Gaussian Process Gauss–Newton for non-parametric simultaneous localization and mapping

“We show that for linear-Gaussian systems, the discrete-time formulation is implementing the continuous-time one exactly, under certain special conditions (Tong et al., 2013; Barfoot et al., 2014).”

“In Barfoot et al.'s methods, the sensor motion is modeled as constant velocity (Dong and Barfoot 2012; Anderson and Barfoot 2013a) and with Gaussian processes (Tong and Barfoot 2013; Anderson and Barfoot 2013b; Tong et al. 2013.”

“Though we could jointly estimate the trajectory at the measurement and query times, a better idea is to use GP interpolation after the solution at the measurement times converges [33, 46] (see Section III-D for more details).”

“Furthermore, it has previously been used for SLAM (see, e.g. , [34–38]).”

“The nonparametric approaches view the trajectory as a Gaussian process [3] that we seek to query at times of interest; in this case, the trajectory is still stored using states at discrete times but the GP prior allows smoothing terms between these states to be constructed in a principled manner based on an underlying…”

“…that in recent years, other continuous-time SLAM algorithms for incremental motion estimation (Anderson and Barfoot, 2013; Dong and Barfoot, 2012; Tong, Furgale, and Barfoot, 2013) and global non-rigid registration (Elseberg, Borrmann, and Nüchter, 2013) have emerged following a similar general…”

“We reiterate the fact that if we marginalize out the ˙ p ( t n ) (or ν ( t n ) ) variables and keep only the p ( t n ) variables to represent the trajectory, the W xx block becomes dense (for these priors); this is precisely the approach of Tong et al. (2013).”

“…et al., 2015b; Bibby and Reid, 2010; Fleps et al., 2011; Furgale et al., 2012; Jung and Taylor, 2001; Oth et al., 2013; Tong and Barfoot, 2013; Tong et al., 2012, 2013, 2014) are naturally able to capture more expressive motions (and avoid discontinuities) than those that use linear…”

“Recent work in the robotics [47] and computer vision [48] literature has shown promising results in this direction, but utilizes Gauss– Newton-based approaches that (as demonstrated herein) limit the class of objective functions that can be reliably employed.”

“Recent work in the robotics [47] and computer vision [48] literature has shown promising results in this direction, but utilizes Gauss-Newton-based approaches that (as demonstrated herein) limit the class of objective functions that can be reliably employed.”

“For example, [25] applies a GP-based approach [24] to obtain the robot poses at measurement times in a lidar-based visual odometry system.”

“[24] employed GP to model the continuous movement in SLAM, where Gauss-Netween method is used for state updating.”

“While it is possible to use SLAM methods designed to capture the continuous nature of the scanning process (Bosse and Zlot, 2009; Alismail et al., 2014; Tong et al., 2013), we opt for this registration-based SLAM approach due to its popularity and the impact of internal calibration on performance.”

“[6].”

“…which attaches each frame an independent pose to estimate, our method can also work with other forms of pose trajectories, such as continuous-time trajectories based on Splines [9], [70] or Gaussian Process models [71], [72], which have the capability to compensate the in-frame motion distortion.”

“[23], based on Furgale et al . framework [13], proposed a non-parametric representation for SLAM , where the state is represented as a Gaussian process.”

“[5,6] proposed a Gaussian process (GP) regression approach to solving the STEAM problem.”

“…of all the TGP methods surveyed in this work is presented in Table VI. 1) Early Works: The use of GPs for CT state estimation was first realized by Tong et al. [49], [59], [229], where they presented Gaussian process Gauss–Newton (GPGN), a nonlinear batch optimization method, and applied it to 2-”

“trajectory as a Gaussian process (GP) [28].”

“Similar limitations exist in the Gaussian-process-based representation of the state, described in Tong et al. (2013).”

“Continuous-time batch estimation has been performed both parametrically, using a weighted sum of temporal basis functions [2], and non-parameterically, using a Gaussian-Process [3].”

“For example, in robotic state estimation problems, a structured GP prior may encourage trajectories to follow known system dynamics, e.g. the robot velocity changes smoothly (Barfoot et al 2014; Tong et al 2013).”

“In order to obtain an accurate motion estimate, we were forced to innovate ways of coping with this distortion [2, 3, 26, 27, 28, 29], but we believe that it has been worth the effort to achieve lighting invariance.”

“The constant-velocity GP prior is generated by a LTV-SDE with white noise on the acceleration and has been used in trajectory estimation [13], [16]”

“[9], [10] showed that batch continuous-time estimation can be carried out by representing the trajectory as a Gaussian process.”

“[29] describe the Gaussian Process Gauss Newton (GPGN) algorithm, which uses a finite set of samples of the state at different times as the set of parameters to estimate.”

“While we can achieve the same result using a weight-space approach and a maximum-likelihood objec-tive function (Tong et al., 2013), we opt to provide only the function-space derivation because it is considerably more concise.”

“…(SQP) problem, which has a closed-form solution by taking the derivative with respect to Since the derivation follows the Gauss–Newton approach [61], the covariance can be expressed as the inverse of the Hessian matrix in (10) The Gaussian process x ( t ) ∼ GP ( ¯ x ( t ) , σ ( t k , t )) can…”

“that the robot velocity changes smoothly [5, 32].”

“The Maximum a Posteriori (MAP) estimate of the trajectory can be computed through Gaussian process Gauss-Newton (GPGN) [20].”

“In this paper, the Gaussian-Newton method [26] is used to solve the nonlinear least squares problems. where (cid:49) T k is a small perturbation, J i ( T k ) and H ( T k ) are the Jacobian matrix and Hessian matrix of the error function f i ( T k ) respectively.”

“By governing each kind of state through a specific GP prior, the state estimation can be formulated as a GP regression problem [36].”

“We will not show this derivation here for the sake of brevity, but invite the reader to see [37], [38] for more details.”

“It is compared quantitatively to other estimation techniques in [24, 35].”

“The approach presented in this paper adopts the parametric state representation due to practicality and effectiveness; however, it is worth noting that non-parametric representations, using Gaussian processes, have also been successfully applied in an effort to achieve greater representational power (Barfoot et al., 2…”

“[14] developed Gaussian Process Gauss-Newton as a non-parametric state estimation framework where similarly to [13], the continuous-time reality of robot motion is acknowledged when processing measurements.”

“A widely used approach [17]– [19] is to impose time consistency using Gaussian processes.”

“Therefore, the Gauss-Newton method [12] is frequently used to address such issues in classical Hector SLAM with the second-order Hessian matrix was approximated by the product of the Jacobian matrix.”

“In parallel with B-spline developments, CTME based on temporal Gaussian process (GP) is also active, which—although not exhaustively—can be traced through a series of works [23], [24], [25], [26], [27].”

“in [9]–[11]), the approaches presented therein are developed for networks that assume the conditional independence of specific subsets of the modeled states (e.”

“The MAP estimate of the trajectory can be computed through the GPGN [18].”

“We use a mobile robot dataset similar to the one reported in [2].”

“Furthermore, different from [16], [17], and [18], we prove robust stability of the MHE scheme under mild assumptions.”

“The MAP estimate of the trajectory can be computed through the Gaussian process Gauss-Newton (GPGN) [18].”

“For missing measurement data, [28] formulated two derivations to reflect both the weight-space and function-space approaches from the GP regression literature, and developed an efficient expression based on GPGN to provide state predictions at any times of interest, with a low computational…”

“Simultaneous localization and mapping[1-3] is the key technology for mobile robots to navigate autonomously in an unknown environment.”

“We employ the Gauss-Newton method [31] to address the above optimization problem as follows: where ∆ T is a small disturbance, J i ( T ) and H ( T ) are Jacobian matrix and Hessian matrix of error function, respectively.”

“In this work, we employ the Gauss–Newton method [26] to solve the problem.”

“This belief is obtained by modeling the object trajectory with a Gaussian Process (GP) prior and applying a fixed-lag GP smoother [20–22] to noisy position-only measurements, yielding a Gaussian estimate of the object’s planar position and latent velocity.”

“Within robotics, Tong et al. (2012, 2013) made the initial connection between GP regression and continuous-time estimation for general batch estimation problems (i.e., arbitrarily connected graph structures such as Simultaneous Localization and Mapping (SLAM)).”

“An alternative trajectory modeling approach, proposed by [22], employs sparse GP to represent continuous-time trajectories, modeling system dynamics as white-noise-on-acceleration (WNOA) to preserve sparsity.”

“However, early GP methods required solving high-dimensional, dense inverse matrix for interpolation, incurring high computational costs unsuitable for real-time SLAM [19].”

“Tong et al. [9] first proposed a GP Gauss-Newton optimization for continuous-time state estimation.”

“According to the linear prediction equations [18, (50)], the structure of ˆ X b is as follows: first, ˆ X b is estimated via GP regression as then ˆ X d is computed as a linear combination of ˆ X b , with weights determined explicitly by b T , d T and b δ T,d , i.e., which ˆ X d is clearly…”

“Other approaches have also been suggested to address BO-SLAM, including GraphSLAM [11] and Gaussian Process (GP) regression [12].”

“Initially employed for the rigid body motion representation and state estimation [20], [21], [22], [23], the GP has subsequently been extended to motion planning for robot arms [24], [25], [26], [27].”