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Environment perception is a crucial ability for robot’s interaction into an environment. One of the first steps in this direction is the combined problem of simultaneous localization and mapping (SLAM). A new method, called G-SLAM, is proposed, where the map is considered as a set of scattered points in the continuous space followed by a probability that states the existence of an obstacle in the subsequent point in space. A probabilistic approach with particle filters for the robot’s pose estimation and an adaptive recursive algorithm for the map’s probability distribution estimation is presented.
Key feature of the G-SLAM method is the adaptive repositioning of the scattered points and their convergence around obstacles. In this paper the goal is to estimate the best robot trajectory along with the probability distribution of the obstacles in space. For experimenta purposes a four wheel rear drive car kinematic model is used and results derived from real case scenarios are discussed.
The problem of Simultaneous Localization And Mapping (SLAM) is vital in case of autonomous robots and vehicles navigating in unknown environments.
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Usually the map consists of a sequence of features (or landmarks), each one of which represents the position of an obstacle or a part of an obstacle (i.e. a big obstacle can be represented by many features). The Extended Kalman Filter (EKF) was extensively used in the SLAM problem [2], but it has the disadvantage that the computational cost increases significantly with the number of features. Since then, many probabilistic approaches have proposed including the Montemerlo’s solution to stochastic SLAM, FastSLAM 1,0 and 2,0 [4–7], Grid-based SLAM [8,9], Dual-FastSLAM [10], DP-SLAM[11], L-SLAM [12,13], etc.
In mobile robots 2-D maps are often sufficient, especially when a robot navigates on a flat surfac and the sensors are mounted so that they capture only a slice of the world.
Instead of occupancy grid maps with fine-grained grid defined over the continuous space, in this paper a set of scattered points in the continuous space is used. It is presented a new method, called G-SLAM [14], where the map is considered as a set of scattered points in the continuous space followed by a probability that states the existence of an obstacle in the subsequent point in space. In addition to [14] we have previously presented, in this paper it is presented the mathematical formulation and derivation of the problem and the experiments and the results are enhanced with more state of the art SLAM methods.
A probabilistic approach based on particle filters is used for the robot’s pose estimation. In the robot’s pose estimation the time series of controls, the time series of the measurements and the latest estimation of the probabilistic map are involved. A recursive algorithm for the map’s probability distribution estimation is used for the map update procedure. The proposed method generates new hypothetical points of features in space which are ubsequently tested whether they correspond to real obstacles or not. That is why we call it G-SLAM for Generative-SLAM.
Key feature of the G-SLAM method is the adaptive repositioning of the scattered map’s points that results in a convergence of all the points around obstacles. The final map resulted from the G-SLAM method exhibits high density of weighted points around the obstacle and a subsequent high sparsity in the space which is free of obstacles. These weighted points represent the probability distribution of the obstacles in the continuous space. This method fits on problems where a detailed map is needed with low computational resources.
This paper is organized as follows. In section 1 the probabilistic analysis of the combined SLAM problem in terms of recursively computed probability distributions which estimates the probabilistic map and the robot’s trajectory along with the G-SLAM method are presented. In section 2.1 the model of the robotic system which consist of the robot’s kinematic model and the distance-bearing measurement model is described. In section 2.3 experimental results from real case scenario are discussed.
The SLAM problem is formulated probabilistically, aiming to estimate both the robot's trajectory and the map's structure based on sensor measurements and control inputs. The key equations governing this process include the SLAM posterior, pose prediction using particle filters, map update procedures, and importance factor calculation for pose estimation refinement. The core of G-SLAM lies in its adaptive mechanism for repositioning scattered points around obstacles based on probabilistic distributions, ensuring a detailed and accurate representation of the environment.
Notations:
While most SLAM methods are trying to estimate the robot’s pose st and the map Θ at timestamp t, in this paper our goal is to estimate the whole time series st and the map Θ using the observation time series zt and the control time series ut. In probabilistic terms this posterior is expressed as:
Prob(st, Θ|ut, zt) (1)
Using the definition of the conditional probability, the posterior in equation 1 can be expressed as:
Prob(st, Θ|ut, zt) = Prob(st|zt, ut)
trajectory posterior
Prob(Θ|st, zt, ut)
The two factors of the equation 2 correspond to the robot’s trajectory posterior and the map’s posterior respectively. The calculation of these two factors is discussed bellow.
The left posterior of the equation 2 refers to the estimation of the pose time series given the map and the time series of the observation and the controls. The calculation of this posterior is done using the technique of particle filtering. The proposal distribution for the particles will be the posterior p(st|zt−1, ut), thus the drawing process for each particle i evolves only the previous state sit−1 and the current control input ut.
sit ∼ p(st|sit−1, ut) (3)
The proposal distribution is generated from the posterior Prob(st|st−1, ut) using the robot’s kinematic model f and of course a random sample of the control’s input noise eit.
This procedure creates a cloud of N particles, all representing a possible pose of the robot. It is noteworthy that each particle carries out its own estimation of the map which is independent from the other particles’ maps. The estimation of the pose timeseries is not done yet since particle filter’s importance factors have to be calculated. The calculation of the importance factors is discussed below in the section 1.5.
The rightmost factor of the equation 2 refers to the estimation of the map given the time series of the observation and the controls. The map consists of a set of scattered points in space θ and each one is associated with a probability that the point θ is an obstacle. The distribution of this probabilistic map can be represented by the following posterior.
pkt = Prob(θk|st, ut, zt) (5)
The equation 5 gives the probability that the feature θk is an obstacle given the observations zt and the controls ut. By the definition of the conditional probability, the posterior 5 is expressed as:
pkt =
Prob(zt, θk|st, ut, zt−1)
Prob(zt|st, ut, zt−1)
Using the law of total probability for all θj the denominator becomes:
pkt =
Prob(zt, θk|st, ut, zt−1)
∑j Prob(zt, θj|st, ut, zt−1)
The posteriors of the numerator and the denominator have the same form:
Prob(zt, θk|st, ut, zt−1) =
Prob(zt|st, ut, zt−1, θk)Prob(θk|st, ut, zt−1)
In the rightmost posterior we note that θk is independent of the control input ut and the pose st due to the absence of the measurement zt. Thus the above expression becomes:
Prob(zt|st, ut, zt−1, θk)Prob(θk|st−1, ut−1, zt−1) =
Prob(zt|st, ut, zt−1, θk)pkt−1 (8)
Equations (7) and (8) imply the recursion:
pkt =
Prob(zt|st, ut, zt−1, θk)pkt−1
∑j Prob(zt|st, ut, zt−1, θj)p
Since zt is independent from previous observations, control inputs and previous robot’s positions the equation (9) is simplified as:
pkt =
Prob(zt|st, θk)pkt−1
∑j Prob(zt|st, θj)p
In order to compute the recursion (10) we need to calculate the quantity qkt = Prob(zt|st, θk). In case that the probability distribution function of the measurement noise is given by function dz(z), then this quantity can be calculated by:
qkt = dz (g(st, θk)) (11)
The distribution as proposed in equation (3) is only the proposal distribution. Using the simulation technique of particle filtering the target distribution is calculated as: target distribution = proposal distribution ∗ importance factor.
Through the target distribution the best estimation for the robot’s pose st is calculated.
wit =
target distribution
proposal distribution
=
p(st,i|zt, ut)
p(st,i|zt−1, ut)
Using the Bayes Theorem the equation (13) is simplified as:
wit ∝
p(st,i|zt−1, ut)p(zt|st,i, ut, zt−1)
p(st,i|zt−1, ut)
wit ∝ p(zt|st,i, ut, zt−1)
So the importance factor is proportional to the posterior p(zt|st,i, ut, zt−1) which is already calculated in the map update section 1.4 as the denominator of equations (6) or (10).
Since the set of particles St = {st1, ..., stM} is finite, the 'cloud' of particles is growing as the time increases, which can lead to the degeneracy of the algorithm. Thus a resampling technique is necessary. In this paper and on the experiments that took place, the technique of Residual Systematic Resampling (RSR) is used [15].
The proposed method, G-SLAM is based on a technique that generates stochastically new scattered points that are added into the map. This stochastic generation is based on the current particle and the current observation. Then the update procedure updates the map by updating each point’s probability, while afterwards the 'meaningless' features are removed from the map set. The sensor’s noise is responsible for the stochastic nature of this procedure. This addition of scattered point into the map set is achieved using a drawing procedure which is described bellow. Afterwards the extended map is updated and the updated points with low probabilities are removed. The small probability in a map point, states that this point in space is unlikely to be an obstacle. Algorithms of this type converges as are discussed in [16]. In the context of this paper, map update procedure converges to high probabilities in map points near obstacles. Removing all low probability map points, the parts of space which are free of obstacles are also free of map points while on the other hand the parts of space with obstacles gathers all the scattered points around them.
The G-SLAM method can be described abstractly in six steps:
Steps 1,3,5,6 are already discussed in sections 1.3,1.4 and 1.5, while generation and removal of map points are discussed bellow.
The existing map can be easily updated using equation (12). For every map point θi it is calculated the probability of the measure zt to correspond to this point in map using the equation (11) and then it is multiplied to the previous map’s point probability pit−1. It is noteworthy that the denominator of the equation (12) is just a normalization factor.
The stochastic addition of new points into the map is achieved based on the observation zt and the current pose of each particle. For every observation zt a drawing procedure takes place in order to generate a set of M new map points that represents the sensor’s measurement probability distribution.
ẑmt ∼ N (zt; zt, Rt)
where Rt is the covariance matrix of the sensor’s noise.
Every element ẑmt is given a probability:
q̂m = dz(ẑmt )
where dz(.) represents the probability density function of the measurement’s noise.
These elements ẑmt are unlikely to correspond to a map point θ ∈ Θ. Thus in order to update the map it is necessary to create new map points θ̂m = g(st, ẑmt ) in the map Θ. But also, in order to proceed with the update, it is necessary to calculate each point’s probability p̂mt−1. In the G-SLAM method the calculation of the probability p̂mt−1 is done numerically using interpolation methods. The pre-updated map contains points and their subsequent probabilities at time t− 1. These points in space are interpolated with θ̂m in order to estimate the subsequent p̂mt−1 probability. Afterwards and using the equation (12) the new map points are updated and added to the map set Θ. This procedure augments the probabilistic map with M new points and their probabilities.
As already discussed, update procedure returns an augmented map with more map points than previously had. Some of those points might have near zero probability meaning that the probability of an obstacle existence in this point in space is highly unlikely.
In G-SLAM a map point removal procedure takes place after map update procedure in order to remove all the meaningless map points. So all points θi which their probabilities pit are less than a predefined thresshold pit < pthr are removed from the map set. Using this technique it is prevented the overpopulation of the set Θ and the features θ tends to gather near obstacles.
Experiments were conducted using a dataset from the University of Sydney, involving a four-wheel rear-drive car equipped with a laser sensor. The results demonstrate G-SLAM's effectiveness in generating detailed probabilistic maps and accurately estimating the robot's trajectory. The method's performance is compared with other SLAM approaches, showing G-SLAM's advantages in terms of map detail and computational efficiency.
The system's kinematic model and sensor measurement model are described, providing the foundation for G-SLAM's pose estimation and map generation processes. The experiments highlight G-SLAM's ability to produce a probabilistic map that accurately represents the environment, with scattered points densely populating areas around obstacles.
G-SLAM presents a novel and efficient approach to the SLAM problem, offering detailed environmental mapping with reduced computational resources. Its adaptive mechanism for repositioning scattered points around obstacles enhances the map's accuracy and detail. Future work will explore the extension of G-SLAM to dynamic environments and its application to real-world robotic platforms, further validating the method's effectiveness and adaptability.
Generative Simultaneous Localization and Mapping (G-SLAM). (2024, Feb 18). Retrieved from https://studymoose.com/document/generative-simultaneous-localization-and-mapping-g-slam
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