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Grasp Analysis As Linear Matrix Inequality Problems Li Han, Jeff C IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 2000 663 Grasp Analysis as Linear Matrix Inequality Problems Li Han, Jeff C. Trinkle, and Zexiang X. Li, Member, IEEE Abstract—Three fundamental problems in the study of grasping Of the remaining problems, the three of interest in this paper and dextrous manipulation with multifingered robotic hands are are the force closure problem, the force feasibility problem, and as follows. a) Given a robotic hand and a grasp characterized by a the force optimization problem, which have mainly been solved set of contact points and the associated contact models, determine if the grasp has force closure. b) Given a grasp along with robotic (with a handful exception discussed below) after conservatively hand kinematic structure and joint effort limit constraints, deter- linearizing the contact friction models. Our main objective here mine if the fingers are able to apply a specified resultant wrench is to develop efficient solution techniques for these fundamental on the object. c) Compute “optimal” contact forces if the answer to nonlinear problems in a unified mathematical framework devel- problem b) is affirmative. oped through the theories of linear matrix inequalities (LMIs) In this paper, based on an early result by Buss et al., which trans- forms the nonlinear friction cone constraints into positive definite- and convex programming. Informally, these problems can be ness constraints imposed on certainy symmetric matrices, we fur- stated as follows. ther cast the friction cone constraints into linear matrix inequali- 1) Force Closure Problem—Given the locations of the con- ties (LMIs) and formulate all three of the problems stated above tact points on the object and the hand, the corresponding as a set of convex optimization problems involving LMIs. The latter problems have been extensively studied in optimization and con- friction models, and the kinematic structure of the hand, trol communities. Currently highly efficient algorithms with poly- determine if every object load in can be balanced.1 nomial time complexity have been developed and made available. 2) Force Feasibility Problem—Given the locations of the We perform numerical studies to show the simplicity and efficiency contact points on the object and the hand, the corre- of the LMI formulation to the three grasp analysis problems. sponding friction models, the kinematic structure of the Index Terms—Convex programming, grasp analysis, force clo- hand, the actuator limits, and known external load on the sure, force optimization, friction cones, linear matrix inequalities. object and hand, determine if the load can be balanced. 3) Force Optimization Problem—Given a grasp force I. INTRODUCTION problem that has passed the feasibility test in item 2 above, determine the “optimal” actuator efforts and T HAS been recognized for some time that robotic systems corresponding contact forces. equipped with multifingered hands have great potential I These three problems will collectively be referred to as grasp for performing useful work in various environments. This analysis problems. One may note that these problems also arise recognition is evidenced by the hundreds of research papers in the study of foot-step planning and force distribution by (see [1]–[12] and references therein for further details) on grasp multilegged robots [14]. Other applications of these problems analysis, synthesis, control, design, and related topics and the can be found in fixturing, cell manipulation by multiple laser large number of mechanical hands built for both robotic and probes, and the control of satellites with multiple unidirectional prosthetic research. Despite the huge effort, many unsolved thrusters. As for grasp synthesis problems which address theoretical and practical problems remain. how to generate grasps of certain desired properties, several approaches based on grasp force properties such as force Manuscript received December 17, 1998; revised May 15, 2000. This paper closure and optimal forces have been proposed. Therefore, the was recommended for publication by Associate Editor A. Bicchi and Editor A. solution techniques to grasp analysis problems discussed in this De Luca upon evaluation of the reviewers’ comments. This work was supported paper can also be applied to grasp synthesis and other relevant in part by the National Science Foundation under Grant IRI-9619850, the Texas Advanced Research Program under Grant 999903-078, the Texas Advanced applications. Technology Program under Grant 999903-095, Sandia National Laboratories, and the Research Grants Council of Hong Kong under Grant HKUST6221/99E A. Related Previous Work and Grant CRC98/01.EG02. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of The major difficulty associated with the three grasp analysis Energy under Contract DE-AC04-94AL85000. This paper was presented in part at the 1999 International Conference on Robotics and Automation, Detroit, MI, problems has been the nonlinearity of the commonly accepted May 1999. contact friction models: point contact with friction (PCWF) and L. Han is with Institute for Complex Engineered Systems, Department of soft-finger contact (SFC). The quadratic nature of both models Electrical and Computer Engineering , Carnegie Mellon University, Pittsburgh, PA 15213-3890 USA (e-mail: [email protected]). has been experimentally verified [2], [5] for some common J. C. Trinkle is with Sandia National Laboratory, Albuquerque, NM materials. For the force closure problem, there exist theorems 87185-1004 USA (e-mail: [email protected]), on leave from the Department [15], [8], [7] for general grasps, which consist of arbitrary of Computer Science, Texas A&M University, College Station, TX 77843 USA. numbers and types of contact points. Due to the difficulty Z. X. Li is with Department of Electrical and Electronic Engineering, HongKong University of Science and Technology, Clear Water Bay, Kowloon, 1There exist other definitions of force closure, e.g., the one without taking the HongKong (e-mail: [email protected]). hand structure into account [7], which was adopted in our earlier publication Publisher Item Identifier S 1042-296X(00)11650-7. [13] and handled similarly as the one in this paper. 1042–296X/00$10.00 © 2000 IEEE 664 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 16, NO. 6, DECEMBER 2000 of handling the nonlinear models, the force closure theorems B. Our Results [8], [11], have been specialized for the grasps characterized In this paper, based on the BHM observation and a detailed by the number of contact points and the associated contact analysis of the structure of the symmetric positive definite ma- models and expressed in geometric terms such as antipodal trices arising from the friction cone constraints, we cast the fric- positions. Even with these specialized theorems, the analysis tion cone constraints into LMIs and formulate the basic grasp and synthesis of frictional force closure grasps has mainly been analysis problems as a set of convex optimization problems in- studied by linearizing the friction cone constraints and then volving LMIs [24]. The latter problems have been extensively applying linear programming techniques. Similar approaches studied in optimization and control communities. Recently the [14], [3], [4] have also prevailed in the study of grasp force efficient algorithms with polynomial time complexity [25], [24] feasibility and optimization problems. have been developed and made available. We used these algo- While simplifying the three grasp analysis problems, the lin- rithms to perform numerical studies that showed the simplicity earized model and linear programming approach have the fol- and efficiency of the LMI formulation to the three grasp anal- lowing disadvantages. 1) The friction cone must be approxi- ysis problems. mated conservatively, to avoid the possibility of finding solu- tions that satisfy the linearized model, but violate the nonlinear II. PROBLEM REVIEW model (false positive). Unfortunately, a conservative lineariza- tion, may cause the linear analysis to yield false negative re- Consider an object grasped by a multifingered robotic hand sults, (e.g., the linear model implies no force closure, when it with contacts between the object and the links of the fingers exists). 2) The orientations of the tangent plane directions in and the palm. The grasp map, , transforms applied the contact frame affect the results of grasp analysis, which vi- finger forces expressed in local contact frames to resultant ob- olates the usual assumption of isotropic Coulomb friction. 3) ject wrenches Small perturbations in the parameters describing the grasp (ge- (1) ometry, physics, and kinematics) can produce large variations in the solutions of the linear programs. The nonsmoothness of where is the contact wrench of solutions of linear programming methods [16] poses difficulty the grasp, and is the independent wrench intensity for optimization-based grasp synthesis and real-time control ap- vector of finger .2 In order for the grasp to be maintained, the plications. (4) Increasing the number of facets in the linearized resultant generalized contact force must balance the external friction models will increase the running time unacceptably for (possibly dynamic) load experienced by the object. Thus we real-time applications. have The problems just discussed, can be alleviated to a large extent by retaining the nonlinear
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