Jijie Xu

On quality functions for grasp synthesis, fixture planning, and coordinated manipulation

Planning a proper set of contact points on a given object/workpiece so as to satisfy a certain optimality criterion is a common problem in grasp synthesis for multifingered robotic hands and in fixture planning for manufacturing automation. In this paper, we formulate the grasp planning problem as optimization problems with respect to three grasp quality functions. The physical significance and properties of each quality function are explained, and computation of the corresponding gradient flows is provided. One noticeable property of some of these quality functions is that the optimal solutions are also force-closure grasps if they do exist for the given object. Furthermore, when specialized to two-fingered or three-fingered grasps on a spherical object, the optimal solutions become the familiar antipodal grasp, or the symmetric grasp, respectively. Thus, by following the gradient flows with arbitrary initial conditions, the optimal grasp synthesis problem is solved for objects with smooth geometries manipulated by hands with any number of fingers. Also, note that our solutions do not involve linearization of the friction cones. We discuss two simplified versions of these problems when real-time solutions are needed, e.g. coordinated manipulation of a robotic hand with contact points servoing. We give simulation and experimental results illustrating validity of the proposed approach for optimal grasp planning. Note to Practitioners: This paper presents three new quality functions for comparing and planning grasps and fixtures. These measures improve on the traditional measure of force closure. We propose a method for computing the optimal solutions of these functions, and a method for reducing their computation time through reasonable simplification/approximation. Preliminary experiments with a three-fingered robotic hand demonstrate that the proposed functions can be used to optimize the grasp quality during manipulation/manufacturing, and keep the optimal grasp configuration once it is reached. However, we only obtain the local optimal solutions for the functions without simplification except for some special cases. We also assume that the object/workpiece is ideally rigid in all three functions. In future research, we will improve these limitations through a compliance model.

On geometric algorithms for real-time grasping force optimization

Grasping force optimization with nonlinear friction constraints is a fundamental problem in dextrous manipulation with multifingered robotic hands. Over the last few years, by transforming the problem into convex optimization problems on Riemannian manifolds of symmetric and positive definite matrices, significant advances have been achieved in this area. Five promising algorithms: two gradient algorithms, two Newton algorithms, and one interior point algorithm have been proposed for real-time solutions of the problem. In this paper, we present in a unified geometric framework, the derivation of these five algorithms and the selection of step sizes for each algorithm. Using the geometric structure of the affine-scaling vector fields associated with the optimization problem, we prove that some of these algorithms have quadratic convergence properties, and their continuous versions are exponentially convergent. We evaluate the performance of these algorithms through simulation and experimental studies with the Hong Kong University of Science and Technology (HKUST) three-fingered hand. This study will facilitate selection and implementation of grasping force optimization algorithms for similar applications.

A general approach for optimal kinematic design of parallel manipulators

This paper deals with the problem of optimal geometry design of parallel manipulators. In order to reduce the main drawbacks of parallel manipulators, relatively small workspace and more singularities, two requirements, workspace and condition number, are considered. The design problem is thus formulated to find a parallel mechanism such that its Cartesian workspace contains a prescribed workspaces with good condition numbers in it. By observing that those requirements can be locally cast into Linear Matrix Inequalities (LMIs), we formulate the design problem locally as a convex optimization problem subject to LMIs with a max-det function as its objective function. Hence, at each node of discretized space of design parameters, there is an LMI-based convex optimization problem. A two-level algorithm can be applied to solve for a set of optimal design parameters: (1) Discretize the space of design parameters into a set of discrete nodes; (2) At each node the Newton algorithm is applied to solve the max-det optimization problem. By comparing all the locally optimal costs, we can obtain a corresponding set of globally optimal design parameters correspondingly. Simulation results verify the effectiveness of the proposed approach.

Force Analysis of Whole Hand Grasp by Multifingered Robotic Hand

Under a whole hand grasp, it may not be possible to generate grasping forces in all directions. Thus, the traditional techniques developed based on fingertip contacts is inadequate. In this paper, we decompose the contact force space into four orthogonal subspaces, each with a clear physical interpretation. Based on linear matrix inequalities (LMIs) representations of grasping constraints, we address and formulate the active force closure and the active grasp feasibility problems as LMI feasibility problems. Combining the effects of both active and passive forces, we propose a new cost index for the whole hand grasping force optimization problem. We further simply the force optimization problem for a whole hand grasp, which is active force closure.

On quality functions for grasp synthesis and fixture planning

Planning a proper set of contact points on a given object/workpiece so as to satisfy a certain optimality criterion is a common problem in grasp synthesis for multi-fingered robotic hands and in fixture planning for manufacturing automation. We formulate the grasp-planning problem as optimization problems with respect to several grasp quality functions. For real-time computation, a simplified min-analytic-center problem is proposed. Simulation and experimental results illustrate the validity of the proposed approach for optimal grasp planning.

Automatic real-time grasping force determination for multifingered manipulation: theory and experiments

This article deals with the problem of real-time grasping force optimization for multifingered manipulation. Based on a review of existing approaches, the BHM and the HTL algorithms, a need for a strictly possible initial solution is found to be a common problem. Two approaches, the max-det approach and min-max approach, are proposed to resolve this problem. The first approach, although efficient in most cases, suffers from the problem of singularity. The latter can resolve the singularity problem, but is relatively slow compared with the former. The two approaches are combined in real implementations to efficiently compute a strictly initial solution, which is in turn used in the HTL algorithm (also the BHM algorithm). The whole algorithm is shown to be fast, fully automatic, and applicable to a wide class of manipulation tasks irrespective of the number of fingers and also the geometry of the manipulated objects. Experiments on the HKUST hand demonstrate the convergence and speed of the algorithm.

Development and implementation of NURBS interpolator with look-ahead technique

With recent advances in high accuracy and high speed machining, the NURBS interpolator has shown significant effect on dealing with the free form curves and surfaces. The present study aims at developing the speed-controlled interpolator and implementing the real-time hardware. However, the system vibration at sharp corner is unavoidable, it reduces the tracking accuracy. This paper proposes and implements a NURBS interpolation algorithm with look-ahead technique to generate smooth trajectory command under the corner error constraints, the average feedrate is increased and so as to shorten the motion time. Experimental results indicate that the proposed NURBS interpolation algorithm is able to provide a satisfactory performance.

Two-Degree-of-Freedom Based Cross-Coupled Control for High-Accuracy Tracking Systems

Recent work recognizes that cross-coupled control (CCC) can significantly improve the accuracy of contour tracking in biaxial systems. However, its complicated to apply CCC to arbitrary contour because of extra requirements to calculation and switching the cross-coupled gains. In addition, since most of CCC are based on the PID controlled loops of the individual axes, performances are conserved in some sense. In this paper, we propose a structure for arbitrary regular contours by efficiently determining the cross-coupling gains for CCC with the two-degree-of-freedom (2DOF) controlled to the single axes. Furthermore, an approach for stability analysis of the CCC is posed. Experimental results for a two-axial motion system indicate that the proposed structure eliminates the contouring error significantly.

Kinematic synthesis of parallel manipulators: a Lie theoretic approach

This paper provided a unified geometric framework for kinematic analysis and synthesis of parallel manipulators. We gave a strict definition on motion types of a mechanism based on distributions on a Lie group. We derived conditions for parallel manipulators with Lie subgroup motions using the intersection of the permissible velocity spaces, or the direct sum of the constraint force spaces of each subchain, and the integration theory on a Lie group. Several practical examples were studied in detail to verify our approach.

A study on geometric algorithms for real-time grasping force optimization

In this paper we propose several strategies for selecting such a step size according to the properties of each algorithm and a method for searching a valid initial point. By investigating the structure of the affine-scaling vector fields associated with the optimization problem, we give a detailed convergence analysis of these algorithms. Simulation and experimental results show the different performance of these algorithms from computation time and convergence rates.