Spatial link inertia

A number of important recurrence relations in the articulated-body fwmula-tion may be derived by analyzing a simple two-link articulated body as shown in Figure 4.3. This articulated body is made up of links 1 and 2 with spatial link inertias of Ii and I2, respectively. The two links are connected by a general joint, joint 2, which is characterized by its motion space, < 2> Recall from Chapter 2 that 2 represents the free modes of this joint In this analysis, it is assumed that the two bodies are initially at rest and experience no applied joint forces or torques on or about 2 (t2 = 0). The tip force, f, is zero b ause the chain is unconstrained. The only spatial force vector exerted on the articulated body is fi, which is applied to link 1, the handle of the articulated body. For the moment, we will assume that the physical parameters describing the links and joints, including the spatial link inertias, are given in absolute coordinates. 

Methods III and IV have reduced computational complexities of 0(N). This is a significant improvement over the first two algorithms. Method IV is the most efficient q>proach for A when V > 6, and for A" when N >7. Recall, however, that Method IV is based on the explicit knowledge of the spatial link inertia of the base member, Iq. This assumption may not be the most appropriate in all cases. If the base is fixed to the inertial firame, then the 0 N) solution of Method III may be used. This approach is more efficient than Method II for A and A" when N > 12. 

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. 

The spatial composite-rigid-body inertia of links i through N,K, was defined in Equation 3.36 as ... 

It can easily be shown that K,+i has the same mathematical fonn as the spatial inertia of a single rigid body. Thus, the congruence transformation of this matrix also requires (49 multiplications, 49 additions) as described above. The addition of I, requires only an extra 10 additions, if we consido the symmetry and form of Ki and I<. Note that the bottom right submatrix of is simply the diagonal matrix of the composite mass of links t through N. Thus, since this composite... 

It may be shown that this procedure for ccmiputing the spatial composite-rigid-body inertia is exactly equivalent to the procedure used by Walker and Orin in [42]. That is, if the composite mass, composite center of mass, and composite moment of inertia matrix are computed for links i through iV, they may be combined to obtain the spatial matrix K,-. After studying this equivalent approach, howev, it appears that the congruence transformation method given hoe is more efficioit... 

Because this approach is based on the application of spatial unit forces at the end effector, we will call it the Unit Force Method. The complete algorithm for the Unit Force Method (Method II) for calculating the ( rational space inertia matrix of a serial iV-link manipulator is given in Table 4.3. Note, once again. 

Feathmtone defines the articulated-body inertia of a link as the 6 x 6 matrix which relates a spatial force applied to the link and the spatial acceleration of the link, taking into account the dynamics of the rest of the articulated body. This relationship is linear, and for link t, it may be written as follows [9] ... 

In Equation 4.51, L acts as a spatial transformation which prqjagates the spatial accel tion vector, a,- i, across joint i. We will call a matrix which transforms spatial vectcs s across actuated joint structures a spatial articulated tran ormation. In general, an articulated transformation is a nonlinear function of the articulated-body inertia and is a dimensionless 6x6 matrix. Featherstone calls the articulated transformation, L,, the acceleration propagator [9]. It relates the spatial acceleration of one link of an articulated body to the spatial acceleration of a neighbraing link in the same articulated body (ignoring bias... 

Given the recursive dynamic equations for a constrained chain, we will now begin the development of a linear recursive algcxithm fw A the inverse operational space inertia matrix of a single chain. First, we will define a new quantity, (A ) an inertial matrix which relates the spatial acceleration of a link and the propagated spatial contact force exoted at the tip of the same link. We may write a defining equation for this matrix (at link t) as follows ... 

By definition, the operational space inertia matrix of an t-link manipulatcM , A, is the matrix which relates the spatial acceleration of link i and the spatial force vector exoled at the tip of link i. Thus, we may write ... 

In this simple recursion, the operational space inertia matrix of the base member, Ao, is propagated across joint 1 by La > a new spatial articulated transformation which is very similar in form to the acceloation propagator of the previous section. The propagated matrix is combined with Ii, the spatial inertia of link 1 to form Ai, the operational space inertia matrix of the two-link partial chain comprised of links 0 and 1. Note the similarity between this recursive procedure and the structural recursion used to derive the Structurally Recursive Method (Method I) in Ch t 3. 

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