Inertia joint space

Although the details may be quite different, every research effort in the area of dynamic simulation faces a common task — the efficient and accurate solution of the Direct Dynamics problem. In the development of algorithms for Direct Dynamics, two basic approaches have emerged for both open- and closed-chain systems. The first utilizes the inversion of the x manipulator joint space inertia matrix to solve for the joint accelerations. More accurately, the accelerations are found via the solution of a system of linear algebraic equations, but the... 

This book is organized into two parts. The first part addresses the efficient computation of manipulator inertia matrices, both joint space and operational space. Grrresponding algorithms may be found in Chapters 3 and 4, respectively. Ibe second part of this book presents efficient dynamic simulation algorithms for closed-chain robotic systems and is comprised of Chapters S and 6. 

In general, the inertia matrix of a manipulator defines the relationship between certain forces exerted on the system and some corresponding acceleration vector. This relationship is of great importance both in real-time control and in the simulation of multibody systems. In the control realm, for example, the inertia matrix has been used to decouple robot dynamics so that control schemes may be more effectively tq>plied [19]. This may be accomplished either in joint space O operational space, since the inertia matrix may be defined in eith domain. The inertia matrix has also been used in the analysis of collision effects [43]. In addition to its use in control applications, the inertia matrix is an explicit and integral part of certain Direct Dynamics algorithms which are used to solve the simulation problem for manipulators and other multibody systems [2, 8, 31, 33,42]. 

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. 

In another recent effort [2S], Li uses the theory of Lagrangian mechanics to formulate the dynamic equations of a manipulator. Similar to Lee and Lee [24] above, this formulation includes an algorithm for computing the elements of the Joint space inertia matrix. In this approach, Li is able to further reduce the required computations fw the inertia matrix, making this algorithm the most efficient serial algorithm prior to the present wwk. The computational complexity is 0 N ) and the equations are applied to revolute and/or prismatic Joint configurations only. 

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. 

A comparison of this result with Equation 3.1 leads to the identification of the one-link manipulator joint space inertia matrix as ... 

Thus, the joint space inertia matrix fa the t-link manipulator is ... 

This is the basic recursive matrix equation for the manipulator joint space inertia matrix. 

If the summation for each element as given in Equation 3.31 is expanded and the terms are regrouped, the joint space inertia matrix for a three-link manipulator, H3, may be written ... 

Rible 3.7 Number of Computations for the Joint Space Inertia Matrix... 

Methods III and IV have reduced computational complexities of 0(N ). Method III is the most efficient tqrproach for the joint space inertia matrix of all those listed for N < 6, and it computes the manipulator Jacobian matrix as well. Method IV, although the simultaneous Jacobian computation has been eliminated, is the most efficient tqrproach for computing the joint space inertia matrix for N >6. Thus, these two methods together provide the most efficient calculation of H for all N. 

The operational space inertia matrix. A, like its joint space counterpart, H, is an inertial quantity which defines the dynamic relationship between certain forces exerted on a manipulator and a corresponding acceleration vector, hi the case of the joint space inertia matrix, H, the forces of interest are the actuator joint forces and torques, and the corresponding acceleration is the joint acceleration vector. On the other hand, A relates the spatial vector of faces and moments exerted at the tip a end effector and the spatial acceleration of this same point. Matrix a vector quantities which are defined with respect to the end effector are often said to be in workspace or operational space coordinates. Hence, A is called the operational space inertia matrix of a manipulator. 

The relationship between the operational space inertia matrix. A, and its joint space counterpart, H, may be established by investigating the relationship between the dynamic equations of their respective fomulations. The joint space dynamic equations of motion fa a single chain may be written [19] ... 

Perhaps the simplest and most obvious way to compute the opmtional space inertia matrix (or its inverse) is by the explicit inversion and multiplication of the Jacobian and joint space inertia matrices as shown in Equations 4.19 and 4.20. We will call this the Explicit Inversion/Multiplication Method. Although we will see that this is not the most efficient approach, it is in standard use today and may serve as a benchmark for new computational approaches. 

Methods I and II both require 0 N ) scalar operations for either A or A. Method I, based on the oiginal formulation of Khatib, is always less efficient than Method II. Method II, despite the order of its computational complexity, is the most efficient proach for A when N < 6, and for A when N <7. Note that both Methods I and II require the computation of the Jacobian and joint space inertia matrices. 

A similar 0(N ) method, presented by Angeles and Ma in [2], uses the concept of an orthogonal complement to construct the joint space inertia matrix. The Cholesky decomposition of this matrix is used in solving the appropriate linear system for the joint accelerations. The computational complexity of this algoithm is slightly better than that in [42], but the algorithm is still not the most efficient It, too, is restricted to configurations of simple revolute and prismatic joints. 

The quantities Hjb, Ck, Gt, and Jjb are, of course, the joint space inertia, cen-tripet /Coriolis, gravity, and Jacobian matrices fix chain k, respectively. They are all functions of the general joint position and rate vectors, qjb and q ,. Recall that the base of each chain is the sujqwit surface, and the tip of each chain... 

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. 

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