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Uncertainty in motion planning. Behrouz Haji Soleimani Dr. Moradi. Outline. What is uncertainty? Some examples Solutions to uncertainty Ignoring uncertainty Markov Decision Process (MDP) Stochastic Motion Roadmap A detailed example Conclusion. What is uncertainty?.
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Uncertainty in motion planning Behrouz Haji Soleimani Dr. Moradi
Outline • What is uncertainty? • Some examples • Solutions to uncertainty • Ignoring uncertainty • Markov Decision Process (MDP) • Stochastic Motion Roadmap • A detailed example • Conclusion
What is uncertainty? • Uncertainty in sensing the current state of the robot and workspace is not known with certainty • Predictability the future state of the robot and workspace cannot be deterministically predicted even when the current state and future actions are known
Uncertainty in sensing • It is not the world that is imperfect, it is our knowledge of it
Predictability • Uncertainty in workspace • Uncertainty in goal location • Dynamic environments with moving obstacles • Uncertainty in robot’s motion
Uncertainty example • A robot with imperfect sensing must reach a goal location among moving obstacles (dynamic world)
Uncertainty example • Robot created at Stanford’s ARL Lab to study issues in robot control and planning in no-gravity space environment air thrusters gas tank air bearing
Markov Decision Process (MDP) • MDP is a general approach to considering uncertainty • Determines model of the environment • Descretizes state space • Requires explicitly defining transition probabilities between states • We can use dynamic programming to solve the MDP
Stochastic Motion Roadmap • Combines a roadmap representation of configuration space with the theory of MDP’s • Maximizes the probability of success • Uses sampling to • learn the configuration space (represented as states) • learn the stochastic motion model (represented as state transition probabilities) • Discretizes state space • Discretizes actions
Stochastic Motion Roadmap • Learning Phase • Selecting random sample of discrete states • Sample the robot’s motion model to build a Stochastic Motion Roadmap (SMR) • Calculating transition probabilities for each action • Query Phase • Specify initial and goal states • Roadmap is used to find a feasible path • Possibly optimizing some criteria such as minimum length
Maximizing probability of success • build an n × n transition probability matrix P(u) for each u U • For each tuple (s, t, p) , we set equals the probability of transitioning from state s to state t given that action u is performed
Maximizing probability of success • It is an MDP and has the form of the Bellman equation Where and It can be optimally solved using infinite horizon dynamic programming
Conclusion • Uncertainty has a great effect on successfully reaching the goal • MDP can consider uncertainty in the model • SMR combines PRM and MDP to handle uncertainty • SMR maximizes the probability of success • SMR makes balance between path safety and minimum length • Continuous actions in SMR is still an open question
References • [1] R. Alterovitz, T. Simeon, and K. Goldberg, “The Stochastic Motion Roadmap: A Sampling Framework for Planning with Markov Motion Uncertainty” 2007 • [2] R. Alterovitz, M. Branicky, and K. Goldberg, “Constant-curvature motion planning under uncertainty with applications in image-guided medical needle steering,” in Workshop on the Algorithmic Foundations of Robotics, July 2006. • [3] R. Alterovitz, A. Lim, K. Goldberg, G. S. Chirikjian, and A. M. Okamura, “Steering flexible needles under Markov motion uncertainty,” in Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS), Aug. 2005, pp. 120–125. • [4] B. Bouilly, T. Simeon, and R. Alami, “A numerical technique for planning motion strategies of a mobile robot in presence of uncertainty,” in Proc. IEEE Int. Conf. on Robotics and Automation (ICRA), Nagoya, Japan, May 1995.
