Motion Control Design for an Omnidirectional Mobile Robot Subject to Velocity Constraints

1Departamento de Ingenieŕıa Eléctrica y Electrónica, Instituto Tecnológico de Sonora, Campus Náinari, Antonio Caso 2266, Villa ITSON, 85130 Ciudad Obregón, SON, Mexico 2Departamento de Electrónica y Telecomunicaciones, CICESE, Carretera Ensenada-Tijuana No. 3918, Zona Playitas, 22860 Ensenada, BC, Mexico 3Departamento de Posgrado, CIDETEC, Instituto Politécnico Nacional, Sección de Mecatrónica, Juan de Dios Bátiz s/n, Nueva Industrial Vallejo, Gustavo A. Madero, 07700 México, DF, Mexico 4Universidad Autónoma del Carmen, Facultad de Ingenieŕıa, Calle 56 No. 4, Benito Juárez, 24180 Ciudad del Carmen, CM, Mexico


Introduction
Control of mobile robots has received great attention in last decades.The interest is motivated by the different potential applications in which these systems are very useful (e.g., indoor/outdoor navigation in wide areas).The fundamental control problems addressed have been regulation and trajectory tracking, and several control techniques have been proposed to successfully solve them (see, e.g., [1]).Also, the demands for mobile robots with high mobility have increased in the last years.Since the conventional mobile robot does not have high mobility due to its nonholonomic constraint, the omnidirectional mobile robot (OMR) has attracted the attention, because of its ability to move simultaneously, and independently in an arbitrary direction in the horizontal plane (translational motion) and with different orientation (rotational motion).Hence, the OMR is a kind of holonomic robot [2].Because of its Euler-Lagrange dynamics, the recognized "classical" control techniques for robot manipulators [3] have been adapted for controlling this kind of mobile robot.Recently, intensive and notable research has been devoted to this device (see, e.g., [1,[4][5][6] and references therein).
To implement the control strategies, a localization system is required to estimate the actual wheeled mobile robot positions and velocities [7], which are mainly based on the odometric data, the robot kinematics, and the Kalman filter.Nevertheless, since there exists a maximum sampling time to reconstruct the velocity of the wheels, the odometric data could have errors if the Nyquist-Shannon criteria are not satisfied.Hence, the important issue to consider during the control design is the inclusion of the physical limitations of the system, such as bounds in velocities and/or torques.For instance, high velocities developed by the mobile robot can result in a higher probability of wheel slides and deviations from the desired trajectory.In practice, the mobile robot has bounded velocities and torques which should never be exceeded.If the control strategy is designed without considering these constraints, the system could exhibit poor performance and even instability [8].Many control strategies in robotics have been proposed without considering explicitly the physical limits of the system.One of the underlying assumptions is that the robot task can be smoothly tailored in space and scaled in time so as to fit to the system limitations.However, simply scaling of the task commands could recover motion feasibility but may no longer satisfy the tracking performance [9].The above description requires off-line planning of the desired trajectory.This may result to be impractical in some applications, for example, when the desired trajectory is available while the robot is in motion or under unpredictable situations in human-robot interaction systems, in which the speed and acceleration magnitude become high during a short time interval.Online trajectory scaling schemes have been proposed in [10][11][12][13], mainly to satisfy torque limits.Moreover, simple hardware and/or software saturation of the commands results in the lack of execution of the desired motion [14].
Control of robots with bounded velocities and torques has proved to be challenging problems, and these have been addressed mainly for robot manipulators.For instance, on one hand, interesting contributions for robot manipulators with bounded torques are reported in [8][9][10][11][12][13][14][15][16][17].On the other hand, control of robot manipulators with bounded velocities is considered in [18][19][20][21][22][23][24][25].Control of mobile robots with bounded torques is reported in [26] and that with velocity constraints is reported in [27][28][29][30][31].However, among all these results, a common characteristic is that the considered system is subject to different operation conditions, and its performance is deteriorated when the signals are saturated.Also, some of them do not completely guarantee the given limits during the system transitory stage, even when the initial conditions of the states are within the specified bounds.A formal stability analysis is missed in many of these works, and with few exceptions, the results are mainly valid for set-point regulation or for the case in which the kinematic model of the system is considered.
In this paper, we focus our attention on the motion control problem of the OMR subject to velocity constraints.Different from that reported in [27][28][29][30][31], the solution proposed here consists in the redesign of a well-known tracking controller to achieve global asymptotic tracking with desired bounded velocities, considering the dynamic model of the system.The proposed control scheme includes explicitly the desired bounds for the developed Cartesian velocities, which in turn bound the maximal velocities developed by the wheels.These bounds are freely set by the user considering the physical limits of the device.The proposal includes a tracking controller (TC) combined with a passive nonlinear controller (PNC) through a negative feedback connection structure.The TC is used to achieve the desired tracking performance, while the PNC is used to ensure bounded velocities.By using Lyapunov theory and passivity tools, global asymptotic tracking with desired bounded velocities is proved.Simulations and experimental results are provided to show the effectiveness of the proposed control strategy.
The rest of the paper is organized as follows.In Section 2, the dynamics of the considered mobile robot is given, and the problem formulation is stated.The main result of the proposed control system, and its stability analysis, is presented in Section 3. Simulations and real-time experiments are given in Section 4. Finally, some conclusions and future work ideas are given in Section 5.

Problem Formulation
2.1.Dynamic Model.Consider the OMR shown in Figure 1, whose dynamics is described by Σ:  () q +  (, q ) q =  () , where  = [  ,   ,   ] ⊤ is the configuration vector, which includes linear and angular positions of the center of mass of the OMR with respect to the inertial frame (  ,   ), q is the velocity vector, q is the acceleration vector, and  ∈ R 3 is the control input which includes the torques applied to the wheels.() ∈ R 3×3 is the inertia matrix, (, q ) q ∈ R 3 is the Coriolis and centrifugal force vector, and () ∈ R 3×3 is the input matrix.These are given as ] . (2 The system parameters correspond to those of a physical prototype and are listed in Table 1.The dynamics of the OMR has the following properties [1]. (P1) The inertia matrix () and its inverse are symmetric and positive definite for all .The inverse of the inertia matrix is bounded for all  by a positive constant  as ‖(()) −1 ‖ ≤ .

Problem Statement.
Suppose that there exists a controller   which makes the system Σ to asymptotically track a desired trajectory   () (i.e., by choosing for (1) the input  =   , the closed-loop system dynamics behaves such that lim  → ∞ () =   ()).However,   has been designed without considering the velocity constraints in the system.So, we are interested in redesigning the given tracking controller to achieve asymptotic tracking with desired bounded velocities.
For  ∈ {1, 2, 3}, let q  denote the th velocity of the system Σ.Also, let the negative and positive constants q   and q   denote the desired lower and upper bounds, respectively, for the th velocity.These bounds are freely set by the user considering the physical capabilities of the system.Define the following complementary sets: Then, it is said that q  is within specifications, if and only if q  ∈ I  , while it is out of specifications if and only if q  ∈ O  .Therefore, the controller redesign to be developed must be capable of achieving asymptotic tracking with desired bounded velocities, either q  ∈ I  or q  ∈ O  .More precisely, the following objectives must be achieved: lim for  ∈ {1, 2, 3} and some finite time  ≥ 0.
Throughout the paper, it is assumed that all variables in the system are measurable, and its parameters are known.

Control Design
3.1.Tracking Controller.Given the desired vector of reference   (), define the tracking errors: and consider for the system Σ the "classical" computed torque controller [1,3]: where   ,   are constant positive definite diagonal matrices, which are chosen accordingly to the desired tracking performance (see, e.g., [3]).It is well known that the closedloop dynamics, obtained by placing ( 8) in ( 1) with  =  CT , is described by the decoupled second-order linear system: whose stability properties are set through the control gains   and   .This tracking controller allows Σ to achieve asymptotic tracking ( 4)-( 5), but not with bounded velocities (6).

A Static Nonlinear Function.
Let  : R 3  → R 3 be a continuous nonlinear vector function of the form whose elements, denoted by   ( q  ), are defined as follows: for  ∈ {1, 2, 3}.Note that the proposed function is a combination of a dead zone and a saturation, with a smooth transition interval defined by the small positive constant Δ  .Figure 2 shows the shape of   ( q  ).
Let  be a positive definite diagonal matrix.Its elements, denoted by   , are constant or possibly time-varying.In either case, they are strictly positive for all time; that is, The entries   are yet to be defined.
Proof.By definition of   ( q  ), the product q    ( q  ) is negative semidefinite for all q  , and  is positive definite (12), so Since the product is positive semidefinite, it is passive.
Proposition 2. Consider the velocity error q defined in (7), the diagonal matrix  given in (12), and the nonlinear function ( q ) defined in ( 10)- (11).Then, the product − q ⊤ ( q ) is passive if the desired velocity bounds are set freely by the user taking into account the relation: Proof.Let us write the product − q ⊤ ( q ) in the form Then the following cases are analysed.
(i) If q  ∈ I  , and if q  ∈ [ q   +Δ  , q   −Δ  ], then   ( q  ) = 0 (see (11) and Figure 2), and therefore the th element of ( 15) is zero; that is, Table 2: Positivity of the th element of (15).(14).Then, with aid of Table 2, it is concluded that the th element of ( 15) is positive definite; that is, As in the previous case, Table 2 helps us to conclude that the th element of ( 15) is positive definite; that is, ( 17) is satisfied again.
From the previous analysis, one has that ( 15) is either zero or positive definite for all q  .Then, Since the product is positive semidefinite, it is passive.

Control
Redesign.Now, to bound the velocities of the OMR, the following nonlinear controller is proposed: where the elements of the diagonal matrix  are defined as where   is a positive constant, and Note that ( 12) is satisfied with this definition of   .
To achieve asymptotic tracking with desired bounded velocities, the foregoing controller must be used simultaneously with the tracking controller.To this aim, the following bounded-velocity controller for trajectory tracking is finally proposed for the system Σ: where  CT is exactly the computed torque controller given in (8), used to asymptotically track the desired reference, while  LV is the nonlinear controller (19), used to keep the velocities within its limits.
Theorem 3. Given the OMR dynamics (1), together with its properties, and a smooth desired reference q  () to be tracked, consider desired bounds for the velocities to be developed by the mobile robot which are freely set considering (14).Then, taking for Σ the control input  =  BV , where  BV is given in (22), global asymptotic tracking with bounded velocities is achieved.
Proof.Take for Σ the control input  =  BV (i.e., place (22) in ( 1)) to obtain the closed-loop dynamics: which represents the system , shown in Figure 3, consisting of the negative feedback connection of  1 and  2 .These systems are described by with the connections where  ≡ 0. Note that  1 corresponds to (9).Hereafter, the proof is presented in two steps.First, global asymptotic tracking is proved by showing that the system  is output strictly passive and zero-state observable (see, e.g., [32,33] for definitions, properties, and results of dissipative systems).Second, bounded velocities are proved by showing that the set I  is attractive.
Passivity.We start by proving that both systems  1 and  2 are passive.
(i) First, consider the system  1 of the feedback connection shown in Figure 3 and described in (24).Take the time derivative of the (radially unbounded) positive definite storage function: along  1 to obtain The foregoing equation clearly states that  1 is output strictly passive.Moreover,  1 is zero-state observable (see Definition 6.5 in [32]), since q = 0 is the unique solution which can stay in the set for   positive definite.
(ii) Second, for the static nonlinear function described by  2 in (24), and connected to  1 by the negative feedback connection shown in Figure 3, its inputoutput product is From Proposition 2, it is clear that  2 is passive.
Then, since  1 and  2 are both passive systems, the whole negative feedback connection  is passive (Theorem 6.1 in [32]).Note that the storage function ( 26) is still valid for system  with the connections (25).The time derivative of the storage is so  is output strictly passive.
Bounded Velocities.Considering (23), the system is under the action of the proposed controller.Then, if q  ∈ I  with q  ∈ [ q   + Δ  , q   − Δ  ], one has that   ( q  ) = 0, which implies that  LV = 0, and q  is driven only with the tracking controller (i.e.,  =  CT ).As a consequence, asymptotic tracking of the reference will be achieved, as it was already proved, and the velocities are bounded.However, the tracking controller could make q  increase rapidly its speed towards the desired bound q  .But inasmuch as q  approaches the bounds, the magnitude of  LV also increases rapidly in accordance with   ( q  ), causing q  not to leave the set I  .In case that q  ∈ O  , one has that q  is driven fastly, in finite time, to the set I  .
The above description corresponds to the attractive behaviour of I  , which is ensured from (23), and it is now proved.To this aim, define Note that   = 0 implies that q  ∈ I  .The dynamics of   is thus described by Take the time derivative of along (32) to obtain By definition,   is positive definite, so we focus on the analysis of the product q  q  .Considering ( 7) and ( 21), the closed-loop dynamics ( 23) can be written in the form and then the product q  q  is written as Since q  ∈ O  , it is true that and using (20), one has that     q and going back to (34), one finally has which states that   = 0 is reached in finite time (see, e.g., [34]).Moreover, (39) states that the th velocity and acceleration have opposite signs, which proves that q  will not leave I  .Therefore, I  is attractive, and the velocities will remain bounded, satisfying (6).

Simulations and Real-Time Experiments
The aim of this section is to illustrate the effectiveness of the proposed control scheme (22) via numerical simulations and real-time experiments.These consist in the tracking control of an OMR with desired bounded velocities.With the purpose to include a reference to better illustrate the capabilities of the proposal ( 22), the simulations and experiments are also developed for the OMR with the computed-torque controller (8), which allows the system to be controlled without bounding its velocities.Consider the OMR described in Section 2.1, and let the desired smooth trajectory be a "Daisy, " whose parametric equations are [6] The trajectory has an external radius of   (  +   ) and is centred at the point (in the Cartesian space)  = 0.In all the graphics to be presented, to distinguish the system performance under the action of the different controllers, the following subscripts are used in the signals: CT for the computed-torque controller (8), and BV for the proposed bounded-velocity trajectory tracking controller (22) 4.1.Simulations.Simulations were developed in MAT-LAB/Simulink with a fixed-step Euler method at 1 ms.The initial condition of the OMR was (0) = [0 0 0 0 0 0] ⊤ , and the desired tracking performance was specified, following [3], through   = diag{ 2  }, and   = diag{2  }, where  = 0.85 is the desired damping ratio and   = 5 [rad/s] is the desired undamped natural frequency.
Figures 4 to 7 show the system behaviour under the action of the CT and BV controllers.For the CT controller, asymptotic tracking is completely achieved in approximately 2.5 s (see Figures 5 and 6, dashed-dot lines), and note that the system develops maximal velocities of about 1.5 rad/s (see Figure 6, dashed-dot lines).Then, for the OMR under the CT controller, it is clear that the desired velocity constraints (dotted lines) are not satisfied with the given tracking performance.However, with the proposed redesign (22), the BV controller, both asymptotic tracking and desired bounded velocities are completely achieved (see Figures 5 and 6, solid lines).This is achieved at the cost of greater convergence times, which is natural as the velocities do remain constrained to the desired maximal values.During the transitory stage, note that the velocity tracking errors are reduced as the velocities remain bounded (with the BV controller, the system velocities are kept closer to the desired velocities to be tracked).Also, as shown in Figure 7, a benefit of having bounded velocities is to require reduced control efforts (because the proposed BV controller dissipates the excess of energy in the system when the velocities remain bounded).

Real-Time Experiments.
Real-time experiments were carried out in the mobile robot prototype shown in Figure 8, which has been designed to maximize its dexterity (see [35]).The dynamics of the system, as well as its parameters, correspond to those given in Section 2.1.
Figures 9 to 12 show the real-time system behaviour under the action of the CT and BV controllers.It can be seen in Figure 11 that the CT controller alone does not allow the system to satisfy the desired velocity bounds.This is not the case for the BV controller, which in fact behaves similar to the CT controller when all the velocities are within their limits, but away of the bounds (see Figures 11 and 12).Note that, for both controllers, convergence to the reference trajectory is achieved at almost the same instant of time, but with bounded errors (see Figures 10 and 11).Among other reasons, these errors could have been produced by the inaccuracy of the dynamic model used to derive the control scheme.The experimental results are very similar to those obtained by numerical simulations and are consistent to the theoretical development.

Conclusions and Future Work
A solution to achieve global asymptotic tracking with desired bounded velocities for an omnidirectional mobile robot was proposed.It consisted in the redesign of a tracking controller to take into account, explicitly, the velocity constraints of the system.The stability analysis was developed using Lyapunov and passivity tools.Simulation and experimental results showed the effectiveness of the proposal.The ideas proposed here can be extended to use other tracking controllers to further improve the system performance.This can be done immediately with those allowing the preservation of the passivity properties of the closed-loop system.Also, the extension can be done to bound other important variables in the system, like positions, accelerations, and torques.Among others, applications of these ideas in parallel robotics, aerial vehicles, and human-robot interaction are visualized.

Figure 1 :
Figure 1: Diagram of the holonomic omnidirectional mobile robot.

Figure 4 :
Figure 4: Simulation results of the task space trajectories described by the OMR under the action of the CT and BV controllers.

Figure 5 :
Figure 5: Simulation results of the positions and tracking errors of the OMR under the action of the CT and BV controllers.

Figure 6 :
Figure 6: Simulation results of the velocities and tracking errors of the OMR under the action of the CT and BV controllers.

Figure 7 :
Figure 7: Simulation results of the torques applied to the OMR under the action of the CT and BV controllers.

Figure 8 :Figure 9 :
Figure 8: Physical prototype of the omnidirectional mobile robot.

Figure 10 :
Figure 10: Experimental results of the positions and tracking errors of the OMR under the action of the CT and BV controllers.
Angular velocity tracking error

Figure 11 :
Figure 11: Experimental results of the velocities and tracking errors of the OMR under the action of the CT and BV controllers.

Figure 12 :
Figure 12: Experimental results of the torques applied to the OMR under the action of the CT and BV controllers.

Table 1 :
Parameters of the omnidirectional mobile robot.