As robots evolve they are becoming increasingly “smarter”, but still can not and do not learn the way humans learn. For example, humans have evolved to move by walking or running; however, humans can learn how to move by other means (e.g. swimming, skateboarding, riding bikes, etc.). I don’t know of any current robot that can learn to move by a different means than what it was design for.

A robot might be able to learn to swim if it were able to map how well it moves in the water to every position it can be in and every action it can take for each of the those positions. The problem with getting a robot to be able to do this mapping is that there are too many possible positions (or states) it can be in and too many different actions it can take. It would never have enough swimming lessons to be able to build an accurate map. In Machine Learning this is known as the “Curse of Dimensionality”.

To mitigate the curse of dimensionality and allow robots to learn how to move on their own we have proposed a novel learning algorithm that learns actions based on boundary conditions and not system states. The details of this work will be available soon through conference and journal publications. Some primarily results can be seen in the video below.

Investigators

• Rowland O’Flaherty

The idea behind this project is to route multiple robots to service spatially distributed requests at specified time instants, while optimizing some criterion, e.g. the total distance travelled. The routing problem is similar to the well known m-TSP or the vehicle routing problem, except that it is temporally constrained in that every request must be met at a particular time instant.

By incorporating these temporal constraints, a notion of directionality appears in the otherwise NP-hard routing problem, and thus, it can be converted to an assignment problem solvable in polynomial time.

In general, we discuss the feasibility aspects of constrained and unconstrained versions of such a routing problem, derive results on the minimum number of robots required, as well as find the corresponding routes that the robots must take in order to solve such a problem.

The problem of spatio-temporal routing is musically inspired in that it requires a bunch of robots to reach a series of planar positions at specified time instants, much like a musician that uses multiple fingers to play a series of notes on an instrument at specified time instants. As a motivating example, we present the Robot Music Wall.

A Motivating Example: The Robot Music Wall

Consider a two-dimensional magnetic-based surface (wall) with a grid of strings in different pitches that generate sound when plucked, as illustrated in the figure above. Distinct positions on the wall correspond to distinct sound frequencies, i.e. distinct notes of an instrument. Multiple robots with the ability to traverse the wall can reach these positions and pluck at the strings above them. In other words, we have a musically instrumented wall where a robot can effectively “play” a musical note by reaching its corresponding position on the wall and plucking the string above it.

With this set-up, we can interpret any piece of music consisting of a series of notes to be played at specified time instants, as a series of corresponding spatio-temporal requests (timed positions) on the music wall. We call such a series a Score, which contains positions that must be reached at specified time instants. Moreover, we might even require that multiple positions are reached simultaneously, akin to a musician that has to play multiple notes of an instrument simultaneously with different fingers. By routing multiple robots to service such timed positions, we can effectively “play” the piece of music associated with them on the wall.

The Robot Music Wall is also an application for group based leader-follower control.

Investigators:

• Smriti Chopra
• Magnus Egerstedt

Related Publications:

S. Chopra and M. Egerstedt. Multi-Robot Routing under Connectivity Constraints. IFAC Workshop on Estimation and Control of Networked Systems, Santa Barbara, CA, Sept. 2012.

S. Chopra and M. Egerstedt. Multi-Robot Routing for Servicing Spatio-Temporal Requests: A Musically Inspired Problem. IFAC Conference on Analysis and Design of Hybrid Systems, Eindhoven, Netherlands, June 2012.

• Puppet Choreography with Motion Programs,
• Imitating Human Motions, and
• Distributed Protocols for Coordinating Puppets.

Puppet Choreography with Motion Programs

Much like Hollywood and Broadway writers, puppeteers write scripts that decribe the elements that make up their performance. These elements include stage setups, characters, timing, and actions. Consequently, puppeteers use these detailed scripts to control the characters (puppets) in an expressive way while simultaneously adhereing to the structure and timing of the play.

Since the plays are described by discrete actions based on a clock mechanism, they may be modeled by a MDL, specifically the variation MDLp. Our research focuses on automating the process of translating puppet play specifications in MDLp to a sequence of valid puppet control modes. Not only are we interested in the specificaiton of these plays, we also wish to optimize the control sequence based on a model of a real puppet in order to create more realistic motion on the real platform.

Check out a movie of our joint work with Northwestern University!

Imitating Human Motions

What does it mean when people say that two motions are “similar?” This is an important question in the area of human-robot interactions, where robots must interpret human movements in order to act in an equivalent manner. Alternatively, suppose we would like a puppet to do “the same thing” as a human. The puppet has different degrees of freedom, different dimensions, and different dynamics from the human: What is the appropriate way to compare puppet motions to human ones, or to control the puppet so it does “the same thing?” This work addresses these questions from two different directions:

1. by directly measuring the preferences of human judges.In this approach, the idea is that when people compare alternatives, they are implicitly making reference to an objective function “inside their heads;” the goal, then, is to reverse-engineer this objective function based on peoples’ expressed preferences.
   
2. by using similarity measures based on homeomorphisms.In this second approach, we “warp” one motion in time and space so that it resembles another motion, and use the amount of warping as our measure of dissimilarity.
   

Distributed Protocols for Coordinating Puppets

In this work, we consider the problem of coordinating multiple pendula attached to mobile bases. In particular,the pendula should move in such a way that their motion is synchronized, which calls for two problems to be solved simultaneously, namely a constrained optimal control problem for each pendulum, and a constrained agreement problem across the network of pendula. Novel ways of manipulating the initial conditions in the consensus equation are employed to deal with the latter of these problems.

Investigators

• Patrick Martin
• Peter Kingston
• Rahul Chipalkatty
• Magnus Egerstedt

In collaboration with Northwestern University:

• Elliot Johnson
• Todd Murphey

and the Center for Puppetry Arts:

• John Ludwig
• Jason Hines

Publications

• P. Kingston, M. Egerstedt. Comparing Apples and Oranges through Partial Orders: An Empirical Approach. To appear atAmerican Control Conference, June 2009.
• P. Martin and M. Egerstedt. Timing Control of Switched Systems with Applications to Robotic Marionettes. To appear inJournal of Discrete Event Dynamic Systems: Theory and Applications, 2009.
• P. Martin and M. Egerstedt. Optimization of Multi-Agent Motion Programs with Applications to Robotic Marionettes.Hybrid Systems: Computation and Control, pp. 262-275, Springer-Verlag, San Francisco, USA, April 2009.
• R. Chipalkatty, M. Egerstedt, and S. Azuma. Multi-Pendulum Synchronization Using Constrained Agreement Protocols.ROBOCOMM, Odense, Denmark, March 2009.
• P. Martin and M. Egerstedt. Optimal Timing Control of Interconnected, Switched Systems with Applications to Robotic Marionettes. Workshop on Discrete Event Systems, Gothenburg, Sweden, May 2008.
• T. Mehta and M. Egerstedt. Multi-Modal Control Using Adaptive Motion Description Languages. Automatica. Accepted for publication. To appear 2008.
• M. Egerstedt, T.Murphey, and J. Ludwig. Motion Programs for Puppet Choreography and Control. Hybrid Systems: Computation and Control, Springer-Verlag, pp. 190-202, Pisa, Italy April 2007.

Sponsors

US National Science Foundation

The primary goal for this research is to develop a method of dynamic role assignment and formation control for multi-robot systems for rotationally and translationally invariant formations. Most previous work has treated assignment and formation synthesis as two separate problems.

Our approach is to determine formation pose as part of the assignment algorithm. This means robots will dynamically determine the formation pose in addition to role assignment. This will allow us to apply techniques such as the hungarian alrogithm even if the formation pose is not known in advance. Furthermore, determining the formation pose during the assignment phase yields a unique point in space to which each robot is assigned. This way, formation synthesis and assignment are accomplished simultaneously without having to introduces additional control laws such as distance based formation control.

Initially, the formation translation and rotation is unknown, nor do the robots know to what role they are assigned in the formation. The goal of the algorithm is to have a network of mobile robots build formations while minimizing the total distance traveled by all robots. The only information available to each robot are the relative positions of all other robots. There is no communication between robots. Since each robot has access to the same information (relative positions between robots), they independently come to the same conclusion.

Robots know their position via an overhead camera system that tracks each robot and broadcasts its position over wifi.

Investigators

• Ted Macdonald

The typical approach in the receding horizon framework is to choose a fixed time horizon over which to predict the unknown variables and obtain the optimal control input. If we had perfect estimates we could then make the time horizon as large as possible subject to factors such as computation speed, convergence, stability, and satisfaction of terminal constraints. However, when we do not have perfect predictions, a time horizon which is too large may be detrimental in that the effect of the poor estimate is amplified over a longer time period. Likewise, if we choose our look-ahead horizon to be too small then the solution may not count on the benefits of the underlying optimal control solution due to the fact that it does not look far into the future to see what is actually optimal.Receding horizon control strategies is a potential remedy to this problem in that they utilize the usefulness of the optimal control strategies while at the same time adding a certain element of feedback into the system. Instead of depending on the model to come up with the entire trajectory, receding horizon strategies only compute the trajectory over a given time horizon and then take a single step along that trajectory. While receding horizon methods are able to capture the desirable benefits of optimal solutions, there is an inherent trade-off between using a large horizon to capture the optimal solution and a short horizon to decrease the detrimental effects of poor predictions.

To find the best look-ahead horizon, ideally, we would like to look at how well our current prediction will compare with future values, but since this is a non-causal problem we must formulate a causal approximation. We make the assumption that our past ability to predict the state will reflect on our future ability to predict the state. Therefore, we look at our previous performance and gauge the look-ahead horizon accordingly.

The figure above shows an example of the time horizon adaptation in a navigation example. Here, the robot has limited information about the surrounding environment. To help the robot adapt to the unknown environment we use two behaviors, avoid obstacle and go-to-goal, to allow the robot to traverse towards the green circle. However, it is not immediately apparent how to choose the weights, so we use a receding horizon strategy to adapt the weights at run time. As mentioned above, it is not clear what time horizon should be used and so we also adapt the horizon according to how well we have been doing at prediction the robot trajectory. On the right-hand side of the figure it is evident that both the behavior weights and the time horizon need to change significantly to best navigate the environment.

Investigators

• Greg Droge
• Magnus Egerstedt

Publications

• G. Droge, M. Egerstedt. Adaptive Time Horizon Optimization in Model Predictive Control. American Control Conference, 2011 .
• Droge, Greg; Egerstedt, Magnus; , “Adaptive look-ahead for robotic navigation in unknown environments,” Intelligent Robots and Systems (IROS) , 2011 IEEE/RSJ International Conference on , vol., no., pp.1134-1139, 25-30 Sept. 2011