Modeling and Hierarchical Tracking Control of Tri-Wheeled Mobile Robots
Pu-Sheng Tsai, Li-Sheng Wang, and Fan-Ren Chang
Abstract—After exploring the structure of the dynamics derived by using the Appell equation, we propose a hierarchical tracking controller for a tri-wheeled mobile robot in this paper. With appropriately chosen privi-leged variables, the reduced equations are decoupled from the kinematic equations associated with the underlying nonholonomic constraints. This special character of the system makes it possible to separate the design into three levels: motion planning, kinematic, and dynamic. In the proposed scheme, a fuzzy inference engine in the kinematic level is used to update the desired trajectory computed in the motion-planning level. An adaptive sliding-mode controller is then adopted to track the new reference values of privileged variables in the dynamic level, which subsequently drives the nonprivileged variables. Simulation results show the effectiveness of such a tracking-control scheme, which concurrently takes kinematics and dy-namics into consideration. All system variables can be tracked asymptot-ically to their desired values, which are assured by the skew-symmetric property of the Appell equation.
Index Terms—Hierarchical tracking control, nonholonomic constraint, privileged coordinates, reduced Appell equation, skew-symmetric prop-erty.
I. INTRODUCTION
For a vehicle or a mobile robot moving with nonsliding rolling wheels, problems associated with nonholonomic constraints naturally arise. To deal with the motion planning of a nonholonomic system, the subject of nonholonomic motion planning (NMP) has been studied extensively [1], in which various algorithms [2], [3] have been de-veloped to generate a feasible trajectory satisfying the nonholonomic constraints. Moreover, the kinematic equations are sometimes treated as a control problem, and one tries to design the so-called kinematic controller, in which discontinuous feedback [4], [5], time-varying feedback [6], [7], or hybrid feedback [8] may be used. However, ignoring the mass and the moment of inertia of the system, which appear in the dynamic equations, may lead to the inability of following the generated trajectory effectively.
In robot motion control, torque commands enter through the dy-namical equations; therefore, it is desirable to design the controller based upon the combined kinematic and dynamic equations. In [9], a kinematic controller and a neural network computed torque controller are integrated to stabilize a nonholonomic mobile robot in which uncertainty exists. The point stabilization problems were solved in [10] and [11] by first transforming the kinematic equation into a skew-symmetric chained form, and then designing the adaptive controller for the combined system. In [12], a robust damping control scheme is proposed. Nevertheless, most of the previous works adopted Lagrangian formulation, in which Lagrange’s multipliers are included
Manuscript received March 31, 2005; revised November 4, 2005. This paper was recommended for publication by Associate Editors N. Sarkar and B. J. Yi and Editors S. Hutchinson and L. Parker upon evaluation of the reviewers’ com-ments. The work was supported in part by the National Science Council of the Republic of China (Taiwan) under Grants NSC-93-2212-E-002-026 and NSC-93-2213-E-002-052. This paper was presented in part at the 44th IEEE Conference on Decision and Control, Seville, Spain, December 2005.
P.-S. Tsai and F.-R. Chang are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: elaine@cc.chit. edu.tw; frchang@ac.ee.ntu.edu.tw).
L.-S. Wang is with the Institute of Applied Mechanics, National Taiwan Uni-versity, Taipei, Taiwan, R.O.C. (e-mail: wangli@iam.ntu.edu.tw).
Digital Object Identifier 10.1109/TRO.2006.878964
in the dynamical equation to deal with the kinematic constraints. The complexity of the problem is then raised, and the failure of the kinematic constraints may occur in numerical computation. While the projection method may be used to reduce the number of equations, a systematic approach which explores the intrinsic structure of the system may lead to a favorable form of the reduced equations. The methodology proposed in this paper serves as an example, and, based on which, the controller design may be made more effective.
Among many fundamental equations in mechanics, such as the D’Alembert–Lagrange equation, the Gibbs–Appell equation, and the Boltzmann–Hamel equations, etc., cf. [13], the Jourdain variational equation [14] in virtual velocities is more suitable to be used to treat the kinematic (velocity) constraints. In a recent paper [15], the Jourdain equation was applied to systematically derive the Appell equation, which is analogous to the equation obtained by using Kane’s approach [16]. In the method, the generalized coordinates are divided into two sets, one for privileged coordinates, and the other for nonprivileged coordinates. It was further noted that if the system is reducible, in the sense that the nonprivileged variables do not appear in the Appell equation, the reduced dynamical equation is decoupled from the kinematic equation.
By extending the results of [15], we establish in Section III a sys-tematic and structural procedure to derive the reduced dynamical equa-tions in matrix form. The equaequa-tions of motion of a tri-wheeled mobile robot described in Section II are then derived. For coupled multibody systems, this approach reduces the complexity significantly over con-ventional methods, such as Lagrangian formulism. Moreover, if the system is reducible, we may design a controller for the reduced dy-namics to steer the privileged coordinates without interlacing with the kinematics. Based on this structure, the proposed hierarchical tracking control discussed in Section IV is composed of three levels. On the top, according to mission requirements, NMP methods are employed to generate a feasible path where the kinematic constraints are satis-fied. The desired trajectories of the privileged variables are, however, changed in the middle level by a kinematic compensator, according to the current configuration of the system. The updated values are then fed into the dynamic controller in the bottom level. Equipped with a good kinematic compensator and dynamic controller, the proposed scheme can produce an effective tracking controller for nonholonomically con-strained mechanical systems.
After suitably choosing the privileged velocities, the system of a tri-wheeled mobile robot is rendered reducible, and the above-described methodology is applicable. A fuzzy logic system is used to implement the kinematic compensator, and an adaptive sliding-mode controller is adopted in the dynamic level to accommodate uncertain parameters. The latter is possible due to the fact that the reduced Appell equation satisfies the skew-symmetric property, as proved in Section III. Simu-lation results presented in Section V demonstrate the performance of the proposed controller, which can indeed drive the system variables defined in Section IV to follow the desired trajectory asymptotically.
II. PROBLEMDESCRIPTION
The problem to be solved here is the tracking of a desired trajectory for a tri-wheeled mobile robot moving on a horizontal plane, as de-picted in Fig. 1. The system may be modeled by a platform (body 4) with massmp, widthw, and length l, attached by two fixed rear wheels (body 1 and 2) and one steering front wheel (body 3) of the same mass mwand radiusa, cf. Fig. 2. The masses of the rim of the wheel and the edge of the platform are denoted bym0wandm0p, respectively. Let 2b denote the distance between the two rear wheels, and f,rbe the respective distances from the mass center of the platformC4 to that 1552-3098/$20.00 © 2006 IEEE
Fig. 1. Three-wheeled mobile robot.
Fig. 2. Configuration of the mobile robot.
of front wheelC3and the center of the rear axleQr. The contacts be-tween the wheels and the plane are assumed to be pure rolling without side slipping.
LetfEx; Ey; Ezg be an inertial frame fixed to the plane. The trans-lational motion of bodyi (i = 1; 2; 3; 4) may be described by the po-sition of its mass center
rC
i = xiEx+ yiEy+ ziEz: (1) For the rotational motion, type (3-1-2) Eulerian angles are used such that the inertial frame is rotated tofi0i; j0i; k0i = Ezg by the heading anglei, then tofi00i = ii0; j00i; k00ig by the camber angle i, and finally to the body-fixed framefexi; eyi = j00i; ezig by the spin angle 'i. If the rotation is time-varying, the rates of change of the Eulerian angles are related to the angular velocity!!!iby
!!
!i= _iEz+ _ ii0i+ _'ij00i; i = 1; 2; 3; 4: (2) From the geometry of the system, the moments of inertia can be found to be
Ji= 12m0wa2i0ii0i+ m0wa2j0ij0i + 12m0
wa2k0ik0i; i = 1; 2; 3 (3) J4= 13m0pw2i40i04+ 13m0pl2j04j04+ 13m0p(w2+l2)k04k04: (4)
The six variables(xi; yi; zi; i; 'i; i) are used here to describe the configuration of bodyi, and therefore, there are 24 coordinates to be specified for the system. However, due to physical constraints, the number may be reduced. Since the motion is horizontal, we have z1 = z2 = z3 = a and z4 = h. The assumption that the platform is kept horizontal yields'4 = 4 = 0, and hence, the body frame fex
4; ey4; ez4g coincides with fi004; j004; k004g and fi04; j04; k04g. If the wheels are properly aligned, the camber angles for the wheels are zeros, i.e., 1 = 2 = 3 = 0, so that fi00i; j00i; k00ig coincides with fi0i; j0i; k0jg, fori = 1; 2; 3. Moreover, it is assumed that the vehicle is driven by the front wheel. Additional constraints on the heading angles of the plat-form and rear wheels are thus imposed as1 = 2 = 4( ). The geometry of the interconnected bodies (cf. Fig. 2) further requires that
x1= x30 cos 0 b sin ; y1= y30 sin + b cos ; x2= x30 cos + b sin ; y2= y30 sin 0 b cos ; x4= x30 cos + rcos ; y4= y30 sin + rsin where = r+ f. Finally, the conditions that the wheels roll without slipping is realized by the following velocity constraints:
_x1= a _'1cos ; _y1= a _'1sin ; _x2= a _'2cos ; _y2= a _'2sin ;
_x3= a _'3cos( + ); _y3= a _'3sin( + ) (5) where (= 30 ) denotes the steering angle of the front wheel.
Due to the geometric constraints, the previous set of six velocity constraints is not independent. In term of the coordinates ofQr, i.e., xr = x3 0 cos and yr = y3 0 sin , which shall be used to describe the motion, the set of constraints in (5) can be converted to the following five independent ones:
a _'1+ 2b _ = a _'2 (6)
_xrsin 0 _yrcos = 0 (7)
_xrcos + _yrsin 0 b _ = a _'1 (8) _xrsin( + ) 0 _yrcos( + ) 0 _ cos = 0 (9) _xrcos( + ) + _yrsin( + ) + _ sin = a _'3: (10) Note that Condition (6) can be integrated to yield a geometric con-strainta'1+ 2b = a'2+ (a constant). As a result, the system is subject to 18 geometric constraints and 4 nonholonomic constraints. The dimension of the system is, hence, dropped to six, and we choose (xr; yr; ; '1; '3; ) to be the generalized coordinates, due to their ap-pearances in (7)–(10). In these generalized coordinates, the angular ve-locities in (2) can be expressed as
!! !1= _Ez+ _'1j001 !! !2= _Ez+ ( _'1+ 2b _=a)j002 !! !3= ( _ + _ )Ez+ _'3j003 !! !4= _Ez: (11)
Based on the above framework, the desired trajectory to be tracked may be specified by(xrd(t); yrd(t); d(t); '1d(t); '3d(t); d(t)), for t 2 (0; tf). To achieve the objective of trajectory tracking, two control torques 1 and 2 are exerted on the spin and steering angle of the front wheel, respectively. The torques affect the motion through the dynamical equations, which shall be derived below by using the Appell equation.
III. REDUCEDAPPELLEQUATION A. Reduced Appell Equation for Coupled Rigid Bodies
Consider a mechanical system consisting ofN interconnected rigid bodies. The motion of bodyj with mass mj may be described by the translation of its mass centerCj,rCj(t) = [xj(t)yj(t)zj(t)]T,j = 1; 2; . . . ; N, and a rotation about Cj represented by8j(t) 2 SO(3), the special orthogonal group, satisfying8j1 8cj = 1, det 8j = 1 (in dyadic notation, cf. [17]). Here8cjdenotes the conjugate of8j, and1 is the identity dyadic. For time-varying rotations with angular velocity !!
!j, we have
_8j = !!!j2 8j = (1 2 !!!j) 1 8j: (12) If the system is subject toK independent geometric constraints, the dimension of the system becomesn = 6N 0 K, and we may be able to express the position and attitude of each body in terms of the generalized coordinatesq = [q1; q2; . . . ; qn]T, i.e.,rCj = ^rCj(q), 8j = ^8j(q). By taking time derivatives, we obtain
_rC j = n r=1 @^rC j @qr(q) _qr; _8j = n r=1 @ ^8j @qr(q) _qr: (13) From (12) and the second equation in (13), the!!!j can be expressed as
!!!j= n r=1 Axial @@q^8j r(q) 1 ^8 c j(q) _qr (14)
whereAxial[1] denotes the axial vector of a skew-symmetric dyadic (tensor).
If the system is further constrained byL independent nonholonomic conditions, the generalized velocities _qr,r = 1; . . . ; n, may be related by
n r=1
Dsr ^rC(q); ^888(q) _qr = 0; s = 1; 2; . . . ; L: (15) LetD 2 RL2ndenote the matrix formed by the coefficientsDsr. Due to the independence of the constraints, the matrixD is of full rank, so that the degree of freedom of the system becomesm = n 0 L. One may now choosem independent generalized velocities, say _yk, k = 1; 2; . . . ; m, which shall be called the privileged velocities, so that some column operations may be performed to decomposeD into two parts: a nonsingular matrixD02 RL2L, andD002 RL2m. Denoting the remaining generalized velocities as_zs,s = 1; 2; . . . ; L, termed the nonprivileged velocities, (15) can be rewritten as
_zs= m k=1 Bsk_yk; s = 1; 2; . . . ; L; Bsk= 0 L h=1 (D0)01 shD00hk : (16) Conceptually, the generalized coordinates refer to independent coor-dinates corresponding to the geometric constraints, whereas the privi-leged velocities are the independent velocities with respect to the non-holonomic constraints. The term “privileged coordinate” has been used in [18] to represent independent coordinates in the set of Lagrangian coordinates and quasi-coordinates.
Under this setting, the generalized velocities can be expressed as _qr=
m k=1
Ark_yk; r = 1; 2; . . . ; n: (17) By substituting (17) into (13) and (14), we obtain
_rC j = m k=1 jk(y; z) _yk; !!!j = m k=1 $jk(y; z) _yk (18) where jk= n r=1 @^rC j @qrArk $jk= n r=1 Axial @@q^8j r(q) 1 ^8 c j(q) Ark (19)
are functions ofy and z in general. Taking the time derivative of (18), the accelerations can be expressed as
rC j = m k=1 ( _ jk_yk+ jkyk); !_!!j = m k=1 ( _$jk_yk+ $jkyk): (20) Now the Appell equation derived in [15] is invoked to establish the equations of motion, fork = 1; 2; . . . ; m
N j=1
mjrCj(q; _q; y) 0 Faj 1 jk(q)+
Jj1 _!!!j+ !!!j2 Jj1 !!!j0 Taj (q; _q; y) 1 $jk(q) = 0: (21) Here,Faj denotes the total applied force,Tajis the total applied moment aboutCj, and the moment of inertia aboutCj can be expressed in dyadic form as Jj= B r0 j1r0j 10rj0r0j dmj= B 12r0 j c1 12r0j dmj (22)
wherer0jdenotes the relative position vector fromCjto the position of the mass elementdmj. In this final form, this set of Appell equations is, in fact, equivalent to the so-called Kane equation in the literature, cf. [19], which is deemed very useful in dealing with complex multibody systems.
If (21) contains only the privileged variables, the system is called re-ducible, in the sense that the dynamic equation for the privileged vari-ables is decoupled from the kinematic conditions (16). Such decoupling paves the way for the design of hierarchical controller proposed in this paper.
B. Reduced Appell Equation in Matrix Form
For a reducible system, the Appell equation (21) can be expressed in matrix form as
M(y)y + C(y; _y) _y = T(y) (23)
where the components of the matrices are Mrs= N j=1 (mj jr1 js+ $jr1 Jj1 $js) (24) Crs= N j=1 mj_ js1 jr+ _$js1 Jj1 $jr + m k=1 $jr1 ($jk2 Jj1 $js) _yk (25) Tr= N j=1 Fja1 jr+ Taj1 $jr : (26) This set of equations possesses the following intrinsic properties.
Lemma 1: M(y) is an m 2 m positive-definite symmetric matrix. Lemma 2: The matrix55 = _5 M(y) 0 2C(y; _y) is skew-symmetric.
Proof: From (24) and (25), we obtain 5rs= N j=1 (mj_ jr1 js0 mj jr1 _ js) + N j=1 ( _$jr1 Jj1 $js0 $jr1 Jj1 _$js) + N j=1 $jr1 _Jj1 $js0 2 N j=1 m k=1 $jr1 ($jk2 Jj) 1 $js_yk: (27) The time derivative ofJj can be derived from (22) as
_Jj= B (1 2 !!!j) 1 1 2 r0j 0 1 2 r0j 1 (1 2 !!!j) c1 1 2 r0j dmj + B 1 2 r0 j c1 (1 2 !!!j) 1 1 2 r0j 0 1 2 r0 j 1 (1 2 !!!j) dmj = B (1 2 !!!j) 1 1 2 r0j c1 1 2 r0j dmj 0 B 1 2 r0 j c1 1 2 r0j 1 (1 2 !!!j)dmj = !!!j2 Jj0 Jj2 !!!j (28)
in which the formula _r0j = !!!j2 r0j has been used. Furthermore, the substitution of (18) into the previous formula gives rise to
_Jj= m k=1
($jk2 Jj0 Jj2 $jk) _yk: (29) Equation (27) is then rewritten as
5rs= N j=1 (mj_ jr1 js0 mj jr1 _ js) + N j=1 ( _$jr1 Jj1 $js0 $jr1 Jj1 _$js) 0 N j=1 m k=1 $jr1 ($jk2 Jj+Jj2$jk)1$js_yk: (30) SinceJjis symmetric and$jk2 Jj+ Jj2 $jkis skew-symmetric, it follows that5 is skew-symmetric.
C. Dynamical Equations of Tri-Wheeled Mobile Robot
For the robot depicted in Fig. 1, the nonholonomic conditions in (7)–(10) may be rewritten in the form of (15), with the 426 coefficient matrixD given by
D =
sin 0 cos 0 0 0 0
cos sin 0b 0a 0 0
cos( + ) sin( + ) sin 0 0a 0 sin( + ) 0 cos( + ) 0 cos 0 0 0
:
The velocity corresponding to the sixth column, i.e., _ , must be chosen as a privileged velocity, otherwiseD0in (16) becomes singular. Since the tri-wheeled mobile robot is driven by the front wheel, one may choose _'3 as the other privileged velocity, such that _y = [ _'3 _ ]T. According to this setting, (16) for the nonprivileged velocities _z =
[ _xr _yr _ _'1]T can be derived, which leads to (17) with coefficient ma-trixA given by
A =
a cos cos a sin cos asin cos 0 bsin 1 0
0 0 0 0 0 1
T
wherea= a= and b = b=.
Substituting the previous expression ofA into (19) and applying (6), the vector ikare found to be
1' = a cos (cos 0 bsin )Ex
+ a sin (cos 0 bsin )Ey; 1 = 0 2' = a cos (cos + bsin )Ex
+ a sin (cos + bsin )Ey; 2 = 0 3' = a cos( + )Ex+ a sin( + )Ey;
3 = 4 = 0 4' = a cos cos 0 r sin sin Ex
+ a sin cos + r cos sin Ey:
On the other hand, the$ikin (19) can be derived from (11) as $1' = asin k001+ (cos 0 bsin )j001; $1 = 0 $2' = asin k002+ (cos + bsin )j002; $2 = 0 $3' = asin k003+ j003; $3 = k003
$4' = asin k004; $4 = 0:
Here, the indexk for the privileged velocity is changed to the original symbol to enhance the readability.
From the physical description in Section II, the applied forces and torques acting on each rigid body are given by
Fa1 = Fa2= Fa3 = 0mwgEz; Fa4= 0mpgEz; Ta
1 = 0; Ta2 = 0; Ta3 = 1j003+ 2k003; Ta4 = 0: Substituting the moments of inertia dyadic given in (3) and (4) and the vectors ikand$ikinto (24), (25), and (26), the reduced Appell equation of tri-wheeled mobile robot in matrix form (23) is deduced withT(y) = B(y) and
M(y) = a2I1sin2 + I2cos2 + I3 aImsin
aImsin Im
C(y; _y) = 0:5(2aI10 I2) _ sin 2 0 aIm_ cos 0 B(y) = 1 0 asin 1 I1= 3Im+ Ip+ 4(b=a)2Im+ 2mwb2+ mp2r ; Im= m0wa2=2 I2= 4Im+ 2mwa2+ mpa2 I3= (mwa2+ 2Im); Ip= mp0(w2+ l2)=3: (31) It can be checked directly that Lemmas 1 and 2 are indeed satisfied.
While one may adopt the Lagrangian formulation to obtain the equa-tions of motion of the same form, the previous process of derivation is much more systematic, and reveals the intrinsic structure of a complex multibody system subject to nonholonomic constraints. Note that the nonprivileged variablesz do not appear in (31). Hence, the system of tri-wheeled mobile robots is reducible, after the subtle selection of the set of privileged variables.
Fig. 3. Block diagram of hierarchical control design.
IV. DESIGN OFHIERARCHICALCONTROLLER A. General Methodology
The reduced Appell equation (21) and the kinematic equation (16) together govern the dynamics of the tri-wheeled robot. Since the former equation fory are decoupled from the latter one, a controller to steer y follow the desiredyd(t) may be designed individually. However, the desiredzd(t) may not be tracked if the initial condition is not perfect, or disturbances occur during the motion without taking (16) into con-sideration. To take advantage of the decoupling feature of the system, a hierarchical tracking control scheme is proposed, with its block dia-gram being shown in Fig. 3.
On the top level, the motion planner produces the desired trajectory qd(t) = [xrd(t) yrd(t) d(t) '1d(t) '3d(t) d(t)]T based on task re-quirements and the conditions of constraints. The tracking ofzd(t) is achieved by changing the desired privileged velocitiesud(= _yd) to uc= ud+ 1u, where 1u is computed through a kinematic compen-sator in the middle level. The updated desired privileged coordinates are then found,yc = yd+ 1y, where1y = 1u, which is fed along withucto the dynamical controller in the bottom level to drive the mobile robot.
It is noted that the information of all generalized coordinates are needed in the middle level, while only those of the privileged ones are required in the bottom level. This feature due to decoupling makes it possible to run the hierarchical controller at different sampling rates. In particular, for the mobile robot system, the kinematic compensator may use slower sensors such as the global positioning system (GPS) to locate the robot, and faster sensors such as the encoders may be imple-mented with the dynamic controller. It shall be shown later by simu-lation that the proposed hierarchical controller with different sampling rates is feasible.
We now apply the concept of the hierarchical control to drive the tri-wheeled mobile robot along a desired trajectory. In terms ofu1 =
_'3andu2 = _ , the kinematic equation becomes _xr= au1cos cos _yr= au1sin cos _ = au1sin _ = u2:
(32)
It can be checked that the above system with system variables (xr; yr; ; ) is differentially flat with two output variables (xr; yr), cf. [20]. For givenxrd(t), yrd(t), we have d = tan01( _yrd= _xrd),
d = tan01(!d=vd), where the reference values for the linear velocity and the angular velocity are, respectively
vd= _x2rd+ _yrd2 ; !d= yrd_x_xrd2 0 xrd_yrd rd+ _yrd2 :
The kinematic compensator is then realized by a fuzzy logic system to infer the compensations1v; 1!. Since v = a _'3cos and ! = (a=) _'3sin , a new set of reference values for the privileged variables are then computed as
uc= 1v + va d 2 + (1! + !a d) 2; d dt tan01 (1! + !d) (1v + vd) (33) yc= 1v + va d 2 + (1! + !a d) 2; tan01(1! + !d) (1v + vd) (34)
which is fed into a sliding-mode controller to track the corrected desired values, cf. Section IV-C.
B. Fuzzy Logic Compensator in Kinematic Level
In steering a mobile robot, the position and the heading (namely, the posture) of the platform are the key variables to be tracked. The deviation of the current posture to the desired one can be represented by, cf. Fig. 4 e e e = cos sin 0 0 sin cos 0 0 0 1 xrd0 xr yrd0 yr d0 : (35)
Denote the length of the vector from the current position to the de-sired one byde = e2+ 2e, and the corresponding angle by#e = tan01(
e=e). The latter shall be termed the line-of-sight angle, and is different from the heading angle errore. To reduce these errors, so that the nonprivileged coordinates in the set of system variables can be tracked, a fuzzy logic system is designed to serve as a kinematic
Fig. 4. Current versus desired configuration of the mobile robot.
Fig. 5. Membership functions of input and output variables.
compensator. Similar ideas have been used in [21], among others, in dealing with the autonomous vehicle navigation problems.
In the design, the sets of input and output variables are selected to be (de,#e,e) and (1v, 1!), respectively. According to drivers’ ex-periences, the membership functions for these variables after scaling by suitable factors are chosen as those shown in Fig. 5. The symbols Z, P, N, S, M, B represent zero, positive, negative, small, medium, big, respectively. Based on these membership functions, a total of 2823 in-ference rules are used to find the intermediate values of1v and 1!, as
listed in Table I. The final compensations are determined by invoking the defuzzification process with the center-of-gravity scheme.
It is noted that botheand#eare taken into account in the design. This is because if onlyeis considered, the line-of-sight angle can not be adjusted and the vehicle may trace a trajectory parallel to the refer-ence one. On the other hand, if only#eis included, the compensated value1! may exhibit a chattering phenomenon, which may lead to zig-zag motions of the vehicle. The concept of line-of-sight angle has been used frequently in proportional navigation for ballistic missiles [22]. Its inclusion in the algorithm here can significantly reduce the steady-state errors.
C. Adaptive Sliding-Mode Control in Dynamic Level
After scaling the compensations1v, 1! obtained from the fuzzy system, a new set of desired privileged variables is computed from (33) and (34), respectively. They are to be tracked by exerting in the re-duced dynamics (23), which is rewritten as
_y = u; M(y) _u + C(y; u)u = B(y) (36) where it is assumed that some parameters such as masses, moments of inertia, and physical specifications, etc., are not known exactly.
To design a controller accommodating the uncertain parameters such that the tracking errors ~u(t) u(t) 0 uc(t) and ~y(t) y(t) 0 yc(t) approach zero asymptotically as t ! 1, the idea of adaptive sliding-mode control [23] is applied. In the method, the sliding variable s is defined as
s = [s1 s2]T = ~u + 333~y = u 0 us (37) whereus= [us1us2]T = [ _'3s _ s]T = uc0 333~y and 33 is a positive-3 definite matrix. Let222 = [a2I1I2 I3 Im]T be the vector containing uncertain parameters. It can be checked that
M(y) _us+ C(y; u)us= 111(y; u; us; _us)222 (38) where the regressor matrix111 in the above linear parametric form [11] is given by the equation shown at the bottom of the page.
From (36) and (38), the evolution equation of the sliding variable is found to be
M(y)_s + C(y; u)s = B(y) 0 111(y; u; us; _us)222: (39) The problem is then converted to find , which depends on the unknown parameter vector222, such that s ! 0. Let ^22 denote the estimate of 22 2.2 By adopting the following control and adaptive laws:
= B#(y) 111(y; _y; u
s; _us) ^222 0 Kss _^222 = 00000111(y; _y; u1
s; _us)Ts
(40)
whereKsand000 are positive-definite matrices, and B#is the left in-verse ofB, one can show by Lyapunov analysis [24] that u ! ucand y ! ycasymptotically. It is seen that the nonprivileged variables in
_us1sin2 +0:5us1u2sin 2
_us1cos2
00:5us1u2sin 2 _us1 a_us2sin
TABLE I
FUZZYRULES FORKINEMATICCOMPENSATOR
z do not appear in (40), and hence, the dynamic controller can operate without the information ofz.
V. SIMULATIONRESULTS
To examine the effectiveness of the proposed hierarchical control strategy, computer simulations were performed. The system parameters of a large mobile robot shown in Fig. 2 are selected asa = 0:3 m, r= 0:5 m, f = 0:75 m, l = 1:75 m, w = 1:5 m, mp = 20 kg, m0p = 6 kg, mw= 1 kg, m0w= 2 kg. The desired trajectory (xrd(t); yrd(t)) is first obtained by finding a cubicB-spline curve, cf. [25], passing through 11 intermediate points, {(0,0), (2,9), (5,16), (9,20), (13,18), (14.5,15), (17,11), (22,13), (22,20), (17,22), (13,18)}. Next, the desired values of other system variables(d(t); d(t)) are computed. Based on the current status of the vehicle, the fuzzy inference engine discussed in Section IV-B with rules given in Table I is then invoked. The scale of1v is proportional to vd with factor 0.6, while that of1! is in-versely proportional tovdwith factor 5. The compensationsucandyc are then computed from (33) and (34), and fed into the sliding-mode tracking controller described in Section IV-C. The parameters in the control scheme and adaptive law (40) are chosen by testing asKs = diagf10; 10g, 3 = diagf4; 4g, 0 = diagf20; 10; 100; 1000g to as-sure appropriate convergence rates. In the adaptive scheme, the initial condition ^222(0) = [0:1 0:1 0:1 0:001]T is used, which is different from the true value222true= [1 2:3 0:3 0:09]T. Moreover, we set the initial posture asxr(0) = 10; yr(0) = 0, and (0) = 0o, comparing with the desired values (0; 0; 45o).
For the above-described scenario, simulations were performed such that the kinematic compensator, the dynamic controller, and the system simulator were executed in different sampling rates: 10 Hz, 100 Hz, and 1 kHz, respectively. The results are shown in Figs. 6 and 7. In Fig. 6, the solid line is the desiredB-spline curve, and the shaded block line shows the tracking performance. It is observed that while the initial condition is significantly away from the desired configuration in both position and heading, the proposed hierarchical controller can successfully steer the mobile robot back to the desired trajectory. The tracking errors of the system variables(xr; yr; ; ) are plotted in Fig. 7. It is seen that not only the errors of the privileged coordinates, but also those of the non-privileged coordinates, in the set of system variables are driven asymp-totically to zero. Furthermore, simulations were also conducted in the presence of sensor errors, such as 10 cm in position and 5in angle, for which the hierarchical controller works quite well.
Fig. 6. B-spline tracking trajectory.
VI. CONCLUSIONS
In this paper, a hierarchical tracking controller was designed for a tri-wheeled mobile robot based on the model established using Appell’s equation. Taking advantage of the decoupling feature of the system, all the system variables can be steered to their respective desired values, even if there are some unknown parameters and measurement errors. Incorporating the practical driving experiences in the design of fuzzy rules, the reference trajectory for the dynamic controller is changed through the kinematic compensator. The skew-symmetric property of the reduced Appell equation proved in Section III assures that the selected adaptive controller can asymptotically steer the privileged variables. This methodology successfully integrates the kinematic constraints and the dynamics to generate practical control command to the mechanical system. Based on our design, it is possible to run the controller and the compensator at different sampling rates, as shown in Section V, which may not be feasible for the previous designs of two-stage controllers, such as in [10] and [11].
The success of the hierarchical design proposed here relies heavily on the decoupling of the dynamical equations from the kinematic con-straints. The separation of coordinates in the early stage into privileged versus nonprivileged ones is essential to the design. Using of the struc-tured form of the Appell equation not only reduces the complexity in the modeling of coupled mechanical systems, but also helps to estab-lish a suitable model for control. This shows the merit of closely in-tegrating the methods of mechanics and those of controls in robotics. One may also adopt other types of controllers, such as a neural net-work controller or robust controller, to steer the privileged variables or to compute the compensations. Applications of the proposed method to more complicated mechanical systems using various types of con-trollers are currently under investigation.
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Stability Analysis of a Vision-Based Control Design for an Autonomous Mobile Robot Jean-Baptiste Coulaud, Guy Campion, Georges Bastin, and
Michel De Wan
Abstract—We propose a simple control design allowing a mobile robot equipped with a camera to track a line on the ground. The control algo-rithm, as well as the image-processing algoalgo-rithm, are very simple. We dis-cuss the existence and the practical stability of an equilibrium trajectory of the robot when tracking a circular reference line. We then give a com-plementary analysis for arbitrary reference lines with bounded curvature. Experimental results confirm the theoretical analysis.
Index Terms—Control-design analysis, mobile robot, path tracking, vi-sual servoing.
I. INTRODUCTION
The problem addressed in this paper is the feedback-control design allowing a mobile robot to track a line on the ground using visual feedback. There exist a lot of image-processing algorithms extracting
Manuscript received September 8, 2005; revised January 23, 2006. This paper was recommended for publication by Associate Editor D. Prattichizzo and Ed-itor L. Parker upon evaluation of the reviewers’ comments. This paper was pre-sented in part at the 16th IFAC World Congress, Prague, Czech Republic, July 2005. Color versions of Figs. 1–12 are available online at http://ieeexplore.org. This paper presents research results of the Belgian Programme on Interuniver-sity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its authors.
The authors are with the CESAME/UCL, B1348 Louvain-La-Neuve, Belgium (e-mail: coulaud@inma.ucl.ac.be; campion@inma.ucl.ac.be; bastin@ inma.ucl.ac.be; dewan@inma.ucl.ac.be).
Color versions of Figs. 1–12 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2006.878934
