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The goal of our research is to build machines which exploit their natural dynamics to achieve extraordinary agility and efficiency. We believe that this challenge involves a tight coupling between mechanical design and underactuated nonlinear control, and that tools from machine learning and optimal control can be used to produce this coupling when classical control techniques fail. Our projects include minimally-actuated dynamic walking on moderate terrain, quadrupedal locomotion on extreme terrain, fixed-wing acrobatics, flapping-winged flight, and feedback control for fluid dynamics.

The Robot Locomotion Group is a part of the CSAIL Center for Robotics.

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Locomotion Group Paper and Multimedia News  

Lyapunov Analysis of Rigid Body Systems with Impacts and Friction via Sums-of-Squares
by Michael Posa and Mark Tobenkin and Russ Tedrake

Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Sums-of-squares (SOS) based methods for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, which can play a powerful role in motion planning and control design. Here, we present a method for applying sums-of-squares verification to rigid bodies with Coulomb friction undergoing discontinuous, inelastic impact events. The proposed algorithm explicitly generates Lyapunov certificates for stability, positive invariance, and reachability over admissible (non-penetrating) states and contact forces. We leverage the complementarity formulation of contact, which naturally generates the semialgebraic constraints that define this admissible region. The approach is demonstrated on multiple robotics examples, including simple models of a walking robot and a perching aircraft.

Camera-ready version to appear in HSCC 2013.

A Direct Method for Trajectory Optimization of Rigid Bodies Through Contact

by Michael Posa and Cecilia Cantu and Russ Tedrake

Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning through contact that eliminates the requirement for an a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem (LCP) for forward simulation, the proposed algorithm leverages Sequential Quadratic Programming (SQP) to naturally resolve contact constraint forces while simultaneously optimizing a trajectory and satisfying nonlinear complementarity constraints. The method scales well to high dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high speed bipedal running on the FastRunner platform.

This is an extended and revised version of our WAFR paper. It is under review. Comments welcome.

Complexity of Ten Decision Problems in Continuous Time Dynamical Systems

by Amir Ali Ahmadi and Anirudha Majumdar and Russ Tedrake

We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudo-polynomial time) algorithm unless P=NP: local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, invariance of a ball, invariance of a quartic semialgebraic set under linear dynamics, local collision avoidance, and existence of a stabilizing control law. We also extend our earlier NP-hardness proof of testing local asymptotic stability for polynomial vector fields to the case of trigonometric differential equations of degree four.

To appear in ACC 2013

A numerical algebraic geometry approach to regional stability analysis

by Frank Permenter and Charles W. Wampler and Russ Tedrake

We explore region of attraction (ROA) estimation for polynomial systems via the numerical solution of polynomial equations. Computing an optimal, stable sub-level set of a Lyapunov function is first posed as a polynomial optimization problem. Solutions to this optimization problem are found by solving a polynomial system of equations using techniques from numerical algebraic geometry. This system describes KKT points and singular points not satisfying a regularity condition. Though this system has exponentially many solutions, the proposed method trivially parallelizes and is practical for problems of moderate dimension and degree. In suitably generic settings, the method solves the underlying optimization problem to arbitrary precision, which could make it a useful tool for studying popular semidefinite programming based relaxations used in ROA analysis.

To appear in ACC 2013

{L2}-Gain Optimization for Robust Bipedal Walking on Unknown Terrain

by Hongkai Dai and Russ Tedrake

In this paper we seek to quantify and explicitly optimize the robustness of a control system for a robot walking on terrain with uncertain geometry. Geometric perturbations to the terrain enter the equations of motion through a relocation of the hybrid event guards which trigger an impact event; these perturbations can have a large effect on the stability of the robot and do not fit into the traditional robust control analysis and design methodologies without additional machinery. We attempt to provide that machinery here. In particular, we quantify the robustness of the system to terrain perturbations by defining an L2 gain from terrain perturbations to deviations from the nominal limit cycle. We show that the solution to a periodic dissipation inequality provides a sufficient upper bound on this gain for a linear approximation of the dynamics around the limit cycle, and we formulate a semidefinite programming problem to compute the L2 gain for the system with a fixed linear controller. We then use either binary search or an iterative optimization method to construct a linear robust controller and to minimize the L2 gain. The simulation results on canonical robots suggest that the L2 gain is closely correlated to the actual number of steps traversed on the rough terrain, and our controller can improve the robot’s robustness to terrain disturbances.

To appear in ICRA 2013

Locomotion Group News  

May 9, 2013. Award. Ani Majumdar and Amir Ali Ahmadi's paper on nonlinear control design along trajectories just won the Best Paper Award at ICRA 2013. Congratulations!

April 10, 2013. Award. Mike Posa and Mark Tobenkin's paper on SOS Verification of Rigid Bodies through Contact won the Best Paper Award at the 16th International Conference on Hybrid Systems: Computation and Control. Congratulations!

May 20, 2012. Award. Russ is the recipient of the 2012 Ruth and Joel Spira Award for Distinguished Teaching.

May 18, 2012. Thesis Defense. John Roberts has successfully defended his thesis on Control of Fluid-Body Systems via Real-Time PIV. Congraulations John!

March 26, 2012. In the news. Our work on flapping flight and perching was featured in the article "A flapping of wings" in this week's issue of Science Magazine. Photo by Jason Dorfman.



June 2, 2011. Award Finalist. Jacob Steinhardt's RSS 2011 paper on stochastic verification was a finalist for the conference Best Student Paper Award. Congratulations Jacob.

May 25, 2011. RSS Workshop. We are co-organizing a workshop at RSS 2011 on "integrated planning and control". As a part of the workshop, we will give a short tutorial on LQR-Trees and Sums-of-Squares Verification for Feedback Motion Planning, which will include tutorial software.

May 12, 2011. Award. Jacob Steinhardt has been awarded the 2011 Robert M. Fano UROP (Undergraduate Research Opportunities Program) award for his outstanding work as an undergraduate researcher. Congratulations Jacob!

May 12, 2011. Award. Hongkai Dai has been awarded the 2011 Frederick C. Hebbie Teaching Award for his outstandng performance as the TA for 6.832 this spring. Congratulations Hongkai!

April 5, 2011. Award. Andy Barry has been awarded an NSF Graduate Research Fellowship. Congratulations Andy!

April 4, 2011. News. CSAIL has posted a short news item about our MURI project.

February 13, 2011. News. Russ has accepted a courtesy appointment with the MIT Department of Aeronautics and Astronautics.

December 4, 2010. Software. We have posted example code for SOS verification of finite-time invariance (e.g. "funnels") along trajectories.

November 28, 2010. Slideshow. A random collection of images from our group lab space appeared in the Sunday edition of the New York Times (Business Day).

August 12, 2010. Video. Lecture videos from the 2010 Dynamic Walking meeting are now available.

July 20, 2010. Award. Rick Cory was named the 2010 Boeing Engineering Student of the Year. Congratulations Rick!! You can watch Boeing's video here.

July 20, 2010. News. Our work on perching was spotlighted on the MIT homepage.

July 16, 2010. MURI. A team with members from MIT, Carnegie Mellon, New York University, Harvard University, and Wageningen was selected for funding on an ONR MURI for UAVs flying through dense forest and urban environments.

July 8, 2010. Dynamic Walking 2010. The dynamic walking meeting was held at MIT. Videos of most of the presentations will be available in the next few days.

June 30, 2010. Talk. Russ gives the Early Career Spotlight Talk at RSS 2010.