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Intell Robot 2021;1(2):131-50. 10.20517/ir.2021.09 © The Author(s) 2021.
Open Access Research Article

Autonomous navigation in unknown environment using sliding mode SLAM and genetic algorithm

Departamento de Control Automatico, National Polytechnic Institute, Mexico City 07360, Mexico.

Correspondence Address: Prof. Wen Yu, Departamento de Control Automatico, National Polytechnic Institute, Mexico City 07360, Mexico. E-mail: yuw@ctrl.cinvestav.mx

    This article belongs to the Special Issue Evolutionary Machine Learning for Engineering Applications
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    Academic Editors: Simon X. Yang, Hao Zhang | Copy Editor: Yue-Yue Zhang | Production Editor: Yue-Yue Zhang

    © The Author(s) 2021. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

    Abstract

    In this paper, sliding mode control is combined with the classical simultaneous localization and mapping (SLAM) method. This combination can overcome the problem of bounded uncertainties in SLAM. With the help of genetic algorithm, our novel path planning method shows many advantages compared with other popular methods.

    1. Introduction

    1.1. Autonomous navigation in unknown environment

    Autonomous navigation (AN) has three jobs[1].

    1. Perception: Mapping from signal to information is the perception of AN[2]. Its algorithms can use human thought[3], intelligent methods[4], optimization[5], probability methods[6], and genetic algorithms[7].

    2. Motion planning: It has three classes, namely graph methods such as a roadmap[8], random sampling[9], and grid[10].

    3. Localization and mapping: In unknown environments, sensors, actuators, and maps may have big uncertainties.

    Path planning (PP) can be performed under the following conditions:

    (1) The environment is known. PP is an optimization problem[1113].

    (2) The environment is partially known. PP can find new objects during navigation[14,15].

    (3) The environment is totally unknown. PP depends on the navigation and has a recursive solution[1618].

    Simultaneous localization and mapping (SLAM) can be used in unknown environments[19] or in partially unknown environments[20]. SLAM[21] uses the current position to construct a map, and it can be classified into feature-based[22], pose-based[23], appearance-based[24], and variants[25].

    The most popular SLAM uses Kalman filter[21] for Gaussian noise. Nonlinear SLAM uses extended Kalman filter (EKF)[26], where the noise assumptions are not satisfied[27]. EKF-SLAM applies linearization[28].

    1.2. Related work

    Few AN uses SLAM. Visual SLAM uses several cameras[29]. AN can use both SLAM and GPS signals[30]. Robots can avoid moving obstacles using neural networks[31]. Swarm optimization helps robots follow an object[32]. Neural networks help robots construct the navigation path[33]. The optimal path is considered in the sense of trajectory length, execution time, or energy consumption.

    Genetic algorithms (GA) have been developed recently[34,35]. They are easy to use for optimization in non-deterministic cases[36], uncertainty models[37], and robust cases[38]. GA can be in form of ant-based GA[39,40], cell decomposition GA[41], potential field GA[42], ant colony[43], and particle swarm optimization[44]. Finite Markov chain is a theory tool for GA[45,46].

    1.3. Our work

    In this paper, we try to design AN in an unknown environment in real time. The contributions are as follows:

    (1) Sliding mode SLAM: The robustness of this SLAM is better than other SLAM models in bounded noise.

    (2) GA SLAM: We use roadmap PP and GA to generate the local optimal map.

    (3) Comparisons and simulations with other SLAM models were made by using a mobile robot[47].

    2. Sliding mode slam

    SLAM gives the robot position and environment map at the same time. At time k, the state is where (xk, yk) is the position and θk is the orientation of the robot. are landmarks, with the ith landmark. We assume the true location is time-invariant.

    Xk has two parts: the robot and the landmarks The state equation is

    where f () is the robot dynamics, Wk is the noise, and uk is the robot control. Since is not influenced by motion noise, the noise is [Wk, 0]T.

    Zk is defined as the position between the robot and the landmark, whose model is

    where h() is the geometry and vik is the noise. Here, wk and vik are not Gaussian noises. We assume wk and vik are bounded.

    To estimate xk in Equations (1) and (2), EKF is needed. We linearize the state model in Equation (1) and the observation model in Equation (2) as

    where is the estimation of xk.

    Prediction. The estimation is based on past states, control, and landmarks:

    where R1 is the covariance of Wk, R1 = E {[WkE (Wk)] [Wk – E (Wk)]T}.

    Correction. The new state is based on predicted states, landmarks, and current observations:

    The motivations of using sliding mode modification to the EKF bases SLAM based are the following:

    (1) The noises wk and in Equations (1) and (2) are not Gaussian.

    (2) There are linearization error terms, and in Equation (3), and the traditional EKF-based methods do not work well for these errors.

    We use the sliding mode method to estimate the robot state and the landmark

    Sliding modes have a number of attractive features, and thus have long been in use for solving various control problems. The basic idea behind design of system with sliding mode is the following two steps: (1) a sliding motion in a certain sense is obtained by an appropriate choice of discontinuity surfaces; and (2) a control is chosen so that the sliding modes on the intersection of those discontinuity surface would be stable. A general class of discontinuous control u (x, t) is defined by the following relationships:

    where the functions u+ (x, t) and u (x, t) are continuous.

    The function S (x) is the discontinuity surface (subspace). The objective of the sliding mode control is to design some switching strategy of the continuous control u+ (x, t) and u (x, t), such that

    In this paper, the sliding surface is defined by the SLAM estimation error as

    Here, the discontinuity surface is e (k) = [e1… en]. We consider the following positive definite function,

    where P is diagonal positive definite matrix, P = PT > 0. The derivative of V is

    The motion e (k) satisfies

    where sgn (0) = 0, then (10) is

    because P = diag {pi}, pi > 0, and ei × sgn (ei) = |ei|

    Thus, By Barbalat’s lemma[48], the estimation error is e (k) → 0.

    The classical SLAM in Equations (4) and (5) is modified by the sliding surface in Equation (11). The sliding mode control can be regarded as a compensator for Equation (4):

    where ρ is a positive constant. The correction step is the same as EKF in Equation (5). The sliding mode SLAM is shown in Figure 1. Here, the estimation error, e (k), is applied to the sliding surface to enhance the robustness in the prediction step with respect to the noise and disturbances.

    Figure 1. Sliding mode simultaneous localization and mapping.

    It is the discrete-time version of Equation (6). We give the stability analysis of this discrete-time sliding mode SLAM at the end of this section.

    For the mobile robot, the sliding mode SLAM can be specified as follows. We define a critical distance dmin to limit the maximal landmark density. It can reduce false positives in data association and avoid overload with useless landmarks. If the new landmark is far from the other landmarks on the map, then the landmark is added; otherwise, it is ignored. If the distance between the new landmark Xk+1 = [xm+1,ym+1] and the others is bigger than dmin, it should be added into xk, i.e.,

    It can be transformed into an absolute framework as

    The nonlinear transformation function T also applies to the uncertainties. We approximate the transformation T by the linearization. Pk can be expressed as

    where

    with

    For the motion part, we use the Ackerman vehicle model [49]

    where Wk is the process noise, vk is the linear velocity, αk is the steering angle, Tk is the sample time, and ba is the distance between the front and the rear wheels.

    At the beginning of map building, the vector k only contains the robot states without landmarks. As exploration increases, the robot detects landmarks and decides if it should add these new landmarks to the state.

    where x, y, z, and m are defined in Equations (1) and (2),

    We exploit the same property in the sliding SLAM. The landmarks with fewer corrections are removed from the state vector.

    where is the compensator, and

    This sliding SLAM algorithm is given in the following algorithm.

    The discrete-time sliding mode SLAM in Equation (19) can be written as

    where

    The correction step for k+2 is the same as EKF:

    where

    The error dynamic of this discrete-time sliding mode observer is

    where is bounded uncertainty, and Kk is the gain of EKF in Equation (21).

    The next theorem gives the stability of the discrete-time sliding mode SLAM.

    Theorem 1If the gain of the sliding mode SLAM is positive, then the estimation error is stable, and the estimation error converges to

    where is the gain of EKF in Equation (21),

    Proof 1Consider the Lyapunov function as

    wherePk+2is the prior covariance matrix in Equation (21), andPk+2> 0. From Equation (22),

    Because the last term on the right side of Equation (25) is

    where is the maximum eigenvalue of

    The second term of Equation (25) is

    whereKkis the gain of EKF in Equation (5). In view of the matrix inequality

    which is valid for any X,Yand for any positive definite matrix 0 < Λ = ΛTthe first term of Equation (27) is

    We apply the sliding mode compensation in Equation (20) to the second term of Equation (27):

    where lij are the elements of the matrixWhen the the orientation θk is not big sin θk 0, cos θk 0,

    Thus, the second term of Equation (27) is negative.

    The first term on the right side of Equation (25) has the following properties:

    (I - KkCk) is invertible, and we have

    According to EKF,

    Thus,

    By the following matrix inversion lemma,

    where Γ and Ω are two non-singular, matrices,

    Using Equation (32) and defining L = (1 –KkCk),

    Now,

    Hence,

    Combining the last term of Equation (29) with the first term on the right side of Equation (25),

    where and

    Combining Equation (26), the first term of Equation (29), and Equation (34),

    Thus,

    whereIf

    then Vk+1– Vk ≤ 0, ||e (k)|| decreases. Thus, ||e (k)|| converges to Equation (23).

    3. Genetic algorithm and slam for path planning

    Path planning is one key problem of autonomous robots. Here, the map is built by the sliding mode SLAM:

    • The obstacle set is defined by Bobs(t).

    • The position is xr(t), Bfree(t) = B\Bobs(t).

    • The path planning is f (x (t), xs, xT),xs = xr(t).

    The previous map is Bfree(t), which requires the path f(x(t),xs,xT).

    We assume the previous map is obstacle-free, the initial point is xs, the target point is xTB,

    the obstacle is is the shape of the robot, and A(zr) is the area of the robot. The objective of the path planning is to find a path f(x,xs,xT)Bfree that allows the robot to navigate.

    D is defined as the search space. We use the GA to find an optimal trajectory f (x, xs, xT), such that

    Here, we use stochastic search for GA, and each iteration includes: reproduction or selection, crossing or combination, and mutation. The population is with m being the size of the population that represents the possible solutions:

    (1) Every chromosome has a solution in D.

    (2) Crossing the chromosomes. An intersection in and belongs to D, such that then,

    where and are the next generation from two compatible chromosomes by crossing.

    (3) Mutation. It replace a number of chromosomes by chromosomes in D.

    The mutation operation is calculated by the fitness of each chromosome, where n is the number of mutations and fit uses the

    Euclidean distance. The total fitness is Therefore, the probability of selection pi, of a chromosome Si for i = 1, 2,…, n is

    An optimal solution pM in D is mutated by

    In the mutation operation, an optimal solution with pM = 1 is a global solution if n ∞. To prove the convergence, we use a Markov chain, as shown in Figure 2. Each chromosome can move from Qij to the state Qi(j+1). The moving probability is ρji,ik > 0, i = 1, 2, …,n, k,j = 1, 2, …,m.

    Figure 2. Roadmap genetic algorithm model as a finite Markov chain.

    The operators, selection, crossing, and mutation create P(k) with pk. It preserves the best chromosomes of P(k – 1). P(k + 1) in the population P(k) can be regarded as the Markov transition:

    Theorem 2If GA for the roadmap is an elitist process, then the probability of p*in D is exponential.

    Proof 2The iteration Q1 is changed with the chromosomes when genetic process is elitist,

    where n is the size of the population. If for all α, βD, there is 0 < τ ≤ msuch that Hτ (α, β) ≥ ∊ > 0, then

    This implies that, given certain state Qt, the probability of transition in time t between t and t + m is at least ∊,

    Without loss of generality, the transition in the iteration k + 1 is

    Using Equation (42), we have

    where then

    Since 0 <≤ 1, the algorithm converges exponentially to p* in: population size n and iteration number k.

    The algorithm of the SLAM-based roadmap GA for the path planning is as follows.

    4. Autonomous navigation

    Our AN uses both sliding mode SLAM (Algorithm 1) and the roadmap GA method (Algorithm 2). The autonomous navigation algorithm is:

    The PP needs the map, robot position, and target. This information is given by the sliding mode SLAM algorithm. When the algorithm falls into a local solution, we use the ComputeOutsideLocal function to provide another Sn which is outside the local zone.

    5. Comparisons

    In this section, we use several examples to compare our method with the three other recent methods: the polar histogram method for path planning[50], the grid method for path planning[51], and SLAM with extended Kalman filter [52].

    5.1. Simulations

    The following simulations were implemented in partially unknown and completely unknown environments. The size of the environments was 100 m × 100 m, in which a solution was sought to find a trajectory from the initial point xs to the target point xT. The sliding mode gains were selected as ρ = diag ([0.1, …, 0.1]).

    In the partially unknown environments, Bobs(0) ≠ θ. The path planning solution p* was partial because the environment Bobs(t) was variant in time. Figure 3 shows a partial solution p* from an initial point xs to the objective point xT for the partially unknown environment. Figure 4 shows the overall result of the robot navigation from point xs to point xT with the robust SLAM algorithm combined with the GA.

    Figure 3. Sliding mode simultaneous localization and mapping.

    Figure 4. Autonomous navigation using sliding mode simultaneous localization and mapping and genetic algorithm method.

    Here, the SLAM algorithm was used to construct the environment and find the position of the robot. At the beginning of navigation in the partially unknown environment, there was a planned trajectory of navigation through the GA algorithm; however, if an obstacle was found in the planned trajectory, the GA algorithm needed to be used to search for a new trajectory within the built environment by the SLAM, BSLAM. The planned trajectory belonged to the set of obstacles that prevent reaching the goal, p* (Bobs(t); therefore, it was necessary to look for a new trajectory using RGA that allowed reaching the goal.

    For the completely unknown environments, Bobs(0) = θ. In these environments, the SLAM algorithm was required to know the environment BSLAM and the position of the robot; in this way, when an obstacle was found that contained the planned trajectory, a new trajectory with the GA algorithm was searched on the map

    BSLAM until the target point was reached the results obtained are shown in Figure 5. When we used the polar histogram method for path planning [50], only the local solutions could be found [Figure 6].

    Figure 5. Sliding mode simultaneous localization and mapping and genetic algorithm in complete unknown environment.

    Figure 6. Polar histogram method in complete unknown environments.

    Now, we compare the path lengths with the polar histogram method. The following density of the obstacles give the navigation complexity. The environment is free of obstacles when dobs = 0. The whole environment is occupied by the obstacles when dobs = 1. The index for the trajectory error is

    We use the averages of the path length. The obstacles density is defined as

    The path length is defined as

    The averages of the path lengths of our RA and the polar histogram are shown in Figure 7. When the density of obstacles was bigger, the path length of the polar histogram grew more quickly than that of ours. When the obstacle density dobs was 0.3, E[lsub, Navigation1] = 1.053, E[lsub, Navigation2] = 1.152.

    Figure 7. The average path length: (A) sliding mode simultaneous localization and mapping and genetic algorithm; and (B) polar histogram.

    Next, we compare our method with the grid method[51]. The comparison results are shown in Figure 8. For the task of navigating the robot or system in partially unknown or completely unknown environments, the SLAM algorithm was used to construct the environment and know the position of the robot. At the beginning of navigation in the partially unknown environment, there was a planned trajectory of navigation through the GA algorithm; however, if an obstacle were found in the planned trajectory, the GA algorithm needed to be used to search for a new trajectory within the built environment by the SLAM, BSLAM.

    Figure 8. Sliding mode simultaneous localization and mapping (gray) and grid method (black)

    The size of the environments was 100m× 100 m, in which a solution was sought to find a trajectory from the initial point xs to the target point xT.Figure 9 shows a path planning based on the proposed methods to find a solution f*. Here, 60 local targets were generated xi with the search space D generated by the trajectories of the local targets that do not intersect with the set of obstacles; thus, it became an optimization problem to find an optimal path.

    Figure 9. Sliding mode simultaneous localization and mapping based path planning (bold) and grid method (gray).

    In an environment with previously generated obstacles of 100 m × 100 m, 120 possible targets were randomly generated xi; therefore, the search space D would be all trajectories g(xi,xj) that do not intersect with the set of obstacles, where the roadmap genetic algorithm solved the problem of optimization to find a solution to the problem of path planning. For the problem presented above, we found that the proposed algorithm converged in 40 iterations. For these results, 100 tests were performed for each number of iterations and, as shown in Figure 10, the roadmap genetic algorithm converged with greater probability within 40 iterations.

    Figure 10. Performance of the genetic algorithm.

    5.2. Application

    The Koala mobile robot by K-team Corporation 2013 was used to validate our sliding mode SLAM. This mobile robot has encoders and one laser range finder. The position precision is less than 0.1 m.

    The objective of this autonomous navigation is to force the robot to return to the starting point. The sliding mode SLAM was compared with SLAM with extended Kalman filter (EKF-SLAM) [52]

    The initial covariance matrices are zero. The parameters of the algorithm are

    Since the robot moves in the environment with bounded noise (see Figure 11), the noises are not Gaussian. Two different conditions are considered: (1) Koala robot pre-processes off-line the sensors data to reduce 90% noises; and (2) the computer uses sliding mode SLAM on-line.

    Figure 11. The environment of the autonomous navigation.

    For the first case, Figure 12 shows that sliding mode SLAM and EKF-SLAM are similar. Both sliding mode SLAM and EKF-SLAM work well for the case with less noise. The robot can return to the starting point, and the map is constructed correctly.

    Figure 12. Results of extended Kalman filter simultaneous localization and mapping and sliding mode simultaneous localization and mapping with small noises.

    For the second case, Figure 13 gives the results of EKF-SLAM. As can be seen, the robot cannot return to the starting point and the map is not exactly the same as the real one with EKF-SLAM because EKF-SLAM is sensitive to non-Gaussian noises.

    Figure 13. Results of extended Kalman filter simultaneous localization and mapping with noises.

    Figure 14 shows the results with SM-SLAM. Under the same bounded noises, SM-SLAM works very well, because of the sliding mode technique.

    Figure 14. Results of sliding mode simultaneous localization and mapping with noise.

    To compare the errors, we define the average of the Euclidean errors as

    where NT is the data number; and are real values for robot position and orientation; and xk, yk, and ϕk are estimations of them. Figure 15 shows the errors of EKF-SLAM and sliding mode SLAM. Obviously, the errors of EKF-SLAM increase quickly. Robots have better estimation in long distances with sliding mode SLAM.

    Figure 15. Direction estimation errors of sliding mode simultaneous localization and mapping (SLAM) and EKF-SLAM. EKF: Extended Kalman filter.

    6. Conclusion

    Navigation in unknown environments is a big challenge. In this paper, we propose sliding mode SLAM with genetic algorithm for path planning. Both sliding mode and GA can work in unknown environments. Convergence analysis is given. Two examples were applied to compare our model with other models, and the results show that our algorithm is much better in unknown environments.

    Declarations

    Authors’ contributions

    Revised the text and agreed to the published version of the manuscript: Ortiz S, Yu W

    Availability of data and materials

    Not applicable.

    Financial support and sponsorship

    None.

    Conflicts of interest

    Both authors declared that there are no conflicts of interest.

    Ethical approval and consent to participate

    Not applicable.

    Consent for publication

    Not applicable.

    Copyright

    © The Author(s) 2021.

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    Cite This Article

    Ortiz S, Yu W. Autonomous navigation in unknown environment using sliding mode SLAM and genetic algorithm. Intell Robot 2021;1(2):131-50. http://dx.doi.org/10.20517/ir.2021.09

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