Multirobot rendezvous with bearing-only or range-only measurements
- Ronghao Zheng^{1}Email author and
- Dong Sun^{1}
https://doi.org/10.1186/s40638-014-0004-5
© Zheng and Sun; licensee Springer. 2014
Received: 12 April 2014
Accepted: 5 August 2014
Published: 1 October 2014
Abstract
This paper studies distributed rendezvous strategies for multiple nonholonomic wheeled mobile robots with the aim of testing their practicality on real robots. We investigate control strategies which use just bearing-only or range-only measurements and do not need inter-robot radio communication to share the measurements. For the bearing-only case, two control laws proposed in our previous study are recalled and adapted. For the range-only case, rendezvous control laws for a two-robot system are proposed first and it is shown analytically a two-robot system achieves rendezvous globally under these control laws. Then the range-only-based control laws are extended to multirobot systems. Monte Carlo simulations indicate that a multirobot system achieves practical convergence under the range-only-based control laws. Experimental results illustrate the applicability and performance of the proposed control strategies for multiple wheeled-robot systems.
Keywords
1Background
Recent theoretical and technological advances have spurred a broad interest to develop practical multirobot systems [1]-[5]. For mobile robots, navigation skill is one of their fundamental capabilities. Different navigation strategies are appropriate for different contexts. Different sensor types and sensing modules are used depending on the application scenarios. However, a problem of common interests and practical significance is how to perform tasks with less information and simpler sensors, such as using binary sensors, or using bearing-only or range-only sensors. This problem has attracted much attention in the multirobot community because an advantage of cooperation of teams of robots is that simple agents are able to perform complex tasks through mutual cooperation.
Bearing-only-based navigation is an approach to simplify the sensing system and is very useful in some cases. For example, for mobile robots equipped with single omnidirectional camera, or radar and sonar operating in passive listening mode, it is a practical requirement to design control strategies based only on measurements of bearings [6],[7].
Range-only-based navigation is another approach and it is useful especially in those scenarios when the robots can only sense the intensity of the signals emitted by other robots, or, for example, underwater applications often use acoustic equipments to measure ranges by registering the time of flight of an echo request and reply. Generally speaking, the control design and analysis become more challenging when each robot can only measure distances to other robots without bearing information. This approach has a lot of applications, such as localization and mapping [8],[9], formation control [10],[11], and target enclosing [12],[13]. If a robot has knowledge of its location in a coordinate frame, then under some persistent excitation condition, the robot may be able to get an estimation of the positions of other robots. This idea has been exploited in works such as [12],[14]. However, in this paper we are interested in those scenarios where there is no global localization system so that absolute position information is not available.
Recently, a lot of researchers have studied the use of range-only-based technique to address the target tracking problem. This task becomes more challenging when wheeled mobile robots are used. Because of the nonholonomic constraints, wheeled mobile robots have restrictions in mobility and typically cannot be controlled by linear controllers. For example, recently, Matveev et al. [15] proposed a sliding mode control law to drive wheeled mobile robots towards a target and circumnavigate the target at a predefined distance. In [16], a two-phase switched logic-based control strategy was proposed.
Inspired by the research on bearing-only and range-only navigation, in this paper we consider another common task, i.e., the rendezvous problem in which multiple nonholonomic mobile robots are required to meet at a single point (see [17],[18] and references therein). Rendezvous control is very useful in a variety of applications for multirobot systems. For example, a group of robots can be deployed to collect samples from a region, and after the task is finished, they are required to get together so that we can collect them and transport them to a new place. However, most existing rendezvous strategies require that every robot knows both bearing and range of its neighboring robots, which restricts many practical applications. In this paper, we study distributed rendezvous strategies using just either bearing-only or range-only measurements. The bearing-only-based control schemes are adapted from our previous work [19] and experimentally validated in this paper. Range-only-based control schemes are also developed. While target tracking often assumes that the pursuer is more maneuverable than the target, in the rendezvous application all robots employ a same control strategy and have same maneuver. This makes the control more difficult especially when only range information is available. The proposed control schemes are then experimentally implemented and validated on a group of wheeled mobile robots.
The work presented in this paper is an extension of our previous work reported in a conference version [20]. Although using a similar idea, the range-only controllers in this paper are redesigned. The first difference is, in [20], that to prove the convergence of the distance between the robot pair, the forward velocity assumes an infinite bound. However, in this paper, a more practical design is provided, which allows both forward and angular velocities to lie in bounded intervals. The second difference is that the convergence of the multirobot system under the range-only controllers is verified by using Monte Carlo simulations. The third difference is that the experimental platform is improved and a behavior-based collision avoidance algorithm is used, which allows us to use more robots in our experiments and allows us to test the performance of the proposed controllers in more realistic scenarios.
The rest of the paper is structured as follows. In ‘Methods’ section, ‘Problem description’ subsection introduces the problem studied in this paper; ‘Bearing-only controllers’ subsection presents two bearing-only-based control schemes adapted from our previous work; ‘Range-only controllers’ subsection proposes two range-only-based control schemes and proves their global stability for the two-robot case. These two control schemes are then naturally extended to the multirobot case. In ‘Results and discussion’ section, Monte Carlo simulations are then carried out to test the convergence of the range-only-based control schemes in multirobot systems. ‘Results and discussion’ section also discusses the experimental detail and results under the proposed control schemes. Concluding remarks are given in ‘Conclusions’ section.
2Methods
2.1 Problem description
Here, control signals v_{ i }∈[−v_{max},v_{max}] and ω_{ i }∈[−ω_{max},ω_{max}] are robot i’s forward and angular speeds, respectively, and v_{max} and ω_{max} (both positive) are bounds for forward and angular speeds, respectively. From (1), it can be seen that the mobile robot is subjected to the nonholonomic constraint ${\stackrel{\u0307}{y}}_{i}cos{\theta}_{i}-{\stackrel{\u0307}{x}}_{i}sin{\theta}_{i}=0$.
Our goal is to design distributed control schemes (v_{ i },ω_{ i }) to get all the N nonholonomic robots to congregate at a common location. With the distributed architecture, the controller of robot i only uses locally measurable information without a common reference frame, i.e., global position information is unavailable. In this paper, we consider two types of measurements. The first one is bearing-only measurement, i.e., each robot can only measure the bearing angles of the detectable robots in its local frame. The second is range-only measurement, i.e., each robot is able to measure only its distances from other robots that it can detect.
In this paper, we assume that the interaction between robots is bidirectional, i.e., if robot i can detect robot j then it can also be detected by robot j and we say that robots i and j are neighbors. This assumption is reasonable, for example, in the case where all the robots use omnidirectional sensors with identical parameters. We then represent the bidirectional interaction topology among robots with an undirected graph $\mathcal{G}=(V,E)$ where V is the node set with each node corresponding to each robot and E is the edge set such that (i,j)∈E implies that robots i and j are neighbors. We denote the set of robot i’s neighbors as ${\mathcal{N}}_{i}$, i.e., ${\mathcal{N}}_{i}=\left\{j\right|(j,i)\in E\}$. A graph is said to be connected if there is a path between every distinct nodes. If for any two distinct nodes i and j there is an edge connecting them, is said to be completely connected.
2.2 Bearing-only controllers
We adapt the following control scheme from [19]. For each robot i,
Here $\left|{\mathcal{N}}_{i}\right|$ denotes the cardinality of the neighbor set ${\mathcal{N}}_{i}$. When is connected, we can see that $\left|{\mathcal{N}}_{i}\right|\ge 1$ for all i.
The convergence of a N-robot system under Controller 1 can be proved by using a Lyapunov-based method and is formally stated by the following result.
Theorem 1 ([19], Theorem 1)
A system of N mobile robots described by (1) rendezvous under Controller 1 provided that is connected. Moreover, the energy function $V:=\frac{1}{2}\sum _{i=1}^{N}\sum _{j\in {\mathcal{N}}_{i}}{\rho}_{\mathit{\text{ij}}}^{2}$ keeps decreasing until the robots achieve rendezvous.
When the interaction topology between robots are completely connected, another bearing-only control scheme is proposed in [19] and adapted here. For each robot i,
Here, Δ α_{ i }∈[0,2π) is defined to be the central angle of the smallest circular sector of robot i which contains all the vectors (cosα_{ ij }, sinα_{ ij }) for $j\in {\mathcal{N}}_{i}$, and α_{i+} and α_{i−} are defined to the bearing angles which correspond to the two radii enclosing robot i’s circular sector (see Figure 1).
The convergence of a N-robot system under Controller 2 is formally stated by the following theorem.
Theorem 2 ([19], Theorem 4)
A system of N mobile robots described by (1) rendezvous under Controller 2 provided that is completely connected, and the perimeter of the convex hull defined by the positions of robots keeps decreasing until the robots achieve rendezvous.
The idea behind Controller 2 can be explained as follows. When Δ α_{ i }≤π, robot i is located at a vertex or on an edge of the convex hull defined by the positions of all robots. In that case, a robot tries to shorten the distances to its neighbors which are also at the vertices or on the edge; otherwise, it just keeps stationary. In this way, the perimeter of the convex hull will shrink to a point and the robots can meet each other at that point.
Remark 1
Using the pseudo-linearization technique, it can be proven that under Controllers 1 and 2, the meeting point is located in a bounded region which is determined by the robots’ initial postures [19].
2.3 Range-only controllers
In this subsection, we propose two range-only control schemes to drive wheeled mobile robots to rendezvous. We first investigate the two-robot case and prove its global convergence. The control schemes are then generalized to deal with the N-robot case.
Note that (4) is valid when ρ_{12}≠0.
Both robots are assumed to have access to only the current distance ρ_{12}(t) and calculate its derivative ${\stackrel{\u0307}{\rho}}_{12}\left(t\right)$ by using a memory unit. The bearing information, i.e., α_{12} and α_{21}, is not available. In the following, we propose two control schemes based on ρ_{12}(t) and ${\stackrel{\u0307}{\rho}}_{12}\left(t\right)$, which drive two robots to rendezvous, i.e., ρ_{12}(t)→0 as t→+∞.
The first range-only control scheme we propose is
In our case, the parameter x of $f(x,\stackrel{\u0304}{x})$ is nonnegative and $\stackrel{\u0304}{x}$ is positive.
The rationale of Controller 3 can be explained by observing that when a robot finds itself approaching another robot, i.e., it is moving in the right direction, it rotates slowly (at the speed ω_{ s }) to try to keep on that direction as long as possible; otherwise, it rotates fast (at the speed ω_{ f }) to try to get to the right direction.
From (6c), we can see that β is a constant under Controller 3.
Theorem 3
Design Controller 3 such that
k_{ v }ρ_{ v }≤v_{max}, ρ_{ v }=ρ_{ ω }, and
k_{ ω }ω_{ f }≤ω_{max}, ω_{ f }>ω_{ s }>2k_{ v }/k_{ ω }.
A two-robot system described by (1) rendezvous under Controller 3 provided cosβ≠0.
Proof
It is clear that when k_{ v }ρ_{ v }≤v_{max} and k_{ ω }ω_{ f }≤ω_{max}, both v and ω are in their admissible intervals.
Because $0<(\frac{{k}_{\omega}{\omega}_{f}}{2{k}_{v}}+cos\beta )(\frac{{k}_{\omega}{\omega}_{s}}{2{k}_{v}}-cos\beta )<(\frac{{k}_{\omega}{\omega}_{f}}{2{k}_{v}}-cos\beta )(\frac{{k}_{\omega}{\omega}_{s}}{2{k}_{v}}+cos\beta )$, we get lnρ(t_{k+1})− lnρ(t_{ k })<0, i.e., ρ(t_{k+1})<ρ(t_{ k }); that is, ρ(t) decreases in every period in which α(t) decreases by 2π. Because α(t) keeps decreasing, ρ(t) will eventually approach 0. (ii) Case II: cosβ<0. The proof is similar to (i).
Using a similar idea as Controller 3, the second range-only controller is proposed. In Controller 4, instead of applying a switch controller on the angular velocity ω_{ i }, we apply the switching control to the forward velocity v_{ i }. Our second range-only controller is
where k_{ f }>k_{ s }>0.
The convergence of ρ under Controller 4 is stated in the following result.
Theorem 4
Design Controller 4 such that
k_{ f }ρ_{ v }≤v_{max}, ρ_{ v }=ρ_{ ω }, and
A two-robot system described by (1) rendezvous under Controller 4 provided cosβ≠0.
Proof.
thus, ρ(t_{k+1})<ρ(t_{ k }). (ii) Case II: cosβ<0. The proof is similar to (i).
To make Controllers 3 and 4 work, we need to avoid the case of cosβ=0. One way to do this is to introduce some random behavior, e.g., stop moving for a while, into the controllers if the robots detect that ρ_{ ij } keeps constant. The introduction of random behavior might also be helpful when the system suffers slow convergence, i.e., | cosβ| is too small.
We generalize both Controllers 3 and 4 to the N-robot case. We call them Controllers 3e and 4e in this paper. To simplify the controller design of ω_{ i }, here we set ρ_{ ω }=+∞.
where ω_{ f }>ω_{ s }>0.
where k_{ f }>k_{ s }>0.
Work is still ongoing to discover the convergence results for the N-robot case. However, in ‘Results and discussion’ section, it is shown by means of both Monte Carlo simulations and experiments that these two range-only-based control schemes perform well provided that the interaction topology among the robots is connected.
3Results and discussion
3.1 Monte Carlo simulations
Parameters used in the Monte Carlo simulations
Parameter | Value |
---|---|
N | 3,…,10 |
k _{ v } | 1 |
k _{ ω } | 1 |
ρ _{ v } | v _{max} |
W | [10,100] |
x_{ i }(0) | [0,W] |
y_{ i }(0) | [0,W] |
θ_{ i }(0) | [0,2π) |
d _{ c } | $\sqrt{{W}^{2}/N}$ |
v _{max} | [0.9,1.1] |
ω _{max} | [5,6] |
ω _{ s } | (2v_{max},3] |
ω _{ f } | (ω_{ s }+3,ω_{max}] |
$\stackrel{\u0304}{\omega}$ | (2v_{max},3] |
k _{ s } | [0.4,0.5] |
k _{ f } | [0.8,1.0] |
The topology is generated in the following way: If the distance between a pair of robots is less than d_{ c }, these two robots are connected to each other. We then test the connectivity of the resulting , if is connected, e.g., the corresponding Laplacian has a rank of N−1, then we continue the evaluation; otherwise, this evaluation is abandoned and a new evaluation is carried out. To avoid using an over-connected , we chose ${d}_{c}=\sqrt{{W}^{2}/N}$ in the simulations. In each evaluation, we record the trajectories of the robots during 0 to 200 s. The sampling time T_{s} is set to be 0.1 s. It is tempting to chose a large ω_{max}, which causes no problem for a continuous-time system. However, a large ω_{max} can cause a discrete-time realization of the system to be unstable. We found that keeping ω_{max}<10×2π/T_{s} provides enough stability margin.
The outcome of each evaluation is the normalized average distance d(t) to the centroid of the robot team, i.e., $d\left(t\right)=\frac{\sum _{i=1}^{N}\parallel \left({x}_{i}\right(t)-\stackrel{\u0304}{x}(t),{y}_{i}(t)-\stackrel{\u0304}{y}(t\left)\right)\parallel}{\sum _{i=1}^{N}\parallel \left({x}_{i}\right(0)-\stackrel{\u0304}{x}(0),{y}_{i}(0)-\stackrel{\u0304}{y}(0\left)\right)\parallel}$, where $\stackrel{\u0304}{x}\left(t\right)=\sum _{i=1}^{N}{x}_{i}\left(t\right)/N$ and $\stackrel{\u0304}{y}\left(t\right)=\sum _{i=1}^{N}{y}_{i}\left(t\right)/N$. We define the ratio of convergence as σ=d(200 s).
For the Controller 3e, we recorded that the average ratio of convergence is 3.0561% and maxσ=58.395%. For the Controller 4e, we recorded that the average ratio of convergence is 24.683 % and maxσ=83.808%. Both results imply the Controllers 3e and 4e lead to practical convergence.
3.2 Experimental platform
3.3 Collision avoidance
In the theoretical analysis, the robots are treated as moving points. However, real robot is never a point, and therefore collision avoidance cannot be negligible when several robots work together in a region. Collision avoidance itself, especially for nonholonomic mobile robots, is a challenging problem (see [23] and references therein). In our experiments, a behavior-based algorithm is adopted, which is straightforward and computationally effective.
Notice that ${\stackrel{~}{v}}_{i}$ may not be achieved due to saturation of the actuator.
Since the working area is bounded, the robots should also avoid collision with the boundaries and keep inside the working area. To do that, once detecting the boundaries, a robot stops moving forward and rotates at the maximal angular speed for a short time span until its forward direction (the direction of v_{ i }) points into the bounded area again.
3.4 Experimental results
Parameters used in the experiments
Parameter | Value |
---|---|
v _{max} | 6 cm/s |
ω _{max} | 2.43 rad/s |
d _{ s } | 15 cm |
d _{ c } | 9 cm |
d _{min} | 12 cm |
k _{ v } | 1 |
k _{ ω } | 1 |
ρ _{ v } | 6 cm/s |
ω _{ s } | 0.63 rad/s |
ω _{ f } | 1.94 rad/s |
$\stackrel{\u0304}{\omega}$ | 1.46 rad/s |
k _{ s } | 0.3 |
k _{ f } | 1.0 |
3.4.1 Controller 1
In our program, the routines run periodically based on a Windows timer. Sometimes the routines may run out of the time slot allocated then the positions of the robots cannot be recorded. This explains the missing parts of the trajectories. In analysis, for example, calculating the evolution of the energy function, we recover the missing parts of the trajectories through linear interpolation.
3.4.2 Controller 2
3.4.3 Controller 3e
To apply Controllers 3e and 4e, ${\stackrel{\u0307}{\rho}}_{\mathit{\text{ij}}}\left(k\right)$ is approximated by [ρ_{ ij }(k)−ρ_{ ij }(k−1)]/T_{s} where T_{s} is the sampling period. The measurement error of ρ_{ ij } may introduce a rapid switching in the control signal. To prevent that, we use a simple filter ${\rho}_{\mathit{\text{ij}}}\left(k\right)=\sum _{l=0}^{n}{a}_{l}{\widehat{\rho}}_{\mathit{\text{ij}}}(k-l)$ to reduce the measurement error effect. In our experiment, we choose n=2 and a_{0}=0.8,a_{1}=a_{2}=0.1.
3.4.4 Controller 4e
It is not a surprising result that Controllers 3e and 4e take longer time to rendezvous than Controllers 1 and 2 do because Controllers 3e and 4e use only scalar information while Controllers 1 and 2 use vector information. By comparing the performances of Controllers 3e and 4e, it can be seen that a switching control of the angular speed (Controller 3e) performs better than the one of the forward speed (Controller 4e). By comparing the experimental results, those shown in Figures 10 and 12 for example, we find that Controller 3e reduces the number of oscillations of ρ_{ ij } so it leads to faster convergence. This result also coincides with the observation from the Monte Carlo simulations.
4Conclusions
In this paper, we study control schemes for driving a group of wheeled robots with nonholonomic constraints to a common location. The proposed control schemes use only the measurements of local bearing angles or only the measurements of distances among the robots. Our purpose is to examine whether the theoretical results obtained for bearing-only and range-only control schemes could be applied in practice to a real multirobot system. To this end, experiments are conducted on a team of e-puck robots. Given that there are unmodeled dynamic delays in the system due to sensing and information processing and the switching of controllers to deal with collision avoidance which are not accounted for in the theoretical analysis, the presented results are very positive.
However, these results are still preliminary. Further research will include developing a more sophisticated method of collision avoidance and implementing the proposed control schemes on more realistic scenarios, such as direct and dynamic interaction topology among mobile robots. Another research topic is to consider sensors with limited field-of-view and bounded range. We also plan to implement the proposed control schemes on mobile robots which have self-localization capability, such as those used in [3],[5], and develop the local localization technique for multirobot systems [24],[25]. There are still many open questions in regards to the convergence of a N-robot system. Monte Carlo simulations indicate that Controllers 3e and 4e achieve practical convergence, but a formal proof of this assertion remains an open problem.
Declarations
Acknowledgements
The work of RZ is supported through the Hong Kong PhD Fellowship Scheme. The work is also supported by a grant from the Research Grants Council, Hong Kong Special Administrative Region, China, under Grant CityU 119612 and a Collaborative Research Fund with Project no. CUHK6/CRF/13G.
Authors’ Affiliations
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