Direct Dynamics closed-chain

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. 

As mentioned above, more efficient algorithms are needed to make real-time dynamic simulation a reality. This need is particularly great fw robotic systems with multiple chains and closed kinematic loops. Thus, a fundamental goal of this book is the development of better and more efficient algorithms for the dynamic simulation of multiple chain robotic systems. In particular, solutions to the Direct Dynamics problem fw simple closed-chain mechanisms are investigated. 

The second quantity of interest, the operational space inertia matrix (O.S.I.M.) of a manipulator, is a newer subject of investigation. It was introduced by Khatib [19] as part of the operational space dynamic formulation, in which manipula-Ux control is carried out in end effector variables. The operational space inertia matrix defines the relationship between the gen lized forces and accelerations of the end effectw, effectively reflecting the dynamics of an actuated chain to its tip. This book will demonstrate its value as a tool in the development of Direct Dynamics algorithms for closed-chain configurations. In addition, a number of efficient algorithms, including two linear recursive methods, are derived for its computation. 

One of the first Direct Dynamics algorithms for single closed-chain robotic mechanisms is presented by Orin and McGhee in [33]. This algorithm is based on the in a matrix invasion aj roach. The dynamic equations of motion for the chain are augmented with kinematic constraint equations at the tip of the... 

The qren-chain tom, 4open> turd the force vector coefficient, fl, are known from Stq) 1, and f is known from Step 2. The computational complexity of this step is 0 N). With this step, the Direct Dynamics problem has been completely solved for this single closed chain system. 

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. 

In developing an efficient algwithm for the dynamic simulation of simple closed-chain mechanisms, we are naturally led to consider the relationship between the physical structure of the robotic system and the computational structure of the desired algorithm. Intuitively, it seems tqyparent that the structural parallelism present in a simple closed-chain mechanism should lead to computational parallelism in the solution of the Direct Dynamics problem for that mechanism. 

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. 

The triplication of Equation 6.42 to ev actuated chain in the simple closed-chain mechanism results in a complete solution to the Direct Dynamics problem for this robotic system. The next state positions and velocities may be computed by integrating the appropriate quantities for each chain and the reference member. As discussed in Chapter 5, small amounts of negative position and rate feedback may be employed to countra t the drift which is a result of the integration process. 

In this paper, an algorithm for dynamic simulation based on the concept of velocity transformations is presented. This algorithm may be applied to the analysis of open and closed-chain systems. The equations of motion for open chain systems are derived using a direct velocity transformation, called open chain velocity trarvrformation. Closed chain systems are analyzed in two steps. First, they are converted into open chain systems by removing some joints and the open chain velocity transformation is applied then, the closed loop conditions are imposed through a second velocity transformation. The implementation of the proposed algorithm was carried out on a SGI 4D/240 workstation and the results obtained for a series of illustrative examples are presented. 

Molecular mechanisms for stress-softening are also discussed. It is shown that this phenomenon is not related to the chain slippage or to a conversion of a "hard" adsorbed phase to a soft one. The obtained results assume that the stress-softening in silicon rubbers is caused by two possible reasons changes in the positions of filler particles relative to the direction of stretching at the first deformation and by a re-distribution of the topological hindrances. It is shown that the tensile strength at break as a fiinction of temperature is closely related to the chain dynamics at the elastomer-filler interface. 

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