Many useful identities pertaining to quaternion multiplications are generalized. Among them multiplicative commutativity is the most powerful. Since quaternion space includes the 3D vector space, the physical quantities related to rotations, such as angular displacement, velocity, acceleration, and momentum, are shown to be vector quaternions, and their expressions in quaternion space are derived. These kinematic and dynamic differential equations are further shown to be invertible due to the fact that they are written in quaternion space, and the highest order term of the rotation parameters can be expressed explicitly in closed form.>

With the aid of graph theory it is possible to combine the topological information of a constrained rigid-body system with the mathematical formulation of the equations of motion in a direct, systematic, and consistent procedural fashion that lends itself to a straightforward computer implementation. Such a graph-theoretical formulation for the dynamics analysis and simulation of three-dimensional constrained rigid-body systems is presented. It is shown that through the graph-theoretic approach, the topological information contributes directly to the systematic formulation procedure and also to an efficient, directly implementable computer algorithm. This algorithm can be implemented in a recursive fashion that reduces the computational cost substantially. The main components of the numerical procedure can be traced back directly to the system graphs of the formulation phase with very little effort. The resulting system of differential equations is left in its implicit form and solved numerically.>

The problem of finding the relative orientation between the reference frames of a line-mounted sensor and the link is formulated as a kinematic equation of the form A/sub a/A/sub x/=A/sub x/A/sub b/, which has to be solved for the rotational transformation matrix A/sub x/ given the transformations A/sub a/ and A/sub b/. This equation can be transformed to its equivalent form in terms of quaternion and then simplified to a well-structured linear system of equations of the form Bx=0. Since B is rank-deficient, the solution is not unique. The generalized-inverse method using singular-value decomposition (SVD) is applied. Although the solution is reached using the analysis of SVD, the SVD is derived symbolically; therefore, the actual implementation of SVD is not required. A method for obtaining a unique solution is proposed where a system of nonlinear equations is solved using Newton-Raphson iteration. The iteration is simplified by a dimension-reduction technique that provides a set of closed-form formulas for solving the resulting linear system of equations.>

The graph-theoretic models of a prismatic joint, a revolute joint, and an open kinematic chain are presented. The Denavit-Hartenberg representation of linkages is encompassed within the general framework of graph-theoretic system theory. The final mathematical model derived by this formalism is a system of differential and algebraic equations.

A set of mixed differential and algebraic equations (DAEs) which arises in the simulation of a robot manipulator is solved simultaneously using implicit integration. The dimension of the DAEs which have to be solved by LU factorization at each integration step can be reduced to the number of degrees of freedom by exploring the special structure of the Jacobian matrix of DAEs. The independent and dependent generalized coordinates are determined directly from the system topology. The simulation of a 6-R manipulator is given as an example.

The discipline of linear graph theory, which has been applied extensively to the analysis of electrical networks, is now extended to encompass general three-dimensional isolated rigid-body systems. The graph-theoretic models of a rigid body and its relevant components are discussed elaborately to pave the way for further studies in multibody systems. Utilizing the analysis based on system topology, the mathematical model of a mechanical system is derived by means of a hybrid formulation in which Euler parameters (or normalized quaternions) are incorporated to define the orientations of rigid bodies. A single body system serves as an illustrative example.