Intelligence & Robotics

Open Access Research Article

Departamento de Control Automatico,

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

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© 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.

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.

Autonomous navigation, sliding mode, SLAM, genetic algorithm

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^{[11–13]}.

(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^{[16–18]}.

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]}.

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]}.

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]}.

SLAM gives the robot position and environment map at the same time. At time *k*, the state is where *(x _{k}, y_{k})* is the position and

**X**_{k} has two parts: the robot and the landmarks The state equation is

where **f** () is the robot dynamics, **W**_{k} is the noise, and **u**_{k} is the robot control. Since is not influenced by motion noise, the noise is [W_{k}, 0]^{T}.

**Z**_{k} is defined as the position between the robot and the landmark, whose model is

where **h**() is the geometry and **v*** ^{i}_{k}* is the noise. Here,

To estimate **x**_{k} 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 **x**_{k}.

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

where *R*_{1} is the covariance of **W**_{k}, R_{1} = E {[**W**_{k} – *E* (**W**_{k})] [**W**_{k} – E (**W**_{k})]^{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 **w**_{k} 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

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

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

Here, the discontinuity surface is *e (k) =* [*e*_{1}*… e _{n}*]. We consider the following positive definite function,

where *P* is diagonal positive definite matrix, *P = P ^{T} >* 0. The derivative of

The motion *e (k)* satisfies

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

because *P = diag* {*pi*}, *pi >* 0, and *e _{i} × sgn* (

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.

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 *d*_{min} 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 X_{k+1} = [*x _{m+}*

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. **P**_{k} can be expressed as

where

with

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

where **W**_{k} is the process noise, *v _{k}* is the linear velocity,

At the beginning of map building, the vector **X̂**_{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 **X̂**_{k+2} is the same as EKF:

where

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

where is bounded uncertainty, and **K**_{k} is the gain of EKF in Equation (21).

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

**Theorem 1***If 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 1***Consider the Lyapunov function as*

*where***P**_{k+2}*is the prior covariance matrix in Equation (21), and***P**_{k+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*

*where***K**_{k}*is the gain of EKF in Equation (5). In view of the matrix inequality*

*which is valid for any X,Y* ∊ *and for any positive definite matrix* 0 < Λ = Λ^{T} ∊ *the first term of Equation (27) is*

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

*where l _{ij} are the elements of the matrix*

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

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

(*I* - **K**_{k}*C*_{k}) *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 –***K*** _{k}C_{k}*),

*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*,

*where**If*

*then V _{k}*

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 *B _{obs}*(

• The position *is x _{r}*(

• The path planning is *f* (*x* (*t)*, *xs, x _{T}*)

The previous map is *B _{free}*(

We assume the previous map is obstacle-free, the initial point is *x _{s}*, the target point is

the obstacle is is the shape of the robot, and *A*(*z _{r}*) is the area of the robot. The objective of the path planning is to find a path

*D* is defined as the search space. We use the GA to find an optimal trajectory *f (x, x _{s}, x_{T})*, 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 *p _{i}*, of a chromosome

An optimal solution *p ^{M}* in

In the mutation operation, an optimal solution with *p ^{M}* = 1 is a global solution if

The operators, selection, crossing, and mutation create *P*(*k*) with *p ^{k}.* It preserves the best chromosomes of

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

**Proof 2***The iteration Q _{1} is changed with the chromosomes when genetic process is elitist*,

*where n is the size of the population. If for all α, β* ∊ *D, there is* 0 < τ ≤ *m**such that H ^{τ} (α, β)* ≥ ∊

*This implies that, given certain state Q _{t}, 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.

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 *S _{n}* which is outside the local zone.

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]}.

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 *x _{s}* to the target point

In the partially unknown environments, *B _{obs}*(0) ≠ θ. The path planning solution

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, *B _{SLAM}.* The planned trajectory belonged to the set of obstacles that prevent reaching the goal,

For the completely unknown environments, *B _{obs}*(0)

*B _{SLAM}* until the target point was reached the results obtained are shown in Figure 5. When we used the polar histogram method for path planning

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

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 *d _{obs} =* 0. The whole environment is occupied by the obstacles when

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 *d _{obs}* was 0.3,

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, *B _{SLAM}.*

The size of the environments was 100m× 100 m, in which a solution was sought to find a trajectory from the initial point *x _{s}* to the target point

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 *x _{i;}* therefore, the search space D would be all trajectories

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.

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

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

where *N _{T}* is the data number; and are real values for robot position and orientation; and

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.

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

Availability of data and materialsNot applicable.

Financial support and sponsorshipNone.

Conflicts of interestBoth authors declared that there are no conflicts of interest.

Ethical approval and consent to participateNot applicable.

Consent for publicationNot applicable.

Copyright© The Author(s) 2021.

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DOI

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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