We consider an agent searching for a set of objects in a partially known environment in which exteroceptive feedback is extremely limited. For instance, we can think of an agent equipped with a range sensor that returns a positive response when in direct contact with an object and no response otherwise. Consider agent living in a Rectangle world, see Figure 1, in which is located two objects. The agent’s uncertainty of its location and that of the objects is encoded by probability distributions P(⋅), which at initialization are known as the agent’s prior beliefs.

As the agent explores the world, it integrates all sensing information at each time step and updates its prior beliefs to posteriors (the result of the prior belief after integrating motion and sensory information). To the author’s knowledge, current SLAM methods are limited in that they consider only uncertainty induced by sensing inaccuracies inherent in the sensor and motion models. The reason for this widespread limitation is that during the theoretical foundation of SLAM, a period that is referred to as the classic age (1986–2004) (Cadena et al., 2016) (this period gave the three paradigms discussed above), the application domains considered the signal noise in range sensor, cameras, and odometry measurements. The features were always considered to be measurable except when occluded. In our setting as the sensory information is strictly haptic, we can confidently assume no measurement noise.

In the search task illustrated in Figure 1, the prior beliefs and sparse measurements are the two sources of the agent’s uncertainty. The absence of a positive measurement (detection of an object of instance) is known as negative information (Thrun, 2002; Hoffman et al., 2005; Thrun et al., 2005, p. 313). Thus, SLAM methodologies that use the Gaussian error between the predicted and estimated position of features, such as in the case of EKF-SLAM and Graph-SLAM, will not perform well in this setting as no measurements of the features positions are available until our agent “bumps” into a feature.

In addition to the negative sensing information, the original beliefs depicted in Figure 1 are non-Gaussian and multimodal. We make no assumption regarding the form of the beliefs. This implies that the joint distribution can no longer be parameterized by a Multivariate Gaussian. This is an assumption made in many SLAM algorithms, notably EKF-SLAM, and allows for a closed form solution to the state estimation problem. Without the Gaussian assumption, no closed form solution to the filtering problem is feasible. Using standard non-parametric methods (Kernel Density, Gaussian Process, Histogram, etc.) to represent or estimate the joint distribution becomes unrealistic after a few dimensions or additional map features. FastSLAM could be a potential candidate, however, as it parameterizes the position uncertainty of the agent by a particle filter and each particle has its own copy of the map the memory demands become quickly significant and in addition has been shown that FastSLAM can degenerate with time regardless of the number of particles used (Bailey et al., 2006). For planning purposes, we would also want to have a single representation of the marginals. The box below summarizes the desirable attributes and assumptions for our filter.

In a wide range of Artificial Intelligence (AI) applications the agent’s beliefs are discrete. This non-parametric representation is the most unconstraining but comes at a cost. The parameterization of the belief’s joint distribution grows at the rate of a double exponential. We propose a Bayesian State Space Estimator (BSSE) which delivers the same filtered beliefs as a traditional filter but without explicitly parametrizing the joint distribution. We refer to our novel filter as the Measurement Likelihood Memory Filter (MLMF). It keeps track of the history of measurement likelihood functions, referred to as the memory, which have been applied to the joint distribution. The MLMF filter efficiently processes negative information. To the author’s knowledge, there has been little research on the integration of negative information in an SLAM setting. Previous work considered the case of active localization (Hoffmann et al., 2006). The incorporation of negative information is useful in many contexts and in particular in Bayesian Theory of Mind (Bake et al., 2011), where the reasoning process of a human is inferred from a Bayesian Network and in our own work (de Chambrier and Billard, 2013) where we model the search behavior of an intentionally blinded human. In such a setting, much negative information is present and an efficient belief filter is required. Our MLMF is thus applicable to the SLAM and AI community in general and to the Cognitive Science community, which models human or agent behaviors through the usage of Bayesian state estimators.

By using this new representation, we implement a set of passive search trajectories through the state space and demonstrate, for a discretized state space, that our novel filter is optimal with respect to the Bayesian criteria (the successive filtered posteriors are exact and not an approximation with respect to Bayes rule). We provide an analysis of the space and time complexity of our algorithm and prove that it is always more efficient even when considering worst-case scenarios. Finally, we consider an Active-SLAM setting and evaluate how constraining the size of the number of memorized likelihood functions impact the decision-making process of a greedy one-step look-ahead planner.

The rest of the document is structured as follows: in section 2, we overview the Bayes filter recursion and apply it to a simple 1D search scenario for both a discrete and Gaussian parametrization of the beliefs. We demonstrate that discrete parametrization gives the correct filtered beliefs but at a very high cost and that the EKF-SLAM fails to provide the adequate solution. In section 3, we introduce the Measurement Likelihood Memory Filter and overview its parametrization. In section 4, we derive the computational time and space complexity of the MLMF. Section 5 describes additional assumptions made with respect to the MLMF to make it scalable (scalable-MLMF). In section 6, we numerically evaluate the time complexity of the scalable-MLMF and check the assumption we made for it to be scalable. We investigate the filter’s sensitivity with respect to its parameters in an Active-SLAM setting.

In EKF-SLAM, the joint density p(A_t, O|Y _0:t, u_1:t; μ_t, Σ_t) is a multivariate Gaussian function parametrized by a mean μ_t and covariance Σ_t. For simplicity, we considering a 1D search of one object. In this setting, the mean of the agent μat∈R and object are μ_o ∈ ℝ are 1D Cartesian positions (1)μt=[μat,μo]T∈ℝ(2×1) Σt=[ΣaΣaoΣoaΣo]∈ℝ(2×2).

The measurement Y_t ∈ [0, 1] of the object by the agent is simulated by a radial basis function as is the measurement function (equation (2)) (2)Y^t=exp−12σ2||At−O||2.

By setting the variance σ² to be small, we can approximate a smooth binary sensor. EKF-SLAM assumes that the measurement is corrupted by Gaussian noise, ϵ ∼𝒩(0, R), giving the likelihood function: (3)p(Yt|At,Ot)=1|2πR|12 exp −12 Yt−Y^t TR−1 Yt−Y^t  where the covariance, R, encompasses the uncertainty in the measurement. The elements of the covariance matrix capture the measurement error between the true Y and expected measurement Y^. As the joint distribution is parameterized by a single Multivariate Gaussian, a closed form solution to the filtering equations exists, called the Kalman Filter (Durrant-Whyte and Bailey, 2006).

In the search task, the agent can only perceive the object when in direct contact. This results in the observation Y_t being mostly always equal to Y^. As a result, the agent’s uncertainty of his location and of the object only changes once the object is found, which is unhelpful during a search. This is because the magnitude of the error between the true and expected measurement e=Yt−Y^t directly affects the joint distribution and is zero during most of the search. Figure 3 illustrates this for the 1D search. The multivariate Gaussian parameterization of the joint and likelihood function only guarantees a dependency between the agent and object random variables when there is a positive sighting of the landmarks. This can be seen in Figure 3B, where the component Σ_ao is 0 most of the time implying A_t ╨ O|Y, which is undesirable. The EKF parameterization in this form is ill suited for search tasks in which negative information is predominant.

In histogram-SLAM, the joint distribution is discretized, and each bin has a parameter, P(A_t = i, O = j|Y _0:t, u_1:t; θ) = θ^(ij) and all sum to one Σ_ijθ^(ij) = 1. For shorthand notation, we will write P(A_t, O|Y _0:t, u_1:t) instead of P(A_t = i, O = j|Y _0:t, u_1:t; θ). The probability distribution of the agent’s position is given by marginalizing the object random variable: (4)P(At|Y0:t,u1:t;θa)=∑j=1|o|P(At,O=j|Y0:t,u1:t; θ).

The converse holds true for the object’s marginal, that is the summation would be over the agent’s variable. We consider a similar 1D search task as the one used for EKF-SLAM. The world of the agent and object is discretized to 10 states, producing a joint distribution with 100 parameters, see Figure 4 (Top) for an illustration of the joint distribution. An observation occurs, as before, only if the agent enters in contact with the object, making the measurement and expected measurement binary Y_t ∈ {0, 1}. Equation (5) is the form of the likelihood function when the object is sensed by the agent (5)P(Yt=1|At,O)=1if At=O0if At≠O.

Figure 4 (Bottom left) illustrates the likelihood function in the case of no contact Y_t = 0. When there is no contact with the object all the parameters of the joint distribution in the black regions become zero, which we refer to as the dependent states A ∩ O of the joint distribution. The white states are the independent states A⊖O, they are not changed by the likelihood function. The values of the joint distribution in the independent states will be unchanged by the likelihood function: P_⊖(A_t, O|Y_0:t, u_1:t) ∝ P_⊖(A_t, O|Y_0:t−1, u_1:t).

When the object is detected (Bottom right) the likelihood constrains all non-zero values of the joint distribution to be in states i = j, which in the case of a 2-dimensional joint distribution is a line. The sparsity of the likelihood function will be key to the development of the MLMF filter.

Figure 5 illustrates the evolution of the joint distribution in a 1D example by applying the filtering Bayesian recursion equations (7) and (8). The agent and object’s true positions (unobservable) are in state 6 and 1. The agent moves three steps toward state 10. At each time step, as the agent fails to sense the object, the likelihood function P(Y_t = 0|A_t, O) (Figure 4, Bottom left) is applied. As the agent moves toward the right, the motion model shifts the joint distribution toward state 10 along the agent’s dimension, i (note that state 1 and 10 are wrapped). As the agent moves to the right more, joint distribution parameters become zero. The re-normalization by the evidence (denominator of equation (8)), which increases the value of the remaining parameters, is equal to the sum of the probability mass, which was set to zero by the likelihood function. Thus, the values of the parameters of the joint distribution that fall on the pink line in Figure 5 (green line also, but only for first time slice) become zero, and their values are redistributed to the remaining non-zero parameters.

The inconvenience with histogram-SLAM is that its time and space complexity is exponential as the joint distribution is discretized and parametrized by θ^(ij). Instead, we propose a new filter, MLMF, which we formally introduce in the next section. This filter achieves the same result as the Histogram filter but without having to parameterize the values of the joint distribution, thus avoiding the exponential growth cost.

Figure 8 (Left) illustrates the volume occupied by the joint distribution for a space with N states. Histogram-SLAM would require N³ parameters for the joint distribution P(A, O⁽¹⁾, O⁽²⁾; θ) and 3 N parameters to store the marginals. In general, for M random variables N^M + M N parameters are necessary, giving an exponential space complexity 𝒪(N^M).

For MLMF-SLAM, each random variable requires two sets of parameters, θ and θ ∗ (see Table 1). Given M random variables, the initial number of parameters is M(2N). At every time step the likelihood memory function increments by one measurement and offset, (Y_t, l  = 0) (Algorithm 1). Given a state space of size N, there can be no more than N different measurement functions (one for each state). In the worst-case scenario, the number of memory likelihood function parameters Ψ_0:t, equation (15), will be N. The total number of parameters is M(2N) + N, which gives a final worst-case space complexity linear in the number of random variables, 𝒪(M N).

For histogram-SLAM, the computational cost is equivalent to that of the space complexity, 𝒪(N^M), since every state in the joint distribution has to be summed to obtain all the marginals.

For MLMF-SLAM, every state in the joint distribution’s state space that has been changed by the likelihood function has to be summed, see Figure 7. As a result, the computational complexity is directly related to the number of dependent states |A ∩ O|. In Figure 7, this corresponds to states where i = j and there are N out of a total N² states for that joint distribution. Figure 8 (Left) illustrates a joint distribution with N³ states. The dependent states |A ∩ O⁽¹⁾ ∩ O⁽²⁾| are those which are within the blue and red planes (where the likelihood evaluates to zero) and comprise N² states each, giving a total of 2N² − N dependent states (negative is to remove the states we count twice at the intersection of the blue and red plane).

The likelihood term P(Y_t|A_t, O⁽¹⁾) evaluates states to zero, which satisfies (i = j, ∀k), as the measurement of object O⁽¹⁾ is independent of object O⁽²⁾. With 3 objects, the joint distribution would be P(A_t = i, O⁽¹⁾ = j, O⁽²⁾ = k, O^(l) = l) then the likelihood P(Y_t|A_t, O⁽¹⁾) evaluated to zero for (i = j, ∀k, ∀l) which would mean N³ dependent states. In general, for M random variables the computational cost is (M − 1) N^M−1 which gives 𝒪(N^M−1) as opposed to the histogram-SLAM’s 𝒪(N^M). The computation complexity in this setup is still exponential but to the order M − 1 as opposed to M, which nevertheless quickly limits the scalability as more objects are added.

Computing the value of a dependent state (i, j, k) in the joint distribution required evaluating equation (12), which contains a product of N likelihood functions, in the worst-case scenario. However, the likelihood functions are not overlapping and binary. As a result, the complete product does not have to be evaluated since only one likelihood function will affect the state (i, j, k). Thus, evaluating equation (12) yields a cost of 𝒪(1).

We measured the time taken by the motion-measurement update loop, as a function of the number beliefs and number of states per belief. We started with 100 states per belief and gradually increase it to 10,000,000 over 50 steps. Each of the 50 steps treated 2–25 objects. Figure 11 (Left) illustrates the computational cost as a function of number of states and objects. For each state-object pair 100, motion-measurement updates were performed. Most of the trials returned time updates below 1 Hz. Figure 11 (Right) shows the computational cost as a function of the number of states plotted for 6 different filter runs with a different number of objects. As the number of states increases exponentially so does the computational cost. Note the cost increases at the same rate as the number of states, which increase at a rate exponential in the number of random variables, meaning that the computational complexity is linear with respect to the number of states. This result is in agreement with the asymptotic time complexity.

In section 5, we made the assumption (for scalability reasons) that the objects’ beliefs are independent of one another. This assumption is validated by comparing the MLMF filter on three random variables, an agent and two objects, with the ground truth, which we obtain from the standard histogram filter. For each of the three beliefs (the agent and two objects), 100 different marginals were generated and the true locations (actual position of the agent and objects) were sampled. Figure 12 (Top left) illustrates one instance of the initialization of the agent and object marginals with their associated sampled true position. The agent carries out a sweep of the state space for each of the marginals and the policy is saved and run with the scalable-MLMF filter. In the first experiment, we assumed that the objects are completely independent and that there was no transfer of information between the pair-wise joint distributions. In the second and third experiments, there is an exchange of information as described in Algorithm A2. Here, we compare the effect of using the product of the agent’s marginals, equation (24), with the average of the marginals, equation (25). We expect the average of the agent’s marginal to yield a result closer to the ground truth as the marginal of the agent P(A_t|Y _0:t, u_1:t) at initialization is the same as the ground truth (the histogram-SLAM’s). As for the marginal of the objects P(O⁽ⁱ⁾|Y _0:t), we expect the difference between them to be independent of whether the product or average of the agent’s marginal is used. This results from Algorithm A2. When an object i is sensed all the corresponding agent marginals P(A^(j)|u_1:t) are set equal to P(A⁽ⁱ⁾|u_1:t) and not to P(A_t|Y _0:t, u_1:t). This is a design decision of our information transfer heuristic. There are many other possibilities but this is one of the simplest. For each of the 100 sweeps, the ground truth is compared with the scalable-MLMF using the Hellinger distance (equation (26)) (26)H(P,Q)=12  P−Q 2 which is a metric that measures the distance between two probability distributions. Its value lies strictly between 0 (the two distributions are identical) and 1 (no overlap between them). Figure 12 shows the kernel density distribution of the Hellinger distances taken at each time step for all 100 sweeps. In the Top left of the figure, for the case when no transfer of information is a applied, all the marginals are far from the ground truth. This results from the introduction of the independence assumption, necessary to scale the MLMF. Figure 12 (Bottom) shows the results for difference between the product and average of the agents marginals. As expected, there is no difference between the objects’ marginals when considering both methods (product and average) with respect to the ground truth. The predominant difference occurs in the agent’s marginal P(A_t|Y _0:t, u_1:t). This is also expected and prompted the introduction of the average method instead of the product.

The scalable-MLMF information exchange heuristic will not lead to any of the objects marginals probability mass being falsely removed during the information transfer, which is close to a winner-take-all approach in terms of beliefs. When object i is sensed its associated agent marginal is set to all other agent–object joint pairs, which results in the information accumulated in the jth agent marginals being replaced by the ith.

The memory measurement likelihood function P(Y _0:t|A_t, O, u_1:t; Ψ_0:t) is parameterized by the history of all the measurement likelihood functions that have been applied to the joint distribution since initialization. As detailed earlier, there can be no more than |Ψ_0:t| ≤ N different measurement likelihood functions added to memory. In the case of a very large state space, this might be cumbersome. We investigate how restricting the memory size, the number of parameters |Ψ_0:t|, can impact on the decision process in an Active-SLAM setting. Given our setup, a breadth-first search in the action space is chosen with a one-time step horizon, making it a greedy algorithm. The objective function utilized is the information gain of the beliefs after applying an action (equation (27)) (27)ut=arg maxutHP(At−1,O|Y0:t−1,u1:t−1)−EYt HP(At,O|Y0:t,u1:t).

For each action the filter is run forward in time and all future measurements since we cannot know ahead of time the actual measurement. The information gain is the difference between the current entropy (defined by H{⋅}) and the future entropy after the simulated motion and measurement update. The action with the highest information gain is subsequently selected. This is repeated at each time step. Figure 13 illustrates the environment setup for a 1D and 2D case. The agent’s task is to find the objects in the environment.

For the 2D search, we consider three different initializations (single-Gaussian, four-Gaussian, Uniform) for the agent’s belief where there are two objects to be found. Ten searches are carried out for each of the three initializations of the agent’s beliefs. The agent’s true location, for each search, is sampled from its initial belief, and the objects’ locations (red and green squares in Figure 13) are kept fixed throughout all searches. Each search is repeated for 18 different memory sizes ranging from 1 to N (the number of states). For the 1D search case, one object is considered since adding more objects makes the search easier and the interest lies in the memory effects of the search and not the search itself. In Figures 14 and 15, we report on the time taken to find all objects with respect to a given memory size, which is shown as the percentage of the total number of states. In the 1D search case, the time variability taken to find the object converges when the memory size is at 60% of the original state space. As for the 2D search with 2 beliefs (agent and 1 object), the convergence depends on the agent’s initial belief. For the 1-Gaussian (green line), all searches take approximately the same amount of time after a memory size of 9%. As for the remaining two initializations, convergence is achieved at 48%. The same holds true for the case of 3 beliefs (agent and 2 objects).

In the 2D searches, the memory size has a less impact on the time taken to find the objects than in the 1D (which is a special search case). Only when the memory size is less than 6% is there a significant change. We conclude that at least in the case of the greedy one-step look-ahead planner that is frequently used in the literature, the size of the memory seems not to be a limiting factor in terms of the time taken to accomplish the search.