Simple closed-chain mechanism

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. 

Figure 1.2 Models of Simple Closed-Chain Mechanisms...

EFFICIENT DYNAMIC SIMULATION OF SIMPLE CLOSED-CHAIN MECHANISMS... 

In this chapter, an efficient soial algorithm for the dynamic simulation of simple closed-chain mechanisms is derived. The development of this algorithm... 

A brief review of previous work related to the dynamic simulation of simple closed-chain mechanisms is given in the second section of this chapter. A model for simple closed-chain mechanisms is described in the third section, and the basic problem statement is also discussed. The dynamic equations of motion for the individual chains and the common reference member are summarized in the fourth section. 

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. 

Multiple chain robotic systems can take many forms, some of them quite complex. Simple closed-chain mechanisms are a subset of multiple chain systems with specific structural characteristics. In this section, a model for simple closed-chain mechanisms is described, and the nature of the simulation problem for these mechanisms is discussed. 

The structure of a simple closed-chain mechanism is characterized by m actuated chains which support a single common reference member [31]. The supporting chains are assumed to be serial-link chains, firee of intmial closed loops. Therefore, the removal of the reference membo breaks all closed Io( in the system. Each chain may have an arbitrary numbo of links and degrees of freedom. The... 

As was mentioned biefly in Chapt 1, th arc two basic types of simple closed-chain mechanisms called Type 0 and lype 1, respectively [31]. These two types are defined based on the nature of the interactions which occur between the links of the chain and the reference member or support surface. For both types, the support surface acts as the base of each chain. We will refa to the link which interacts with the support surface as link 1, and the link which intmcts with the reference member will be called the last link or end effector (link N). The far end of link N is the tip of the chain. 

Figure 6.2 illusuiates a typical Type 1 simple closed-chain mechanism which may be used to model multilegged vehicles. For a Type 1 mechanism, the first link of each chain int cts with the support surface through an unpowered... 

Figure 6.2 Example of a Type 1 Simple Closed-Chain Mechanism...

Each chain in the simple closed-chain mechanism is governed by the dynamic equations of motion fw a single chain. These are ... 

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. 

We may now solve for ao ftom this linear system of algebraic equations using any linear system solver. Note that the characteristic system mabix is only 6x6 and represents the effective operational space inotia of the complete simple closed-chain mechanism as seen by the reference member. 

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. 

The coefficient matrix of ao in Equation 6.51 rq>resoits the effective reference member at the origin of its own coordinate system. The operational space inertia of the reference member is just its spatial inertia matrix, lo. Note that the operational space inertias of the augmented chains (acting in parallel on the reference member) add in a simple sum. This is a general rule for inertia matrices. For actuated chains connected in series, the combination rule is not as simple. In this case, extended versions of the recursive algorithms of Chapter 4 may be applied. 

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. 

To determine the total number of scalar operations required to simulate the entire simple closed-chain mechanism, the number of operations required for a single chain is simply multiplied by m, the numb of chains, and added to... 

In this chapter, a general and efficient dynamic simulation algorithm for simple closed-chain mechanisms was derived. The algorithm is tq>plicable to both lype... 

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