Simple closed-chain mechanism computations

While considerable effort has been spent studying the simulation problem for single closed chains, fewer results are available for more complex multiple chain robotic systems. Existing algorithms for simple closed-chain mechanisms are, in general, difficult to apply and/or computationally inefficient Some relevant... 

In [31], Oh and Orin extend the basic method of Orin and McGhee [33] to include simple closed-chain mechanisms with m chains of N links each. The dynamic equations of motion for each chain are combined with the net face and moment equations for the reference membo and the kinematic constraint equations at the chain tips to form a large system of linear algebraic equations. The system unknowns are the joint accelerations for all the chains, the constraint fcwces applied to the reference memba, and the spatial acceleration of the reference member, lb find the Joint accelerations, this system must be solved as a whole via standard elimination techniques. Although this approach is sbmghtforward, its computational complexity of 0(m N ) is high. 

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. 

More specifically, in a simple closed-chain mechanism, the m actuated chains act on the reference member in parallel, and their motion is coupled with that of the reference member. If the reference member is removed, the chains may function independently. Computationally, the physical removal of the reference member corresponds to solving for the forces which it ex on each chain. Once these forces are known, the system is equivalent to a group of independent chains with known tip forces. The general joint accelerations may then be computed for each chain separately. Given enough processors (at least one per chain), the computations for each chain may be carried out simultaneously. 

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 method outlined above is quite straightforward and illustrates some features of a parallel computational structure as discussed previously. Of course, the illustrated example represents a special case. We will now develop a similar approach fex a general simple closed-chain mechanism which, although the notation becomes a bit more complicated, also exhibits the parallel computational characteristics we expect. 

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. 

Like the dynamic simulation algorithm fw a single closed chain, the algorithm developed here for simple closed-chain mechanisms may also be presented as a series of steps. In this case, five computational steps are required, and they are as follows ... 

In the develc ment of the simulation algorithm in the previous section, the objective was to decouple the simple closed-chain mechanism by computing the spatial force vectors exerted by the chains on the reference membo. The spatial tip forces computed in that algorithm are real, measurable forces, associated with the general jdnts which connect the reference membo and each chain tip. Once these forces are known, the chains are effectively decoupled from the refnence member, and the general joint accelerations may be computed for each chain separately. 

In this secdon, we will consid the computadonal requirements of the dynanuc simuladon algorithm for simple closed-chain mechanisms presented in Secdon 6.S. First, the number of scalar q>eradons required fw each chain of the mechanism will be tabulated, followed by the number of operadons required to compute the spadal acceleration of the reference member. The computadonal complexity of the complete algorithm will then be discussed. The parallel implementation of this algorithm will also be consid ed. 

Given the computations required for each individual chain, the number of scalar operations needed to compute the spatial acceleration of the reference membo, ao, is given in Table 6.3. Equation 6.38 is used to obtain the solution, which requires 0(m) spatial additions and a single 6x6 symmetric linear system solution. Thus, the number of opmtions required for ao is a function only of m, the number of chains in the simple closed-chain mechanism. The example of three chains (m s 3) is given in the last two columns of this table. 

The purpose of this book is to present computationally efficient algorithms for the dynamic simulation of closed-chain robotic systems. In particular, the simulation of single closed chains and simple closed-chain mechanisms (such as multilegged vehicles or dexterous hands) is investi ted in detail. In conjunction with the simulation algorithms, efficient algorithms are also derived for the computation of the joint space and operational space inntia matrices of a manipulator. These two inertial quantities are important factors in a variety of robotics applications, including both simulation and control. 

The book may be organized into two parts. Part one addresses the efficient computation of the joint space and operational space inotia matrices. Four algorithms are presented for the computation of each inertia matrix. Part two of the book presents the dynamic simidation algorithms which are develt red for single closed chains and simple closed-chain mechanisms, respectively. 

Although a single closed chain is a simple example of a closed-chain robotic mechanism, its real-time dynamic simulation is not trivial. The dynamics of the chain must be combined with the kinematic constraints which are imposed by the tip contact. In general, both the contact forces at the tip and the joint accelerations must be computed to completely solve the system. 

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