Rigid-body method

The simplest type of rigid-body method involves simply transferring the backbone conformation of the core of the protein from a single template to the unknown protein. An alternative is to construct a framework by averaging the structures from a number of protein templates Each template can be given a weight related to its sequence similarity to the unknown target [Srinivasan et al 1996]... 

Although rigid body methods can be effective for simulations of rigid models, we focus here on methods that apply effectively also to flexible models. Therefore, this chapter is not concerned with the methods of rigid body dynamics, either in the form of the older and more problematic Euler angles method, or as represented by the more recent and more effective quaternion methods. 

The Structurally Recursive Method is then expanded, and a second, non-recursive algorithm fw the manipulator inertia matrix is derived from it A finite summation, which is a function of individual link inertia matrices and columns of the propriate Jacobian matrices, is defined fw each element of the joint space inertia matrix in the Inertia Projection Method. Further manipulation of this expression and application of the composite-rigid-body inertia concept [42] are used to obtain two additional algwithms, the Modified Composite-Rigid-Body Method and the Spatial Composite-Rigid-Body Method, also in the fourth section. These algorithms do make use of recursive expressions and are more computationally efficient. 

In the sixth section, the computational requirements for the methods presented here are compared with those of existing methods for computing the joint space inertia matrix. Both general and specific cases are considered. It is shown that the Modified Composite-Rigid-Body and Spatial Composite-Rigid-Body Methods are the most computationally efficient of all those compared. 

Walker and Orin [42] present one of the most familiar and efficient approaches for the computation of the inertia matrix in the so-called Composite-Rigid-Body Method [9]. This algorithm utilizes the concq>t of composite-rigid-body inertias to simplify the calculation of the manipulator inotia matrix. The computational complexity of this approach, 0(N% is significantly reduced compared to those described above, but the restriction to revolute and/w prismatic Joints remains. 

Parallel computation methods have also been investigated for the Joint space inertia matrix. Amin-Javaheri and Orin [1], as well as Fijany and Bejczy [10], have achieved bett performance by developing parallel and/or pipelined algorithms. In both cases, the parallel forms are based to a great extent on the serial Composite-Rigid-Body Method of Walker and Orin [42], and, of course, the improvement in performance is dependent on an increased number of processes. 

Four algorithms for computing the joint space inertia matrix of a manipulator are presented in this section. We begin with the most physically intuitive algorithm the Structurally Recursive Method. Development of the remaining three methods, namely, the Inertia Projection Method, the Modified Composite-Rigid-Body Method, arid the Spatial Composite-Rigid-Body Method, follows directly from the results of this tot intuitive derivation. 

Table 3.3 Algwithm for the Modified Composite-Rigid-Body Method...

The number of scalar operations required by each of the four methods presented in this chapto have been calculated explicitly. As a specific example. Table 3.6 lists the computations required by the Modified Composite-Rigid-Body Method for the case of an A -link manipulator with simple revolute and prismatic joints. Note that the computations listed for the transformation matrix, % i, correspond to the use of two screw transformations as discussed in the previous section. 

---