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2018, IFAC-PapersOnLine
…
6 pages
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This work presents a control strategy for a leader-follower formation distance-based model of two single order kinematic model of robots. It uses feedback linearization techniques to derive the control action on the follower robot to maintain a desired distance and orientation with respect to the leader robot. Besides, it is shown that a suitable selection of the desired parameters make the formation behave as a rigid body. The control strategy is applied in a laboratory environment with two omnidirectional robots to show its performance.
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In
Multi-robot systems have been the subject of research for some time now but still there is the need to develop more robust and efficient algorithms to control the coordination of such complex systems. Coordination is usually managed by defining and controlling specific formations of robots (e.g. Oh et al. (2015)). Some formations have a special robot named the leader, sometimes more than one, and the rest of the pack follows them somehow to achieve tracking of a desired trajectory while keeping the selected formation structure.
Some works defined the connection between leader and follower using cartesian coordinates. Desai et al. (2001) used the distance and relative bearing information between leader and followers to model the formation and define a control strategy. Several closed-loop control strategies of such formations have been reported in the literature. For example, Lyapunov techniques have been applied to design a framework for multiagent control in Mastellone et al. (2008). Ying and Xu (2015) and Zhang and Li (2015) have applied Artificial Potential Fields to achieve tracking of the leader while avoiding obstacles at the same time. In Zhao J. González-Sierra gratefully acknowledge the financial support from CONACYT 266524. E. Hernandez-Martinez gratefully acknowledge the financial support from Universidad Iberoamericana F401029.
et al. (2017) sliding mode techniques has been applied to control a dynamic model of a leader-follower formation of kinematically unicycle robots. Backstepping techniques were applied in Vallejo-Alarcón et al. (2015) to include the dynamic model of the follower. Adaptive control to take into account model disturbances like wheel slip have been considered in Cai et al. (2012). On the other hand, Huang et al. (2006) also analyzes sensory and control strategies for heterogeneous formations where the leader has more capabilities than the followers. Tsiamis et al. (2015) have used leader-follower formation control to carry objects with no explicit communication between the robots. Network traffic and urban traffic control problems have also been presented in Dimirovski et al. (2001) and Boel et al. (2015) respectively.
A similar approach to this work has been recently presented in Liu et al. (2018) where the distance and angle dynamics are defined in terms of the leader and follower control actions. While the approach presented in Liu et al. (2018) introduces unicycle dynamics and have distance and orientation been measured visually, subjected to sensor constraints, the aim of this work is to obtain a leaderfollower distance-based model of two omnidirectional mobile robots and analyze the behavior of the formation structure by the application of feedback linearization techniques to define the control actions of the followers. In this sense, the main contribution relies that it is shown that, for a suitable selection of the desired parameters, the formation behave as a rigid body with the advantage that the robots are not physically connected.
The paper is organized as follows. Section 2 presents the problem to work with. In section 3 the control strategy is described in detail and theoretical results are shown. Laboratory experiments with real robots are presented in section 4. Finally, conclusions and future work are outlined in section 5.
Let N = {R F , R L } be a group of two omnidirectional mobile robots moving in a horizontal plane with kinematic model given byξ
(1) where the sub-index L corresponds to the leader agent and F is the follower, R(θ i ) is the rotation matrix defined by
is the state vector with x i , y i ∈ R as the position in the plane of the i−th agent, θ i ∈ R is the orientation with respect to the horizontal axis and
is the control input vector with v xi ∈ R as the longitudinal velocity, v yi ∈ R is the lateral velocity and w i ∈ R is the angular velocity. Fig. 1. Leader-follower scheme represented by two omnidirectional robots.
Figure 1
The problems of interest are two. In a first step, a dynamic model that describes the motion of the agents (Fig. 1) as a function of the distance and angles between them has to be developed, i.e. • lim t→∞ (α − α * ) = 0 where α * is the desired angle.
• lim t→∞ (θ F − θ * F ) = 0 where θ * F is the desired orientation of the follower.
Note that the control objective is that the robots maintain a certain distance between them, where the follower robot is placed on a desired point of the circle centered at the position of the leader robot, with radius d * , according to the combination of α * and θ * F .
Based on Fig. 1, the distance d and the angle α are given by
where
The time-derivative of (2) is given byḋ
withḋ
Substituting (4) into (3) and considering that d x = d cos(θ F − α), and
where γ = θ L − θ F + α and η = [d α θ F ] is the state vector. Note that matrix B is non-singular since det(B) = 1 d , therefore it is possible to define a static state feedback control in order to linearize system (5), as follows
where p d is a desired polynomial defined by
with k d , k α and k θ F as positive design gains. Proof. The dynamics of the error coordinates are given byė =η −η * ,
where e = [e d e α e θ F ] and
is the desired vector. Substituting (5), (6) and 7into 8 By this instance, we have proved that the omnidirectional robots maintain a desired distance and a desired angle between them but it is interesting to know how the follower agent will behave. To solve this, the controls (6)- 7are substituted into equation 1, obtaining the following closed-loop system for the followeṙ
(10) From system (10) many cases can arise depending on the values of the desired polynomial given in (7). In the following, this work is focused in some special cases. Theorem 2. From (10), suppose that
then, the system (10) is reduced, in scalar representation, toẋ
From the above expression, note thatẋ
On the other hand, from Fig. 2, the velocity relationship between any two points of a rigid body can be expressed by
Figure 2
Motion equations of two points of a rigid body.
where v F and v L are the linear velocity vectors of points F and L, respectively, w is the angular velocity vector, r F/L is the position vector of point F with respect to point L, θ B is the angle of the rigid body with respect to the horizontal axis,θ F andθ L are the orientations of the velocities of points F and L, respectively andd is the distance between F and L. Considering a two dimensional space, therefore, the motion equation of rigid body (12) has to satisfied the following expression
Note that, from Fig. 2
Comparing (13) with (11a) and (11b), it is possible to notice that the system, composed by two omnidirectional robots, can achieve the same kinematic behavior and performance of a rigid body. Moreover, it is important to point out that, from (11a)-(11b), the follower agent is placed in a circle with radiusd and it is located in the circumference according to the expression θ L −ᾱ, and, from (11c), both agents have the same angular velocity.
Remark 3. A special case arises whenᾱ = 0. In this sense, θ * F = θ L and therefore,θ * F = w L . With this particular choice, (11) is rewritten aṡ
This implies that the whole rigid body has the same orientation of the leader, i.e. θ B = θ L . Corollary 4. From (10), suppose that d * =d ∈ R, α * = θ * F = θ L and consider that the system (5) has reached the steady state, i.e. lim t→∞ e d = 0. Furthermore, when t → ∞, then θ F and α converges to θ L , hence, the leader-follower scheme, represented by two omnidirectional robots, will emulate the marching control behavior with static horizontal position vectors, i.e. θ B = 0.
Proof. Since d * is constant, then lim t→∞ e d = 0. Moreover, hence, (7) is reduced to
, then, the system (10) is reduced, in scalar representation, toẋ
This means that the follower has the same linear and angular velocity of the leader. Integrating, one obtains the position of the follower given by
, and, taking into account that lim t→∞ e d = 0, θ F and α converges to θ L when t → ∞, therefore the position of the follower is given by
This result implies that the follower will keep a horizontal line with respect to the leader.
The approach is tested using the two mobile robots with four mechanum wheels shown in the Fig. 3. The robots are actuated by servomotors Dynamixel AX − 12W, and controlled by a microcontroller NXP R model LP C1768 with Bluetooth communication to a PC computer. The position and orientation of the robots were measured by a Vicon R motion capture system composed by 6 cameras model Bonita R . The motion capture measures within an available workspace area of 3 × 7 meters. Note in Fig. 3 that the reflective markers were placed on the top of the robots in order to be identified by the Vicon R system. The control algorithm runs at a 117ms rate with a resolution of ±5mm. According to Fig. 4, the velocities of each wheel can be calculated by
Figure 3
Omnidirectional wheeled mobile robots.
Figure 4
where L = 10cm, l = 5cm and r = 2.75cm. Note that the equation 14can be adequated according to the number of wheels of the robot, for example, for the three omnidirectional wheeled mobile robot. The leader robot is controlled to follow a circled-shape trajectory with radius equal to 0.7m oriented to the velocity vector of the trajectory, i.e. the desired values for the leader are given by
where m x = 0.7 sin 2π 20 t and m y = 0.7 cos 2π 20 t .
Thus, the control law for the leader robot is established as
The trajectories of the robots are shown in Fig. 5(a), comparing the simulation signals (continuous lines), versus the experimental results (dashed lines). The red color represents the leader robot, whereas the blue color is the follower robot. The robots are also drawn at four different time instances to show the desired behavior. Note that the follower robot converges to the desired distance, oriented to the angle of the leader robot θ L . It is visualized in the trajectories of the axes X and Y and the orientation angles shown in the Fig. 5(b), and the convergence of the errors
Figure 5
Trajectories of the state variables x i , y i and θ i .
given by the Fig. 5(c). The control inputs are depicted in the Fig. 5(d). Noise effects in the trajectories appear due to the non modelled friction of the wheels on the floor, sensor measurements and actuator errors, among others. Observe that the errors converge to zero. Note that for the definition of α * and θ * F , the possible posture of the follower robot is in a horizontal straight line, oriented to the angle of the leader robot.
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