(ICSC), 2017 6th International Conference on Systems and Control 7-9 may 2017
Stereo Vision as a Sensor for SLAM Based Smooth Variable Structure Filter with an Adaptive Boundary Layer Width ALLAM Ahmed (1), TADJINE Mohamed (2), NEMRA Abdelkrim (3) and KOBZILI Elhaouari (4)
Abstract— The autonomous navigation task of a mobile robot depends on its ability of localization and owning a description about its environment. To deal with these requirements, robots need to be equipped with Simultaneous Localization and Mapping (SLAM) module. There are many ways to approach this problem, mostly based on the sequential probabilistic approach, based around extended Kalman filter (EKF) or the Rao-Blackwellized particle filter. The SLAM has been already approached using a new alternative filter which is the Smooth Variable Structure Filter (SVSF). This earlier is a predictor corrector estimator based on sliding mode control and estimation concepts. The first version of SVSF uses a predefined Boundary Layer Width vector and don’t require covariance derivation. We propose in this paper to use a new version of SVSF to solve the SLAM problem based on an adaptive (optimal) boundary layer matrix. It has been demonstrated that the (ASVSF) is stable and very robust face modeling uncertainties and noises and keeps a compromise between robustness and accuracy. Visual SVSF-SLAM and ASVSF-SLAM are implemented, validated with experimentation and compared with EKF-SLAM algorithm. The comparison of simulation results proofs the efficiency, accuracy and the robustness of ASVSF-SLAM comparing to the other algorithms, while the experimental results show that ASVSF-SLAM is the less accurate.
I. INTRODUCTION The localization using GPS (Global Positioning System) benefits from its limited positioning error, but the inaccuracy and the impermanence presence of the GPS signal makes this localization method not suitable for autonomous robots. The ability of a mobile robot to localize itself is critical to its autonomous operation and navigation. A robot that navigates using maps must be able to accurately localize itself. This problem is known as simultaneous localization and mapping SLAM, and has attracted a lot of attention in the robotics community.
applications [1], outdoor robotic applications [2], underwater applications and aerial applications [3]. Currently, cameras are becoming competitive alternatives due to their low cost and the rich information content they provide. Many vision based SLAM systems [4-6], have been developed, and have shown their remarkable performance on mapping and localization in real-time. There are many ways to approach the problem of visual SLAM, mostly based on the sequential probabilistic approach, based around extended Kalman filter (EKF) [7] or the Rao-Blackwellized particle filter [8]. The use of the classical solution to SLAM based on the Extended Kalman Filter EKF-SLAM, displays several shortcomings such as quadratic complexity and sensitivity to false associations, problems due to the linearization of nonlinear motion and observation models [9]. Nonlinearity and errors modeling can be a significant problem for EKF-SLAM and leads to inevitable and sometimes dramatic inconsistency in solutions. While EKF-SLAM and FastSLAM are the two most popular solution methods, newer alternatives, which offer much potential, have been proposed, including the use of the unscented Kalman filter (UKF) proposed by Julier and Uhlmann in SLAM [10].Unlike the (EKF), the (UKF) uses a set of chosen samples to represent the state distribution. The UKF-SLAM avoids the calculation of the Jacobian and Hessian matrices but also obtain higher approximation accuracy with the unscented transformation (UT). However, for high-dimensional systems, the computation load is still heavy; thus, the filter converges slowly. The cubature Kalman filter(CKF) and adaptive cubature Kalman filter (ACKF) based on a cubature transform which is more accurate and its complexity computation is lower than the one of the (UKF) were used in SLAM [11,12].
The SLAM is the process of concurrently building a map of the environment and using that map for estimating the position of the robot. This presents a chicken and egg situation: an accurate map is necessary for localization, and accurate localization is necessary to build a map. Consequently, several successful applications of SLAM algorithms have been developed for indoor robotic
However, the above filters are all based on the frame work of the Kalman filter (KF); it can only achieve a good performance under the assumption that the complete and exact information of the process model and the noise distribution which has to be known as a prior. But in practice, the prior noise statistic is usually unknown totally, and the process and observation models may be not well known or contain modeling uncertainties, thus, the state estimation will have large errors, or, even, the filters will be possible to diverge.
A. ALLAM is with the National Polytechnic School, Algiers, Algeria. (e-mail: ahmedallam900@ gmail.com). M. TADJINE is with the National Polytechnic School, Algiers, Algeria. (e-mail:
[email protected]). A. NEMRA is with the Military Polytechnic School, Algiers, Algeria. (e-mail:
[email protected]). E. KOBZILI is with the National Polytechnic School, Algiers, Algeria. (e-mail: kobzili_elhaouari @yahoo.fr).
To overcome some of these limitations, [13] proposed to use a new alternative which is the Smooth Variable Structure Filter (SVSF) as fusion technique for SLAM based stereo vision. As seen in [13], this earlier offers more accuracy, robustness and less-cost computation to the robot pose and map estimation comparing to the EKF.
DOI: 10.1109/ICoSC.2017.7958700
In this work, we describe our efforts to investigate a new version of the SVSF based data fusion [14,15]. We propose an Adaptive (Optimal) SVSF implementation of 3D visual SLAM algorithm using both robot wheels encoders and stereo vision system. Our motivations are improving the process accuracy, offering more robustness, stability to the robot pose and map estimation and quantifying the process accuracy using covariance matrices. We begin with a brief description of how we obtain measurements of 3D landmarks from images taken by stereo vision system and how we track landmarks over time, we give the observation model used. We give an overview of the motion model of our robot, a summering of the SVSF and ASVSF estimation processes, and then we explain how to use ASVSF for SLAM problem with the derivation of some equations, followed by some simulations where we compare SLAM based ASVSF with both SVSF-SLAM and EKFSLAM algorithms. We provide experimental results from our system on outdoor environment. Finally, we draw some conclusions, some final remarks and perspectives. II. IMAGE FEATURES AND STEREO VISION SENSOR MODEL In order to execute SLAM the robot needs to be able to choose and track appropriate reference or landmarks in the environment over time to localize itself. We opt for a 3D point landmark (or feature based) approach, where the map is a collection of 3D landmark locations. Since we use a stereo vision sensor (two synchronized cameras), the landmarks are as primary visual features (corners, edges…), which must be extracted from the acquired images over the time. A. Feature detection and matching In the literature several techniques to extract robust and distinctive features from image, the two most well-known are: the scale invariant feature transform (SIFT) [16] and the Speeded Up Robust Feature (SURF) algorithm [17] which is a fast version of SIFT. Once we have visual features from two images (right and left), a nearest neighbor search can be performed on the descriptors to find matches. For each of the matching SURF points, we then calculate the 3D position in the camera frame which results in a 3D landmark measurement which represents the 3D coordinates of the landmarks with respect to the robot frame Fig. 1. Figure 1. System geometry.
found again at the current time step. All the new features that have not yet been observed are subsequently added to the map. Visual features detection, matching and tracking are illustrated in Fig. 2. Figure 2.
Features detection, matching and tracking.
Now that we have stereo image features that can be tracked over time, we can even convert into 3D landmarks ant track them. B. Observation model From Fig. 1(b) we see that the robot pose can be described with the state vector,
The model representing the robot frame coordinates of an individual landmark according to its global frame coordinates ( ) and the robot configuration is called the direct model observation and will be noted:
[
](
)
In reality the observation (measurement) is subject to noise. The real observation model is given in the following model: ̃
Where is the measurement noise, characterized by covariance matrix . Regarding the dynamic nature of the SLAM algorithm, new observed landmark must be initialized prior to be added to the state vector [18]. The observation model stated in (3) gives three equations for three dimension variable . The 3D coordinates of a new landmark( ) with respect to the global framework are initialized by solving (3) as follows:
(a) Stereo vision system geometry (b) Robot configuration
To track landmarks over time, we can match the descriptors of a new measurement to the descriptors of the features in the map. Due to the increasing computational complexity with an increasing number of features, we remove all the old features from the map that have not been
[
]
( )
C. Process Model We proceed to model the movement of the robot and the noise associated with it. We assume that the robot is
DOI: 10.1109/ICoSC.2017.7958700
operating in planar environments Fig. 1, whose kinematic state, or pose, is summarized by three variables (1). Where T is the sample period, we assume that the robot is controlled by two velocities: a translational velocity and a rotational velocity at sample . We note the control input at sample by . The kinematic model of the robot is given as,
*
+
+
*
The model (8) states the kinematic for an ideal, noise-free robot. In reality, robot motion is subject to noise, slip or lift. The actual velocities differ from the measured ones by odometer sensor. We will model this difference by a random variable, let: *
+
*
+
III. OVERVIEW OF THE EKF-SLAM ALGORITHM We base our approach to the EKF-SLAM algorithm on the method stated in [18], where our approach is similar in concept. The EKF-SLAM algorithm can be divided into two parts: a control update (prediction) where the robot motion is predicted and measurement update using the landmarks observation. For more details refer to [7], [18]. IV. THE SMOOTH VARIABLE STRUCTURE FILTER A. The SVSF with fixed boundary layer width Introduced on 2007, the smoothing variable structure filter (SVSF) is relatively a new filter [19]. It is a new form of predictor-corrector estimator based on sliding mode control and estimation concepts. Mostly used for state and parameter estimation of dynamic system whether linear or nonlinear, accurate fault detection, estimation and tracking applications [20], it has been demonstrated that the (SVSF) is extremely robust and stable to modeling uncertainties and noise [19],[21].Essentially this method uses a switching gain to converge the estimates to within a boundary of the true state values (i.e., existence subspace) as shown in Fig. 3. The basic estimation concepts of the SVSF [19].
(̂
⁄
)
⁄
The corresponding predicted measurements ⁄ and the a priori measurement errors may be calculated: ⁄ ̂
̂
⁄
⁄
̂
⁄
⁄
The (SVSF) guarantees bounded-input bounded-output (BIBO) stability and the convergence of the estimation process by using the Lyapunov stability condition. The derivation of the (SVSF) gain and its stability conditions can be found in [19], [21]. The (SVSF) gain (14) is a function of , , the smoothing boundary layer width , the ⁄ ⁄ memory or convergence rate matrix given by user (with ), as well as the linearzed measurement matrix . The (SVSF) gain matrix is defined as:
The measured velocity equals the true velocity plus some small, additive error (noise) [7].
Figure 3.
̂
̃
set of equations [19],[21]. The a priori state is calculated as follows:
[(| |
⁄
|)
⁄
|
(̅
)][
⁄
(
⁄
)]
Where ° signifies Schur (or element-by-element) multiplication, the superscript + refers to the pseudo inverse of a matrix and ̅ is a diagonal matrix constructed from the smoothing boundary layer vector , such that: ̅
[
]
Note that is the number of measurements, the saturation function of (14) is defined by: ⁄
(̅
⁄
)
,
⁄
⁄
⁄
⁄ ⁄
⁄
⁄
The updated state estimates ̂
̂
⁄
is calculated as:
⁄ ⁄
⁄
Finally, the updated measurement estimate and ⁄ the a posteriori measurement errors are calculated, ⁄ and are used in later iterations: ⁄
̂
̂
⁄
⁄ ⁄
Note that the process estimation needs to be initialized, ⁄
The original form of the (SVSF) as presented in [19] did not include covariance derivations. The estimation process of (SVSF) is iterative and may be summarized by the following
⁄
The vector is defined as an upper-bound on the uncertainties present in the estimation process (i.e., modeling errors, magnitude of noise, etc.) defined by a vector denoted [19]. The vector is selected in condition to satisfy .
DOI: 10.1109/ICoSC.2017.7958700
B. The SVSF with an adaptive boundary layer width An augmented form of the (SVSF) which includes a full derivation of covariance matrices was presented later in [14, 15]. A more well-defined boundary layer will yield more accurate estimates. In [15] derives a solution for an optimal (time varying) boundary layer width by minimizing the trace of the a posteriori covariance matrix. We will denote (ASVSF), the Adaptive (SVSF) that use (time varying) boundary layer that will be noted . (
[
])
[
]
The estimation process of (ASVSF) is summarized in equations (23 to 30) [14,15]: ̂
(̂
⁄
)
⁄
⁄
̂
⁄
⁄
( )(
) [ [(|
|
⁄
|)
⁄
|
|)]
|
|
((
̂
(|
)
̂
⁄
)] [
⁄
(
⁄
⁄
)]
map vector containing the entire landmarks coordinate at time , is the number of the current landmarks, and is the landmark coordinates in the map vector at time . As the robot moves, the state vector changes according to the standard noise-free kinematic model (8). In SLAM this motion model is extended to the augmented state vector: ̂
⁄
Only the first three elements are updated with control input. Landmarks supposed fix, remain where they are. In our approach, we obtain at each sample a set of m observations denoted by , where each observation consists of the robot frame coordinates of the observed landmark. The observation model is deduced from (3): ̂
⁄
⁄
⁄
Where is the index of an individual landmark observation in , and is the index of its associated landmark in the system state vector at time . Even the (ASVSF) process requires linear observation model, the non-linear observation function is approximated using a first degree Taylor expansion [15]. Since h depends only on two elements of the state vector, the robot pose and the location of the associated landmark , we can write the Jacobian of calculated with respect to [ ] as follows:
|
⁄
⁄
⁄
The ASVSF-SLAM requires covariance matrices, and uses an adaptive (optimal) smoothing boundary layer matrix .
⁄
̂
⁄
[
V. THE SLAM BASED ADAPTIVE SVSF ALGORITHM In this section, we will apply both the SVSF and the ASVSF estimators to SLAM, which yields SVSF-SLAM and ASVSF-SLAM algorithms. We compare the new algorithms with the famous EKF-SLAM algorithm. The SVSF-SLAM based stereo-vision sensor was already implemented and well explained in [13]. So, the main objective of this section is to investigate the use of ASVSF with adaptive boundary layer matrix to solve the SLAM problem, in order to improve the stability, robustness and the accuracy of the process estimation against uncertainties and modeling errors. In addition to estimating the robot pose , the ASVSFSLAM algorithm estimates the coordinates of the landmarks observed along the way. For convenience, we will call the state vector comprising robot pose and the map the combined state vector, and denote this vector . The combined vector is given by ( ) . Where is the robot pose at time , is the
];
]
[
We denote is the rate convergence matrix with ). The execution procedure of the ASVSFSLAM algorithm can be divided into three major steps: A) Initialization In SLAM, the initial pose is taken to the origin of the coordinate system. The ASVSF-SLAM algorithm requires the initialization of the a posteriori measurements error vector and initial covariance. is the number of initial landmarks. ̅ (
;
̅
)
; [
(
)
]
B) Control update (prediction) 1. ̂ 2.
̃ . ⁄
⁄
.
DOI: 10.1109/ICoSC.2017.7958700
anticlockwise as it is shown in Fig. 4. At each time step we add some noise unscented and correlative to the input command and to the measurements coordinates vector, in order to simulate the uncertainty. The table below Table-I, presents the applied noises characteristics.
C) Measurement update 3.For all landmarks observations ̂ 4. ̂ ̂
.
5. If landmark j is seen before (correspondence found) ̂
6.
⁄
7.
⁄
8. (
*
+
TABLE I.
NOISES CHARACTERISTICS USED IN SIMULATION.
Iterations
[
(
⁄
)(
) [
⁄
*(|
9. | 10. 11.
. // a priori measurement error.
⁄
|)
(|
⁄
((
)+ [
⁄
[
(
⁄
)
̂
12. ⁄ 13. End If
.
14. Use (7) to initialize the new landmark
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)
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]
*
+
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]
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+
*
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. )
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̂
)
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.
15. Add to the system state vector ̂ . 16. Use (19) to initialize . 17. Add to . 18. // the total number of current landmarks. 19. End For 20. , ⁄ ⁄
(
]
In order to verify the effectiveness, accuracy and robustness of the new ASVSF-SLAM algorithm proposed in this paper comparing with the SVSF-SLAM and EKFSLAM, five simulations were done with different environments noise, and then the root mean square errors (RMSE) of the estimated results were compared. We used, ;
.
To illustrate the solution of the new SLAM based ASVSF algorithm comparing with the SVSF-SLAM and EKFSLAM, we present the followings results. Figure 4. SLAM implementation with different filters.
̂ 21. Return
,
⁄
,
.
From the lines 10 and 11 of the algorithm given above, we can note that only the observation model is linearized, there were no need to linearize the motion model as in (EKF) estimation process, while the linearized motion model matrix is not used directly in the computation of the ASVSF gain, it’s used to compute the adaptive boundary layer matrix , this earlier will be used for ASVSF gain computation, so linearization errors won’t affect significantly the process estimation. We note the unknown vector as an upper-bound on the uncertainties present in the estimation process (i.e., modeling errors, magnitude of noise, etc.)[13],[15]. In SVSFSLAM [13], the vector is constant and selected in condition to satisfy , in case where is too much greater then , the SVSF-SLAM guarantees robustness in the detriment of the process accuracy, while in the ASVSFSLAM, it makes a compromise between robustness and precision by selecting an optimal matrix .
Figure 5. Comparisons of the RMSEs of the X and Y axis, orientation and estimated robot position.
VI. RESULTS A. Simulation Results The simulation environment is a 16m x 14 m area includes the movement path and60 static landmark points. The mobile robot moves along a random path from
DOI: 10.1109/ICoSC.2017.7958700
First, consider Fig.4, which shows the result of position estimated after the robot has traveled along the real trajectory. The red path (odometer only) is obviously incorrect. On the other hand the ASVSF-SLAM corrected path is consistently following the reference path butter than that of the SVSF-SLAM and EKF-SLAM. Fig. 5, shows Comparisons of the RMSEs of the X and Y axis, robot orientation and robot position estimated errors, in which we see significantly the accuracy and the robustness of the ASVSF-SLAM comparing to the other algorithms. Figure 6. Comparaison of the Map estimation RMSEs.
source oriented object, programmed with C++ enabling to users to implement applications (e.g. Path tracking, SLAM…etc.). We used OpenCV 1.1 library for images treatment and the Armadillo 3.8 library for all matrices calculation. We implemented the several algorithms onboard of the robot in C++ language, then using PC user via wi-fi connection, we can control the robot and acquire data of the experience. Figure 9. The experimental platform Pioneer 3-AT.
Wi-Fi Connection
Embedded PC
User PC
From Fig 6, we learn that the RMSE values of the estimated map of the ASVSF-SLAM algorithm is smaller than that of the others algorithms, so we can note that the ASVSF-SLAM performs significantly better than the other algorithms in terms of estimation accuracy and robustness.
Experiences are done on outdoor environment in a plane area situated in face a building as it shown in Fig. 10. Figure 10. Images from the envirnement of experience.
Figure 7. Comparisons of thethe aposteriori measurement errors.
Figure 8. Evolution of the diagonal elements of the adaptive boundary layer matrix .
From Fig. 11, we can see that the ASVSF-SLAM corrected paths is not consistently following the reference path which is represented by a set of signs “+” in magenta, those latter were marked by user during the experience. Figure 11. Experiments results obtained Outdoor.
Finally, consider Fig. 7 and Fig. 8 which show the evolution of the aposteriori measurement error vector and the diagonal elements of the boundary layer matrix, we can see that the magnitude of the error in case of ASVSF is much smaller than that of SVSF, since this error is the index of process performance estimation. In the other side, we show how the adaptive boundary layer matrix evaluating with time according to the environment noise traveled by the robot, so in sort guarantying the optimal robot position and map estimation. B. Experimental results The experimental platform used to validate our algorithm is a robot Pionner 3-AT, which consists of a 4-wheeled skidsteering mobile robot that has an embedded PC (windows XP). The robot benefits of an ARIA soft-platform (Advanced Robot Interface for Applications), which is an interface open
The table-II below, shows the RMSE of the robot pose estimation using the different algorithms. Contrary to the simulation results, we see that the ASVSF-SLAM gives the less precise estimation, even for several experiences, while the robot position precision using EKF and SVSF are so close.
DOI: 10.1109/ICoSC.2017.7958700
TABLE II.
RMSE(m)
RMSE ESTIMATION BY THE DIFFERENT ALGORITHMS. EKF-SLAM
SVSF-SLAM
ASVSF-SLAM
0.221
0.213
0.27
Fig. 12 shows a sketch of SLAM using ASVSF, we can remark the map as a set of 3D points.
an UAV (i.e. quad-copters), in order to overcome or to reduce the effects of data-association problem. REFERENCES [1] [2]
Figure 12. Sketch of ASVSF-SLAM showing the constructed map. [3] [4]
[5]
[6]
[7]
[8] Figure 13. Variation of the diagonal elements of the adaptive boundary layer matrix .
[9]
[10]
[11]
In Fig. 13, we show the variation of the diagonal elements of adaptive boundary layer matrix, those latter are limited to the value 30 in order to keep the process estimation stable against using big values of .
[12]
[13]
VII. CONCLUSION In this work, we have developed, implemented and validated onboard of an experimental platform Pionner 3-AT an adaptive smoothing variable structure filter SLAM algorithm, which is based on odometer and 3D observation acquired from a stereo-camera. It was proposed in order to improve the stability, robustness and accuracy of the estimation process comparing to SVSF-SLAM, in order to solve the problem of the unknown noise statistic characteristics of the system in real world and to deal with the parameters uncertainties. Simulation results showed that ASVSF-SLAM algorithm enables robust and more precise navigation using an adaptive boundary layer matrix comparing to the (EKF & SVSF)-SLAM, while experiment results show the contrary, this is due to the sensitivity of ASVSF to the false data association during measurement update stage. As future work, we expect to use the ASVSF filter combined with Rao-Blackwellized particle filter and to use robust techniques of features tracking to solve the SLAM for
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[21]
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