a general contact force analysis of an under actuated finger in robot hand grasping

International Journal of Advanced Robotic Systems ARTICLE A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping Regular Paper Xuan Vinh Ha1, Cheolkeun Ha1* and Dang Khoa Nguyen1 School of Mechanical Engineering, University of Ulsan, Ulsan, Republic of Korea *Corresponding author(s) E-mail: cheolkeun@gmail.com Received 31 October 2014; Accepted 07 December 2015 DOI: 10.5772/62131 © 2016 Author(s) Licensee InTech This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract This paper develops a mathematical analysis of contact forces for the under-actuated finger in a general underactuated robotic hand during grasping The concept of under-actuation in robotic grasping with fewer actuators than degrees of freedom (DOF), through the use of springs and mechanical limits, allows the hand to adjust itself to an irregularly shaped object without complex control strat‐ egies and sensors Here the main concern is the contact forces, which are important elements in grasping tasks, based on the proposed mathematical analysis of their distributions of the n-DOF under-actuated finger The simulation results, along with the 3-DOF finger from the ADAMS model, show the effectiveness of the mathematical analysis method, while comparing them with the measured results The system can find magnitudes of the contact forces at the contact positions between the phalanges and the object Keywords Under-actuated Mechanism, Robotic Gripper, Contact Forces, Grasping, Kinetostatic Analysis Introduction Several researchers have investigated different types of devices for grasping and handling unstructured objects Such a device must adapt itself to the shape being grasped An isotropic gripper that provides uniform contact pres‐ sure is introduced in [1], while the closest gripper to the human finger required more than ten actuators and sensors [2] Many dexterous hands that have several actuators can be mentioned, such as the Utah/MIT hand [3], the Stanford/ JPL Salisbury’s hand [4], the Belgrade hand revisited at USC [5] and the DLR hand [6] As one example of research in the robotic hand field, J.A Corrales et al developed the kinematic, dynamic and contact models of a three-fingered robotic hand (Barrett‐ Hand) in order to obtain a complete description of the system required for manipulation tasks [7] Another study by R Rizk et al introduced the grasp stability of an isotropic under-actuated finger, which is made by two phalanges, and uses cams and tendon for actuation [8] They also presented a study of the internal forces devel‐ oped in the transmission chains G Dandash presents the design of a three-phalanx, pseudo-isotropic, underactuated finger with anthropomorphic dimensions Two cams were used to ensure grasping was as isotropic as possible [9] Additionally, for a multi-fingered telemanipulation system, Angelika Peer et al presented a point-to-point mapping algorithm, which depends largely on the object identification process and the estimation of human intention It allows the system to map fingertip Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 motions of a human hand to a three-finger robotic gripper, known as the BarrettHand [10] The dexterity can also be obtained by under-actuation, which consists of equipping the finger with fewer actuators than the number of DOF Thus, Thierry Laliberté et al addressed the simulation and design of under-actuated mechanical hands to grasp a wide variety of objects with large forces in industrial tasks Architectures of 2-DOF under-actuated fingers are proposed and their behaviour is analysed through a simulation tool A design of a threefingered hand is then proposed using a chosen finger [11] The design of a 3-DOF finger is also discussed with stability of grasp, equilibrium and ejection problems in [12] In [13], Dalibor Petkovic et al investigate a kinetostatic model of a design for an under-actuated robotic gripper with fully distributed compliance Given the highly non-linear system and complicated mathematical model, an approxi‐ mated adaptive neuro-fuzzy inference system (ANFIS) is proposed for forecasting the gripper contact forces Moreover, Lionel Girglen et al analysed several common differential mechanisms modelled as basic force transmis‐ sion, such as a movable pulley, seesaw mechanism, fluidic T-pipe, and planetary and bevel gear differentials A mathematical method to obtain the output force capabili‐ ties of connected differential mechanisms is presented and two types of under-actuated robotic hands are introduced in [14] In [15], a fundamental basis of the analysis of underactuated fingers with a general approach is established This method proposes two matrices that describe the relationship between the input torque of the finger actua‐ tor(s) and the contact forces on the phalanges Another approach, LARM Hand, which includes three fingers, was designed for anthropomorphic behaviour Marco Ceccarelli et al [16] proposed the grasping adapta‐ tion of a 1-DOF anthropomorphic finger mechanism in LARM Hand by using flexible links and/or under-actuated mechanisms with additional spring elements or flexural joints For a flexible mechanism, flexible links and joints were represented through lumped spring elements, while the under-actuated mechanism was obtained by substitut‐ ing a crank of the original four-bar linkage with a dyad, whose links are connected by a spring element In addition, a new finger mechanism with an active 1-DOF was investigated to improve an existing prototype of LARM Hand with a torsional spring at a rotational joint, while a sliding joint is used for the linear spring to achieve a flexible link [17] The proposed mechanism is not simple since it is composed of seven links, one slider and two springs In addition, it is requested to be sized within a finger body with human-like size and to operate with an anthropomor‐ phic grasp behaviour The introduction of two new matrices in [15] allows the system to calculate the contact forces on the phalanges through the input torque of the finger actuator in the case of full-phalanx grasping However, in the case of fewerthan-n phalanx grasping, it is difficult to obtain the contact Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 forces based on the relationship between the input torque of the finger actuator and the contact forces on phalanges This paper proposes a general mathematical analysis of the distributions of contact forces for the under-actuated finger in the case of full-phalanx grasping, while taking into account cases of fewer-than-n phalanx grasping The simulation results, with the 3-DOF finger model from the ADAMS environment, show the effectiveness of the mathematical analysis method, while comparing with the measured results The system can find magnitudes of the contact forces at the contact positions between the finger phalanges with the object The remainder of this paper is organized in eight sections The related works are introduced in Section Section reviews the original analysis of an n-DOF under-actuated finger Section proposes the general contact force analysis of an under-actuated finger Simulation set-up is intro‐ duced in Section Section shows the simulation results Discussion is mentioned in Section Finally, Section presents the conclusions Related works Contact forces play an important role in grasping tasks of the Robot Hand Contact forces depend mainly on the actuator torque and the torque transmission ratio between under-actuated joints R Rizk et al analysed the contact forces of the under-actuated finger, which is made by two phalanges and uses cams and tendon for actuation It allows authors to determine the grasp stability of the finger and the efforts exerted on the passive elements, respective‐ ly [8] In [9], G Dandash presents a method to compute the contact forces in a pulley-tendon finger It is a matter of establishing a balance between the powers at equilibrium In another study, Dalibor Petkovic et al proposed an approximated ANFIS for forecasting the gripper contact forces of the under-actuated robotic gripper with fully distributed compliance because of the highly non-linear system and complicated mathematical model [13] Several researchers have applied the mathematical analysis to calculate the contact forces in designing the gripper systems A mathematical analysis to obtain the contact force of the under-actuated gripper was considered in [14] In [15], the authors propose two matrices that describe the relationship between the input torque of the finger actua‐ tor(s) and the contact forces on the phalanges In addition, Wu LiCheng et al proposed a static analysis method to obtain the contact forces and a Jacobian matrix of the proposed finger mechanism with an active 1-DOF to improve the existing prototype of LARM Hand [17] In another approach, the numerical simulation in ADAMS’ environment is used to characterize the functionality of the new prototype, which is a new under-actuated finger mechanism for LARM hand [18] Recently, the sensor technique has been widely developed and applied in the robotic field Tactile sensors are devices providing pressure data and often surficial distribution of the latter on the sensors; i.e., localization In [19], the sensor feedbacks from force/torque sensors and tactile sensors were used to implement and validate the robust grasp primitive for a three-finger BarrettHand In another study, Lionel Birglen et al implemented the tactile sensors on the MARS prototype finger’s phalanges to control underactuated hands, as shown in Figure The behaviour of under-actuated fingers can be substantially enhanced with tactile information [20] L i = the length of the ith phalanx = the length of the first driving bar of the ith four-bar linkages bi = the length of the ith under-actuation bar ci = the length of the second driving bar of the ith four-bar linkages θi = the rotating angle of the ith phalanx with respect to the base θia = the rotating angle of the first driving bar of the ith four- bar linkages with respect to the base ψi = the angle between Oi Pi' and Oi Pi T = the torque of the actuator at the first joint T si , i > = the spring torque of the ith joint F i = the contact force on the ith phalanx ki = the contact position on the ith phalanx In previous research [15], Lionel Birglen et al analysed and discussed the stability of the grasp – i.e., equilibrium and ejection phenomenon, achieving stable grasps and phalanx In previous research [15], Lionel Birglen et al analysed and force distribution, and avoiding weak last phalanges that ejection phenomenon, achieving stable grasps and phalanx cannot ensure sufficient force to secure the grasp cannot ensure sufficient force to secure the grasp Pn xn n Figure MARS’ finger equipped with tactile sensors n cn 1 Tsn On Ln b3 yn kn Fn Review of the original analysis of an under-actuated finger F3 Ts k3 3.1 General n-DOF, one degree of actuation (DOA) finger O3 3a a 3 c2 P3 P3' y3 Figure illustrates the type of under-actuated n-phalanx finger considered in this section and all important param‐ eters The actuation wrench T is applied to the input of the L2 F2 finger and transmitted to the phalanges through suitable mechanical elements, such as four-bar linkages A simple kinetostatic model for the fully adaptive finger with compliant joints can be obtained by adding springs to every joint of the finger The torque spring T si in the joint Oi is used to keep the finger from incoherent motions Passive elements are used to kinematically constrain the finger and ensure the finger adapts to the shape of the object being grasped A grasp state is defined as the set of the geometric configurations of the finger and the contact locations on the phalanges, which are necessary to characterize the behav‐ iour of the finger Important parameters are denoted as follows: x3 L3 3 k2 T s2 O2 y2 F1 x2 b2 2  2a a2 c1  P2' P2 L1 x1 k1 y1 T1 O1 b1 1 1a a1 P1 Figure Geometric and force parameters of under-actuated n-DOF finger Figure Geometric and force parameters of under-actuated n-DO Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping 3.2 Static analysis of under-actuated n-DOF fi Even though three phalanges are normally used for robot f with four-bar linkages for general static analysis The finge 3.2 Static analysis of under-actuated n-DOF finger Even though three phalanges are normally used for robot fingers, this section considers a general n-DOF, 1-DOA finger with four-bar linkages for general static analysis The finger model is illustrated in Figure To determine the distributions of the contact forces that depend on the contact point location and the joint torques inserted by springs, we proceed with a static modelling of the finger Additionally, the friction must be ignored and the grasping object has to be fixed Equating the input and the output virtual powers of the finger [15] yields: T T wa = F T v (1) where T is the input torque vector by the actuator and springs, ωa is the corresponding velocity vector, F is the contact force vector, and v is the projected velocity vector of the contact points; i.e., (2) where K i is the stiffness of the torsional spring located at joint Oi , and Δi , i > is the difference between the current and initial angles of the joint Oi Thus, the projected velocities can be simply expressed as a product of a Jacobian matrix J T and the derivative vector v = JTq& q& = J awa In the under-actuated finger model, the four-bar linkage is used to transmit the actuator torque to each phalanx, while the principle of transmission gives the angular velocity ratio of four-bar linkage, known as Kennedy’s Theorem ' [21-22] With the ith four-bar linkage Oi Pi Pi+1 Oi+1, we have: c ( L sin(q i + 1a - y i + ) - sin(q i - q ia + q i + 1a - y i + ) ) q&i = q&ia + q&i + 1a i i ( Li sin(q i - q ia ) + ci sin(q i - q ia + q i + 1a - y i + ) ) T ; i.e., éq&1 ù é X1 ê& ú ê êq ú ê êq& ú = êK K ê 3ú ê êLú ê 0 êq& ú ê 0 ë nû ë é X1 ê ê0 J a = êK K ê ê0 ê0 ë a ii = ki (5) q& = J awa (10) K ù ú X2 K ú K K K ú ú K Xn - ú K úû ci ( Li sin(q i + 1a - y i + ) - sin(q i - q ia + q i + 1a - y i + ) ) ( Li sin(q i - q ia ) + ci sin(q i - q ia + q i + 1a - y i + ) ) cn -1 ( Ln -1 sin(q n - y n ) - an -1 sin(q n -1 - q n -1a + q n - y n ) ) an -1 ( Ln -1 sin(q n -1 - q n -1a ) + cn -1 sin(q n -1 - q n -1a + q n - y n ) ) (11) (12) (13) and X i is a function that is used to transmit the actuator torque to the ith phalanx Finally, from Equations (1), (3) and (7), we obtain: and j -1 ỉ j a ij = k j + Lk cos ỗỗ q m ữữ , k =i è m= k +1 ø Xi = Xn - = where ù éq&1a ù K úê ú ú êq&2 a ú X2 K K K K ú êq&3 a ú , or úê ú K Xn - ú ê L ú K úû êë q&n úû where projected velocities can be obtained in a lower triangular form: (4) (9) From Equations (8) and (9), Equation (7) can be described by Equation (10) as: (3) K 0ù ú K 0ú K 0ú ú K Kú K kn úû (8) From the last four-bar linkage, On−1Pn−1Pn On , we have: As illustrated in Figure 2, the Jacobian matrix J T of the é k1 0 ê k a ê 12 JT = êa 13 a 23 k3 ê êK K K êa ë 1n a n a n (7) c ( L sin(q n - y n ) - an -1 sin(q n -1 - q n -1a + q n - y n ) ) q&n -1 = q&n -1a + q&n n -1 n -1 an -1 ( Ln -1 sin(q n -1 - q n -1a ) + cn -1 sin(q n -1 - q n -1a + q n - y n ) ) é vyc ù éq&1a ù é ù é F1 ù T1 ê ú ê& ú ê ú ê ú ê vyc ú êq a ú êTs = - K Dq ú ê F2 ú ê ú T = êTs = - K 3Dq ú ,wa = êq&3 a ú , F = ê F3 ú and v = ê vyc ú ê ú ê ú ê ú êLú L êL ú ê ú êLú ê ú ê q& ú êT = - K Dq ú êF ú n nû ë sn ë nû ë nû ëê vycn ûú of the phalanx joint coordinates θ˙ = θ˙ 1, θ˙ 2, θ˙ 3, , θ˙ n Through differential calculus, one can also relate the vector to the derivatives of the phalanx joint coordinates defined previously with an actuation Jacobian matrix J a : i< j Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 (6) F = JT-T J a-TT (14) Dummy Text and X i is a function that is used to transmit the actuator torque to the ith pha (1), (3) and (7), we obtain: F  JTT J aT T (14) which is the equation that provides a practical relationship a1 [mm] 28.5 between the actuator torques and contact forces Equation which is the equation that provides a practical b1 [mm] relationship 54.0 (14) is valid if and only if k1k2k3 kn ≠ 0, which is the condition a2 [mm] 22.0 between the actuator torques b [mm] 38.0 c [mm] is the condition 14.5 cof [mm] 10.0 for the J m (14) is valid if and onlyJ aifcannot k1k2 kbe3 singular; kn  , which singularity J T matrix of singularity for the T 2 however, the finger may contact the object in the case that ψ2 [degree] 42.5 [mm] however, the finger may contact the objectLin the case57.5that fewer-than-n phalanges are to fewer-than-n phalanges are touching the object That L [mm] 37.8 ψ3 [degree] 90.0 assumption leads toleads the singularity the J T matrix, such assumption to theofsingularity of the JT matrix, such that Equation (14) cannot perfo L [mm] that Equation (14) cannot perform 34.5 Table The identified parameters of the 3-DOF finger 3.3 Stability of the grasp of the 3-DOF under-actuated finger 3.3 Stability of the grasp of the 3-DOF under-actuated finger the geometric parameters of the mechanism, one can obtain We will now analyse the stability of the grasp of the underfinalof stability of the grasp Hence, the choice offinger the the actuated 3-DOF finger The geometric and force parame‐ We will now analyse the stability of the grasp the under-actuated 3-DOF The ge design parameters is a very important issue when obtain‐ ters under-actuated 3-DOF finger are described in Figure 3, under-actuated 3-DOF finger are described Figure 3, while real structure ing in stable grasps and a properits distribution of the forcesdesign is s while its real structure design is shown in Figure The among the phalanges identified parameters of the finger are on Table illustrated identified parameters ofillustrated the finger are on Table The parameters, illustrated in Figure 3, will now be discussed The length of the phalanges - i.e., L1, L2 and L3 are fixed from comparison with other existing fingers, simulations and experimentation with a finger model on objects to be grasped The remaining parameters are , bi , ci and ψi In order to reduce the number of independent variables, some relationships between these parameters are imposed, while the number of variables is reduced to two It was clearly shown that the behaviour of the fingers is mainly dictated by the ratios Ri = / ci , i = 1, [11] In [12], Thiery Lalibeté et al referred to the global performance index to evaluate the criteria that was used to determine the performance of the fingers The graph of the global performance index was a function of R1 and R2 An effective finger, including the stable grasps, could then be chosen among the best values From our finger design, R1 and R2 are approximately and 2.2, respectively (which corre‐ spond approximately to the R1 and R2 in [12]) P3 3 c L3 3 C3 b2 O3 Ts C2 k3 F3 P2 L2 2 F2 k2 Ts  2a a2 2 c1 P2' O2 b1 C1 F1 L1 k1 1 T1 a1 1a P1 O1 Figure Geometric and force parameters of under-actuated 3-DOF finger Figure Geometric and force parameters of Figure The structure design of the under-actuated finger Firstly, the behaviour of the finger is largely determined by its geometry, prescribed at the design stage Depending on Secondly, the mechanical limit allows a pre-loading of the spring to prevent any undesirable motion of the second and third phalanges due to its own weight and/or inertial effects, as well as to prevent hyperflexion of the finger The set of the contact situations pair (k, θ) corresponds to the stable part of the space; namely, the space of contact configurations, where k = k1, k2, k3 T and θ = θ1, θ2, θ3 T A under-actuated 3-DOF finger contact situation pair, which affects a stable grasp, corre‐ sponds to a vector F where no component is negative If springs are neglected, expressions of the latter vectors become most simple θ1 is obviously absent from the expressions because rotation about this axis leaves the mechanism in the same kinematic configuration (the finger is rotated as one single rigid body) It can also be shown that signs of elements are independent of k1 ; the proof is, however, more cumbersome and relies on the general inverse calculus by means of co-factors [15] Coming back to our issue, the set of parameters presented in Table (which corresponds approximately to the parameters used in prototypes of under-actuated hands [15]), taking into account the mechanical joint limits, < θ2 < 90o , < θ3 < 90o and < ki < L i , i = 2, 3, the volume of the stable three-phalanx grasps is approximately 32% of the Figure The structure design of the under-actuated finger Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping a1 [mm] 28.5 a2 [mm] whole space of contact configurations Similarly, the design presented in [11] insists on the mechanical joint limits of < θi < 90o to avoid the latter type of ejection Furthermore, one should remember that full-phalanx grasps correspond only to a part of the whole possible grasps That is, fewerthan-full phalanx grasps can also be stable [15] Mechanical limits are key elements in the design of under-actuated fingers when considering stability issues, because they limit the shape adaptation to reasonable configurations (thus avoiding ejection) The general contact force analysis of under-actuated finger 4.1 Case of n-DOF, 1-DOA finger According to Lionel Birglen et al [15], in order for a lessthan-n phalanx grasp to be stable, every phalanx in contact with the object should have a strictly positive correspond‐ ing force Actually, the contacts appear not only with all phalanges, but also with fewer-than-n phalanges in object grasping The corresponding generated forces for phalang‐ es not in contact with the object should be zero, since the latter forces can also be seen as the external forces needed to counter the actuation torque However, calculating contact forces in the case of fewer-than-n phalanges touching the object by using Equation (14) can be a problem because of the singularity of the J T matrix This section tries to solve that problem by proposing a general method to determine the distributions of contact forces in all cases of gripper behaviours in object grasping In order to that, we assume that the stability of the grasp must be satisfied in all cases From Equations (1), (3) and (7), we also obtain: JTT F = J a-TT (15) From Equation (15), the component J a−T T on the right side is the torque vector τ = τ1, τ2, τn T at all joints of the finger (where τi is the torque at the ith joint) relating to the actuator, spring torques and functions of torque transmission between actuation and phalanges, as follows: Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 é k1 0 ê êa 12 k2 JTT F = êa 13 a 23 k3 ê êK K K êa ë 1n a n a n T K ù é F1 ù ú ê ú K ú ê F2 ú K ú ê F3 ú ú ê ú K Kú êLú K kn úû êë Fn úû (17) From Equations (16-17), the general Equation (15) then becomes Equation (18): é k1 0 ê k a ê 12 êa 13 a 23 k3 ê êK K K êa ë 1n a n a n T K ù é F1 ù ét ù ú ê ú ê ú K ú ê F2 ú êt ú K ú ê F3 ú = êt ú ú ê ú ê ú K Kú êLú êLú K kn úû êë Fn úû êët n úû (18) Equation (18) shows that the torque τi at the ith joint of the finger is calculated with respect to the contact forces vector F and parameters αij in Equation (19): n åa j =i F = t i , a ii = ki ij j (19) In the case of fewer-than-n phalanges touching the object (e.g., when the ith phalanx is not touching the object), the parameters αij in Equation (19) are not relevant and F i is zero As this means that Equation (19) is not suitable for this condition, we not need to consider this equation to compute the torque τi in the case of the ith phalanx not touching the object As mentioned above, in order to calculate the contact forces vector F in Equation (18), except for F i , we use the following process: • Neglect the ith column in the matrix J T because all parameters αij , j = n not exist • Neglect the ith row in the matrix J T because all parame‐ ters α ji , j = n relate to F i = on the left side • Remove the F i = element of the force vector F on the left side • Neglect the τi element at the ith joint of the torque vector -T ét ù é X1 K ù é T1 ù ê ú ê ú ê ú ú êTs ú êt ú ê X2 K êt ú = J -TT = êK K K K K ú êTs ú ê 3ú a ê ú ê ú K Xn - ú ê L ú êLú ê0 ê ú ê0 0 K úû êëTsn úû ë ët n û T1 é ù ê ú T X T s2 1 ê ú ê Ts - X2Ts + X1X2T1 ú ú , Ts1 º T1 =ê K ê ú ê ú n -1 n -1 é ù êT + ê( -1)n - j X T ú ú Õ i si ê sn å j =1 ê i= j úû úû ë ë The left component in Equation (15) can be expressed as: τ on the right side After neglecting the ith column and ith row, the J T matrix dimension is reduced by n − × n − 1, while J T is guaranteed (16) not to be singular Consequently, Equation (18) can be used to calculate the contact forces, except for F i The above process is also used in the case of more than one phalanx not touching the object 4.2 The case of 3-DOF, 1-degree-of actuation finger In case of the under-actuated 3-DOF, Equation (14) is valid if and only if k1k2k3 ≠ 0, which is the condition of singularity for the J T matrix, as shown in Figure 5a However, the finger can contact the object in the case of one or two phalanges of the finger not touching the object, as shown in Figures 5b,5c and 5d F2 = Ts - X1T1 ( k3 + L2C )(Ts - X 2Ts + X1X 2T1 ) k2 k2 k3 (22) Ts - X2Ts + X1X 2T1 k3 (23) In order to calculate the contact forces, F 1, F and F 3, in the F3 = grasping object, we must separate the behaviours between the finger and object into four cases: Case 2: The proximal and distal phalanges contact with the object, which means the parameter k2 does not exist, while F is zero, as illustrated in Figure 5b From Equation (20), (18) the second column and row in the J T matrix relating to the medial phalanx are removed, as well as the elements F and τ2 = T s2 − X 1T in the F and τ vectors Equation (20) then becomes: e ith joint of the contact forces 9): (a) é k1 ê ë0 (b) (19) ù k3 + L1C 23 + L2C ù é F1 ù é T1 úê ú = ê ú k3 F T X T X X T + s2 1û û ë û ë s3 (24) with F and F then calculated using Equations (25) and (23), respectively hing the object the object), the levant and Fi is not suitable for r this equation ith phalanx not e, in order to (c) Equation (18), Figure Four of finger grasping Figure Four casescases of finger grasping : F1 = as shown in Figure 5c From Equation (20), the first column and row in the J T matrix relating to the proximal phalanx (d) Equation (18), we then derive a practical relationship J T because all Equation (18), we then derive a practical relationship between the actuator torques and contact forces by  on the left between the actuator torques and contact forces by Equation (20) Equation (20) t of the torque w, the J T matrix while J T is ently, Equation forces, except he case of more   é k1 k2k2 + L1C 2k3 k3L2+C3L1C 23 +F2L2C ù é F1 ù ê k3 k + L C F3  ú ê F ú k2 3 ê0 úê 2ú T1  ê0  ú ê F3 ú k ë ûë û τ vectors Equation (20) then becomes: é k2 ê ë0 (20)  T XT   é s 1T1  Ts3  ê X 2Ts  X1X 2T1  ù ú =ê Ts - X1T1 ú ê s - X2Ts + X1X 2T1 ú From ëT Equation (20), the ûthree contact forces, F1 , F2 (20) F are zero, as shown in Figure 5d From Equation (20), the first and second column and row in the J T matrix relating and F3 , are computed by using Equations (21), (22) and quation (14) is he condition of in Figure 5a t in the case of t touching the and F3 , in he behaviours es:  to the proximal and medial phalanges are removed, as well as the elements F 1, F 2, τ1 and τ2 in the F and τ vectors Equation (20) then becomes: k3 F3 = Ts - X2Ts + X1X 2T1 (27) (k3  L1C23  L2C3 )(Ts  X 2Ts  X1 X 2T1 ) (21) k31T1 ) T ( k + L1C )(Ts -k1X F1 = - k1 (k2  L1C2 )(kk13k2 L2C3 )(Ts3  X 2Ts  X1 X 2T1 ) - (26) Case 4: Finally, the distal phalanx contacts the object, which means the parameters k1 and k2 not exist, while F and are computed by using Equations (21), (22) and (23), T (k  L1C2 )(Ts  X1T1 ) F1   respectively k kk er ù k3 + L2C ù é F2 ù é Ts - X1T1 úê ú = ê ú k3 û ë F3 û ëTs - X2Ts + X1X 2T1 û respectively (23), respectively From Equation (20), the three contact forces, F 1, F and F 3, , F2 are removed, as well as the elements F and τ1 in the F and F and F are calculated by using Equations (22) and (23), k2  L1C2 k3  L1C23  L2C3   F1  k1 0   (25) Case 3: The medial and distal phalanges contact the object, meaning that the parameter k1 does not exist and F is zero, 1: All threephalanges phalanges of of the thethe object, CaseCase 1: All three thefinger fingercontact contact object, x J T because all which means that , as shown in Figure 5a From k k k  From which means that k1k2k3 ≠ 0, as shown in Figure 5a ce vector F on T1 ( k3 + L1C 23 + L2C )(Ts - X 2Ts + X1X 2T1 ) k1 k1k3 X T +X X T ) ( k3 + L1C 23 + L2C )(k1Tks23k2 s2 k1k3 Set-up (21) T  X T1 (k3  L2C3 )(Ts  X 2Ts  X1 X 2T1 ) F2  (sk2 + L1 C )( k + L C )(T - X T + X X T ) sk3 k s2 + k2 3 k1k2 k3 F3 is calculated by using Equation (23) (22) 5.1 Gripper model set-up Since the complexity of products has been increasing, in order to increase competition in production, the require‐ Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping ment of the product development cycle times ought to be reduced Therefore, building a hardware prototype for testing has taken the majority of time for launching new product The simulation technique based on the virtual prototype is proposed as an approach that significantly reduces manufacturing cost and time, compared to the traditional build-and-test approach The virtual prototyp‐ ing approach is an integrating software solution that consists of modelling a mechanical system, simulating and visualizing its 3D motion behaviour under real world operating conditions, and refining and optimizing the design through iterative design studies The advantages of this simulation technique consist of conceiving a detailed model that is used in a virtual experiment similar to one in a real scenario Virtual measurements of parameters and components of the mechanical model can also be carried out conveniently Figure shows the creation of a virtual prototype for testing and simulating the gripper system The Computer- Aided Design (CAD) drawing of the adaptive gripper was designed by a company in the Republic of Korea (SOLIDWORKS) The ADAM/View is the tool of the virtual platform, which is used for analysing, optimising and simulating the kinematic and dynamic behaviour of the mechanical system under real operating conditions Normal Force: IMPACT Function model Stiffness 1.0*108 [N/m] Force Exponent 1.5 Damping 1.0*104 [N/m] Penetration Depth 1.0*10-8 [N/m] Friction Force: Coulomb friction Static Coefficient 0.7 Dynamic Coefficient 0.5 Stiction Transition Velocity 0.1 [m/s] Friction Transition Velocity 1.0 [m/s] Table The identified contact parameters in the ADAMS model Constructing a control system for the virtual gripper model is necessary for co-simulation of the two separate simula‐ tion programs into a whole system The control design is developed based on ADAMS/Control and MATLAB/ Simulink To export the virtual mechanical model of the gripper from ADAMS to the MATLAB environment, the input and output variables are firstly defined in the ADAMS model The input signals are theusing forces that Figure Block diagram of ADAM gripper model creation thecontrol Matlab/Simuli Figure Block diagram of ADAM gripper model creation using the Matlab/ the servomotors of gripper fingers Meanwhile, the output Simulink Environment signals are the measured parameters of gear angle, screw The virtual prototyping platform includes software tools, such as CAD (S speed, joint angles and contact forces Subsequently, this The virtual prototyping platform includes software tools, model is exported to MATLAB/Simulink In the MATLAB ADAMS andPROENGINEER), MATLAB/Simulink The CAD software is used to create the CATIA, such as CAD (SOLIDWORKS, environment, a.mfile and an adams_sys are created The MSC ADAMS and MATLAB/Simulink The CAD software system This model includes the rigid parts with the shape and adams_sys presents the non-linear MSC/ADAMS modeldimension is used to create the geometric model of the gripper with mass inputs and outputs In thisproperties paper, the ADAMS mechanical system This model includes the rigid partsabout containing information and inertia of finger these rigid pa model has a torque input and, 10 outputs, as shown in with the shape and dimension of the physical prototype to the ADAM/View environment a file format, such Figure using The ADAMS block is created basedas onStep the (CATIA model, as well as containing information about mass and information from the.mfile inertia properties of these rigid parts The CAD geometry ADAM/View is the tool of the virtual platform, which is used for analysin model is then exported to the ADAM/View environment The material types of all finger elements and the object, andasdynamic the mechanical system under real operating co using a file format, such Step (CATIA) behaviour or Parasolid.x_t of shown in Figure 5, are declared by dry aluminium Then, gear  angle screw  speed 1 1a T1 2  2a 3 Fm1 Fm Fm3 Figure The ADAM block of finger in adams_sys Figure The ADAM Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 block of finger in adams_sys Normal Force: IMPACT Function model includes two closed-loop controls: a low-level closed which controls the finger position to follow the desired position, is going to be sto loop control for the finger’s position angle (gear ang touch the object in finger grasping At that time, the CFD will issue a switch signa the low-level closed-loop control, the PID controller to the second process (force control process) In the second process, the FPID Conto designed for high-level closed-loop control because touching the object based on the desired contact force F3 e2 For the force control system, the tuning FPID Contro from the Contact Force Detector (CFD) block, where applied in Figure The inputs of the CFD  F as shown  , , , , , F the measured contact forces and motor torque, while As described in Section 4, the distal phalanx of the fi Therefore, the contact force on the distal phalanx ( F3 k p2 e2 de2 ki kd e1 ki1 de1 kd Figure The diagram of the simulated control system for one finger 2a meas p k p1 e1 1a In finger control strategy, there are two control proce finger The torque input of the ADAMS model (  ) is which controls the finger position to follow the desir Figure The diagram of the simulated control system one finger touch the object infor finger grasping At that time, the the contact feature parameters between phalanges and object are chosen suitably according to the material types under real world operating conditions Table shows identified contact feature parameters at which the ADAMS contact behaviour resembles the real world contact behaviour 5.2 The simulated control system langes Assecond describedprocess in Section (force 4, the distal phalanx of the to the control process) In the s finger always contacts the object in four cases of finger touching the object based on the desired contact forc grasping Therefore, the contact force on the distal phalanx (F 3) is chosen to control for the force control system in four cases F3 In finger control strategy, there are two control processes F e2 The first control process is used for the position angle of k p2 finger The torque input of the ADAMS model (τ ) is e2 ki provided by the de torque outputkd 2(τp ) of this process This  1 , 1a ,  ,  a ,  Since properly designed under-actuated mechanisms perform shape adaptation “automatically”, no motor process, which controls the finger position to follow the e coordination is needed Before performing a grasp, the desired position, is1 going to be stopped when the  p distal k geometry of the object should be determined and the hand phalanx starts to touch the objectp1in finger grasping At that ki1 e1 should adjust itself to this geometry by orienting the time, the CFD will de issue a switchksignal to control the Switch d1 fingers To orient the Figure fingers, a9.simple trajectory is gener‐ block to switch to the second process (force control proc‐ The tuning FPID diagram ated to a prescribed position and the gear motor follows ess) In the second process, the FPID Controller will this trajectory with a PD/PID position control In order to control the distal finger touching the object based on the set the grasping force on the object, a maximum motor desired contact force Figure The diagram of the simulated control system torque is set to a desired value The relationship between the force on the object and the torque of the motor is obtained using the proposed method to determine contact forces In the finger control approach, an integration of position and force control methods for one finger is applied Figure shows a diagram of the simulated control system As shown in this figure, the position control system for the finger includes two closed-loop controls: a low-level closed-loop control for motor speed (screw speed) and a high-level closed-loop control for the finger’s position angle (gear angle) based on measured motor speed and gear angle feedbacks For the low-level closed-loop control, the PID controller is applied Meanwhile, the tuning fuzzy PID (FPID) Controller is designed for high-level closedloop control because of the non-linear system For the force control system, the tuning FPID Controller is also used, based on the calculated contact force feedback from the Contact Force Detector (CFD) block, where the proposed method to determine contact forces in Section is applied as shown in Figure The inputs of the CFD block are rotating angles of phalanges and driving bars, as well as the measured contact forces and motor torque, while the outputs are three calculated contact forces on three pha‐ Figure The tuning FPID diagram Figure The tuning FPID diagram Figure 10 Membership functions of inputs |e| Figure 10 Membership functions of inputs |e| Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping for o Figure 10 Membership functions of inputs |e| Figure 10 Membership functions of inputs |e| where μ(e) and μ(de) are membership values with respect i to input variables, while μout is the membership value with respect to the output variable at the ith rule The centroid de-fuzzification method is used to convert the aggregated fuzzy, which is set to a crisp output value In this case, because the membership functions for the fuzzy output partitions are in Singleton form, the outputs of fuzzy tuners are calculated as: Figure 11 Membership functions of inputs |de| Figure 11 Membership functions of inputs |de| Figure 11 Membership functions of inputs |de| 25 yout = åm i =1 i out 25 i × yout å mouti (29) i =1 i is the output value of the ith rules, which can be where yout Figure 12 Membership functions of the outputs kp, ki and kd determined in Figure 12, while the output of the fuzzy tuner yout is kp, ki or kd These output values of the fuzzy tuners are then substituted into Equation (30) to compute three K i and K d , asthat follows: Figure 12 Membership functions of the outputs k p , k i and k d The detailed FPID controller is shown in Figure Fromparameters, this figure,Kitp , can be seen there are three fuzzy tuners for the The detailed FPID controller is shown in Figure From this K K , and Two input signals are needed for each fuzzy the absolute three output parameters: K p d i Theitdetailed FPID controller shown in tuners Figurefor From this figure, it can be seen thattuner there[23]; are namely, three fuzzy tuners for th can be seen that there are is three fuzzy figure, ( ) K K k K K = + p p p are p max ptuner error | e | and derivative error | de | Triangle and trapeze membership functions then utilized to create the fuzzy K K , and Two input signals are needed for each fuzzy [23]; namely, the absolute three output parameters: K the three output parameters: K p ,p K d dand K i iTwo input ( ) K K k K K = + (30) input partitions Here, five membership functions (VS, S, M, B and B) representing the five input states (very small, i i i i max i signals for each fuzzy tuner namely, thetrapeze membership errorare| eneeded | and derivative error | de | [23]; Triangle and functions are then utilized to create the fuzzy ) inputs’ membership Kd = Kd Details K d + kd ( Kd max | e | andbig | de | Triangle and absolute error derivative error small, medium, and very big), respectively, are used for the controller of -the fuzzy input partitions Here, five membership functions (VS, S, M, B and B) representing the five input states (very small, trapeze membership functions are then utilized to create functions are shown in Figures 10 and 11 (a and b parameters are two constants that are determined in experiment medium, big and very respectively, are used for the controller Details of the fuzzy inputs’ membership the small, fuzzy input partitions Here, five big), membership functions simulation) where Kare , K pmax , K imin,that K imaxareand K d min, K d max are pmin areB)shown in Figures 10input and 11 (a and b parameters two constants determined in experiment (VS,functions S, M, B and representing the five states (very the ranges of K , K and K , respectively p i d small, small, medium, big and very big), respectively, are simulation) There are three outputs from the three fuzzy tuners, kp, ki and kd, with the outputs having ranges from to Singleton used for the controller Details of the fuzzy inputs’ mem‐ membership functions areFigures then used for11 the partitions results Figure 12 shows five membership functions (VS, S, bership functions are shown in and (afuzzy and b output There are three outputs from with the 10 three fuzzy tuners, kp6 , (very kSimulation i and kd, with the outputs having ranges from to Singleton M, B and VB) corresponding the five output states small, small, medium, big and very big), respectively parameters are two constants that are determined in membership functions are then used for the fuzzy output partitions Figure 12 shows five membership functions (VS, S, We now separate the behaviours between finger and object experiment simulation) M, B and VB) corresponding with the five output states (very medium, big and into foursmall, cases, small, as illustrated in Figure very In all big), cases,respectively the There are three outputs from the three fuzzy tuners, kp, ki |de| distal phalanx is always the last finger in contact with the kp,ki,kranges d and kd, with the outputs having from to Singleton object Therefore, Firstly, the VS S Mthe process follows B four steps:VB membership functions are then used for the fuzzy output |de| input torque of the ADAMS model is issued to move the kp,k i,kmembership d partitions Figure 12 shows five functions (VS, VS/VS/VS VS/VS/VS VS/S/VS VS/S/VS VS/S/VS VS finger; secondly, the virtual force sensors in the ADAMS VS S M B VB S, M, B and VB) corresponding with the five output states model is generated during finger grasping; thirdly, the (very small, small, medium, big and respectively M/S/S M/VS/S S/S/VS S/M/VS S/M/VS S very big), VS/VS/VS VS/VS/VS VS/S/VS VS/S/VSwith VS system will inspect how manyVS/S/VS phalanges in contact the object andM/M/S decide which case of finger behaviour B/S/M B/M/M M/B/S M/B/VS will |e| M M/VS/S M/S/S S/S/VS S/M/VS S/M/VS S|de| be used to determine the contact forces; and, finally, the kp,ki,kd VS S B VB VB/M/B VB/B/B M/B/M M/VB/M M/VB/S forces B M proposed method will to start to calculate contact B/S/M B/M/M M/M/S M/B/S M/B/VS |e| M VS VS/VS/VS VS/VS/VS VS/S/VS VS/S/VS VS/S/VS after contact between the distal phalanx and object VB/VB/VB VB/VB/VB VB/VB/B VB/VB/B VB/VB/B VB S M/VS/S M/S/S VB/M/B S/M/VS VB/B/B M/B/M M/VB/M M/VB/S BS/S/VS S/M/VS In this paper, two simulations are used to apply the Rule table B/M/M of the fuzzy tuners M/B/S M B/S/M M/M/S M/B/VS |e| Table proposed method to determineVB/VB/B the contact forces between VB/VB/VB VB/VB/VB VB/VB/B VB/VB/B VB B VB/M/B VB/B/B M/B/M M/VB/M M/VB/S the phalanges and the object In the first simulation, the The design rules of the fuzzy tuners are shown in Table The MAX-PROD formula is chosen as the main strategy for Table Rule table of the fuzzy tuners VB VB/VB/VB VB/VB/VB VB/VB/B VB/VB/B VB/VB/B torque input is constant, while contact forces in each case the implication process: are calculated by the proposal method based on inputs, Table Rule table of the fuzzy tuners The design rules of the fuzzy tuners are shown in Table The MAX-PROD formula is chosen as the main strategy for such as rotating angles of phalanges and driving bars, the the implication process: measured contact forces and motor torque input The The design rules of the fuzzy tuners are shown in Table results are then compared with measured contact forces The MAX-PROD formula is chosen as the main strategy for from the ADAMS model to prove the correctness of the the implication process: proposed method The second simulation is to apply the position and force control approaches in order to evaluate i m out = max ( m ( e ) × m (de ) ) (28) the convergence and stability of the system Figure 12 Membership functions of the outputs kp, ki and kd 10 Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 6.1 The first simulation results In the first simulation, there are three input torques for each case: T = 1.0 Nm , T = 1.5 Nm and T = 2.0 Nm The contact forces of four cases are shown in Figures 13 to 24 In these figures, the measured contact forces (dash dot lines) are obtained from the force measurement functions in the ADAMS/View environment, while the calculated contact forces (solid lines) are computed by using the proposed method for determining the distributions of contact forces in Section As demonstrated by our simulation results, in case 1, three phalanges contact the object From Figures 13 to 15, with three input torques, 1.0[Nm], 1.5[Nm] and 2.0[Nm], the calculated contact forces are very close to the three meas‐ ured values For instance, with the input torque T =1.0[Nm], the calculated and measured contact forces F and F m1 are 10.1289 [N] and 10.0779 [N] at proximal phalanx, F and F m2 are 5.7138 [N] and 5.6525 [N] at medial phalanx, and F and F m3 are 7.8316 [N] and 7.7661 [N] at distal phalanx, respectively The closing between the calculated and measured contact forces proves that the proposed method can be used to determine the contact forces accurately However, there are significant errors between the measured and calculated contact forces because of the effect of the mass of gripper elements and the frictions in the ADAMS model Therefore, it can see that the three calculated and measured contact forces are also suitable with the input torques T = 1.5 Nm and T = 2.0 Nm , even though the finger is a non-linear system In case 2, the proximal and distal phalanges make contact with the object, which means that there is no contact force between the medial phalanx and object In turn, the measured contact force F m2 on the medial phalanx is zero, and the calculated force F is also set to be zero, as illus‐ trated in Figures 16 to 18 From these above figures, it can see that the calculated contact forces on the proximal and distal phalanges, F (14.3496[N], 20.7475[N], 27.0532[N]) and F (8.1763[N], 12.6521[N], 17.3404[N]), are still close to the measured values, F m1 (14.1518[N], 20.3435[N], 26.7318[N]) and F m3 (8.1505[N], 12.6104[N], 17.2194[N]) It can be seen that the proposed method still determines the contact forces precisely in the case of fewer-than-n phalanx grasping Figures 19 to 24 show the results of cases and Analyses of these two cases are similar to case Finally, as demonstrated in four cases, the simulation results show that the proposed method is very effective for determining the contact forces in the case of fewer pha‐ langes touching the object in finger grasping 6.2 The second simulation results In the second simulation, the integrated control system, which combines the position and force control processes in finger grasping, is used Firstly, the performance of the position control process is based on the desired angle position inputs until the distal phalanx touches the object, after which the system switches to the force control process In the force control process, the contact force feedback is the calculated contact force on the distal phalanx, which is made using the proposed method There are two desired contact forces on distal phalanx in four cases of gripper behaviour: F 3d = 10 N and F 3d = 14 N As demonstrated by our second simulation results, in case 1, three phalanges contact the object Figure 25 shows the desired and real angle position of the finger It can be seen that the desired value is 1.4 [rad] while the real value comes up to 0.7 [rad] because the finger is prevented by the object During the time from to 0.65 [s], the real angle is close to the desired value, thereby proving that the position control process works well and is stable Figures 26 and 27 show the calculated forces from CFD and the measured contact forces from the ADAMS model, as well as comparing with the desired contact forces For instance, in Figure 26, the calculated force F (dash line), after three oscillations, goes to the steady state and is close to the desired contact force, F 3d = 10 N (cyan line) It proves that the system is very stable and convergent, and its stability and convergence are also expressed, while the desired input F 3d changing from 10[N] to 14[N], as shown in Figure 27, F is still close to the F 3d Meanwhile, the three measured contact forces, F m1, F m2 and F m3, from the ADAMS model are still close to the three calculated contact forces, F 1, F and F 3, in case with F 3d = 10 N and F 3d = 14 N , as illustrated in Figures 26 and 27 It shows that the proposed method is still precise in the control application, even though the finger is a non-linear system In case 2, the proximal and distal phalanges contact with the object, it means that there is no contact force between the medial phalanx and the object The desired and real angle positions of the finger are shown in Figure 28 The real value comes up to 0.95 [rad] because the finger is prevented by the object During the time from to [s], the real angle is close to the desired value, which proves that the position control process still works well and with stably The calculated forces F (dash line) still go to the steady state and are close to the desired contact force, F 3d = 10 N and F 3d = 14 N (cyan line), while the measured contact forces F m1 are also close to the calculated contact forces F 1, as shown in Figures 29 and 30 It proves that the system is still stable and convergent, and that the proposed method still determines the contact forces precisely in the case of fewer-than-n phalanx grasping Figures 30 to 36 show the results of cases and Analyses of these two cases are similar to the case In total, the second simulation results show that the system can perform stably, while the proposed method is very effective for estimating the contact forces as well as in control applications in the case of fewer phalanges touch‐ ing the object in finger grasping, as demonstrated by four cases Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping 11 F1[ N ] F [ N ] 8 t  0.631s t  0.631s F3[ N ] F [ N ] F2 [ N ]F [ N ] F2 [ N3] F1[ N ] 0.0 t  0.798s 15 10 0.0   F3[ NF2] [ NF3] [ N ]F2 [ N ]F1[FN2 []N ] F1[ N ] F1[ N ] 12 0.3   0.6 0.9 0.6s t  0.631 0.3 0.9 1.2 1.5   1.2 0 0.0 0.0 F3[ N ] 0.0 0 0.0 0.0 0.30.3 0.3 0.6 0.6 0.9 0.9 0.6 0.9 1.5 1.2 Measured Force Calculated Force 0.3 0.3 0.6 0.6 0.9 Time [s] 0.9 06 Figure 13 Contact forces with parameters 0.0 1.2 1.2 0.3 0.6 1.2 1.5 1.2 Measured Force Calculated Force T = 1.0 Nm , k1 = 0.037 m , k2 = 0.018 m and k3 = 0.018 m Time in case[s] 0.9 1.2 Measured Force t  0.798s 15 10 0.0 0.6 15 0.4 100.0 0.6 0.2 0.3 0.6 0.9 1.2 1.5 1.5 t  0.798s 0.4 0.0 0.0 90 0.2 0.0 0.6 0.0 0.40.0 39 0.2 06 0.0 0.0 0.0 F3[ N ]   0.3 0.6 0.9 1.2 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 Time [s] 0.9 1.2 1.5 0.9 Figure60.0 16 Contact 0.3 forces with parameters 0.6 Measured Force Calculated Force Measured Force1.5 1.2 Calculated Force T0.9= 1.0 Nm 1.2 , k = 0.048 1.5 m Figure 16 Contact forces with parameters T1  1.0[ Nm]1, k1  0.048[m] and k3  0.020[ Time [s] and k33= 0.020 m in case Measured Force Calculated Force Calculated Force 24 Figure 13 Contact forces with parameters T1  1.0[ Nm] , k1  0.037[m] , k2  0.018[ ] in case m] and tk3   0.018[  0.3 0.484 s mwith Figure 16 Contact forces 0.6parameters 0.9 T1  1.0[ Nm 1.2 ] , k1  0.048[ 1.5 m] and k3  0.020 18 0.0 15 Time [s] 0.9 1.2 10 15 0.0 10 12 0.0 412 F3[ N ]F [ N ] F2[ N ] F [ N ]   08 0.0 12 t  0.393s 0.3 0.3 0.3 F3[ N ] 0.3 0.3 12 24 0.484ms] k2  0.018[ and k3t0.018[ 18m]16 Figure Contact forces 120.0 0.3 0.6 24 t  0.484 s 0.4 18   in case with parameters T1  1.0[ Nm] , k1  0.048[m] and k3  0.020[ 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 Time [s] 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5   0.6 0.9 0.6 0.9 0.6 0.9 1.2 1.2 1.5 1.5 1.2 1.5 80 0.0 12 0.0 Time [s] F3[ NF]2 [FN3[] N ]F2 [ N ] F1F[2N[ N ] ] F1[ N ] F1[ N ] 0.6 Figure 13 Contact forces with parameters T1  1.0[ Nm] , k1  0.037[m] , F1[ N ]   0.3 t  0.393s 0.6 0.9 0.6 0.9 Time [s] 1.2 1.5 Measured Force Calculated Force 1.2 Measured Force Calculated Force 1.5 0.0 0.6 12 0.2 0.4 0.0 0.0 0.2 0.0 0.6 12 0.0 0.4 0.0 12 0.2 08 0.0 0.0 0.0 12 80.0 F3[ N ] 0.0 0.9 Measured Force Calculated Force 1.2 Measured Force1.5 Calculated Force Figure 17 Contact forces with parameters T1  1.5[ Nm] , k1  0.048[m] and k3  0.020 Time [s] Measured Force Calculated Force 0.0 0.3 0.6 0.9 1.2 1.5 Figure 17 tContact forces with parameters T = 1.5 Nm , k1 = 0.048 m Figure 14 Contact forces with parameters T = 1.5 Nm , k1 = 0.037 m ,  0.343 s0.018[   0.018[ 27 0.0 0.6parameters 1.2] , k  0.048[ 1.5 m] and k  0.020 Time [s] Figure 14 Contact forces with parameters T1  1.5[ Nm] , k1  0.037[m] , k2Figure  m ] and k3 0.3forces mwith ] in case 0.9 T1  1.5[ Nm 17 Contact Time [s] and k183 = 0.020 m in case k2 = 0.018 m and k3 = 0.018 m in case F [N ] F1[ N ] 16   24 t  k0.343 s   279m] and k1  0.037[m] , k2  0.018[  0.018[ m] in case Figure 17 Contact forces with parameters T1  1.5[ Nm] , k1  0.048[m] and k3  0.020[ t  0.295s 16 0.0 12 F [ N ] F [ N ]F2 [ N ] F3[ N ] 80 0.0 12 0.0 15 0.0 10 15 10 0.0 0.0 0.3 0.3 0.3 0.3 0.3 0.3 F3[ N ] F3[ N ]F2F[ 3N[ N ] ]F2 [ N ] FF12[[NN]] F1[ N ] F1[ N ] 24   Figure t 14  0.295 s forces with parameters T1  1.5[ Nm] , Contact   0.6 0.9 0.6 0.9 0.6 0.9 0.6 0.9 0.6 0.6 Time [s] 0.9 0.9 1.2 1.2 1.2 1.2 1.5 1.5 1.5 1.5 Measured Force Calculated Force Measured Force 1.5 1.2 Calculated Force 1.2 1.5 18 0.0 0.6 27 0.40 180.0 0.6 0.2 0.4 0.0 0.0 0.0 0.2 0.6 15 0.0 0.4 10 0.0 0.2 15 0.0 100.0 0.0 15 100.0 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 Time [s] 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 t  0.343s Measured Force Calculated Force Measured Force Calculated Force Time [s] Figure 18 Contact forces with parameters T1  2.0[ Nm] , kForce Measured  0.048[m] and k3  0.020 0.0 Calculated Force 1.2 1.5 Figure 18 Contact forces with parameters T1  2.0[ Nm] , k1  0.048[m] and k3  0.020 Time [s] Figure 15 Contact forces with parameters T1  2.0[ Nm] , k1  0.037[m] , k2  0.018[m] and k3  0.018[m] in case Time [s] Figure 15 Contact forces with parameters T = 2.0 Nm , k = 0.037 m , Figure 18 Contact forces with parameters T = 2.0 Nm , k = 0.048 m 1 T  2.0[ Nm]1, k  0.037[m] , k  0.018[m] and k  0.018[ m] in case Figure 15 Contact forces with parameters forces with parameters T1  2.0[ Nm] , k1  0.048[m] and k3  0.020[ 1 Figure 18 Contact k2 = 0.018 m and k3 = 0.018 m in case 12 Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 and k3 = 0.020 m in case 0.4 0.6 0.2 t  0.264s 0.4 0.6 0.0 0.0 0.2 0.4 t0.3 0.264s 0.0 0.2 0.0 62 0.0 0.0 640 0.0 12 42 20 0.0 124 0.0 1280 0.0 0.4   0.6 0.9 1.2 0.6 0.9 0.3 0.6 0.9 1.2 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 Time [s] 1.2 1.5 0.3 0.3 0.6 1.5 1.5 Measured Force Calculated Force 0.2 0.6 0.0 0.40.0 0.6 0.6 0.2 0.4 0.4 0.0 0.2 0.20.0 0.6 F12[ N ] F1[ N ]       F1[ N ] t  0.264s   t  0.218s t  0.218s 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.0 40.0 0.2 0.3 0.6 0.9 0.0 0.0 0.0 0.3 0.3 0.6 0.6 0.9 0.9 0.0 0.0 0.40.0 0.0 0.6 0.2 0.4 F3[ N ]F3[ N ] F2F[3N[ N ]F]2[ N ] 0.6 F3[ NF]3[ N ] FF2[ [NN]F]2 [ N ] F [F 1[ NF] [ N ] F [ N ] N] 1   0.3 t  0.218s 1.2 Measured Force1.5 Calculated Force 1.2 1.2 1.5 1.5 [ N] ]F1[ N ] F [ N ] F3[ N ]F3[ N ] FF2 [[NN]F ] [ N ] F2F[1N F1[ N ] Time [s] Measured Force Measured Force Calculated Force Calculated Force Figure0 22 Contact forces with parameters T = 1.0 Nm and Measured Force 0.0 T = 1.0 Figure 19 Contact 0.3 forces with parameters Nm , k = 0.023 m 0.6 0.9 1.2 1.5 22 Contact Nm k3 1.5 0.018[m] in case T1  1.0[ Nm ] , k2  0.023[ m] and Figure k3  0.017[ m] in case Figure 19 Contact forces with parameters with 40.0 0.3forces 0.6parameters 0.9 T1  1.0[ 1.2] andForce Measured Time [s] Calculated Force Time [s] k = 0.018 m in case and k03 = 0.017 m in case 3 Calculated Force 0.0 0.3 0.6 0.9 1.2 1.5 Time [s] 0.3 s 0.6 0.9 1.2 1.5   0.6   0.6 0.0 t  0.128 t  0.163s Time [s] k3  0.017[ m] in case Figure 19 Contact forces with parameters T1  1.0[ Nm] , k2  0.023[m] and Figure 0.4 0.4 22 Contact forces with parameters T1  1.0[ Nm] and k3  0.018[ m] in case 0.0 0.4 0.6 0.0 0.2 0.4 0.0 0.2 0.0 93 t  0.163s 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.0 60.0 0.0 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 15 10 0.0 15 00 0.0 100.0 0.3 0.6 0.9 0.3 0.3 0.6 0.6 0.9 0.9 155 1.2 Measured Force1.5 Calculated Force 1.2 1.2 1.5 1.5 Measured Force Calculated Force Time [s] 100 0.0   0.0 0.40.0 0.6 0.6 0.2 0.4 0.4 0.0 0.2 0.20.0 0.6 0.0 0.0 0.40.0 0.0 0.6 15 0.2 0.4 10 0.0 0.0 0.2 15 0.0 0.0 100.0 [ NF] [ N ] F3[ N ]F3[ N ] FF [3N ]   FF21[ N ] F1[ N ] 0.219 Contact forces with parameters T1  1.0[ Nm] , k2  0.023[m] and k3 0.2  0.017[m] in case Figure   0.6 t  0.163s Figure forces   0.6 22 Contact t  0.128 s with parameters T1  1.0[ Nm] and k3  0.018[m] in case 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 0.3 0.3 0.6 0.6 t  0.128s Time [s] 0.9 0.9 1.2 Measured Force1.5 Calculated Force 1.2 1.2 1.5 1.5 Measured Force Calculated Force 15 10 0.3 0.6 0.9 1.2 Measured Force1.5 0.0 0.3forces with 0.6parameters 0.9 T1  1.5[ 1.2] andForce 1.50.018[ m] in case 23 Contact Nm k3  Measured Figure 20 Contact forces with parameters T1  1.5[ Nm] , k2 Force  0.023[m] andFigure k3  0.017[ m] in case Time [s] Calculated Time [s] Calculated Force 0.0 0.6 0.3 0.6 0.9 1.2 0.6 0.0 1.5   Figure 23 t  Contact parameters T = 1.5 Nm and [s] 0.095s forces withTime   20 Contact Figure forces Nm , k2 = 0.023 m t  0.114 s with parameters Time [s]T =T1.5 Figure 23 Contact forces Figure 20 Contact forces with parameters 0.017[ m] in case with parameters T1  1.5[ Nm] and k3  0.018[m] in case  1.5[ Nm] , k2  0.023[m] andkk3=0.4 0.4 0.018 m in case and k3 = 0.017 m in case 0.6 0.9 1.2 1.5 F1[ N ] F1[ N ] 0.3 0.2 0.2 0.6 Figure 23 Contact forces with parameters T1  1.5[ Nm] and k3  0.018[m] in case   0.6 20 Contact Figure forces in case ]0.095 s t  0.114 s with parameters T1  1.5[ Nm] , k2  0.023[m] and  k3  0.017[t m   0.4 0.0 0.0 0.2 0.0 0.0 60.0 21 93 14 60 0.0 21 0 14 0.0 0.0 217 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 0.3 0.3 0.6 0.9 0.6 Time [s] 0.9 t  0.114 s 1.2 Measured Force1.5 Calculated Force 1.2 1.2 1.5 1.5 Measured Force Calculated Force 0.0 0.4 0.0 0.6 0.6 0.2 0.4 0.4 0.0 0.20.0 0.2 0.6 [ NF] [ N ] FF21[[N N]] F1[ N ] F3[ N ]F3[ N ] FF [3N ] F3[ N ] F3[ N ] F2 [FN[]NF]2 [ N ] FF1[[NN]] F1[ N ] 0.0 0.4 0.0 0.6 0.29   0.0 0.0 0.4 0.0 0.0 0.6 0.2 21 0.4 14 0.0 0.20.0 21 0.0 0.0 140.0 0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5 0.3 0.3 0.6 0.6 0.9 0.9 1.2 1.2 1.5 1.5 0.3 0.6 0.9 0.3 0.3 0.6 0.6 t  0.095s 21 Time [s] 0.9 0.9 1.2 Measured Force1.5 Calculated Force 1.2 1.2 1.5 1.5 Measured Force Calculated Force 14 14 Figure k3  0.017[ m] in case 0.021 Contact 0.3forces with 0.6 parameters 0.9 T1  2.0[ Nm 1.2 ] , k2  0.023[ 1.5 m] andFigure 0.0 0.3 0.6 0.9 T  2.0[ 1.2 ] and Measured Force Nm k3 1.5  0.018[m] in case 24 Contact forces with parameters Time [s] Time [s] Measured Force Calculated Force 0.0 0.3 0.6 0.9 1.2 1.5 Calculated Force 0.0 0.3 0.6 Time [s] 0.9 1.2 1.5 Figure 21 Contact forces with parameters T1  2.0[ Nm] , k2  0.023[m] andFigure k3  0.017[ m] in case with parameters T1  2.0[ Nm] and k3  0.018[m] in case Time [s] 24 Contact forces Figure 24 Contact forces with parameters T = 2.0 Nm and Figure 21 Contact forces with parameters T = 2.0 Nm , k2 = 0.023 m Figure 24 Contact forces with parameters T1  2.0[ Nm] and k3  0.018[m] in case Figure 21 Contact forces with parameters T1  2.0[ Nm] , k2  0.023[m] andk k=3 0.018  0.017[ ] in case m case and k3 = 0.017 m in case Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping 13 Angle Angle Position [rad] Angle Position Position [rad] [rad] 1.2 1.5 1.2 1.5 1.2 1.2 0.9 0.9 0.6 0.6 0.6 0.9 0.6 0.6 0.3 0.3 0.3 0.6 0.3 0.3 0.6 0.3 0.6 Time [s] 0.9 0.9 1.2 1.2 Time [s] Figure 0.0 25 The desired and real angle positions of finger in case 0.3 0.6 0.9 1.2 1.5 1.5 Figure 25 The desired and realTime angle[s]positions of finger in case 16 16 Figure 25 The desired and real angle positions of finger in case 16 Contact Forces [N] Contact Forces Contact Forces [N] [N] 12 12 12 8 4 00.0 0.0 0.3 0.3 0.6 0.6 0.9 F1 calculated force F2 calculated calculatedforce force F1 F3 calculated calculatedforce force F2 Fd3 desired force F3 calculated force F1 calculated force Fd3 desired force Fm1 measured force F2 calculated force Fm1 measured force Fm2 measured force F3 calculated force Fm2 measured force Fm3desired measured force Fd3 force Fm3 measured measured force force Fm1 1.2 0.3 0.0 0.00.0 0.0 1.5 Figure 25 The desired and real angle positions of finger in case 0.0 Desired Angle Pos Real Angle Pos 1.2 0.9 0.9 0.9 1.2 0.0 0.3 0.0 0.0 0.0 Desired Angle Pos Desired Angle Real Angle PosPos Real Angle Pos Angle Angle Position Position [rad] [rad] Angle Position [rad] 1.5 1.5 1.5 Desired Angle Pos Real Angle Pos Desired Angle Pos Real Angle Pos Desired Angle Pos Real Angle Pos 0.3 0.3 0.6 0.6 0.9 Time [s] 0.9 Time [s] 1.2 1.2 1.5 1.5 Figure 0.028 The desired and real angle positions of finger in case Figure0.0 28 The desired and real of finger 0.3 0.6angle positions 0.9 1.2 in case 1.5 Figure 28 The desired and real angle of finger in case Time positions [s] F1 calculated force F1 calculated force F2 desired calculated force Figure 28 The and real angle positions of finger in case 16 F2 calculated force F3 calculated force 16 F3 calculated force F3 desired force F1 calculated force F3 force F1 measured force F2desired calculated force 16 F1 measured F2 measured force F3 calculatedforce force 12 F2 force F3 measured force F3measured desired force 12 F3 measured force F1 measured force F2 measured force 12 F3 measured force Contact Contact Forces Forces [N][N] Contact Forces [N] 1.5 8 4 00.0 0.0 1.5 Time [s] 0.9 Fm2 measured force1.5 1.2 Time [s] Fm3 measured force 0.3 0.3 0.6 0.6 0.9 Time [s] 0.9 Time [s] 1.2 1.2 1.5 1.5 Figure 29 The calculated and measured contact forces of finger with the force feedb 0.0 The calculated 0.3 0.6 0.9 forces of1.2 1.5 measured contact finger with the Figure 26.The Thecalculated calculated contact of finger with theFigure force29 feedback controland inand case 1, F3d[s]  10[ N ] forces 0.026 0.3 0.6measured 0.9 Time Figure 29 The calculated measured contact of finger with the force feedb andand measured contact forcesforces of1.2 finger with 1.5 the Figure Time [s] contact forces of finger with theforce Figure 26 The calculated and measured force feedback control in case F  10[ N ] control in case 2, F 1, = 10 N d 25feedback 3d force feedback control in case 1, F 3d = 10 N F1 calculated force 25 8 4 00 0.0 0.0 0.3 0.3 F1 calculated force F1 calculated force F2 calculated force F2 calculated force F3 calculated force F3 calculated force F1 calculated force Fd3 desiredforce force Fd3 desired F2 calculated force Fm1 measured force Fm1 measured force F3 calculated force Fm2 measured force Fm2 measured force Fd3 forceforce Fm3desired measured Fm3 measured force Fm1 measured force 0.6 0.9 Fm2 measured 1.2 1.5 force1.5 0.6 0.9 1.2 Time [s] Time [s] Fm3 measured force Contact Contact Forces Forces [N][N] Contact Forces [N] ContactContact ForcesForces [N] Contact Forces[N] [N] F1 calculated force Figure 29 The F2 calculated calculated and forcemeasured contact forces of finger with the force feed Figure 26 The calculated and measured contact forces of finger with the force feedback in case F2 control calculated force1, F3d  10[ N ] 16 F3 calculated force 16 25 F3 calculated force 20 F3 force F1desired calculated force 20 F3 force F1 measured force F2desired calculated force F1 16 F2 measured force F3measured calculatedforce force F2 force 12 F3 measured force 20 F3measured desired force 12 15 F3 measured force F1 measured force 15 F2 measured force 12 F3 measured force 15 10 10 10 5 05 00.0 0.0 0.3 0.3 0.6 0.6 0.9 Time [s] 0.9 Time [s] 1.2 1.2 1.5 1.5 0 Figure and0.6measured 0.9 contact forces with the force feedb 0.030 The calculated 0.3 1.2 of finger 1.5 0.0 0.3 0.6 measured 0.9contact forces 1.2 of finger 1.5with the force Figure feedback controlinincase case 1,  14[ 30 The calculated and measured contact Figure27 27 The The calculated calculated and and measured forces of finger with the Figure force feedback control 1,Time F3F 14[ N ]N ] forces of finger with the force feedb d[s] Time [s] contact d3 Figure 27 The calculated and measured contact forces of finger with the Figure 30 The calculated and measured contact forces of finger with the Figure 30 The control calculated and 1, measured forces of finger with the force feed Figure 27 The calculated and measured contact forces of finger with the force feedback in case F3d  14[ contact N] force feedback control in case 1, F 3d = 14 14 N Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 force feedback control in case 2, F 3d = 14 N Angle Position [rad][rad] Angle Position Angle Position [rad] 1.2 1.5 0.9 Real Angle Pos 1.2 1.2 1.5 Desired Angle Pos Real Angle Pos Angle Position [rad] Angle Position [rad] Angle Position [rad] 1.2 1.5 1.5 Desired Angle Pos Real Angle PosPos Desired Angle 0.6 0.3 0.3 0.6 0.3 0.6 0.3 0.3 0.6 0.3 0.6 Time [s] Time [s] 0.9 1.2 1.5 0.9 1.2 1.5 Figure desired andand realreal angleangle positions of finger case 3in case Figure31 31.The The desired positions of in finger 0.0 Figure0.031 The desired and real of finger in case 0.3 0.6 angle positions 0.9 1.2 1.53 F1 calculated force F2 calculated force F1 16 F3 calculated calculated force force F2 calculated force3 Figure 31 The desired and real angle positions of case F3finger desiredinforce F3 force F1 calculated measured force F3 desired force 12 F2calculated measured force F1 force 16 F1 measured force F3calculated measuredforce force F2 12 F2calculated measuredforce force F3 F3desired measured force F3 force Time [s] Forces [N] [N] Contact Forces ContactContact Forces [N] 16 F1 measured force F2 measured force F3 measured force 12 48 04 0.0 0.0 Desired Angle Pos Real Angle Pos 0.6 0.9 0.6 0.6 0.9 0.0 0.0 0.3 0.0 0.0 Desired Angle Pos Desired Angle Pos Real Angle Pos Real Angle Pos 0.9 1.2 0.9 0.9 1.2 0.3 0.6 0.3 0.6 Time [s] 0.9 1.2 1.5 0.9 1.2 1.5 0.0 0.0 0.3 0.0 0.0 0.3 0.3 0.6 0.9 Time [s] 0.9 Time [s] 0.6 1.2 1.5 1.2 1.5 Figure 34 The desired and real angle positions of finger in case Figure 34 The desired and real angle positions of finger in case 0.0 Figure and 0.6 real angle positions of 1.2 finger in case 0.3 0.9 1.5 15 0.0 34 The desired Time [s] 15 Figure 34 The desired and real angle positions of finger in case 12 15 12 Contact Forces ContactContact Forces [N] Forces [N] [N] 1.5 1.5 12 9 6 30.0 0.0 0.3 0.3 F1 calculated force F2 calculated force F3 F1 calculated force calculated force F3 F2 desired force force calculated F1 F3 measured force calculated force F2 measured desiredforce force F1F3 calculated force F3 measured force measured force F2F1 calculated force measured force F3F2 calculated force 0.6 0.9 1.2 1.5 measured F3F3 desired force force Time [s] F1 measured force 0.6 0.9 1.2 F2 measured force 1.5 Time [s] measured force with and measured contactF3forces of finger Figure 35 The calculated Time [s] feedback control in case 3, F Figure 32 The calculated and measured contact forces of finger with the force 3d  10[ N ] the force feedb 0.035 The calculated 0.3 0.9 1.5the Figure and 0.6 measured contact forces 1.2 of finger with 0.032 The calculated 0.3 0.6 0.9 forces of 1.2finger with1.5 Figure 35 The calculated andTime measured contact forces of finger with the force fe Figure and measured contact the Figure 32 The calculated and measured force feedback control in4,case F3[s] Time [s] contact forces of finger with theforce dN 10[ N ] feedback control in case F 3, = 10 force feedback control in case 3, F 3d = 10 N 3d 16 16 Figure 35 The calculated and measured contact forces of finger with the force feed Figure 32 The calculated and measured contact forces of finger with the force feedback control in case 3, F3d  10[ N ] 16 12 16 12 16 12 12 12 12 48 04 0.0 0.0 Contact [N] Contact Forces [N] Forces Contact Forces [N] Contact Forces [N] [N] Contact Contact Forces [N] Forces 16 0.3 0.6 Time [s] 0.9 F1 calculated force F2 calculated force F1 F3 calculated force F2 force F3 calculated desired force F3 F1 calculated measured force F1 calculated force F2 desired measured force F3 force F2 calculated force F3calculated F1 measuredforce force F3 F2desired measured force F3 force 1.2 F3measured measured force1.5 F1 force F2 measured force 1.2 1.5 F3 measured force 8 40.0 0.3 F1 calculated force F2 calculated force F3 calculated force F3 F1 desired force force calculated F1 F2 measured force calculated force F2 F3 measured force calculated force F1 calculated force F3 F3 measured desiredforce force F2 calculated force measured force F3F1 calculated force 0.6 0.9 1.2 1.5 measured F3F2 desired force force Time [s] measured force F1F3 measured force F2 measured force 0.9 1.2 1.5 the and0.6 measured contact forces of finger F3 measured forcewith 0.036 The calculated 0.3 Figure force feedb Time [s] Time [s] feedback control in case 3, F 33 The calculated and measured contact forces of finger with the force Figure  14[ N ] d 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.3 0.6 0.9 1.2 1.5 Time [s] Discussion Time [s] 36 The calculated and measured forces of finger with the force fe Figure 33 The calculated and measured contact forces of finger with the Figure force feedback control in case 3, F3d  14[contact N] 0.3 0.6 0.9 Figure 33 The calculated and measured contact forces of finger with the Figure 36 The calculated and measured contact forces of finger with the Figure 36 The control calculated and3,measured contact forces of finger with the force feed Figure 33 The calculated and measured contact forces of finger with the force force feedback in case F d N 14[ N ] feedback control in case 4, F =314 force feedback control in case 3, F 3d = 14 N Discussion Discussion 3d Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping 15 Discussion The introduction of two new matrices in [15] allows the system to calculate the contact forces on the phalanges through the input torque of the finger actuator in the case of full-phalanx grasping Configurations of the finger leading to stable grasps are considered by using these two matrices However, in order for a less-than-n phalanx grasp to be stable, every phalanx in contact with the object should have a strictly positive corresponding force Actually, the contacts appear not only with all phalanges, but also with fewer-than-n phalanges in object grasping The corre‐ sponding generated forces for phalanges not in contact with the object should be zero, since the latter forces can also be seen as the external forces needed to counter the actuation torque Moreover, calculating contact forces in the case of fewer-than-n phalanges touching the object by using Equation (14) can be a problem because of the singularity of the J T matrix The proposed method in this paper, then, solves the above special case A general mathematical analysis of the distributions of contact forces for the under-actuated finger was presented in the case of full-phalanx grasping, while taking into account cases of fewer-than-n phalanx grasping Lionel Birglen et al believed that static analysis can help refine under-actuated finger designs in term of geometric parameters in order to achieve stable grasps and phalanx force distribution, avoiding weak last phalanges that cannot ensure sufficient force to secure the grasp [15] Furthermore, with regard to the finger design process, the proposed method provides designers with a tool to select motor specifications (e.g., motor torque) and evaluate the object grasping forces, as well as provide the sensor-based contact force feedbacks for control strategies As mentioned in Section 5, the simulation technique based on the virtual prototype significantly reduces manufactur‐ ing cost and time compared to the traditional build-andtest approach The virtual prototyping approach is an integrating software solution that consists of modelling a mechanical system, simulating and visualizing its 3D motion behaviour under real world operating conditions, as well as refining and optimizing the design through iterative design studies The advantages of this simulation technique consist of conceiving a detailed model, which is used in a virtual experiment similar to a real scenario Virtual measurements of parameters and components of the mechanical model can also be carried out conveniently In light of the above reasons, authors have decided to choose the ADAM/View software to simulate the underactuated finger This includes real world operating condi‐ tions, such as material finger, friction parameters of joints, contact parameters (stiffness, force exponent, damping ratio and penetration depth) between phalanges and object From the virtual prototyping process, the real system will be manufactured based on the simulation results The proposed method in this paper offers good simulation results in determining the contact forces and control 16 Int J Adv Robot Syst, 2016, 13:14 | doi: 10.5772/62131 application The system is stable and convergent There‐ fore, the proposed method can be applied in the real time experiment Javier Felip et al implemented and validated the robust grasp primitive for the BarrettHand gripper based on the sensor feedbacks from torque/torque and tactile sensors [19] In order to control under-actuated hands, the MARS prototype finger’s phalanges has been equipped with Force Sensing Resistors (FSR) to allow experimental testing of the added value of tactile sensing, as shown in Figure [20] In our gripper system, the potentiometers are installed at the phalanx joints to get the phalanx angles, while tactile sensors provide the contact positions between phalanges and object, while the torque sensors are also applied to measure the input torques These sensors provide all the parameters needed to apply the proposed method to estimate the contact forces, which will be used to force feedbacks in the finger control strategies In turn, this provides a low-cost, high perform‐ ance and easy-to-use operation system Given the nonlinear system, intelligent control approaches have been developed, such as the sliding mode controller (SMC) or the Fuzzy Logic Controller (FLC) In gripper control strategy, tactile sensing is of the utmost importance By using these sensors, one can design a closed-loop force controller and, for example, detect whether the grasping forces are on the edge of vanishing and, in the process, resume the grasping actuation Moreover, the robustness in grasp task is not only achieved by designing sensorbased controllers, but also by combining several controllers with different optimisation goals This will be done in real time experimentation in future studies Conclusions This paper presents a mathematical analysis to determine the distribution of contact forces for the under-actuated finger in general grasping cases of an under-actuated robotic hand Due to the importance of the contact forces, the proposed method for static analysis of the distributions of the contact forces focuses on the n-DOF under-actuated finger The simulation results, with the 3-DOF underactuated finger from the ADAMS model, show the effec‐ tiveness of the mathematical analysis method, as well as comparing the measured results with, especially, the stability and convergence in control application The system can find magnitudes of the contact forces at the contact positions between the finger phalanges with the object Acknowledgements This work(2015.07-2015.12) is the result of a study on the "Leaders INdustry-university Cooperation" Project, supported by the Ministry of Education, Science & Tech‐ nology(MEST) In addition, this work (2013.03-2014.11) was supported by Development Program of Local Science Park, funded by Ulsan Metropolitan City and MSIP(Min‐ istry of Science, ICT and Future Planning) 10 References [1] Hirose, S and Umetani, Y (1978) The develop‐ ment of soft gripper for the versatile robot hand Mechanism and Machine Theory: 13, 351-358 [2] Birglen, L and Gosselin, C (2004) Optimal Design of 2-Phalanx Under-actuated Fingers Proceedings of the International Conference on Intelligent Manipulation and Grasp: 110–116 [3] Jacobsen, S.C., Iversen, E.K., Knutti, D.F., Johnson, R.T and Biggers, K.B (1986) Design of the UTAH/MIT dextrous hand IEEE International Conference on Robotics and Automation: 1520– 1532 [4] Salisbury, J.K and Craig, J J (1982) Articulated hands: Force control and kinematic issues Interna‐ tional Journal of Robotics Research: 1, 4–17 [5] Bekey, G.A., Tomovic, R and Zeljkovic, I (1999) Control Architecture for the Belgrade/USC Hand in Dextrous Robot Hands Springer-Verlag, New York [6] Butterfass, J., Grebenstein, M., Liu H and Hirzinger G (2001) DLR-hand II: Next generation of a dex‐ trous robot hand IEEE International Conference on Robotics and Automation: 1, 109–114 [7] Corrales, J.A., Jara, C.A and Torres, F (2010) Modelling and simulation of a Multi-fingered Robotic Hand for Grasping Tasks International Conference on Control, Automation, Robotics and Vision: 1577-1582 [8] Rizk, R., Krut, E and Dombre, E (2007) Graspstability analysis of a two-phalanx isotropic under‐ actuated finger Proceedings of the International Conference on Intelligent Robots: 3289-3294 [9] Dandash, G., Rizk, R., Krut, E and Dombre, E (2011) A pseudo-isotropic three-phalanx underac‐ tuated finger World Congress in Mechanism and Machine Science: 1-8 [10] Peer, A., Einenkel, S and Buss, M (2008) Multifingered telemanipulation - mapping of a human hand to a three finger gripper IEEE International Symposium on Robot and Human Interactive Communication: 465 - 470 [11] Laliberté, T and Gosselin, C.M (1998) Simulation and design of underactuated mechanical hands Mechanism and Machine Theory: 33, 39–57 [12] Laliberté, T., Birglen, L and Gosselin, C (2002) Underactuation in robotic grasping hands Journal of Machine Intelligence and Robotic Control: 4, 1-11 [13] Petković, D., Pavlović, N.D., Žarko Ćojbašić, Z and Pavlović, N.T (2013) Adaptive neuro fuzzy estima‐ tion of underactuated robotic gripper contact forces Expert Systems with Applications: 40, 281–286 [14] Birglen, L and Gosselin, C.M (2006) Force analysis of connected differential mechanisms: Application to grasping International Journal of Robotics Research: 25, 1033-1046 [15] Birglen, L and Gosselin, C.M (2004) Kinetostatic analysis of underactuated fingers IEEE Transac‐ tions on Robotics and Automation: 20, 211-221 [16] Ceccarelli, M., Tavolieri, C and Lu, Z (2006) Design Considerations for Underactuated Grasp with a one D.O.F Anthropomorphic Finger Mechanism International Conference on Intelligent Robots and Systems: 1611 – 1616 [17] Wu, L., Carbone, G and Ceccarelli, M (2009) Designing An Underactuated Mechanism for a Active Dof Finger Operation Mechanism and Machine Theory: 44, 336-348 [18] Ceccarellia, M and Zottolaa, M (2015) Design and simulation of an underactuated finger mechanism for LARM Hand Robotica: Available on CJO 2015 doi:10.1017/S0263574715000648, 15 pages [19] Felip, J and Morales, A (2009) Robust sensor-based grasp primitive for a three-finger robot hand IEEE / RSJ International Conference on Intelligent Robots and Systems: 1811 - 1816 [20] Birglen, L., Gosselin, C.M and Birglen, L (2005) Fuzzy Enhanced Control of an Underactuated Finger Using Tactile and Position Sensors IEEE International Conference on Robotics and Automa‐ tion: 2320 - 2325 [21] Norton, R.L (1992) Design of Machinery New York: McGraw-Hill, Inc [22] McCarthy, J.M (2000) Geometric Design of Linkag‐ es New York: Springer-Verlag [23] Xuan Vinh Ha, Cheolkeun Ha and Jewon Lee, (2013) Fuzzy Vector Field Orientation Feedback ControlBased Slip Compensation for Trajectory Tracking Control of a Four Track Wheel Skid-Steered Mobile Robot International Journal of Advanced Robotic Systems: 10, pages 15 Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under-actuated Finger in Robot Hand Grasping 17 ... under- actuated n-DOF finger Figure Geometric and force parameters of under- actuated n-DO Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under- actuated Finger in. .. Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under- actuated Finger in Robot Hand Grasping 13 Angle Angle Position [rad] Angle Position Position [rad] [rad]... of inputs |e| Xuan Vinh Ha, Cheolkeun Ha and Dang Khoa Nguyen: A General Contact Force Analysis of an Under- actuated Finger in Robot Hand Grasping for o Figure 10 Membership functions of inputs