A diagonal equation _ + C; = for robot dynamics is developed by combining recent mass matrix factorization results 1 7 with classical Lagrangian mechanics. Diagonalization implies that at each xed time instant the equation at each joint is decoupled from all of the other joint equations. The equation involves two important variables: a v e ctor of total joint rotational rates and a corresponding vector of working joint moments. The nonlinear Coriolis term C; depends on the joint angles and the rates . The total joint rates are related to the relative joint-angle rates _ by a linear spatial operator m mechanized b y a b ase-to-tip spatially recursive algorithm. The total rate k at a given joint k re ects, in a very unique sense the total rotational velocity about the joint, and includes the combined e ects from all the links between joint k and the manipulator base. This di ers from the more traditional joint-angle rates _ which only re ect the relative, as opposed to total, rotation about the joints. Similarly, the working moments =`T are related to the applied moments T by the spatial operator`= m ,1 mechanized by a tipto-base spatially recursive algorithm. The working moment k at a given joint k is that part of the applied moment Tk which does actual mechanical work, while its other part a ects only the non-working internal constraint forces. The diagonal equations are obtained by using the recently developed 1 mass matrix factorization M=mm of the system Lagrangian. The diagonalization is achieved in velocity space. This means that only the velocity variables _ are replaced with the new variables , while the original con guration variables are retained. The new joint velocity variables can be viewed as time-derivatives of Lagrangian quasi-coordinates, similar to those of classical mechanics. The velocity transformations are shown to always exist for tree-like, articulated multibody systems, and they can be readily implemented using the spatially recursive ltering and smoothing methods 1,4,7 advanced by the authors in recent years. Mass Matrix Factors Diagonalize Lagrange's EquationsThe main new result in this paper is a diagonalized equations of motion _ + C; = , which embodies in a simple, elegant, diagonal equation the complete dynamical behavior of robotic manipulators, while simultaneously exploiting the computational e ciency of the spatially recursive ltering and smoothing algorithms of 4, 7 to conduct necessary velocity coordinate transformations. The diagonal equations of motion result by combining Lagrangian mechanics with the mass matrix factorization

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There has been a growing interest in the development of new and efficient algorithms for multibody dynamics in recent years. Serial rigid multibody systems form the basic subcomponents of general multibody systems, and a variety of algorithms to solve the serial chain forward dynamics problem have been proposed. In this paper, the economy of representation and analysis tools provided by the spatial operator algebra are used to clarify the inherent structure of these algorithms, to identify those that are similar, and to study the relationships among the ones that are distinct. For the purposes of this study, the algorithms are categorized into three classes: algorithms that require the explicit computation of the mass matrix, algorithms that are completely recursive in nature, and algorithms of intermediate complexity. In addition, alternative factorizations for the mass matrix and closed form expressions for its inverse are derived. These results provide a unifying perspective, within which these diverse dynamics algorithms arise naturally as a consequence of a progressive exploitation of the structure of the mass matrix. Serial Chain oft €(R 9lx9l n .91 r p (k) r v (k) Link/ Joint Properties Nomenclature! = mass matrix for the serial chain = number of links in the serial chain = total number of motion DOF for the serial chain = number of positional DOF of the kth joint = number of motion DOF of the kth joint = joint matrix for the kth joint = moment of inertia of the Ath linkabout G k = vector from 0^ to 0^_ i = spatial inertia of the Ath link about G k = mass of the Ath link = reference location of the Ath joint on the Ath link = reference location of the Arth joint on the (k + l)th link = vector from 0* to the center of mass of the Arth link = joint motion parameters for Ath joint = configuration variables for Ath joint Forces and Velocities a(k) €(R 6 = Coriolis and centrifugal spatial acceleration at 0* b(k) € (ft 6 = gyroscopic spatial force at G k F(k) € (R 3 = linear force of interaction between the (k + l)th and kth links at 0* l(k,k-\) M(Ar) m(k)

The Newton-Euler inverse mass operator (NEIMO) method for internal coordinate molecular dynamics (MD) of macromolecules (proteins and polymers) leads to stable dynamics for time steps about 10 times larger than conventional dynamics (e.g., 20 or 30 fs rather than 1 or 2 fs for systems containing hydrogens). NEIMO is practical for large systems since the computation time scales linearly with the number of degrees of freedom N (instead of the N 3 scaling for conventional constrained MD methods). In this paper we generalize the NEIMO formalism to the Nosé (and Hoover) thermostat to derive the Nosé and Hoover equations of motion for constrained canonical ensemble molecular dynamics. We also examined the optimum mass, Q, determining the time scale (τ s ) for exchange of energy with the heat bath for NEIMO-Hoover dynamics of polymers. We carried out NEIMO-Hoover simulations on the amorphous polymers poly(vinyl chloride) and poly-(vinylidene fluoride), where we find that time steps of 20-30 fs lead to stable dynamics (10 times larger than for Cartesian dynamics). The computational efficiency of the NEIMO canonical MD method should make it a powerful tool for MD simulations of macromolecular materials. IntroductionFor studies of the conformations and dynamics of polymers and proteins, it is often useful to simplify the description of the systems by constraining such structural properties as bond lengths and bond angles so that the focus can be on the dihedral angles distinguishing the conformations. 1-4 With such constraints the number of degrees of freedom (dof) drops from 3N (where N is the number of atoms) to N. Thus, for a polyethylene polymer, C p H 2p+2 , 3N ) 9p + 6, while the number of torsional dof is N ) p -1. In addition to simplifying the analysis, such constrained molecular dynamics (MD) can allow significantly increased time steps.With constrained MD, Newton's equation of motion becomes where θ denotes the vector of the generalized coordinates (e.g. torsional angles), t denotes the vector of generalized forces (e.g., torques), M denotes the mass matrix (moment of inertia tensor), and C includes the Coriolis forces. The dynamics of motion is obtained by solving (1) for the acceleration and integrating to obtain new velocities and coordinates. The problem here is that M is a N × N matrix, and hence the calculation of M -1 in (2) involves a computational cost scaling as N 3 . Since the other parts of the calculation generally scale linearly in N, calculation of M -1 can become the dominant cost in the dynamics of large systems. Thus for 1001 atoms (p ) 333), it is necessary to invert a 332 × 332 matrix every step of the dynamics, costing more than the rest of the dynamics. Using the cell multipole method, 5 MD has been demonstrated 4 to be practical for 10 6 atoms (p ) 333 333). However, inverting the 333 332 × 333 332 matrix would be clearly impractical.To solve this problem we use the Newton-Euler inverse mass operator (NEIMO) approach to calculate the θ 2 of (2) directly without going through the step of ex...