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The motion control of a wheeled mobile robot
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The motion control of a wheeled mobile robot夽 A.S. Andreyev, O.A. Peregudova Ul’yanovsk State University, Russia
a r t i c l e
i n f o
Article history: Received 30 September 2014 Available online xxx
a b s t r a c t The problems of stabilizing controlled motions and tracking the trajectory of a mobile robot with three roller-carrying wheels are investigated. The novelty of the results lies in the construction of control actions that solve the stabilization and tracking problems for a wide variety of programmed motions and trajectories of the robot taking into account the non-linearity and non-stationary nature of the system and its unknown mass–inertia characteristics. © 2016 Elsevier Ltd. All rights reserved.
Numerous papers (see, for example, Refs 1–8) have been devoted to investigating the motion of robots with “omnidirectional” wheels. Rollers, whose axes of rotation lie in the plane of the wheels, are attached to the wheels of a robot, enabling it to move in any direction without preliminary turning, which significantly increases the manoeuvrability of the robot. A dynamic model of a robot controlled by DC motors is considered. A corresponding simulation was previously given,1 and the problem of stabilizing steady motions was also investigated. In particular, the problem of stabilizing controlled motions of the robot has been considered.4–6,8 A control which stabilizes the programmed motion of a robot, but ensures a high rate of convergence only for small deviations from this motion has been designed, based on the calculated torque method.5 An adaptive control law,6 whose shortcoming is the complexity of the practical implementation of the algorithms devised for calculating the regulation coefficients, has been proposed. Various control laws for the kinematic model of a robot2,8 have been constructed without taking the dynamics into account. The investigations performed below are based on the use of the direct Lyapunov method and its extensions.9,10 The results obtained are distinguished by the simplicity of the verification of the stabilization and tracking conditions, as well as by the applicability of control laws with large initial deviations from the programmed motion and inexactly known mass–inertia parameters of the system.
1. Formulation of the problem of stabilizing the programmed motion and tracking the trajectory of a robot The equations of controlled motion of a mobile robot with three omni wheels (Fig. 1) in the horizontal plane under the action of torques developed by DC electric motors when there is no slipping of the wheels have the form1
(1.1) Here and are the coordinates of the centre of the robot platform in the stationary Cartesian coordinate system O; is the angle of rotation of the platform about the vertical measured from the axis; u1 , u2 and u3 are the control voltages supplied to the electric motors; a is the distance from the centre of the platform to the centre of each wheel; the constant h > 0 is determined by the torque coefficient of the
夽 Prikl. Mat. Mekh., Vol. 79, No. 4, pp. 451–462, 2015. E-mail addresses: [email protected] (A.S. Andreyev), [email protected] (O.A. Peregudova). http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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Fig. 1.
counter electromotive force c and the radius of a wheel r: h = 3c /(2r2 ); m, md and I are the components of the mass–inertia parameters of the system, which are defined in the form
where r1 is the radius of inertia of a wheel about its axis of rotation, m0 is the mass of the platform, m1 is the mass of a wheel, and 0 and 1 are, respectively, the radii of inertia of the platform and a wheel about the vertical axis, passing through their centre of mass. Suppose there are three functions 0 (t), 0 (t) and 0 (t), which are bounded and twice continuously differentiable for all t ≥ 0, and 1 1 2 1 2 suppose there are positive constants max , 1max , max , max , max , 2max , and max , which are such that the following inequalities hold for all t ≥ 0:
(1.2) We will first examine the statement of the problem of stabilizing the programmed motion of a robot (1.3) which can be accomplished according to Eqs (1.1) under the action of the programmed control
where
We introduce the deviations from the programmed motion (1.4) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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To solve the problem of stabilizing programmed motion (1.3), it is necessary to find a continuous feedback control law (1.5) which is such that programmed motion (1.3) of system (1.1) will be uniformly asymptotically stable. We will now formulate the problem of tracking the trajectory of the robot when its mass–inertia characteristics are inaccurately known by considering equations of motion (1.1) of the robot after making the replacement (1.6) Here m and I are unknown components of the mass of the platform and its moment of inertia, which satisfy the constraints
and we will assume that
The problem of tracking trajectory (1.3) of the robot is as follows: it is required to find control (1.5) and to indicate the constraints imposed on the parameters of the system and the trajectories under which, for a certain number ␥ > 0 (the tracking error), an instant of time t* = t*(␥) > 0 can be found, which is such that for all t ≥ t* the inequality (1.7) will hold, if at the initial instant of time t = t0 ≥ 0 the values xj (t0 ) and x˙ j (t0 ) belong to a certain vicinity of the zero point
2. Solution of the problem of stabilizing the programmed motion To solve the problem of stabilizing programmed motion (1.3) of system (1.1), we will seek control law (1.5) in the form of the continuous function
(2.1) Here
> 0, and i ≥ 0 (i = 1, 2) are certain constants. In the notation adopted the following theorem regarding stabilization of the programmed motion of a robot holds. Theorem 2.1.
Suppose the constants 1 and 2 in expressions (2.1) are such that the following inequalities hold: (2.2)
Then control (2.1) solves the problem of stabilizing programmed motion (1.3) of system (1.1). Proof.
In deviations (1.4) we write system (1.1) when expressions (2.1) are taken into account in the form
(2.3) For system (2.3) we take the positive-definite Lyapunov function in the form
(2.4) We calculate its derivative with respect to time by virtue of system (2.3): (2.5) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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This function will be quadratic form with respect to the variables x˙ 1 , x˙ 2 , x˙ 3 , if condition (2.2) holds. It is not difficult to a negative-definite prove that the set V˙ = 0 does not contain solutions of the limiting system for (2.3) apart from the zero solution, xj = x˙ j = 0 (j = 1, 2, 3). The limiting system is obtained when the functions ˙ (t), ˙ (t) and ˙ (t) are replaced in system (2.3) by the corresponding limiting ∗
∗
0
0
0
functions ˙ 0 (t), ˙ ∗0 (t) and ˙ 0 (t), which are defined as follows for the sequence tk → +∞:
Using the theorem of uniform asymptotic stability,9 we find that the zero solution
of system (2.3) is uniformly asymptotically stable. Thus, control (2.1) ensures stabilization of programmed motion (1.3) of system (1.1). Remark 2.1. The control law found (Eqs (2.1)) solves the problem of stabilizing programmed motion (1.3) for any initial deviations, i.e., it ensures uniform asymptotic stability of the zero solution of the system under deviations (2.3) as a whole. 3. Tracking a robot trajectory A control law in the form of a relay function. To solve the problem of tracking the assigned trajectory (1.3), we will first seek control law (1.5) in the form of the relay function
(3.1) where > 0, i > 0 (i = 1, 2) and j > 0 (j = 1, 2, 3) are constants. We introduce the notation
The following theorem regarding the tracking of trajectory (1.3) holds. Theorem 3.1.
Suppose we can find the constants
which are such that the following inequalities hold:
(3.2)
Then control (3.1) solves the problem of tracking trajectory (1.3) of system (1.1) under replacement (1.6) with an a priori assigned tracking error, i.e., the deviations from the tracked trajectory tend asymptotically to zero: xj (t) → 0 when t → +∞ (j = 1, 2, 3). The set of initial perturbations (3.3) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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satisfies the constraints
(3.4) Proof.
We introduce the variables (3.5)
Then, under deviations (1.4) and (3.5), after making replacement (1.6) and taking into account relations (3.1), we write system (1.1) in the form
(3.6) We take the Lyapunov vector function in the form
(3.7) and estimate its right-hand derivative with respect to time from system (3.6):
Consider the behaviour of Lyapunov vector function (3.7) along the solution of system (3.6) that satisfies the initial condition
(3.8) When inequality (3.2) holds, we obtain the estimate
(3.9) It follows from inequalities (3.9) that there is an instant of time t* > 0, which is such that the following inequalities will hold for all t ≥ t*:
Hence we find that V1 → 0 and V2 → 0 when t → +∞. In other words, control (3.1) solves the problem of tracking trajectory (1.3) of system (1.1), (1.6) with any a priori assigned tracking error. Remark 3.1. Control (3.1) has a discontinuous form, as a consequence of which high-frequency fluctuations (chatter) appear in the system when it is constructed in practice, due to the imperfect nature of the switching systems, the influence of perturbations and the presence Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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of small delays in the transmission of the signals. A continuous analogue of control (3.1), which provides a way to reduce this chatter, is found in the next subsection. A control law in the form of a continuous function. To solve the problem of tracking the assigned trajectory (1.3), we will seek control law (1.5) in the form of the continuous function
(3.10) where
(3.11) The saturation function sat(z) with a high angle of inclination of the switching line (when the constant ␥ is sufficiently small) is a continuous approximation of the relay function. Thus, the choice of a control law in the form (3.10), on the one hand, enables us to reduce the amplitude of the fluctuations appearing when the control law is implemented in practice compared to the relay law and, on the other hand, it ensures the property of robustness towards variations in the parameters of the system. The following theorem regarding the tracking of an assigned robot trajectory holds (the notation adopted in the formulation of Theorem 3.1 is used). Theorem 3.2. Suppose the conditions of Theorem 3.1 hold and suppose, in addition, there is a number ␥, which is such that 0 < ␥ < min{␦1 , ␦2 } and the following inequalities hold:
(3.12) Then control (3.10) solves the problem of tracking trajectory (1.3) of the robot with a tracking error that does not exceed the number ␥. Then the set of initial perturbations (3.3) satisfies constraints (3.4). Proof. Introducing the variables (3.5) and taking into account expressions (3.10), we obtain a system of equations in the deviations (1.4) and (3.5), which is similar to system (3.6), where the sign functions are replaced by the corresponding saturation functions (3.11). When inequalities (3.12) hold, for the values V1 and V2 which are such that
we obtain the estimate
Hence it follows that there is an instant of time t* = t*(␥) > 0, which is such that inequality (1.7) will hold at all t ≥ t*. Remark 3.2. Theorems 3.1 and 3.2 remain valid when the error of the wheel mass measurements is non-zero, i.e., when m1 = m10 + m1 , where m1 is an unknown component of the wheel mass, which is such that
and m10 < m10 . Then, in conditions (3.2), (3.3) and (3.14), (3.15) of Theorems 3.1 and 3.2 the parameter md must be replaced by the value of md + md0 , where
Theorems 3.1 and 3.2, which have been proved, enable us to solve tracking problems for a broad class of non-stationary trajectories of a wheeled robot under fairly easily tested conditions and thereby shorten the calculation time.
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Fig. 2.
4. Numerical simulation results Application of control law (2.1). Control law (2.1) was used in the numerical simulation of the controlled motion of a robot with the following values of the parameters: (4.1) The programmed motion considered was the motion when the centre of mass of the robot moves rectilinearly with the constant velocity V and the platform rotates about the centre of mass with the constant angular velocity : (4.2) A control which ensures this motion was found for stationary motion (4.2) of the robot, but the issue of stabilization was not investigated.1 The following constraint on the control parameters 1 and 2 was found using Theorem 2.1: (4.3) √ Analysing condition (4.3), we note that if the programmed velocity of the centre of mass of the robot is bounded by the inequality V < 2 2ah/md , the control parameters 1 and 2 can be set equal to zero, and stabilization of motion (4.2) will be ensured for any positive value of . Figure 2 shows the results of a numerical simulation for the values (4.1) of the parameters of the robot and (4.4) The following control parameters were selected from condition (4.3):
The integration time interval was 40 s. The initial deviations from the programmed motion found are equal to ±2. The dashed lines are graphs of the components of the programmed motion, and the solid lines are graphs of the actual motion of the robot. Figure 3 shows graphs of the time dependence of the components of control (2.1). Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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Fig. 3.
Remark 4.1. Condition (4.3) remains valid for programmed motions with the sign-variant velocities V and , i.e., when there is reversal of the direction of motion and rotation. Application of control laws (3.1) and (3.10). Laws (3.1) and (3.10) were used in a numerical simulation of the motion of a robot with parameter values (4.1) and
Fig. 4.
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Fig. 5.
for the tracked trajectory (4.5) Motion (4.5) of the robot is cylindrical precession, under which the centre of mass of the robot moves uniformly along a circle of radius R, and the platform rotates uniformly about the centre of mass. It was shown that such robotic motion exists when there is no control, and the value of the radius R was found.1 We will consider the solution of the problem of tracking trajectory (4.5) using Theorems 3.1 and 3.2. Figure 4 shows graphs of the tracked trajectory (the dashed curves) and the actual trajectory (the solid curves) under control (3.1) (the upper part of Fig. 4) and control (3.10) (the lower part). The values of the parameters of the tracked trajectory were chosen as (4.4) and R = 1 m. Using the conditions of Theorems 3.1 and 3.2, we selected the following values of the control parameters, which solve the problem of tracking trajectory (4.5):
It can be seen that under control (3.1) the centre of mass of the robot asymptotically approaches the tracked trajectory and that under control (3.10) it reaches the tracked trajectory with the tracking error ␥ within a finite time. The advantage of control (3.10) over control (3.1) is the fact that the energy expenditure required to achieve this effect under continuous law (3.10) is reduced compared with that under relay law (3.1). Figure 5 presents graphs of the time dependence of control actions (3.10). It can be concluded that during the time t = 8 s, the control actions will be restricted in absolute value to 4 N. Calculations showed that the components of control (3.1) are restricted in absolute value to 17.8 N over the entire integration time, equal to 30 s. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-33082, 15-01-08482). References 1. Martynenko YuG, Formal’skii AM. On the motion of a mobile robot with roller-carrying wheels. J Comp Systems Sci Intern 2007;46(6):976–83. 2. Velasco-Villa M, Del-Muro-Cuellar B, Alvarez-Aguirre A. Smith-predictor compensator for a delayed omnidirectional mobile robot. In: Proc 15th Mediter Conf Control and Automation. 2007. p. 1–6. 3. Zobova AA, Tatarinov YaV. The dynamics of an omni-mobile vehicle. JAMM 2009;73(1):8–15. 4. Liu Y, Zhu JJ, Williams II RL, Wu J. Omni-directional mobile robot controller based on trajectory linearization. Robotics Autonom Syst 2008;45(5):461–79. 5. Vazques JA, Velasco-Villa M. Path-tracking dynamic model based control of an omnidirectional mobile robot. In: Proc 17th World Congr. The International Federation of Automatic Control. 2008. p. 5365–70. 6. Huang NS, Tsai CC. Adaptive trajectory tracking and stabilization for omnidirectional mobile robot with dynamic effect and uncertainties. In: Proc 17th World Congr. The International Federation of Automatic Control. 2008. p. 5383–8. 7. Martynenko YuG. Stability of steady motions of a mobile robot with roller-carrying wheels and a displaced centre of mass. JAMM 2010;74(4):436–42. 8. Karavayev YuL, Trefilov SA. Deviation-based discrete control algorithm for an omni-wheeled mobile robot. Nelin Dinamika 2013;9(1):91–100. 9. Andreyev AS. On the asymptotic stability and instability of the zero solution of a non-autonomous system. JAMM 1984;48(2):154–60. 10. Andreyev AS, Peregudova OA. On the method of comparison in asymptotic-stability problems. Dokl Phys 2005;50(2):91–4.
Translated by P.S.
Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
ARTICLE IN PRESS
JAMM-2287; No. of Pages 9
Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The motion control of a wheeled mobile robot夽 A.S. Andreyev, O.A. Peregudova Ul’yanovsk State University, Russia
a r t i c l e
i n f o
Article history: Received 30 September 2014 Available online xxx
a b s t r a c t The problems of stabilizing controlled motions and tracking the trajectory of a mobile robot with three roller-carrying wheels are investigated. The novelty of the results lies in the construction of control actions that solve the stabilization and tracking problems for a wide variety of programmed motions and trajectories of the robot taking into account the non-linearity and non-stationary nature of the system and its unknown mass–inertia characteristics. © 2016 Elsevier Ltd. All rights reserved.
Numerous papers (see, for example, Refs 1–8) have been devoted to investigating the motion of robots with “omnidirectional” wheels. Rollers, whose axes of rotation lie in the plane of the wheels, are attached to the wheels of a robot, enabling it to move in any direction without preliminary turning, which significantly increases the manoeuvrability of the robot. A dynamic model of a robot controlled by DC motors is considered. A corresponding simulation was previously given,1 and the problem of stabilizing steady motions was also investigated. In particular, the problem of stabilizing controlled motions of the robot has been considered.4–6,8 A control which stabilizes the programmed motion of a robot, but ensures a high rate of convergence only for small deviations from this motion has been designed, based on the calculated torque method.5 An adaptive control law,6 whose shortcoming is the complexity of the practical implementation of the algorithms devised for calculating the regulation coefficients, has been proposed. Various control laws for the kinematic model of a robot2,8 have been constructed without taking the dynamics into account. The investigations performed below are based on the use of the direct Lyapunov method and its extensions.9,10 The results obtained are distinguished by the simplicity of the verification of the stabilization and tracking conditions, as well as by the applicability of control laws with large initial deviations from the programmed motion and inexactly known mass–inertia parameters of the system.
1. Formulation of the problem of stabilizing the programmed motion and tracking the trajectory of a robot The equations of controlled motion of a mobile robot with three omni wheels (Fig. 1) in the horizontal plane under the action of torques developed by DC electric motors when there is no slipping of the wheels have the form1
(1.1) Here and are the coordinates of the centre of the robot platform in the stationary Cartesian coordinate system O; is the angle of rotation of the platform about the vertical measured from the axis; u1 , u2 and u3 are the control voltages supplied to the electric motors; a is the distance from the centre of the platform to the centre of each wheel; the constant h > 0 is determined by the torque coefficient of the
夽 Prikl. Mat. Mekh., Vol. 79, No. 4, pp. 451–462, 2015. E-mail addresses: [email protected] (A.S. Andreyev), [email protected] (O.A. Peregudova). http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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Fig. 1.
counter electromotive force c and the radius of a wheel r: h = 3c /(2r2 ); m, md and I are the components of the mass–inertia parameters of the system, which are defined in the form
where r1 is the radius of inertia of a wheel about its axis of rotation, m0 is the mass of the platform, m1 is the mass of a wheel, and 0 and 1 are, respectively, the radii of inertia of the platform and a wheel about the vertical axis, passing through their centre of mass. Suppose there are three functions 0 (t), 0 (t) and 0 (t), which are bounded and twice continuously differentiable for all t ≥ 0, and 1 1 2 1 2 suppose there are positive constants max , 1max , max , max , max , 2max , and max , which are such that the following inequalities hold for all t ≥ 0:
(1.2) We will first examine the statement of the problem of stabilizing the programmed motion of a robot (1.3) which can be accomplished according to Eqs (1.1) under the action of the programmed control
where
We introduce the deviations from the programmed motion (1.4) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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To solve the problem of stabilizing programmed motion (1.3), it is necessary to find a continuous feedback control law (1.5) which is such that programmed motion (1.3) of system (1.1) will be uniformly asymptotically stable. We will now formulate the problem of tracking the trajectory of the robot when its mass–inertia characteristics are inaccurately known by considering equations of motion (1.1) of the robot after making the replacement (1.6) Here m and I are unknown components of the mass of the platform and its moment of inertia, which satisfy the constraints
and we will assume that
The problem of tracking trajectory (1.3) of the robot is as follows: it is required to find control (1.5) and to indicate the constraints imposed on the parameters of the system and the trajectories under which, for a certain number ␥ > 0 (the tracking error), an instant of time t* = t*(␥) > 0 can be found, which is such that for all t ≥ t* the inequality (1.7) will hold, if at the initial instant of time t = t0 ≥ 0 the values xj (t0 ) and x˙ j (t0 ) belong to a certain vicinity of the zero point
2. Solution of the problem of stabilizing the programmed motion To solve the problem of stabilizing programmed motion (1.3) of system (1.1), we will seek control law (1.5) in the form of the continuous function
(2.1) Here
> 0, and i ≥ 0 (i = 1, 2) are certain constants. In the notation adopted the following theorem regarding stabilization of the programmed motion of a robot holds. Theorem 2.1.
Suppose the constants 1 and 2 in expressions (2.1) are such that the following inequalities hold: (2.2)
Then control (2.1) solves the problem of stabilizing programmed motion (1.3) of system (1.1). Proof.
In deviations (1.4) we write system (1.1) when expressions (2.1) are taken into account in the form
(2.3) For system (2.3) we take the positive-definite Lyapunov function in the form
(2.4) We calculate its derivative with respect to time by virtue of system (2.3): (2.5) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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This function will be quadratic form with respect to the variables x˙ 1 , x˙ 2 , x˙ 3 , if condition (2.2) holds. It is not difficult to a negative-definite prove that the set V˙ = 0 does not contain solutions of the limiting system for (2.3) apart from the zero solution, xj = x˙ j = 0 (j = 1, 2, 3). The limiting system is obtained when the functions ˙ (t), ˙ (t) and ˙ (t) are replaced in system (2.3) by the corresponding limiting ∗
∗
0
0
0
functions ˙ 0 (t), ˙ ∗0 (t) and ˙ 0 (t), which are defined as follows for the sequence tk → +∞:
Using the theorem of uniform asymptotic stability,9 we find that the zero solution
of system (2.3) is uniformly asymptotically stable. Thus, control (2.1) ensures stabilization of programmed motion (1.3) of system (1.1). Remark 2.1. The control law found (Eqs (2.1)) solves the problem of stabilizing programmed motion (1.3) for any initial deviations, i.e., it ensures uniform asymptotic stability of the zero solution of the system under deviations (2.3) as a whole. 3. Tracking a robot trajectory A control law in the form of a relay function. To solve the problem of tracking the assigned trajectory (1.3), we will first seek control law (1.5) in the form of the relay function
(3.1) where > 0, i > 0 (i = 1, 2) and j > 0 (j = 1, 2, 3) are constants. We introduce the notation
The following theorem regarding the tracking of trajectory (1.3) holds. Theorem 3.1.
Suppose we can find the constants
which are such that the following inequalities hold:
(3.2)
Then control (3.1) solves the problem of tracking trajectory (1.3) of system (1.1) under replacement (1.6) with an a priori assigned tracking error, i.e., the deviations from the tracked trajectory tend asymptotically to zero: xj (t) → 0 when t → +∞ (j = 1, 2, 3). The set of initial perturbations (3.3) Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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satisfies the constraints
(3.4) Proof.
We introduce the variables (3.5)
Then, under deviations (1.4) and (3.5), after making replacement (1.6) and taking into account relations (3.1), we write system (1.1) in the form
(3.6) We take the Lyapunov vector function in the form
(3.7) and estimate its right-hand derivative with respect to time from system (3.6):
Consider the behaviour of Lyapunov vector function (3.7) along the solution of system (3.6) that satisfies the initial condition
(3.8) When inequality (3.2) holds, we obtain the estimate
(3.9) It follows from inequalities (3.9) that there is an instant of time t* > 0, which is such that the following inequalities will hold for all t ≥ t*:
Hence we find that V1 → 0 and V2 → 0 when t → +∞. In other words, control (3.1) solves the problem of tracking trajectory (1.3) of system (1.1), (1.6) with any a priori assigned tracking error. Remark 3.1. Control (3.1) has a discontinuous form, as a consequence of which high-frequency fluctuations (chatter) appear in the system when it is constructed in practice, due to the imperfect nature of the switching systems, the influence of perturbations and the presence Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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of small delays in the transmission of the signals. A continuous analogue of control (3.1), which provides a way to reduce this chatter, is found in the next subsection. A control law in the form of a continuous function. To solve the problem of tracking the assigned trajectory (1.3), we will seek control law (1.5) in the form of the continuous function
(3.10) where
(3.11) The saturation function sat(z) with a high angle of inclination of the switching line (when the constant ␥ is sufficiently small) is a continuous approximation of the relay function. Thus, the choice of a control law in the form (3.10), on the one hand, enables us to reduce the amplitude of the fluctuations appearing when the control law is implemented in practice compared to the relay law and, on the other hand, it ensures the property of robustness towards variations in the parameters of the system. The following theorem regarding the tracking of an assigned robot trajectory holds (the notation adopted in the formulation of Theorem 3.1 is used). Theorem 3.2. Suppose the conditions of Theorem 3.1 hold and suppose, in addition, there is a number ␥, which is such that 0 < ␥ < min{␦1 , ␦2 } and the following inequalities hold:
(3.12) Then control (3.10) solves the problem of tracking trajectory (1.3) of the robot with a tracking error that does not exceed the number ␥. Then the set of initial perturbations (3.3) satisfies constraints (3.4). Proof. Introducing the variables (3.5) and taking into account expressions (3.10), we obtain a system of equations in the deviations (1.4) and (3.5), which is similar to system (3.6), where the sign functions are replaced by the corresponding saturation functions (3.11). When inequalities (3.12) hold, for the values V1 and V2 which are such that
we obtain the estimate
Hence it follows that there is an instant of time t* = t*(␥) > 0, which is such that inequality (1.7) will hold at all t ≥ t*. Remark 3.2. Theorems 3.1 and 3.2 remain valid when the error of the wheel mass measurements is non-zero, i.e., when m1 = m10 + m1 , where m1 is an unknown component of the wheel mass, which is such that
and m10 < m10 . Then, in conditions (3.2), (3.3) and (3.14), (3.15) of Theorems 3.1 and 3.2 the parameter md must be replaced by the value of md + md0 , where
Theorems 3.1 and 3.2, which have been proved, enable us to solve tracking problems for a broad class of non-stationary trajectories of a wheeled robot under fairly easily tested conditions and thereby shorten the calculation time.
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Fig. 2.
4. Numerical simulation results Application of control law (2.1). Control law (2.1) was used in the numerical simulation of the controlled motion of a robot with the following values of the parameters: (4.1) The programmed motion considered was the motion when the centre of mass of the robot moves rectilinearly with the constant velocity V and the platform rotates about the centre of mass with the constant angular velocity : (4.2) A control which ensures this motion was found for stationary motion (4.2) of the robot, but the issue of stabilization was not investigated.1 The following constraint on the control parameters 1 and 2 was found using Theorem 2.1: (4.3) √ Analysing condition (4.3), we note that if the programmed velocity of the centre of mass of the robot is bounded by the inequality V < 2 2ah/md , the control parameters 1 and 2 can be set equal to zero, and stabilization of motion (4.2) will be ensured for any positive value of . Figure 2 shows the results of a numerical simulation for the values (4.1) of the parameters of the robot and (4.4) The following control parameters were selected from condition (4.3):
The integration time interval was 40 s. The initial deviations from the programmed motion found are equal to ±2. The dashed lines are graphs of the components of the programmed motion, and the solid lines are graphs of the actual motion of the robot. Figure 3 shows graphs of the time dependence of the components of control (2.1). Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002
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Fig. 3.
Remark 4.1. Condition (4.3) remains valid for programmed motions with the sign-variant velocities V and , i.e., when there is reversal of the direction of motion and rotation. Application of control laws (3.1) and (3.10). Laws (3.1) and (3.10) were used in a numerical simulation of the motion of a robot with parameter values (4.1) and
Fig. 4.
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Fig. 5.
for the tracked trajectory (4.5) Motion (4.5) of the robot is cylindrical precession, under which the centre of mass of the robot moves uniformly along a circle of radius R, and the platform rotates uniformly about the centre of mass. It was shown that such robotic motion exists when there is no control, and the value of the radius R was found.1 We will consider the solution of the problem of tracking trajectory (4.5) using Theorems 3.1 and 3.2. Figure 4 shows graphs of the tracked trajectory (the dashed curves) and the actual trajectory (the solid curves) under control (3.1) (the upper part of Fig. 4) and control (3.10) (the lower part). The values of the parameters of the tracked trajectory were chosen as (4.4) and R = 1 m. Using the conditions of Theorems 3.1 and 3.2, we selected the following values of the control parameters, which solve the problem of tracking trajectory (4.5):
It can be seen that under control (3.1) the centre of mass of the robot asymptotically approaches the tracked trajectory and that under control (3.10) it reaches the tracked trajectory with the tracking error ␥ within a finite time. The advantage of control (3.10) over control (3.1) is the fact that the energy expenditure required to achieve this effect under continuous law (3.10) is reduced compared with that under relay law (3.1). Figure 5 presents graphs of the time dependence of control actions (3.10). It can be concluded that during the time t = 8 s, the control actions will be restricted in absolute value to 4 N. Calculations showed that the components of control (3.1) are restricted in absolute value to 17.8 N over the entire integration time, equal to 30 s. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-33082, 15-01-08482). References 1. Martynenko YuG, Formal’skii AM. On the motion of a mobile robot with roller-carrying wheels. J Comp Systems Sci Intern 2007;46(6):976–83. 2. Velasco-Villa M, Del-Muro-Cuellar B, Alvarez-Aguirre A. Smith-predictor compensator for a delayed omnidirectional mobile robot. In: Proc 15th Mediter Conf Control and Automation. 2007. p. 1–6. 3. Zobova AA, Tatarinov YaV. The dynamics of an omni-mobile vehicle. JAMM 2009;73(1):8–15. 4. Liu Y, Zhu JJ, Williams II RL, Wu J. Omni-directional mobile robot controller based on trajectory linearization. Robotics Autonom Syst 2008;45(5):461–79. 5. Vazques JA, Velasco-Villa M. Path-tracking dynamic model based control of an omnidirectional mobile robot. In: Proc 17th World Congr. The International Federation of Automatic Control. 2008. p. 5365–70. 6. Huang NS, Tsai CC. Adaptive trajectory tracking and stabilization for omnidirectional mobile robot with dynamic effect and uncertainties. In: Proc 17th World Congr. The International Federation of Automatic Control. 2008. p. 5383–8. 7. Martynenko YuG. Stability of steady motions of a mobile robot with roller-carrying wheels and a displaced centre of mass. JAMM 2010;74(4):436–42. 8. Karavayev YuL, Trefilov SA. Deviation-based discrete control algorithm for an omni-wheeled mobile robot. Nelin Dinamika 2013;9(1):91–100. 9. Andreyev AS. On the asymptotic stability and instability of the zero solution of a non-autonomous system. JAMM 1984;48(2):154–60. 10. Andreyev AS, Peregudova OA. On the method of comparison in asymptotic-stability problems. Dokl Phys 2005;50(2):91–4.
Translated by P.S.
Please cite this article in press as: Andreyev AS, Peregudova OA. The motion control of a wheeled mobile robot. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2016.01.002