Dynamic Analysis and Control Synthesis of Grasping and ......Dynamic Analysis and Control Synthesis of Grasping and Slippage of an Object Manipulated by a Robot Shahram Hadian Jazi,

Advanced Robotics 22 (2008) 1559–1584www.brill.nl/ar

Full paper

Dynamic Analysis and Control Synthesis of Grasping andSlippage of an Object Manipulated by a Robot

Shahram Hadian Jazi, Mehdi Keshmiri ∗ and Farid Sheikholeslam

Isfahan University of Technology, Isfahan 8415683111, Iran

Received 26 February 2008; accepted 28 May 2008

AbstractGrasping an object by a cooperating system such as multi-fingered hands and multi-manipulator robotic sys-tem has received much attention. Research has focused on analysis of force-closure grasps and the synthesisof optimal grasping, when there is no slipping condition. Although the control system is designed to keepthe contact force in the friction cone and avoid the slipping condition, slippage can occur for many reasons.In this research, dynamics analysis and control synthesis of a manipulator moving an object on a horizontalsurface using the contact force of an end-effector are performed considering the slipping condition. Equalityand inequality equations of frictional contact conditions are replaced by a single second-order differentialequation with switching coefficients in order to facilitate the dynamic modeling. Accuracy of this modelingis verified by comparing the results of the model with those of SimMech. Using this modeling of friction,a set of reduced order form is obtained for equations of motion of the system. A new method is proposedto control the object motion and the end-effector undesired slippage based on the reduced form. Finally,performance of the method is evaluated both numerically and experimentally.© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2008

KeywordsGrasping, contact modeling, slippage control, frictional contact, undesired slipping, cooperating system

1. Introduction

Grasping an object in a cooperating system such as multi-fingered hands and multi-ple robots is an important issue for researchers. During a full multi-fingered manip-ulation cycle, grasp planning arises on several occasions, such as when an object isfirst picked up. Grasp analysis and synthesis are fundamental problems in the studyof grasp planning. Many papers can be found on testing and planning force-closuregrasps. In grasp analysis, most of the research has been focused on finding appropri-ate conditions for force-closure grasps. Previously, Salisbury and Roth developed

* To whom correspondence should be addressed. E-mail: [email protected]

© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2008 DOI:10.1163/156855308X360776

1560 S. H. Jazi et al. / Advanced Robotics 22 (2008) 1559–1584

several different types of finger contacts and showed which finger configurations al-low complete immobilization of the gripped object relative to the fingers, and alsoallow for the manipulation of the object by the fingers while maintaining the grasp,using screw theory [1]. Mishra et al. proposed necessary and sufficient conditionsfor force-closure grasp with friction point contacts (FPCs) [2]. Bicchi translatedthe force-closure problem into the stability of an ordinary differential equation [3].With the linearization of the friction cone, Liu developed a ray-shooting-based algo-rithm using the duality of polytopes [4]. Zheng and Qian enhanced the ray-shootingapproach proposed by Liu to complete the exactness, increase the efficiency andextend the scope [5]. Using linear matrix inequality representation of nonlinearfriction cone constraints, Han et al. reformulated the force-closure problem as thefeasibility problem of a semi-definite or max-det problem and used an interior pointalgorithm for it [6]. Thus, the general problem of determining if a grasp is a force-closure grasp is considered to be completely solved.

Having sufficient conditions for force closure, grasp synthesis deals with optimalgrasping. This synthesis consists of (i) determination of the optimality criteria, and(ii) derivation of methods and algorithms for computing contact locations with re-spect to the optimality criteria and subject to accessibility constraints. Park and Starrpresented a simple and efficient algorithm to find an optimum force-closure graspof a planar polygon using a three-fingered robot hand. The optimum grasp was de-fined as a grasp that has the minimum value of a heuristic function [7]. Mishra et al.proposed an algorithm for computing force-closure grasps for polyhedral objectsunder FPCs [2]. Tung and Kak presented a new theorem and an algorithm for thefast synthesis of two-fingered force-closure grasps for arbitrary polygonal objects.The polygonal objects were allowed to be of arbitrary shape and each edge of thepolygon was allowed to have different frictional characteristics [8]. Liu presentedan efficient algorithm for computing all n-finger form-closure grasps on a polygonalobject based on a new sufficient and necessary condition for form-closure. With thisnew condition, it is possible to transfer the problem of computing the form-closuregrasp in R

3 to one in R1 [9]. Based on the geometric condition of the closure prop-

erty, Zhu and Ding presented a numerical test to quantify how far a grasp is fromlosing form/force closure. They also developed an iterative algorithm for computingoptimal force-closure grasps [10]. Morales et al. addressed the problem of design-ing a practical system able to grasp real objects with a three-fingered robot hand.They presented a general approach for synthesizing two- and three-finger grasps onplanar unknown objects using visual perception [11]. Al-Gallaf presented a novelneural network for dexterous hand-grasping inverse kinematics mapping used inforce optimization. He showed that the proposed optimization is globally conver-gent to the optimal grasping force [12]. Zhu et al. introduced the Q distance andadopted the radius of the largest ball contained in the convex hull of the primitivewrenches as a quality measure [13, 14]. Platt et al. presented an algorithm for grasp-ing an unknown object by cooperating robots. They found proper contact pointssuch that resultant couple and force on the object are zero, and controllability of nor-

S. H. Jazi et al. / Advanced Robotics 22 (2008) 1559–1584 1561

mal force control or tangential velocity control was maximized [15]. Miyabe et al.analyzed object grasping with elastic arms. They used optimized contact velocityto find forces that can grasp object [16]. Li et al. presented a novel algorithm forcomputing three-finger force-closure grasps of two-dimensional (2-D) and three-dimensional (3-D) objects. New necessary and sufficient conditions for 2-D and3-D equilibrium and force-closure grasps were deduced, and a corresponding algo-rithm for computing force-closure grasps was developed [17]. Liu et al. proposeda complete and efficient algorithm for searching form-closure grasps of n hard fin-gers on the surface of a 3-D object represented by discrete points [18]. All of theabove research considers no slippage in grasping and the control system tries tokeep control forces within in the friction cone.

Zheng et al. addressed dynamic and control analysis of a three-fingered hand ma-nipulating and regrasping an object in 3-D space. They allowed one of the fingersto slide on a predefined path on the object surface to change its grasp location [19].Cole et al. consider control of the sliding motion of the fingertip of a two-fingeredhand along the object surface and position and orientation control of the object si-multaneously. They assumed that only one specific finger slides on a predefinedpath on the object surface. Their work is useful for regrasping an object held in ahand [20]. Kao and Cutkosky compared theoretical and experimental sliding mo-tions for a sheet of paper or similar objects on a planar surface, manipulated by atwo-fingered hand, using static equilibrium equations [21]. Chong et al. proposeda motion/force planning algorithm for multi-fingered hands manipulating an objectof an arbitrary shape using both rolling and sliding contacts. They used a nonlinearoptimization approach to calculate the joint velocities and contact forces at eachstep of time [22].

Although the above research considers slippage in object regrasping analysis, theslippage should be completely predefined. The finger which slides on the object,starting time and duration of slippage, and sliding path are known in advance. Thismeans dynamic and control analysis of undesired slippage still remains undiscussedin the literature. It can occur during the grasping maneuver due to many reasons,such as changes in the object geometry, mass, inertia and coefficient of friction ordealing with an unknown object. As an example, one can assume the practical casewhen a cooperative system manipulates a dirty object or manipulates an object in adirty circumstance. In such a case, the coefficient of friction between end-effectorsand object can change.

In this research, dynamic analysis and control synthesis of a manipulator mov-ing an object on a surface using contact forces and considering undesired slippingconditions have been performed. The system under consideration is a platform forfurther extension to grasping analysis of a cooperative manipulator, carrying anobject while slipping conditions can occur. In Section 2, problem definition andassumptions are given, and general formulation and equations of motion for thesystem are provided. In Section 3 dynamics of friction are formulated. In order tocontrol sliding of the object, as well as its path tracking, two types of controllers


have been proposed in Sections 4 and 5. Section 6 provides the numerical results.An experimental setup built for implementing the method practically is describedin Section 7. Comparison of the experiment results with those of the numericalsimulation are also provided in this section.

1.1. Dynamic Analysis

The system under consideration is shown in Fig. 1. It consists of a two-link rigidmanipulator that moves an object (B) on the horizontal surface. Contact between themanipulator and object is assumed to be point contact which can be moved alongthe object surface.

The whole motion is assumed to be in the vertical plane and the following as-sumptions are taken into account as well:

• The contact point on the end-effector remains fixed. Note that without this as-sumption, an additional kinematics problem must be considered.

• The object cannot rotate or its inertia momentum is zero.

• The coefficient of friction on the top of the object is greater than the bottom.

• There is no uncertainty in the system.

Using the above assumptions, the motion of the whole system can be describedby the following equations:

Mq + h = Bτ − JTF1 (1)

M0xb + g0 = GF (2)

Hu(F1, xs) = 0 (3)

Hd(F2, xb) = 0, (4)

where M, h, B and J are the well-known matrices and vectors of the two-link ma-nipulator and:

Figure 1. Schematic of the system under consideration.


M0 =[

mb

0

]

g0 =[

0mbg

](5)

G = [ I2 I2 ]

F =[

F1F2

].

F1 and F2 are the forces exerted on the object by end-effector and ground, respec-tively, xb and xs are the movements of the object on the horizontal surface andend-effector on the object, respectively, and mb is the mass of the manipulated ob-ject. Hu and Hd are functions which model friction on the surfaces of the object.They are described in the next section.

System motion is a kind of constrained motion and the constraint equations canbe written as

xe = xb + xs(6)

ye = const.,

where end-effector position, xe and ye, can be written as:

xe = l1 cos(q1) + l2 cos(q1 + q2)(7)

ye = l1 sin(q1) + l2 sin(q1 + q2).

Substituting for xe and ye from (7) in (6) and differentiating with respect to time,the constraint equations can be written as the following velocity form:

Aq q + Axxb + As xs = 0, (8)

where:

Aq = J, Ax = As =[−1

0

]. (9)

2. Modeling the Contact Forces

Friction appears at the physical interface between two surfaces when two bod-ies are in contact, which is strongly influenced by contamination. There is awide range of physical phenomena that cause friction. These include elastic andplastic deformations, fluid mechanics and wave phenomena, and material sci-ences (see Refs [23, 24]). A survey of models of friction analysis can be foundin Ref. [25].


Figure 2. Coulomb friction model.

Figure 3. Free body diagram for a moving object on a surface.

Assuming the Standard Coulomb friction model without stiction (Fig. 2) with μ

as the coefficient of friction (μs = μk = μ), the friction force exerted on a bodyfrom the contacting surface (Fig. 3) can be written as:{

Ft = −μFn sign(v) if v �= 0|Ft | � μFn if v = 0 and v = 0Ft = 0 if v = 0 and v �= 0,

(10)

where v is the speed of the body relative to the surface, and Ft and Fn are frictionand normal forces, respectively. Fn is assumed to be a positive value.

Note that the second equation in (10) describes three different conditions: start-ing forward motion, starting backward motion and stationary condition. We canreformulate the above conditions in a single equation:

α1v + α2Ft + α3μFn = 0, (11)

where αi (i = 1,2,3) are state-dependent coefficients calculated from Table 1.When there is more than one choice for αi (i = 1,2,3), we have to choose one

set and check the selection for consistency with the result from dynamic analy-sis.

For the simulation and control purposes, detecting the zero velocity is a problemdue to numerical issues. A remedy for this can be found in the model presentedby Karnopp [26]. It was developed to overcome the problem and to avoid switch-ing between different state equations for sticking and sliding. The model defines azero-velocity interval, |v| < δv. For velocities within this interval the internal state


Table 1.Values for αi (i = 1,2,3) in different conditions

v = 0

v �= 0 v− �= 0 v− = 0

αi Movement Motion reversing No Motion No Motion Start forward Start backward

α1 0 0 1 1 0 0α2 1 1 0 0 1 1α3 sign(v) 0 0 0 1 −1

Figure 4. Free body diagram for the object in Fig. 1.

of the system (the velocity) may change and be non-zero, but the output of theblock is maintained at zero by a dead zone. The drawback with this model is thatit is so strongly coupled with the rest of the system and it is not always explicitlygiven. In fact this is the reason that we have to choose one option and check for itsconsistency. This model is utilized in the presented research.

Let us consider the object shown in Fig. 1. Its free body diagram is given inFig. 4. The contact conditions can be formulated by:

Hu(F1, xs) = α1xs + α2F1x + α3μ1F1y = 0 (12)

Hd(F2, xb) = β1xb + β2F2x + β3μ2F2y = 0, (13)

where αi and βi (i = 1,2,3) are calculated from Table A.1 in Appendix A.The results of the above modeling are compared with those of the SimMech

toolbox of MATLAB software which uses differential-algebraic equations. Surpris-ingly, we obtained the same results.

3. Control Synthesis — Conventional Approach

In the conventional approach of grasping analysis the controller is designed suchthat the manipulator exerts the requested force on the object and satisfies the noslipping condition. It is worth mentioning that, in most of the existing research,


researchers solve the problem for the case that there is no slippage. Although wehave used the same approach in this section, we extend the method and previousworks for the case that the end-effector slips on the object.

Consider equations of motion for the system (2) and let the desired trajectory ofthe object be xdes

b (t). If the acceleration of the object is chosen as:

xb = xdesb + Kvbeb + Kpbeb, (14)

where Kvb and Kpb are positive constants and eb = xdesb − xb, and the following

resultant force is applied on the object:

Fres = GF = M0(xdesb + Kvbeb + Kpbeb) + g0, (15)

then the object’s motion is governed by:

eb + Kvbeb + Kpbeb = 0. (16)

This guarantees asymptotic stability of the tracking.Now one has to decompose the resultant force into the exerted forces on the

object by the manipulator and horizontal surface, and then control the robot in sucha way to ensure that the calculated forces for the manipulator are implemented.

Due to the redundancy in the driving forces of the object, decomposition of theresultant force leads to the following optimization problem:

Minimize ‖Fdes‖such that:

Fres = GFdes

eTN1Fdes

1 � η1‖Fdes1 ‖

eTN2Fdes

2 = η2‖Fdes2 ‖ (17)

eTN1Fdes

1 > 0

eTN2Fdes

2 > 0,

where ηi = 1√1+μ2

i

and eNi is inward normal direction in the ith contact point.

In (17) we have used Fdes instead of F since the exerted forces by the manipulatorand the horizontal surface can differ from the calculated ones. Contact stability canbe deteriorated once the manipulator cannot exert the desired forces. In this casethe end-effector will slip on the object.

Since the end-effector forces must be controlled in all directions, the usual hybridposition/force control cannot be used. Thus, we design the controller of the manip-ulator such that the desired forces are exerted by the end-effector and no slippagecondition is satisfied in the contact point, i.e.:

xe = xb. (18)


If we divide the input torques in (1) into two parts, τ = τ e +τf , where τ e and τf

are responsible for satisfying the no slippage condition (18), and exerting the cal-culated force, Fdes

1 on the object, respectively, we can compute τf from the staticequilibrium condition:

Bτf = JTFdes1 , (19)

and τ e from the free motion of manipulator (Mq + h = Bτ e). In this research afeedback linearization method is used to control the free motion of the manipulator.

One must note that using this approach cannot always generate any arbitrarypair of F1 and xe [27]. However, since xe is somehow the result of F1, the aboveapproach can result in the desired objectives. The fundamental structure of thiscontroller is shown in Fig. 5.

Two different strategies can be imagined to control slipping of the end-effectoron the object. In the first strategy we set the velocity of the object as the desiredvelocity for the end-effector and the contact point position for its desired position,i.e.:

xdese = xb + xs

(20)xdese = xb.

This means in each instant we try to stop slipping, without trying to return thecontact point to its original position. In the second strategy the end-effector alwaystries to return to its original position on the object, i.e.:

xdese = xb. (21)

In Appendix B, it is shown that the first strategy always results in a steady-stateerror in velocity, while the second strategy will conclude to zero error as soon asthe desired force objective is met by the controller. This advantage is shown in thenumerical analysis as well.

Figure 5. System description in the conventional approach.


4. Control Synthesis — Proposed Approach

As can be seen, the conventional approach is an open-loop controller in force con-trol and a closed-loop controller in position control. Since avoiding slippage inthis approach depends on the error in the exerted force, one can expect fluctua-tion around the zero-slipping condition. Also we cannot perform stability analysisfor this method.

Now a new approach in designing the controller is proposed. In this approachwe convert the constrained equations of motion into two inputs, two outputs, a non-constrained motion and design the controller for the new sets of equations.

Let us rewrite (12) and (13) as:

α1xs + D1F1 = 0 (22)

β1xb + D2F2 = 0, (23)

where:

D1 = [α2 α3μ1 ](24)

D2 = [β2 β3μ2 ].Calculating F1, F2 and q from (1), (2) and the derivative of (8), respectively, andsubstituting them in (22) and (23), one can write the following reduced order form:

M{

xb

xs

}+ h = Bτ , (25)

where:

M =[

D1J−TMA−1q Ax α1 + D1J−TMA−1

q As

β1 + D2M0 − D2J−TMA−1q Ax −D2J−TMA−1

q As

]

h =[ −D1J−TMA−1

q b − D1J−ThD2g0 + D2J−TMA−1

q b + D2J−Th

](26)

B =[−D1

D2

]J−TB

b = −(Aq q + Axxb + As xs). (27)

Note that all of the above matrices and vectors depend on q, xb and xs , and theirderivatives.

Equation (25) relates input and output of the system. In order to use this equationto design the controller, one has to take care of the internal stability of the systemand to show that M is invertible.

Theorem 1. The system of Fig. 1 represented by (25) is internally stable, if it isinput–output stable.


Proof. Assuming input–output stability means that τ , xb and xs are bounded. Sincex-position of the end-effector can be written as:

xe = xb + xs, (28)

it means xe and its derivatives are also bounded. Considering this fact and the kine-matics relation between q and end-effector velocity, and noting that q appears intrigonometric equations, results that q is bounded. Using (1) and (2), one can showthat F1 and F2 are also bounded.

Theorem 2. M is invertible if and only if:

β1I2 �∫ t0+T

t0

MdMTd dt � β2I2, (29)

for all t0, where Md = M(xdesb , xdes

s ). β1, β2 and T are positive constant scalars.Proof is given in Appendix C.

Remark. Since the determinant is a continuous function of the matrix elements, fora matrix whose elements are a continuous function of a parameter such as time thedeterminant will also be a continuous function of time. Thus, in any interval thatdeterminant is non-zero the matrix is invertible.

Since we are facing a multi-phase dynamic system, we propose an appropriatemulti-phase controller such that it complies with the system.

As far as slipping is concerned, the system undergoes the following four phases:

(1) Slipping on both sides.

(2) Slipping only on the bottom.

(3) Slipping only on the top.

(4) No slipping on either side.

Each of these four phases covers part of the xb–xs plane. As shown in Fig. 6,phase 1 covers the whole plane excluding the axes. Phases 2 and 3 cover xb and xs

axes, respectively, excluding the origin. Phase 4 covers only the origin.When the system is in phase 1, the controller must prevent slipping in the upper

side of the object and keep the object moving such that it tracks the desired trajec-tory. In phase 2, there is no slipping in the upper side of the object and the controllermust only ensure that the object tracks the desired trajectory. Phase 3 and 4 are infact non-desirable phases, and the controller must enforce the system to leave thesephases and enter in phase 1 or 2.

Using this strategy, the controller pushes the system working always in phase 1or 2, except for the limited period of time where the object stops and the controllertries to move it. Hence, any stable and convergent control law for phase 1 and 2guarantees stability and convergence of the whole motion. We can guarantee that in


Figure 6. Four phases of the system in the xb–xs plane.

phase 3 and 4 the controller puts the object in motion in a limited time and differentforce condition from the time that the object stops.

The following control law has been used for phase 1 and 2, for the sake of nu-merical simulation:

M(Xdes + Kv e + Kpe) + h = Bτ , (30)

where:

Xdes ={

xdesb

0

}(31)

e ={

eb

es

}=

{xdesb − xb

0 − xs

}. (32)

Kp and Kv are two positive definite matrices that regulate frequency and speed oftracking convergence.

Therefore, input vector, τ , can be calculated as:

τ = B+M(Xdes + Kv e + Kpe) + B+h + (I − B+

B)y, (33)

where B+ is the Moore–Penrose pseudoinverse of B and y is an arbitrary vector.Substituting (33) in (1) and eliminating F1 and F2 and q from (1), (2) and the

derivative of (8), the following equation is obtained:

MX + h − BB+

M(Xdes + Kv e + Kpe) − BB+

h − B(I − B+

B)y = 0. (34)

Since B(I − B+

B) = 0, (34) can be rewritten as:

MX + h − BB+

M(Xdes + Kv e + Kpe) − BB+

h = 0, (35)

defining:

P = BB+, (36)


and performing some matrix operation, one can rewrite (35) as:

(I − P)(MX + h) − PML(e) = 0, (37)

where:

L(e) = e + Kv e + Kpe. (38)

In phase 1 D1 �= 0 and D2 �= 0. Hence, B is invertible and B+ = B−1, so:

P = I, (39)

and then:

ML(e) = 0. (40)

L(e) becomes continuously zero and it guarantees the error convergence, sincethe matrix M is full rank.

Physical interpretation: the proposed controller controls slippage on the upperside and reduces the object motion error by increasing the normal force exertedon the object by the end-effector. This causes a change in the tangential forces,F1x and F2x , such that the object tracks the desired trajectory and the end-effectorslippage on the object diminishes.

In phase 2, the first row of B becomes zero. It can be shown that in this phase:

P =[

0 00 1

], (41)

and:

(I − P)(MX + h) ={

xs

0

}. (42)

Since there is no sliding in the upper face, xs = 0 and we can write:

(I − P)(MX + h) = 0. (43)

Using the above relation in (37) leads to:

PML(e) =[

0m21(eb + kvbeb + kpbeb) + m22(es + kvs es + kpses)

]= 0. (44)

Note that in this phase xs = xdess = 0 and xs = xdes

s = 0. So we will have:

m21(eb + kvbeb + kpbeb) + m22kpses = 0. (45)

Selecting kps = 0 in this phase guarantees the tracking error of the object convergesto zero.

Since the dimension of null space of B is not zero, y can have infinite choices.It allows us to compute τ1 and τ2 such that the norm of contact forces becomes


minimum. Hence, for this case we solve the following problem:

Minimize ‖F(y)‖such that

τ = B+[M(Xdes + Kv e + Kpe) + h] + (I2 − B+B)y (a)JTF1 = −(Mq + h) + Bτ (b)F1 + F2 = M0xb + g0 (c)Aq q + Axxb = b (d)MX + h = Bτ (e)[1 σμ1 sign(xdes

b ) ]F1 = 0, (f)

(46)

where F = [F1F2

].

σ � 1 ensures F1 remains inside its related friction cone and ensure no slippingin the upper surface. The analytic solution for this optimization problem is given inAppendix D.

In phase 3, the second row of B becomes zero and the equations of motion are asfollows:

m12xs + h1 = b11τ1 + b12τ2(47)

xb = 0.

Now, if τ is calculated similar to phase 1 and instead of D2 = 0, which causessingularity in B, D′

2 = [1 μ2 sign(xdesb )] is used, the control law similar to phase 1

increases the normal force on the object. Therefore after a limited period of time itcan put the object on motion. This phenomenon can be described as follows. Sincethe object is in a static condition:

|�F1y | = |�F2y ||�F1x | = μ1|�F1y | (48)

|�F1x | = |�F2x |.The physical condition for object coefficients of friction, μ1 > μ2, ensures that

the contact force between object and horizontal surface moves to the edge of thefriction cone and brings the object to the motion starting position. In this position,D2 is not zero any more and practically the controller switches to phase 1.

In phase 4, B becomes zero, i.e., D1 = 0 and D2 = 0. Similar to phase 3, theobject does not move and the controller must try to regulate the actuators such thatthe object leaves the static condition. In a similar way, if D′

1 = [1 μ1 sign(xdesb )]

and D′2 = [1 μ2 sign(xdes

b )] are used instead of D1 = 0 and D2 = 0, the object ispushed to the motion starting position and the system switches to the other phases1 or 2.

There are two points that must be mentioned here regarding this proposed ap-proach. (i) There might be some cases that D1 = D2 = [1 0]. In this case the signof xs and xb changes simultaneously and B is once again singular. This is not the


case that we should worry about, since it rarely occurs and the system does not stayin this condition. (ii) It is clear that we need feedback of acceleration to compute αi

and βi (i = 1,2,3) in the control law.

5. Numerical Results

Numerical results for both the conventional and proposed control approaches aregiven and compared in this section. The physical parameters listed in Table 2 areused for numerical purposes.

The object is assumed to track the following desired trajectory:

xdesb =

{0.0256 0 < t < 10 1 � t < 6−0.0256 6 � t < 7 (49)

xdesb (0) = 1.366, xdes

b (0) = 0.

In order to simulate the slipping phenomenon, we assume that during motion,the coefficient of friction of object and horizontal surface changes from its nominalvalue:

μ2 = μ2 if 0 � t � 0.5 and t > 2(50)

μ2 = 0.15 if 0.5 < t � 2.

Note that control laws are calculated using nominal values.Accuracy of the results of dynamical modeling is shown in Fig. 7 by plotting the

difference between the results of the present model and the SimMech model for xb,q1 and q2.

Superiority of the second tracking strategy (21) to the first one (20) for slippagecontrol is shown in Fig. 8 by comparing the results of two strategies for xs and xb.

Performances of the two approaches are compared in Fig. 9. The controller pa-rameters in both approaches are similar.

Robustness of the controllers is studied numerically with reducing the mass andlength parameters by 20% in the controller with respect to those of the system.Results are shown and compared in Fig. 10. It can be seen that the controller per-formance in the conventional approach is affected seriously, while the proposedapproach can adjust itself to the new conditions very well.

Table 2.Numerical values for parameters

m1 m2 L1 L2 mb μ1 μ2

1 kg 1 kg 1 m 1 m 2.5 kg 0.25 0.1


Figure 7. Difference between numerical results from the present dynamic model and the SimMechmodel. (a) Object position. (b) Joint velocities.

Figure 8. Comparison of obtained results from two strategies using the conventional approach((20 and (21)). (a) Sliding velocity of end-effector on object surface. (b) Object velocity.

6. Experimental Setup and Results

An experimental setup is established to verify the method in practice. The setupand its model, in INVENTOR, are shown in Fig. 11. It consists of a two-link ma-nipulator, a slider block, a slider railway, a control board and a longitudinal motionmeasurement encoder (1–5 in Fig. 11a). A flat surface that can freely rotate with re-spect to the second link is used as the end-effector in order to satisfy the non-rollingcondition (Fig. 11b). In order to enforce the slippage that happens in the experiment,the railway is prepared such that its coefficient of friction with the slider block ischanged in midway (35 < xb < 40). The end-effector position and its derivativesare measured using joint sensors. Similar measurement for the slider is carried oututilizing the longitudinal motion encoder.


Figure 9. Comparison of results obtained from two control approaches. (a) Tracking error. (b) Objectvelocity. (c) End-effector movement on the object surface. (d) Sliding velocity of the end-effector onthe object surface. (e) Time history of τ1. (f) Time history of τ2.


Figure 10. Comparison of results obtained from two different control approaches when the modelparameters differ from the actual parameters. (a) Tracking error. (b) Object velocity. (c) End-effectormovement on the object surface. (d) Sliding velocity of the end-effector on the object surface. (e) Timehistory of τ1. (f) Time history of τ2.


Figure 11. Experiment setup. (a) Real setup. (b) Setup model in INVENTOR.

Table 3.Physical parameters values of the experiment setup

m1 m2 mb L1 L2 μ1 μ2

1.82 kg 1.45 kg 1 kg 0.23 m 0.2964 m 0.6 0.3

The following trajectory is set as the desired trajectory for the slider:

xdesb =

{−0.0256 0 < t < 10 1 � t < 6,0.0256 6 � t < 7

xdesb (0) = 0, xdes

b (0) = −0.27. (51)

The physical parameters of the system are given in Table 3.Figure 12 compares the experimental results with those of the model for the

above-mentioned parameters. As it is seen, the end-effector slides on the block inthe initial and middle phases of motion. The initial phase slippage is due to thedifference between parameters in the controller and the system, while the middleslippage is due to the changes in the friction coefficient between the slider and itsrailway. The controller stops these slippages and tries to perform the tracking ob-jective in the remaining time. The slider velocity obtained from the simulation andexperiment are compared with the desired one in Fig. 12a. Figure 12b compares theend-effector slippage velocity on the block. The motor torques calculated in the sim-ulation are compared with those obtained from the experiment in Fig. 12c and 12d.Note that the experimental results are filtered by a low-pass filter in MATLAB. Fig-ure 12e and 12f shows the effects of this filtering on the experimental results forblock velocity as well as end-effector slippage velocity. The performance of thecontroller in controlling the end-effector slippage in practice and the accuracy ofthe dynamical modeling is clearly seen from these experimental results.


(a) (b)

(c) (d)

(e) (f)

Figure 12. Comparison of experimental and simulation results. (a) Object velocity. (b) End-effectorslippage velocity. (c) Time history of τ1. (d) Time history of τ2. (e) Filtered and unfiltered objectvelocity. (f) Filtered and unfiltered end-effector slippage velocity.

7. Conclusions

The sliding phenomenon in grasping an object by a manipulator is studied in thispaper, considering a typical problem of moving an object by a manipulator. In or-


der to formulate and simulate the dynamics of the system, equality and inequalityequations of contact conditions are replaced by a single second-order differentialequation with switching coefficients. This kind of formulation allows us to synthe-size the controller analytically. The accuracy of the model is verified by comparingits results with those of SimMech of MATLAB software.

The conventional control method in grasping of an object is modified for thecases that the end-effector of the manipulator slides on the object by includingthe movement of the end-effector on the object and its velocity in the control law.Although the modified method can simulate and control sliding of the manipulatoron the object, it does not show good performance, especially in the case that themodel parameters differ from the actual parameters. Since the method is a hybridfeedforward and feedback control method, it is very difficult to prove the stabilityof the method.

A new method is proposed by reducing the dynamical model to an input–outputreduced form. The governing equation in the new form is in the conventionalform, Mx + h = Bτ , with state-dependent matrices and vectors switching fromone phase of motion to another. It is shown that any stable control law for thismodel guarantees internal stability of the system. In order to control the slippageof the end-effector on the object and force the object to track the desired trajectory,a multi-phase controller is proposed.

Performance and robustness of the proposed method are numerically comparedwith the conventional method. Also, an experimental setup is established andused to evaluate the controller performance practically. Both simulation and ex-perimental results show the excellent performance of the controller in controllingend-effector slippage while it still pushes the object to track its desired trajectory.Although the convergence speed was not discussed, it can be changed by changingthe controller parameters, such as Kp and Kv . Practically, this convergence speedcan be realized from simulation results, as well as from experimental results.

References

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2. B. Mishra, J. T. Schwartz and M. Sharir, On the existence and synthesis of multifinger positivegrips, Algorithmica 2, 541–558 (1987).

3. A. Bicchi, On the closure properties of robotic grasping, Int. J. Robotics Res. 14, 319–334 (1995).4. Y. H. Liu, Qualitative test and force optimization of 3-D frictional form closure grasps using linear

programming, IEEE Trans. Robotics Automat. 15, 163–173 (1999).5. Y. Zheng and W. H. Qian, An enhanced ray-shooting approach to force-closure problems,

J. Manufact. Sci. Eng. 128, 960–968 (2006).6. L. Han, J. C. Trinkle and Z. X. Li, Grasp analysis as linear matrix inequality problems, IEEE

Trans. Robotics Automat. 16, 663–674 (2000).7. Y. C. Park and G. P. Starr, Grasping synthesis of polygonal objects using a three-fingered robot

hand, Int. J. Robotics Res. 11, 163–184 (1992).


8. C. P. Tung and A. C. Kak, Fast construction of force-closure grasps, IEEE Trans. Robotics Au-tomat. 12, 615–626 (1996).

9. Y. H. Liu, Computing n-finger form-closure grasps on polygonal objects, Int. J. Robotic Res. 19,149–158 (2000).

10. X. Zhu and H. Ding, Computation of force-closure grasps: an iterative algorithm, IEEE Trans.Robotics 22, 172–179 (2006).

11. A. Morales, P. J. Sanz, A. P. del Pobil and A. H. Fagg, Vision-based three-finger grasp synthesisconstrained by hand geometry, Robotics Autonomous Syst. 54, 496–512 (2006).

12. E. A. Al-Gallaf, Multi-fingered robot hand optimal task force distribution: neural inverse kinemat-ics approach, Robotics Autonomous Syst. 54, 34–51 (2006).

13. X. Y. Zhu, H. Ding and J. Wang, Grasp analysis and synthesis based on a new quantitative measure,IEEE Trans. Robotics Automat. 19, 942–953 (2003).

14. X. Y. Zhu and J. Wang, Synthesis of force-closure grasps on 3-D objects based on the Q distance,IEEE Trans. Robotics Automat. 19, 669–679 (2003).

15. R. Platt Jr, A. H. Fagg and R. A. Grupen, Nullspace composition of control laws for grasping, in:Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Lausanne, pp. 1717–1723 (2002).

16. T. Miyabe, M. Yamamo, A. Konno and M. Uchiyama, An approach toward a robust object recov-ery with flexible manipulators, in: Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems,Maui, HI, pp. 907–912 (2001).

17. J. W. Li, H. Liu and H. G. Cai, On computing three-finger force-closure grasps of 2-D and 3-Dobjects, IEEE Trans. Robotics Automat. 19, 155–161 (2003).

18. Y. H. Liu, M. L. Lam and D. Ding, A complete and efficient algorithm for searching 3-D form-closure grasps in the discrete domain, IEEE Trans. Robotics 20, 805–816 (2004).

19. X. Z. Zheng, R. Nakashima and T. Yoshikawa, On dynamic control of finger sliding and objectmotion in manipulation with multi-fingered hands, IEEE Trans. Robotics Automat. 16, 469–481(2000).

20. A. A. Cole, P. Hsu and S. S. Sastry, Dynamic control of sliding by robot hands for regrasping,IEEE Trans. Robotics Automat. 8, 42–52 (1992).

21. I. Kao and M. R. Cutkosky, Comparison of theoretical and experimental force/motion trajectoriesfor dextrous manipulation with sliding, Int. J. Robotics Res. 12, 529–534 (1993).

22. N. Y. Chong, D. Choi and Il. H. Suh, A generalized motion force planning strategy for multi-fingered hands using both rolling and sliding contacts, in: Proc. IEEE/RSJ Int. Conf. on IntelligentRobots and Systems, Yokohama, pp. 113–120 (1993).

23. E. Rabinowicz, Friction and Wear of Materials, 2nd edn. Wiley, New York, NY (1995).24. F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids: Part II. Oxford University

Press, Oxford (1964).25. B. Armstronghélouvry, P. Dupont and C. C. Canudas de Wit, A survey of models, analysis tools

and compensation methods for the control of machines with friction, Automatica 30, 1083–1138(1994).

26. D. Karnopp, Computer simulation of slip–stick friction in mechanical dynamic systems, J. Dyn.Syst. Meas. Control 107, 100–103 (1985).

27. Y. Nakamura, Advanced Robotics: Redundancy and Optimization, Addison-Wesley, Reading, MA(1991).

28. C. T. Chen, Introduction to Linear System Theory, Holt, Rinehart and Winston, New York, NY(1970).

29. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs,NJ (1974).

S. H. Jazi et al. / Advanced Robotics 22 (2008) 1559–1584 1581A

ppen

dix

A.

Val

ues

ofα

ian

dβ

i(i

=1,

2,3)

Tabl

eA

.1.

Val

ues

forα

T=

[α 1α

2α

3]a

ndβ

T=

[β 1β

2β

3]u

nder

diff

eren

tcon

ditio

ns

xs=

0

xs�=

0x− s

�=0

x− s

=0

xb

Mov

emen

tM

otio

nre

vers

ing

No

Mot

ion

No

Mot

ion

Star

tfor

war

dSt

artb

ackw

ard

xb

�=0

mov

e-α

T=

[01

sign

(xs)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]m

ent

βT

=[0

1β

T=

[01

βT

=[0

1β

T=

[01

βT

=[0

1β

T=

[01

sign

(xb)]

sign

(xb)]

sign

(xb)]

sign

(xb)]

sign

(xb)]

sign

(xb)]

xb

=0

x− b

�=0

mot

ion

αT

=[0

1si

gn(x

s)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]re

vers

ing

βT

=[0

10]

βT

=[0

10]

βT

=[0

10]

βT

=[0

10]

βT

=[0

10]

βT

=[0

10]

noα

T=

[01

sign

(xs)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]m

otio

nβ

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]x− b

=0

noα

T=

[01

sign

(xs)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]m

otio

nβ

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]β

T=

[10

0]st

art

αT

=[0

1si

gn(x

s)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]fo

rwar

dβ

T=

[01

1]β

T=

[01

1]β

T=

[01

1]β

T=

[01

1]β

T=

[01

1]β

T=

[01

1]st

art

αT

=[0

1si

gn(x

s)]

αT

=[0

10]

αT

=[1

00]

αT

=[1

00]

αT

=[0

11]

αT

=[0

1−

1]ba

ckw

ard

βT

=[0

1−

1]β

T=

[01

−1]

βT

=[0

1−

1]β

T=

[01

−1]

βT

=[0

1−

1]β

T=

[01

−1]


B. Error Analysis for Strategies in (20) and (21)

Let us define eq = qc − q and use a typical feedback linearization form for positioncontrol:

Bτ e = M(qc + Kvq eq + Kpqeq) + h. (B.1)

Substituting (B.1) and (19) in (1) we can write:

Mq + h = M(qc + Kvq eq + Kpqeq) + h + JT(Fdes1 − F1). (B.2)

Hence, the governing equation of the tracking error is obtained as:

eq + Kvq eq + Kpqeq = −M−1JTeF1, (B.3)

assuming no uncertainty in the modeling.In the first strategy, (20), since xdes

e = xb + xs , at each instant qc = q. Henceequation (B.3) is rewritten as:

eq + Kvq eq = −M−1JTeF1 . (B.4)

For the second strategy qc is not necessarily equal to q and the tracking erroris computed from (B.3). Comparing (B.3) and (B.4) shows the advantage of thesecond strategy. For example, once we have a constant term in the right-hand sideof (B.3) and (B.4) in the steady-state phase of the system, (B.4) results in constanterror in xs , while xs vanishes according to (B.3) and (21).

C. Proof of Theorem 2

Notice that columns or rows of M are functions of time and according to theoremof [28], M is full row rank and then invertible on [t1 t2], if and only if:

� =∫ t2

t1

MMT

dt, (C.1)

is a non-singular matrix.� is the Hermitian matrix, so it is a symmetric semi-positive definite matrix.

Thus, if is also non-singular, it will be positive definite and regarded to Rayliegh–Ritz theorem:

λ1I2 � � � λ2I2, (C.2)

where positive scalars λ1 and λ2 are minimum and maximum eigenvalues of � ,respectively, and vary with time t1 and t2. If we choose t1 = t0 and t2 = t0 + T anddefine β1 = supt0∈(−∞,+∞) λ1(t0, t0 + T ) and β2 = inft0∈(−∞,+∞) λ2(t0, t0 + T ),then β1 and β2 are constant scalars and we can write for all t0:

β1I2 �∫ t0+T

t0

MMT

dt � β2I2. (C.3)


Since we have shown that velocity and position tracking errors converge to zero,the above condition will be met if the desired trajectory satisfies:

β1I2 �∫ t0+T

t0

MdMTd dt � β2I2, (C.4)

where Md = M(xdesb , xdes

s ).

D. Analytical Solution to (46)

Using (46e) we can write:

xb = Kbτ + λb, (D.1)

where Kb and λb are the first row of M−1B and −M−1h, respectively. Substitut-ing (D.1) in (46d), q is calculated as:

q = −A−1q AxKbτ − A−1

q Axλb + A−1q b. (D.2)

Substituting (D.1) and (D.2) in (46b) and (46c), we obtain:

F1 = K1τ + λ1, F2 = K2τ + λ2, (D.3)

where:

K1 = J−T(B + MA−1q AxKb), K2 = M0Kb − K1 (D.4)

λ1 = J−TMA−1q Axλb − J−TMA−1

q b − J−Th, λ2 = M0λb + g0 − λ1. (D.5)

Defining S = B+[M(Xdes +Kv e+Kpe)+ h] and using (46a), (D.3) can be rewrittenas:

F1 = K′1y + λ′

1, F2 = K′2y + λ′

2, (D.6)

where:

K′1 = K1(I2 − B+B), K′

2 = K2(I2 − B+B)(D.7)

λ′1 = K1S + λ1, λ′

2 = K2S + λ2.

If we define:

Kf =[

K′1

K′2

], λf =

[λ′

1λ′

2

], (D.8)

and use (46f), then (D.6) can be reformed as:

Q

[F1y

F2x

F2y

]= Kf y + λf , where Q =

⎡⎢⎣

−σ1μ1 sign(xdesb ) 0 0

1 0 00 1 00 0 1

⎤⎥⎦ . (D.9)

As one can see, Q is full column rank, so:[F1y

F2x

F2y

]= Q+Kf y + Q+λf . (D.10)


Now if we choose y as:

y = −(Q+Kf )+Q+λf , (D.11)

F will have its minimum norm with respect to y [29].

About the Authors

Shahram Hadian Jazi received his BS and MS degrees in Mechanical Engineer-ing in 1998 and 2000, respectively, both from Isfahan University of Technology(IUT), Isfahan, Iran, where he is currently working towards his PhD degree. Hisresearch interests are robotics (analysis, design and manufacturing) and controlof dynamical systems. He is an active member of the Dynamic and Robotic Re-search Group of IUT and has been involved in several projects of the group. He isa member of the ASME.

Mehdi Keshmiri received his BS and MS degrees in Mechanical Engineering,both from Sharif University of Technology, Tehran, Iran, in 1986 and 1989, re-spectively, and his PhD from McGill University, Montreal, Canada, in 1995. Since1995, he has been with the Department of Mechanical Engineering at Isfahan Uni-versity of Technology, Isfahan, Iran. His research interests are system dynamics,control of dynamical systems and vibration, and also he is very active in the fieldof science park development and technology management.

Farid Sheikholeslam received his BS degree in Electronics from Sharif Univer-sity of Technology, Tehran, Iran, in 1990. He also received his MS degree inCommunications and a PhD in Electrical Engineering from Isfahan Universityof Technology (IUT), Isfahan, Iran, in 1994 and 1998, respectively. Since 1999 hehas been with the Department of Electrical and Computer Engineering at IUT. Hisresearch interests are robotics, control algorithms, fuzzy systems, neural networks,stability analysis and nonlinear systems.

Documents

Dynamic Analysis and Control Synthesis of Grasping and ......Dynamic Analysis and Control Synthesis of Grasping and Slippage of an Object Manipulated by a Robot Shahram Hadian Jazi,