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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 3, MAY 2003
A Dual Neural Network for Redundancy Resolution of Kinematically Redundant Manipulators Subject to Joint Limits and Joint Velocity Limits Yunong Zhang, Member, IEEE, Jun Wang, Senior Member, IEEE, and Youshen Xia
Abstract—In this paper, a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators. Physical constraints such as joint limits and joint velocity limits, together with the drift-free criterion as a secondary task, are incorporated into the problem formulation of redundancy resolution. Compared to other recurrent neural networks, the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons. The dual neural network is shown to be globally (exponentially) convergent to optimal solutions. The dual neural network is simulated to control the PA10 robot manipulator with effectiveness demonstrated. Index Terms—Drift-free, dual neural network, joint limits, joint velocity limits, kinematically redundant manipulators.
I. INTRODUCTION
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INEMATICALLY redundant manipulators are those having more degrees of freedom (DOF) than required for position and orientation [1]. The redundancy of such manipulators including intrinsical redundancy and functional redundancy can be utilized to avoid obstacles [2], singularities [3], [4] and to optimize various performance criteria [5]–[7], as well as conduct the end-effector motion task. Since redundant manipulators have more DOF than necessary for position and orientation, multiple solutions exist. As a result, the redundancy of inverse kinematic mappings complicates the manipulator control problem considerably, in addition to the nonlinearity. To take full advantage of the redundancy, various computational schemes have been developed. Conventionally, the general solution of redundancy resolution is obtained by the pseudoinverse formulation as one minimum-norm particular solution plus a homogeneous solution [8], [9]. Most of current researchers have applied the pseudoinverse technique to formulate and resolve the redundancy by considering different optimization criteria, such as least-square joint velocities, singularity avoidance, obstacle avoidance and task priority control. However, among those techniques, the physical constraints such as joint limits and joint velocity limits are usually not taken Manuscript received September 24, 2001; revised January 3, 2003. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4165/98E. Y. Zhang and J. Wang are with the Department of Automation and ComputerAided Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong (e-mail: [email protected]; [email protected]). Y. Xia is with the Department of Applied Mathematics, Nanjing University of Post and Telecommunications, Nanjing, China. Digital Object Identifier 10.1109/TNN.2003.810607
into account. If these physical constraints are not considered, a saturation may occur. As a result, the tracking error may increase considerably, not to mention the physical damage possibly caused when a commanded joint or joint velocity hits its physical bound. For the joint-constrained inverse kinematics, numerical algorithms of redundancy resolution have been developed with physical constraints included. For example, the Jacobian matrix was augmented in [5] by incorporating the joint and velocity constraints and a routine predicting which constraints might be violated was required. The joint and velocity constraints were considered also in [10]. By applying the Gram–Schmidt orthogonalization procedure, a formulation was developed to express the general solution in terms of the redundant joint velocities only. Cheng et al. [11] formulated the constrained kinematic redundancy problem into a quadratic programming (QP) form. A compact QP method, using Gaussian elimination with partial pivoting, was finally developed in [12] to improve the computational efficiency. However, the numerical solutions to redundant inverse kinematics are in general computationally intensive. With more physical constraints and additional tasks considered, the computation process requires considerable time that may hinder the on-line applications, especially in high-DOF sensor-based robotic systems. In recent years, as parallel distributed computational models, neural networks have been developed for the redundancy resolution of robot manipulators, e.g., [13]–[17]. In particular, two neural networks, namely the pseudoinverse network [18] and the linear-programming neural network [19], were applied to the minimum infinity norm kinematic control in [15]. The obtained redundancy solution explicitly minimized joint velocities in the minimum infinity-norm sense. A two-layered primal-dual neural network was presented in [16] to online minimize the weighted joint velocity. To reduce network complexity and increase computational efficiency, an early dual-neural-network model [17] was then proposed for kinematic control of redundant manipulators. In the aforementioned schemes, it is assumed implicitly that there exists no joint limits or joint velocity limits when solving the redundancy resolution problem. But physical limits do exist in almost any robotic system. If a solution exceeds such mechanical joint limits or joint velocity limits and locks there, the desired motion path may become impossible to accomplish. In this paper, our attention is focused on the design and analysis of a general dual-neural-network approach to online redundancy resolution of physically constrained manipulators.
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The dual neural network is proven to be globally convergent. The proposed dual neural network scheme is simulated on the 7-DOF PA10 robot arm for the primary task of end-effector trajectory tracking and simultaneously to eliminate the drift phenomenon [20] as an additional task. Its effectiveness and efficiency are then demonstrated. The remainder of this paper is organized into five sections. Section II provides the background information and the problem formulation of the drift-free redundancy resolution for physically constrained manipulators. Section III presents the proposed dual neural-network approach to online drift-free redundancy resolution of constrained manipulators. The theoretical results on global (exponential) convergence and position error estimation are given in Section IV. Section V illustrates and discusses simulation results of the dual neural network and the PA10 manipulator to show their operating characteristics and performance. Section VI concludes this paper with final remarks. II. PROBLEM FORMULATION A. Drift-Free Inverse Kinematics In a robot manipulator, the end-effector position and orientain the Cartesian space is related to the joint tion vector space by a forward kinematics equation (1) is joint variable vector and is a continuous where nonlinear mapping function with a known structure and parameters for a given manipulator. The relation between the Cartesian velocity and the joint velocity can be obtained by differentiating (1) (2) is the Jacobian matrix defined as . In a redundant manipulator, since , (1) and (2) are both undetermined and hence admit infinite number of solutions. The conventional pseudoinverse-type solution to the differential inverse kinematics problem (2) is generally formulated as one minimum-norm particular solution plus a homogeneous solution [8]. That is
where
(3) is the pseudoinverse of and is an where arbitrary vector and can be selected by using different optimization criteria, such as singularity avoidance, obstacle avoidance and task priority control. However, it is shown in [20] that the pseudoinverse or pseudoinverse-type solutions (3) are generally not repeatable in the sense that a closed path of the end-effector does not yield a closed path in joint space. Such joint angle drift is undesirable for cyclic motion control, since the manipulator does not necessarily return to its initial joint configuration after tracking a closed path in the task space. Of course, for the repeatability requirement, the manipulator configuration can be readjusted with a suitable self-motion at the end of every motion cycle, but this
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would be inefficient and unwanted. The analytic test using Lie bracket condition is given in [21] and [22], which determines whether or not a given pseudoinverse control strategy possesses the drift-free property, especially for the planar three-link manipulator case [21]. To make the kinematic control repeatable [8], a projection term is added to the pseudoinverse term which generates null-space velocities to minimize a secondary criterion. In [11], Cheng et al. proposed an inverse-kinematic control scheme by solving a quadratic program which minimizes the joint displacements between the current states and the initial states for the remedy of the drift problem. In the QP formulation, the objective function to be minimized is with
(4)
and is a positive parameter to scale the magnitude of the manipulator response to joint displacement. However, with more subtask criteria and physical constraints considered, the redundancy resolution becomes very time-consuming either by computing the pseudoinverse-type solution or solving the augmented QP problem. The requirement of real-time computation in sensor-based robotic systems further requests the development of much more efficient parallel-processing schemes as alternatives to replace numerical algorithms for online kinematic control of redundant manipulators. B. Joint Limits Conversion Besides the drift problem in repetitive motion, the pseudoinverse control (3) does not take account of physical limits of manipulators. As all manipulators are physically limited in their joints and joint velocities, it is more realistic to consider constrained kinematic control. Let us consider the joint limits and joint velocity limits simultaneously, with the superscripts and , respectively, denoting the upper and lower limits/bounds. Since the manipulator redundancy is resolved at velocity ] has to be converted into level, the limited joint range [ a dynamically updated joint-velocity bound constraint, e.g., (5) is selected such that where the critical coefficient there appears a deceleration when the robot arm enters the critor , while the intensity coeffiical region is used to scale the feasible region of . The cocient efficient is selected such that the feasible region of made by joint limits conversion is not smaller than the original one made by joint velocity limits; that is, is selected not less than . Note that large values of may cause joint deceleration quickly when the manipulator approaches its joint limits. ] can thus be Equation (5) and joint velocity limits [ , where the combined into this bound constraint th elements of and are defined, respectively, as
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Therefore, the physically constrained drift-free redundancy resolution problem can be formulated as
The quadratic program (6) can be reformulated as minimize
minimize
subject to
subject to
(8)
(6)
It is worth mentioning that actuator limits such as joint acceleration/torque limits are also important factors affecting robot performance [9], [11]. Such a feature of torque limit avoidance can be achieved by extending the proposed method to the acceleration level, e.g., [23]. C. Neural-Network Approaches By the duality theory [24], the dual problem of the primal quadratic program (6) is
where noting the
, identity matrix, we define
and with
de-
At any time instant, the above QP problem may be viewed as a parametric optimization problem. Now let us analyze the optimal solution to (8). By the Karush– Kuhn–Tucker condition [24], we know that is a solution to (8) if and only if there exists the dual decision variable such that and
minimize if
subject to where , and are the vectors of the dual decision variables. According to the design methodology [19], [25], [26], a primal-dual neural network can be developed by minimizing the duality gap via gradient decent direction. The dynamical equations of the primal-dual neural network are described below [26]
(7) where
is the state vector representing the estimated , and with if if if
The symbol in hereafter denotes the Euclidean norm of a vector or the Frobenius norm of a matrix. The positive capacitive parameters and are used to scale the convergence rate of the primal-dual neural network. It can be seen that the primal-dual network (7) is composed of two connected layers of neurons and that the network architecture is much complicated in terms of the nonlinear dynamics with third-order elements of . A recurrent neural networks called the Lagrangian neural network can be similarly developed [14], [27], [28], but the Lagrangian network with joint limits included has four layers of neurons. The hardware complexity of such a neural-network implementation may increase substantially as the number of neurons and layers increase. III. DUAL NEURAL NETWORK In this section, we propose a dual neural network approach with a reduced network complexity and increased computational efficiency to the real-time drift-free redundancy resolution of physically constrained manipulators.
if if which is equivalent to the system of piecewise linear [29]–[31], where equations is called the projection operator and defined as if if if
.
Therefore, is a solution to (8) if and only if there exists the such that and dual vector . That is
which gives rise to the proposed dual neural network for solving (8) with the following dynamic equation and output equation: (9) (10) is a capacitive design parameter used to scale the where convergence of the proposed network and the network output represents the estimated . The dynamic equation described in (9) shows that the dual neurons neural network is composed of only one layer of and without using any analog multiplier or penalty parameter. Compared to the prime-dual neural network (7), the dynamics of dual neural network is piecewise linear without any high-order nonlinear term. Consequently, the architecture of the dual neural network is much simpler than that of the primal-dual network. In a circuit realizing the dual neural network, the piecewise-linear might be implemented by using an activation function operational amplifier. In the robot control process, the desired
ZHANG et al.: DUAL NEURAL NETWORK FOR REDUNDANCY RESOLUTION
velocity vector is input into the dual network and simultaand , namely neously the network outputs the signals . the estimated joint velocity vector If ignoring the physical constraints for comparison purposes, we can simplify the dual neural network (9) for solving the “unconstrained” drift-free redundancy resolution problem as fol, and thus lows. That is, , then the dual neural network (9) is , which becomes the early simplified as dual neural-network model [17].
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Therefore, it follows from (11) that
(12) In addition, by Lemma 1, we have
(13) Then, adding (12) and (13) yields
IV. CONVERGENCE RESULTS In this section, we prove the global convergence and exponential convergence of the proposed dual neural network for kinematics control of redundant manipulators and estimate the position tracking error. Definitions about global convergence and exponential convergence of a neural system are present in [17]. The exponential convergence implies that such a system converges arbitrarily fast. Besides, the following lemma about projection property is often used in optimization literature [32]–[34]. is a closed Lemma 1 [29]: Assume that the set convex set, then the following two inequalities hold:
Defining follows from the above inequality that
, it
and, thus,
(14) Now we choose a Lyapunov function candidate as (15)
where
is a projection operator defined as .
It is clear that the set is a closed convex set and satisfies the above projection property. The convergence results of the proposed dual neural network (9) for constrained inverse kinematics is thus obtained as follows. Theorem 1: Starting form any initial state, the dual neural which denetwork (9) is convergent to an equilibrium point is pends on the initial state of the trajectory and an optimal solution to the inverse-kinematics QP problem (6). Moreover, the exponential convergence of the dual network can be achieved under a mild condition, i.e., (17). Proof: To show the convergence property, the following numbered inequalities are first derived. At any equilibrium point , we have this inequality [17], [31], [34] (11) which can be obtained by discussing the following three cases: , , Case 1) If for some , then ; , , Case 2) If for some and , then and thus ; , , Case 3) If for some and , then and, thus, .
is symmetric positive definite and . Clearly, is positive if and iff ) for definite (i.e., taken in the domain (i.e., the attraction region , since, in view of (14), of ). is negative definite in
where matrix
(16) if , iff in . Thus, it and follows that the dual neural network (9) is asymptotically conis clearly the optimal vergent to , of which the resulting solution to (6) in view of the Karush–Kuhn–Tucker optimality condition. Furthermore, we show the global exponential converand again. It follows from gence by reviewing , where (15) that are, respectively, the maximal and minimal . Clearly, and are proeigenvalues of portional to the capacitive parameter . Moreover, in view that for any in and amounts to , the following mild condition is presented , [35], [36]:
(17)
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In addition, by analyzing the linear/saturation cases of , the range of is (0,1]. Therefore from (14) and (17), we have
where is proportional to the reciprocal of ca, pacitive parameter . Thus we have and hence , , which completes the proof on exponential convergence of the proposed dual network. Remark 1: Without loss of generality, the above derivation is attained based on the existence of inverse kinematics solutions. is proportional to Since the exponential convergence rate , we can expedite the convergence of dual neural network and are, thus, sufficiently fast by decreasing . time-varying in a time scale sufficiently slower than that of (9). and within some Specifically, starting from any the maximal variation of ( , ) is small time interval sufficiently small, while the proposed dual neural network has been asymptotically convergent to the corresponding theoretwith a sufficiently small relative error. In the ical solution finite-time path-following task, the worst case of and , can be estimated on average as and , respectively, where , depends on the capacitive and , we can parameter . In view of estimate the inverse-kinematics joint configuration deviation as
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TABLE I JOINT LIMITS AND VELOCITY LIMITS OF PA10 MANIPULATOR
matrix such that has all the poles located on the left half complex plane, the error dynamics converges to zero. V. SIMULATION RESULTS The Mitsubishi PA10 manipulator (portable general purpose intelligent arm) has 7 DOF (three rotation axes and four pivot axes). The mechanical configuration and coordinate system, together with other specifications, of the PA10 redundant manipand joint velocity ulator can be found in [14]. Joint limits are shown in Table I. limits In this section, we will apply the dual neural network to drift-free redundancy resolution of physically constrained PA10 manipulator. Simulation has been performed for the path-following task that the end-effector of PA10 manipulator move along a given circle or straight-line in the three-dimensional workspace. In this study, only the positioning of the end-effector is considered, the Jacobian matrix is thus 3 7 in dimension, and the degree of redundancy is 4. A. Circular Motion
where can be made arbitrarily small by decreasing , namely, increasing the convergence rate of (9). Based on the Taylor se, the position tracking error ries expansion of (1), is bounded by the function with less , where the coefficient can be made arbitrarily small than by decreasing the capacitive parameter . From the above position error estimation, we know that the error can be made small by decreasing . Moreover, though a can lessen the position error too, it may cause high small joint velocity or acceleration. The ensuing simulation results will verify the soundness of the proposed error estimation. Remark 2: It is worth pointing out that the above derivation and estimation are effective on the condition that external disturbance does not exist. In a real system, model disturbance and computational round-off error always exist, the feedback control thus should be applied. One way is the closed-loop motion , the end-effector morate control [37]–[39]. Instead of tion rate can be given as
where and are respectively the commanded position and velocity vector. By appropriately choosing the diagonal gain
In this Section, first we show the redundancy resolution results without considering joint physical constraints and drift-free criterion. Specifically, the proposed dual neural disabled and the drift-free coefficient network (9) with , (i.e., the simplified dual network) is applied to the PA10 . robot arm. The capacitive parameter The desired motion of the PA10 end-effector is a circular path cm and the revolute angle about -axis, with radius . The task time of the motion is 10 s and the iniin ratial joint variables dians. Fig. 1 illustrates motion trajectories of the PA10 manipulator when its end-effector moving along a circle in the three-dimensional workspace and correspondingly the transient of the joint variables. Although the maximal Cartesian position and velocity tracking errors are less than 1.5 10 mm and has ex6.0 10 mm/sec respectively, the joint variable ceeded its mechanical range [ 2.6831, 2.6831] and thus the solution becomes inapplicable. If such a solution is directly apfinally locked at poplied to the PA10 manipulator with sition 2.6831 rad, the tracking error increases considerably, in addition to the physical damage possibly caused. Hence the path-following task fails. Moreover, as seen form Fig. 1, the solution is not repeatable in the sense that the final state of the PA10 manipulator does not coincide with its initial state; i.e., , and . Hence an
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Fig. 1. End-effector of PA10 manipulator moves along a circle in three-dimensional workspace without considering joint constraints and drift-free criterion, where the dark lines denote initial and final states of the manipulator, respectively.
Fig. 2. End-effector of PA10 manipulator moves along a circle in three-dimensional workspace with joint constraints and drift-free criterion considered, where the dark lines denote initial and final states of the manipulator, respectively.
inefficient and undesirable readjustment is needed for the cyclic motion control. In summary, the above simulation results show that physical constraints and drift-free criterion are in general worthy considering in the repetitive path-following tasks. Now let us consider the joint limits in Table I and the drift-free criterion. Specifically, the proposed dual neural network, with , , and , is applied to PA10 for the same circular-path following task. Fig. 2 shows the three-dimensional motion trajectories of the PA10 manipulator and the corresponding joint variables. Fig. 3 illustrates the transient behaviors of the proposed dual neural network and the PA10 manipulator, including the joint velocity variables in Fig. 3(a), the dual decision variables in Fig. 3(b) and the Cartesian position error and velocity error depicted in Fig. 3(c) and (d), has never respectively. As seen in Fig. 2, the joint variable exceeded the mechanical range [ 2.6831, 2.6831] and the solution is repeatable in the sense that the initial state and final state of PA10 manipulator coincide with each other. It follows from Fig. 3(a) that no joint velocity variable exceeds its limits in Table I. In addition, the maximal position and velocity errors are less than 8 10 and 6 10 mm/s, respectively. Namely, in subplots of Fig. 3(c) and (d), , , and denote the components of tracking position error , respectively, along the ,
and axes of the base frame and similarly , and denote, respectively, the , , and -axis components of tracking velocity error at the end-effector of the PA10 robot arm. The circular-path following experiments demonstrate the capability of the proposed dual neural network for online resolving the drift-free redundancy of physically constrained manipulators. B. Straight Line In this section, the PA10 manipulator is controlled to move forward and return backward along a straight line, like a reciprocating spot-welding task. The straight line of length 2.5 m, at every motion cycle, starts from the PA10 initial state and shall finally return to the initial state. Angles of the desired straight line making with , , and planes are, respectively, rad, rad and rad. The duration of the path-following task at every motion cycle is specified as 7.0 s. For comparison, the drift-free inverse kinematics problem is first solved without considering joint physical constraints, as shown in Fig. 4. Clearly, the solution is not acceptable in pracrad at time tice, since hits its mechanical limit
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(a)
(b)
(c)
(d)
Fig. 3.
Transient of the dual neural network and the PA10 manipulator moving along a circle.
Fig. 4.
End-effector of PA10 manipulator moves along a straight line with only the drift-free criterion considered.
s and and 4.54 s.
also hits its joint velocity limits at
The dual neural network is then applied to the physically constrained PA10 manipulator for drift-free redundancy resolution.
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(a)
(b)
Fig. 5. End-effector of PA10 manipulator moves along a straight line with joint constraints and drift-free criterion considered.
Fig. 6. Transients of the dual neural network and the PA10 manipulator moving along a straight line.
Parameters of the dual network are selected same as the circular . Simulation results are depicted in Figs. 5 example except and 6. The transient behaviors of joint variables and joint velocity variables are depicted in Fig. 5. Compared to Fig. 4(left),
the joint variable in Fig. 5 has never exceeded its upper limit 1.7637 rad and all the other joint variables remain in their limited ranges. Compared to Fig. 4(right), the joint velocity variincluding , as shown in Fig. 5(b), are kept always ables
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within their mechanical limits. The maximal position and velocity errors are less than 4 10 and 10 mm/s, respectively. Observed from other simulations, the maximal tracking error decreases rapidly when the capacitive parameter decreases. The above simulation results substantiate the efficacy of the neural-network approach to online drift-free redundancy resolution for physically constrained manipulators. VI. CONCLUDING REMARKS The proposed one-layer dual neural network provides a new parallel distributed computational approach to online drift-free redundancy resolution for physically constrained redundant manipulators. Compared with the supervised-learning neural network approaches to robot kinematic control, the present approach eliminates the need of off-line training and guarantees fast convergence due to the exponential convergence. Compared with other recurrent neural networks, the proposed dual neural network is able to resolve manipulator redundancy under physical constraints such as joint limits and joint velocity limits. Moreover, the dynamic equation of the dual neural network is piecewise linear and does not contain any high-order nonlinear term and, thus, the architecture is much simple. Simulation results based on PA10 robot manipulator also demonstrate the efficacy of dual neural network for real-time kinematic control of joint-constrained redundant manipulators. REFERENCES [1] L. Sciavicco and B. Siciliano, Modeling and Control of Robot Manipulators. London, U.K.: Springer-Verlag, 2000. [2] A. A. Maciejemski and C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robot. Res., vol. 4, no. 3, pp. 109–117, 1985. [3] T. Yoshikawa, “Manipulability of robot mechanisms,” Int. J. Robot. Res., vol. 4, no. 2, pp. 3–9, 1985. [4] L. Sciavicco and B. Siciliano, “A solution algorithm to the inverse kinematic problem for redundant manipulators,” IEEE J. Robot. Automat., vol. 4, pp. 403–410, Aug. 1988. [5] M. Vukobratoric and M. Kircanski, “A dynamic approach to nominal trajectory synthesis for redundant manipulator,” IEEE Trans. Syst., Man, Cybern., vol. SMC-14, pp. 580–586, 1984. [6] Y. Nakamura, H. Hanafusa, and T. Yoshikawa, “Task-priority based redundancy control of robot manipulators,” Int. J. Robot. Res., vol. 6, no. 2, pp. 3–15, 1987. [7] I. D. Walker and S. I. Marcus, “Subtask performance by redundancy resolution for redundant robot manipulators,” IEEE J. Robot. Automat., vol. 4, pp. 350–354, June 1988. [8] A. Liegeois, “Automatic supervisory control of the configuration and behavior of multibody mechanisms,” IEEE Trans. Syst., Man, Cybern., vol. SMC-7, pp. 868–871, Dec. 1977. [9] O. Khatib and A. Bowling, “Optimization of the inertial and acceleration characteristics of manipulators,” in Proc. IEEE Int. Conf. Robot. Automat., vol. 4, 1996, pp. 2883–2889. [10] T. M. Abdel-Rahman, “A generalized practical method for analytic solution of the constrained inverse kinematics problem of redundant manipulators,” Int. J. Robot. Res., vol. 10, no. 4, pp. 382–395, 1991. [11] F.-T. Cheng, T.-H. Chen, and Y.-Y. Sun, “Resolving manipulator redundancy under inequality constraints,” IEEE J. Robot. Automat., vol. 10, pp. 65–71, Feb. 1994. [12] F.-T. Cheng, R.-J. Sheu, and T.-H. Chen, “The improved compact QP method for resolving manipulator redundancy,” IEEE Trans. Syst., Man, Cybern., vol. 25, pp. 1521–1530, Nov. 1995. [13] Z. Mao and T. C. Hsia, “Obstacle avoidance inverse kinematics solution of redundant robots by neural networks,” Robotica, vol. 15, pp. 3–10, 1997.
[14] J. Wang, Q. Hu, and D. Jiang, “A Lagrangian network for kinematic control of redundant manipulators,” IEEE Trans. Neural Networks, vol. 10, pp. 1123–1132, Sept. 1999. [15] H. Ding and J. Wang, “Recurrent neural networks for minimum infinity-norm kinematic control of redundant manipulators,” IEEE Trans. Syst., Man, Cybern. A, vol. 29, pp. 269–276, Jan. 1999. [16] W. S. Tang and J. Wang, “A recurrent neural network for minimum infinity-norm kinematic control of redundant manipulators with an improved problem formulation and reduced architectural complexity,” IEEE Trans. Syst., Man, Cybern. B, vol. 31, pp. 98–105, Feb. 2001. [17] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. B, pt. Part B, vol. 31, pp. 147–154, Feb. 2001. [18] J. Wang, “Recurrent neural networks for computing pseudoinverses of rank-deficient matrices,” SIAM J. Sci. Comput., vol. 18, no. 5, pp. 1479–1493, 1997. [19] Y. Xia, “A new neural network for solving linear programming problems and its application,” IEEE Trans. Neural Networks, vol. 7, pp. 525–529, Mar. 1996. [20] C. Klein and C. Huang, “Review of pseudoinverse control for use with kinematically redundant manipulators,” IEEE Trans. Syst., Man, Cybern., vol. SMC-13, pp. 245–250, 1983. [21] T. Shamir and Y. Yomdin, “Repeatability of redundant manipulators: Athematical solution of the problem,” IEEE Trans. Automat. Contr., vol. 33, pp. 1004–1009, Nov. 1988. [22] R. G. Roberts and A. A. Maciejewski, “Nearest optimal repeatable control strategies for kinematically redundant manipulators,” IEEE Trans. Robot. Automat., vol. 8, pp. 327–337, May 1992. [23] Y. Zhang and J. Wang, “A dual neural network for constrained torque optimization of kinematically redundant manipulators,” IEEE Trans. Syst., Man, Cybern. B, vol. 32, pp. 654–662, Oct. 2002. [24] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming—Theory and Algorithms. New York: Wiley, 1993. [25] Y. Xia and J. Wang, “Neural network for solving linear programming problems with bounded variables,” IEEE Trans. Neural Networks, vol. 6, pp. 515–519, Mar. 1995. [26] W. S. Tang and J. Wang, “A primal-dual neural network for kinematic control of redundant manipulators subject to joint velocity constraints,” in Proc. 6th Int. Conf. Neural Information Processing, vol. 2, 1999, pp. 801–806. [27] S. Zhang and A. G. Constantinides, “Lagrange programming neural networks,” IEEE Trans. Circuits Syst., vol. 39, pp. 441–452, July 1992. [28] A. Cichocki and R. Unbehauen, Neural Network for Optimization and Signal Processing. Chichester, U.K.: Wiley, 1993. [29] D. P. Bertsekas, Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1989. [30] O. L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods,” J. Optim. Theory Applicat., vol. 22, no. 2, pp. 465–485, 1979. [31] W. Li and J. Swetits, “A new algorithm for solving strictly convex quadratic programs,” SIAM J. Optim., vol. 7, no. 3, pp. 595–619, 1997. [32] J.-S. Pang, “A posterior error bounds for the linearly-constrained variational inequality problem,” Math. Oper. Res., vol. 12, pp. 474–484, 1987. [33] J.-S. Pang and J.-C. Yao, “On a generalization of a normal map and equation,” SIAM J. Contr. Optim., vol. 33, pp. 168–184, 1995. [34] Y. Xia and J. Wang, “A general methodology for designing globally convergent optimization neural networks,” IEEE Trans. Neural Networks, vol. 9, pp. 1331–1343, Nov. 1998. , “Global exponential stability of recurrent neural networks for [35] solving optimization and related problems,” IEEE Trans. Neural Networks, vol. 11, pp. 1017–1022, July 2000. [36] , “Global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights,” IEEE Trans. Automat. Contr., vol. 46, pp. 635–638, Apr. 2001. [37] D. E. Whitney, “Resolved motion rate control of manipulators and human prostheses,” IEEE Trans. Man-Machine Syst., vol. MMS-10, pp. 47–53, 1969. [38] J. Y. S. Luh, M. W. Walker, and R. P. C. Paul, “Resolved acceleration control of mechanical manipulators,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 468–474, Mar. 1980. [39] M. Kircanski and N. Kircanski, “Resolved rate and acceleration control in the presence of actuator constraints,” IEEE Contr. Syst. Mag., vol. 18, pp. 42–47, Jan. 1998.
ZHANG et al.: DUAL NEURAL NETWORK FOR REDUNDANCY RESOLUTION
Yunong Zhang (S’02–M’03) received the B.E. and M.E. degrees, both in automatic control engineering, from the Huazhong University of Science and Technology and the South China University of Technology, China, in 1996 and 1999, respectively, and the Ph.D. degree in 2003 in mechanical and automation engineering from the Chinese University of Hong Kong, Hong Kong. His research interests include nonlinear systems, robotics, and neural networks.
Jun Wang (S’89–M’90–SM’93) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from Dalian University of Technology, Dalilan, China, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH. He was an Associate Professor at the University of North Dakota, Grand Forks. He is currently a Professor of automation and computer-aided engineering at The Chinese University of Hong Kong, Hong Kong. His current research interests include neural networks and their engineering applications. Dr. Wang is an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B.
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Youshen Xia received the B.S. and M.S. degrees, both in computational mathematics and applied software, from the University of Nanjing, Nanjing, China, in 1982 and 1989, respectively. He received the Ph.D. degree in computational intelligence from the Chinese University of Hong Kong, Hong Kong, in 2000. He has been an associate Professor with the Department of Applied Mathematics, Nanjing University of Posts and Telecommunications, China, since 1995. His research interests include neural-network design, theory, and applications.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 3, MAY 2003
A Dual Neural Network for Redundancy Resolution of Kinematically Redundant Manipulators Subject to Joint Limits and Joint Velocity Limits Yunong Zhang, Member, IEEE, Jun Wang, Senior Member, IEEE, and Youshen Xia
Abstract—In this paper, a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators. Physical constraints such as joint limits and joint velocity limits, together with the drift-free criterion as a secondary task, are incorporated into the problem formulation of redundancy resolution. Compared to other recurrent neural networks, the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons. The dual neural network is shown to be globally (exponentially) convergent to optimal solutions. The dual neural network is simulated to control the PA10 robot manipulator with effectiveness demonstrated. Index Terms—Drift-free, dual neural network, joint limits, joint velocity limits, kinematically redundant manipulators.
I. INTRODUCTION
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INEMATICALLY redundant manipulators are those having more degrees of freedom (DOF) than required for position and orientation [1]. The redundancy of such manipulators including intrinsical redundancy and functional redundancy can be utilized to avoid obstacles [2], singularities [3], [4] and to optimize various performance criteria [5]–[7], as well as conduct the end-effector motion task. Since redundant manipulators have more DOF than necessary for position and orientation, multiple solutions exist. As a result, the redundancy of inverse kinematic mappings complicates the manipulator control problem considerably, in addition to the nonlinearity. To take full advantage of the redundancy, various computational schemes have been developed. Conventionally, the general solution of redundancy resolution is obtained by the pseudoinverse formulation as one minimum-norm particular solution plus a homogeneous solution [8], [9]. Most of current researchers have applied the pseudoinverse technique to formulate and resolve the redundancy by considering different optimization criteria, such as least-square joint velocities, singularity avoidance, obstacle avoidance and task priority control. However, among those techniques, the physical constraints such as joint limits and joint velocity limits are usually not taken Manuscript received September 24, 2001; revised January 3, 2003. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4165/98E. Y. Zhang and J. Wang are with the Department of Automation and ComputerAided Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong (e-mail: [email protected]; [email protected]). Y. Xia is with the Department of Applied Mathematics, Nanjing University of Post and Telecommunications, Nanjing, China. Digital Object Identifier 10.1109/TNN.2003.810607
into account. If these physical constraints are not considered, a saturation may occur. As a result, the tracking error may increase considerably, not to mention the physical damage possibly caused when a commanded joint or joint velocity hits its physical bound. For the joint-constrained inverse kinematics, numerical algorithms of redundancy resolution have been developed with physical constraints included. For example, the Jacobian matrix was augmented in [5] by incorporating the joint and velocity constraints and a routine predicting which constraints might be violated was required. The joint and velocity constraints were considered also in [10]. By applying the Gram–Schmidt orthogonalization procedure, a formulation was developed to express the general solution in terms of the redundant joint velocities only. Cheng et al. [11] formulated the constrained kinematic redundancy problem into a quadratic programming (QP) form. A compact QP method, using Gaussian elimination with partial pivoting, was finally developed in [12] to improve the computational efficiency. However, the numerical solutions to redundant inverse kinematics are in general computationally intensive. With more physical constraints and additional tasks considered, the computation process requires considerable time that may hinder the on-line applications, especially in high-DOF sensor-based robotic systems. In recent years, as parallel distributed computational models, neural networks have been developed for the redundancy resolution of robot manipulators, e.g., [13]–[17]. In particular, two neural networks, namely the pseudoinverse network [18] and the linear-programming neural network [19], were applied to the minimum infinity norm kinematic control in [15]. The obtained redundancy solution explicitly minimized joint velocities in the minimum infinity-norm sense. A two-layered primal-dual neural network was presented in [16] to online minimize the weighted joint velocity. To reduce network complexity and increase computational efficiency, an early dual-neural-network model [17] was then proposed for kinematic control of redundant manipulators. In the aforementioned schemes, it is assumed implicitly that there exists no joint limits or joint velocity limits when solving the redundancy resolution problem. But physical limits do exist in almost any robotic system. If a solution exceeds such mechanical joint limits or joint velocity limits and locks there, the desired motion path may become impossible to accomplish. In this paper, our attention is focused on the design and analysis of a general dual-neural-network approach to online redundancy resolution of physically constrained manipulators.
1045-9227/03$17.00 © 2003 IEEE
ZHANG et al.: DUAL NEURAL NETWORK FOR REDUNDANCY RESOLUTION
The dual neural network is proven to be globally convergent. The proposed dual neural network scheme is simulated on the 7-DOF PA10 robot arm for the primary task of end-effector trajectory tracking and simultaneously to eliminate the drift phenomenon [20] as an additional task. Its effectiveness and efficiency are then demonstrated. The remainder of this paper is organized into five sections. Section II provides the background information and the problem formulation of the drift-free redundancy resolution for physically constrained manipulators. Section III presents the proposed dual neural-network approach to online drift-free redundancy resolution of constrained manipulators. The theoretical results on global (exponential) convergence and position error estimation are given in Section IV. Section V illustrates and discusses simulation results of the dual neural network and the PA10 manipulator to show their operating characteristics and performance. Section VI concludes this paper with final remarks. II. PROBLEM FORMULATION A. Drift-Free Inverse Kinematics In a robot manipulator, the end-effector position and orientain the Cartesian space is related to the joint tion vector space by a forward kinematics equation (1) is joint variable vector and is a continuous where nonlinear mapping function with a known structure and parameters for a given manipulator. The relation between the Cartesian velocity and the joint velocity can be obtained by differentiating (1) (2) is the Jacobian matrix defined as . In a redundant manipulator, since , (1) and (2) are both undetermined and hence admit infinite number of solutions. The conventional pseudoinverse-type solution to the differential inverse kinematics problem (2) is generally formulated as one minimum-norm particular solution plus a homogeneous solution [8]. That is
where
(3) is the pseudoinverse of and is an where arbitrary vector and can be selected by using different optimization criteria, such as singularity avoidance, obstacle avoidance and task priority control. However, it is shown in [20] that the pseudoinverse or pseudoinverse-type solutions (3) are generally not repeatable in the sense that a closed path of the end-effector does not yield a closed path in joint space. Such joint angle drift is undesirable for cyclic motion control, since the manipulator does not necessarily return to its initial joint configuration after tracking a closed path in the task space. Of course, for the repeatability requirement, the manipulator configuration can be readjusted with a suitable self-motion at the end of every motion cycle, but this
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would be inefficient and unwanted. The analytic test using Lie bracket condition is given in [21] and [22], which determines whether or not a given pseudoinverse control strategy possesses the drift-free property, especially for the planar three-link manipulator case [21]. To make the kinematic control repeatable [8], a projection term is added to the pseudoinverse term which generates null-space velocities to minimize a secondary criterion. In [11], Cheng et al. proposed an inverse-kinematic control scheme by solving a quadratic program which minimizes the joint displacements between the current states and the initial states for the remedy of the drift problem. In the QP formulation, the objective function to be minimized is with
(4)
and is a positive parameter to scale the magnitude of the manipulator response to joint displacement. However, with more subtask criteria and physical constraints considered, the redundancy resolution becomes very time-consuming either by computing the pseudoinverse-type solution or solving the augmented QP problem. The requirement of real-time computation in sensor-based robotic systems further requests the development of much more efficient parallel-processing schemes as alternatives to replace numerical algorithms for online kinematic control of redundant manipulators. B. Joint Limits Conversion Besides the drift problem in repetitive motion, the pseudoinverse control (3) does not take account of physical limits of manipulators. As all manipulators are physically limited in their joints and joint velocities, it is more realistic to consider constrained kinematic control. Let us consider the joint limits and joint velocity limits simultaneously, with the superscripts and , respectively, denoting the upper and lower limits/bounds. Since the manipulator redundancy is resolved at velocity ] has to be converted into level, the limited joint range [ a dynamically updated joint-velocity bound constraint, e.g., (5) is selected such that where the critical coefficient there appears a deceleration when the robot arm enters the critor , while the intensity coeffiical region is used to scale the feasible region of . The cocient efficient is selected such that the feasible region of made by joint limits conversion is not smaller than the original one made by joint velocity limits; that is, is selected not less than . Note that large values of may cause joint deceleration quickly when the manipulator approaches its joint limits. ] can thus be Equation (5) and joint velocity limits [ , where the combined into this bound constraint th elements of and are defined, respectively, as
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Therefore, the physically constrained drift-free redundancy resolution problem can be formulated as
The quadratic program (6) can be reformulated as minimize
minimize
subject to
subject to
(8)
(6)
It is worth mentioning that actuator limits such as joint acceleration/torque limits are also important factors affecting robot performance [9], [11]. Such a feature of torque limit avoidance can be achieved by extending the proposed method to the acceleration level, e.g., [23]. C. Neural-Network Approaches By the duality theory [24], the dual problem of the primal quadratic program (6) is
where noting the
, identity matrix, we define
and with
de-
At any time instant, the above QP problem may be viewed as a parametric optimization problem. Now let us analyze the optimal solution to (8). By the Karush– Kuhn–Tucker condition [24], we know that is a solution to (8) if and only if there exists the dual decision variable such that and
minimize if
subject to where , and are the vectors of the dual decision variables. According to the design methodology [19], [25], [26], a primal-dual neural network can be developed by minimizing the duality gap via gradient decent direction. The dynamical equations of the primal-dual neural network are described below [26]
(7) where
is the state vector representing the estimated , and with if if if
The symbol in hereafter denotes the Euclidean norm of a vector or the Frobenius norm of a matrix. The positive capacitive parameters and are used to scale the convergence rate of the primal-dual neural network. It can be seen that the primal-dual network (7) is composed of two connected layers of neurons and that the network architecture is much complicated in terms of the nonlinear dynamics with third-order elements of . A recurrent neural networks called the Lagrangian neural network can be similarly developed [14], [27], [28], but the Lagrangian network with joint limits included has four layers of neurons. The hardware complexity of such a neural-network implementation may increase substantially as the number of neurons and layers increase. III. DUAL NEURAL NETWORK In this section, we propose a dual neural network approach with a reduced network complexity and increased computational efficiency to the real-time drift-free redundancy resolution of physically constrained manipulators.
if if which is equivalent to the system of piecewise linear [29]–[31], where equations is called the projection operator and defined as if if if
.
Therefore, is a solution to (8) if and only if there exists the such that and dual vector . That is
which gives rise to the proposed dual neural network for solving (8) with the following dynamic equation and output equation: (9) (10) is a capacitive design parameter used to scale the where convergence of the proposed network and the network output represents the estimated . The dynamic equation described in (9) shows that the dual neurons neural network is composed of only one layer of and without using any analog multiplier or penalty parameter. Compared to the prime-dual neural network (7), the dynamics of dual neural network is piecewise linear without any high-order nonlinear term. Consequently, the architecture of the dual neural network is much simpler than that of the primal-dual network. In a circuit realizing the dual neural network, the piecewise-linear might be implemented by using an activation function operational amplifier. In the robot control process, the desired
ZHANG et al.: DUAL NEURAL NETWORK FOR REDUNDANCY RESOLUTION
velocity vector is input into the dual network and simultaand , namely neously the network outputs the signals . the estimated joint velocity vector If ignoring the physical constraints for comparison purposes, we can simplify the dual neural network (9) for solving the “unconstrained” drift-free redundancy resolution problem as fol, and thus lows. That is, , then the dual neural network (9) is , which becomes the early simplified as dual neural-network model [17].
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Therefore, it follows from (11) that
(12) In addition, by Lemma 1, we have
(13) Then, adding (12) and (13) yields
IV. CONVERGENCE RESULTS In this section, we prove the global convergence and exponential convergence of the proposed dual neural network for kinematics control of redundant manipulators and estimate the position tracking error. Definitions about global convergence and exponential convergence of a neural system are present in [17]. The exponential convergence implies that such a system converges arbitrarily fast. Besides, the following lemma about projection property is often used in optimization literature [32]–[34]. is a closed Lemma 1 [29]: Assume that the set convex set, then the following two inequalities hold:
Defining follows from the above inequality that
, it
and, thus,
(14) Now we choose a Lyapunov function candidate as (15)
where
is a projection operator defined as .
It is clear that the set is a closed convex set and satisfies the above projection property. The convergence results of the proposed dual neural network (9) for constrained inverse kinematics is thus obtained as follows. Theorem 1: Starting form any initial state, the dual neural which denetwork (9) is convergent to an equilibrium point is pends on the initial state of the trajectory and an optimal solution to the inverse-kinematics QP problem (6). Moreover, the exponential convergence of the dual network can be achieved under a mild condition, i.e., (17). Proof: To show the convergence property, the following numbered inequalities are first derived. At any equilibrium point , we have this inequality [17], [31], [34] (11) which can be obtained by discussing the following three cases: , , Case 1) If for some , then ; , , Case 2) If for some and , then and thus ; , , Case 3) If for some and , then and, thus, .
is symmetric positive definite and . Clearly, is positive if and iff ) for definite (i.e., taken in the domain (i.e., the attraction region , since, in view of (14), of ). is negative definite in
where matrix
(16) if , iff in . Thus, it and follows that the dual neural network (9) is asymptotically conis clearly the optimal vergent to , of which the resulting solution to (6) in view of the Karush–Kuhn–Tucker optimality condition. Furthermore, we show the global exponential converand again. It follows from gence by reviewing , where (15) that are, respectively, the maximal and minimal . Clearly, and are proeigenvalues of portional to the capacitive parameter . Moreover, in view that for any in and amounts to , the following mild condition is presented , [35], [36]:
(17)
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In addition, by analyzing the linear/saturation cases of , the range of is (0,1]. Therefore from (14) and (17), we have
where is proportional to the reciprocal of ca, pacitive parameter . Thus we have and hence , , which completes the proof on exponential convergence of the proposed dual network. Remark 1: Without loss of generality, the above derivation is attained based on the existence of inverse kinematics solutions. is proportional to Since the exponential convergence rate , we can expedite the convergence of dual neural network and are, thus, sufficiently fast by decreasing . time-varying in a time scale sufficiently slower than that of (9). and within some Specifically, starting from any the maximal variation of ( , ) is small time interval sufficiently small, while the proposed dual neural network has been asymptotically convergent to the corresponding theoretwith a sufficiently small relative error. In the ical solution finite-time path-following task, the worst case of and , can be estimated on average as and , respectively, where , depends on the capacitive and , we can parameter . In view of estimate the inverse-kinematics joint configuration deviation as
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 3, MAY 2003
TABLE I JOINT LIMITS AND VELOCITY LIMITS OF PA10 MANIPULATOR
matrix such that has all the poles located on the left half complex plane, the error dynamics converges to zero. V. SIMULATION RESULTS The Mitsubishi PA10 manipulator (portable general purpose intelligent arm) has 7 DOF (three rotation axes and four pivot axes). The mechanical configuration and coordinate system, together with other specifications, of the PA10 redundant manipand joint velocity ulator can be found in [14]. Joint limits are shown in Table I. limits In this section, we will apply the dual neural network to drift-free redundancy resolution of physically constrained PA10 manipulator. Simulation has been performed for the path-following task that the end-effector of PA10 manipulator move along a given circle or straight-line in the three-dimensional workspace. In this study, only the positioning of the end-effector is considered, the Jacobian matrix is thus 3 7 in dimension, and the degree of redundancy is 4. A. Circular Motion
where can be made arbitrarily small by decreasing , namely, increasing the convergence rate of (9). Based on the Taylor se, the position tracking error ries expansion of (1), is bounded by the function with less , where the coefficient can be made arbitrarily small than by decreasing the capacitive parameter . From the above position error estimation, we know that the error can be made small by decreasing . Moreover, though a can lessen the position error too, it may cause high small joint velocity or acceleration. The ensuing simulation results will verify the soundness of the proposed error estimation. Remark 2: It is worth pointing out that the above derivation and estimation are effective on the condition that external disturbance does not exist. In a real system, model disturbance and computational round-off error always exist, the feedback control thus should be applied. One way is the closed-loop motion , the end-effector morate control [37]–[39]. Instead of tion rate can be given as
where and are respectively the commanded position and velocity vector. By appropriately choosing the diagonal gain
In this Section, first we show the redundancy resolution results without considering joint physical constraints and drift-free criterion. Specifically, the proposed dual neural disabled and the drift-free coefficient network (9) with , (i.e., the simplified dual network) is applied to the PA10 . robot arm. The capacitive parameter The desired motion of the PA10 end-effector is a circular path cm and the revolute angle about -axis, with radius . The task time of the motion is 10 s and the iniin ratial joint variables dians. Fig. 1 illustrates motion trajectories of the PA10 manipulator when its end-effector moving along a circle in the three-dimensional workspace and correspondingly the transient of the joint variables. Although the maximal Cartesian position and velocity tracking errors are less than 1.5 10 mm and has ex6.0 10 mm/sec respectively, the joint variable ceeded its mechanical range [ 2.6831, 2.6831] and thus the solution becomes inapplicable. If such a solution is directly apfinally locked at poplied to the PA10 manipulator with sition 2.6831 rad, the tracking error increases considerably, in addition to the physical damage possibly caused. Hence the path-following task fails. Moreover, as seen form Fig. 1, the solution is not repeatable in the sense that the final state of the PA10 manipulator does not coincide with its initial state; i.e., , and . Hence an
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Fig. 1. End-effector of PA10 manipulator moves along a circle in three-dimensional workspace without considering joint constraints and drift-free criterion, where the dark lines denote initial and final states of the manipulator, respectively.
Fig. 2. End-effector of PA10 manipulator moves along a circle in three-dimensional workspace with joint constraints and drift-free criterion considered, where the dark lines denote initial and final states of the manipulator, respectively.
inefficient and undesirable readjustment is needed for the cyclic motion control. In summary, the above simulation results show that physical constraints and drift-free criterion are in general worthy considering in the repetitive path-following tasks. Now let us consider the joint limits in Table I and the drift-free criterion. Specifically, the proposed dual neural network, with , , and , is applied to PA10 for the same circular-path following task. Fig. 2 shows the three-dimensional motion trajectories of the PA10 manipulator and the corresponding joint variables. Fig. 3 illustrates the transient behaviors of the proposed dual neural network and the PA10 manipulator, including the joint velocity variables in Fig. 3(a), the dual decision variables in Fig. 3(b) and the Cartesian position error and velocity error depicted in Fig. 3(c) and (d), has never respectively. As seen in Fig. 2, the joint variable exceeded the mechanical range [ 2.6831, 2.6831] and the solution is repeatable in the sense that the initial state and final state of PA10 manipulator coincide with each other. It follows from Fig. 3(a) that no joint velocity variable exceeds its limits in Table I. In addition, the maximal position and velocity errors are less than 8 10 and 6 10 mm/s, respectively. Namely, in subplots of Fig. 3(c) and (d), , , and denote the components of tracking position error , respectively, along the ,
and axes of the base frame and similarly , and denote, respectively, the , , and -axis components of tracking velocity error at the end-effector of the PA10 robot arm. The circular-path following experiments demonstrate the capability of the proposed dual neural network for online resolving the drift-free redundancy of physically constrained manipulators. B. Straight Line In this section, the PA10 manipulator is controlled to move forward and return backward along a straight line, like a reciprocating spot-welding task. The straight line of length 2.5 m, at every motion cycle, starts from the PA10 initial state and shall finally return to the initial state. Angles of the desired straight line making with , , and planes are, respectively, rad, rad and rad. The duration of the path-following task at every motion cycle is specified as 7.0 s. For comparison, the drift-free inverse kinematics problem is first solved without considering joint physical constraints, as shown in Fig. 4. Clearly, the solution is not acceptable in pracrad at time tice, since hits its mechanical limit
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(a)
(b)
(c)
(d)
Fig. 3.
Transient of the dual neural network and the PA10 manipulator moving along a circle.
Fig. 4.
End-effector of PA10 manipulator moves along a straight line with only the drift-free criterion considered.
s and and 4.54 s.
also hits its joint velocity limits at
The dual neural network is then applied to the physically constrained PA10 manipulator for drift-free redundancy resolution.
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(a)
(b)
Fig. 5. End-effector of PA10 manipulator moves along a straight line with joint constraints and drift-free criterion considered.
Fig. 6. Transients of the dual neural network and the PA10 manipulator moving along a straight line.
Parameters of the dual network are selected same as the circular . Simulation results are depicted in Figs. 5 example except and 6. The transient behaviors of joint variables and joint velocity variables are depicted in Fig. 5. Compared to Fig. 4(left),
the joint variable in Fig. 5 has never exceeded its upper limit 1.7637 rad and all the other joint variables remain in their limited ranges. Compared to Fig. 4(right), the joint velocity variincluding , as shown in Fig. 5(b), are kept always ables
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within their mechanical limits. The maximal position and velocity errors are less than 4 10 and 10 mm/s, respectively. Observed from other simulations, the maximal tracking error decreases rapidly when the capacitive parameter decreases. The above simulation results substantiate the efficacy of the neural-network approach to online drift-free redundancy resolution for physically constrained manipulators. VI. CONCLUDING REMARKS The proposed one-layer dual neural network provides a new parallel distributed computational approach to online drift-free redundancy resolution for physically constrained redundant manipulators. Compared with the supervised-learning neural network approaches to robot kinematic control, the present approach eliminates the need of off-line training and guarantees fast convergence due to the exponential convergence. Compared with other recurrent neural networks, the proposed dual neural network is able to resolve manipulator redundancy under physical constraints such as joint limits and joint velocity limits. Moreover, the dynamic equation of the dual neural network is piecewise linear and does not contain any high-order nonlinear term and, thus, the architecture is much simple. Simulation results based on PA10 robot manipulator also demonstrate the efficacy of dual neural network for real-time kinematic control of joint-constrained redundant manipulators. REFERENCES [1] L. Sciavicco and B. Siciliano, Modeling and Control of Robot Manipulators. London, U.K.: Springer-Verlag, 2000. [2] A. A. Maciejemski and C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robot. Res., vol. 4, no. 3, pp. 109–117, 1985. [3] T. Yoshikawa, “Manipulability of robot mechanisms,” Int. J. Robot. Res., vol. 4, no. 2, pp. 3–9, 1985. [4] L. Sciavicco and B. Siciliano, “A solution algorithm to the inverse kinematic problem for redundant manipulators,” IEEE J. Robot. Automat., vol. 4, pp. 403–410, Aug. 1988. [5] M. Vukobratoric and M. Kircanski, “A dynamic approach to nominal trajectory synthesis for redundant manipulator,” IEEE Trans. Syst., Man, Cybern., vol. SMC-14, pp. 580–586, 1984. [6] Y. Nakamura, H. Hanafusa, and T. Yoshikawa, “Task-priority based redundancy control of robot manipulators,” Int. J. Robot. Res., vol. 6, no. 2, pp. 3–15, 1987. [7] I. D. Walker and S. I. Marcus, “Subtask performance by redundancy resolution for redundant robot manipulators,” IEEE J. Robot. Automat., vol. 4, pp. 350–354, June 1988. [8] A. Liegeois, “Automatic supervisory control of the configuration and behavior of multibody mechanisms,” IEEE Trans. Syst., Man, Cybern., vol. SMC-7, pp. 868–871, Dec. 1977. [9] O. Khatib and A. Bowling, “Optimization of the inertial and acceleration characteristics of manipulators,” in Proc. IEEE Int. Conf. Robot. Automat., vol. 4, 1996, pp. 2883–2889. [10] T. M. Abdel-Rahman, “A generalized practical method for analytic solution of the constrained inverse kinematics problem of redundant manipulators,” Int. J. Robot. Res., vol. 10, no. 4, pp. 382–395, 1991. [11] F.-T. Cheng, T.-H. Chen, and Y.-Y. Sun, “Resolving manipulator redundancy under inequality constraints,” IEEE J. Robot. Automat., vol. 10, pp. 65–71, Feb. 1994. [12] F.-T. Cheng, R.-J. Sheu, and T.-H. Chen, “The improved compact QP method for resolving manipulator redundancy,” IEEE Trans. Syst., Man, Cybern., vol. 25, pp. 1521–1530, Nov. 1995. [13] Z. Mao and T. C. Hsia, “Obstacle avoidance inverse kinematics solution of redundant robots by neural networks,” Robotica, vol. 15, pp. 3–10, 1997.
[14] J. Wang, Q. Hu, and D. Jiang, “A Lagrangian network for kinematic control of redundant manipulators,” IEEE Trans. Neural Networks, vol. 10, pp. 1123–1132, Sept. 1999. [15] H. Ding and J. Wang, “Recurrent neural networks for minimum infinity-norm kinematic control of redundant manipulators,” IEEE Trans. Syst., Man, Cybern. A, vol. 29, pp. 269–276, Jan. 1999. [16] W. S. Tang and J. Wang, “A recurrent neural network for minimum infinity-norm kinematic control of redundant manipulators with an improved problem formulation and reduced architectural complexity,” IEEE Trans. Syst., Man, Cybern. B, vol. 31, pp. 98–105, Feb. 2001. [17] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. B, pt. Part B, vol. 31, pp. 147–154, Feb. 2001. [18] J. Wang, “Recurrent neural networks for computing pseudoinverses of rank-deficient matrices,” SIAM J. Sci. Comput., vol. 18, no. 5, pp. 1479–1493, 1997. [19] Y. Xia, “A new neural network for solving linear programming problems and its application,” IEEE Trans. Neural Networks, vol. 7, pp. 525–529, Mar. 1996. [20] C. Klein and C. Huang, “Review of pseudoinverse control for use with kinematically redundant manipulators,” IEEE Trans. Syst., Man, Cybern., vol. SMC-13, pp. 245–250, 1983. [21] T. Shamir and Y. Yomdin, “Repeatability of redundant manipulators: Athematical solution of the problem,” IEEE Trans. Automat. Contr., vol. 33, pp. 1004–1009, Nov. 1988. [22] R. G. Roberts and A. A. Maciejewski, “Nearest optimal repeatable control strategies for kinematically redundant manipulators,” IEEE Trans. Robot. Automat., vol. 8, pp. 327–337, May 1992. [23] Y. Zhang and J. Wang, “A dual neural network for constrained torque optimization of kinematically redundant manipulators,” IEEE Trans. Syst., Man, Cybern. B, vol. 32, pp. 654–662, Oct. 2002. [24] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming—Theory and Algorithms. New York: Wiley, 1993. [25] Y. Xia and J. Wang, “Neural network for solving linear programming problems with bounded variables,” IEEE Trans. Neural Networks, vol. 6, pp. 515–519, Mar. 1995. [26] W. S. Tang and J. Wang, “A primal-dual neural network for kinematic control of redundant manipulators subject to joint velocity constraints,” in Proc. 6th Int. Conf. Neural Information Processing, vol. 2, 1999, pp. 801–806. [27] S. Zhang and A. G. Constantinides, “Lagrange programming neural networks,” IEEE Trans. Circuits Syst., vol. 39, pp. 441–452, July 1992. [28] A. Cichocki and R. Unbehauen, Neural Network for Optimization and Signal Processing. Chichester, U.K.: Wiley, 1993. [29] D. P. Bertsekas, Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1989. [30] O. L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods,” J. Optim. Theory Applicat., vol. 22, no. 2, pp. 465–485, 1979. [31] W. Li and J. Swetits, “A new algorithm for solving strictly convex quadratic programs,” SIAM J. Optim., vol. 7, no. 3, pp. 595–619, 1997. [32] J.-S. Pang, “A posterior error bounds for the linearly-constrained variational inequality problem,” Math. Oper. Res., vol. 12, pp. 474–484, 1987. [33] J.-S. Pang and J.-C. Yao, “On a generalization of a normal map and equation,” SIAM J. Contr. Optim., vol. 33, pp. 168–184, 1995. [34] Y. Xia and J. Wang, “A general methodology for designing globally convergent optimization neural networks,” IEEE Trans. Neural Networks, vol. 9, pp. 1331–1343, Nov. 1998. , “Global exponential stability of recurrent neural networks for [35] solving optimization and related problems,” IEEE Trans. Neural Networks, vol. 11, pp. 1017–1022, July 2000. [36] , “Global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights,” IEEE Trans. Automat. Contr., vol. 46, pp. 635–638, Apr. 2001. [37] D. E. Whitney, “Resolved motion rate control of manipulators and human prostheses,” IEEE Trans. Man-Machine Syst., vol. MMS-10, pp. 47–53, 1969. [38] J. Y. S. Luh, M. W. Walker, and R. P. C. Paul, “Resolved acceleration control of mechanical manipulators,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 468–474, Mar. 1980. [39] M. Kircanski and N. Kircanski, “Resolved rate and acceleration control in the presence of actuator constraints,” IEEE Contr. Syst. Mag., vol. 18, pp. 42–47, Jan. 1998.
ZHANG et al.: DUAL NEURAL NETWORK FOR REDUNDANCY RESOLUTION
Yunong Zhang (S’02–M’03) received the B.E. and M.E. degrees, both in automatic control engineering, from the Huazhong University of Science and Technology and the South China University of Technology, China, in 1996 and 1999, respectively, and the Ph.D. degree in 2003 in mechanical and automation engineering from the Chinese University of Hong Kong, Hong Kong. His research interests include nonlinear systems, robotics, and neural networks.
Jun Wang (S’89–M’90–SM’93) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from Dalian University of Technology, Dalilan, China, and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH. He was an Associate Professor at the University of North Dakota, Grand Forks. He is currently a Professor of automation and computer-aided engineering at The Chinese University of Hong Kong, Hong Kong. His current research interests include neural networks and their engineering applications. Dr. Wang is an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B.
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Youshen Xia received the B.S. and M.S. degrees, both in computational mathematics and applied software, from the University of Nanjing, Nanjing, China, in 1982 and 1989, respectively. He received the Ph.D. degree in computational intelligence from the Chinese University of Hong Kong, Hong Kong, in 2000. He has been an associate Professor with the Department of Applied Mathematics, Nanjing University of Posts and Telecommunications, China, since 1995. His research interests include neural-network design, theory, and applications.
