Asymptotic control for wheeled mobile robots with driftless constraints

Robotics and Autonomous Systems 43 (2003) 29–37

Asymptotic control for wheeled mobile robots with driftless constraints Hyun-Sik Shim a , Yoon-Gyeoung Sung b,∗ a b

Road Traffic Information System (ROTIS) Inc., 656-3, Seongsu 1-ga, Seongdong-gu, Seoul 133-110, South Korea Department of Mechanical Engineering, Chosun University, 375 Soesuk-dong Dong-gu, Gwangju, South Korea Received 4 June 2001; received in revised form 1 December 2002

Abstract In this paper, an asymptotic control with driftless constraints is proposed for wheeled mobile robots based on empirical practice. The equations of motion of a designed mobile robot are derived. In order to obtain the desired performance for high velocity and consecutive motions, the motion requirements are defined and then an asymptotic control is developed by using the kinematic equations of the wheeled mobile robot. The convergence of the asymptotic control is presented by using Lyapunov stability theory. Even if a nonideal velocity controller is employed for driving wheels, it is shown that the numerical performance of the asymptotic control is effective and robustness with the appropriate selection of control parameters. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Wheeled mobile robot; Asymptotic control; Lyapunov stability theory; Driftless constraints

1. Introduction It is recognized that the stabilization of wheeled mobile robots with driftless constraints to an equilibrium state is quite difficult. As shown by Brockett [1], nonholonomic systems cannot be stabilized via smooth state feedback. The stabilization problem for driftless systems represents a challenge for nonlinear control theory because the linearization of the system is not controllable. Therefore, one is forced to rely on the use of strongly nonlinear techniques to stabilize the system. Results in asymptotic stability typically rely on the use of discontinuous feedback, time-varying feedback, or a combination of the two. Even if a variety of control algorithms [2–4] for the nonholonomic mobile robot have been designed, they are not readily resulting in fast and consecutive performance of mo∗

Corresponding author. E-mail address: [email protected] (Y.-G. Sung).

tions in realistic applications. For the realistic implementations, it is difficult to get good performance due to the limitations of motor output, velocity controller performance, system uncertainty etc. For instance, no matter how the velocity controller is constructed, it is difficult to design an ideal velocity controller because of the existence of velocity and acceleration limits of the actuator. Crowley [5] developed a locomotion control system with a three-layered structure by employing the concept of a “virtual vehicle”. The concept of the virtual structure is useful to construct a control algorithm which is independent of the robot. In its command system, independent control of linear and rotational motions is possible such that the trajectories of smooth curves could be obtained. Kanayama et al. [6] presented the use of reference and current configurations for vehicle control, the use of a local error coordinate system, and a PI control algorithm for linear and rotational velocities. A novel control algorithm for deter-

0921-8890/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-8890(02)00361-5

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mining a robot’s linear and rotational velocities based on Lyapunov theory is described in [7]. The algorithm has a convergence issue so that if the reference linear velocity is zero, convergence cannot be achieved. Aicardi et al. [8] presented a control algorithm which uses polar coordinates and its stability was proven with the Laypunov stability theory. For convenient analysis and motion control design, the algorithms [7,8] were reported with numerical simulations. Recently, Yang and Kim [11] presented an algorithm with sliding mode control. The algorithm showed a little zigzag movement of the mobile robot so that it took time to arrive at the final location. Astolfi [9] presented a control mechanism with an exponential stabilization of the kinematic and dynamic model of a simple wheeled mobile robot by using a discontinuous, bounded, time invariant, state feedback control law. He evaluated it under modeling error and noisy measurement conditions. However, it was necessary to evaluate Astolfi’s result in the case of high velocity and acceleration, due to the zigzag motion of a mobile robot, caused by output noise, instead of noisy measurements. The development of a control algorithm in this paper is empirically induced by making robot kinematics at the controller design stage as simple as possible. Therefore, an asymptotic controller is proposed for autonomous mobile robots with high velocity operations. On the contrary to previous approaches to control design, the control algorithm is designed by focusing on the motion requirements of the robot’s desired trajectory and physical limitations. Then, the stability of the proposed controller is addressed by using the Lyapunov stability theory and its effectiveness and robustness are numerically demonstrated with the worst case scenario of velocity controllers. The remainder of the paper is organized as follows. In Section 2, the notations are defined and the system equations are presented. Section 3 addresses empirical motion requirements and the convergence of the proposed control algorithm. In Section 4, numerical evaluation illustrates the effectiveness and robustness of the control law. Finally, the conclusion follows.

Fig. 1. Top view of the mobile robot structure.

kinematic equations, at first, consider a mobile robot located on a 2D plane in which a global Cartesian coordinate system is defined in Fig. 1. It has two driving wheels fixed to the axis that passes through C. The two fixed wheels are operated independently by motors. There are global coordinates X–Y and local coordinates x–y attached to the center of the robot body as shown in Fig. 2. The robot has 3 degrees of freedom in its relative positioning which is represented by the configuration p = [x, y, θ]T ,

(1)

where the heading direction θ is taken in a counterclockwise direction from the X-axis to x-axis. The

2. System equations with driftless constraints The kinematic and dynamic equations are presented with nonintegrable kinematic constraints. For

Fig. 2. Mobile robot configuration (subscript “c” refers to the “current state”).

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angle θ indicates the orientation of the robot and the wheels. The set (x, y) represents the robot position. In Fig. 1, the vector (xc , yc , θc )T corresponds to the current configuration of the robot. The robot motion control is given with its linear velocity v and angular velocity ω. The robot kinematics are defined by the Jacobian matrix J(θ):   cos θ 0   v p˙ =  sin θ 0  = J(θ)q. (2) ω 0 1

H(pc )q˙ c + G(pc , qc ) = τ,

As with the mobile robot configurations in the control implementation, there are two configurations such as the reference pr = (xr , yr , θr )T which is the target configuration and the current configuration pc = (xc , yc , θc )T which is the true configuration at the current moment. By transforming the position of the robot in Cartesian coordinate into polar coordinate, the configuration error is expressed as   re cos φe   pec = R(θ)(pr − pc ) =  re sin φe  , (3) θe

A control structure is composed of an asymptotic controller and a velocity controller as shown in Fig. 3. If a reference configuration pr is given, the controllers command the mobile robot to take the reference configuration. The asymptotic controller makes the robot proceed to the reference configuration through a velocity command qm = (vm , ωm )T . Then, the velocity controller makes the robot follow the command qm . At this instance, the velocity controller is affected by the dynamics of the mobile robot. In this paper, an asymptotic controller is presented under the assumption that the velocity controller is provided using an asymptotic control scheme [12]. If the velocity controller is robust to the environmental effects such as friction, unknown disturbances and motor saturation, it could assume that a command velocity becomes a robot’s velocity: qm = qc . Finally, pc is obtained by using Eq. (2) and the integration of time.

where re > 0 is the distance error, θe = θr − θc the orientation error with respect to pc , and φe the angle measured from the robot principle axis to the distance error vector re . The configuration velocity error dynamics [8] with definition of pec = (re , φe , θe )T is expressed as   −vc cos φe vc   sin φe  . p˙ ec =  −ωc + (4) re −ωc Using Euler–Lagrangian formulation [10], the dynamic equations are expressed as

(5)

where H2×2 is the inertia matrix, G2×2 the matrix to centrifugal and Coriolis forces, τ = (τL , τR )T the torque vector applied to the left and right driving wheels, and qc = (vc , ωc )T . Eq. (5) are employed to numerically simulate the mobile robot.

3. Asymptotic controller design

3.1. Requirements for motion trajectory The design requirements based on the configuration velocity error dynamics Eq. (4), to follow a moving trajectory or a velocity command, are presented as follows:

Fig. 3. Schematic control diagram (Jacobian matrix J of Eq. (2)).

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(1) If the moving distance to the reference configuration is large, then move fast. Reduce speed as it comes closer. (2) It should take a least arriving time to arrive at a final configuration. (3) If the angle error φe is large, the robot’s linear velocity should be reduced so that the robot will have a minimum moving distance. (4) If the reference configuration is far-off, the distance error needs to be reduced first. (5) The robot should meet a reference configuration immediately. These requirements require the motion controller to generate an optimal moving trajectory with a minimum configuration error. The motion controller is designed using the vector ( re φe θe )T and the velocity command to the robot is generated based on the above requirements. For the first and second requirements, a linear velocity controller with vm = kr re is considered. This equation is similar to the equation proposed by Aicardi et al. [8]. When the distance is very large, the velocity command will be very high and the robot velocity will be saturated. Instead of re , a new variable re is introduced to take care of such a situation: π  r e ≥ Rm ,  , 2  re = πre   , otherwise. 2Rm A ‘sine’ relation instead of a linear one is employed so that the velocity becomes higher even with a smaller re : vm = kr sin re . For the third specification, the angle error φe is used in controlling the linear velocity. To apply an inversely proportional relationship for φe , the following equation is employed: vm = kr sin re cos φe . The following variables are defined for a similar purpose:  1, −Φm ≤ φe ≤ Φm ,   d = −1, φe < −π + Φm or φe > π − Φm ,   0, otherwise,

Fig. 4. Control parameter assignment to the robot’s location.

 π  φe − π, φe > ,   2  φe = φe + π, φe < − π ,   2   φe , otherwise,  π  θe − π, θe > ,   2  θe = θe + π, θe < − π ,   2   θe , otherwise, where Φm is an arbitrary constant between 0 and π/2. By introducing d, φe and θe , it is possible to make the robot move back and forth because the front and rear sides of a mobile robot are not distinguishable. Thus, the linear velocity controller law with newly assigned variables as in Fig. 4 is proposed as vm = kr d sin re cos φe ,

(6)

where kr is a positive constant with the unit of velocity and d refers to the direction of the robot so that the robot moves in a forward direction with d = 1 and in a backward direction with d = −1. Because there is no difference between the front and rear sides of a mobile robot, only the regions re ≥ 0, −π/2 ≤ φe ≤ π/2 and −π/2 ≤ θe ≤ π/2 are considered. For both the distance error re and angle error φe , four regions are classified as in Fig. 5.

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3.2. Stability The following theorem summarizes the asymptotic control which has a variable control structure according to the robot’s error ranges. Theorem 1. Consider the kinematics of a mobile robot given by Eq. (2). The control laws in Eqs. (6) and (7) drive the robot to converge to an allowable error boundary. Proof. Choose a Lyapunov function candidate as follows: V(t) = 21 re2 + (1 − cos φe ) + (1 − cos θe ), Fig. 5. Four regions to re and φe . Region 0 (re ≤ Re ), Region 1 (−Φm ≤ φe ≤ Φm and Re ≤ re ≤ Ra ), Region 2 (−Φm ≤ φe ≤ Φm and re ≥ Ra ) and Region 3 (Φm < φe ≤ π/2, −π/2 ≤ φe < −Φm and re > Re ).

In order to regulate the angle and orientation errors, the angular velocity is uniquely employed in this paper. Based on the design concept above, a relation equation could be expressed as ωm = kθ ( sin φe + sin θe ), where kθ is a positive constant with the unit of angular velocity. When the reference configuration is at a distance, the angle error φe should be considered more to meet the fourth specification. As the robot is approaching the reference configuration, the orientation error θe will be the important factor to meet the fourth requirements. Therefore, the angular velocity control law is deliberately proposed as ωm = kθ ( sin φe + (1 − ) sin θe ),

(7)

where a variable  is defined as  0, re ≤ Re ,      0.5 − α Sgn( sin θe ) =  Sgn( sin θe + sin φe ), Re < re < Ra ,     1, otherwise, where 0 < Re ≤ Ra ≤ Rm , 0 < α < 0.5 and, the function Sgn(x) is 1 if x is positive and is −1 otherwise.

(8)

where −π/2 ≤ φe ≤ π/2 and −π/2 ≤ θe ≤ π/2. Substituting Eqs. (6) and (7) for qc with the assumption that qm = qc , the following equation is derived: V˙ (re , φe , θe ) = r˙e re + φ˙ e sin φe + θ˙ e sin θe sin 2 φe =−re vc cos φe + vc − ωc sin φe − ωc sin θe re kr d sin re =− cos φe (re2 cos φe − sin 2 φe ) re −kθ sin 2 φe  − kθ sin 2 θe (1 − ) −kθ sin φe sin θe .

(9)

Case 1 (In the case of Region 3, Φm < φe ≤ π/2, −π/2 ≤ φe < −Φm and re > Re ). In Region 3, since d = 0 and  = 1, it results in vm = 0 and ωm = kθ  sin φe as control commands so that the derivative of the Lyapunov function is only written as the function of φe and the last term of Eq. (9) is eliminated: V˙ (φe ) = −kθ sin 2 φe ≤ 0. As a result, the robot in Region 3 will move to another region. Case 2 (In the case of Region 2, −Φm ≤ φe ≤ Φm and re ≥ Ra ). Since d = 1 and  = 1 in Region 2, the controller commands are the functions of re and φe to be controlled. Therefore, the derivative of the Lyapunov function is given as

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kr sin re cos φe (re2 cos φe − sin 2 φe ) re −kθ sin 2 φe kr sin re ≤− cos φe (re2 cos Φm − sin 2 Φm ) re −kθ sin 2 φe ,

V˙ (re , φe ) = −

where it is not the function of θe . Since re , sin re and cos φe are positive and re is greater than ( sin Φm tan Φm )1/2 in Region 2 in the case of cos Φm ≤ ((1 + (R2e /4))1/2 − (Re /2)), then V˙ (re , φe ) < 0. Thus, re and φe will converge toward zero so that the robot in Region 2 moves to Region 1 or Region 3. Case 3 (In the case of Region 1, −Φm ≤ φe ≤ Φm and Re ≤ re ≤ Ra ). Since d = 1 and  = 0.5 − α Sgn( sin θe ) Sgn( sin θe + sin φe ), the control commands are functions of re , φe and θe . The derivative of the Lyapunov function is kr sin re cos φe (re2 cos φe − sin 2 φe ) re −kθ ( sin φe + sin θe )

V˙ (re , φe , θe ) = −

×( sin φe + β sin θe ), where β = (1 − )/. The first term in the above equation is negative as shown in Case 2. To prove that the second term on the right-hand side is negative, it is classified into four cases:

is stable. As the planes (0, φe , −φe )T and (0, φe , sin −1 (−(1/β) sin φe ))T are invariant sets where V˙ = 0, all the system phase trajectories converge toward one of the points asymptotically as per the local invariant set theorem [13]. To show that the unique convergent point is the origin (0, 0, 0)T among the points in the planes (0, φe , −φe )T and (0, φe , − sin −1 (−(1/β) sin φe ))T , the proposed control laws are substituted into the configuration error equation (4): r˙e = −d cos φe (kr sin re ) cos φe , φ˙ e =

d sin φe cos φe (kr sin re ) − kθ  sin φe re −kθ (1 − ) sin θe ,

θ˙ e = −kθ sin φe  − kθ sin θe (1 − ).

(11) (12)

From Eqs. (10)–(12), it is shown that none of the points, except (0, 0, 0)T , is an equilibrium point. Therefore, the errors converges to (0, 0, 0)T asymptotically and the robot moves to Region 0. Case 4 (In the case of Region 0, re ≤ Re ). If  = 0, the motion controller command is the function of re and θe . In the region, θe will converge to 0. However, it cannot guarantee the convergence of re and φe . Therefore, by the proposed motion control law, a robot located at any four regions will move to Region 0 with an orientation allowed within that region.

(i) If sin θe ≥ 0 and sin φe + sin θe ≥ 0, the second term is negative since β > 1 and sin φe + β sin θe > 0. (ii) If sin θe < 0 and sin φe + sin θe ≥ 0, it is negative since β < 1 and sin φe + β sin θe > 0. (iii) If sin θe ≥ 0 and sin φe + sin θe < 0, it is negative since β < 1 and sin φe + β sin θe < 0. (iv) If sin θe < 0 and sin φe + sin θe < 0, it is negative since β > 1 and sin φe + β sin θe < 0. Therefore, since the second term is negative in all cases except at planes (0, φe , −φe )T and (0, φe , sin −1 (−(1/β) sin φe ))T , V˙ is negative semi-definite function in Region 1. This means that the error trajectories converge toward a subset of the planes (0, φe , −φe )T or (0, φe , sin −1 (−(1/β) sin φe ))T . Since V is a locally positive definite function and V˙ is a negative semi-definite function, this system

(10)

Fig. 6. Error ranges of Re .

H.-S. Shim, Y.-G. Sung / Robotics and Autonomous Systems 43 (2003) 29–37 Table 1 Physical parameters of a mobile robot Symbol

Value

Description

M (kg) D (cm) r (cm) m (g)

0.5 3.7 1.5 10

Mass of the body Diameter of the body Radius of drive wheels Mass of drive wheels

It is noted that the proposed algorithm could not guarantee the convergence of re within Re . Re can be set within an allowable error range which in turn depends on Φm . Fig. 6 shows the nonconvergent error regions according to the robot positions. If Φm is π/2, the error is infinite, and if φm is zero, the error is zero. By using Φm < π/2, we can limit the maximum error range to ( sin Φm tan Φm )1/2 and it reduces to zero according to ( sin φe tan φe )1/2 .

4. Numerical simulation The motion controller is numerically evaluated for its effectiveness and robustness. In the numerical simulation with Eq. (5), there are two cases of ideal

Fig. 7. Motion trajectories with an ideal velocity controller (kr = 120, Φm = 70◦ , Rm = 20, kθ = 10, Ra = 20).

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velocity controller which provides qm = qc and no limitation on speed and nonideal velocity controller which has speed limitation and qm = qc . In the case of the ideal velocity controller, it is not necessary to have a velocity controller. In the case of the nonideal velocity controller, the adaptive velocity controller [14] for the mobile robot in Fig. 1 is used with the physical parameters in Table 1. The speed of both wheels is limited to 125 cm/s with a sampling time 33 ms. In the numerical simulation, the mobile robot moves from the current configuration pc = (0, 0, 0)T to a reference configuration pr . In Fig. 7, the motion trajectories of the robot are shown in respect to several reference configurations which has the unit (cm, cm, deg) in an ideal velocity controller. The reference configurations pr are assigned in a counterclockwise direction as (40, 0, −45◦ )T , (30, 30, 0◦ )T , (0, 40, 90◦ )T , (−30, 30, 135◦ )T , (−30, −30, 0◦ )T , (0, −40, 0◦ )T , and (30, −30, 90◦ )T . In Fig. 8, the motion trajectories of the robot are presented with the same reference configurations as the ideal case and the maximum speed of the two

Fig. 8. Motion trajectories with a nonideal velocity controller and the speed limit of two robot wheels set to 125 cm/s (kr = 120, Φm = 70◦ , Rm = 20, kθ = 10, Ra = 20).

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5. Conclusion

Fig. 9. Motion trajectory comparison with ideal and nonideal velocity controllers without the satisfaction of kr + Lkθ ≤ Vmax (kr = 60, Φm = 70◦ , Rm = 20, kθ = 10, Ra = 20).

In this paper, an asymptotic control for wheeled mobile robot was proposed with the convergence proof by using the Lyapunov stability theory. The control law was derived based on the empirical motion requirements for the desired trajectory of the mobile robots in order to obtain fast and consecutive movements. It was shown that the behavior of the motion controller asymptotically converges to the reference trajectory. In the numerical simulation, it was shown that the performance of the asymptotic control is effective without zigzag movement in both ideal and nonideal velocity controllers for two driving wheels with the appropriate selection of control parameters.

Acknowledgements robot wheels is limited to 125 cm/s. It is shown that the motion controller satisfies control objectives under a nonideal situation without zigzag motion. Due to the unsatisfaction of the speed limit condition with regard to control parameter selection, the motion trajectories with the speed constraint are not smooth unlike the case of the ideal velocity controller. Therefore, if the wheel speed does not satisfy kr + Lkθ ≤ Vmax , the motion trajectories in Fig. 9 are not smooth unlike Fig. 10 which meets the condition of control parameter selection.

Fig. 10. Motion trajectory comparison with ideal and nonideal velocity controllers with the satisfaction of kr +Lkθ ≤ Vmax and the speed limit (kr = 60, Φm = 70◦ , Rmax = 20, kθ = 10, Rarea = 20).

This study was supported (in part) by research funds from Chosun University, 2002.

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Hyun-Sik Shim Received his B.S., M.S., and Ph.D. degrees in Electrical Engineering, Korea Advanced Institute of Science and Technology, 1991, 1993, 1998, respectively. He is working for LG Road Traffic Information System. His research interests are mobile robots and intelligent traffic control. He is member of IEEE, KITE, and ICASE.

Yoon-Gyeoung Sung Received his B.S., M.S., and Ph.D. degrees in Mechanical Engineering, Yeungnam University in 1982. Yonsei University in 1985. University of Alabama in 1997, respectively. Currently, He is at one of faculties in Chosun University. His research interests are robotics, micro/opto-mechatronics, smart structure, and nonlinear control. He is a member of ASME, KSME, and ICASE.