Direct Dynamics joint

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

Physical motion is common to most situations in which the human functions and is therefore fundamental to the analysis of performance. Parameters such as segment position, orientation, velocity, and acceleration are derived using kinematic or dynamic analysis or both. This approach is equally appropriate for operations on a single joint system or linked multibody systems, such as is typically required for human analysis. Depending on the desired output, foreword (direct) or inverse analysis may be employed to obtain the parameters of interest. For example, inverse dynamic analysis can provide joint torque, given motion and force data while foreword (direct) dynamic analysis uses joint torque to derive motion. Especially for three-dimensional analyses of multijoint systems, the methods are quite complex and are presently a focal point for computer implementation [Allard et al., 1994]. 

In this chapter we have examined forward and inverse dynamics approaches to the study of human motion. We have outlined the steps involved in using the inverse approach to studying movement with a particular focus on human gait. This is perhaps the most commonly used method for examining joint kinetics. The forward or direct dynamics approach requires that one start with knowledge of the neural command signal, the muscle forces, or, perhaps, the joint torques. These are then used to compute kinematics. 

In the area of simulation, the fundamental problem to be solved is called Forward ot Direct Dynamics. Solution of this problem requires the detmnina-tion of the joint motion which results from a given set of applied jmnt torques and/or forces. Again, the present state joint positions and rates arc known. A Direct Dynamics algorithm calculates the joint accelerations which result from the application of the given actuatm forces. Once the acceloations are known, integration is used to determine the next state joint rates and positions for the simulation. 

Although the details may be quite different, every research effort in the area of dynamic simulation faces a common task — the efficient and accurate solution of the Direct Dynamics problem. In the development of algorithms for Direct Dynamics, two basic approaches have emerged for both open- and closed-chain systems. The first utilizes the inversion of the x manipulator joint space inertia matrix to solve for the joint accelerations. More accurately, the accelerations are found via the solution of a system of linear algebraic equations, but the... 

In an attempt to circumvent the computational complexity of matrix inversion, some researchers are pursuing solutions for the joint accelerations which have a linear recursive form. The inversion of the in tia matrix is explicitly avoided. The resulting linear recursive algorithms have a reduced computational complexity which is 0(N). This is the second basic solution approach to the Direct Dynamics problem, and it has been rqjplied to serial open chains [3,7], single closed chains [22], and some more genoal multibody systems [4, 37]. It is believed that the structure of linear recursive algorithms may also facilitate their implementation on parallel computer systems. 

When considering the Direct Dynamics problem, t,, the vector of driving forces, is known for every joint. The appended vecbv of applied forces for the entire system is called r. The unknowns to be found are the joint accelerations for the system, 4, and all the constraint fcxces, r (if desired). 

In general, the inertia matrix of a manipulator defines the relationship between certain forces exerted on the system and some corresponding acceleration vector. This relationship is of great importance both in real-time control and in the simulation of multibody systems. In the control realm, for example, the inertia matrix has been used to decouple robot dynamics so that control schemes may be more effectively tq>plied [19]. This may be accomplished either in joint space O operational space, since the inertia matrix may be defined in eith domain. The inertia matrix has also been used in the analysis of collision effects [43]. In addition to its use in control applications, the inertia matrix is an explicit and integral part of certain Direct Dynamics algorithms which are used to solve the simulation problem for manipulators and other multibody systems [2, 8, 31, 33,42]. 

Given the p sent state of the manipulator, q and q, the input joint actuator traques and forces, r, and the motion of the base, the open-chain term, ijopen. is completely defined. An appropiate Direct Dynamics algorithm for < )en-chain manipulator may be used to detomine its value. With the same givoi information, the coefficient of f in the constrained term, G, is also known. The efficient computation of G was discussed in Chapter 4. 

As was true for the joint accelerations, if the present state, driving actuator tffl ques and/or forces, and motion of the base are known, the open-chain term, Xopenf is known. Its value may also be determined using an tq>pr(piate Direct Dynamics algorithm for ( n-chain manipulators. Because the joint positions are assumed known, the inverse operational space inertia matrix, A is defined. The efficient computation of A and its inverse was the primary topic of Ch ter 4. 

In the first step, the Direct Dynamics problem is solved for an q)en-chain configuration of the system. That is, the joint acceleration and end effectra acceleration vectors are computed assuming that the contact force vector is zero. The coefficients of f in Equations 5.6 and 5.14, fi and A" , are also computed. These matrices are both functions of the joint positions only. 

As in the single closed chain case, the open-chain terms, (qt)open and (Xik)open, are completely defined for each chain given the present state genial joint positions and rates, qt and qt, the applied graeral joint torques/forces in the free directions, n, and the motion of the supprat surface. Any appropriate open-chain Direct Dynamics algorithm may be used to calculate these terms. Because the general joint positions are known, fit and Aj are also defined. The efficient computation of fit and for a single serial-link chain was discussed in detail in Chapter 4. 

With ao known, we may also solve explicitly for the spatial tip force fit, jk = 1,..., m, using Equation 6.12. Thus, the motion of the refnence membo and the spatial force exerted at the tip of each chain are completely defined, and the simple closed-chain mechanism is effectively decoupled. Each manipulator may now be treated as an independent chain with a known spatial tip force. The joint accelerations for each chain may be computed separately using an r pro xiate Direct Dynamics algorithm and then integrated to obtain the next state. 

Contribution to the fundamental science will influence development of macrscopic kinetics, classical equilibrium thermodynamics, and joint application of these disciplines to study the macroworld. The capabilities of kinetic analysis will be surely expanded considerably, if traditional kinetic methods that are reduced to the analysis of trajectory equations are sup>plemented by novel numerical methods. The latter are to be based on consideration of continuous sequences of stationary processes in infinitesimal time intervals. The problems of searching for the trajectories, being included into the subject of equilibrium thermodynamics, would make deserved the definition of this discipline as a closed theory that allows the study of any macroscopic systems and processes on the basis of equilibrium principles. Like the equilibrium analytical mechanics of Lagrange the thermodynamics may be called the unified theory of statics and dynamics. Joint application of kinetic and thermodynamic models further increases the noted potential advantages of the discussed directions of studies. 

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As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

Uncertainties of the conventional parameters of H-atoms have been addressed since the early applications of X-ray charge density method. Support from ND measurements appears to be essential, because the neutron scattering power is a nuclear property (it is independent of the electronic structure and the scattering angle). The accuracy of nuclear parameters obtained from ND data thus depends mainly on the extent to which dynamic effects (most markedly thermal diffuse scattering) and extinction are correctable. Problems associated with different experimental conditions and different systematic errors affecting the ND and XRD measurements have to be addressed whenever a joint interpretation of these data is attempted. This has become apparent in studies which aimed either to refine XRD and ND data simultaneously [59] (commonly referred to as the X+N method), or to impose ND-derived parameters directly into the fit of XRD data (X—N method) [16]. In order to avoid these problems, usually only the ND parameters of the H-atoms are used and fixed in the XRD refinement (X-(X+N) method). 

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