A continuous model of Coulomb friction is used with a tangent space formulation of differential algebraic equations of motion for simulation of multibody dynamic model problems. Characteristics of the model problems studied are similar to those encountered in broad classes of multibody systems, without the associated geometric and analytical complexities. An implicit trapezoidal numerical solution algorithm is used to simulate dynamic response that includes the onset of stiction, its progression, and its termination, avoiding stiff behavior that is reported in the literature when index 3 formulations are used. Analytical criteria for stiction are derived for a three mass Coulomb friction model problem that defines the onset of and departure from stiction events with redundant equations of constraint. The tangent space formulation with implicit trapezoidal integration is applied to this analytical model to compute dynamic response, determine ranges of constraint forces that may occur during periods of stiction, and demonstrate that dynamic response is a discontinuous function of model parameters when stiction occurs. Accuracy of the continuous model of Coulomb friction is established, through comparison of results with those of the analytical model. Cartesian coordinate models of higher dimension are presented for three and four mass model problems that encounter a higher degree of redundancy in constraints during periods of stiction. Simulation of the Cartesian coordinate models, which have characteristics similar to more general multibody systems, yields accurate solutions, without any indication of stiffness in the tangent space equations of motion. Methods successfully demonstrated in model problems provide a foundation for simulation of spatial multibody dynamic systems with friction.

The Maggi and Kane equations of motion are valid for systems with only nonholonomic constraints, but may fail when applied to systems with holonomic constraints. A tangent space ordinary differential equation (ODE) extension of the Maggi and Kane formulations that enforces holonomic constraints is presented and shown to be theoretically sound and computationally effective. Numerical examples are presented that demonstrate the extended formulation leads to solutions that satisfy position, velocity, and acceleration constraints for holonomic systems to near computer precision.

A formulation for modeling and simulation of friction effects in spatial multibody systems is presented. Constraint reaction forces on rigid bodies that are connected by joints that support friction are derived as functions of Lagrange multipliers, using D’Alembert’s principle. Friction forces acting on bodies are calculated as a function of joint geometry, constraint reaction forces that are functions of Lagrange multipliers, and relative velocities at constraint contact points that are determined by system kinematics. Friction forces are implemented in index 0 differential-algebraic equations of motion that are solved numerically using explicit and implicit numerical integration methods. Spatial examples are presented, yielding accurate results and demonstrating that the systems are not stiff, even in the presence of friction and stiction.

Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

A recently published treatment of nonredundant manipulator kinematics and dynamics on differentiable manifolds is extended to kinematically redundant manipulators. Analysis at the configuration level shows that forward kinematics and dynamics of redundant manipulators are identical to that for nonredundant manipulators. The manifold-based inverse kinematics formulation that is presented for redundant manipulators, in contrast, yields parameterizations of set-valued inverse kinematic mappings at the configuration level, where sharper results are obtained than those presented in the literature using velocity formulations. Explicit expressions are derived for set-valued inverse kinematic mappings for both serial and nonserial (called compound) kinematically redundant manipulators, as functions of vectors of arbitrary parameters. Parameterizations are presented for both manipulator regular configuration manifolds and self-motion manifolds, the latter comprised of sets of inputs that map into the same output. It is shown that kinematically redundant configuration manifolds and self-motion differentiable manifolds are distinctly different and play complementary roles in redundant manipulator kinematics. Computational methods are presented for evaluation of set-valued inverse kinematic mappings, without problem-dependent ad hoc analytical manipulations. Redundant serial and compound manipulator examples are presented to illustrate computation of set-valued inverse kinematic mappings and the use of self-motion manifold mappings in obstacle avoidance applications. Differentiation of configuration level inverse mappings yields inverse velocity and acceleration mappings as functions of time-dependent arbitrary parameters that play a central role in manipulator dynamics and control.

An inverse kinematic mapping for redundant serial manipulators is presented at the configuration level, for which periodic manipulator operational trajectories map into periodic input trajectories, i.e., for which serial manipulators are cyclic. The inverse kinematic mapping defines a differentiable manifold on which output and self-motion coordinates comprise operational coordinates that represent manipulator redundant degrees-of-freedom. The inverse kinematic mapping and differentiable manifold are defined in analytical form and a computational method for their evaluation is presented. Numerical examples are presented to illustrate the validity of the formulation on subsets of manipulator regular configuration space, or on the entire space.