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Procedia Engineering 41 (2012) 923 – 931

International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)

Dynamic Modeling and Control of a Multi-Fingered Robot Hand for Grasping Task Rim Boughdiriab* , Habib Nasserb , Hala Bezinea , Nacer K. M’Sirdib, Adel M.Alimia, Aziz Naamaneb a REGIM: REsearch Group on Intelligent Machines, University of Sfax, National Engineering School of Sfax (ENIS), BP 1173, Sfax, 3038, Tunisia b LSIS, CNRS UMR 7296. Dom. Universitaire St Jérôme, Avenue Escandrille Normandie–Niemen 13397 Marseille France

Abstract Multi-fingered robot hands have been one of the major research topics since holding an object and manipulating it, are crucial functionalities of several robotic systems, including service robots, industrial robots and wheel-type mobile robots. In this paper, we consider the problem of model-based control for a multi-fingered robot hand grasping an object with known geometrical characteristics. Two main topics are presented. Firstly, this paper attempts to derive a mathematical model of the dynamics of a designed multi-fingered robot hand with five fingers with twenty DOF (three for each finger, two for the thumb and six for the wrist) which grasps a rigid object. We exploit our methodology based on the Lagrange formulation, which is presented in previous work, to identify the parameters of dynamic models of hand-object system. These models are instrumental for the design problems of control for dynamic stable grasping that is discussed on second part. Finally, several simulation results demonstrate the controller performance based on the derived model.

© 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre of Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. Open access under CC BY-NC-ND license.

Keywords: Multi-fingered robot hand; Dynamic Modeling; Model-based control; Simulation; Coupling dynamics.

Nomenclature Σp Σo Σi po ηo ωo qi

Task coordinate system velocity Object coordinate system fixed on the object i-th fingertip coordinate system fixed on the i-th fingertip Position vector of the origin of the object coordinate system Σo with respect to Σp Orientation vector of the origin of the object coordinate system Σo with respect to Σp Angular velocity vector of the origin of the object coordinate system Σo with respect to Σp Joint angle vector of i-th finger

1. Introduction Human hands are a great challenge and fascinate many roboticists since they present tremendous skill and versatility. In fact, human hands have richness architecture due to the important degree of freedom of the fingers, the large number of its

* Corresponding author. Tel.: +216-97-839-810; fax: +216-74-677-545. E-mail address: [email protected]

1877-7058 © 2012 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.07.264

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muscles, the extremely sensitive sensors and the inseparable couple that forms with the brain. Therefore the grasping task is one of the most complex functions reproduced by a robotic system. It involves the use of mechanical systems that must have capabilities of adaptation and perception of the environment. Hence, the integration of sensors is necessary for the location of the fingertips contact position, the determination of the grasp configuration, the measurement of the force applied on the object. Such mechanical systems need to control both the end-effector position and the forces interacting between the robot hand and the object. The control system design is highly complicated due to the imposed constraints, and the large number of actuators that exceed the mobility of the system. Thus, the main control problems in grasping and manipulation tasks are related to the inherent high nonlinearity and coupling in its dynamics. Therefore the development of control strategy is especially challenging. A great many control schemes have been developed in the literature to achieve stable grasp. Despite the diversity of approaches, it is possible to classify most of the design procedures as based on three major approaches: model-based controllers [1,2,3,4,5,6], adaptive controllers [7,8,9] and controllers based on intelligent methods [10,11,12,13]. The model-based multi-fingered robot hand control, purpose of this paper, guarantees an accurate finger positioning. Such controllers suppose the knowledge of the mathematical model parameters which describes the behavior of the system. The research structure presented in this paper is novel in the sense that, the dynamics of the system is generated taking into account the coupling between the fingers and only the important dynamic influence, since in actual works, each finger of multi-fingered robot hands is considered as an independent manipulator. Basically, the model-based control methods can be classified into the following two types: The hybrid position/force control including the computed torque control approach which controls simultaneously the force in some directions and the position in other directions [1,2,3] and the other is impedance control which controls the force indirectly through specification of the mechanical impedance of the grasped object against external forces [4,5,6].In this paper we used the impedance control for the fingers because it has distinct advantages over the hybrid position/force control in the sense that the problem of the large overshoot in positional response caused by the delay in changing the control mode doesn’t exist. Moreover the stability is usually guaranteed. This article is organized as follows. In section 1, a brief literature review related to the control of grasping and manipulation by multi-fingered robots hands is presented. Then in section 2, we exploit our methodology presented in previous work to determine the dynamics equations of the grasped object and the hand. Section 3 describes the proposed control design. In order to prove the validity of the generated dynamic model and the controller performance, simulation results for five-fingered hand grasping a spherical abject is illustrated in Section 4. Finally, Section 5 concludes the paper. 2. System modeling Dynamic modeling is fundamental for the study of model-based controller design. In this section, we derive effective dynamic equations of an articulated robot hand grasping an object by our methodology presented in [14]. Consider a robotic hand with four fingers namely, index finger, middle finger, ring finger and pinky finger with three joints on each finger, the thumb with two effective joints and a wrist, manipulating a rigid object. Each joint represents one DOF, so each finger has three DOF and the thumb has two DOF. The wrist has six DOF. Fig.1 describes the hand object system with different coordinate frames. Fig.2 presents the kinematic model of the specified hand. Fciy Fci

Fcix Fciz

Fcm o

Fci

Fc1

finger 5 1

1DOF 1DOF

cfi

Fe

Object

1DOF

1DOF 1DOF

p

finger 1 6DOFs

finger i Fig. 1. Hand-Object system and coordinate frames

Fig. 2. Kinematic model of the specified hand

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Some assumptions are made to reduce the complexity of the system and facilitate the dynamic formulation by taking into account only the important dynamic influence. The following assumptions are: x x x x x

The hand has a modular structure. The whole hand is divided into two sections: five fingers and a supporting palm. The palm is considered as a rigid body with 6 DoF. Each finger is considered as a rigid body that has a rotational movement with the palm. Deformation of the fingertips is not considered. The contact is punctual and located in the fingertip.

In this work, the formalism of Lagrange is used to study the general structure of the models, to determine the different parameters and to verify their properties. So in order to compute the global dynamic model we need to define the different energies and the corresponding matrices. The inertia matrixes are directly related to the expression of the kinetic energy by the following equation: Ec

n n

1 2 ¦ ¦ M i, j q i q j

(1)

i 1j 1

The inertia matrix is symmetric. We have one solution to determine each element of this matrix. To use certain properties (1) used in control of mechanical systems, the matrix C must satisfy the relation: N q, q M q  2C q, q such that matrix N is antisymmetric. A solution to determine C is to use the Christoffel symbols of the first kind [15]. This is directly implemented in symbolic computation:

C i, j

n § w M i, j w M i,k w M i,k   ¦ ¨¨ wqj w qi k 1© w q k

· qk ¸¸ ¹ 2

(2)

The effects of gravity vector G is calculated as the gradient of the potential energy with respect to the position vector q.

Gq ’ E p q

(3)

In order to generate the dynamic model, we created a configuration file that defines the entire mechanism (frames, parameters, contacts) and we will use it in Maple software [16]. The hand nominal model is developed assuming that the contact is permanent and reduced to a single point for each fingertip. The dynamic equations of the object and robot fingers can be described respectively by:

M o r o Xo  Co ro , ro X o G o ro Fo  Fe

(4)

T M hnd q q  Chnd q, q q G hnd q W hnd  J hnd Fc

(5)

Where Mo Є R6×6 , Co Є R6×6 and Go Є R6×1 , denotes respectively the inertia matrix of the object, the centrifugal and Coriolis forces, and the gravitational force. Fo Є R6×1 is the resultant force applied to the object by the fingers and Fe Є R6h1 is the other possible external force applied on the object. Mhnd(q) Є R20×20 is the inertia matrix of the robot fingers, Chnd(q,q ሶ) Є R20×20 is the Coriolis and centrifugal matrix of the robot fingers and Ghnd(q) Є R20×1 is the gravity force term of the robot fingers. τhnd Є R20×1 is the input joint torques. Jhnd Є R6h6 is the kinematic Jacobian matrix at contact point Ci. The object velocity is given by:

Xo

>p

T o

ZTo

@

T

T o K o ro

(6)

To is a transformation matrix between the original object velocity representation using angular velocity vector ¹o and that representation using roll, pitch and yaw angle.

T o K o

ªI 3 0 º «0 », T r¼ ¬

Tr

0 sin T º ª1 «0 cos )  sin ) cosT » » « «¬0 sin ) cos ) cosT »¼

(7)

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T o K o # I 6

Note that in this paper we consider that ) # 0 , so we have

X o # ro

and

Each robot finger applies a force Fci Є R3h1 through the contact point Ci to the object. Then the resultant force and moment, fo, ηo Є R3h1, applied to the object by multi-fingered hands are given by: 5

¦ F ci

fo no



Fo



Fo

(8)

i 1

¦  poci u F ci 5

i 1

>f

T o

nTo

@

(9)

T

(10)

5

¦ W i F ci

(11)

i 1

Where Wi is a grasp matrix at the contact point Ci expressed by: Wi

ªI 3 º 6u3 «> » ƒ @ u ¬ p oci ¼

(12)

Where I3 Є R3h3 is the identity matrix and [pocih] Є R3h3 is a skew-symmetric which expresses the cross product form of p. Fo

(13)

W Fc

Where W= [W1 W2 W3 W4 W5] and Fc= [Fc1 Fc2 Fc3 Fc4 Fc5]T 3. Proposed controller The problem involves the determination of the required set of input joint torques such that the object tracks a defined trajectory and the contact force converge to desired value. The following assumptions are made to design hand control law: x

The contact between the fingertips and the object is assumed to be punctual without slip or contact break. So the contact point velocity υc=GT υo is equal to the fingertips velocities υF=Jhnd q ሶ

>

x x



@



T 1 q J hnd GT T o ro  GT T o  G T o ro  J hnd q ro=[po ,ηo ] , υo, qi , q ሶi, Fci are measurable. Desired trajectory of the object rod=[podT, ηodT ]T, r oሶ d and r oሷ d are time continuous and bounded. T

rod , ropd, roppd

Fintd

(14)

T T

ro, rop

Generator of desired external force of the object

τ

Fcd Generator of desired contact force

Hand controller

Fc Fingers

q, qp, Fc Fig. 3. Multi-fingered robot hand control architecture

Object

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Grasping is performed in two steps: hand moving and fingers closing. Therefore the conception of the Hand controller, shown in Fig.3, is based on the combination of a PD with constant gravity compensation of approach for maintaining a predefined position and orientation of the wrist and an impedance control approach for the control of the fingers. 3.1. PD control with constant gravity compensation The proposed control is aimed at regulating the position and the orientation of the wrist to desired values. Such controller has a simple structure and an easy implementation. Moreover, if the control gains are carefully chosen, the PD control with constant gravity compensation present an acceptable performance for trajectory tracking with the asymptotic stability. The PD control with constant gravity compensation is expressed as: W wrist G [  k p e  k v e (15) e [  [d





Where ξ denotes the wrist position and orientation vector, [ d is the desired position and orientation vector, e is the tracking error, e denotes the derivative of the tracking error and τwrist is the applied torque. kp, kv are both symmetric and typically diagonal gains and G(ξ ) is the vector of gravitational forces. 3.2. Impedance controller The objective of the typical impedance control approach is to realize the following desired object impedance property.

M D ro  DD ro  ror  K D ro  ror F e

(16)

Where MD , DD, and KD are, respectively, the desired inertia, damping coefficient, and stiffness matrix. These matrixes are diagonal, positive definite and symmetric. A simple way of choosing these matrixes is to determine the diagonal elements in such a way that the impedance is made small in the force control directions so that the object can comply with constraints and can adjust the contact force properly, and it is made large in the position control directions so that the position error is corrected with high feedback gain. ror is the position of the reference frame to which the desired impedance is attached. We substitute (16) into (4), the resultant force Fo needed to realize the desired object impedance property is given by:

Fo

M o  M D ro  DD ro  ror  K D ro  ror  H o ro  Go

(17)

The algorithm of the impedance control can be resumed in the following three steps [17]: i. Calculate the resultant force Fo that realizes the desired object impedance (16), using (17). Note that the desired external force applied to the object is computed based on the dynamic parameters of the object. ii. Determine the contact force Fc that produces Fo obtained in (i) and satisfies the friction condition at the contact points and by adjusting the desired internal force parameter. From equation (13):





d F c G  F o  I 15  G  G F int

(18)

Where G+ is the pseudo-inverse of G .The first term of (18) produces the resultant external force Fo and the second term represents the internal grasping force that does not affect the object motion. iii. Calculate the finger command torque ´hnd that produces Fc obtained in (ii), using (5) and (14). Physically, it is noticed that the computation in step (i) is for the object, that in step (ii) is for the contact points, and that in step (iii) is for the fingers.Our challenge was to describe the multi-fingered hand dynamics in terms of blocs and find a common area that connects the two subsystems studied. These models have almost the same major components. Thus, we have identified two different subsystems as follows: translations and rotations of the palm and the finger dynamics. Afterwards we have splited up each part in the form of blocs. Hence the possibility of applying our observers and robust control and switch easily between different systems. Furthermore, the calculation in step (iii) can be decomposed into that for each finger without considering that the parameters of the dynamic model of the robot hand as diagonal of each finger.

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4. Simulation results and discussions An example of simulation is performed to show the effectiveness of the proposed architecture where the dynamic parameters of the object and the robot fingers are identified by our methodology presented in previous work. The proposed control law was implemented and evaluated for the case in which five multi-fingered hand grasp a spherical object of radius of 0.07 m and mass of 0.2kg. The initial values of the parameters of simulation are set to zero. We suppose that the hand is fully extended and the initial object position is located at the center of the palm. For the control of the wrist, the parameters of the control law are set to kp=[1000 1000 1000 1000 1000 1000], kv=[100 100 100 100 100 100] to achieve the desired position and orientation. The desired trajectory of the object is given by 5 order polynomial in time with initial rod(0)= [0,0.1,0.14,0,1.57,0]T(m, rad) and final rod(0.5)=[-0.03,0.1,0.11,0,1.92,0]T (m, rad). The magnitude of Fintd is set to 1 N for the Thumb and 0.5 N for the others fingers. We assume that the desired internal force is directed along the normal to the contact surface. Fig. 4 and Fig. 5 show the object’s position and orientation errors. From the figures it can be seen that the proposed controller enforces the object asymptotically to converge to the desired trajectory since the object’s position and orientation errors converge to zero. Fig. 6 shows that the contact force error of the thumb converge to zero.

Fig. 4. Object position errors

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Fig. 5. Object orientation errors

Fig. 6. Conact force error of one finger

Moreover, the results obtained bellow has been validated graphically with a 3D simulator HandGrasp developed in our laboratory [18]. Joints positions and contact forces are saved in a file data from Matlab Simulink to serve as an input for the 3D Simulator.

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Initial posture

Intermediate posture

Final posture Fig. 7. Simulation 3D results for the spherical object It is observed that both numerical and 3D simulation results described above prove that the model-based controller can achieve high positioning performance. So the derived model is also validated and it can predict the dynamic of the grasp task. 5. Conclusion A dynamic model of a multi-fingered robot hand based on the formalism of Lagrange has been presented. Since many researchers derive the dynamic model of the hand considering the dynamic model of a manipulator three DOF each finger separately, the model derived in this paper takes into account the coupling between the fingers. The model, because of its computational simplicity and its new decoupled structure, is used for the simulation of modelbased controllers for multi-fingered robot hand achieving a grasping task and the diagnostic. The obtained results are quite satisfactory and show the validity of the model. As future works, we will attempt to design controllers based on intelligent methods for the grasping task. Acknowledgements The authors would like to acknowledge the financial support of this work by grants from General Direction of Scientific Research (DGRST), Tunisia, under the ARUB program.

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