Grasping-Force Optimization for Multifingered Robotic Hands Using a ...
IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004
[16] J. Ponce and B. Faverjon, “On computing three-finger force closure grasp of polygonal objects,” IEEE Trans. Robot. Automat., vol. 11, pp. 868–881, Dec. 1995. [17] J. Ponce, S. Sullivan, A. Sudsang, J.-D. Boissonnat, and J.-P. Merlet, “On computing four-finger equilibrium and force-closure grasps of polyhedral objects,” Int. J. Robot. Res., vol. 16, no. 1, pp. 11–35, 1997. [18] Y. H. Liu, “Computing n-finger form-closure grasps on polygonal objects,” Int. J. Robot. Res., vol. 19, no. 2, pp. 149–158, 2000. [19] D. Kirkpatrick, B. Mishra, and C. Yap, “Quantitative Steinitz’s theorem with applications to multi-fingered grasping,” Discr. Computat. Geom., vol. 7, no. 3, pp. 295–318, 1992. [20] C. Ferrari and J. Canny, “Planning optimal grasps,” in Proc. IEEE ICRA, 1992, pp. 2290–2295. [21] B. Mirtich and J. Canny, “Easily computable optimum grasps in 2-D and 3-D,” in Proc. IEEE ICRA, 1994, pp. 739–747. [22] G. Mantriota, “Communication on optimal grip points for contact stability,” Int. J. Robot. Res., vol. 18, no. 5, pp. 502–513, 1999. [23] L. Mangialardi, G. Mantriota, and A. Trentadue, “A three-dimensional criterion for the determination of optimal grip points,” Robot., Comput. Integ. Manuf., vol. 12, no. 2, pp. 157–167, 1996. [24] Z. X. Li and S. Sastry, “Task-oriented optimal grasp by multifingered robot hand,” IEEE J. Robot. Automat., vol. 4, pp. 32–44, Feb. 1988. [25] A. F. Stappen, C. Wentink, and M. H. Overmars, “Computing immobilizing grasps of polygonal parts,” Int. J. Robot. Res., vol. 19, no. 5, pp. 467–479, 2000. [26] G. M. Bone and Y. Du, “Multi-metric comparison of optimal 2-D grasp planning algorithms,” in Proc. IEEE ICRA, 2001, pp. 3061–3066. [27] I.-M. Chen and J. W. Burdick, “A qualitative test for n-finger forceclosure grasps on planar objects with applications to manipulation and finger gaits,” in Proc. IEEE ICRA, 1993, pp. 814–820. [28] D. Ding, Y. H. Liu, J. Zhang, and A. Knoll, “Computation of fingertip positions for a form-closure grasp,” in Proc. IEEE ICRA, 2001, pp. 2217–2222. [29] Y. H. Liu, “Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming,” IEEE Trans. Robot. Automat., vol. 15, pp. 163–173, Feb. 1999. [30] X. Y. Zhu and J. Wang, “Synthesis of force-closure grasps on 3-D objects based on the Q distance,” IEEE Trans. Robot. Automat., vol. 19, pp. 669–679, Aug. 2003. [31] R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970. [32] N. Xydas and I. Kao, “Modeling of contact mechanics and friction limit surfaces for soft fingers in robotics, with experimental results,” Int. J. Robot. Res., vol. 18, no. 8, pp. 941–950, 1999. [33] R. D. Howe and M. R. Cutkosky, “Practical force-motion models for sliding manipulation,” Int. J. Robot. Res., vol. 15, no. 6, pp. 557–572, 1996. [34] R. M. Murray, Z. X. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994.
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Grasping-Force Optimization for Multifingered Robotic Hands Using a Recurrent Neural Network Youshen Xia, Jun Wang, and Lo-Ming Fok Abstract—Grasping-force optimization of multifingered robotic hands can be formulated as a problem for minimizing an objective function subject to form-closure constraints and balance constraints of external force. This paper presents a novel recurrent neural network for real-time dextrous hand-grasping force optimization. The proposed neural network is shown to be globally convergent to the optimal grasping force. Compared with existing approaches to grasping-force optimization, the proposed neural-network approach has the advantages that the complexity for implementation is reduced, and the solution accuracy is increased, by avoiding the linearization of quadratic friction constraints. Simulation results show that the proposed neural network can achieve optimal grasping force in real time. Index Terms—Grasping-force optimization, multifingered robotic hands, recurrent neural networks.
I. INTRODUCTION Multifingered robot hands have become of great interest in robotics due to their advantages over conventional grippers in tasks requiring dexterous manipulation [1]–[3]. Robotic hands are more complex in construction and require more sophisticated analysis to determine the grasping force to be exerted on an object without dropping or breaking the object [4], [5]. One of important issues on the optimal control of robotic hands is the grasping force-optimization problem. The problem for multifingered robotic hands is concerned with determining the least grasping force from each finger exerted on a three-dimensional (3-D) rigid object without slipping or losing the grasp. The grasping-force optimization is to maximize the efficiency and dexterity of robotic manipulations. The objective of the force optimization is to minimize the magnitude of grasping force from every finger applied to an object, while holding the object against slippage and balancing external forces due to gravity, acceleration, and others. Various optimization approaches have been proposed to achieve the optimal force distribution. Nakamura et al. [6] proposed using Lagrange multipliers to find the optimal forces. Sinha and Abel [7] proposed solving the nonlinear friction constraints by using the successive quadratic programming method. Kumar and Waldron [8] presented a class of suboptimal algorithms for contact-force optimization. These methods are computationally intensive because of the nonlinearity of the friction-cone constraint. Thus, the grasping-force optimization problem is usually solved by using linearization of the friction cone followed by linear or quadratic programming [9]–[12]. Linearization methods, however, may result in the obtained grasp force satisfying the linearized friction constraints, but violating the nonlinear constraints, and they increase the number of facets in the linearized friction cone resulting in excessive computational effort. Buss et al. [13] reformulated the contact force-optimization problem as a semidefinite programming problem, and solved it by using a projection gradient-flow algorithm, which requires an initial point
Manuscript received July 19, 2002; revised January 7, 2003. This paper was recommended for publication by Associate Editor C. Melchiorri and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4174/00E. The authors are with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TRA.2004.824946 1042-296X/04$20.00 © 2004 IEEE
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satisfying the nonlinear friction constraints. Han et al. [14] cast the friction cone constraints further into linear matrix inequalities and solved the grasp-force problems by using the interior point algorithm. Moreover, Liu et al. [15] gave a min 0 max method for automatic generation of the initial points. To perform highly dexterous robotic manipulations, the result of grasping-force optimization must be used in motion-control law, and thus, the optimal force-distribution problem should be solved in real time. Parallel and distributed approaches to grasping-force optimization for dexterous robotic manipulations are deemed necessary and desirable. In recent years, neural network approaches have demonstrated their great promise for optimization. Neural networks for constrained optimization problems have been widely explored for real-time applications [16]–[18]. Reported results of numerous investigations have shown many advantages over the traditional optimization algorithms, especially in real-time applications. Moreover, neural networks have been applied to solve various problems in robotics [19]–[21]. For example, Ding and Wang [19] demonstrated the effectiveness of using recurrent neural networks for torque optimization in robotic manipulation. Xia and Wang [20] proposed a dual neural network for kinematic control of redundant robot manipulators. In this paper, we propose a novel recurrent neural network for real-time grasping-force minimization of multifingered robotic hands. We first reformulate the force-optimization problem. Based on the derived equivalent formulation, we then present a novel neural network for grasping-force optimization. The proposed neural network has a low complexity for parallel implementation, and is shown to be globally convergent to the optimal grasping force. Furthermore, the present neural network is capable of taking into account the nonlinearity of the friction constraints, so that the solution accuracy is ensured.
To derive a neural network model for solving (1), we first give an equivalent relationship between the solution to (1) and the solution to the following formulation:
Qx + q + rg(x)y 0 GT z = 0 + (y + g (x)) 0 y = 0 Gx = 0fext
T where g (x ) = [g1 (x); . . . ; gm (x)] , where gi (x) = x2i1 + x2i2 0 i xi3 ; (y)+ = ([y1 ]+ ; . . . ; [ym ]+ )T ; and + = maxf0; yi g; rg(x) = [rg1 (x); . . . ; rgm (x)] is an [y i ] 3m 2 m matrix. Lemma 1: x3 is a solution to (1) if and only if there exist nonzero vectors y 3 and z 3 such that (x3 ; y 3 ; z 3 ) satisfies (2).
Proof: (1) can be rewritten as minimize subject to
0fext :
(3)
1
2
According to the well-known saddle-point theorem [23], x3 is a solum and z 3 2 R6 such that tion to (3) if and only if there exists y 3 2 R+ 3 m m 6 3 for any (x; y; z ) 2 R 2 R+ 2 R ; (x ; y 3 ; z 3 ) satisfies
L(x3 ; y; z ) L(x3 ; y3 ; z 3 ) L(x; y3 ; z 3 )
2
2 Rm j y 0g. It implies that Gx3 = 0fext and
3 T 3 T T 3 (x ) Q (x ) + q x + y g (x )
12 (x3 )T Q(x3 ) + qT x + (y3 )T g(x3 ) 21 xT Qx + qT x + (y3)T g(x) 0 (Gx + fext )T z3 :
From the first inequality above, it follows that (y
0 y3 )T (0g(x3 )) 0 8y 0:
On the other side, let
(x) = xT Qx + qT x + (y3 )T g(x) 0 (Gx + fext )T z 3 : 1
2
Then the right inequality above implies
(x ) (x 3 )
8x 2 R3m :
It follows that r (x3 ) = 0. Then
Qx3 + q + rg(x3 )y3 0 GT z 3 = 0:
1
f (x) = xT Qx + qT x 2 subject to Gx = 0fext x2i1 + x2i2 i xi3 ; (i = 1; . . . ; m) minimize
L(x; y; z ) = xT Qx + qT x + yT g(x) 0 (Gx + fext )T z:
1
Consider a multifingered robot hand grasping a single object in a 3-D workspace with m point contacts between the grasped object and the fingers [12]. The problem of the grasp-force optimization is to find a set of contact forces such that the object is held at the desired position and external forces are compensated. So, a grasping force xi is applied by each finger to the object to hold the object without slippage and to balance any external forces. To ensure nonslipping at a contact point, the grasping force xi should satisfy x2i1 + x2i2 i x2i3 , where i > 0 is the friction coefficient at finger i, and xi1 ; xi2 , and xi3 are components of contact force xi in the contact coordinate frame. Besides the form-closure constraints, to balance any external wrench fext to maintain a stable grasp, each finger must apply a grasping force xi = 623m [xi1 ; xi2 ; xi3 ] to the object such that Gx = 0fext , where G 2 R T is the grasp transformation matrix, and x = [x1 ; . . . ; xm ] 2 R3m is the grasping force vector. On the other side, to avoid crushing the object, the contact forces should be as small as possible. In this paper, we consider the quadratic objective function of contact forces. So, the optimal grasping force with holding the object can be formulated as the following optimization problem with linear and nonlinear constraints:
1 T x Qx + qT x 2 g(x) 0; Gx =
Define the Lagrangian function of (3) as
m = fy where R+
II. PROBLEM FORMULATION FOR OPTIMAL GRASPS
(2)
So, x3 is a solution to (3) if and only if (x3 ; y 3 ; z 3 ) satisfies (1)
where q 2 R3m and Q is a 3m 2 3m symmetric and positive definite matrix. The above quadratic objective function represents the minimum weighted norm of joint torque vectors and the minimum norm force when q = 0.
Qx3 + q + rg(x3 )y3 0 GT z 3 = 0 3 T 3 (y 0 y ) (0g (x )) 0 8y 0 3 Gx = 0fext : From the projection theorem [22], it can be seen that the above formulation is equivalent to (2).
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it follows
y(t) = e0(t0t ) y(t0 ) + e0t
Fig. 1. Block diagram of the proposed neural network in (4).
III. NEURAL NETWORK MODEL Based on the equivalent formulation in (2), we propose a recurrent neural network for solving (1), with its dynamical equation given by
0Qx 0 q 0 rg(x)y + GT z d = (4) (y + g(x))+ 0 y dt 0Gx 0 fext where x 2 R3m ; y 2 Rm ; z 2 R6 , and > 0 is a scaling parameter. x y z
The dynamical equation (4) can be easily realized by a recurrent neural network with a simple structure, as shown in Fig. 1, where = 1. The circuit realizing the proposed neural network consists of 4m + 6 integrators, m processors for gi (x); m processors for rgi (x); 9m2 + 12m connection weights, and some summers. Compared with the conservative linearization method [12], the proposed method can avoid the disadvantage in increasing the number of facets in the linearized friction cone, and has a better solution accuracy. Furthermore, the proposed method can be guaranteed to be globally convergent, as shown in the next section. IV. CONVERGENCE ANALYSIS In this section, we prove the global convergence of the proposed neural network for grasping-force optimization. We first give a lemma. Lemma 2: [22]. Let be a closed convex set. Then
(v 0 P (v))T (P (v) 0 x) 0; v 2 Rl ; x 2
kP (u) 0 P (v)k ku 0 vk; v; u 2 Rl where P (x) = arg minv 2 kx 0 v k and k 1 k denotes l2 norm. Theorem 1: 1) For any initial point u(t0 ) = (x(t0 ); y (t0 ); z (t0 )) 2 R4m+6 , there exists one unique continuous solution trajectory u(t) = (x(t); y(t); z(t)) for (4). Moreover, y(t) 0, provided that y(t0 ) 0. m 2 R6 is an equilibrium point of (4), 2) If (x3 ; y 3 ; z 3 ) 2 R3m 2 R+ then x3 is an optimal solution to (1). Proof: 1) Note that (y + g (x))+ is locally Lipschitz continuous, then the right term of (4) is locally Lipschitz continuous. According to the local-existence theorem of ordinary differential equations, there exists a unique continuous solution u(t) of (4) for (t0 ; T ). Next, for the given initial point u0 with y (t0 ) 0
t t
dy dt
dy dt
+y
+ y = (y + g(x))+ es ds =
t
t
es (y + g(x))+ ds
t t
es (y + g(x))+ ds
since (y + g (x))+ 0; y (t) 0. 2) Clearly, the equilibrium point of (4) satisfies (2). From Lemma 1, it follows that x3 is an optimal solution to (1) when (x3 ; y 3 ; z 3 ) 2 m 2 R6 is an equilibrium point of (4). R3m 2 R+ We now establish our main results concerning the global convergence of the proposed neural network in (4). Theorem 2: The proposed neural network in (4) is stable in the sense of Lyapunov and globally convergent to the Karush–Kuhn–Tucker (KKT) point u3 = (x3 ; y 3 ; z 3 ), where x3 is the optimal grasping-force vector. Proof: This proof is divided into four parts. First, we show that the proposed neural network in (4) is stable in the sense of Lyapunov. Let u(t) be the solution of (4) with the initial point u(t0 ) = (x(t0 ); y(t0 ); z (t0 )) 2 R4m+6 , where y(t0 ) 0. From Proposition 1, it follows that u(t) = (x(t); y (t); z (t)) satisfies y(t) 0. We define a Lyapunov function
E (u ) = E 0 (u ) +
1 (kx 0 x 3 k 2 + ky 0 y 3 k 2 + kz 0 z 3 k 2 ) 2
where
E 0 (u ) =
1 (kQx + q + rg(x)y 0 GT zk2 + kGx + f k2 ) ext 2 1 +g(x)T [(y + g(x))+ 0 y] 0 2 k(y + g(x))+ 0 yk2 :
We have
dE0 (u) du
= F (u) 0 (rF (u) 0 I )(P (u 0 F (u)) 0 u)
where P (u) = [x; (y )+ ; z ]T ; rF (u) denotes the Jacobian matrix of F , and Qx + q + rg(x)y 0 GT z
F (u ) =
Let r(u)
dE (u) dt
0 g (x ) Gx + fext
:
= P (u 0 F (u)) 0 u, then
= rE (u)T du dt = ( F (u ) + u 0 u 3 ) T r (u ) + kr (u )k 2 0 r (u )T r F (u )r (u )
rE (u) is the gradient of E . In the equality of Lemma 2, let = u 0 F (u) and let x = u3 , we get
where
v
(r(u) + u 0 u3 )T (0r(u) 0 F (u)) 0: It follows that
dE (u) dt
0 F (u ) T (u 0 u 3 ) 0 r (u )T r F (u )r (u ) 0 :
Second, we show that E (u) is radially unbounded. In the equality of Lemma 2, let v = u 0 F (u) and let x = u, we obtain
0 F (u )T r (u ) k r (u )k 2 : It follows that E (u) ku 0 u3 k2 =2. Thus E (u) is positive definite and is radially unbounded. Therefore, for any initial point u0 2 R3m 2 m 2 R6 , there exists a convergent subsequence fu(t )g such that R+ k limk!1 u(tk ) = u^, where dE (^u)=dt = 0.
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Third, we prove that dE(^ u)=dt = 0 if and only if u ^ is a KKT point. By Theorem 1, it can be seen that y^ 0. Since the Jacobian matrix Q+
rF (u) =
Om2m O62m
G
Om26 O626
is positive semidefinite, where Om2m ; O62m ; Om26 ; O626 are zero matrices, F (u) is monotone. Then T
F (u) (u
0 u3 ) F (u3 )T (u 0 u3 ) 0:
0 x3 )T (Q^x + q + rg(^x)^y 0 GT z^) + (^y 0 y3 )T (0g(^x)) 3 T +(^ z 0 z ) (G^ x + fext ) = 0:
3 T It follows that (^ y 0 y ) g(^ x) = 0 and
0 u3 ) = 0 rF (^u)r(^u) = 0:
F (^ u) (^ u (r(^ u))
T
T x) = (y 3 )T g(x3 ) = 0: (^ y ) g(^
Therefore, u ^ = (^ x; y ^; z ^) is a KKT point of (3). Finally, define a Lyapunov function again
It can be seen that r(^ u)
rF (^u)r(^u) = p(x; z)T rg(^x)p(x; z)
T
k 0 u^k2 =2:
^ E(u) = E0 (u) + u
T ^ and where p(x; z) = Q^ x + q + rg(^ x) 0 G z m J (^ x) = Q +
i=1
y ^i
^ ^ u) = 0 and thus, limk!1 E(u(t ^ u) = 0. For 8 > Then E(^ k )) = E(^ 0, there exists q > 0 such that for all tk tq , we have
r2 gi (^x):
^ ^) < : E(u(t k ); u
Since J (^ x) is positive definite Q^ x+q+
Similar to the previous analysis, we have dE(u)=dt t tq
rg(^x)^y 0 GT z^ = 0: 0
T
Q^ x + q + g(^ x)^ y G z ^=0 3 3 3 T 3 Qx + q + g(x )y G z =0
rg(^x)^y 0 (Qx3 + q + rg(x3 )y3 ) GT (^z 0 z3 ):
dE(u) dt
It follows that
r 0
Q^ x+q+
0 r
r 0 0r 0 r
3
0
0
0
0
r
0 y^):
Because Qx + q + rg(x)^ y is strictly monotone (^ x
(^ x
0
0z
)+
rg(x
3
)(y
3
0 y^)] > 0:
(5)
On the other hand, we have 3 Qx + q +
rg(x3 )^y 0 GT z^ = rg(x3 )(^y 0 y3 ) + GT (z3 0 z^):
Then (^ x
E(u(t))
0 E(u(t0 )) 0
E(u(t))
E(u(t0 )) 0
t t
t
t
r(u(t))
r(u(t))
ku(t) 0 u3 k2 2E(u(t0 )) 0 2
Thus 3
0r(u)T rF (u)r(u) 0
T
T
rF (u(t))r(u(t)) ds
rF (u(t))r(u(t)) ds:
Since E(u) ku 0 u3 k2 =2 where u3 is a KKT point of (1)
0 x3 )T [Q(^x 0 x3 ) + rg(^x)^y 0 rg(x3 )^y)] > 0: 3 T T x ) [G (^ z
Then for
where r(u) is defined in Theorem 2. Then
3
g(^ x)^ y (Qx + q + g(x )^ y) T z z 3 ) + g(x3 )(y 3 y^) = G (^ 3 3 3 T x x ) + g(^ x)^ y g(x )^ y )] (^ x x ) [Q(^ 3 T T 3 3 T 3 3 z z ) + (^ x x ) g(x )(y = (^ x x ) G (^
0.
fore, the proposed neural network in (4) is globally convergent to a KKT point of (1). Finally, we analyze the the convergence rate of the trajectory. From the proof of Theorem 2, it can be seen that for any > 0
0
we get Q^ x+q+
^ ^ ku(t) 0 u^k 2E(u(t)) E(u(t q )) 2: Thus limt!1 ku(t) 0 u ^k = 0, and hence, limt!1 u(t) = u ^. There-
Assume that x ^ = 6 x3 . Using
r r
(6)
3 equations (5) and (6) constitute a contradiction. So x ^ = x and G^ x= T 3 0fext . Now, F (^u) (^u 0 u ) = 0 implies (^ x
It follows that dE(^ u)=dt = 0 if and only if T
0 x3 )T (Qx3 + q + rg(x3 )^y 0 GT z^) 0 3 T 3 3 T 3 ^)] 0 (^ x 0 x ) [rg(x )(^ y 0 y ) + G (z 0 z 3 T T 3 3 3 (^ x 0 x ) [G (^ z 0 z ) + rg(x )(y 0 y ^)] 0 (^ x
rg(x) 0GT
r
m y 2 g (x) i i=1 i T g(x)
0r
Since
0 x3 )T (Qx3 + q + rg(x3 )^y 0 GT z^) 3 T 3 3 T 3 ^)]: y 0 y ) + G (z 0 z = (^ x 0 x ) [rg(x )(^
t t
r(u(t))
T
rF (u(t))r(u(t)) ds:
Therefore, the convergence rate of the neural network in (4) increases as increases. This point will be further illustrated in the next section. V. SIMULATION RESULTS In this section, to show the effectiveness of the proposed neural network, we present some simulation results for grasping-force optimization. Consider a minimum norm force objective function f (x) = kxk2 =2. In this example, a polyhedral object with M = 0:1 kg is grasped by a three-fingered robotic hand, shown in Fig. 2. The grasping model is defined by the position vector of three grasp points: p1 = [0; 1; 0]T ; p2 =
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Fig. 2.
553
Three-finger grasp example. Fig. 4.
Convergence of the optimal force for the three-fingered robotic hand.
Fig. 5.
Convergence of the energy function with = 100.
shown in Fig. 3 with a constant velocity v = 0:2 m/s. A time-varying external wrench applied to the center of mass of the object is
0
fext = [0; fc sin((t)); M g + fc cos((t)); 0; 0; 0]T
Fig. 3. Motion of the three-finger grasp.
0
T ; and p = [0; 1; 0]T . The friction coefficient of each 3 finger is i = 0:6, and the grasp transformation matrix is [1; 0:5; 0]
0
0
1
0
0
GT =
0 0
01
01 0
01 0
1
0
0
0
0
1
01 01
0
0
0
0
0
01
0
0
0
0
0 1 0 0 :5
1
0
0
0
0
0
0 :5
0
0
0
1
1
0
0
0
0
0
01 0
V (x; y; z ) =
0
01
0
where g = 9:8 m/s2 ; 2 [0; 2 ], and fc = M v 2 =r . We use the proposed neural network in (4) to find the optimal force. Fig. 4 shows the convergence of the optimal force. Let the energy function of (4) be
:
To study grasping-force optimization of the multifingered robotic hand, we consider the time-varying external wrench applied on it. Let the robotic hand move along a circular trajectory of radius r = 0:5 m,
1 2
kx + r g (x )y 0 G T z k 2 1 2 + 2 1 + k(y + g (x)) 0 y k + kGx + fext k : 2 2
Figs. 5 and 6 show the convergence rate of V (x; y; z ) based on (4) with two different . Finally, for comparisons, we use the linearization method [21], where the friction cone is linearized into a 20-side polyhedral convex cone inscribed in the friction cone. The grasping-force optimization problem is then solved by the primal-dual neural network [18]. Furthermore, the Matlab optimization toolbox is employed for calculating the Euclidean norm of the optimal force at some time instants. Fig. 7 shows the transients of the Euclidean norm of the optimal force from the proposed neural network, the primal-dual network, and
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Fig. 6. Convergence of the energy function with = 1000.
Fig. 7. Transient of Euclidean norm of optimal forces using three different methods.
Matlab optimization toolbox. It can be seen that the proposed neural network obtained the smallest Euclidean norm of the optimal force. VI. CONCLUDING REMARKS This paper presents a novel neural network for real-time graspingforce optimization of 3-D dexterous manipulation using multifingered robotic hands. The proposed neural network has a simple structure with a low model complexity, and avoids linearizing the friction constraints such that the solution accuracy is greatly enhanced. Moreover, the proposed neural network is shown to be globally convergent to the optimal grasping force. Therefore, the proposed neural networks are conceived to be capable of solving the time-dependent grasping force optimization problems autonomously in real time, and suitable for optimal force control of dexterous manipulations using multifingered robotic hands in uncertain environments. REFERENCES [1] M. R. Cutkosky, “On grasp choice, grasp model, and design of hands for manufacturing task,” IEEE Trans. Robot. Automat., vol. 5, pp. 269–279, Apr. 1989. [2] Z. Li, P. Hsu, and S. Sastry, “Grasping and coordinated manipulation by a multifingered robot hand,” Int. J. Robot. Res., vol. 8, pp. 33–50, 1989.
[3] R. Murray, Z. X. Li, and S. Sastry, A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994. [4] A. Bicchi, “On the closure properties of robotic grasping,” Int. J. Robot. Res., vol. 14, no. 4, pp. 319–334, 1995. [5] D. Prattichizzo, J. K. Salisbury, and A. Bicchi, Contact and Grasp Robustness Measures: Analysis and Experiments, O. Khatib and K. Salisbury, Eds. Berlin, Germany: Springer-Verlag, 1996. [6] Y. Nakamura, K. Nagai, and T. Yoshikawa, “Dynamics and stability in coordination of multiple robotic mechanisms,” Int. J. Robot. Res., vol. 8, no. 2, pp. 44–61, 1989. [7] P. R. Sinha and J. M. Abel, “A contact stress model for multifingered grasps of rough objects,” IEEE Trans. Robot. Automat., vol. 8, pp. 7–22, Feb. 1992. [8] V. Kumar and K. Waldron, “Suboptimal algorithms for force distribution in multifingered grippers,” IEEE Trans. Robot. Automat., vol. 5, pp. 252–257, Apr. 1989. [9] J. Kerr and B. Roth, “Analysis of multifingered hands,” Int. J. Robot. Res., vol. 8, no. 4, pp. 3–17, 1986. [10] C. A. Klein and S. Kittivatcharapong, “Optimization force distribution for the legs of a walking machine with friction cone constraints,” IEEE Trans. Robot. Automat., vol. 6, pp. 73–85, Feb. 1990. [11] F. T. Cheng and D. E. Orin, “Efficient algorithm for optimal force distribution—The compact dual LP method,” IEEE Trans. Robot. Automat., vol. 6, pp. 178–187, Apr. 1990. [12] Y. H. Liu, “Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming,” IEEE Trans. Robot. Automat., vol. 15, pp. 163–173, Apr. 1999. [13] M. Buss, H. Hashimoto, and J. B. Moore, “Dextrous hand grasping force optimization,” IEEE Trans. Robot. Automat., vol. 12, pp. 406–417, June 1996. [14] L. Han, J. C. Trinkle, and Z. X. Li, “Grasp analysis as linear matrix inequality problems,” IEEE Trans. Robot. Automat., vol. 16, pp. 663–674, Dec. 2000. [15] G. F. Liu, J. J. Xu, and Z. X. Li, “Automatic real-time grasping-force determination for multifingered manipulation: Theory and experiments,” in Proc. IROS, 2002, pp. 1676–1680. [16] M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Trans. Circuits Syst., vol. 35, pp. 554–562, May 1988. [17] A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing. London, U.K.: Wiley, 1993. [18] Y. Xia, H. Leung, and J. Wang, “A projection neural network and its application to constrained optimization problems,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 447–458, Apr. 2002. [19] H. Ding and J. Wang, “Recurrent neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. A, vol. 29, pp. 269–276, Feb. 1999. [20] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. B, vol. 31, pp. 147–154, Jan. 2001. [21] L. Fok and J. Wang, “Two recurrent neural networks for grasping-force optimization of multifingered robotic hands,” in Proc. IEEE Int. Joint Conf. Neural Networks, Honolulu, HI, 2002, pp. 35–40. [22] D. Kinderlehrer and G. Stampcchia, An Introduction to Variational Inequalities and Their Applications. New York: Academic, 1980. [23] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. New York: Wiley, 1993.
[16] J. Ponce and B. Faverjon, “On computing three-finger force closure grasp of polygonal objects,” IEEE Trans. Robot. Automat., vol. 11, pp. 868–881, Dec. 1995. [17] J. Ponce, S. Sullivan, A. Sudsang, J.-D. Boissonnat, and J.-P. Merlet, “On computing four-finger equilibrium and force-closure grasps of polyhedral objects,” Int. J. Robot. Res., vol. 16, no. 1, pp. 11–35, 1997. [18] Y. H. Liu, “Computing n-finger form-closure grasps on polygonal objects,” Int. J. Robot. Res., vol. 19, no. 2, pp. 149–158, 2000. [19] D. Kirkpatrick, B. Mishra, and C. Yap, “Quantitative Steinitz’s theorem with applications to multi-fingered grasping,” Discr. Computat. Geom., vol. 7, no. 3, pp. 295–318, 1992. [20] C. Ferrari and J. Canny, “Planning optimal grasps,” in Proc. IEEE ICRA, 1992, pp. 2290–2295. [21] B. Mirtich and J. Canny, “Easily computable optimum grasps in 2-D and 3-D,” in Proc. IEEE ICRA, 1994, pp. 739–747. [22] G. Mantriota, “Communication on optimal grip points for contact stability,” Int. J. Robot. Res., vol. 18, no. 5, pp. 502–513, 1999. [23] L. Mangialardi, G. Mantriota, and A. Trentadue, “A three-dimensional criterion for the determination of optimal grip points,” Robot., Comput. Integ. Manuf., vol. 12, no. 2, pp. 157–167, 1996. [24] Z. X. Li and S. Sastry, “Task-oriented optimal grasp by multifingered robot hand,” IEEE J. Robot. Automat., vol. 4, pp. 32–44, Feb. 1988. [25] A. F. Stappen, C. Wentink, and M. H. Overmars, “Computing immobilizing grasps of polygonal parts,” Int. J. Robot. Res., vol. 19, no. 5, pp. 467–479, 2000. [26] G. M. Bone and Y. Du, “Multi-metric comparison of optimal 2-D grasp planning algorithms,” in Proc. IEEE ICRA, 2001, pp. 3061–3066. [27] I.-M. Chen and J. W. Burdick, “A qualitative test for n-finger forceclosure grasps on planar objects with applications to manipulation and finger gaits,” in Proc. IEEE ICRA, 1993, pp. 814–820. [28] D. Ding, Y. H. Liu, J. Zhang, and A. Knoll, “Computation of fingertip positions for a form-closure grasp,” in Proc. IEEE ICRA, 2001, pp. 2217–2222. [29] Y. H. Liu, “Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming,” IEEE Trans. Robot. Automat., vol. 15, pp. 163–173, Feb. 1999. [30] X. Y. Zhu and J. Wang, “Synthesis of force-closure grasps on 3-D objects based on the Q distance,” IEEE Trans. Robot. Automat., vol. 19, pp. 669–679, Aug. 2003. [31] R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970. [32] N. Xydas and I. Kao, “Modeling of contact mechanics and friction limit surfaces for soft fingers in robotics, with experimental results,” Int. J. Robot. Res., vol. 18, no. 8, pp. 941–950, 1999. [33] R. D. Howe and M. R. Cutkosky, “Practical force-motion models for sliding manipulation,” Int. J. Robot. Res., vol. 15, no. 6, pp. 557–572, 1996. [34] R. M. Murray, Z. X. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994.
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Grasping-Force Optimization for Multifingered Robotic Hands Using a Recurrent Neural Network Youshen Xia, Jun Wang, and Lo-Ming Fok Abstract—Grasping-force optimization of multifingered robotic hands can be formulated as a problem for minimizing an objective function subject to form-closure constraints and balance constraints of external force. This paper presents a novel recurrent neural network for real-time dextrous hand-grasping force optimization. The proposed neural network is shown to be globally convergent to the optimal grasping force. Compared with existing approaches to grasping-force optimization, the proposed neural-network approach has the advantages that the complexity for implementation is reduced, and the solution accuracy is increased, by avoiding the linearization of quadratic friction constraints. Simulation results show that the proposed neural network can achieve optimal grasping force in real time. Index Terms—Grasping-force optimization, multifingered robotic hands, recurrent neural networks.
I. INTRODUCTION Multifingered robot hands have become of great interest in robotics due to their advantages over conventional grippers in tasks requiring dexterous manipulation [1]–[3]. Robotic hands are more complex in construction and require more sophisticated analysis to determine the grasping force to be exerted on an object without dropping or breaking the object [4], [5]. One of important issues on the optimal control of robotic hands is the grasping force-optimization problem. The problem for multifingered robotic hands is concerned with determining the least grasping force from each finger exerted on a three-dimensional (3-D) rigid object without slipping or losing the grasp. The grasping-force optimization is to maximize the efficiency and dexterity of robotic manipulations. The objective of the force optimization is to minimize the magnitude of grasping force from every finger applied to an object, while holding the object against slippage and balancing external forces due to gravity, acceleration, and others. Various optimization approaches have been proposed to achieve the optimal force distribution. Nakamura et al. [6] proposed using Lagrange multipliers to find the optimal forces. Sinha and Abel [7] proposed solving the nonlinear friction constraints by using the successive quadratic programming method. Kumar and Waldron [8] presented a class of suboptimal algorithms for contact-force optimization. These methods are computationally intensive because of the nonlinearity of the friction-cone constraint. Thus, the grasping-force optimization problem is usually solved by using linearization of the friction cone followed by linear or quadratic programming [9]–[12]. Linearization methods, however, may result in the obtained grasp force satisfying the linearized friction constraints, but violating the nonlinear constraints, and they increase the number of facets in the linearized friction cone resulting in excessive computational effort. Buss et al. [13] reformulated the contact force-optimization problem as a semidefinite programming problem, and solved it by using a projection gradient-flow algorithm, which requires an initial point
Manuscript received July 19, 2002; revised January 7, 2003. This paper was recommended for publication by Associate Editor C. Melchiorri and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4174/00E. The authors are with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TRA.2004.824946 1042-296X/04$20.00 © 2004 IEEE
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satisfying the nonlinear friction constraints. Han et al. [14] cast the friction cone constraints further into linear matrix inequalities and solved the grasp-force problems by using the interior point algorithm. Moreover, Liu et al. [15] gave a min 0 max method for automatic generation of the initial points. To perform highly dexterous robotic manipulations, the result of grasping-force optimization must be used in motion-control law, and thus, the optimal force-distribution problem should be solved in real time. Parallel and distributed approaches to grasping-force optimization for dexterous robotic manipulations are deemed necessary and desirable. In recent years, neural network approaches have demonstrated their great promise for optimization. Neural networks for constrained optimization problems have been widely explored for real-time applications [16]–[18]. Reported results of numerous investigations have shown many advantages over the traditional optimization algorithms, especially in real-time applications. Moreover, neural networks have been applied to solve various problems in robotics [19]–[21]. For example, Ding and Wang [19] demonstrated the effectiveness of using recurrent neural networks for torque optimization in robotic manipulation. Xia and Wang [20] proposed a dual neural network for kinematic control of redundant robot manipulators. In this paper, we propose a novel recurrent neural network for real-time grasping-force minimization of multifingered robotic hands. We first reformulate the force-optimization problem. Based on the derived equivalent formulation, we then present a novel neural network for grasping-force optimization. The proposed neural network has a low complexity for parallel implementation, and is shown to be globally convergent to the optimal grasping force. Furthermore, the present neural network is capable of taking into account the nonlinearity of the friction constraints, so that the solution accuracy is ensured.
To derive a neural network model for solving (1), we first give an equivalent relationship between the solution to (1) and the solution to the following formulation:
Qx + q + rg(x)y 0 GT z = 0 + (y + g (x)) 0 y = 0 Gx = 0fext
T where g (x ) = [g1 (x); . . . ; gm (x)] , where gi (x) = x2i1 + x2i2 0 i xi3 ; (y)+ = ([y1 ]+ ; . . . ; [ym ]+ )T ; and + = maxf0; yi g; rg(x) = [rg1 (x); . . . ; rgm (x)] is an [y i ] 3m 2 m matrix. Lemma 1: x3 is a solution to (1) if and only if there exist nonzero vectors y 3 and z 3 such that (x3 ; y 3 ; z 3 ) satisfies (2).
Proof: (1) can be rewritten as minimize subject to
0fext :
(3)
1
2
According to the well-known saddle-point theorem [23], x3 is a solum and z 3 2 R6 such that tion to (3) if and only if there exists y 3 2 R+ 3 m m 6 3 for any (x; y; z ) 2 R 2 R+ 2 R ; (x ; y 3 ; z 3 ) satisfies
L(x3 ; y; z ) L(x3 ; y3 ; z 3 ) L(x; y3 ; z 3 )
2
2 Rm j y 0g. It implies that Gx3 = 0fext and
3 T 3 T T 3 (x ) Q (x ) + q x + y g (x )
12 (x3 )T Q(x3 ) + qT x + (y3 )T g(x3 ) 21 xT Qx + qT x + (y3)T g(x) 0 (Gx + fext )T z3 :
From the first inequality above, it follows that (y
0 y3 )T (0g(x3 )) 0 8y 0:
On the other side, let
(x) = xT Qx + qT x + (y3 )T g(x) 0 (Gx + fext )T z 3 : 1
2
Then the right inequality above implies
(x ) (x 3 )
8x 2 R3m :
It follows that r (x3 ) = 0. Then
Qx3 + q + rg(x3 )y3 0 GT z 3 = 0:
1
f (x) = xT Qx + qT x 2 subject to Gx = 0fext x2i1 + x2i2 i xi3 ; (i = 1; . . . ; m) minimize
L(x; y; z ) = xT Qx + qT x + yT g(x) 0 (Gx + fext )T z:
1
Consider a multifingered robot hand grasping a single object in a 3-D workspace with m point contacts between the grasped object and the fingers [12]. The problem of the grasp-force optimization is to find a set of contact forces such that the object is held at the desired position and external forces are compensated. So, a grasping force xi is applied by each finger to the object to hold the object without slippage and to balance any external forces. To ensure nonslipping at a contact point, the grasping force xi should satisfy x2i1 + x2i2 i x2i3 , where i > 0 is the friction coefficient at finger i, and xi1 ; xi2 , and xi3 are components of contact force xi in the contact coordinate frame. Besides the form-closure constraints, to balance any external wrench fext to maintain a stable grasp, each finger must apply a grasping force xi = 623m [xi1 ; xi2 ; xi3 ] to the object such that Gx = 0fext , where G 2 R T is the grasp transformation matrix, and x = [x1 ; . . . ; xm ] 2 R3m is the grasping force vector. On the other side, to avoid crushing the object, the contact forces should be as small as possible. In this paper, we consider the quadratic objective function of contact forces. So, the optimal grasping force with holding the object can be formulated as the following optimization problem with linear and nonlinear constraints:
1 T x Qx + qT x 2 g(x) 0; Gx =
Define the Lagrangian function of (3) as
m = fy where R+
II. PROBLEM FORMULATION FOR OPTIMAL GRASPS
(2)
So, x3 is a solution to (3) if and only if (x3 ; y 3 ; z 3 ) satisfies (1)
where q 2 R3m and Q is a 3m 2 3m symmetric and positive definite matrix. The above quadratic objective function represents the minimum weighted norm of joint torque vectors and the minimum norm force when q = 0.
Qx3 + q + rg(x3 )y3 0 GT z 3 = 0 3 T 3 (y 0 y ) (0g (x )) 0 8y 0 3 Gx = 0fext : From the projection theorem [22], it can be seen that the above formulation is equivalent to (2).
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it follows
y(t) = e0(t0t ) y(t0 ) + e0t
Fig. 1. Block diagram of the proposed neural network in (4).
III. NEURAL NETWORK MODEL Based on the equivalent formulation in (2), we propose a recurrent neural network for solving (1), with its dynamical equation given by
0Qx 0 q 0 rg(x)y + GT z d = (4) (y + g(x))+ 0 y dt 0Gx 0 fext where x 2 R3m ; y 2 Rm ; z 2 R6 , and > 0 is a scaling parameter. x y z
The dynamical equation (4) can be easily realized by a recurrent neural network with a simple structure, as shown in Fig. 1, where = 1. The circuit realizing the proposed neural network consists of 4m + 6 integrators, m processors for gi (x); m processors for rgi (x); 9m2 + 12m connection weights, and some summers. Compared with the conservative linearization method [12], the proposed method can avoid the disadvantage in increasing the number of facets in the linearized friction cone, and has a better solution accuracy. Furthermore, the proposed method can be guaranteed to be globally convergent, as shown in the next section. IV. CONVERGENCE ANALYSIS In this section, we prove the global convergence of the proposed neural network for grasping-force optimization. We first give a lemma. Lemma 2: [22]. Let be a closed convex set. Then
(v 0 P (v))T (P (v) 0 x) 0; v 2 Rl ; x 2
kP (u) 0 P (v)k ku 0 vk; v; u 2 Rl where P (x) = arg minv 2 kx 0 v k and k 1 k denotes l2 norm. Theorem 1: 1) For any initial point u(t0 ) = (x(t0 ); y (t0 ); z (t0 )) 2 R4m+6 , there exists one unique continuous solution trajectory u(t) = (x(t); y(t); z(t)) for (4). Moreover, y(t) 0, provided that y(t0 ) 0. m 2 R6 is an equilibrium point of (4), 2) If (x3 ; y 3 ; z 3 ) 2 R3m 2 R+ then x3 is an optimal solution to (1). Proof: 1) Note that (y + g (x))+ is locally Lipschitz continuous, then the right term of (4) is locally Lipschitz continuous. According to the local-existence theorem of ordinary differential equations, there exists a unique continuous solution u(t) of (4) for (t0 ; T ). Next, for the given initial point u0 with y (t0 ) 0
t t
dy dt
dy dt
+y
+ y = (y + g(x))+ es ds =
t
t
es (y + g(x))+ ds
t t
es (y + g(x))+ ds
since (y + g (x))+ 0; y (t) 0. 2) Clearly, the equilibrium point of (4) satisfies (2). From Lemma 1, it follows that x3 is an optimal solution to (1) when (x3 ; y 3 ; z 3 ) 2 m 2 R6 is an equilibrium point of (4). R3m 2 R+ We now establish our main results concerning the global convergence of the proposed neural network in (4). Theorem 2: The proposed neural network in (4) is stable in the sense of Lyapunov and globally convergent to the Karush–Kuhn–Tucker (KKT) point u3 = (x3 ; y 3 ; z 3 ), where x3 is the optimal grasping-force vector. Proof: This proof is divided into four parts. First, we show that the proposed neural network in (4) is stable in the sense of Lyapunov. Let u(t) be the solution of (4) with the initial point u(t0 ) = (x(t0 ); y(t0 ); z (t0 )) 2 R4m+6 , where y(t0 ) 0. From Proposition 1, it follows that u(t) = (x(t); y (t); z (t)) satisfies y(t) 0. We define a Lyapunov function
E (u ) = E 0 (u ) +
1 (kx 0 x 3 k 2 + ky 0 y 3 k 2 + kz 0 z 3 k 2 ) 2
where
E 0 (u ) =
1 (kQx + q + rg(x)y 0 GT zk2 + kGx + f k2 ) ext 2 1 +g(x)T [(y + g(x))+ 0 y] 0 2 k(y + g(x))+ 0 yk2 :
We have
dE0 (u) du
= F (u) 0 (rF (u) 0 I )(P (u 0 F (u)) 0 u)
where P (u) = [x; (y )+ ; z ]T ; rF (u) denotes the Jacobian matrix of F , and Qx + q + rg(x)y 0 GT z
F (u ) =
Let r(u)
dE (u) dt
0 g (x ) Gx + fext
:
= P (u 0 F (u)) 0 u, then
= rE (u)T du dt = ( F (u ) + u 0 u 3 ) T r (u ) + kr (u )k 2 0 r (u )T r F (u )r (u )
rE (u) is the gradient of E . In the equality of Lemma 2, let = u 0 F (u) and let x = u3 , we get
where
v
(r(u) + u 0 u3 )T (0r(u) 0 F (u)) 0: It follows that
dE (u) dt
0 F (u ) T (u 0 u 3 ) 0 r (u )T r F (u )r (u ) 0 :
Second, we show that E (u) is radially unbounded. In the equality of Lemma 2, let v = u 0 F (u) and let x = u, we obtain
0 F (u )T r (u ) k r (u )k 2 : It follows that E (u) ku 0 u3 k2 =2. Thus E (u) is positive definite and is radially unbounded. Therefore, for any initial point u0 2 R3m 2 m 2 R6 , there exists a convergent subsequence fu(t )g such that R+ k limk!1 u(tk ) = u^, where dE (^u)=dt = 0.
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Third, we prove that dE(^ u)=dt = 0 if and only if u ^ is a KKT point. By Theorem 1, it can be seen that y^ 0. Since the Jacobian matrix Q+
rF (u) =
Om2m O62m
G
Om26 O626
is positive semidefinite, where Om2m ; O62m ; Om26 ; O626 are zero matrices, F (u) is monotone. Then T
F (u) (u
0 u3 ) F (u3 )T (u 0 u3 ) 0:
0 x3 )T (Q^x + q + rg(^x)^y 0 GT z^) + (^y 0 y3 )T (0g(^x)) 3 T +(^ z 0 z ) (G^ x + fext ) = 0:
3 T It follows that (^ y 0 y ) g(^ x) = 0 and
0 u3 ) = 0 rF (^u)r(^u) = 0:
F (^ u) (^ u (r(^ u))
T
T x) = (y 3 )T g(x3 ) = 0: (^ y ) g(^
Therefore, u ^ = (^ x; y ^; z ^) is a KKT point of (3). Finally, define a Lyapunov function again
It can be seen that r(^ u)
rF (^u)r(^u) = p(x; z)T rg(^x)p(x; z)
T
k 0 u^k2 =2:
^ E(u) = E0 (u) + u
T ^ and where p(x; z) = Q^ x + q + rg(^ x) 0 G z m J (^ x) = Q +
i=1
y ^i
^ ^ u) = 0 and thus, limk!1 E(u(t ^ u) = 0. For 8 > Then E(^ k )) = E(^ 0, there exists q > 0 such that for all tk tq , we have
r2 gi (^x):
^ ^) < : E(u(t k ); u
Since J (^ x) is positive definite Q^ x+q+
Similar to the previous analysis, we have dE(u)=dt t tq
rg(^x)^y 0 GT z^ = 0: 0
T
Q^ x + q + g(^ x)^ y G z ^=0 3 3 3 T 3 Qx + q + g(x )y G z =0
rg(^x)^y 0 (Qx3 + q + rg(x3 )y3 ) GT (^z 0 z3 ):
dE(u) dt
It follows that
r 0
Q^ x+q+
0 r
r 0 0r 0 r
3
0
0
0
0
r
0 y^):
Because Qx + q + rg(x)^ y is strictly monotone (^ x
(^ x
0
0z
)+
rg(x
3
)(y
3
0 y^)] > 0:
(5)
On the other hand, we have 3 Qx + q +
rg(x3 )^y 0 GT z^ = rg(x3 )(^y 0 y3 ) + GT (z3 0 z^):
Then (^ x
E(u(t))
0 E(u(t0 )) 0
E(u(t))
E(u(t0 )) 0
t t
t
t
r(u(t))
r(u(t))
ku(t) 0 u3 k2 2E(u(t0 )) 0 2
Thus 3
0r(u)T rF (u)r(u) 0
T
T
rF (u(t))r(u(t)) ds
rF (u(t))r(u(t)) ds:
Since E(u) ku 0 u3 k2 =2 where u3 is a KKT point of (1)
0 x3 )T [Q(^x 0 x3 ) + rg(^x)^y 0 rg(x3 )^y)] > 0: 3 T T x ) [G (^ z
Then for
where r(u) is defined in Theorem 2. Then
3
g(^ x)^ y (Qx + q + g(x )^ y) T z z 3 ) + g(x3 )(y 3 y^) = G (^ 3 3 3 T x x ) + g(^ x)^ y g(x )^ y )] (^ x x ) [Q(^ 3 T T 3 3 T 3 3 z z ) + (^ x x ) g(x )(y = (^ x x ) G (^
0.
fore, the proposed neural network in (4) is globally convergent to a KKT point of (1). Finally, we analyze the the convergence rate of the trajectory. From the proof of Theorem 2, it can be seen that for any > 0
0
we get Q^ x+q+
^ ^ ku(t) 0 u^k 2E(u(t)) E(u(t q )) 2: Thus limt!1 ku(t) 0 u ^k = 0, and hence, limt!1 u(t) = u ^. There-
Assume that x ^ = 6 x3 . Using
r r
(6)
3 equations (5) and (6) constitute a contradiction. So x ^ = x and G^ x= T 3 0fext . Now, F (^u) (^u 0 u ) = 0 implies (^ x
It follows that dE(^ u)=dt = 0 if and only if T
0 x3 )T (Qx3 + q + rg(x3 )^y 0 GT z^) 0 3 T 3 3 T 3 ^)] 0 (^ x 0 x ) [rg(x )(^ y 0 y ) + G (z 0 z 3 T T 3 3 3 (^ x 0 x ) [G (^ z 0 z ) + rg(x )(y 0 y ^)] 0 (^ x
rg(x) 0GT
r
m y 2 g (x) i i=1 i T g(x)
0r
Since
0 x3 )T (Qx3 + q + rg(x3 )^y 0 GT z^) 3 T 3 3 T 3 ^)]: y 0 y ) + G (z 0 z = (^ x 0 x ) [rg(x )(^
t t
r(u(t))
T
rF (u(t))r(u(t)) ds:
Therefore, the convergence rate of the neural network in (4) increases as increases. This point will be further illustrated in the next section. V. SIMULATION RESULTS In this section, to show the effectiveness of the proposed neural network, we present some simulation results for grasping-force optimization. Consider a minimum norm force objective function f (x) = kxk2 =2. In this example, a polyhedral object with M = 0:1 kg is grasped by a three-fingered robotic hand, shown in Fig. 2. The grasping model is defined by the position vector of three grasp points: p1 = [0; 1; 0]T ; p2 =
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Fig. 2.
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Three-finger grasp example. Fig. 4.
Convergence of the optimal force for the three-fingered robotic hand.
Fig. 5.
Convergence of the energy function with = 100.
shown in Fig. 3 with a constant velocity v = 0:2 m/s. A time-varying external wrench applied to the center of mass of the object is
0
fext = [0; fc sin((t)); M g + fc cos((t)); 0; 0; 0]T
Fig. 3. Motion of the three-finger grasp.
0
T ; and p = [0; 1; 0]T . The friction coefficient of each 3 finger is i = 0:6, and the grasp transformation matrix is [1; 0:5; 0]
0
0
1
0
0
GT =
0 0
01
01 0
01 0
1
0
0
0
0
1
01 01
0
0
0
0
0
01
0
0
0
0
0 1 0 0 :5
1
0
0
0
0
0
0 :5
0
0
0
1
1
0
0
0
0
0
01 0
V (x; y; z ) =
0
01
0
where g = 9:8 m/s2 ; 2 [0; 2 ], and fc = M v 2 =r . We use the proposed neural network in (4) to find the optimal force. Fig. 4 shows the convergence of the optimal force. Let the energy function of (4) be
:
To study grasping-force optimization of the multifingered robotic hand, we consider the time-varying external wrench applied on it. Let the robotic hand move along a circular trajectory of radius r = 0:5 m,
1 2
kx + r g (x )y 0 G T z k 2 1 2 + 2 1 + k(y + g (x)) 0 y k + kGx + fext k : 2 2
Figs. 5 and 6 show the convergence rate of V (x; y; z ) based on (4) with two different . Finally, for comparisons, we use the linearization method [21], where the friction cone is linearized into a 20-side polyhedral convex cone inscribed in the friction cone. The grasping-force optimization problem is then solved by the primal-dual neural network [18]. Furthermore, the Matlab optimization toolbox is employed for calculating the Euclidean norm of the optimal force at some time instants. Fig. 7 shows the transients of the Euclidean norm of the optimal force from the proposed neural network, the primal-dual network, and
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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004
Fig. 6. Convergence of the energy function with = 1000.
Fig. 7. Transient of Euclidean norm of optimal forces using three different methods.
Matlab optimization toolbox. It can be seen that the proposed neural network obtained the smallest Euclidean norm of the optimal force. VI. CONCLUDING REMARKS This paper presents a novel neural network for real-time graspingforce optimization of 3-D dexterous manipulation using multifingered robotic hands. The proposed neural network has a simple structure with a low model complexity, and avoids linearizing the friction constraints such that the solution accuracy is greatly enhanced. Moreover, the proposed neural network is shown to be globally convergent to the optimal grasping force. Therefore, the proposed neural networks are conceived to be capable of solving the time-dependent grasping force optimization problems autonomously in real time, and suitable for optimal force control of dexterous manipulations using multifingered robotic hands in uncertain environments. REFERENCES [1] M. R. Cutkosky, “On grasp choice, grasp model, and design of hands for manufacturing task,” IEEE Trans. Robot. Automat., vol. 5, pp. 269–279, Apr. 1989. [2] Z. Li, P. Hsu, and S. Sastry, “Grasping and coordinated manipulation by a multifingered robot hand,” Int. J. Robot. Res., vol. 8, pp. 33–50, 1989.
[3] R. Murray, Z. X. Li, and S. Sastry, A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994. [4] A. Bicchi, “On the closure properties of robotic grasping,” Int. J. Robot. Res., vol. 14, no. 4, pp. 319–334, 1995. [5] D. Prattichizzo, J. K. Salisbury, and A. Bicchi, Contact and Grasp Robustness Measures: Analysis and Experiments, O. Khatib and K. Salisbury, Eds. Berlin, Germany: Springer-Verlag, 1996. [6] Y. Nakamura, K. Nagai, and T. Yoshikawa, “Dynamics and stability in coordination of multiple robotic mechanisms,” Int. J. Robot. Res., vol. 8, no. 2, pp. 44–61, 1989. [7] P. R. Sinha and J. M. Abel, “A contact stress model for multifingered grasps of rough objects,” IEEE Trans. Robot. Automat., vol. 8, pp. 7–22, Feb. 1992. [8] V. Kumar and K. Waldron, “Suboptimal algorithms for force distribution in multifingered grippers,” IEEE Trans. Robot. Automat., vol. 5, pp. 252–257, Apr. 1989. [9] J. Kerr and B. Roth, “Analysis of multifingered hands,” Int. J. Robot. Res., vol. 8, no. 4, pp. 3–17, 1986. [10] C. A. Klein and S. Kittivatcharapong, “Optimization force distribution for the legs of a walking machine with friction cone constraints,” IEEE Trans. Robot. Automat., vol. 6, pp. 73–85, Feb. 1990. [11] F. T. Cheng and D. E. Orin, “Efficient algorithm for optimal force distribution—The compact dual LP method,” IEEE Trans. Robot. Automat., vol. 6, pp. 178–187, Apr. 1990. [12] Y. H. Liu, “Qualitative test and force optimization of 3-D frictional form-closure grasps using linear programming,” IEEE Trans. Robot. Automat., vol. 15, pp. 163–173, Apr. 1999. [13] M. Buss, H. Hashimoto, and J. B. Moore, “Dextrous hand grasping force optimization,” IEEE Trans. Robot. Automat., vol. 12, pp. 406–417, June 1996. [14] L. Han, J. C. Trinkle, and Z. X. Li, “Grasp analysis as linear matrix inequality problems,” IEEE Trans. Robot. Automat., vol. 16, pp. 663–674, Dec. 2000. [15] G. F. Liu, J. J. Xu, and Z. X. Li, “Automatic real-time grasping-force determination for multifingered manipulation: Theory and experiments,” in Proc. IROS, 2002, pp. 1676–1680. [16] M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Trans. Circuits Syst., vol. 35, pp. 554–562, May 1988. [17] A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing. London, U.K.: Wiley, 1993. [18] Y. Xia, H. Leung, and J. Wang, “A projection neural network and its application to constrained optimization problems,” IEEE Trans. Circuits Syst. I, vol. 49, pp. 447–458, Apr. 2002. [19] H. Ding and J. Wang, “Recurrent neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. A, vol. 29, pp. 269–276, Feb. 1999. [20] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. B, vol. 31, pp. 147–154, Jan. 2001. [21] L. Fok and J. Wang, “Two recurrent neural networks for grasping-force optimization of multifingered robotic hands,” in Proc. IEEE Int. Joint Conf. Neural Networks, Honolulu, HI, 2002, pp. 35–40. [22] D. Kinderlehrer and G. Stampcchia, An Introduction to Variational Inequalities and Their Applications. New York: Academic, 1980. [23] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. New York: Wiley, 1993.
