A Criteria-Based Approach to Grasp Synthesis

A Criteria-Based Approach to Grasp Synthesis R. D. Hester, M. Cetin, C. Kapoor and D. Tesar Robotics Research Group The University of Texas at Austin Mail Code R9925, Austin, TX 78712-1100 Recently, NASA-JSC1 designed a robotic hand to be used as the end-effector of the Robonaut, a dual arm robot to be used in space station extra-vehicular activities. The Robonaut Hand [13] is a 12-degrees of freedom (dof), five-fingered, anthropomorphic grasping device based on the 95th percentile man's hand. The Robonaut Hand design includes a forearm, wrist with pitch and yaw capability, palm, palm dof, two primary fingers, two secondary fingers and a thumb finger (See Figure 1). Robotics Research Group (RRG) at UT Austin was asked to study the grasp synthesis problem for the NASA-JSC hand. This paper presents the approach developed and applied for the grasp synthesis problem of the NASA-JSC hand. In Section 2, the common definitions used in grasp synthesis are introduced and the previous studies in grasp planning are surveyed with an emphasis on analytical approaches. In Section 3, the approach to the grasp synthesis problem is explained. Although the approach presented was applied only to the NASA-JSC hand, it is general and should apply to any arbitrary dof robotic hand. In Section 4, a case study is introduced and the results are discussed. Section 5 contains the summary and conclusions for future work. Abstract This study introduces a criteria-based methodology for grasp synthesis. The method allows using multiple performance criteria both at the finger and the hand levels, which are used to generate a preliminary grasp and an optimum grasp, respectively. The approach offers reduced complexity by decomposing grasp synthesis into manageable phases. A case study using a physical hand demonstrates the effectiveness of the method. 1.0 Introduction Grasp synthesis can be defined as constructing a set of contact locations and a corresponding hand configuration for a given set of task attributes associated with a target object. Grasp synthesis, also known as grasp planning, has enjoyed the attention of researchers. The approaches to grasp planning can be classified as empirical and analytical depending on the procedure used to construct a target grasp. The viability of these approaches has been tested in a number of cases with the physical hands designed and popularized in grasp research [11],[14]. 2.0 Preliminaries The study of grasps can be categorized into two broad groups. Empirical approaches mimic human grasping by using heuristics to select one of a set of standard hand shapes that best conforms to task requirements and the target object geometry [4][8][12]. Analytical approaches, on the other hand, determine the contact locations on the object and the hand configuration that satisfy task requirements through kinematic and dynamic formulations. The method discussed in this paper is an analytical approach and the remainder of this section reviews analytical approaches to grasping. 2.1 Definitions This section introduces various definitions used in the rest of the paper. Most of these definitions can Figure 1 NASA-JSC Robonaut Hand 1 National Aeronautics and Space Administration – Johnson Space Center 1 be found in [6]. The following assumptions are made for the grasp synthesis approach 1. The grasp is limited to a precision (fingertip) grasp 2. The object and finger links are rigid bodies. 3. Accurate models of the object and the hand exist. 4. Relative positioning of the wrist frame with respect to the object frame is available and fixed. 5. No rolling or slipping occurs at the contact points. Task objectives must be defined to guide the grasp synthesis process. The expected loads on the object can be characterized by a task wrench space { 2 2   ν 2 ν y ν 2 ω2 ωy ω2 B = (ν x ,L,ω z ) ∈ R 6 , x2 + 2 + z2 + 2x + 2 + 2z ≤ β  δ1 δ 2 δ 3 δ 4 δ 5 δ 6   0  U1  δ 1   M  ≡ U∆U T ,U ∈ R 6 , δ ≥ 0 B = [U1 , L,U 6 ] O i i   δ 6  U 6   0 (1) where Ji is the Jacobian of the ith finger. The grip map G describes the resultant force on the object due to forces at the contacts  f c1    Fo = G  M , G = [G1 L G n ]  fc   n (3) where G + is the transpose of the generalized inverse of the grip map. The contacts between the fingertips and object are assumed point contacts with friction. The friction cone FC represents the forces at a contact. T { } f12 + f 22 ≤ µf 3 , f 3 ≥ 0 (4) Many grasp analysis methods utilize a linearized form of the friction cone µ µ   LFC ci =  f ∈ R 3 : f 12 ≤ f 3 , f 22 ≤ f 3 , f 3 ≥ 0 2 2   (8) 2.2 Previous Work The computational intensity of grasp synthesis stems from two factors. Analysis of any particular grasp is complex and the solution space of feasible grasps is typically very large. Shimoga [11] surveyed current grasp synthesis algorithms available in literature and categorized the popular grasp properties into five basic classes: force closure, equilibrium, dexterity, stability and dynamic response. The methods that were reviewed focus on the analysis of an arbitrary grasp rather than the grasp synthesis. Force closure, the ability to resist arbitrary loads, is a necessary but not a sufficient condition for an admissible grasp. Ferrari [2] and Pollard [10] use convexity theory to verify force closure and calculate a “grasp quality” factor, GQ. This “grasp quality” measure furnishes a means for comparing force closure grasps. GQ is a function of the contact locations on the object. The hand configuration need not be known. In addition to force closure, the grasp must satisfy the equilibrium requirements. External loads must be balanced without exceeding friction limits at the contacts or torque limits at the finger joints. Finding the contact forces that satisfy all the constraints is complex since the contact force solution is the sum of the forces that balance the external load and the internal grasping forces for which there is a multiplicity of solutions. Kerr [5], where Gi is the grip map of the i-th contact. The grasp matrix Gh defines the velocity of the object with respect to joint velocity and is a function of the grip map and hand Jacobian FC ci = f ∈ R 3 : (7) where δi is the relative importance ratio of motion in the direction Ui in the object frame. The twist ellipsoid is centered on the object frame origin. (2) T Vo = G hθ&, G h = G + J h (6) The task wrench space W defines the hyperplanes orthogonal to the wrench space axes that bound the set of task loads. Hsu, Li and Sastry [3] represent the task motion requirements with a task twist ellipsoid: A grasp is fully defined by the hand Jacobian, grip map and contact model. The hand Jacobian Jh describes the contact motion with respect to joint velocity 0 J 1  x& C = J h (θ )θ&, J h (θ ) =  O   J n   0 } W = ( f1 ,L,τ 3 ) ∈ R 6 , f i ≤ f i max , τ i ≤ τ i max , i = 1,2,3 (5) The linearized friction cone LFC defines the largest pyramid enclosed by the friction cone and independently constrains the friction terms. 2 Shimoga [11] formulate the solution for the contact forces as an optimization problem in which the internal grasping forces are minimized. These methods assume a single load condition and repetitive analysis is necessary for sets of external loads. Dexterity criteria provide a means to evaluate different grasps for a particular task. Shimoga [11] presents a list of dexterity measures that involve the grasp matrix Gh. Most of the reviewed criteria had been developed for redundant manipulators, but their extensions to multi-fingered hands were discussed. Stability, the ability to reject load disturbances, is not essential to grasp synthesis and therefore not included in this study. In fact, Nguyen [9] determined that any 3D force closure grasp could be made stable. Dynamic response, a subset of stability analysis, is also not considered. Hsu, Li and Sastry [3][7] present a method for grasp synthesis based on specifying the grasp map G and hand Jacobian Jh so that their performance measure is maximized. The task ellipsoids, grasp map and hand Jacobian are used to develop structured quality measures in the twist space and wrench space. The grasp analysis methods mentioned so far evaluate the whole grasp by examining the grasp matrix Gh or the grip map G, and the hand Jacobian Jh. This complicates grasp synthesis since it expands the solution space of feasible grasps by the product of the feasible finger solutions. We attempt to avoid this complexity by deconstructing the grasp analysis to the finger level and reducing the solution space to the sum of the feasible finger solutions. (internal) grasping forces. Therefore, force criteria at the finger level can evaluate the ability to generate forces at the contact, but provide no information on the overall effect on the object. A hand level criterion that is an extension of a finger level force criterion must be combined with a force criterion of the grip map G to provide complete information about the grasp. In this study, we propose a two-phase grasp synthesis methodology that incorporates the advantages of grasp deconstruction (Figure 2). The preliminary grasp phase constructs an initial grasp by selecting the optimum finger configurations based on finger level criteria. This initial grasp serves as the starting point for the second phase. The Grasp optimization phase then searches the grasp solution space for a locally optimum configuration. The objective function for this phase is composed of hand level criteria and GQ, which is a force criterion for the grip map. Environment Constraints Wrist and Object Position Hand Constraints DH Parameters Fwd Kinematics Hand Optimization Contact point, Contact normal Object Model Finger Level Criteria Initial Hand Configuration Preliminary Grasp Grasp Optimization Hand Level Contact Model Grasp Optimization 3.0 Criteria-based Grasp Planning Deconstructing grasp synthesis to the finger level provides two computational advantages: • Motion and force criteria use the finger Jacobian Ji and contact map Gi. • Grasp synthesis is reduced to finding the best configuration of each finger. The former advantage reduces computational intensity by using smaller matrices. More importantly, the latter advantage reduces the size of the search space from the product of the number of finger solutions to their sum. The rigid body assumptions allow the grasp to be fully deconstructed in the velocity domain. The desired object motion is mapped to motion at a contact through Gi. The finger motion criteria can then be evaluated individually using the contact motion requirements. The grasp cannot be deconstructed in the force domain since the forces at a contact are a combination of the external loads and the inter-finger Task Wrench/ Twist Space Task Requirements Contact Points, Hand Configuration Grasp Equilibrium Contact Points Hand Config. Joint Torques Figure 2 Two-Phase Grasp Synthesis In the rest of the section, we discuss the finger level criteria incorporated in this study, their extension to hand level criteria, and the grasp quality measure GQ. The implementation of these criteria in the preliminary grasp and grasp optimization phases is also presented. 3.1 Velocity Transmission 3 The velocity transmission ratio is a measure of how effectively the joint velocities of a manipulator are conveyed to end-effector velocity. This ratio has been defined in robotics literature [1][15] as follows: ( ( ) α = eiT JJ T −1 ei ) −1 / 2 point contacts with friction, are bounded by the friction cone FC. These directions are approximated using the vectors, Ufc,j (j=1,…,4), aligned with the four edges of the linearized friction pyramid LFC. The force transmission ratio for the direction Ufc,j is expressed as follows: (9) ( ( ( Α j = U Tj Gi J i J iT ) −1 GiTU j ) −1 / 2 ( ) B j = U Tfc , j JJ T U fc , j where ei is the direction of interest for velocity transmission in the manipulator base frame and J is the Jacobian matrix of the manipulator. In the case of grasp synthesis, the directions of interest are defined by the columns Uj (j=1…6) of the twist ellipsoid matrix U in (8). The object velocities in these directions can be transformed to velocities at a contact by the transpose of the contact map Gi. The velocity transmission ratio for the direction Uj is expressed as follows: ) −1 / 2 (14) The force transmission ratio for a finger is constructed from the sum of Bj2. 6 η ftr ,i = ∑ B 2j (15) j =1 This criterion is equivalent to the velocity term of the Task Compatibility Index proposed by Chiu [1]. The velocity transmission ratio for the hand is simply the weighted sum of the finger criteria. (10) n η ftr = ∑ wiη ftr ,i where Ji is the finger Jacobian matrix. The velocity transmission ratio for a finger is constructed as the sum of Aj2 weighted by the relative importance ratios δi of the twist ellipsoid directions. (16) i =1 The weights wi are assigned the relative importance of force transmission through the individual fingers. 6 ηvtr ,i = ∑ δ j Α 2j (11) 3.3 Grasp Quality An intuitive definition of the grasp quality factor is that it is a measure of how hard the fingers must squeeze the grasped object in order to resist external loads and maintain a grasp. The grasp of an object can be characterized as a linear combination of the primitive wrenches imposed by a unit force at each of the n contacts in a grasp. For point contacts with friction, four unit forces aligned with the edges of the linearized friction pyramid are defined. j =1 This criterion is equivalent to the velocity term of the Task Compatibility Index proposed by Chiu [1]. The velocity transmission ratio for the hand is simply the weighted sum of the finger criteria. n ηvtr = ∑ wiηvtr ,i (12) i =1 The weights wI are assigned the relative importance of velocity transmission through the individual fingers. 4n  GS =  w | w = ∑ α i wi , α i ≥ 0, i =1  ( ( )) β = eiT JJ T ei i =1 i  ≤ 1, f ci = 1  (17) The convex hull of the grasp wrench space GS forms the bound of external loads that can be resisted by unit forces at the contacts. The grasp quality factor GQ is the reciprocal of the amount by which the grasp wrench space GS is scaled so that it encloses the task wrench space W. 3.2 Force Transmission Similar to the velocity transmission criterion of the previous section, the force transmission ratio a measure of how effectively joint torques are transformed into forces at the end-effector. This criterion has been defined in robotics literature [1][15] as follows: −1 / 2 4n ∑α (13) where ei is the direction of interest in the manipulator base frame and J is the manipulator Jacobian matrix. The directions of interest for a finger, in the case of k m = min( k ) | W ∈ kGS (18) GQ = k m−1 (19) Interested readers should refer to Pollard [10] and Ferrari [2] who extensively discuss the calculation and the derivation of GQ. 4 τ  JTR = ∑  i τ i max   i =1 n 3.4 Preliminary Grasp The preliminary grasp phase aims to construct a preliminary grasp by • determining the accessible surface areas of the object for each finger, • selecting the best configuration for each finger based on finger-related criteria, and The surface of the given object is partitioned by laying a grid on the surface. The grid dimensions are chosen by considering the size of the search space in accordance with the actual fingertip contact area. The finger configuration sets are created by determining the accessible points on the grid. The finger configurations are then ranked according to the following selection criterion: η f ,i = γη ftr ,i + (1 − γ )η vtr ,i (22) where τimax is the torque limit of the ith joint. JTR represents the joint torques needed to grasp a given object, and thus is related to the power requirements for a grasp synthesis. Lower values of JTR are desired for which the torque requirements on the joints are decreased. For this case study, the task wrench space consisted of unit forces in the object frame. Second grasp property, grasp matrix velocity transmission, is defined for motion related tasks along the major axis of the twist ellipsoid B. It is given by ( ( T vtr (Gh ) = U major GhGhT (20) ) −1 U major ) −1 / 2 (22) where Umajor is the direction of greatest importance of the motion ellipsoid B. For this case study, U was aligned with the object reference frame and (δ1,…,δ6)=(1,10,1,1,1,1). After laying a grid on the cylinder object, all the collision free, force closure grasps (3240 configurations) were identified. The scatter plots in Figures 4 and 5 show the variation of the grasp properties JTR and vtr(Gh) with respect to ηftr and ηvtr for all these configurations. Also displayed in these plots are the logarithmic trend lines for the scattered data. Figure 3 shows that Joint Torque Ratio has a decreasing trend with respect to the increasing value of (ηftr. x GQ) where γ is the parameter that determines if the task is motion-oriented (γ<0.5) or force-oriented (γ>0.5). The finger configurations that maximize the selection criterion form the initial grasp. 3.5 Grasp Optimization The resulting hand configuration of the preliminary phase is chosen as a starting configuration for the final search routine. A directsearch is conducted on the grid surface by considering the hand level criteria. The search is controlled by the following objective function: η g = γη ftr GQ + (1 − γ )η vtr 2 (21) 1 The task selection parameter γ is the same as for the finger selection criterion. The search ends when ηg reaches a local maximum. The calculation intensity of the search phase is significantly reduced since the hand level criteria ηftr and ηvtr are algebraic sums of the previously calculated finger criteria. The convex hull analysis to determine GQ is the only complex computation necessary for each grasp evaluation. 0.9 0.8 0.7 JTR 0.6 0.5 0.4 0.3 0.2 0.1 0 0 4.0 Evaluation of the Proposed Method In order to justify using the performance criteria of the previous section to get desirable grasp properties, the grasp synthesis problem for a cylinder object is considered. The results in this section show the variation of two grasp properties with respect to varying force and velocity transmission values. First grasp property, Joint Torque Ratio (JTR), is defined for force related tasks. Joint Torque Ratio for an n-dof hand is given by 0.5 1 1.5 2 2.5 3 ηftr x GQ Figure 3 Torque Ratio vs. Force Transmission Figure 4 shows that vtr(Gh) has an increasing trend with respect to the increasing value of ηvtr. These results indicate that by optimizing the hand level criteria ηftr and ηvtr, the desired grasp properties for the velocity and force related tasks are enhanced. The final criteria values, and therefore the 5 [2] grasp properties, also depend on the search routine used in the final phase. [3] 0.57 0.56 0.55 vtr(Gh) 0.54 [4] 0.53 0.52 0.51 [5] 0.50 0.49 0.48 40 45 50 [6] 55 η vtr [7] Figure 4 Grasp Matrix Velocity Transmission 5.0 Summary and Conclusions A synthesis approach has been introduced for grasp planning problem of robotic hands. The approach tries to decompose the synthesis problem into two interconnected problems in finger and hand levels, preliminary and final grasp, respectively. At each level, a grasp configuration is selected using different performance criteria. In the case of preliminary grasp, the selected configuration is fed into the final grasp phase, which constitutes a starting point for the grasp optimization. The finger and hand level criteria are connected, so the two-phase approach is justified. The methodology has also been implemented as an algorithm to be used in the grasp planning of the NASA-Robonaut hand. Although the computational requirements/properties of the algorithm is not in the scope of this study, near real-time performance has been observed and the final local optimum of the algorithm has been satisfactory (under the trend lines presented in Section 4.0). Future work should focus on developing better search techniques than the direct-search employed in this study so that the efficiency of the algorithm can be enhanced. Acknowledgements This research was conducted under the NASA Grant no. NAG9-809. We thank Dr. H. Aldridge of NASA-JSC for his comments and discussions during this project. We also acknowledge the efforts of M. Pryor and C. Cocca of RRG at UT. [8] [9] [10] [11] [12] [13] [14] [15] References [1] S. Chiu, “Task Compatibility of Manipulator Postures”, Int. J. of Robotics Research, Vol. 7, No. 5, 1988, pp. 13-21. 6 C. Ferrari and J. Canny, “Planning Optimal Grasps”, Proc. 1992 Int. Conf. on Robot. and Auto., pp. 2290-2295. P. Hsu, Z. Li and S. Sastry, “ On Grasping and Coordinated Manipulation by a Multifingered Robot Hand”, Proc. 1988 Int. Conf. on Robot. and Auto., pp. 384-389. T. Iberall, et.al. “ Knowledge-Based Prehension: Capturing Human Dexterity”, Proc. 1988 Int. Conf. on Robot. and Auto., pp. 82-87. J. Kerr and B. Roth, “ Analysis of Multifingered Hands”, Int. J. of Robotics Research, Vol. 4, 1986, pp.3-17. R. M. Murray, Z. Li and S. S. Sastry, “A Mathematical Introduction to Robotic Manipulation”, CRC Press, Boca Raton, (1994). Z. Li and S. Sastry, “Task Oriented Optimal Grasping by Multifingered Robot Hands”, Proc. 1987 Int. Conf. on Robot. and Auto., pp. 389393. H. Liu, T. Iberall and G. Bekey, “The Multidimensional Quality of Task Requirements for Dextrous Robot Hand Control”, Proc. 1989 Int. Conf. on Robot. and Auto., pp. 452-457. V. Nguyen, “ Constructing Stable Grasps in 3D”, Proc. 1987 Int. Conf. on Robot. and Auto., pp. 234-239. N. S. Pollard, “Parallel Methods for Synthesizing Whole-Hand Grasps from Generalized Prototypes”, MIT AI Lab. Tech. Rep. AI-TR 1464, (1994). K. B. Shimoga, “ Robot Grasp Synthesis Algorithms: A Survey”, Proc. 1996 Int. Conf. on Robot. and Auto., pp. 230-266. R. Tomovic, G.Bekey and W. Karplus, “A Strategy for Grasp Synthesis with Multifingered Hands”, Proc. 1987 Int. Conf. on Robot. and Auto., pp. 83-89. C. Lovchik and M. Difler, “The Robonaut Hand: A Dextrous Robotic Hand for Space”, submitted to the 1999 Int. Conf. on Robot. and Auto.. G. Hirzinger, “Status Report 1997”, Institute of Robotics and System Dynamics, DLR German Aerospace Res. Est., (1997). M.J. Van Doren and D. Tesar, "Criteria Development To Support Decision Making Software for Modular, Reconfigurable Robotic Manipulators", M.S.E. Thesis, University of Texas at Austin.