Grasping and manipulation of deformable objects

Grasping and manipulation of deformable objects based on internal force requirements Sohil Garg1 & Ashish Dutta2 1 2

Dept. of Mechanical Engg., Indian Institute of Technology Guwahati, Amingao, Guwahati, India Dept. of Mechanical Engg., Indian Institute of Technology Kanpur, Kanpur 208 016, India

Abstract: In this paper an analysis of grasping and manipulation of deformable objects by a three finger robot hand has been carried out. It is proved that the required fingertip grasping forces and velocities vary with change in object size due to deformation. The variation of the internal force with the change in fingertip and object contact angle has been investigated in detail. From the results it is concluded that it is very difficult to manipulate an object if the finger contact angle is not between 30 o and 70 o, as the internal forces or velocities become very large outside this range. Hence even if the object is inside the work volume of the three fingers it would still not be possible to manipulate it. A simple control model is proposed which can control the grasping and manipulation of a deformable object. Experimental results are also presented to prove the proposed method. Keywords: Manipulation, grasping, fingertip

1. Introduction The analysis of mechanical fixtures dates back to the work of Reuleaux in 1875. In the area of robotics, work on grasping started with the research of Asada and Hanafusa and Saliabury’s three finger hand. In the last few decades a large number of papers have appeared in the robotics literature, a detailed review of which is given in Bicchi and Kumar have observed that ‘much of the work on grasping ignores the kinematics of the fingers’. Reuleaux problem of form closure focuses on the geometry of the object and arrangements of the contacts around the object. It is difficult to analyze force closure without considering the kinematics of the fingers. The problem is most acute in the case of grasping of deformable objects as the kinematics is related to the object size, which changes with deformation. Hence the motivation of this paper is to prove that the internal forces that are required for grasping and manipulation of an object changes with the size of an object when it deforms, even when the object weight (external force moment on the object) might be constant. A detailed analysis of the grasping of deformable object by a three finger hand was carried out and it is proved that the internal forces required to grasp a deformable object vary before and after deformation, because the size of the object changes. The variation of the grasping force with the contact angle and size of an object was also examined. The variation of finger contact velocities with fingertip contact angle has also been derived. It is proved that it is very difficult to manipulate an object if the finger and object contact angle is not between 30 and 70 degrees,

because the forces required for gasping or velocities for trajectory tracking become very large. A simple FEM model has been developed to relate the object deformation with the fingertip force, based on this an impedance based hybrid control model was developed to control the gripper. The results obtained from the analysis are important because it is proved that even in cases where the object to be manipulated is inside the work volume of the hand, forces or velocities required to grasp and manipulate can become very large. It is also shown that the very basic process of designing a gripper based on the work volume of the fingers may not always be suitable for handling deformable objects. 2. Literature review Cutkosky [6] showed that grasp stability is a function of the fingertip contact models and small changes in the grasp geometry. Mason and Salisbury [7] gave conditions for complete restraint of an object in terms of internal forces. Kerr and Roth [8] proposed a method to determine the optimal internal forces based on approximated frictional constraint at the object and fingertip. Meer et al. [9] and Patton et al. [10] designed a system, which focuses on force control versus deformation control. In these systems the robot manipulator is designed to control the deformation of the object. Howard and Bekey [11] developed a generalized learning algorithm for handling of 3-D deformable objects in which prior knowledge of object attributes is not required. They used neural network with mass, spring and damping constant as input and the force needed to grasp the object as output

International Journal of Advanced Robotic Systems, Vol. 3, No. 2 (2006) ISSN 1729-8806, pp. 107-114

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International Journal of Advanced Robotic Systems, Vol. 3, No. 2 (2006)

of the network. Ghaboussi et al.[12] used FEM for the representation of deformable objects. Wakamatsu et al. [13] analyzed stable grasping of deformable objects based on the concept of bounded force closure. They showed that stability of deformable object grasp depends on contact friction and the shape at the grasping contact point. Hirai et al. [14] proposed a robust control law for grasping and manipulation of deformable objects. They developed a control law to grasp and manipulate a deformable object using a real time vision system. Vision based systems would fail if the fingers are embedded and covered by the deformed object. Also in 3D cases it would be very difficult to observe the displacements and rotations of the markers as proposed in this paper. Taylor [15] and Henrich [16] have also studied manipulation of deformable objects by considering grasping and manipulation separately. Ono [17] have considered the unfolding of cloth. Xydas et al. [18] have studied soft finger contact mechanics using finite element analysis and experiments. In section 3 the relation between the external force acting on the object and internal force required for grasping is explained. The variation of internal forces with contact angle and object size is explained in section 4. While in section 5 the variation of finger tip velocity and object velocity is examined. The conclusions are given in section 6. 3.1 Grip matrix and internal forces When a body is grasped by a multi finger hand then each contact can exert a system of wrenches or twists on it. Figure 2 shows the direction of forces acting at the three fingertips for a point contact with friction. In this figure the vector (xi, yi, zi) denotes the direction of fingertip forces and i = 1, 2, 3 are the finger numbers.

can be applied to a body, W = [w1 w2 . . . wn] gives the net wrench which can be applied. The solution for C is given by: C = Cm + Ci Cm = W#F Ci = Nλ

(2) (3) (4)

where Cm is a particular solution to equation (1) and it gives the motion causing forces, Ci is a homogenous solution to equation (1) and it represents the internal force on the object, N is an orthonormal basis spanning the null space of W, W# is the generalized inverse of W, λ is an arbitrary free variable, which determines the magnitude of the internal force. Ci represents a set of contact and frictional forces which can be applied to the object without disturbing its equilibrium. It results in no net force on the object. The general practice followed in deciding the internal force, is to take the maximum value of ‘λ’ to satisfy friction conditions [8]. This results in a very high value of the internal force for any position of contact angle or deformation. If this approach is followed then in case of gasping deformable objects the fingers will deform the object to the maximum amount. This may damage the object or the fingers of the robot. This approach will also fail when the internal forces becomes very large, which might not be supported by the actuators. To determine the finger contact wrenches which must be exerted in order to apply a desired net force on a grasped object the following relation is used. Fc = G–T Fo

(5)

where Fc is an n-vector of contact wrench intensities for the n contact wrenches, Fo is an n-vector with the first six elements being the net wrench (external force) applied to the object and the last (n-6) elements being the magnitudes of the internal forces. It is defined by Fo = [fx fy fz mx my mz λ1 . . . λn-6]T G is known as grip matrix and it gives the relation between object velocity vector and contact velocity vector. G-T is formed by augmenting the (6×n) W matrix with the (n-6), n-element basis vectors of the homogenous solution. It is given by:

Fig. 1. The position of each axis at each fingertip. The relation between the net wrench system which can be applied to the object and contact wrench intensity for a system shown in Fig. 2. is given by: (1) F = WC where C is a vector of fingertip forces corresponding to each wrench intensity, F is the net external wrench which

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⎡ ⎤ W ⎢ ⎥ − − − − − −⎥ ⎢ -T G = ⎢ NT ⎥ .....(6) 1, h ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ T ⎢⎣ N n −6,h ⎥⎦ This matrix will be square and invertible. The (n-6) Ni,h are linearly independent and span the (n-6) dimensional

Sohil Garg & Ashish Dutta / Grasping and manipulation of deformable objects based on internal force requirements

null space of W. G is a square matrix having dimension and rank equal to the number of finger contact wrenches acting on the object. G needs to be recomputed when the size of the object or the contact angle changes. The relation between vector of twist intensities per unit time occurring along the wrench axes, Vc, and Vo is given by equation (7). The first six components of which represent the body’s linear and angular velocity and the remaining (n-6) γis are its velocities resulting from deformation of the body is given by: Vc = G–T Vo

(7)

where Vo = [vx vy vz ωx ωy ωz γ1…γn-6]T is a vector of object velocity. Vc = [vx1 vy1 vz1 . . . . . vxn vyn vzn]T is a vector of finger contact velocities for n contacts. Using equation (7) the set of finger velocities required to have a desired set of body velocities can be determined. The net wrench on the body is determined by the following matrix,

I I ⎞ ⎛I ⎜ ⎟ W = ⎜L L L ⎟ ⎜ ⎟ M ⎝ ⎠ where I = 3 X 3 identity matrix corresponding to one of the fingers, M = the matrix of distances between the fingertips and object center. For the generalized case shown in Fig. 2, W is a 6 X 9 matrix as given below

⎛ ⎛cos(θ) ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎝ 0 W=⎜ ⎜ ⎜⎛ 0 ⎜⎜ ⎜⎜ 0 ⎜⎜−b*cos(θ) ⎝⎝

0 ⎞ ⎟ 1 0 ⎟ 0 sin(θ)⎟⎠ 0

0 b*sin(θ)⎞ ⎟ 0 x*sin(θ)⎟ −x 0 ⎟⎠

⎛cos(θ) 0 0 ⎞ ⎜ ⎟ ⎜ 0 1 0 ⎟ ⎜ 0 0 sin(θ)⎟ ⎝ ⎠ 0 −b*sin(θ)⎞ ⎛ 0 ⎜ ⎟ 0 0 x*sin(θ) ⎟ ⎜ ⎜b*cos(θ) −x 0 ⎟⎠ ⎝

⎛cos(θ) 0 0 ⎞⎞⎟ ⎜ ⎟ ⎜ 0 1 0 ⎟⎟⎟ ⎜ 0 0 sin(θ)⎟ ⎝ ⎠⎟ ⎟.....(7) ⎟ ⎛0 0 0 ⎞⎟ ⎜ ⎟⎟ ⎜0 0 −y*sin(θ)⎟⎟ ⎜0 y 0 ⎟⎠⎟ ⎝ ⎠

N=

⎡ -0.5550 ⎢ 0.0085 ⎢ ⎢ 0.0000 ⎢ -0.2405 ⎢ 0.0203 ⎢ ⎢ 0.0000 ⎢ 0.7955 ⎢ ⎢ -0.0288 ⎢⎣ -0.0000

0.5260 -0.2640

⎤

-0.3776 -0.6012 ⎥ 0.0000 -0.6867 0.2667 0.0000 0.1607 0.1108 -0.0000

⎥

0.0000 ⎥ 0.3537 ⎥

⎥ ⎥ 0.0000 ⎥ -0.0897 ⎥ ⎥ -0.0565 ⎥ -0.0000⎥ ⎦

0.6577 ....(8)

Initial Values for computing N based on matrix (7) before deformation: b = 1; x = 2; y = 2; Final Values after the deformation of the object: b = 1; x = 1.25; y = 0.5; For the object to be in equilibrium the sum of all the forces and moments in the principal directions must be zero (∑ Fx =0). The null matrix given by (8) has three column vectors and the one satisfying the condition of static equilibrium given by sum of all forces about the three axes should individually be zero is chosen as the internal force vector. We find that the direction of the internal forces with respect to the chosen axes for each finger is correctly represented by the column 1 of the Null vector. From the null matrix, it is quite clear that the component of the force in z direction does not contribute to the internal forces. 4. Analysis of the variation of internal forces with deformation The analysis (using MATLAB) was carried out to study the effect of variation of internal forces with change in object size due to deformation and also change in contact angle.

where, θ = contact angle of the fingertip normal with the x axis. The object moves only on the x-z plane and this has been assumed to simplify the object motion in 2-D. The value of friction coefficient was assumed to be very high (almost 1) as the gripper and the object was made of material having very high friction coefficient like rubber. This would enable forces to be applied in the three principal axis directions. As an example if the contact angle between the object and finger is θ = 800 then the null vector is found using MATLAB as given below:

(a)

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(b)

(b)

(c) Fig.3 Variation of internal force in x direction with size of an object and contact angle for the three fingers.

(c)

The size of the object was decreased from x=0.5 to x=2 and keeping the size of the object constant for each stage (keeping x and y constant), the variation of internal force with contact angle was plotted by finding the null vector column in each case. Figure 3 shows that as the contact angle ‘θ’ increases, the magnitude of the internal force in x direction for the first finger increases, for the second finger, it decreases and for the third finger, it again decreases for a constant object size.

Fig. 4 Variation of internal force with contact angle and size of object in y direction. With deformation of an object the internal force at the first finger decreases, for the second finger it increases and for the third finger it also increases. It is also verified that at each instant the fingertip forces maintain the conditions of ∑ Fx =0 . Figure 4 shows that as contact angle increases for an object of the same size the internal forces in the ydirection decreases for all the fingers. With increase in the deformation of the object, the internal forces increase in magnitude. These forces also maintain the force closure property ∑ Fy =0. As already shown, the z component of the forces does not contribute to the internal force. 5. Variation of finger velocities

(a)

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The variation of velocities has been calculated assuming the following values of the object velocities and using equation (7): vx = 1 , vy = 0 , vz = 1 , ωx = 0 , ωy = 1 , ωz = 0 , γ1 = 1 , γ2 = 0 , γ3 = 0 The deformation will occur mainly in the x direction due to the compressive forces exerted by the internal forces. An FEM analysis was performed which showed that the deformations due to the other forces except forces in the x


direction produce very small deformations. Hence in practical applications it can be assumed that γ2 = 0, γ3 = 0 . This also leads us to believe that simultaneous control of grasping and manipulation would be very difficulty if all deformations caused by all forces are to be considered. The plots showing the variation of the finger contact velocities are given below:

increasing exponentially after θ = 68.750 (≈ 700) approximately. This puts an upper limit to the maximum value of the contact angle that the robot finger should have in order to grasp deformable objects. After this the finger tip velocity tends to become infinite. An important result is that the finger velocity in x direction does not change with deformation for any of the fingers. The Variation of Vyi for the three fingers is shown in Figure 6.

(a)

(a)

(b)

(b)

(c) Fig. 5. Variation of velocities in the x directions.

(c) Fig. 6 variation of velocity in the y direction.

The velocities of the finger tips in the x direction increase with the contact angle ‘θ’ , with their magnitude

The magnitude of the velocities in y direction for all the three fingers decreases with increase in the contact angle.

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Moreover with deformation, the magnitudes of the velocities increase. For the first finger after θ = 34.380 approximately all the forces remain constant. Variation of Vzi for the three fingers is shown in the next set of graphs.

internal forces become very large and it may not be apply such large forces for a given actuator torques.

(a)

(a)

(b)

(b)

5.1 Contact angle approaches 900

(c) Fig. 7. Variation of velocity in the z direction.

(c) Fig.8 variation of internal grasping force when the contact angle approaches 90 degrees.

With increase in the contact angle, the finger velocities in z direction decrease. Also with increase in deformation the finger velocity decreases for the third finger and increases for the first and second finger. After θ=220 the

One of the singular cases is when the contact angle is equal to 900. It is impossible to grasp an object with the robot fingers perpendicular to its surface, the results show sudden break in the magnitudes and the trend of

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variation of the internal forces. In case of finger contact velocities, the grip matrix no longer remains square as ‘θ’ becomes 900, hence making it impossible to compute the velocities. A few graphs showing these cases are given below. 5.2. Contact angle approaches 00 Similar to the case above the internal forces vary sharply as the contact angle approaches 00 and the grip matrix no longer remains square.

grasping and manipulation as force or velocity might become very large. 6. Conclusion In this paper a detailed analysis of grasping of deformable objects by a three finger hand was carried out, and it has been proved that the internal forces required for grasping deformable objects vary with size of object and finger contact angle. In order to clearly understand the variation of the internal force at different parts of the work volume, the variation of the internal force with size of object and contact angle has been plotted. It has been proved that it might not be possible to manipulate an object outside the range of contact angle 30o to 70o, as the velocities become very large. This result is important as even if the object is inside the work volume, it still may not be possible to manipulate it along a desired trajectory.

7. References

(a)

(b) Fig. 9. Sample of variation of initial grasping force for contact angle equals zero. Moreover the grip matrix does not remain square if the deformation exceeds beyond the center point. Hence the analysis is not suitable for deformations exceeding beyond the center point, such as in buckling of a slender beam. From the above analysis we find that the operating region of the robotic gripper for grasping deformable objects so that the fingertip force or velocity does not become very large is between 300 ≤ θ ≤ 700, where ‘θ’ is the angle between the finger and the object. This also proves that simply by maximizing the internal force as is conventionally done [8] might prove dangerous for

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[11] A. M. Howard and G.A. Bekey, “Intelligent learning for Deformable Object Manipulation,” IEEE Transactions on Robotics and Automation. Pp. 15-20. 1999. [12] J. Ghaboussi, E.L. Wilson and J. Isenberg, Finite element for rockjoint and interfaces. ASCE J. 99 (1973) 833-848. [13] Wakamatsu H., Hirai S., Iwata K., “Modeling of Linear Objects considering Bend, Twist and Extensional Deformations”, In Proc. 1995 IEEE Int. Conference on Robotics and Automation, vol.1, pp.433-438, Nagoya, Japan, May 1995. [14] Hirai, S., Tsuboi T., and Wada T., Robust Grasping Manipulation of Deformable Objects. Proc. of the 4th IEEE Symposium on Assembly and Task Planning. Japan May 2001. [15] Tayler, P. M. et al. Sensory Robotics for the Handling of Limp Materials, Springer-Verlag, 1990. [16] Henrich, D and Worn, H. Eds. Robotic Manipulation of Deformable Objects, SpringerVerlag, Advanced manufacturing Series, 2000. [17] Ono, E. et al. “ Strategy for unfolding a fabric piece by cooperative sensing of touch and vision”, Proc. of the IEEE/RSJ Intl. Conf. on Intelligent Robotics Systems, pp 441-445, 1995. [18] Xydas, N et al. “ Study of soft finger contact mechanics using finite element analysis and experiments”, Proc. of the IEEE Intl. cong on robotics and Automation, vol.3, pp. 2179 -2184, 2000. [19] Tsuboi, T, Masubuchi, A. and Hirai , S. “Video frame rate detection of position and orientation of planer motion objects using one sided radon tansform”, Proc of the ICRA , May 2001.

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