Loop detection of mobile robots using interval analysis

Abstract

a b s t r a c tThis paper proposes an original set-membership approach for loop detection of mobile robots in the situation where proprioceptive sensors only are available. To detect loops, the new concepts of the t-plane (which is a two dimensional space with time coordinates) are introduced. Intervals of functions (or tubes) are then used to represent uncertain trajectories and tests are provided in order to eliminate parts of the t-plane that do not correspond to any loop. An experiment with an actual underwater robot is proposed in order to illustrate the principle and the efficiency of the approach.

Proving the existence of loops in robot trajectories

“…Hence, T is a reliable enclosure of T * so that for each t-pair in T, there exist values in the set of measurements that lead to the detection of a feasible loop. Therefore, the following relation is guaranteed: Figure 6 illustrates numerical approximations of T with a SIVIA algorithm [19,2] over several examples. As can be seen, the detection of a potential loop is not a proof of its existence.…”

“…The main contribution of this paper is to provide a reliable existence test that will verify a given loop closure detection. Such test has already been the subject of [2] with a proposition based on the Newton operator [26]. However, this test N is not always able Figure 2: An underwater robot exploring its environment with a single beam echo-sounder.…”

Solving equations and optimization problems with uncertainty

“…Also the efficiency-tuned algorithm presented in this paper can be tweaked into the setting of optimization very easily and thus the binary search avoided. We may assume that both f and o are only given via function values in vertices and simplex-wise Lipschitz constants, and want to approximate inf g−f ≤r max x∈g −1 (0) o(x) for some r. 6 The only difference occurs before each call of Earliest Solution subroutine where we sort the rows of the matrix M and the right-hand side a (n-simplices in the case of primary obstruction) according to their o-filtration value (minimum of o(v) over their vertices v). Also we cut off the columns of the matrix with filtration value larger than r. After the column matrix reduction as described in Appendix B, the desired approximation of the worst-case optimal value is the o-filtration value corresponding to the row of the lowest nonzero element on the right hand side a after the reduction.…”

“…For any such k, Ω(A r k ) is a coset in Z n−1 (X) Ω(A r k ) = x + Z n−1 (X, A r k ; Z) and hence equation (5) reduces to δc = v(x − w), for some w ∈ Z n−1 (X, A r k ; Z), c ∈ C n (X, A r k ; Z 2 ). (6) The crucial property we will use is that v(x − w) is a relative coboundary iff v(x) − v(w) is a relative coboundary: this follows directly from the linearity of the Steenrod square operation H n−1 (X, A; Z 2 ) → H n+1 (X, A; Z 2 ) for n > 3.…”

“…Other instances of uncertain functions come from measurements of physical quantities, such as in medical imaging [12,33,21] or robotics [43,6]. We suppose that potential applications include robust detection of level sets f −1 (a) in medical image processing, analysing robot trajectory based on data obtained from sensors [6,7], or computing the inner approximation of reachable regions of a robotic arm [34]. The algorithm could also be exploited for analysis of functions obtained by regression (say, in machine learning), where the function is chosen to fit some given set of sampled values.…”