The Operational Space Inertia Matrix

In the control domain, A may be used to decouple face and/a motion control about the workspace axes. Later in this boc, we will show that it is also useful in the development of dynamic simulation algoithms fa closed-chain robotic systems. 

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]  

From Equation 4.4, we may solve fa the joint acceloations as follows  

The genoalized joint force vecta, F, is related to the q)erational face vecta, F, as follows [19]  

Combining Equations 4.8 and 4.12, we may express the spatial end effector acceleration vectw as follows  

---

The second quantity of interest, the operational space inertia matrix (O.S.I.M.) of a manipulator, is a newer subject of investigation. It was introduced by Khatib [19] as part of the operational space dynamic formulation, in which manipula-Ux control is carried out in end effector variables. The operational space inertia matrix defines the relationship between the gen lized forces and accelerations of the end effectw, effectively reflecting the dynamics of an actuated chain to its tip. This book will demonstrate its value as a tool in the development of Direct Dynamics algorithms for closed-chain configurations. In addition, a number of efficient algorithms, including two linear recursive methods, are derived for its computation. 

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. 

Because the operationa] space fonnulation for robot dynamics is fairly new, few efficient methods exist fw computing its components. The conceptual framework for the operational space fomulation was presented by Khatib in [17,18,19], where he established a basic definition for the operational space inertia matrix. In [19], Khatib shows that the operational space inertia matrix of a 6 degree-of-freedom manipulator may be computed as follows ... 

Khatib also discusses the operational space formulation for redundant manipulators. In this case, the definition for the operational space inertia matrix... 

Rodriguez, Kreutz, and Jain [37, 38] present a linear recursive algorithm for the operational space inertia matrix, referred to as the operational space mass matrix , as part of an original operate formulation for open- and closed-chain multibody dynamics. In general, this operator apfxoach appears to be quite powerful, especially in matrix factorization and inversion, and with it, the authors... 

If we compare Equation 4.16 with Equation 4.3, we may identify the operational space inertia matrix as follows ... 

From Equation 4.17, we see that the operational space inertia matrix is a function of position only. It is always a 6 x 6 symmetric matrix, independent of N. Note, however, that A is only defined if the bracketed term has full rank, that is, if the product,... 

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. 

Note that when the recursion of Equation 4.129 reaches link N, the operational space inertia matrix of the entire chain. A, is known. That is. 

A review of previous work related to the dynamic simulation of single closed chains is given in the second section of this chapter. The next three sections discuss several steps in the development of the simulation algorithm. In particular, in the third section, the equations of motion for a single chain are used to partition the joint acceleration vector into two terms, one known and one unknown. The unknown term is a function of the contact forces and moments at the tip. The end effector acceloation vector is partitioned in a similar way in the fourth section, making use of the operational space inertia matrix. In the fifth section, two classes of contacts are defined which may be used to model interactions between the end effector and other rigid bodies. Specific examples are provided. 

Note that if chain k is rigidly grasping the reference membo, then the constraint space for this genoal Joint, is the 6 x 6 identity matrix. In this case, also note that 0 1 and ht are identically zero for each chain. If chain k has six degrees of fire om and is not in a singular position, then will be exactly equal to A.k, the operational space inertia matrix for chain k, and the solution for h will be ... 

---