When executing a desired motion, robotic controllers remain largely unable to adapt to unexpected contact events. Similarly, they are unable to leverage the potential benefits that could arise from initiating contact, such as bracing against a wall. A local control policy that, in real-time, can adaptively select from a number of potential contacts will enable safer and more capable robots. A fundamental challenge lies in the complexity of the contacts between robot and object. Frictional contact introduces a host of mathematical challenges to the governing equations of motion, particularly discontinuities which result in a combinatorial number of hybrid modes. While current robots must operate with extreme caution, fearful of unexpected contact, research into fundamental control strategies will enable high-performance robots to fluidly and robustly interact with the world. This project will combine theoretical analysis with experimentation on grasping and dexterous manipulation.
@conference{Aydinoglu2020,
title = {Contact-Aware Controller Design for Complementarity Systems},
author = {Alp Aydinoglu, Victor Preciado, Michael Posa},
year = {2019},
date = {2019-09-15},
booktitle = {Under review},
abstract = {While many robotic tasks, like manipulation and locomotion, are fundamentally based in making and breaking contact with the environment, state-of-the-art control policies struggle to deal with the hybrid nature of multi-contact motion. Such controllers often rely heavily upon heuristics or, due to the combinatoric structure in the dynamics, are unsuitable for real-time control.
Principled deployment of tactile sensors offers a promising mechanism for stable and robust control, but modern approaches often use this data in an ad hoc manner, for instance to guide guarded moves. In this work, by exploiting the complementarity structure of contact dynamics, we propose a control framework which can close the loop on rich, tactile sensors. Critically, this framework is non-combinatoric, enabling optimization algorithms to automatically synthesize provably stable control policies. We demonstrate this approach on three different underactuated, multi-contact robotics problems.},
keywords = {}}
While many robotic tasks, like manipulation and locomotion, are fundamentally based in making and breaking contact with the environment, state-of-the-art control policies struggle to deal with the hybrid nature of multi-contact motion. Such controllers often rely heavily upon heuristics or, due to the combinatoric structure in the dynamics, are unsuitable for real-time control.
Principled deployment of tactile sensors offers a promising mechanism for stable and robust control, but modern approaches often use this data in an ad hoc manner, for instance to guide guarded moves. In this work, by exploiting the complementarity structure of contact dynamics, we propose a control framework which can close the loop on rich, tactile sensors. Critically, this framework is non-combinatoric, enabling optimization algorithms to automatically synthesize provably stable control policies. We demonstrate this approach on three different underactuated, multi-contact robotics problems.
@article{Posa16,
title = {Stability analysis and control of rigid-body systems with impacts and friction},
author = { Michael Posa and Mark Tobenkin and Russ Tedrake},
year = {2016},
date = {2016-06-01},
journal = {IEEE Transactions on Automatic Control (TAC)},
volume = {61},
number = {6},
pages = {1423--1437},
abstract = {Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Methods based on sums-of-squares (SOS) for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, and can additionally be used to automatically synthesize stabilizing feedback controllers. 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 safety 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, a perching aircraft, and control design of a balancing robot.},
keywords = {}}
Many critical tasks in robotics, such as locomotion or manipulation, involve collisions between a rigid body and the environment or between multiple bodies. Methods based on sums-of-squares (SOS) for numerical computation of Lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems, and can additionally be used to automatically synthesize stabilizing feedback controllers. 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 safety 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, a perching aircraft, and control design of a balancing robot.
@inproceedings{Posa13,
title = {Lyapunov analysis of rigid body systems with impacts and friction via sums-of-squares},
author = { Michael Posa and Mark Tobenkin and Russ Tedrake},
year = {2013},
date = {2013-01-01},
booktitle = {The 16th International Conference on Hybrid Systems: Computation and Control (HSCC)},
pages = {63--72},
organization = {ACM},
abstract = {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.},
note = {Winner of the Best Paper Award},
keywords = {}}
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.
Legged robots, bipedal or otherwise, promise the capability to traverse complex indoor and outdoor environments. Despite recent success, most control strategies either require precise footfalls or have been carefully hand-designed. Our aim is to research contact-robust locomotion and design algorithms which, given a robot specification and objective, can design efficient and stable motions over challenging terrain. One aspect to this research is to investigate “mid-level” dynamic models of walking robots, models that expose the relevant centroidal dynamics without the unnecessary complexity of full rigid-body descriptions. A mathematically rigorous approach to these problems, utilizing computation and numerical optimization, will be more extensible to new robots and tasks, and ultimately an enabling technology for legged robots to leave research settings. As part of this project, we will combine simulated experiments with laboratory testing on an Agility Robotics Cassie bipedal robot.
@conference{Chen2020,
title = {Optimal Reduced-order Modeling of Bipedal Locomotion},
author = {Yu-Ming Chen, Michael Posa},
year = {2019},
date = {2019-09-15},
booktitle = {Under review},
journal = {Under review},
abstract = {State-of-the-art approaches to legged locomotion are widely dependent on the use of models like the linear inverted pendulum (LIP) and the spring-loaded inverted pendulum (SLIP), popular because their simplicity enables a wide array of tools for planning, control, and analysis. However, they inevitably limit the ability to execute complex tasks or agile maneuvers. In this work, we aim to automatically synthesize models that remain low-dimensional but retain the capabilities of the high-dimensional system. For example, if one were to restore a small degree of complexity to LIP, SLIP, or a similar model, our approach discovers the form of that additional complexity which optimizes performance. In this paper, we define a class of reduced-order models and provide an algorithm for optimization within this class. To demonstrate our method, we optimize models for walking at a range of speeds and ground inclines, for both a five-link model and the Cassie bipedal robot.},
keywords = {}}
State-of-the-art approaches to legged locomotion are widely dependent on the use of models like the linear inverted pendulum (LIP) and the spring-loaded inverted pendulum (SLIP), popular because their simplicity enables a wide array of tools for planning, control, and analysis. However, they inevitably limit the ability to execute complex tasks or agile maneuvers. In this work, we aim to automatically synthesize models that remain low-dimensional but retain the capabilities of the high-dimensional system. For example, if one were to restore a small degree of complexity to LIP, SLIP, or a similar model, our approach discovers the form of that additional complexity which optimizes performance. In this paper, we define a class of reduced-order models and provide an algorithm for optimization within this class. To demonstrate our method, we optimize models for walking at a range of speeds and ground inclines, for both a five-link model and the Cassie bipedal robot.
@inproceedings{Posa17,
title = {Balancing and Step Recovery Capturability via Sums-of-Squares Optimization},
author = { Michael Posa and Twan Koolen and Russ Tedrake},
year = {2017},
date = {2017-01-01},
booktitle = {Robotics: Science and Systems (RSS)},
abstract = {A fundamental requirement for legged robots is to maintain balance and prevent potentially damaging falls whenever possible. As a response to outside disturbances, fall prevention can be achieved by a combination of active balancing actions, e.g. through ankle torques and upper-body motion, and through reactive step placement. While it is widely accepted that stepping is required to respond to large disturbances, the limits of active motions on balancing and step recovery are only well understood for the simplest of walking models. Recent advances in convex optimization-based verification and control techniques enable a more complete understanding of the limits and capabilities of more complex models. In this work, we present an algorithmic approach for formal analysis of the viable-capture basins of walking robots, calculating both inner and outer approximations and corresponding push recovery control strategies. Extending beyond the classic Linear Inverted Pendulum Model (LIPM), we analyze a series of centroidal momentum based planar walking models, examining the effects of center of mass height, angular momentum, and impact dynamics during stepping on capturability. This formal analysis enables an explicit calculation of the differences between these models, and assessment of whether the simplest models ultimately sacrifice capability, and thus stability, when designing push recovery control policies.},
keywords = {}}
A fundamental requirement for legged robots is to maintain balance and prevent potentially damaging falls whenever possible. As a response to outside disturbances, fall prevention can be achieved by a combination of active balancing actions, e.g. through ankle torques and upper-body motion, and through reactive step placement. While it is widely accepted that stepping is required to respond to large disturbances, the limits of active motions on balancing and step recovery are only well understood for the simplest of walking models. Recent advances in convex optimization-based verification and control techniques enable a more complete understanding of the limits and capabilities of more complex models. In this work, we present an algorithmic approach for formal analysis of the viable-capture basins of walking robots, calculating both inner and outer approximations and corresponding push recovery control strategies. Extending beyond the classic Linear Inverted Pendulum Model (LIPM), we analyze a series of centroidal momentum based planar walking models, examining the effects of center of mass height, angular momentum, and impact dynamics during stepping on capturability. This formal analysis enables an explicit calculation of the differences between these models, and assessment of whether the simplest models ultimately sacrifice capability, and thus stability, when designing push recovery control policies.
@inproceedings{Koolen16,
title = {Balance control using center of mass height variation: limitations imposed by unilateral contact},
author = { Twan Koolen and Michael Posa and Russ Tedrake},
year = {2016},
date = {2016-01-01},
booktitle = {2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids)},
pages = {8--15},
publisher = {IEEE},
abstract = {Maintaining balance is fundamental to legged robots. The most commonly used mechanisms for balance control are taking a step, regulating the center of pressure (ankle strategies), and to a lesser extent, changing centroidal angular momentum (e.g., hip strategies). In this paper, we disregard these three mechanisms, instead focusing on a fourth: varying center of mass height. We study a 2D variable-height center of mass model, and analyze how center of mass height variation can be used to achieve balance, in the sense of convergence to a fixed point of the dynamics. In this analysis, we pay special attention to the constraint of unilateral contact forces. We first derive a necessary condition that must be satisfied to be able to achieve balance. We then present two control laws, and derive their regions of attraction in closed form. We show that one of the control laws achieves balance from any state satisfying the necessary condition for balance. Finally, we briefly discuss the relative importance of CoM height variation and other balance mechanisms.},
note = {Finalist for the Best Oral Paper Award},
keywords = {}}
Maintaining balance is fundamental to legged robots. The most commonly used mechanisms for balance control are taking a step, regulating the center of pressure (ankle strategies), and to a lesser extent, changing centroidal angular momentum (e.g., hip strategies). In this paper, we disregard these three mechanisms, instead focusing on a fourth: varying center of mass height. We study a 2D variable-height center of mass model, and analyze how center of mass height variation can be used to achieve balance, in the sense of convergence to a fixed point of the dynamics. In this analysis, we pay special attention to the constraint of unilateral contact forces. We first derive a necessary condition that must be satisfied to be able to achieve balance. We then present two control laws, and derive their regions of attraction in closed form. We show that one of the control laws achieves balance from any state satisfying the necessary condition for balance. Finally, we briefly discuss the relative importance of CoM height variation and other balance mechanisms.
At a fundamental level, multi-contact motion is inherently ill-conditioned. Think, for example, of a four-legged table or a block of wood falling end-first onto the ground. These motions are not only hybrid, but they betray extreme sensitivity to initial conditions. We are interested in describing multi-contact motion on a number of fundamental levels. Can, via basic principles and experimental data, these motions be predicted? When, as in the case of the chaotic bouncing block, reliable prediction is impossible, can we identify and understand this phenomena? In particular, we aim to exploit the structure of both dynamic and quasi-static notions of non-smooth dynamics. This active project incorporates elements of optimization-based modeling, formal analysis, and machine learning.
@conference{Halm2019,
title = {Modeling and Analysis of Non-unique Behaviors in Multiple Frictional Impacts},
author = {Mathew Halm and Michael Posa},
year = {2019},
date = {2019-06-01},
booktitle = {Robotics: Science and Systems (RSS)},
abstract = {Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. To represent the complexity of frictional contact events, impulsive impact models are especially popular, as they often lead to mathematically and computationally tractable approaches. However, when two or more impacts occur simultaneously, the precise sequencing of impact forces is generally unknown, leading to the potential for multiple possible outcomes. This simultaneity is far from pathological, and occurs in many common robotics applications. In this work, we propose an approach for resolving simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh’s method, naturally capture the set of potential post-impact velocities. We prove that solutions to the presented model must terminate. This is, to the best of our knowledge, the first such guarantee for set-valued outcomes to simultaneous frictional impacts.},
keywords = {}}
Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. To represent the complexity of frictional contact events, impulsive impact models are especially popular, as they often lead to mathematically and computationally tractable approaches. However, when two or more impacts occur simultaneously, the precise sequencing of impact forces is generally unknown, leading to the potential for multiple possible outcomes. This simultaneity is far from pathological, and occurs in many common robotics applications. In this work, we propose an approach for resolving simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh’s method, naturally capture the set of potential post-impact velocities. We prove that solutions to the presented model must terminate. This is, to the best of our knowledge, the first such guarantee for set-valued outcomes to simultaneous frictional impacts.
@inproceedings{Halm2018,
title = {A Quasi-static Model and Simulation Approach for Pushing, Grasping, and Jamming},
author = {Mathew Halm and Michael Posa},
year = {2018},
date = {2018-12-01},
booktitle = {The Workshop on the Algorithmic Foundations of Robotics (WAFR)},
journal = {To appear in the Workshop on Algorithmic Foundations of Robotics (WAFR)},
abstract = {Quasi-static models of robotic motion with frictional contact provide a computationally efficient framework for analysis and have been widely used for planning and control of non-prehensile manipulation. In this work, we present a novel quasi-static model of planar manipulation that directly maps commanded manipulator velocities to object motion. While quasi-static models have traditionally been unable to capture grasping and jamming behaviors, our approach solves this issue by explicitly modeling the limiting behavior of a velocity-controlled manipulator. We retain the precise modeling of surface contact pressure distributions and efficient computation of contact-rich behaviors of previous methods and additionally prove existence of solutions for any desired manipulator motion. We derive continuous and time-stepping formulations, both posed as tractable Linear Complementarity Problems (LCPs).},
keywords = {}}
Quasi-static models of robotic motion with frictional contact provide a computationally efficient framework for analysis and have been widely used for planning and control of non-prehensile manipulation. In this work, we present a novel quasi-static model of planar manipulation that directly maps commanded manipulator velocities to object motion. While quasi-static models have traditionally been unable to capture grasping and jamming behaviors, our approach solves this issue by explicitly modeling the limiting behavior of a velocity-controlled manipulator. We retain the precise modeling of surface contact pressure distributions and efficient computation of contact-rich behaviors of previous methods and additionally prove existence of solutions for any desired manipulator motion. We derive continuous and time-stepping formulations, both posed as tractable Linear Complementarity Problems (LCPs).
Trajectory optimization algorithms utilize nonlinear programming to synthesize locally optimal motions for high-dimensional systems with potentially complex sets of goals and constraints. Our research in this area is targeted toward optimization in contact-rich environments, where the exact sequence of contacts and impacts needed for a motion is initially unknown. Given some number of rigid links or points on the robot which might contact the environment, there are exponentially many possible contact states. For scenarios of even modest complexity, it is computationally intractable to optimize over all such states and sequences. Research into contact-implicit optimization directly incorporates the complementarity structure of contact dynamics, and is capable of reasoning over possible contact sequences explicit enumeration. Additional work has focused on refining and tracking these contact-rich motions by appropriately reasoning about the geometric manifold induced by contact.
@inproceedings{Posa16a,
title = {Optimization and Stabilization of Trajectories for Constrained Dynamical Systems},
author = { Michael Posa and Scott Kuindersma and Russ Tedrake},
year = {2016},
date = {2016-01-01},
booktitle = {International Conference on Robotics and Automation (ICRA)},
abstract = {Contact constraints, such as those between a foot and the ground or a hand and an object, are inherent in many robotic tasks. These constraints define a manifold of feasible states; while well understood mathematically, they pose numerical challenges to many algorithms for planning and controlling whole-body dynamic motions. In this paper, we present an approach to the synthesis and stabilization of complex trajectories for both fully-actuated and underactuated robots subject to contact constraints. We introduce an extension to the direct collocation trajectory optimization algorithm that naturally incorporates the manifold constraints to produce a nominal trajectory with third-order integration accuracy---a critical feature for achieving reliable tracking control. We adapt the classical time-varying linear quadratic regulator to produce a local cost-to-go in the tangent plane of the manifold. Finally, we descend the cost-to-go using a quadratic program that incorporates unilateral friction and torque constraints. This approach is demonstrated on three complex walking and climbing locomotion examples in simulation.},
keywords = {}}
Contact constraints, such as those between a foot and the ground or a hand and an object, are inherent in many robotic tasks. These constraints define a manifold of feasible states; while well understood mathematically, they pose numerical challenges to many algorithms for planning and controlling whole-body dynamic motions. In this paper, we present an approach to the synthesis and stabilization of complex trajectories for both fully-actuated and underactuated robots subject to contact constraints. We introduce an extension to the direct collocation trajectory optimization algorithm that naturally incorporates the manifold constraints to produce a nominal trajectory with third-order integration accuracy---a critical feature for achieving reliable tracking control. We adapt the classical time-varying linear quadratic regulator to produce a local cost-to-go in the tangent plane of the manifold. Finally, we descend the cost-to-go using a quadratic program that incorporates unilateral friction and torque constraints. This approach is demonstrated on three complex walking and climbing locomotion examples in simulation.
@article{Posa14,
title = {A direct method for trajectory optimization of rigid bodies through contact},
author = { Michael Posa and Cecilia Cantu and Russ Tedrake},
year = {2014},
date = {2014-01-01},
journal = {The International Journal of Robotics Research (IJRR)},
volume = {33},
number = {1},
pages = {69--81},
abstract = {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 of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the 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.},
keywords = {}}
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 of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the 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.
@inproceedings{Posa12,
title = {Direct Trajectory Optimization of Rigid Body Dynamical Systems Through Contact},
author = { Michael Posa and Russ Tedrake},
year = {2012},
date = {2012-06-01},
booktitle = {The Workshop on the Algorithmic Foundations of Robotics (WAFR)},
pages = {16},
address = {Cambridge, MA},
abstract = {Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Most state-of-the-art techniques treat the discontinuous dynamics of contact 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 $backslash$empha 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 using three increasingly complex systems: rotating a pinned object with a finger, planar walking with the Spring Flamingo robot, and high speed bipedal running on the FastRunner platform.},
keywords = {}}
Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Most state-of-the-art techniques treat the discontinuous dynamics of contact 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 $backslash$empha 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 using three increasingly complex systems: rotating a pinned object with a finger, planar walking with the Spring Flamingo robot, and high speed bipedal running on the FastRunner platform.
