On Constraint Manifolds of Lorentz Sphere

The expression of the structure equation of a mechanism is significant to present the last position of the mechanism. Moreover, in order to attain the constraint manifold of a chain, we need to constitute the structure equation. In this paper, we determine the structure equations and the constraint manifolds of a spherical open-chain in the Lorentz space. The structure equations of spherical open chain with reference to the causal character of the first link are obtained. Later, the constraint manifolds of the mechanism are determined by means of these equations. The geometric constructions corresponding to these manifolds are studied.

Robust approximate constraint-following control design for permanent magnet linear motor and experimental validation

A robust approximate constraint-following control method based on the fundamental equation of Udwadia–Kalaba approach is designed to control the permanent magnet linear motor. The control scheme consists of three parts: the nominal part to suppress any tendency of deviating from the constraints, the second part to deal with any possible initial condition deviation from the constraint manifold and drive the system toward the constraints, and the robust part to compensate for the effect due to uncertainty. Theoretical proof, numerical simulation, and experimental validation have been done to verify the effectiveness of this control method.

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.

Inverse kinematics-based motion planning for dual-arm robot with orientation constraints

This article proposes an efficient and probabilistic complete planning algorithm to address motion planning problem involving orientation constraints for decoupled dual-arm robots. The algorithm is to combine sampling-based planning method with analytical inverse kinematic calculation, which randomly samples constraint-satisfying configurations on the constraint manifold using the analytical inverse kinematic solver and incrementally connects them to the motion paths in joint space. As the analytical inverse kinematic solver is applied to calculate constraint-satisfying joint configurations, the proposed algorithm is characterized by its efficiency and accuracy. We have demonstrated the effectiveness of our approach on the Willow Garage’s PR2 simulation platform by generating trajectory across a wide range of orientation-constrained scenarios for dual-arm manipulation.

A Double-Layered Artificial Delay-Based Approach for Maneuvering Control of Planar Snake Robots

Uncertainty and disturbance are common in a planar snake robot model due to its structural complexity and variation in system parameters. To achieve efficient head angle and velocity tracking with least computational complexity and unknown uncertainty bounds, a time-delayed control (TDC) scheme has been presented in this paper. A Serpenoid gait function is being tracked by the joint angles utilizing virtual holonomic constraints (VHCs) method. The first layer of TDC has been proposed for stabilizing the VHC dynamics to the origin. Once the VHCs are satisfied, the system is said to be on the constraint manifold. The second layer of TDC has been applied to an output system defined over the reduced order dynamics on the constrained manifold. To establish the robustness of the control approach through simulation, uncertainty in the friction coefficients is considered to be time-varying emulating change in the ground conditions. Simulation results and Lyapunov stability analysis affirm the uniformly ultimately bounded stability of the robot employing the proposed approach.

Singularity Variety of a 3SPS-1S Spherical Parallel Manipulator

In this paper, we study the manifold of configurations of a 3SPS-1S spherical parallel manipulator. This manifold is obtained as the intersection of quadrics in the hypersphere defined by quaternion coordinates and is called its constraint manifold. We then formulate Jacobian for this manipulator and consider its singular. This is a quartic algebraic manifold called the singularity variety of the parallel manipulator. A survey of the architectures that can be defined for the 3SPS-1S spherical parallel manipulators yield a number of special cases, in particular the architectures with coincident base or moving pivots yields singularity varieties that factor into two quadric surfaces.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Holonomic Constraints

A method is presented for formulating and numerically integrating ordinary differential equations (ODEs) of motion for holonomically constrained multibody systems. Tangent space coordinates are defined as independent generalized coordinates that serve as state variables in the formulation, yielding ODEs of motion. Orthogonal dependent coordinates are used to enforce kinematic constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for redefining local coordinates on the constraint manifold, as needed, transparent to the user and at minimal computational cost. The formulation is developed for holonomically constrained multibody models that are based on essentially any form of generalized coordinates. A spinning top with Euler parameter orientation coordinates is used as a model problem to analytically reduce Euler's equations of motion to ODEs. Numerical results using a fourth-order Nystrom integrator verify that accurate results are obtained, satisfying position, velocity, and acceleration constraints to computer precision. A computational algorithm for implementing the approach with state-of-the-art explicit numerical integrators is presented and used in solution of three examples, one planar and two spatial. Performance of the method in satisfying all three forms of kinematic constraint, based on error tolerances embedded in the formulation, is verified.