Trajectory Planning in Task Space for General Manipulators

The time-optimal path following (OPF) problem is to find a time evolution along a prescribed path in task space with shortest time duration. Numerical solution algorithms rely on an algorithm-specific (usually equidistant) sampling of the path parameter. This does not account for the dynamics in joint space, that is, the actual motion of the robot, however. Moreover, a well-known problem is that large joint velocities are obtained when approaching singularities, even for slow task space motions. This can be avoided by a sampling in joint space, where the path parameter is replaced by the arc length. Such discretization in task space leads to an adaptive refinement according to the nonlinear forward kinematics and guarantees bounded joint velocities. The adaptive refinement is also beneficial for the numerical solution of the problem. It is shown that this yields trajectories with improved continuity compared to an equidistant sampling. The OPF is reformulated as a second-order cone programming and solved numerically. The approach is demonstrated for a 6-DOF industrial robot following various paths in task space.

Kinematic redundancy of manipulators is a well-understood topic, and various methods were developed for the redundancy resolution in order to solve the inverse kinematics problem, at least for serial manipulators. An important question, with high practical relevance, is whether the inverse kinematics solution is cyclic, i.e., whether the redundancy solution leads to a closed path in joint space as a solution of a closed path in task space. This paper investigates the cyclicity property of two widely used redundancy resolution methods, namely the projected gradient method (PGM) and the augmented Jacobian method (AJM), by means of examples. Both methods determine solutions that minimize an objective function, and from an application point of view, the sensitivity of the methods on the initial configuration is crucial. Numerical results are reported for redundant serial robotic arms and for redundant parallel kinematic manipulators. While the AJM is known to be cyclic, it turns out that also the PGM exhibits cyclicity. However, only the PGM converges to the local optimum of the objective function when starting from an initial configuration of the cyclic trajectory.

The authors analyse the contour error dynamics in contouring control of dual‐arm robotic manipulators with holonomic constraints. With the modified dynamics of robot, the dynamics of dual‐arm robot with holonomic constraints is symetrically transformed into dynamics error problem by considering the equivalent error method. It becomes the control problem of stabilisation such that it is suitable for robust control approach to improve control performance in terms of contouring control, i.e. contour accuracy with high speed. The proposed method can deal with both analytic and non‐analytic functions of desired path in task space. The experimental results reveal that the proposed method yields excellent performance in comparison to conventional distributed control. As a result, the proposed method can lead to the development of a modified distributed control for high degree of freedom robot manipulators.

This work is concerned with the contouring control problem of robotic manipulators using the method of equivalent errors. The method was previously proposed for multi-axis motion systems with emphasis on the machine tools applications. It is shown in this paper that the method of equivalent errors is equally applicable to the robot manipulators. For robotic systems, a desired path is usually described in terms of the position of the end-effector (task space), but the system dynamics are described in generalized coordinates (joint space). The proposed method utilizes the forward kinematics to transform the desired path in the task space into that in the joint space. As such, the equivalent errors can be easily defined. A two-link robot manipulator contouring a circular path has been numerically studied. It is found that the contouring errors are virtually zero (order of fento meter), verifying the effectiveness of the method.

We introduce and examine the property of repeatability of inverse kinematics algorithms for mobile manipulators. Similarly to stationary manipulators, repeatability of mobile manipulators is defined by requiring that a closed path in the task space should be transformed by the inverse kinematics algorithm into a closed path in the configuration space. In a simply connected, singularity-free region of the task space, a necessary and sufficient condition for repeatability is derived as the integrability condition of a distribution associated with the inverse kinematics algorithm.

Repeatable, extended Jacobian inverse kinematics algorithm for mobile manipulators

In this paper, we address the path planning problem with general end-effector constraints (PPGEC) for robot manipulators. Two approaches are proposed. The first approach is adapted from an existing randomized gradient descent (RGD) method for closed-chain robots. The second approach is radically different. We call it ATACE alternate task-space and configuration-space exploration. Unlike the first approach which searches purely in C-space, ATACE works in both task space and C-space. It explores the task space for end-effector paths satisfying given constraints, and utilizes trajectory tracking technique(s) as a local planner(s) to track these paths in the configuration space. We have implemented both approaches and compare their relative performances in different scenarios. ATACE outperforms RGD in majority (but not all) of the scenarios. We outline intuitive explanations for the relative performances of these two approaches.

This paper concerns an approach for the inverse kinematic resolution of kinematically redundant manipulators when following a desired path. It can be called joint velocity compensation (JVC). Its principle is employing the redundancy to ensure uniform velocity or velocities with desired simple patterns for certain selected joints, while the remaining joints move with variable velocities as required by a given task. As many joints as the order of redundancy can have predefined velocities. From a control point of view it would be easier and more precise to maintain a uniform velocity for a joint rather than one that varies. Compared with other techniques, this approach has the advantages that it conserves the cyclic behaviour of a manipulator in the joint space for a closed path in the task space, less arithmetic operations are involved and the control of the manipulator as a whole becomes easier. Because of the physical and task constraints the choice of these selected velocities are not arbitrary. This paper elaborates the matter on how a desired velocity can be assigned to a joint such that all the constraints are respected and the task is performed as desired. Simulation results of the implementation of this approach to a redundant robot arm with 7-DOF are presented.

The process of connecting task descriptions originating from machine intelligence planning programs to the mechanization of feedback control of robot arms is analyzed. It is shown in this paper that control theories and practices can be extended to a higher level where feedback control of robot arms directly can respond to work space task commands provided that the work space task as a command is given in the form of a closed function of time. A general mathematical procedure using tools from differential geometry is introduced for synthesizing task space motion planning so that the planned motion can be used as a direct input to the robot arm feedback control system to achieve desired robot hand motion. By definition, "intelligent control" is being manifested through robot performance in the task space relative to task space commands. Thus, the capability of implementing feedback control of robot arms directly driven by appropriate task descriptions in the workspace as commands is a step toward intelligent control.

Given a smooth manifold M and a totally nonholonomic distribution $$\Delta \subset TM $$ of rank $$d\ge 3$$ , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map $$F\,{:}\,\gamma \mapsto \gamma (1)$$ defined on the space $$\Omega $$ of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy $$J\,{:}\,\Omega (y)\rightarrow \mathbb R$$ , where $$\Omega (y)=F^{-1}(y)$$ is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets $$\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}$$ . We call them homotopically invisible. It turns out that for $$d\ge 3$$ generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).

ANHOLONOMY IN THE CONTROL OF KINEMATICALLY REDUNDANT MECHANISMS WITH FLEXIBLE COMPONENTS

Delta Robots are used in industry for light weight material handling and sorting. This paper presents a sequence optimizing methodology for a color and size sorting delta robot. It finds the optimum path in the task space to perform an industry emulated scenario. An OpenCV-Python program was developed to sort objects according to their colors and sizes. The static positional coordinates of the objects in the robot workspace are obtained using the program. Genetic algorithm is used for pick-and-place sequence optimization to ensure that the sorting process is performed in the shortest possible path. The static positional coordinates are used to calculate the fitness. Single point crossover and mutation are applied with elitist selection when the current generation evolves to the next generation. The genetic algorithm ensures that the sequence of pick-and-place converges to the highest fitness in minimum number of generations reducing the computational time.

We analyse generic AdS flux backgrounds preserving eight supercharges in $D=4$ and $D=5$ dimensions using exceptional generalised geometry. We show that they are described by a pair of globally defined, generalised structures, identical to those that appear for flat flux backgrounds but with different integrability conditions. We give a number of explicit examples of such "exceptional Sasaki-Einstein" backgrounds in type IIB supergravity and M-theory. In particular, we give the complete analysis of the generic AdS$_5$ M-theory backgrounds. We also briefly discuss the structure of the moduli space of solutions. In all cases, one structure defines a "generalised Reeb vector" that generates a Killing symmetry of the background corresponding to the R-symmetry of the dual field theory, and in addition encodes the generic contact structures that appear in the $D=4$ M-theory and $D=5$ type IIB cases. Finally, we investigate the relation between generalised structures and quantities in the dual field theory, showing that the central charge and R-charge of BPS wrapped-brane states are both encoded by the generalised Reeb vector, as well as discussing how volume minimisation (the dual of $a$- and $\mathcal{F}$-maximisation) is encoded.

This paper deals with the derivation of “first approximations” to the solutions of the Orr‐Sommerfeld equation which are uniformly valid in a full neighborhood of the critical point. To this order the theory is remarkably simple. The essential elements in the theory are all well known from the older heuristic theories but its general structure is substantially different. The uniform approximations are also vastly simpler than the composite approximations obtained recently by the method of matched asymptotic expansions.

Complex networks economy systems engineering in general synergetic structure