward Recursion. The computational complexity of the complete algorithm is 0(N), an improvement over the previous two algorithms. The efficient coordinate tiansfcMmations described in Chapter 3 are utilized in every case. 

Table 4.S Algorithm for the Force Propagation Method (Method III)...

The two tables differ only in the algorithm used to compute the inverse operational space inertia matrix, A and the coefficient fl. In Chapter 4, the efficient computation of these two quantities was discussed in some detail. It was detomined that the Unit Force Method (Method II) is the most efficient algorithm for these two matrices together for N < 21. The Force Propagation Method (Method ni) is the best solution for and fl for AT > 21. The scalar opmtions required for Method II are used in Table 5.1, while those required for Method III are used in Table 5.2. 

Brandi, Johanni, and Otter [3] computes the articulated-body inertia of each link in the chain, starting at the tip and moving back to the base. This same recursion is the first recursion in the Force Propagation Method for computing A. That is, there is an overlap of computations between the solution for the q)en-chain acceleration terms, tjopen and Xopen. and the calculation of the inverse ( rational space inertia matrix, A for this case. This fact was taken into account when the operations were tabulated. The ( rations listed for SI and A in Table 5.2 include only the second recursion for A and the additional opoations needed for SI. The recursion which computes the articulated-body inertias is included in the computatimis for open and x<,pe . 

The efficient computation of fl and A was discussed in detail in Chapt 4. The most efficient method known for the computation of both fl and A for iV < 21 is the Unit Force Method (Method II), which is O(AT ) for an A/ degree-of-freedom manipulator with revolute and/or prismatic joints. For N > 21, the 0(N) Force Propagation Method (Method III) is the most efficient. The use of these two methods will be discussed further in Section 5.1. 

Note that the number of operations listed for fl and A in Table 5.2 is less than the total given for these two quantities in the 0 N) Force Propagation Method in Chapter 4. This reduction was achieved through a little insight First we note that the first recursion in the open-chain Direct Dynamics algorithm of... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

For both the tongue and Elmendorf test methods, it is important to observe the behavior of the specimen as the tear is propagated. In cases where the yams in the test direction are much stronger than the perpendicular yams, it is sometimes difficult or impossible to propagate the tear in the desired direction. In this case, a crosswise tear results. Tear resistance is primarily a function of fabric constmction. Loose, open weaves such as cheesecloth tend to resist tear, whereas tight weaves tend to tear easily. In the open weave, the concentrated force field at the point of tear is dissipated by the compliance of the fabric stmcture to accommodate the stress field, thereby distributing the force over a greater number of yams. 

Ultrasonic absorption is a so-called stationary method in which a periodic forcing function is used. The forcing function in this case is a sound wave of known frequency. Such a wave propagating through a medium creates a periodically varying pressure difference. (It may also produce a periodic temperature difference.) Now suppose that the system contains a chemical equilibrium that can respond to pressure differences [as a consequence of Eq. (4-28)]. If the sound wave frequency is much lower than I/t, the characteristic frequency of the chemical relaxation (t is the... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

Before considering particular test methods, it is useful to survey the principles and terms used in dynamic testing. There are basically two classes of dynamic motion, free vibration in which the test piece is set into oscillation and the amplitude allowed to decay due to damping in the system, and forced vibration in which the oscillation is maintained by external means. These are illustrated in Figure 9.1 together with a subdivision of forced vibration in which the test piece is subjected to a series of half-cycles. The two classes could be sub-divided in a number of ways, for example forced vibration machines may operate at resonance or away from resonance. Wave propagation (e.g. ultrasonics) is a form of forced vibration method and rebound resilience is a simple unforced method consisting of one half-cycle. The most common type of free vibration apparatus is the torsion pendulum. 

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