JOURNAL OF J>CitNCE & TECHNOLOGY * No 79B - 2010 CONTROL ALGORITHMS FOR MANIPULATING AN OBJFXT BY A PAIR OF MINIMUM-DOF ROBOTIC FINGERS CAC THUAT TOAN DIEU KHIEN ROBOT KIEU NGON TAY CO SO BAC TU' DO TOI THIEU TRONG TI lAO TAG CAC VAT Till: Nguyen Pham Thuc Anh Hanoi University of Science and Technology ABSTRACT Recently control of multifingered robotic hands in dexterous manipulating an object becomes a new direction that attracts interest of researchers in robotics Observations in everyday life show that human can grasp a small object securely by using a dual pair of his thumb and index fingers, then rotate it to a desired angle and move it to the given vicinity The purpose of this paper is to present algorithms for concurrent grasping and manipulating a flat sudace object by a pair of minimum-DOF robotic fingers with soft ends in a horizontal plane Firstly, the dynamics of the fingers-object system have been formulated by applying variational Hamilton's principle Secondly, a control framework for concurrent grasping and orientation controlling of the object have been proposed Finally, the effectiveness of the proposed control inputs has been proved by theoretical analysis and reconfirmed by computer simulation results Our research aims to contribute for developing of intelligent robots that work in assembly lines of electronic industry and welfare robots in supporting service disable and elderly people TOM TAT Diiu khiin cdc Robot ngdn tay thwc hien cac hoat ddng linh hoat nhw ngdn tay la mdt cdc ITnh vwc m&i thu hut sw quan tam cua cdc nha nghien ciru Quan sat d&i sdng thw&ng nhat cho thiy ngw&i cd khd nang sir dung cap ddi ngdu gdm hai ngdn tay cai va trd di kep chde mdt vat thi nhd, sau dd quay nd theo cdc true xac dinh vd di chuyin nd t&i cdc vi tri lan can dwgc xdc dinh trw&c Bdi bdo tnnh bay mdt sd thudt todn di diiu khiin Iwc kep mot vat thi va dinh hw&ng nd khdng gian hai chiiu b&i Robot dang hai ngdn tay mim ddi ngdu /Wd hinh ddng Iwc hoc cua he thdng robot-vat thi dwgc thiit lap theo luat Hamilton Tren ca sa dd, thuat todn di diiu khiin ddng th&i viec kep chac chin vdt thi vd quay nd tai hw&ng mong mudn dwgc phat triin Cudi cimg mot sd kit qua md phdng dwgc tnnh bdy di chirng minh tinh hieu qua cua cac thuat toan diiu khiin di xuit Nhirng kit qua dat dwgc cd thi dwgc irng dung cho viec phdt triin cdc Robot thdng minh cdng nghe ldp rdp linh kien dien tir vd dac biet Id cdc Robot hoat ddng ITnh vwc y ti cham sdc ngwai gia va tan tat be helpful for developing of industrial robots working in assembl}' operations and welfare robots in supporting elderly persons I INTRODUCTION It is often observed in everyday life that human hands can implement dexterous manipulation with excellent skill Researches in biology science have pointed out the dexterity of human hands is due to partly two their special characteristics: (1) finger-tips are soft to ensure stable grasping of objects and (2) a thumb finger arranges a dual structure with other fingers, especially with an index finger This fact becomes a challenge for many research works in designing robotic fingers and discovering intelligent control algorithms to manipulate surrounding objects dexterously in analogous manner to human fingers This will II SYSTEM DESCRIPTION A pair of dual single DOF fingers with soft fips grasping an object has been illustrated in Fig.l The index is for the left finger and for the right finger Finger tips are sphericalshaped with radius r, (for /=1,2) and covered by a visco-elastic material Symbols q=(qi,q2)^, [=(Ij, hf m=(mi,myf l=(li,l2J^ denote the vector of joint angles, moments on inertia, masses, and link lengths of fingers respectively The object has parallel surfaces 49 JOURNAL OF SCIENCE & TECHNOLC with mass A/, inertia moment /, height d, and width u' ( ir/ + tr.) fhe svmbiil L staiuls for the distance between two base positicMis o and o' of the two fingers Positions of end-point Ooj of link / are denoted bv VCCIIM' /„, (.\'i,i.y,ii)' Due to the visco-elastic characteristics ol' the soft finger-tips, the cimlacl bclwecn the lingers and surfaces oC the objecl is o\' area-conlacl that supports lo grasp the object securely and manipulate il llcxiblv When the soft lingers grasp the object, there arise deformations with lengths of U/ and h respectively Reproducing forces /, (caused by the deformed areas arc functions of U, in the following form /, = A',Av; ; where K is a silliness coenicicnt The reproducing forces /, direct from the point 0(H to the point Oj, where O, is the center point of the contact area / fhe viscous force acting from Oo, to the point O, can be expressed in the form of c,(.Vv ),\v , where s ( A v J is an V, = Vn,-(';"^,K (1) In the object frame {ObJ, the points /, have a form '">,=[ u ):]','">,.[w,,-};]' (2) where the term } stands for distances between the center points Xi of contact areas and ihc other smlacc point at which the A'-axis of the object frame crosses the object surfaces he term )', must be subject to the constraints that the fingertips purely roll without slipping on the objecl surfaces, or cquivalentlv the velocities dV/dt on the object surfaces equal to that on its corresponding finger-ends (r lv,)df7Vd/, that is dt (3) dt where (/?, + q^ =7r + ( -\)'0 increasing function of Av, The fingers-object setup is confined in a horizontal plane and therefore is not affected by the gravitational force We define the reference frame {oxy} locating at the base position of the left finger and the object coordinate {OXY} frame at its mass center O, position of the mass center of the object in the reference frame by vector 7.=(-Y, v) and the rotational angle of the object by ^ (4) From the above two relations, two important formulas can be obtained /;'•(/,.-/„:) = -(>;-):) (5) where // = [cos^,sin6?],r^'^ = [ - s i n ^ c o s ^ ] III DYNAMICS OF INTERACTION BETWEEN AND AN O B J E C T PHYSICAL FINGERS Dynamics ol" a pair of dual single DOF fingers grasping a 21) object with flat surfaces are considered, fhe kinetic energv A' of the overall svstem can be expressed as follows: t-)(6) fhe potential energy P deformation is described as: P=T Fig I Fingers-object system Next we formulate geometric relations for keeping contacts between two fingers and the object The position of the center points Xi^ of the contact areas in the coordinate frames can be calculated in the following form \f,in)drj of finger-end (7) It should be remarked that eq (5) is nonholomic constraint due to an existence of functions Ax, and eq (3) can not be integrated in time However, eq (3) can be regarded as the SO JOURNAL OF SCIENCE & TECHNOLOGY * No 79B - 2010 form written in terms of infinitesimally small variation of 5x as follows: ^ ' ' ^ dxj desired orientation while keeping the contact stably The time for switching from grasping to manipulating depend on fiexibility and skill of human In the first phase, he can grasp an object without knowing its physical parameters and state parameters such as position and orientation In the second phase, in order to rotate the object to the desired angle, the information about actual rotational angle is necessary With an assumption that the rotational angle of the object can be measured, wc proposed a control framework for two phases: the Hrsl phase 11, ii/i + ii,,,, for stable grasping the object and the second phase u,-u,i\uo, for controlling its orientation The first phase is from starting time to a switching time /, the control input u, Uii+u,,,, aims to formulate stable grasping is: dx,' Thus, the variational form for the Lagrangian L=K-P with external forces of control input // and the non-holomic constraints can be expressed as: \j;^{5L + ^{^x,)A.^,^cSX ^u!Sc}, (8) -A,Kr,-Ax,)^^^]5X)dt^0 d.\ c\ where A, expresses Lagrange multiplies vector A' is defined as The X=(qi.q\x.y.O)^ By applying this principle to the objective system, we can obtain the dynamic equations of two fingers as [1]: h'4\ + Aj'Ji Jcos0^ suit/) -[>^oVv -''i + ^^"i]^ = "i -c.q,-{-\)' (9) I-,q.,_ - Jl^J^f - Wlz'', - '2 + •^^•: ]^_ = ": 10 -YJ,+ + iA,+A,) = iY,-Y,)f, where u,,,, is for torque balance, Cj are damping coefficients, and J„, are Jacobian matrices defined as: (10) YJ, - M'/^ + Ms/l, = Jo,= The tangential constraint forces with the magnitude A, emerge at the center points of the contact areas / in the direction tangential to the object surfaces The normal contact forces/ are functions of deformation Ax, and their derivatives as follows [2]: /=/+^,(Ax,)Ai:, I('',-^v,) 1=1,2 and of the grasping object as: Mjc-{f-f.)cose-{?^+?^)^t) My-{f-fi_)sme (12) r-Ax dq, ' dq, , From the switching time, the control input u, =Uf,+Ue, aims to rotate the object can be designed angle as /cos^^ U,=Uj.,+Ug, = - ( - l ) ' J o fd-(^A, 'ys'm0 J (13) (11) Js:\n0^ -(-1)' Based on the dynamics of the fingersobject systems, sensory-feedback algorithms for grasping and manipulafing an object will be designed in the next section ^^ l^COSt'y • r, + Ax K^^0 ) The closed dynamics have a form: lA^JlrM^-^-^l^-'-'^^iY.-Yf) dq, IV ALGORITHM FOR STABLE GRASPKVG OF AN 2D OBJECT AND CONTROLLING ITS ROTATIONAL ANGLE rM' = -/:v,^|-./>,.^,Ai-, fq, -JlrM Refer to a regular manipulation of human fingers, we can see that there are two phases in his motion: firstly the human being has to grasp the object securely and secondly rotates it to a = 51 -A:V,^ - ^ ^ oq., + JoVv^2^2 (14) rw ^'-^^^{Y,-Y) (15) J O l k N A L OF S( lENCI & TECHNOLOGY • No 7915 - 2UM) Mx -/;A/,+/vV, It is possible to have Mi \ (22) 16) dll , '•^-' r.v A = ; ; ( ; , A \ , 10 -\\\u + y,.\i,- where rw - /• -( /•, ; -CA\,) -I Vv, where /•" ' A, ^-^th=-Y (17) \v //(>;->')- |r//;+^,(Av,)Ax-,^ Av, > ¥, >o and \/' = X r (/('/)-./>/'/ ,\A, as / ->oo Y,-Y._^0 That implies the balance of torque and force on the grasping object and clearly that the stable grasping the object bv means of dual single-DOF soft fingers can be realized in theoretical analysis The second phase is for controlling of rotational angle of the object Taking derivative in time eq (5) leads to = (20) (2( Table J Parameters of control input q'jLr,-q[-Jl,r^+w{^x,)d Desired contact force fd Desired rotational 0j angle Damping coefficient Ci=C2 Gain coefficient Ko Switching time /v and -^r(-^oV;-''i+^^i) + ^/2(-7,L'v -r^ + lyx^) = {M\ +w,)0 >o Table I Physical parameters of the system Link mass w;/=-»/; 0.025 [kgl 6.66E-06[kg.m-] Inertia // /: moment Link lensith 0.040[m] //-/: Object mass \/ 0.03 [kg] Inertia I 7.5E-06[kg.m'] moment Object length h 0.05[ml Object width U/ + 1f; 0.03[m] Object height d 0.0 l[m] Stiffness A 150000[N/m-] cocL Viscous coef V AO >o Next in order to verify the effectiveness of the proposed control scheme for stable grasping of the objecL a computer simulation has been carried out Physical parameters of fingers-object system have been reported in Table and control parameters in Table Vv ^ +(j„,-j)^^ as / >0 }; -r, Since £1 is positive definite with respect to X^(qi.q:.x.y.0)' and X under nonlolomic constraints of eq (3) and relation (5), £; plays a role of Lyapunov function It is possible to conclude that -();-K):.(Z,„-ir)r (24) Since /•,'• is positive definite with respect to V (cfi.q^x.y.Ol' then E^ plays a role of Lyapunov function It is possible to conclude that \q, -> (19) 2(/-,^':) \/| —>0 " ' (23) w here /••, =F, +A'^(»', ^ w,)^0^ I.: A + \T t •^ As combined with etj (1 8), we have It is possible to reali/c that the closed dvnamics ensure that: 1d /•, = - X {c,^/ + ^,(Vv),\v; (18) 2dt I 1.2 (21) 52 UN] [rad] 0.001[msN] 0.3 0.0285 [sec] JOURNAL OF SCIENCE & TECHNOLOG\ * No 79B - 2010 From starting time to a switching time /, (0.0285 second), we apply the control input of eq (12) for stable grasping and after U the control input of eq (13) for controlling of object's rotational angle is applied The response of the overall fingers-object dvnamics governed by the proposed control inputs have been shown in the following Figures i \ : : • ; : ' : : i ' AA- V; \; i • , • : i ' : ; ; 0.1 02 ' • : 03 04 ' ' ' , ' ] ' I ', ', I \ ' 05 06 07 , 02 03 04 05 06 IJmolBecond] 07 08 09 t Fig Deformation AX/ of the left-finger I 0.1 ] =/\A^ - -f 0.8 09 time[ second] / lg-2 Contact force f J atthe leftflnger 1 • ' ' : • ' Vf 0.1 02 0,3 04 05 0.6 time[secondl 0,7 0.8 0.9 Fig Deformation AX2 of the right-finger ; ; 1 • ; ; ; 0.3 04 ; The responses have verified the effectiveness of the proposed control input of eqs (12 ,13) The normal contact f o r c e s / (for /=1,2) converge to the desired v a l u e / / in 0.06 second (Figs 2,3)r"The rotational angle of the object ^converges to the desired value 0d in 0.12 second (Fig 4) The fingers keep contact with the object during whole manipulating time when Ax, is always positive (Figs 5,6) Clearly that the simulation results have verified the effectiveness of the proposed control algorithm : -'.I ) 0.1 06 0.8 09 tinne[second] Fig Contact force f2 at the right finger • 0.15 • 0.1 0.3 0.4 0.5 0.6 0.7 0.9 The noteworthy point that the fingers always maintain the contact with the object during manipulation time It is not a serious matter when the overall system is confined in a horizontal plane, but in a vertical plane, the contact loosing of only one finger will make the object to drop freely Then in the next work, we will develop the proposed control input for manipulating the object under the effect of gravity time[second] Fig The rotational angle of the object 53 JOURNAl O F S( IKN( F "i FIX HNOLOGY * No 79B - 2010 V CONCLUSION The paper dealt with the two problems: (1) stable grasping an 2D fiat surface objecl without an\ i>bjeel sensing information and (2) concurrent reali/.ation '.•'I' secure grasp and orientation ol" the object bv means ol" dual single-DOF fingers with soft ends in a horizontal plane Control frameworks have been found out in dynamic sense and designed Irom joint sensing information so that they become applicable in experiments Theoretical analvsis has proved a convergence of controlled variables in both grasping and orientation control of the object to the desired states The simulation results have reconfirmed the validity o\' the control methods It is interesting to reali/c a dexterity of the simplest fingers with soft-ends The basis of study can be developed for the case of grasping and manipulating an objecl under the effect of gravity REFERENCES S Arimoto P.I".A Nguven II.-Y Han, and /, Doulgeri; "Dynamics and control of a set of dual fingers with soft-tips""; Robotics 2000, 18, Part 1, pp 71-80 Suguru Arimoto; "Dexterity and Control fheory of Multi-fingered Hands: A DifferentialGeometric Approach"; Springer, 2007 P.T.A Nguven, S Arimoto, and H.-Y Han; "Computer simulation of controlled motions of dual fingers with soft-tips grasping an object"; Proc Japan-USA Symposium on Flexible Automation, 2000, pp 1039-1046 P.T.A Nguyen and S Arimoto; "Dexterous manipulation of an object by means of multi-DOF robotic fingers with soft-tips"; J of Robotic System, 17, No.7, 2002, pp 349-362 S Arimoto and P.T.A Nguyen; "Principle of superposition for realizing dexterous pinching motion of a pair of robot fingers with soft-tips"; lEICE Trans, on Fundamental Electronics Communication and Computer Sciences, E84-A, No 1, pp 39-47 Author s address: Nguyen Pham Thuc Anh fel.: (+844) 3869.2306 Email: thucanhnguyen@mail.hut.edu.vn Department of Industrial Automation Hanoi University of Science and Technology No 1, Dai Co Vict Str., Ha Noi, Viet Nam^ 54 ... smlacc point at which the A''-axis of the object frame crosses the object surfaces he term )'', must be subject to the constraints that the fingertips purely roll without slipping on the objecl surfaces,... orientation In the second phase, in order to rotate the object to the desired angle, the information about actual rotational angle is necessary With an assumption that the rotational angle of the object... M''/^ + Ms/l, = Jo,= The tangential constraint forces with the magnitude A, emerge at the center points of the contact areas / in the direction tangential to the object surfaces The normal contact