Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part II Dynamics

The manipulator differentiable manifold formulation presented in Part I of this paper is used to create algorithms for forward and inverse kinematics on maximal, singularity free, path connected manifold components. Existence of forward and inverse configuration mappings in manifold components is extended to obtain forward and inverse velocity mappings. Computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of the four categories of manipulator treated. Manifold parameterizations derived in Part I are used to transform variational equations of motion in Cartesian generalized coordinates to second order ordinary differential equations of manipulator dynamics, in terms of both input and output coordinates, avoiding ad-hoc derivation of equations of motion. This process is illustrated in evaluating terms required for equations of motion of the four model manipulators analyzed in Part I. It is shown that computation involved in forward and inverse kinematics and in evaluation of equations of manipulator dynamics can be carried out in real-time on modern microprocessors, supporting on-line implementation of modern methods of manipulator control.

Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part I Kinematics

Using basic tools of Euclidean space topology and differential geometry, a regular manipulator configuration space comprised of input and output coordinates and conditions that assure existence of both forward and inverse kinematic mappings is shown to be a differentiable manifold, with valuable analytical and computational properties. For effective use of the manifold structure in support of manipulator analysis and control, four categories of manipulator are treated; (1) serial manipulators in which inputs explicitly determine outputs, (2) explicit parallel manipulators in which outputs explicitly determine inputs, (3) implicit manipulators in which explicit input-output relations are not possible, and (4) compound manipulators that require use of mechanism generalized coordinates to characterize input-output relations. Basic results of differential geometry show that differentiable manifolds in each category are naturally partitioned into maximal, disjoint, path connected submanifolds in which the manipulator is singularity free, hence programmable and controllable. Model manipulators in each of the four categories are analyzed to illustrate use of the manifold structure, employing only multivariable calculus and linear algebra. Computational methods for forward and inverse kinematics and construction of ordinary differential equations of manipulator dynamics on differentiable manifolds are presented in part II of the paper, in support of manipulator control.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain $${\mathscr {E}}(M)$$ E ( M ) on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ M ⊆ R d can be defined and study essential properties of p. We prove that if M is a $$C^1$$ C 1 submanifold of $${{\mathbb {R}}}^d$$ R d satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$ E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$ C 2 or if the topological skeleton of $$M^c$$ M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$ C k -submanifold M with $$k\ge 2$$ k ≥ 2 , the projection map is $$C^{k-1}$$ C k - 1 on $${\mathscr {E}}(M)$$ E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) is that M is a $$C^1$$ C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) , then M must be $$C^1$$ C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$ E ( M ) and the topological skeleton of $$M^c$$ M c .

Multibody Dynamics on Differentiable Manifolds

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.

Periodic Perturbations of a Class of Functional Differential Equations

We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.