Fast on-line determination of quasi-optimal grasp configurations based on off-line analysis and parametrization of the wrist position

Fast on-line determination of quasi-optimal grasp configurations based on off-line analysis and parametrization on the wrist position Gerrit Färber, Raúl Suárez IOC-DT-P-2005-1 Gener 2005 Fast on-line determination of quasi-optimal grasp configurations based on off-line analysis and parametrization of the wrist position* Gerrit Färber Raúl Suárez Institut d´Organització i Control de Sistemes Industrials (IOC) Universitat Politècnica de Catalunya (UPC) — Abstract — Determining the optimum configuration of a mechanical hand in order to grasp an object is a computational hard work, mainly due to the large number of degrees of freedom (fingers and wrist), the existence of a large number of solutions, and the constraints imposed by the object or the task to be done. The paper presents an approach to solve this problem under certain conditions, which is then particularized for an anthropomorphic mechanical hand involving 22 degrees of freedom. The configurations of the hand to optimally grasp rectangular parallelepipeds of different sizes with a planar grasp (contact points on the same plane) are determined off-line from the results the wrist position is analyzed and stored as approximated functions of the height and width of the grasped object. This information is used for a fast on-line computation of the hand configuration in a real grasp operation, given the rectangular parallelepiped bounding-box of the part of the object that will be "inside" the hand after the grasp. The approach has been implemented for the mechanical hand MA-I and the paper includes numerical examples. * This work was partially supported by the CICYT projects DPI2004-03104 and DPI2002-03540. 1 TABLE OF CONTENTS I. INTRODUCTION............................................................................................................1 II. ASSUMPTIONS AND APPROACH OVERVIEW............................................2 III. DESCRIPTION OF THE HAND...............................................................................4 IV. DETAILED PROCEDURE..........................................................................................5 A. INITIAL POSITION OF PT-C..........................................................................................................6 B. INVERSE KINEMATICS OF MIDDLE FINGER AND PALM..........................................................7 C. ORIENTATION OF THE WRIST....................................................................................................7 D. INVERSE KINEMATICS OF THUMB ............................................................................................7 E. ORIENTATION OF THUMB..........................................................................................................8 F. ANGLES IN THE FINGERTIPS OF THE MIDDLE FINGER AND THE THUMB .............................8 G. POSITION OF HAND-BASE..........................................................................................................8 H. CONFIGURATION OF INDEX FINGER AND RING FINGER ........................................................9 I. DEFINITION OF A QUALITY MEASURE .....................................................................................9 J. SEARCH FOR AN OPTIMAL CONFIGURATION ........................................................................10 K. PARTITION OF THE WIDTH-HEIGHT SPACE............................................................................10 L. APPROXIMATION OF X, Y AND Z COORDINATES WITH SECOND ORDER SURFACES.......10 V. NUMERICAL RESULTS FOR THE MECHANICAL HAND MA-I ......11 VI. CONCLUSION................................................................................................................12 VII. REFERENCES................................................................................................................13 I. INTRODUCTION The grasp of an object, performed by a mechanical hand, is subject to several considerations like dexterity, equilibrium, stability and dynamic behavior [1]. The quality of the grasp regarding these topics depends on the grasp configuration of the mechanical hand. The use of anthropomorphic mechanical hands has the advantage of allowing redundant solutions, this, however, has the drawback that infinite solutions are possible and the determination of a “good”, if not “optimal”, solution is a computational hard work. Although there are works that, in one way or another, consider the hand kinematics in the grasp computation (for instance [2], [3]), quite frequently, the grasping problem is addressed focused only on the object to be grasped without considering the mechanical hand (for instance [4], [5], [6], [7], among several others). This may result in an excellent set of contact points for the fingers, ensuring a robust grasp, 2 but due to the limitations imposed by the kinematics of a given hand, the grasp may be impossible to be performed. Other works deal with the problem of positioning the hand given the set of grasping contact points, which is equivalent to the inverse kinematics problem. Nevertheless, since the hand kinematics was not considered in the search of the grasp contact points there may be long iterative processes generating, first, a set of contact points for the fingers and, then, trying to find an actual hand configuration to perform the grasp. This may be a time consuming approach, so learning approaches were proposed to do it faster during a real task execution. As an example, the use of neural nets is presented in [8], but strong assumptions are considered to provide robustness to the system, like for instance that the initial orientation of the wrist in the solution search is smaller than 90º (so the hand is already “well” oriented) and the position search space for the wrist position is relatively small. Finding a proper wrist position and orientation is a key point because with this information solving the inverse kinematics of an anthropomorphic mechanical hand is relatively simple. In this context, the work in this paper proposes a new approach to quickly generate a hand configuration, close to the optimum, during on-line real operation. As in the traditional learning approaches, a computational hard work is done off-line, but the difference is that in this approach the knowledge is captured as explicit functions that provide the position of the wrist given the height and width of the object to be grasped. With this information, the online-determination of the whole hand configuration can be done in a simple way. The approach was implemented for the anthropomorphic mechanical hand MA-I developed and built at the Robotics Lab of the IOC. Numerical examples are included in the paper. II. ASSUMPTIONS AND APPROACH OVERVIEW The following assumptions are initially considered in this work (some of them can be easily relaxed to obtain variations of the approach): The contacts between the fingers and the object is a point (the called “hard contact” in [4]), but the fingers can contact the object in different points of the fingertips (note that in the general case this adds 2 “virtual” degrees of freedom per finger to the problem). The fingertips are assumed to be spherical. The hand is anthropomorphic and will perform the most common planar grasp, so the contact points are in the same plane and the thumb will push against the other fingers. For simplicity and without loss of generality, the grasp plane is assumed to be horizontal. The object is considered to be a rectangular parallelepiped, which in any case is a simple bounding box for a real object. This consideration is relevant because the grasp must avoid the collision of the object with the palm and the fingers of the hand. Figure 1 illustrates these considerations showing a simplified schematic of the hand grasping the object. Finally, the optimization criterion for the hand configuration is that all the joints of the hand are as close to the middle of their ranges as possible. 3 In order to provide an overview of the proposed approach we will describe it according to the following steps. Steps to be done off-line: 1. Determination of the optimum configuration of the hand (wrist and fingers) for objects of different height and width. This is done once for a finite number of pairs [height, width] using a search procedure. The starting position of the wrist is determined with a heuristics in order to accelerate the process. The search tries to minimize the distance from the joint positions to the middle of their ranges. 2. Analysis of the wrist positions, which are approximated with simple functions of the object’s height and width. The X, Y and Z-coordinates of the wrist positions are then approximated with a few second order surfaces over proper regions in the domain defined by the variables [height, width]. These functions are stored for their on-line use. Steps to be done on-line: 1. Determination of the object size (or proper bounding box) determining the length of the object that the grasp must be able to accept inside the hand (object height), and the distance between the main planes where the fingers will touch (object width). 2. Identification of the domain containing the pair [height, width] corresponding to the actual object and apply the three corresponding second order functions to obtain the X, Y and Z-coordinates of the wrist position. 3. Solution of the inverse kinematics of the fingers and the orientation of the palm for the wrist position obtained in the previous step. Figure 1. Schematic of the hand grasping an object defined by its height and width (only two fingers are shown). The inset shows the 2 virtual articulations in the fingertips. 4 This approach presents a good balance between the amount of information that has to be stored and the amount of computation that has to be done on-line, being both of them relatively small and quite acceptable. Moreover, from the research point of view, one particular advantage of the approach is that the information obtained offline explicitly shows the behavior of the hand when an optimal grasp is performed, allowing a better understanding of the grasp action and opening new opportunities for further improvements. This is not possible with other techniques, where the knowledge is implicitly stored, like, for instance, in neural nets. The rest of the paper deals with the off-line steps of the approach, using the anthropomorphic mechanical hand MA-I (described below) to introduce basic nomenclature and facilitate explanations. The on-line steps are straightforward and therefore are not discussed in the paper. III. DESCRIPTION OF THE HAND The hand MA-I [9] consists of a palm and four fingers: thumb, index-, middle- and ring-finger, and it is assembled with the wrist of a robot arm with six degrees of freedom (d.o.f.) (Figure 2). The following basic nomenclature for the subindices is used along the paper: H Hand F Finger (to be specified as: T, I, M, R) T Thumb I Index finger M Middle finger R Ring finger Furthermore, the following nomenclature is also used (see Figure 1) (positions are defined in a world reference system unless indicated differently): PF-O Position of contact point finger-object PiF-S Position of the base of articulation i of finger F PF-S Position of the centre of the sphere in the last phalanx (fingertip) of finger F (P10F-S) PF-C Position of the connection finger-palm PH-B Position of the hand-base Θ F-i Angle of articulation i of Finger F DTS-MS Distance between PT-S and PM-S. DTC-MS Distance between PT-C and PM-S. DTC-TS Distance between PT-C and PT-S. XF,YF, ZF Position of PF-S with Θ H-4 to Θ H-6 equal to 0. ( Local position of PF-S with respect to PH-B) In order for the explanation to be independent of the arm manipulator, it will be considered that the position, PH-B, of the hand is determined by three parameters XH-B, YH-B and ZH-B and the orientation of the hand is determined by three virtual joints Θ H-4, Θ H-5 and Θ H-6. The values of XH-B, YH-B, ZH-B and Θ H-4, Θ H-5 and Θ H-6 describe the configuration of the robot wrist and are obtained from the six real joints of the robot arm. The articulations Θ F-7 to Θ F-10 define the angles of the rigid 5 phalanges of the fingers ( Θ F-7 being the abduction-adduction d.o.f.). The articulations Θ F-11 and Θ F-12 are virtual articulations that describe the position of the contact point on the spherical fingertip of each finger (Figure 1). The kinematics chain from PT-O to PM-O can be divided into two individual chains. The first chain is formed by the middle finger together with the palm and the second chain is formed by the thumb. The common point of the two chains is PT-C, where the thumb is connected with the palm. PT-C serves as an auxiliary reference-point of the hand. IV. DETAILED PROCEDURE The detailed description of the proposed off-line procedure is done according to the following steps: 1. Determination of the optimum configuration of the hand (wrist and fingers) for objects of different height and width. For each pair [height, width] within the desired range (in our case: 200x200 mm) the following actions are done. 1.1. Determination of the optimum configuration of the hand considering only the palm, the middle finger and the thumb. Includes the determination of: A. Initial position of PT-C (Heuristic method). B. Inverse kinematics of middle finger and palm. C. Orientation of the wrist. D. Inverse kinematics of thumb. E. Orientation of thumb (angles Θ T-7 and Θ T-8) F. Angles in fingertips of middle finger and thumb (angles Θ F-11 and Θ F-12) G. Position of hand-base PH-B. H. Configuration of Index finger and ring finger. Figure 2. Mechanical Hand MA-I. 6 1.2. Application of a guided search for the ´optimum´ position of PT-C. Includes: I. Definition of a quality measure. J. Search for an optimal configuration. 2. Analysis of the position of the wrist and approximation of the palm position with second order surfaces: K. Partition of the width-height space in approximation regions for the X, Y and Z-coordinate of PT-C. L. Approximation of the solution space with second order surfaces. The following subsections deals with each action from A to L particularized, when necessary, for the hand MA-I (additional details about the derivations of equations can be found in [10]). A. Initial position of PT-C. The point PT-C fixes the position of the hand. The initial point PT-C must ensure that the distances DTC-MS and DTC-TS do not exceed their maximum possible value. The usefulness of knowing this point in advance is the reduction of the computing time in the subsequent search. The positions of PM-O and PT-O can be determined from the object width, and, from them, PM-S and PT-S can be directly obtained (Figure 1). Using a simple heuristic criterion, PT-C is initially determined by the intersection of two circles of radius DTC-MS and DTC-TS centered on PM-S and PT-S, respectively (Figure 3). Both circles lie on a plane, orthogonal to the grasp plane, containing PM-S and PT-S. It must be noted that this is an initial value for PT-C, and since the ranges of the joints were not yet considered, it may produce a solution of the inverse kinematics of the fingers with some joints outside the reachable ranges (in particular, Θ T-7 is likely to be out of range). Figure 3. Determination of the initial position of the hand (PT-C). 7 B. Inverse kinematics of middle finger and palm. In order to simplify the solution it is considered that for the middle finger Θ M-7 = 0, while for the other fingers Θ T-7, Θ I-7 and Θ R-7 have to be properly determined. The configuration of the palm and the fingers can be determined as a function of the distances between PT-C and the points PT-S and PM-S. The finger joints Θ M-8, Θ M-9 and Θ M-10 are coplanar and therefore infinite solutions are possible. In order to determine a solution, it is initially assumed that Θ M-8 = Θ M-9 = Θ M-10 and the angle Θ M-8 is solved from the equation of the Euclidean distance between PT-C and PM-S . Due to the difficulty in inverting this function, it was approximated with a polynomial function ( Θ M-8 could also be obtained using the Newton’s iterative method, however, this results in a series of additional iterative calculations). Using the determined approximated value of Θ M-8, the values of Θ M-9 and Θ M-10 of the middle finger are calculated by solving them from the finger inverse kinematics using simple trigonometric replacements. C. Orientation of the wrist The next step is the determination of Θ H-4 and Θ H-5 according to the current location of PT-C such that the final position of PM-S is located according to the object requirements. The known data are PM-S, PT-C and PM-S| Θ H-4= Θ H-5=0. Then Θ H-4 and Θ H-5 are obtained solving the system: () ( ) 45 45 0 HH HH M STC MS TC PP RRP P −− −− −− ΘΘ −Θ=Θ=− −=⋅⋅ − where 4H R− Θand 5H R− Θare the rotation matrices for the rotating axis of Θ H-4 and Θ H-5. D. Inverse kinematics of thumb The three angles of the thumb Θ T-8, Θ T-9 and Θ T-10 are computed in the same way as in the case of the middle finger to agree with the distance DTC-TS, but without considering the actual orientation of the thumb with respect to the palm. This orientation will be considered later to recalculated Θ T-8, while the obtained Θ T-9 and Θ T-10 are final values. Figure 4. Schematics showing a) the two planes used for the determination of Θ T-7; b) the determination of Θ T-8. 14 [6] J.W. Li, H. Liu and H.G. Cai, 2003, "On Computing Three-Finger Force- Closure Grasps of 2-D and 3-D Objects", Transactions on Robotics and Automation., 19(1), pp.155-161. [7] X. Zhu and J. Wang, 2003, "Synthesis of Force-Closure Grasps on 3-D Objects Based on the Q Distance", Transactions on Robotics and Automation, 19(4), pp.669-679. [8] P.Gorce and N.Rezzoug, 2003, "A method to learn hand grasping posture from noisy sensing information", Robotic, 22, pp.309-318 [9] R. Suárez and P. Grosch, 2004, "Dexterous Robotic Hand MA-I, Software and Hardware Architecture", Intelligent Manipulation and Grasping International Conference, IMG'04, (ISBN 88-900426-1-3), Genova, Italy, pp. 91-96. [10] G. Färber and R. Suárez, 2003, "Determination of the inverse kinematics of a mechanic hand in order to grasp a predetermined object with two or more fingers", Institut d’Organizació i Control de Sistemes Industrials, Barcelona, Spain, Technical report IOC-DT-P-2003-11.