Probabilistic Control of Mobile Robot

We propose the use of Bayesian approach to a reactive robot control in conjunction with a nonlinear filtering scheme known as particle filters. The approach integrates the optimal control from Bayesian framework with one of the path planning methods known as Vector Field Histogram. Doing this ensures the particle filtering method to track an optimal steering direction. In addition, collision avoidance method is inherently embedded into that scheme due to the fast computing power and a simple implementation of these integrated approach. 1. FROM BAYESIAN TO PARTICLE FILTERS The Bayesian approach provides a formal framework such that measurements, parameter estimation errors, and state/control trajectory costs are to be evaluated. The particle filtering methods allow us to carry out Bayesian parameter and state estimation for a quite general class of nonlinear stochastic systems. In order to obtain the estimate of the system, two models are introduced as follows: a dynamic model and a observation model.The dynamic model here describes the evolution of the state of the system with time. On the other hand, the observation model encapsulates the noisy uncertainty present on measuring the current state. To explicitly deal with the uncertainty present in the observations and the dynamics, the Bayesian approach represents its estimate of the system state as a posterior probability density function (pdf ) computed based on all available information. First, a mathematically tractable representation of the system is needed, often called a state-space model. In the case of detecting the obstacles, state could for example be the position and angle of the obstacles. The state of the system at time t is represented by a random variable xt. Assuming that there are T frames of data to be processed, and at time t only data from times 1 . . . t − 1 are available, the measurements at time t are labeled zt and will contain a list of feature measurements. The measurements up to t are denoted Z, Z = {z1, · · · , zt}. The objective of a Bayesian filter is now to find the posterior density p(xt|Z) conditioned over all observations until time t, using Bayes’ formula: p(xt|Z) = p(xt|zt, Z) = p(zt|xt, Z)p(xt|Z) p(zt|Zt−1) = p(zt|xt)p(xt|Z) p(zt|Zt−1) (1) A full description of this standard particle filter is beyond the scope of this paper, but the interested reader is referred to Ryu et al [2006] for further details. For the sake of understanding the algorithm in this paper, it is sufficient to know that each posterior distribution p(xt|Z) is represented by a collection of M particles {ω t , i = 1, 2, ...,M}. It allows that properties of the posterior distribution can be estimated directly from the collection of particles. Suppose we have two conditional probabilities relating to detection of the obstacle given that all we know is range and angle as follows: p(X|Z Range) (2) p(X|Z Angle) (3) We can combine the evidence from Equation (2) and (3) and apply Bayes’ rule on them. Since it is very difficult to estimate conditional probability for evidence combination, we simplify the application of Bayes’ rule to add one evidence at a time, which is called Bayesian updating. The first statement of Equation (4) can be read as the probability of the current state (i.e., detection of obstacle) at time t given that all we know is the range and angle from sensor. Therefore, we can reformulate the above equations as follows: p(xt|Z range, Z angle) = p(xt|Z angle)p(Zt range|Z angle, xt) p(Zt range|Z angle) = p(xt)p(Z t angle|xt)p(Z range|xt) p(Zt range)p(Z t range|xt) = αp(xt)p(Z t angle|xt)p(Z range|xt) (4) where α is normalization factor, p(xt) prior for the transition model, and p(Z angle|xt)p(Z range|xt) observation model. Equation (4) basically is derivation of the following Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 9130 10.3182/20080706-5-KR-1001.4224 rules: conditional independence of each added observation and Bayesian updating rule. It is noted that this simplified form of Bayesian updating only works when the conditional independence relationships hold. 1.1 Observation Likelihood Model The observation likelihood is probabilities that provide a representation for expressing the certainty about active cell (i,j). Therefore, we must have a function, which transfers a laser scanner reading into appropriate probability for each active cell. The following equations are a set of functions which quantify the observation models of Equation (4) into probabilities. p(Z angle|xt) = 1 − ( θ Amax ) p(Z range|xt) = 1 − ( r Rmax ) (5) where Rmax and Amax are the maximum detection range and the scanning angle, respectively. Equation (5) indicates that the higher the observation likelihood is, the closer the obstacles is to the acoustic axis and likewise the nearer the active cell (i,j) is to the origin of the laser scanner.