Robust Adaptive Control of Nonholonomic Mobile Robot With Parameter and Nonparameter Uncertainties

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1 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL [4] P. Coelho and U. Nunes, Lie algebra application to mobile robot control: A tutorial, Robotica, vol. 1, no. 5, pp , 003. [5] P. Coelho, Path following control of wheeled mobile robots in presence of uncertainties, Ph.D. dissertation (in Portuguese), Univ. Coimbra, Coimbra, Portugal, 004. [6] G. Campion et al., Controllability and state feedback stabilization of nonholonomic mechanical systems, in Lecture Notes in Control and Information Science, C.C. dewitet al., Ed. New York: Springer-Verlag, 1991, vol. 16, pp [7] X. Yun and Y. Yamamoto, Stability analysis of the internal dynamics of a wheeled mobile robot, J. Robot. Syst., vol. 14, no. 10, pp , [8] L. E. Dubins, On curves of minimal lenght with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., vol. 79, pp , [9] H. Nijmeijer and A. Schaft, Nonlinear Dynamic Control Systems. New York: Springer-Verlag, [30] K. J. Åström and B. Wittenmark, Computer Controlled Systems: Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, [31] G. Pires and U. Nunes, A wheelchair steered through voice commands and assisted by a reactive fuzzy-logic controller, J. Intell. Robot. Syst., vol. 34, no. 3, pp , 00. Robust Adaptive Control of Nonholonomic Mobile Robot With Parameter and Nonparameter Uncertainties Wenjie Dong and K.-D. Kuhnert Abstract This paper considers the tracking-control problem of a nonholonomic wheeled mobile robot with both parameter and nonparameter uncertainties.a robust adaptive controller is proposed with the aid of the adaptive backstepping technique and the learning ability of neural networks.the proposed controller guarantees that the tracking error converges to a small ball containing the origin.the ball s radius can be adjusted by control parameters.the proposed controller is successfully implemented in our simulator. Index Terms Adaptive control, nonholonomic system, robust control, tracking control, wheeled mobile robot. I. INTRODUCTION In recent years, there has been a growing interest in the design of feedback-control laws for mechanical systems subjected to nonholonomic constraints. Due to Brockett s necessary condition for asymptotic feedback stabilization [1], a nonholonomic system cannot be asymptotically stabilized to a equilibrium by any smooth or continuous pure-state feedback law. To overcome this limitation, several approaches have been proposed in last decade (see [18], [0], and the references therein). Tracking control is another control problem of nonholonomic systems. Based on whether the system is described by a kinematic model or a dynamic model, the tracking-control problem is classified as either a kinematic or a dynamic tracking-control problem. The kinematic tracking-control problem has been studied in recent years. With Manuscript received January 1, 004; revised May 11, 004. This paper was recommended for publication by Associate Editor G. Oriolo and Editor H. Arai upon evaluation of the reviewers comments. W. Dong was with the Department of Electrical and Computer Engineering, University of Siegen, Siegen D-57068, Germany. K.-D. Kuhnert is with the Department of Electrical and Computer Engineering, University of Siegen, Siegen D-57068, Germany. Digital Object Identifier /TRO Fig. 1. Configuration of a wheeled mobile robot. the aid of the linearization technique, local controllers are proposed in [9], [17], and [5]. Based on the dynamic feedback linearization and the differential flatness concept, the dynamic controllers with singular points are proposed in [4] and [10]. In [15], global tracking controllers are proposed for nonholonomic wheeled mobile robots. For chained systems, tracking controllers are proposed in [8] and [16]. The dynamic tracking-control problem of the nonholonomic system has received attention recently, because most practical nonholonomic mechanical systems are dynamic systems and have uncertainties. In [3], Suand Stepanenko study the tracking-control problem of the dynamic nonholonomic systems with unknown inertia parameters, and propose an adaptive controller. Chen et al. discuss the dynamic tracking problem of the uncertain nonholonomic systems, and propose a robust H-infinite controller in []. In [9] and [11], the dynamic tracking problem of a wheeled mobile robot is studied, and neural-networkbased controllers are proposed. In [5], [6], [7], and [4], the dynamic tracking-control problem of nonholonomic systems with uncertainty is discussed. Robust and adaptive controllers are proposed. Most of the results on the dynamic tracking problem of nonholonomic systems are proposed based on the assumption that the kinematics of the system is exactly known, and there are only uncertainties in the dynamics of the system. However, in practice, there are uncertainties in both kinematics and dynamics. In this paper, we study the tracking-control problem of a wheeled mobile robot with uncertainties in both kinematics and dynamics. It is assumed that in the kinematics, there are parameter uncertainties, and in the dynamics, there are parameter uncertainties and nonparameter uncertainties. To solve this tracking-control problem, a robust adaptive neural-network controller is proposed with the aid of backstepping techniques and the approximation property of neural networks. II. PROBLEM STATEMENT Consider a wheeled mobile robot moving on a horizontal plane (Fig. 1). The robot is composed of a rigid body, two fixed rear wheels, and one steering wheel. Two torque motors are equipped in the front wheel for driving and steering. Given a differentiable simple curve (C) defined by one of its points, the unitary tangent vector at this point, and its curvature curv(s), with s being the curvilinear coordinate along the curve, for a point Q in the curve (C), assume that the curvilinear coordinate at Q is s. Let fq; T (s);n(s)g be the Frenet frame on the curve at point Q, and assume that jcurv(s)j < 1=R(8s) where R(> 0) is a constant. If the distance between P and the curve (C) is smaller than R, the position of P is parameterized by (s; d), where d is the coordinate of P along N (s). The robot s configuration is parameterized by q =[q 1 ;q ;q 3 ;q 4 ] T =[s; d; ; ] T, where is the angle between PF and T (s) and is the steering angle of the front /$ IEEE

2 6 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL 005 wheel with respect to the robot body. By the classic law of mechanics, and also the results in [], one has v 1 cos q 3 _q 1 = 1 0 curv(q 1 )q _q = v 1 sin q 3 v1 tan q4 v1 curv(q1) cos (1) q3 _q 3 = 0 l 1 0 curv(q 1)q _q 4 = v M (q)_v + C(q; _q)v + G(q) +U (q; _q) =B(q) () where l is the distance between the two points P and F; v 1 is the velocity of the point P; v is the angular velocity of the steering wheel, M (q) is a bounded positive definite symmetric inertia matrix, C(q; _q)_q is centripetal and Coriolis torques, G(q) is the gravitational torque, U (q; _q) includes the unmodeled dynamics of the system and bounded continuous external disturbance, B(q) is the input matrix, is the control input, and the superscript T denotes the transpose. For (1) (), it is assumed that: 1) in (1), l is not exactly known, i.e., l [l min ;l max ] where l max(> 0) and l min(> 0) are known; ) in (), the matrices M (q);c(q; _q);g(q), and U (q; _q) are unknown; 3) in (), the expression of B(q) is known. In fact, it can be easily derived that B(q) =diag[1=(r cos q 4 ); 1]. InB(q), it is assumed that r is not exactly known, i.e., r [r min;r max] where r min (> 0) and r max (> 0) are known. Control Problem: Given a desired simple curve (C) and a desired velocity v1(t) 3 of the robot, for (1) (), the control problem in this paper is defined as finding a controller, such that lim q(t) = 0 lim q3(t) = 0 lim t!1 t!1 t!1 (v1(t) 0 v3 1(t)) = 0: Remark 1: Noting that the assumption jcurv(q 1)j < 1=R, (1) is well-defined if jq j < R and jq 4 j < =. In the controller design, these conditions will be guaranteed if jq (0)j <Rand jq 4(0)j <=. In (), the following well-known property is satisfied [3]. Property 1: For a suitably defined C(q; _q); ( M _ 0 C) is skewsymmetric. III. BACKSTEPPING DESIGN PROCEDURE To deal with parameter uncertainties and nonparameter uncertainties in (1) (), the adaptive backstepping technique [19] and the learning ability of neural networks [3], [14] are used to design the controller. Assuming jq (0)j < R;jq 3(0)j < =, and jq 4(0)j < =, let u 1 = v 1 cos q 3 =(1 0 curv(q 1 )q );u = v ;b = 1=l; g = [0; 0; 0; 1] T ;g 1 =[1; (1 0 curv(q 1 )q ) tan q 3 ; 0curv(q 1 ); 0] T ;g 3 = [0; 0; (1 0 curv(q 1)q ) tan q 4=cos q 3; 0] T, (1) () can be written as where _q = g 1u 1 + g u + bg 3u 1 (3) M 1 (q)_u + C 1 (q; _q)u + G 1 (q)+u 1 (q; _q) =B 1 (q) (4) Step 1: Introducing 1 = u 1 0 v 3 1 if u 1 were the actual control input, one had 1 0 and u 1 v 3 1. Let z = h(q ) where h(q ) is a smooth monotonic function which maps (0R; R) onto (01; +1), with the first derivative (with respect to q ) strictly larger than a positive real number, and such that h(0) = 0, then _z = v 3 1L g z + u L g z + bv 3 1L g z = v 3 1L g z (5) where L g z j is the Lie derivative of z j along g i. Hereafter, L means Lie derivative in this paper. Introducing z 3 = L g z 0 3 if L g z were the actual control input, one had z 3 0 and L g z 3. Let Lyapunov function to make we choose V = 1 z _V = 0k z v 3 1 where constant k (> 0) is a design parameter. Since L g z is not the control, z So 3 = 0k z (6) _V = 0k z v z z 3 v 3 1: The second term z z 3 v 3 1 will be cancelled at the next step. The closedloop system (5) with (6) is _z = 0k z v z 3 v 3 1 (7) _z 3 = v 3 1L g z 0 k z v k z 3 v bv 3 1L g L g z : (8) Step : Introducing let Lyapunov function z 4 = L g z 0 4 V 3 = 1 z + z (^b 0 b) where constant 1(>0) is a design parameter, then _V 3 = 0k z v z 3 (z 4 + z 0 k z + k z 3 + ^bl g L g z + 4 )v (^b 0 b)( _^b 0 1 v 3 1z 3 L g L g z ): If L g z were the actual control input, one had z 4 0. To make we would choose _^b = 1 _V 3 = 0k z v k 3z 3v 3 1 M 1(q) =9 T (q)m(q)9(q) C 1 (q; _q) =9 T (q)(m(q) 9(q)+C(p; p)9(q)) G 1(q) =9 T (q)g(q) U 1(q; _q) =9 T (q)u(q; _q) B 1 (q) = diag[(1 0 curv(q 1 )q )=(r ); 1] 9(q) = diag[(1 0 curv(q 1)q )= cos q 3; 1]: 4 = 0(k + k 3 )z 3 0 (1 0 k)z 0 ^bl g L g z (9) where 1 = 1 v1z 3 3 L g L g z constant k 3(> 0) is a design parameter.

3 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL Since L g z is not the control, z and we do not use _^b = 1 as an update law in the control. Then _V 3 = 0k z v k 3 z 3v z 3 z 4 v (^b 0 b)( _^b 0 1 ): The third term z 3 z 4 v 3 1 will be cancelled at the next step. The closedloop system (8) with (9) is _z 3 = 0k 3 z 3 v z 4 v z v 3 1 +(b 0 ^b)v 3 1L g L g z _z 4 = v 3 1L 3 g z +(k + k 3 )z 4 v (k k 3 + k k )z 3 v (k 0 k 3 + k 3)z v _^bl g L g z + ^bv 3 1L g L g L g z +(k + k 3 )(b 0 ^b)v 3 1L g L g z + ^bbv 3 1L g L g z + bv 3 1L g L g z + ^bu L g L g L g z : (10) Step 3: Introducing let Lyapunov function then = u 0 5 V 4 = 1 z + z3 + z (^b 0 b) _V 4 = 0k z v k 3 z 3v z 4 [z 3 v v 3 1L 3 g z +(k + k 3 )z 4 v (k k 3 + k k )z 3 v (k 0 k 3 + k 3 )z v _^bl g L g z + ^bv 3 1L g L g L g z + ^b v 3 1L g L g z + ^bv 1L 3 g Lg z + ^b( + 5)L g L g L g z ] (^b 0 b)( _^b 0 1 (z 3 v1l 3 g L g z + z 4 [(k + k 3 ) v1l 3 g L g z + ^bv 1L 3 g L g z + v1l 3 g Lg z ])): If u were the actual control input, one had 0. The update law of ^b is chosen as _^b = 0 1 (^b 0 b 0 ); if ^b (b l ;b u ); or ^b = b l ; 0; or ^b = b u 0 0 1(^b 0 b 0); if ^b = b l ; < 0 or ^b = b u > 0 (11) where b l = 1=l max and b u = 1=l min, constants 1(> 0) and b 0( (b l ;b u )) are design parameters, and is defined by = z 4 v 3 1[(k + k 3 )L g L g z + ^bl g L g z + L g L g z ]: The virtual control 5 is chosen as where 5 =[0k 4 z 4 v z 3 v v 3 1L 3 g z 0 (k + k 3 )z 4 v 3 1 +(k k 3 + k k )z 3v 3 1 +(k 0 k 3 + k 3)z v ^bv 3 1L g L g L g z 0 ^b v 3 1L g L g z 0 ^bv 3 1L g L g z ]=(^bl g L g L g z ) 5= (^b 0 b 0 ) L g L g z tanh z (^b 0 b 0) L g L g z )= ) constants k 4 (> 0) and (> 0) are design parameters, and 5 is the third stabilizing function. Then _V 4 0k zv k 3 z3v k 4 z4v (^b 0 b) (^b 0 b 0) where satisfies = e 0(+1) (i.e., =0:785) [1]. (b 0 b 0) + Remark : By the relation between z and q, boundedness of z ;z 3, and z 4 guarantees that q (0R; R); q 3 (0=;=), and q 4 (0=;=). The update law (11) guarantees ^b [b l ;b u ] all the time. Therefore, 5 is well defined. Since u is not the control input, 6 0. However, we will choose (11) as the final update law of ^b in the control. Then _V 4 0k z v k 3z 3v k 4z 4v z 4^bL g L g L g z (b 0 b 0 ) + : The closed-loop system (10) is _z 4 = 0k 4 z 4 v z 3 v ^bl g L g L g z 1 1 (^b 0 b) (^b 0 b 0 ) + L g L g z _^b (b 0 ^b)[^bl g L g z + L g L g z +(k + k 3 )L g L g z ]v 3 1: Step 4: Since u 1 is not the control input, Let and then =[ 1; ] T =[v 3 1; 5 ] T (1) M 1 _ = B1 0 C 1 0 (M 1 _ + C 1 + G 1 + U 1 ): (13) In (13), M 1; C 1;G 1, and U 1 are unknown. Also in B 1; ris unknown. To deal with the uncertainty of (M 1 _ + C 1 + G 1 + U 1 ), a two-layer neural network with sigmoid base functions will be used to approximate it. Since (M 1 _ + C 1 + G 1 + U 1) is continuous, let s(q; _q; ; _) be an m-vector of continuous sigmoidal functions, according to the approximation property of neural networks [3] M 1(q)_ + C 1(q; _q) + G 1(q)+U 1(q; _q) =w T s(q; _q; ; _) + (14) where w R m is an unknown optimal constant weight vector, and is defined by w := arg min R f sup km 1 (q)_ + C 1 (q; _q) (q;_q;;_) +G 1 (q)+u 1 (q; _q) 0 T s(q; _q; ; _)kg where is a compact region. = [ 1; ] T represents the network reconstruction error corresponding to the optimal weight vector. Generally, increasing the number of adjustable weights (i.e., m) reduces the network reconstruction error. The approximation results for neural networks indicate that if m is sufficiently large, then can be made bounded on a compact region [3], [1]. Given a compact region,we assume that kk ; 8(q; _q; ; _) (15) where is an unknown bound. It is worth noting that the unknown bound is not unique, since any constant which is greater than satisfies (15). To avoid confusion, we define to be the smallest constant, such that (15) is satisfied. With (14), (13) can be written as M 1 (q) _ = B 1 (q) 0 C 1 (q; _q) 0 w T s 0 : Defining c =1=r, and letting ^c; ^w, and ^ be the estimates of c; w, and, respectively, we choose the control law = ^B 01 1 [0K p + ^w T s(q; u) 0 ^ tanh +3] (16)

4 64 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL 005 where K p is a positive definite matrix, and ^B 1 is the value of B 1 corresponding to the estimate ^c (i.e., ^B 1 = diag[^c(1 0 curv(q 1 )q )= ; 1]). 3 and the update laws of ^w and ^ will be determined next. Then M 1 _ = 0Kp 0 C 1 +(^w 0 w) T s 0 ^ tanh Let 0 +3+(B 1 0 ^B 1): V 5 = 1 z + z 3 + z 4 + T M (^b 0 b) + 01 tr then (( ^w 0 w) T (^w 0 w)) ( ^ 0 ) + 01 (^c 0 c) _V 5 = 0k zv k 3 z3v k 4 z4v z 4 L g L g z _^b + z (^b 0 b) b _ 0 z 3u 1L g L g z 0 z 4u ^bl 1 g L g z + L g L g z +(k + k 3 )L g L g z 0 T K p +tr (^w 0 w) T 01 _^w + s T tanh tanh ( ^ 0 ) _^ 0 3 T tanh + T 3 0 k z + k 3 z 3 + k 4z z 4 ^bl g L g L g z + L g L g z (^c 0 c) _^c (1 0 curv(q 1)q ) where 1 is the first element of. If we choose the update law ^b as in (11), the update laws of ^w; ^, and ^c are as follows: _^w = 0 s T 0 1 (^w 0 w 0 ) (17) _^ = 3 T tanh _^c = 0 1( ^ 0 0) (18) (1 0 curv(q 1 )q ) 0 1 (^c 0 c 0 ); if ^c (c l ;c u ) or ^c = c l 1 1 (1 0 curv(q 1 )q ) > 0 or ^c = c u 1 1 (1 0 curv(q 1 )q ) < 0 1 1(1 0 curv(q 1)q ) 0 1 (^c 0 c 0 ); if ^c = c l ; (1 0 curv(q 1 )q ) or ^c = c u ; 0 3= kz + k 3z 3 + k 4z 4 0z 4^bL g L g L g z 4 (19) where (> 0); 3 (> 0); 4 (> 0); w 0 ; 0, and c 0 ( (c l ;c u )) are design parameters, c l =1=r max, and c u =1=r min, then _V 5 0k zv k 3 z3v k 4 z4v (^b 0 b) 0 T K p tr(( ^w 0 w)t (^w 0 w)) ( ^ 0 ) (^c 0 c) (b 0 b 0 ) tr((w 0 w 0 ) T (w 0 w 0 )) ( 0 0 ) (c 0 c0) + + : (0) IV. MAIN RESULTS AND DISCUSSIONS A. Main Results With the aid of the preceding design procedure, one has the following result. Theorem 1: With the controller (16) and the update laws of ^b; ^w; ^, and ^c defined in (11), (17), (18), and (19), respectively, if v1(t) 3 v > 0, then z i ( i 4); ; (^b 0 b); (^c 0 c); (^w 0 w), and ( ^0 ) are uniformly bounded, and exponentially converge to a small ball containing the origin. The radius of the ball can be adjusted by the design parameters. Proof: It can be proved that the modified projection algorithm (11) guarantees that ^b [b l ;b u ], therefore 5 is well defined all the time. With the update law (19), it can be proved that ^c [c l ;c u ]. So the control law (16) is well defined. Therefore, all variables in the system are well defined. Differentiating V 5 with respect to time along the closed-loop system, one has (0). Therefore _V V 5 + (1) where 1 is a positive constant which depends on the control parameters, and = (b 0 b 0 ) tr((w 0 w 0 ) T (w 0 w 0 )) So ( 0 0) V 5 (t) 1(c 0 c0) + + : V 5 (0) 0 1 e 0 t + 1 : z i ( i 4); ; ^b; ^c; ^w, and ^ are uniformly bounded and exponentially converge to a small ball. The radius of the ball can be adjusted by the design parameters i(1 i 4); 1;b 0;c 0;w 0; 0, and. With the aid of the state transformation and Theorem 1, one has the following result. Theorem : With the controller (16) and the update laws ^b; ^w; ^, and ^c defined in (11), (17), (18), and (19), respectively, if jq (0)j < R; jq 3 (0)j 6= =; jq 4 (0)j 6= =, and v 3 1(t) v > 0, then q i ( i 4); (v 10v 3 1 ); (^b0b); (^c0c); (^w0w), and ( ^0 ) are uniformly bounded and converge to a small ball containing the origin. The radius of the ball can be adjusted by the design parameters. Proof: By Theorem 1, z i( i 4); ; (^b 0 b); (^c 0 c); (^w 0 w), and ( ^0 ) are uniformly bounded and exponentially converge to a small ball. By calculation, it can be proved that q i( i 4) and (v 1 0 v 3 1 ) are uniformly bounded and converge to a small ball. B. Discussions If jq (0)j <R, in order to make q (0R; R) all the time, z = h(q ) is introduced in Step 1. With the condition imposed on h(q ),if z is bounded, q (0R; R). Therefore, the definition of point Q is unique, and d is well defined in the control. If R<1, one choice of h(q ) is h(q R )= tan q R : Specially, if curv(s) =0, one can choose h(q )=q.ifjq 3(0)j < = and jq 4 (0)j < =, the proposed controller will make jq 3 j < = and jq 4j <=all the time. If q (0)j R, jq 3(0)j = =, or jq 4 (0)j = =, one can first use an open-loop control law to make the robot move into the region where the proposed controller can be applied, then apply the proposed controller. Unknown parameters b(= 1=l) and c(= 1=r) are updated by the adaptive laws (11) and (19), respectively. They guarantee that ^b [b l ;b u] and ^c [c l ;c u]. The neural network is used to approximate

5 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL Fig.. Response of d. Fig. 4. Response of (v 0 v ). Fig. 3. Response of. Fig. 5. Response of ^b. the unknown dynamics of the system. The weight matrix is updated by the adaptive law (17). The reconstruction error of the neural network is estimated by the adaptive law (18). The control parameters are k i ( i 4); K p ; i (1 i 4); 1 ; ; b 0 ; c 0 ; 0, and w 0. Large values of k i ( i 4) and K p make q i( i 4) and (v 1 0 v1) 3 converge quickly to the small ball. Small values of 01 i 1 (1 i 4) and 1 make the radius of the ball small. Parameters b 0 ;c 0 ; 0, and w 0 also affect the radius of the small ball. If b 0;c 0; 0, and w 0 are close to b; c;, and w, respectively, the tracking error will be small. Therefore, in order to make q i ( i 4) and (v 1 0 v1) 3 converge quickly to the origin, one can make k i(1 i 4); K p; i(1 i 4) large and 1 small. V. SIMULATION In order to verify the validity of the proposed controller, we implement the controller in the simulator available in our lab. The setup of the simulator is comprised of two interconnected computers. One computer, the so-called System-PC, processes the video streams and calculates the control commands for the robot to follow the path. These control commands are then passed to the second computer, the so-called Simulator-PC, which is responsible for the robot and environment simulation. In the experiment, we choose one type of wheeled mobile Fig. 6. Response of ^c. robots and a rocky path to test the controller. Figs. 4 show the responses of d;, and (v 1 0 v 3 1) with a group of control parameters. From the results, it is shown that d;, and (v 1 0 v 3 1) converge to zero during the control. Figs. 5 and 6 show the responses of ^b and ^c. Itis

6 66 IEEE TRANSACTIONS ON ROBOTICS, VOL. 1, NO., APRIL 005 shown that they are bounded. Especially, ^b and ^c do not go through zero. Responses of ^ and ^w also show the boundedness of ^ and ^w. The control inputs calculated from the control law are bounded and not large. They can be realized by typical mobile actuators. These experimental results show that the proposed controller is effective. To test the proposed controller further, we implement it in different robots and different paths. All results show that the controller is effective. VI. CONCLUSION In this paper, the tracking-control problem of a nonholonomic wheeled mobile robot with parameter uncertainties and nonparameter uncertainties is considered. A robust adaptive controller is proposed. The idea in this paper can be applied to control other uncertain nonholonomic dynamics systems. [1] M. M. Polycarpou, Stable adaptive neural network control scheme for nonlinear systems, IEEE Trans. Autom. 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A New Formulation of Visual Servoing Based on Cylindrical Coordinate System Masami Iwatsuki and Norimitsu Okiyama Abstract Image-based visual servoing is a flexible and robust technique to control a robot and guide it to a desired position only by using two-dimensional visual data.however, it is well known that the classical visual servoing based on the Cartesian coordinate system has one crucial problem, that the camera moves backward at infinity, in case that the camera motion from the initial to desired poses is a pure rotation of 180 around the optical axis.this paper proposes a new formulation of visual servoing, based on a cylindrical coordinate system that can shift the position of the origin. The proposed approach can interpret from a pure rotation around an arbitrary axis to the proper camera rotational motion.it is shown that this formulation contains the classical approach based on the Cartesian coordinate system as an extreme case with the origin located at infinity.furthermore, we propose a decision method of the origin-shift parameters by estimating a rotational motion from the differences between initial and desired image-plane positions of feature points. Index Terms Camera retreat problem, cylindrical coordinate system, eye-in-hand robot, visual servoing. I. INTRODUCTION Visual servoing is a flexible and robust control technique of robots using vision in feedback-control loops. It is classified into positionbased approaches and image-based approaches [1], []. It is known that the image-based visual servoing can derive a robot motion directly from two-dimensional (-D) visual data [3]. One of the chief advantages to the image-based visual servoing is that the positioning accuracy of the robot is less sensitive to robot and camera calibration errors and image measurement errors. In most of the image-based visual servo systems, image features are composed with a set of image points, frequently the centroids of regions, thanks to the robustness and implementation ease of feature extraction [1]. Unfortunately, the conventional approach of monocular Manuscript received April 5, 003; revised May 11, 004; June 19, 004; July 19, 004. This paper was recommended for publication by Associate Editor F. Chaumette and Editor S. Hutchinson upon evaluation of the reviewers comments. This paper was presented in part at the IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, October 00. The authors are with the Faculty of Engineering, Hosei University, Tokyo , Japan ( iwatsuki@k.hosei.ac.jp). Digital Object Identifier /TRO /$ IEEE