A Study on Object Grasp with Multifingered Robot Hand Ying LI, Ph.D. Department of Mechanical Engineering Kagoshima University, Japan

2 Introduction Human Arm Robot Hand Elements Finger links sensors actuators function Grasp Manipulation Mechanism control Control Motion Force Design Mechanical analysis Motion planning Mechanical analysis Motion planning

3 Simple tasks such as welding, carrying etc. Object Multifingered robotic hand Dexterous and fine manipulation tasks such as assembling multifingered hand are demanded Gripper 1. Constraining Mechanisms

4 Form-closure Force-closure Force and moment can be applied on the object Object can not be manipulated Object can be manipulated while being grasped Object is totally constrained 2. Grasping Constraints object finger

5 3. Conditions of Force-closure Grasp Finger configuration condition (In 2D case) A set of contact points with Coulomb friction including two different points (3D, three different points ) Non-zero internal finger force condition a set of contact forces (1)The resultant force and moment are zero while the friction conditions is satisfied (2) All of the individual finger forces are not zero simultaneously contact point Finger force

6 4. Finger Force Features Based on Force-closure Resultant force and moment on object are zero Resultant force and moment can resist external force External force existing No external force

7 5. Finger Position Region Features Graspable finger position (external force existing) grasp by internal forces based on force-closure stable exert finger force for resisting external force unstable

8 The Purposes of This Study (1) Determine the graspable finger position region and graspable finger force (2) Find finger positions for stable grasp by an analytical approach. For a given object, based on force-closure theory Finger configuration space

9 Previous Works ● Analyzing the grasp mechanisms and constructing the force-closure and the form-closure About finger force V.-D. Nguyen (1988), X.Markenscoff (1989), G.-C.Park, G.-P. Starr (1992), J.Ponce (1995) ● Computing the positions of fingertips with the force and moment equilibriums when a polyhedral object is grasped T.Omata (1995) About finger position Yun-hui Liu (2000)

10 o resultant force and moment Object Finger force Finger position Variables Nonlinear problem For example Scanning method Huge calculation load Proximate result No an effective method for planning graspable finger position and finger force Analytical method For 3, 4, fingers

11 １． Selecting Edge Candidates Edge candidate Can’t be grasped All the combinations of the contact edges Possible grasp finger force span １ span ２ The Method of This Study Calculation load is reduced force equilibrium

12 span １ span2 Possible Finger force ２． Graspable Finger Position Region Moment equilibrium linearly nonlinear problem linear problem Graspable finger position region For an edge candidate

13 For an known finger configuration Internal finger forces for grasp Finger force components for resisting external force External force acting on the object No external force ＋ ３． Graspable Finger Forces Internal finger forces for grasp Resultant force and moment equilibrium

14 (1)Radius of the biggest inscribed hypersphere of the graspable finger position region (2) Volume of the convex set( 2D, area) including the biggest inscribed hypersphere For obtained finger position region ４． Stable Finger Position Region Two measures for evaluating

15 Assumptions (1) The object is a 2-dimensional polygon with definite geometric shape (2) At least 2 fingertips of a robot hand touch the object by point contact with Coulomb friction Basing on the above assumptions we discuss the finger position region according to the force closure

16 : Internal Force and Moment Equilibrium Conditions Internal Force and Moment Equilibrium Conditions, The edge vectors of the friction cone The magnitudes of the finger force The force equilibrium The moment equilibrium : The position vector of finger : Force closure can be achieved

17 Selecting Edge Candidates The solution exists Force equilibrium The solution doesn’t exist Fingers For a combination of edges edge candidate Solving positive Internal finger force : finger force magnitudes : edge vectors of friction cone

18 Possible finger force A finger configuration Moment equilibrium : span vectors : the parameter of finger forces Permissible region of Both force and moment equilibriums must be used to determine the graspable finger position region Both force and moment equilibriums must be used to determine the graspable finger position region For an edge candidate Finger force solution Convex polyhedral cone

19 Two propositions spans Polyhedral cone of possible finger force Boundaries Analytically linearly Finger position region Determining Finger Position Region Moment equilibrium

20 Vertex position vector Finger position vector : Unit direction vector of edge Finger position parameter For n fingers, Graspable Finger Position Region is defined as the solution set of 1. Defining Graspable Finger Position Region For finger

21 ： expressed by a hyperplane given Finger force parameters Finger position parameters are variables Linear Possible finger forces Finger position vectors substituting Moment equilibrium Non-linear 2. Deriving Boundary Hyperplanes

22 Finger position region Spans of polyhedral cone Possible finger force hyperplane For example Boundary hyperplanes A arbitrary finger force spans

23 Possible finger force between two spans and Finger position region bounded By two hyperplans and proposition １： Moment equilibrium 3. The Finger Position Region Bounded by Two Boundary Hyperplanes 3. The Finger Position Region Bounded by Two Boundary Hyperplanes

24 Convex polyhedron Graspable finger position region by linear method The length limits of object edges rectangular parallelepiped

25 spans Possible finger force Hyperplanes Possible Finger force set formed by spans proposition ２： 4. The Finger Position Region Bounded by Plural Boundary Hyperplanes 4. The Finger Position Region Bounded by Plural Boundary Hyperplanes Finger position region: union set of all of the convex polyhedrons formed by two hyperplanes

26 Graspable finger position region two hyperplanes For example Convex polyhedrons

27 Graspable Finger Force External force acting on the object For an arbitrary finger configuration ＋ No external force Internal finger force and finger force component for resisting external force Internal finger force for grasp

28 Finger force equilibrium And moment equilibrium Finger force equilibrium And moment equilibrium Internal finger force for grasp For example ， selecting a finger configuration substituting In the Case of No External Force is the function of

29 Possible finger force Features of Internal Finger Force for a Finger Configuration Features of Internal Finger Force for a Finger Configuration Polyhedral cone changes depending on finger configuration The magnitudes of the internal forces are no limits

30 Finger force equilibrium moment equilibrium Finger force equilibrium moment equilibrium Graspable finger force External Force Existing For a given finger configuration substituting Internal finger force for grasp Internal finger force for grasp Finger force for resisting external force Finger force for resisting external force For example,

31 Features of Finger Force Component for Resisting External Force Features of Finger Force Component for Resisting External Force For the same finger configuration The term of internal finger force component are same The solution set of finger force component for resisting external force is a convex polyhedron The magnitudes have limits depending on magnitude of external force

32 Determining Stable Finger Position Region Determining Stable Finger Position Region 1. Defining Stable Finger Position Region Not convex The more stable grasp configuration The stable finger position region The convex polyhedron that contains the biggest inscribed Hyper-sphere of the finger position region, and has the larger volume Center of biggest inscribed Hyper-sphere

33 2. How to Solve the Biggest Inscribed Hyper-sphere For the finger position region (Proposition 3) Its biggest inscribed hyper-sphere must lie in a convex polyhedron formed by two boundary hyperplanes and edge limit hyperplanes Convex polydron formed by three hyperplans For example,

The Stable Finger Position Region Solving biggest inscribed hyper-spheres Selecting biggest one from them if same size biggest inscribed hyper-spheres Solving volumes of the convex polyhedrons Selecting the larger one an algorithm is proposed Stable finger position region All the convex polyhedron formed by two hyperplanes

35 Numerical Example For example: edge fingers All combinations of edges Are edge candidates Boundary hyperplanes Possible finger force Finger position region Vertexes are given Friction coefficient is given spans

36 ．．． Boundary hyperplanes Finger position region Finger force Stable finger position The Finger Position Region All convex polyhedron formed by two hyperplanes Polyhedron but not a convex one

37 Determining the Finger forces Internal finger force (polyhedral cone) Internal finger force (polyhedral cone) No external force acting on the object No external force acting on the object A finger configuration

38 Finger force Finger force component for existing external force (convex polyhedron) Finger force component for existing external force (convex polyhedron) External force acting on the object External force acting on the object For the same finger configuration

39 The stable finger position region Determining the Stable Finger Position Region The center set The biggest inscribed Hyper-sphere The biggest inscribed Hyper-sphere

40 The Finger Configuration at the Center Set The Finger Configuration at the Center Set Vertexes of the center set The shortest distance between anyone finger and object vertex is radius (1.768) The shortest distance between anyone finger and object vertex is radius (1.768) center set of the biggest inscribed hyper-sphere The more stable grasps

41 Conclusions An analytical approach for planning the stable grasp on an object is proposed. (1) The grasp edge candidates can be selected before the finger position region is determined. (2) Two propositions were proposed for exactly determining the object finger position region by using linear programming method. (3) For an arbitrary finger configuration, the graspable finger forces were analyzed. (4) Another proposition was proposed for determining the stable finger position region.