Acceleration reference member

An alternate form of the algorithm is also presented which is based on resolving the dynamics of each chain to the reference member. This dynamic analysis leads to a solution for the spatial acceleration of the reference member and the resolved force vectors exerted on it by each chain. Once again, after the resolved forces are completely defined, the closed-chain joint accelerations may be computed for each chain separately. This alternate approach is less efficient than the initial tq>proach discussed above, but it provides a slightly diffoent perspective on the problem which is both useful and interesting. 

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. 

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. 

Thus, we have established an explicit relationship between the spatial tip force, ft, and the spatial acceleration of the reference member, ao. This expression... 

Consider a mechanism with m chains, each with an arbitrary number of degrees of freedom, N. The interaction between each chain tip and the reference member is arbitrary and will be modelled using the general joint model of Chapter 2. To begin, we will derive an explicit relationship between the spatial acceleration of each chain tip and the spatial acceleration of the reference member. The spatial acceleration of the tip of chain k is denoted by xt. The relative spatial acceleration between the tip of chain k and the reference member, x, resolved in the orthogonal vector spaces of the general joint between them, may be written ... 

This equation matches the spatial accelerations at the coupling point between chain k and the reference member, giving an explicit relationship between jtk and ao when the coiqtling is arbitrary. The unknowns in Equation 6.19 are x , Ok, and ao. All other vectors and matrices are known from the initial information given for the simulation problem. 

To decouple the chains and the reference member, we need an explicit mathematical relationship between the spatial force exerted by chain ib on the reference member, f t> and the spatial acceleration of the reference member, ao. Equation 6.19 relates the spatial acceleration of the refnence member and the spatial acceleration of the tip of chain k. We may eliminate Ok, the unknown components of the relative acceleration, by projecting Equation 6.19 onto the constraint space of the corresponding general joint as follows ... 

If the spatial acceleration of the reference member is known, we may find an explicit solution for the unknown force components at the tip of chain k from the following set of linear algelwaic equations ... 

Und these conditions, the spatial tip acceleration of augmented chain k, XJ, is exactly equal to the spatial accel tion of the reference member. That is. 

The only unknown in Equation 6.51 is the spatial acceleration of the reference member, ao. Its solution may be found using any linear system solution method. Note that the coefficient matrix of ao will always be a 6 x 6 matrix. Thus, the computational cost of solving for ao is still constant We may also write the following explicit analytical solution for ao ... 

With the spatial acceleration of the reference member known, the force vector exerted by each augmented chain on the reference memb, f, may be computed using Equation 6.47. The explicit knowledge of f allows us to treat each augmented chain as an independent chain with a known tip force. The general closed-chain joint accelerations for each augmented chain may now be computed using the equation ... 

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. 

In Step 2 of the simulation algorithm, the spatial acceleration of the reference member is calculated using Equation 6.38. For this task, X P and X R must be computed fm each chain. The number of q)erations required to compute P, R, X P, and X R are also listed in Table 6.2. In this case, the numb of opontions is a function of the number of degrees of constraint at the gen joint between the chain tip and the reference member n ). This number can never be greater than six. The computational complexity of these calculations is 0(n ) due to the linear system solution required in the computation of both P and R (see Table 6.1). 

The spatial fcare vecux, f, exited by each chain on the reference member, and the closed-chain joint accelerations for the chain, q, are calculated in Steps 3 and 4 of the simulation algoithm, respectively. The appropriate equations are given in Table 6.1. The q>erations required to calculate these vectors complete the table of computations. The specific number of operations required for the special case of TV = 6 and ne = 3 are also provided in Table 6.2. This value of Tie might correspond to a hard point contact between the manipulator tip and a constraining body or surface when the tip is not slipping. 

Click chemistry refers to the reaction between an azido functional group and an alkyne to form a [3 + 2] cycloaddition product, a 5-membered triazole ring. This reaction has been used for many years in organic synthesis to form heterocyclic rings. Normally, the click reaction requires high temperatures, and this was the main reason that it was not used as a bioconjugation tool. However, it was discovered that in aqueous solutions and in the presence of Cu(I), the reaction kinetics are dramatically accelerated to provide high yields even at room temperature and ambient pressures (Rostovtsev et al., 2002 Tornoe et al., 2002 Sharpless et al., 2005). 

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. 

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