Магистерский проект

1. Магистерский проект Simultaneous localization and mapping Магистрант: Матрунич Сергей Анатольевич Научный руководитель: Зимянин Л.Ф. 2008

2. Краткое содержание • Введение • SLAM используя фильтр Калмана • SLAM используя фильтр для частиц • Исследование : проблема поиска

3. Введение: SLAM SLAM: Simultaneous Localization and Mapping Робот изучает незнакомое, статическое окружение. • Дано: • Система управления роботом • Визуальные датчики Оба источника данных зашумлены. • Оценка: • Положения робота -- localization где Я ? • Детализация окружающего пространства – mapping На что похоже то что вокруг меня?

4. a2 at a1 at-1 … x1 x2 xt xt-1 … x0 o2 o1 ot ot-1 … m Введение: SLAM Допущение Маркова State transition: Observation function:

5. p(xt) p(mt) or mt p(x1) p(m1) or m1 p(xt-1) p(mt-1) or mt-1 … Prior distribution on xt after taking action at Введение: SLAM Method: Sequentially estimate the probability distribution p(xt) and update the map. Prior: p(x0)

6. Represent the distribution of robot location xt(and map mt) by a Normal distribution Введение: SLAM Методы: The robot’s trajectory estimateis a tracking problem 1. Parametric method – Kalman filter Sequentially update μt and Σt 2. Sample-based method – particle filter Represent the distribution of robot location xt(and map mt)by a large amount of simulated samples. Resample xt (and mt) at each time step

7. Location error Map error Введение: SLAM Почему SLAM трудная задача? Robot location and map are both unknown. • The small error will quickly accumulated over time steps. • The errors come from inaccurate measurement of actual robot motion (noisy action) and the distance from obstacle/landmark (noisy observation). When the robot closes a physical loop in the environment, serious misalignment errors could happen.

8. SLAM: фильтр Калмана Корректирующее уровнение: Предположение: Prior p(x0) is a normal distribution Observation function p(o|x) is a normal distribution Тогда: Posterior p(x1), …, p(xt) are all normally distributed. Mean μtand covariance matrix Σt can be derived analytically. Sequentially update μt and Σt for each time step t

9. Assume: State transition Observation function Kalman filter: Propagation (motion model): Update (sensor model): SLAM: фильтр Калмана

10. localization mapping SLAM: фильтр Калмана The hidden state for landmark-based SLAM: Map with N landmarks: (3+2N)-dimentional Gaussian State vector xt can be grown as new landmarks are discovered.

11. Основная идея: • Normal distribution assumption in Kalman filter is not necessary • A set of samples approximates the posterior distribution and will be used at next iteration. • Each sample maintains its own map; or all samples maintain a single map. • The map(s) is updated upon observation, assuming that the robot location is given correctly. SLAM: фильтр для частиц Update equation:

12. Particle filter: Assume it is difficult to sample directly from But get samples from another distribution is easy. We sample from , with normalized weight for each xit as The set of (particles) is an approximation of Resamplextfrom ,with replacement, to get a sample set with uniform weights SLAM: фильтр для частиц

13. Particle filter (cont’d): 0.4 0.3 0.2 0.1 SLAM: фильтр для частиц

14. Choose appropriate Transition probability Choose Then Observation function SLAM: фильтр для частиц

15. Алгоритм: Let state xt represent the robot’s location, 1. Propagate each state through the state transition probability . This samples a new state given the previous state. 2. Weight each new state according to the observation function 3. Normalize the weights, get . 4. Resampling: sample Ns new states from are the updated robot location from SLAM: фильтр для частиц

16. are the expected robot moving distance (angle) by taking action at. Measured distance (observation) for sensor k Map distance from location xtto the obstacle SLAM: фильтр для частиц State transition probability: Observation probability:

17. SLAM: фильтр для частиц • Lots of work on SLAM using particle filter are focused on: • Reducing the cumulative error • Fast SLAM (online) • Way to organize the data structure (saving robot path and map). Maintain single map: cumulative error Multiple maps: memory and computation time • In Parr’s paper: • Use ancestry tree to record particle history • Each particle has its own map (multiple maps) • Use observation tree for each grid square (cell) to record the map corresponding to each particle. • Update ancestry tree and observation tree at each iteration. • Cell occupancy is represented by a probabilistic approach.

18. SLAM: фильтр для частиц

19. Проблема поиска Assumption: • The agent doesn’t have map, doesn’t know the underlying model, doesn’t know where the target is. • Agent has 2 sensors: • Camera: tell agent “occupied” or “empty” cells in 4 orientations, noisy sensor. • Acoustic sensor: find the orientation of the target, effective only within certain distance. • Noisy observation, noisy action.

20. Проблема поиска • Similar to SLAM • To find the target, agent need build map and estimate its location. • Differences from SLAM • Rough map is enough; an accurate map is not necessary. • Objective is to find the target. Robot need to actively select actions to find the target as soon as possible.

21. Проблема поиска • Model: • Environment is represented by a rough grid; • Each grid square (state) is either occupied or empty. • The agent moves between the empty grid squares. • Actions: walk to any one of the 4 directions, or “stay”. Could fail in walking with certain probability. • Observations: observe 4 orientations of its neighbor grid squares: “occupied” or “empty”. Could make a wrong observation with certain probability. • State, action and observation are all discrete.

22. Проблема поиска In each step, the agent updates its location and map: • Belief state: the agent believes which state it is currently in. It is a distribution over all the states in the current map. • The map: The agent thinks what the environment is . • For each state (grid square), a 2-dimentional Dirichlet distribution is used to represent the probability of “empty” and “occupied”. • The hyperparameters of Dirichlet distribution are updated based on current observation and belief state.

23. Belief state update: the set of neighbor states of s where Probability of successful moving from sjto s when taking action a From map representation and with Neighbor of s in orientation j Проблема поиска

24. Предпологаем на шаге t-1, гиперпараметр для S На шаге t, гиперпараметр для S обновляется до and are the posterior after observing o given that the agent is in the neighbor of state s. If the probability of wrong observation for any orientation is p0,then p0 if o is “occupied” 1-p0 if o is “empty” prior can be computed similarly. Проблема поиска Обновление карты (распределение Дирихле):

25. Проблема поиска Предпологаемое изменение: a=“up” a=“right” Map representation update: a=“right” a=“up”

26. Проблема поиска Выберите действие: Each state is assigned a reward R(s) according to following rules: Less explored grids have higher reward. Try to walk to the “empty” grid square. Consider neighbor of s with priority. x x

27. Cпасибо за внимание