Category Archives: 2015

Overview

The motion of physical objects is subject to kinematic and dynamic constraints. For example, a car cannot move sidewise; a bouncing ball must obey the laws of physics. These constraints compound the difficulty of motion planning. We have extended our sampling techniques for path planning to handle objects whose motion is described by a control system, which is a set of differential equations that captures diverse types of kinematic and dynamic constraints. In particular, our implemented algorithm is fast enough to deal with moving obstacles in “real-time”.

Experiments in Simulated Environments

We tested our algorithm on two different systems. The first one consists of two wheeled mobile robots that maintains a direct line of sight as well as a minimum and a maximum distance between them (see the movie below). The second system is a cylindrical robot propelled by eight air-thrusters. It operates on a “frictionless” granite table with moving obstacles and is subject to maximum acceleration bounds due to the limited actuating forces of air-thrusters. Despite the apparent differences in these two systems, our algorithm deals with them in a unified framework.

r2d2

Quicktime movie showing R2-D2 robots
executing motion computed by the motion planner (2.5 MB).
Download the Quicktime viewer

Experiments on Real Robots

We have also experiemented with a hardware implementation of our algorithm on the second system mentioned above at the Stanford University’s Aerospace Robotics Laboratory. The robot floats “frictionlessly” on the granite table using air-bearing. The roughly circular objects on the granite table are moving obstacles. An overhead vision system detects the motion of obstacles. In reponse, the robot computes a collision-free trajectory to the goal state on the fly.

The photos show the motion of the robot in an experiment, in which the robot attempts to move to the goal state at the front of the table.

Initial state Manuever to avoid the incoming obstacle

Moving towards the goal

Wait for the obstacle to pass Goal state

References

  • D. Hsu, R. Kindel, J.C. Latombe, and S. Rock. Randomized kinodynamic motion planning with moving obstaclesInt. J. Robotics Research, 21(3):233–255, 2002.
    PDF

  • D. Hsu, R. Kindel, J.C. Latombe, and S. Rock. Control-based randomized motion planning for dynamic environments. In B.R. Donald and others, editors, Algorithmic and Computational Robotics: New Directions—Proc. Workshop on the Algorithmic Foundations of Robotics (WAFR), pp. 247–264, A. K. Peters, Wellesley, MA, 2000.
    PDF

  • R. Kindel, D. Hsu, J.C. Latombe, and S. Rock. Kinodynamic motion planning amidst moving obstacles. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 537–543, 2000.
    PDF

People

Overview

Computing distance between objects is an important problem in many applications, for example, virtual environment simulation, computer animation, and robot motion planning. Distance information can be used to detect collision, as collision has undesirable consequences in most systems. It can also be used to guide sampling in randomized motion planning algorithms. This project investigates efficient algorithms for collision detection and distance computation. The particular emphasis is on collision detection among moving objects.

H-Walk: Distance Computation for Moving Convex Bodies

The Hierarchical Walk, or H-Walk algorithm, maintains the distance between two moving convex bodies by exploiting both coherence of motion and hierarchical representation. For convex polygons, we have proven that H-Walk improves on the classic Lin-Canny and the Dobkin-Kirkpatrick algorithms. We have also implemented H-Walk for moving convex polyhedra in three dimensions. Experimental results indicate that unlike previous incremental distance computation algorithms, H-Walk scales well with respect to coherence.

Asteriod avoidance

Quicktime movie showing H-Walk in working (0.9 MB).
Download the Quicktime viewer

In our experiments, we used a pair of identical objects. One of them is fixed at the origin, and the other orbits around it and rotates about some randomly chosen axis. The plots below show the performance of H-Walk, as well as that of V-Clip, a very efficient implementation of the Lin-Canny algorithm. In the graphs below, the horizontal axis is some estimate of coherence. The vertical axis is the number of steps that it takes each algorithm to find the closest pair of features. H-Walk can also automatically adjust a parameter in the algorithm in order to respond to changes in the level of coherence and thus maintain the best performance. In the experiments, we kept the parameter fixed in order to see its effect on the algorithm. In the graphs, “h-walk n” stands for H-Walk with the parameter set to n. See the paper for details and more experimental results.

References

  • L.J. Guibas, D. Hsu, and L. Zhang. A hierarchical method for real-time distance computation among moving convex bodiesComputatonal Geometry: Theory and Applications, 15(1-3):51–68, 2000.
    PDF
  • L.J. Guibas, D. Hsu, and L. Zhang. H-Walk: hierarchical distance computation for moving convex bodies. In Proc. ACM Symp. on Computational Geometry, pp. 265–273, 1999.
    PDF

 People

 Acknowledgements

We thank Brian Mirtich for providing us an implementation of the V-Clip alogorithm, which greatly eased our experiments.

PXR Pocket
The binding pocket of hPXR.

Overview

The human pregnane X receptor (hPXR) is a nuclear receptor that binds to various ligands, regulating the breakdown of drugs in the human body. To study drug-drug interactions, we are investigating a method that predicts potential ligand binding conformations in the binding pocket of hPXR.

Binding Conformations from the Geometry of Hydrogen Bonds

The binding pocket of hPXR contains eight polar residues which are key in binding ligands. These polar atoms can potentially share hydrogens with the polar atoms of ligands, forming hydrogen bonds.

A stick for the preferred bond direction from a donor polar atom. Two sticks for the preferred bond directions from an sp2 hybridized acceptor polar atom. A wedge for the preferred bond directions from an sp3 hybridized acceptor polar atom.

Hydrogen bonds have a preferred geometry that depends on the chemical neighborhood of the polar atoms. We model the observed preferred geometry for hydrogen bonding by sticks and wedges. A hydrogen bond is made if two polar atoms are close, and a donor stick is aligned with an acceptor stick or wedge. These geometric constraints are formulated in an optimization problem, whose solutions are ligand conformations that establish at least two hydrogen bonds between a ligand and the binding pocket. We then check the candidate conformations for steric hindrance using a hierarchical collision-detection algorithm and retain the valid conformations in the binding pocket.

References

  • R.-P. Berretty, D. Hsu, L. Kettner, A. Mascarenhas, M.R. Redinbo, and J. Snoeyink. Ligand binding to the pregnane X receptor by geometric matching of hydrogen bonding. In L. Florea and others, editors, Currents in Computational Molecular Biology, pp. 22–23, 2002. The book contains extended poster abstr acts from RECOMB 2002.
    PDF

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Acknowledgements

This work has been partially funded by an NSF-ITR grant.

Overview

Many interesting properties of molecular motion are best characterized statistically by considering an ensemble of motion pathways rather than an individual one. Classic simulation techniques, such as the Monte Carlo method and molecular dynamics, generate individual pathways one at a time and are easily “trapped” in the local minima of the energy landscape. They are computationally inefficient if applied in a brute-force fashion to deal with many pathways. We introduce Stochastic Roadmap Simulation (SRS), a randomized technique for sampling molecular motion and exploring the kinetics of such motion by examining multiple pathways simultaneously.

Stochastic Roadmap Simulation

SRS compactly encode many motion pathways in a directed graph, called a probabilistic conformational roadmap. A node in the graph is a point sampled at random from the conformation space of a molecule. Every path in the roadmap is a potential motion pathway for the molecule. A roadmap thus contains many pathways, with associated probabilities indicating the likelihood that a molecule may follow these pathways. Using tools from the Markov chain theory, we can efficiently obtain kinetic information on the motion of molecules from the roadmap.

We tested SRS by first using it to compute the probability of folding, an important order parameter that measures the “kinetic distance” of a conformation to the native state of a protein. Our computational studies demonstrate that, compared with the Monte Carlo method, SRS obtains more accurate results and reduces the running time by several orders of magnitudes. Problems that required 100 days of computation with the Monte Carlo method were solved in an hour with SRS. Furthermore we proved that, in the limit, SRS converges to the same distribution as the Monte Carlo method.

We then used the probability of folding to estimate the transition state ensemble (TSE) and predicted the rates and the phi-values for protein folding. Of the 16 proteins we studied, comparison with experimental data shows that our method estimates the TSE much more accurately than an existing method based on dynamic programming. This improvement leads to better folding-rate predictions. We also computed the mean first passage time of the unfolded states and show that the computed values correlate with experimentally determined folding rates. The results on phi-value predictions are mixed, possibly due to the simple energy model used in the tests. The comparison with experimental data further validates the SRS method and indicates its potential as a general tool for studying protein folding kinetics.

Predicted folding rates versus the experimentally measured folding rates. The 16 proteins ranged from 56 to 128 amino acids in length.

3CHY sequence of secondary structure formation. Warmer colors indicate higher degree of native contacts attained. The colored bar on the left indicates secondary structures, red for helices and green for strands.

References

  • T.H. Chiang, M.S. Apaydin, D.L. Brutlag, D. Hsu and J.C. Latombe. Predicting Experimental Quantities in Protein Folding Kinetics using Stochastic Roadmap Simulation. In Proc. ACM Int. Conf. on Computational Biology (RECOMB), pp. 410–424, 2006.
    PDF
  • M.S. Apaydin, D.L. Brutlag, C. Guestrin, D. Hsu, J.C. Latombe, and C. Varma. Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motionJ. Computational Biology, 10(3-4):247–281, 2003.
    PDF
  • M.S. Apaydin, D.L. Brutlag, C. Guestrin, D. Hsu, and J.C. Latombe. Stochastic roadmap simulation: An efficient representation and algorithm for analyzing molecular motion. In Proc. ACM Int. Conf. on Computational Biology (RECOMB), pp. 12–21, 2002.
    PDF

People

Overview

Protein conformational changes play a critical role in vital biological functions. Due to noise in data, deteremining salient conformational changes accurately and efficiently is a challenging problem. We have developed an efficient algorithm for analyzing conformational changes of a protein, given its structures in two different conformations. A key element of the algorithm is a statistical test that determines the similarity of two protein structures in the presence of noise. Using data from the Protein Data Bank and the Macromolecular Movements Database, we tested the algorithm on proteins that exhibit a range of different conformational changes. The results show that our algorithm can reliably detect salient conformational changes, including well-known examples such as hinge and shear.

Protein Conformation Flexibility Analysis

pFlexAna analyzes two protein structures in differing conformations and accurately determines residues involved in conformational change. It’s output is a per residue flexibility measure, which also indicates the degree of flexibility in each residue of the protein. The lower this metric, the more flexible the residue.

A salient feature of our method is a principled analysis of statistical noise in structural data, which allows us to reliably distinguish noise from true conformational change. As a result, pFlexAna produces far more accurate results than other methods used to detect conformational changes, as shown in the diagram below. The structure used for this example is the N-lobe of Lactoferrin, which has been shown to undergo hinged conformational changes around residues 90 and 250.

B-factors Torsion angles Sliding RMSD

pFlexAna

Using PDB data, we tested our algorithm on proteins exhibiting a wide range of conformational changes – including TBSV Coat Protein, Lactoferrin, Lactate Dehydrogenase, HIV-1 protease, Adenosylcobinamide Kinase and Aspartate transcarbamoylase. Results for these proteins closely match those reported elsewhere in the literature and are shown below.

Software

The software pFlexAna is an implementation of the algorithm. The source code is availabe for download, and the web server provides direct access to the software.
source code •  documentation • web server

References

  • A. Nigham and D. Hsu. Protein Conformational Flexibility Analysis with Noisy Data. In Proc. ACM Int. Conf. on Computational Biology (RECOMB), 2007.
    PDF

People

Overview

Proper placement of the base of a robot manipulator is an important issue in many robotics applications. For instance, in manufacturing, the base location of a manipulator has a significant impact on the cycle time of tasks such as spot welding and inspection. An automated means to determine the best placement can both increase the throughput of workcells and reduce set-up time. This work investigates methods that employ randomized motion planning techniques to search for the best placement efficiently.

An Iterative Robot Placement Algorithm

Our algorithm computes simultaneously a base location and a corresponding collision-free path that are optimized with respect to the execution time of tasks. The algorithm has been tested on both synthetic examples and real-life CAD data from the automotive industry.

Table

Porsche

Poles

GM cell

Four test scenes (click to enlarge the images).
Porsche Movie
Quicktime movie (1.0 MB).
Download the Quicktime viewer

There is a QuickTime movie for one of the computed examples. The movie clip first shows a PUMA arm at some arbitrary initial base location and executing a path computed by a randomized motion planner, and then shows the manipulator performing the same task after path and base location optimization.

References

  • D. Hsu, J.C. Latombe, and S. Sorkin. Placing a robot manipulator amid obstacles for optimized execution. In Proc. IEEE Int. Symp. on Assembly & Task Planning, pp. 280–285, 1999.
    PDF

People

Acknowledgments

This research has been supported by a grant from SIMA (Stanford Integrated Manufacturing Association), ARO MURI grant DAAH04-96-1-007, a Microsoft Graduate Fellowship, General Motors Corp., and Deneb Robotics Inc.

Overview

Motion planning is concerned with computing collision-free motion for moving objects (e.g., drug molecules inside the binding cavity of a protein, robot manipulators on an assembly line, or articulated models of human arms) in an environment populated with obstacles. The key difficulty of motion planning is its exponential dependence on the number of degrees of freedom of a moving object. The focus of this work is to use random sampling as a fundamental technique for attacking this issue.

Path Planning

In its simplest form, motion planning is a purely geometric problem: given a description of an object and a static environment, the goal is to find a collision-free path to move the object from an initial to a goal configuration. Simple as it may appear, a fast path-planning algorithm is directly useful for applications such as computer animation and virtual prototyping.

The algorithm that we have developed first samples in the neighborhoods of the initial and the goal configuration and then iteratively in the neighborhoods of the newly-sampled configurations, until a path is found. Extensive experimental results indicate that our algorithm is simple to implement and very efficient in practice. The images below show the results on some of the experiments that we have performed.

Alpha

Quicktime movie (2.3 MB).
Download the Quicktime viewer

Expansive Spaces

The analysis of randomized motion planning algorithm is an interesting and difficult question. If there exists a path, does the algorithm always find it? The intuitive explanation for the observed success of randomized motion planning algorithms is that a small number of sampled points can effectively capture the connectivity of the environment. We have introduced the notion of expansiveness to characterize the complexity of the environment. We have proven that in an expansive space, our algorithm quickly finds a collision-free path with high probability, if one exists.

References

  • J. Basch, L.J. Guibas, D. Hsu, and A.T. Nguyen. Disconnection proofs for motion planning. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 1765–1772, 2001.
    PDF
  • D. Hsu. Randomized Single-query Motion Planning in Expansive Spaces. Ph.D. Thesis, Dept. of Computer Science, Stanford University, Stanford, CA, 2000.
    PDF
  • D. Hsu, J.C. Latombe, R. Motwani, and L.E. Kavraki. Capturing the connectivity of high-dimensional geometric spaces by parallelizable random sampling techniques. In P.M. Pardalos and S. Rajasekaran, editors, Advances in randomized parallel computing, pp. 159–182, Kluwer Academic Publishers, Boston, MA, 1999.
    PDF
  • D. Hsu, J.C. Latombe, and R. Motwani. Path planning in expansive configuration spacesInt. J. Computational Geometry & Applications, 9(4-5):495–512, 1999.
    PDF
  • D. Hsu, L.E. Kavraki, J.C. Latombe, R. Motwani, and S. Sorkin. On finding narrow passages with probabilistic roadmap planners. In P.K. Agarwal and others, editors, Robotics: The Algorithmic Perspective: 1998 Workshop on the Algorithmic Foundations of Robotics, pp. 141–154, A. K. Peters, Wellesley, MA, 1998.
    PDF

People

Acknowledgements

We thank Nancy Amato at Texas A&M University and Boris Yamrom at General Electric Corporation for providing us the data used in some of the experiments shown above.

Overview

Several advanced sampling strategies have been proposed in recent years to address the narrow passage problem for probabilistic roadmap (PRM) planning. These sampling strategies all have unique strengths, but none of them solves the problem completely. We investigate general and systematic approaches for adaptively combining multiple sampling strategies so that their individual strengths are preserved. Our preliminary results show that although the performance of individual sampling strategies varies across different environments, the adaptive hybrid sampling strategies that we have constructed perform consistently well. We can also show that, under reasonable assumptions, the adaptive strategies are provably competitive against all individual strategies used.

References

  • D. Hsu, G. Sánchez-Ante, and Z. Sun. Hybrid PRM sampling with a cost-sensitive adaptive strategy. In Proc. IEEE Int. Conf. on Robotics & Automation, 2005.
    PDF

People

Overview

Our motivation is to build autonomous robots that can follow people and recognize their activities. Such capabilities are important in applications such as home care for the elderly, intelligent environments, and iteractive media. In particular, the goal of this project is to develop reliable and efficient motion strategies for an autonomous robot to follow a target and keep it within sensing range. In this research, we investigate the use of greedy strategies using local information to acheive the goal.

A typical target following scenario in an everyday environment, e.g., the school cafeteria.

People walking around create a highly cluttered and dynamic environment in which the robot has to follow the target.Quicktime video (3.4 MB)

Greedy Strategies

Greedy tracking is useful, when very little is known about the environment and the target. Without an a-priori map of the environment, the tracker must acquire local information on the target and the environment using its sensors. In this work, we assume sensors based on a line-of-sight visibility model, e.g. laser range finders, cameras, etc. In particular, we use a SICK LMS200 laser range finder mounted on a robot with differential drive as the tracker in our experiments. The target is a person walking around a crowded environment. The target’s intentions are also unknown in advance, in particular we focus on non-adversarial targets.

Tracker : A SICK Lms-200 laser mounted on a Pioneer-3DX base

The robot’s local information.

Tracking Strategy

The objective of tracking is to keep the target within the tracker’s visibility region at all times. The tracker’s visibility is occluded by the presence of environmental obstacles and the sensing range of the tracker’s visibility sensor. The target can escape the tracker’s view through the gaps in the visibility region, also known as gap edges.

A line-of-sight visibility region has gaps due occlusions and the sensor limits, Range and Field-of-View (FoV). The danger zone around the gap-edge is the region where the target has higher relative vantage than the tracker. The robot balances the short term goal of swinging and long term goal of moving towards the occlusion edge while following the target along a corner. Quicktime video (2.4 MB)

To select actions effectively, the robot must balance between the short-term goal of preventing the immediate loss of the target and the long-term goal of keeping it visible for the maximum duration possible. The tracker does so by intelligently maneuvering the gap-edges away from the target. The ability of the tracker’s maneuvering power, and hence its ability to track the target, depends on its relative position and velocity w.r.t the target towards the gap-edge. This generates a notion of relative-vantage w.r.t the target. We define a region where the target’s ability to escape is greater than the tracker’s ability to eliminate the gap-edge as the danger zone. And so, the target has a higher vantage inside the dangerzone and the tracker outside. Taking this vantage as the objective function, the tracker can optimize its velocity to improve its relative vantage w.r.t the target.

Local Prediction

We predict the target’s motion locally in order to choose actions to give higher weightage to more probable escape points. Although many prediction methods are possible, we use a simple one in which the target’s velocity in the next step is estimated by storing a limited history and extrapolating, and the uncertainty of the target’s velocity estimation is modeled as a probability distribution.

Without prediction the attention of the tracker is spread over all the escape edges. A simple prediction lets the tracker focus on the most probable escape point. The simple probability measure is applied and it generates reliable relative probability of escape of the target through the escape edges Quicktime video (1.6 MB)

2-D Target Tracking

Experiments show that such a tracking strategy performs better than those that just servo onto the target or those that just take the shortest distance to escape of the target into account. Below, the red robot is the target being followed by the green tracker. We see that the vantage tracker is able to track the longest.

Visual Servo Shortest distance based tracker Vantage tracker
The visual servo algorithm does not take into account the changing environment and allows the target to be occluded by the box. Quicktime video (1.7 MB) The vantage tracker explicitly models the visibility relationship of the target, itself and the occlusion due to the box at each instant and successfully follows the target. Quicktime video (3.0 MB)

Stealth Tracking

An advantage of our fomulation for the tracking problem in terms of objective function is the ease with which additional considerations can be accomodated. For example, stealth tracking requires the robot to follow a target without letting itself exposed. In this aspect stealth is actually the inverse of the tracking problem. Our algorithm applies a stealth constraint to the tracking objective function. The stealth constraint requires the robot to stay close to the boundaries of the target’s visibility while keeping the target away from the robot’s visibility boundaries. Due to local sensing, the target’s visibility can only be computed within the robot’s own visibility. We impose a lookout region (L) around the target’s visiblity boundaries in which to constrain the optimization problem.

Lookout regions due to the target’s visibility. Constrained optimization causes the robot to move in a stealthy fashion. Quicktime video (1.7 MB)

Hardware Limitations

The limitations of the hardware also affects the tracking problem. E.g, to cater for the limited field of view (FoV) of a sensing mechanism, the robot must in addition make sure that the target is away from the FoV limits. The tracking strategy considers these limits as pseudo escape edges and tries manipulate these away from the target in a similar fashion to the occlusion edges. To validate our algorithm, we bring the robot to follow a person in the school cafeteria.

The target (red pioneer) is followed in an indoor environment by the robot (green pioneer). The right window shows the visibility and decision making process for the robot. Blue lines denote the escape edges and the green line shows the current decision velocity. Note that when the target marked by T, goes too near the edge, the robot swings to keep it in view. The most critical edge is marked red. Quicktime video(7.4 MB) The canteen environment is inherently complex with the table and chair legs having a similar signature as the target person’s legs. People walking by create visual occlusions and mobility obstructions. Quicktime video (2.2 MB)

Robustness of the vantage tracker

Due to the fast online nature of the vantage tracker, the errors due to uncertainty are bounded, and it is able to recover from temporary occlusions. Such a tracker is also robust to uneven terrains.

Temporary occlusions: A challenging aspect of following the target in a crowd is when someone walks in between the robot and the target. In this video the robot is following the target in green t-shirt when if faces an temporary occlusion by a lady walking across (purple). These temporary occlusions in such a dynamic environment is handled well by the robot. Quicktime video (1.8 MB) Uneven terrain: The robot follows the person successfully in uneven terrains. Notice the high fluctuations in the sensor readings in the bottom right window. The fast recomputation and online nature of the robot helps it in maintaining the target as the robot takes a bumpy ride. Quicktime video (1.5 MB)

Tracking in 3D Environments

A related approach has been developed for tracking in 3-D. Although the basic principles are similar, the 3-D problem is much more difficult. Moreover, there are situations where pure local greedy strategies may not work well.

The consideration of relative vantage in 3-D leads us to vantage zones as shown. The robot tries to manipulate these vantage zones in 3-D leading to tracking motions.

Tracking scenario in 3-D. A single occlusion plane (in blue) being generated due to the obstacle (pink wall). The blue sphere denotes the robot position and the red cube the target. Vantage zone in 3-D. R is the robot, O the occlusion plane, Oe is the lateral edge and g, the occlusion edge. The grey prism represents the obstacle.

Tracking in an urban scenario. A tracking helicopter follows the target in an urban environment using a 3-d scanning laser data points. 3-D Visibility polygon is extracted from the data point cloud. The motion is then generated to optimize risk based on the vantage zones. Quicktime video (3.3 MB)

References

  • T. Bandyopadhyay, D. Hsu, and Ang Jr., M.H.. Motion strategies for people tracking in cluttered dynamic environments. In Proc. Int. Symp. on Experimental Robotics, 2008. To appear.
    PDF
  • T. Bandyopadhyay, Ang Jr., M.H., and D. Hsu. Motion planning for 3-D target tracking among obstacles. In Proc. Int. Symp. on Robotics Research, 2007.
    PDF
  • T. Bandyopadhyay, Y.P. Li, Ang Jr., M.H., and D. Hsu. A greedy strategy for tracking a locally predictable target among obstacles. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 2342–2347, 2006.
    PDF
  • T. Bandyopadhyay, Y.P. Li, Ang Jr., M.H., and D. Hsu. Stealth tracking of an unpredictable target among obstacles. In M. Erdmann and others, editors, Algorithmic Foundations of Robotics VI—Proc. Workshop on the Algorithmic Foundations of Robotics (WAFR), pp. 43–58, Springer, 2004.
    PDF

People

Overview

Our motivation is to build autonomous robots that can follow people and recognize their activities. Such capabilities are important in applications such as home care for the elderly, intelligent environments, and iteractive media. In particular, the goal of this project is to develop reliable and efficient motion strategies for an autonomous robot to follow a target and keep it within the sensor range, despite occlusion by obstacles. We are currently investigating two approaches, POMDPs and greedy strategies.

Experimental setup: a Sick laser mounted on a Pioneer DX3 indoor robot.

POMDP Trackers

Target tracking has two variants that are often studied independently with different approaches: target searching requires a robot to find a target initially not visible, and target following requires a robot to maintain visibility on a target initially visible. We use partially observable Markov decision process (POMDP) to build a single model that unifies target searching and target following. The resulting POMDP policy exhibits interesting tracking behaviors, such as anticipatory moves that exploit target dynamics, information-gathering moves that reduce target position uncertainty, and energy-conserving actions that allow the target to get out of sight, but do not compromise tracking performance.

Some preliminary results are shown in the videos below. The problem setting is motivated by homecare applications. Imagine that an elderly person moves around at home and has a call button to call a robot over for help. The call status stays on for some time and then goes off. If the robot arrives while the call status is on, it gets a reward; otherwise, it gets no reward. Clearly the robot should stay close the person in order to improve the chance of receiving rewards, but at the same time, the robot needs to minimize movement in order to reduce power consumption. So the naive strategy of following right behind the person does not work well. What is interesting about these examples is that the robot manage to “track” the target while the target is outside the robot’s sensor visibility region most of the time.

Quicktime video (1.1 MB)

Quicktime video (1.0 MB)

Quicktime video (2.3 MB)

 

Light blue areas indicate obstacles. The green area around the robot indicate sensor’ visibility region. The various shades of gray show the robot’s belief of the target position. Lighter color indicates higher probability.

References

  • D. Hsu, W.S. Lee, and N. Rong. A point-based POMDP planner for target tracking. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 2644–2650, 2008.
    BibTeX PDF

Greedy Followers

POMDP trackers integrate global information on the target behavior and the environment for optimal decision making. When little is known about the target behavior or the environment, a local greedy strategy is more effective. Key to our algorithm is the definition of a risk function, which tries to capture the target’s ability in escaping from the robot sensors’ visibility region in both short and long terms. To select actions effectively, the robot must balance between the short-term goal of preventing the immediate loss of the target and the long-term goal of keeping it visible for the maximum duration possible. Interestingly, a good comprise can be achieved, using only local information available to the robot’s sensors. By analyzing the local geometry, our algorithm computes a global risk function as a weighted sum of components, each associated with a single visibility constraint. It then chooses an action to minimize the risk locally in a greedy fashion.

As the algorithm uses only local geometric information available to the robot’s visual sensors, it does not require a global map and thus bypasses the difficulty of localization with respect to a global map. Furthermore, uncertainty in sensing and motion control does not accumulate. This improves the reliability of tracking.

This approach can be developed in both 2-D and 3-D. The 3-D case is, however, much more challenging technically, as both the robot and the target gain one more degree of freedom to maneuver and the visibility relationships in 3-D are more complex than those in 2-D. More details can be found here.

Quicktime video (0.6 MB) Quicktime video (1.2 MB)
Quicktime video (0.5 MB) Quicktime video (0.5 MB)
The videos show the same motion of a robot helicopter from two different viewing angles.

References

  • T. Bandyopadhyay, D. Hsu, and Ang Jr., M.H.. Motion strategies for people tracking in cluttered dynamic environments. In Proc. Int. Symp. on Experimental Robotics, 2008. To appear.
    PDF
  • T. Bandyopadhyay, Ang Jr., M.H., and D. Hsu. Motion planning for 3-D target tracking among obstacles. In Proc. Int. Symp. on Robotics Research, 2007.
    PDF
  • T. Bandyopadhyay, Y.P. Li, Ang Jr., M.H., and D. Hsu. A greedy strategy for tracking a locally predictable target among obstacles. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 2342–2347, 2006.
    PDF
  • T. Bandyopadhyay, Y.P. Li, Ang Jr., M.H., and D. Hsu. Stealth tracking of an unpredictable target among obstacles. In M. Erdmann and others, editors, Algorithmic Foundations of Robotics VI—Proc. Workshop on the Algorithmic Foundations of Robotics (WAFR), pp. 43–58, Springer, 2004.
    PDF

Overview

POMDPs provide a principled mathematical framework for planning and decision making under uncertainty. However, POMDPs are often avoided in practice, because solving POMDPs exactly is computationally intractable. Not long ago, the best algorithms could spend hours computing exact solutions to POMDPs with only a dozen states. In recent years, point-based algorithms have made impressive progress in computing approximately optimal solutions for large POMDPs: POMDPs with hundreds of states have been solved in a matter of seconds. This project consists of two main thrusts:

  • understand why the point-based algorithms work well, whether there are sub-classes of POMDPs that are computationally easier, and what complexity measures are suitable for capturing the difficulty of POMDP planning for point-based algorithms, and
  • develop efficient point-based POMDP algorithms by exploting the insights from the theoretical analysis.

Software

  • Dec 15, 2009
  • APPL release 0.91 is available for download here.
  • Feb 15, 2009
  • APPL release 0.3 is available for download here.
  • Jul 17, 2008
  • APPL release 0.2 is available for download here.
  • Apr 15, 2008
  • APPL, our software package for solving POMDPs, is available for download here.

The Covering Number and The Complexity of POMDP Planning

Intuitively, the computational intractability is due to the “curse of dimensionality”: the belief space B used in solving a POMDP typically has dimensionality equal to |S|, the number of states in the POMDP, and therefore the size of B grows exponentially with |S|. As a result, the number of states is often used as an important complexity measure for POMDP planning. However, if the number of states was an accurate measure of the “true” complexity of a POMDP, it would seem surprising that point-based algorithms can obtain (approximate) solutions in seconds in belief spaces of hundreds of dimensions.

We propose to use the covering number as an alternative measure of the complexity of POMDP planning. As expected, not all belief spaces have the same complexity even if they have the same dimenionality. Consider the example below. Although B and B′ are both two-dimensional, it is intuitively clear that the size of B′, which consists only three points, is much smaller than of that B.

The covering number captures several interesting structural properties that drastically reduce the size of the belief space, such as sparsity and factorization. Using the idea of the covering number, we have identified several conditions under which approximate POMDP solutions can be computed efficiently.

SARSOP

One key idea of point-based POMDP algorithms is to sample a set of points from a belief space Band use it as an approximate representation of B, instead of representing B exactly. Some early POMDP algorithms sample the entire belief space B, using a uniform sampling distribution, such as a grid. However, it is difficult to sample a representative set of points from B, due to its large size. More recent point-based algorithms sample only R, the subset of belief points reachable from a given initial point b0 in B, under arbitrary sequences of actions. It is generally believed that R is much smaller than B. Indeed, focusing on R allows point-based algorithms to scale up to larger problems. To push further in this direction, we would like to sample near R*, the subset of belief points reachable from b0 under optimal sequences of actions, as  R* is usually much smaller than R. Of course, the optimal sequences of actions constitute exactly the POMDP solution, which is unknown in advance. The main idea of our algorithm is to compute successive approximations of R* through sampling and converge to it iteratively. Hence the name SARSOP, which stands for Successive Approximations of the Reachable Space under Optimal Policies.

Navigation. Grasping. Target tracking.
Integrated exploration

In simulation, we successfully applied SARSOP to a set of common robotic tasks, including instances of coastal navigation, grasping, mobile robot exploration, and target tracking, all modeled as POMDPs with a large number of states. In most of the instances studied, SARSOP substantially outperformed one of the fastest existing point-based algorithms. The figure below shows the performance of SARSOP on the well-known Tag problem.

References

  • D. Hsu. Towards Large-Scale POMDP Planning for Robotic Tasks. Invited talk at International Conference on Machine Learning (ICML), Workshop on Planning and Acting with Uncertain Models, 2011. Research supported in part by MDA GAMBIT grant R-252-000-398-490, MoE AcRF grant 2010-T2-2-071, and NUS AcRF grant R-252-000-424-112.
    Slides
  • H. Kurniawati, Y. Du, D. Hsu, and W.S. Lee. Motion planning under uncertainty for robotic tasks with long time horizonsInt. J. Robotics Research, 30(3):308-323, 2011.
    PDF
  • S.C.W. Ong, S.W. Png, D. Hsu, and W.S. Lee. Planning under uncertainty for robotic tasks with mixed observabilityInt. J. Robotics Research, 29(8):1053–1068, 2010.
    PDF
  • H. Kurniawati, D. Hsu, and W.S. Lee. SARSOP: Efficient point-based POMDP planning by approximating optimally reachable belief spaces. In Proc. Robotics: Science and Systems, 2008.
    PDF
  • D. Hsu, W.S. Lee, and N. Rong. A point-based POMDP planner for target tracking. In Proc. IEEE Int. Conf. on Robotics & Automation, pp. 2644–2650, 2008.
    PDF
  • D. Hsu, W.S. Lee, and N. Rong. What makes some POMDP problems easy to approximate? In Advances in Neural Information Processing Systems (NIPS), 2007.
    PDF

People

“Movement is the essence of animation.”

– The Technique of Film Animation, J. Halas and R. Manvell

Creating motion for human-like characters is an interesting and important problem. Despite the versatility of human movement, two types of behaviors are essential: locomotion (moving around in an environment) and manipulation (using hands and arms to manipulate objects).

Locomotion

For locomotion, obstacles in the environment can often be tightly bounded by prisms perpendicular to the floor. We can exploit this fact and project the three-dimensional geometry of the environment and the character to a two-dimensional plane. A trajectory in the plane is then computed and tracked by a controller. We have used this idea in the Human Figure Animation Project to control the behavior of a stick figure whose motion is generated by interpolating motion capture data. The combination of motion planning and motion capture techniques renders motion that is both versatile and realistic for character animation.

Manipulating Objects

Manipulating objects

Quicktime movie (2.7 MB)

Manipulating objects with hands and arms is a difficult task. Due to the complex geometric interaction of the character with the environment, keyframe interpolation, which requires lots of user intervention, is too tedious. More advanced methods such as dynamic simulation and space-time constraints are not applicable. In contrast, motion planners are a perfect candidate for this task. It creates collision-free motion with little user intervention and easily handles task-level commands such as “pick up the apple”.

Geometric model of the character made available by Seamless Solutions, Inc.

The emergence of fast randomized motion planners has allowed the motion to be computed at interactive rates. The movie and the images above demonstrate a sequence of movements generated by our motion planner. We specify only three configurations for the arm: the initial configiration, the grasp configuration, and the final goal. The motion planner then automatically generates the collision-free motion without furthur human intervention.

Unfortunately animation generated solely by motion planners often looks “stiff” as the movie clip shows, because motion planners typically operate on simplified models of characters that capture the the functional, but not the aesthetic aspect of movement. A fruitful approach, we believe, is to build a realistic motion model from captured data through machine learning and then synthesize novel motion through motion planning. We are working on techniques to combine model-driven motion planning and data-driven motion capture to create versatile and realistic animation.

Acknowledgements

This research has been supported partically by ARO MURI grant DAAH04-96-1-007 and a Microsoft Graduate Fellowship. We thank Seamless Solutions, Inc. for making available the character model used in the animation.