DAIR Lab | Publications

An up to date list of publications can also be found on Google scholar. When available, the links for each paper will also point to video highlights, code repositories, and presentation recordings.

Preprints

Consensus Complementarity Control for Multi-Contact MPC

Alp Aydinoglu, Adam Wei, and Michael Posa

arXiv preprint arXiv:2304.11259, 2023

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We propose a hybrid model predictive control algorithm, consensus complementarity control (C3), for systems that make and break contact with their environment. Many state-of-the-art controllers for tasks which require initiating contact with the environment, such as locomotion and manipulation, require a priori mode schedules or are too computationally complex to run at real-time rates. We present a method based on the alternating direction method of multipliers (ADMM) that is capable of high-speed reasoning over potential contact events. Via a consensus formulation, our approach enables parallelization of the contact scheduling problem. We validate our results on five numerical examples, including four high-dimensional frictional contact problems, and a physical experimentation on an underactuated multi-contact system. We further demonstrate the effectiveness of our method on a physical experiment accomplishing a high-dimensional, multi-contact manipulation task with a robot arm.
@article{Aydinoglu2023, title = {Consensus Complementarity Control for Multi-Contact MPC}, author = {Aydinoglu, Alp and Wei, Adam and Posa, Michael}, year = {2023}, month = apr, journal = {arXiv preprint arXiv:2304.11259}, youtube = {XBzlyNhKl8w}, arxiv = {2304.11259} }
Integrable Whole-body Orientation Coordinates for Legged Robots

Yu-Ming Chen, Gabriel Nelson, Robert Griffin, Michael Posa, and Jerry Pratt

arXiv preprint arXiv:2210.08111, 2023

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Complex multibody legged robots can have complex rotational control challenges. In this paper, we propose a concise way to understand and formulate a whole-body orientation that (i) depends on system configuration only and not a history of motion, (ii) can be representative of the orientation of the entire system while not be attached to any specific link, and (iii) has a rate of change that approximates total system angular momentum. We relate this orientation coordinate to past work, and discuss and demonstrate, including on hardware, several different uses for it.
@article{Chen2023, title = {Integrable Whole-body Orientation Coordinates for Legged Robots}, author = {Chen, Yu-Ming and Nelson, Gabriel and Griffin, Robert and Posa, Michael and Pratt, Jerry}, journal = {arXiv preprint arXiv:2210.08111}, year = {2023}, month = mar, arxiv = {2210.08111}, youtube = {AZqAWImoAe4} }
Impact-Invariant Control: Maximizing Control Authority During Impacts

William Yang and Michael Posa

arXiv preprint arXiv:2303.00817, 2023

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When legged robots impact their environment, they undergo large changes in their velocities in a short amount of time. Measuring and applying feedback to these velocities is challenging, further complicated by uncertainty in the impact model and impact timing. This work proposes a general framework for adapting feedback control during impact by projecting the control objectives to a subspace that is invariant to the impact event. The resultant controller is robust to uncertainties in the impact event while maintaining maximum control authority over the impact-invariant subspace. We demonstrate the improved performance of the projection over other commonly used heuristics on a walking controller for a planar five-link-biped. The projection is also applied to jumping, box jumping on to a platform 0.4 m tall, and running controllers for the compliant 3D bipedal robot, Cassie. The modification is easily applied to these various controllers and is a critical component to deploying on the physical robot.
@article{Yang2023, title = {Impact-Invariant Control: Maximizing Control Authority During Impacts}, author = {Yang, William and Posa, Michael}, year = {2023}, month = mar, journal = {arXiv preprint arXiv:2303.00817}, arxiv = {2303.00817}, youtube = {_v_CKU47znQ} }
Beyond Inverted Pendulums: Task-optimal Simple Models of Legged Locomotion

Yu-Ming Chen, Jianshu Hu, and Michael Posa

arXiv preprint arXiv:2301.02075, 2023

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Reduced-order models (ROM) are popular in online motion planning due to their simplicity. A good ROM captures the bulk of the full model’s dynamics while remaining low dimension. However, planning within the reduced-order space unavoidably constrains the full model, and hence we sacrifice the full potential of the robot. In the community of legged locomotion, this has lead to a search for better model extensions, but many of these extensions require human intuition, and there has not existed a principled way of evaluating the model performance and discovering new models. In this work, we propose a model optimization algorithm that automatically synthesizes reduced-order models, optimal with respect to any user-specified cost function. To demonstrate our work, we optimized models for a bipedal robot Cassie. We show in hardware experiment that the optimal ROM is simple enough for real time planning application and that the real robot achieves higher performance by using the optimal ROM.
@article{Chen2023b, title = {Beyond Inverted Pendulums: Task-optimal Simple Models of Legged Locomotion}, author = {Chen, Yu-Ming and Hu, Jianshu and Posa, Michael}, year = {2023}, month = jan, journal = {arXiv preprint arXiv:2301.02075}, arxiv = {2301.02075}, youtube = {3tZ32BRIVH4} }
Task-Driven Hybrid Model Reduction for Dexterous Manipulation

Wanxin Jin and Michael Posa

arXiv preprint arXiv:2211.16657, 2022

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In contact-rich tasks, like dexterous manipulation, the hybrid nature of making and breaking contact creates challenges for model representation and control. For example, choosing and sequencing contact locations for in-hand manipulation, where there are thousands of potential hybrid modes, is not generally tractable. In this paper, we are inspired by the observation that far fewer modes are actually necessary to accomplish many tasks. Building on our prior work learning hybrid models, represented as linear complementarity systems, we find a reduced-order hybrid model requiring only a limited number of task-relevant modes. This simplified representation, in combination with model predictive control, enables real-time control yet is sufficient for achieving high performance. We demonstrate the proposed method first on synthetic hybrid systems, reducing the mode count by multiple orders of magnitude while achieving task performance loss of less than 5%. We also apply the proposed method to a three-fingered robotic hand manipulating a previously unknown object. With no prior knowledge, we achieve state-of-the-art closed-loop performance in less than five minutes of online learning.
@article{Jin2023, title = {Task-Driven Hybrid Model Reduction for Dexterous Manipulation}, author = {Jin, Wanxin and Posa, Michael}, journal = {arXiv preprint arXiv:2211.16657}, year = {2022}, month = nov, arxiv = {2211.16657}, youtube = {OvhTOQoagTM}, website = {https://wanxinjin.github.io/td_hybridreduction/} }
Optimization-Based Control for Dynamic Legged Robots

Patrick M Wensing, Michael Posa, Yue Hu, Adrien Escande, Nicolas Mansard, and Andrea Del Prete

arXiv preprint arXiv:2211.11644, 2022

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Over the past decade, we have witnessed rapid growth in the capabilities of legged robots, transitioning from a state of the art where only a few research groups had access to capable platforms, to one where robust locomotion is now common in industry and academic laboratories. This rapid progress has been enabled by a combination of advances across design and control, with optimization-based control strategies playing a central role in many of the milestone demonstrations during this period. Through these advances, current bipeds, quadrupeds, and humanoids can now walk reliably in nominal environments, and these improved capabilities have led to the first practical deployments of legged systems (e.g., the robot Spot from Boston Dynamics). In a world built for legs, the growing scope of these deployments offers a broad opportunity for impact on applications spanning logistics (e.g., delivery), agriculture, and home assistance, among many others.
@article{Wensing2023, title = {Optimization-Based Control for Dynamic Legged Robots}, author = {Wensing, Patrick M and Posa, Michael and Hu, Yue and Escande, Adrien and Mansard, Nicolas and Del Prete, Andrea}, year = {2022}, month = nov, journal = {arXiv preprint arXiv:2211.11644}, arxiv = {2211.11644} }
Set-Valued Rigid Body Dynamics for Simultaneous Frictional Impact

Mathew Halm and Michael Posa

arXiv preprint arXiv:2103.15714, 2021

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Robotic manipulation and locomotion often entail nearly-simultaneous collisions\textemdashsuch as heel and toe strikes during a foot step\textemdashwith outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators commonly lack the accuracy to predict this ordering, and instead pick one with a heuristic. This discrepancy degrades performance when model-based controllers and policies learned in simulation are placed on a real robot. We reconcile this issue with a set-valued rigid-body model which generates a broad set of physically reasonable outcomes of simultaneous frictional impacts. We first extend Routh’s impact model to multiple impacts by reformulating it as a differential inclusion (DI), and show that any solution will resolve all impacts in finite time. By considering time as a state, we embed this model into another DI which captures the continuous-time evolution of rigid body dynamics, and guarantee existence of solutions. We finally cast simulation of simultaneous impacts as a linear complementarity problem (LCP), and develop a probabilistically-complete algorithm for approximating the post-impact velocity set. We demonstrate our approach on several examples drawn from manipulation and legged locomotion.
@article{Halm2021, title = {Set-Valued Rigid Body Dynamics for Simultaneous Frictional Impact}, author = {Halm, Mathew and Posa, Michael}, journal = {arXiv preprint arXiv:2103.15714}, arxiv = {2103.15714}, year = {2021}, month = apr }

Conference and Journal

Real-Time Multi-Contact Model Predictive Control via ADMM

Alp Aydinoglu and Michael Posa

In IEEE International Conference on Robotics and Automation (ICRA), 2022

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We propose a general hybrid model predictive control algorithm, consensus complementarity control (C3), for systems that make and break contact with their environment. Many state-of-the-art controllers for tasks which require initiating contact with the environment, such as locomotion and manipulation, require a priori mode schedules or are so computationally complex that they cannot run at real-time rates. We present a method, based on the alternating direction method of multipliers (ADMM), capable of highspeed reasoning over potential contact events. Via a consensus formulation, our approach enables parallelization of the contact scheduling problem. We validate our results on three numerical examples, including two frictional contact problems, and physical experimentation on an underactuated multi-contact system.
@inproceedings{Aydinoglu2022, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, author = {Aydinoglu, Alp and Posa, Michael}, arxiv = {2109.07076}, month = sep, title = {{Real-Time Multi-Contact Model Predictive Control via ADMM}}, year = {2022}, code = {https://github.com/AlpAydinoglu/coptimal}, youtube = {HyEv-pK9Qfs}, url = {https://ieeexplore.ieee.org/document/9811957}, doi = {10.1109/ICRA46639.2022.9811957} }
Generalization Bounded Implicit Learning of Nearly Discontinuous Functions

Bibit Bianchini, Mathew Halm, Nikolai Matni, and Michael Posa

In Proceedings of The 4th Annual Learning for Dynamics and Control Conference (L4DC), 2022

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Inspired by recent strides in empirical efficacy of implicit learning in many robotics tasks, we seek to understand the theoretical benefits of implicit formulations in the face of nearly discontinuous functions, common characteristics for systems that make and break contact with the environment such as in legged locomotion and manipulation. We present and motivate three formulations for learning a function: one explicit and two implicit. We derive generalization bounds for each of these three approaches, exposing where explicit and implicit methods alike based on prediction error losses typically fail to produce tight bounds, in contrast to other implicit methods with violation-based loss definitions that can be fundamentally more robust to steep slopes. Furthermore, we demonstrate that this violation implicit loss can tightly bound graph distance, a quantity that often has physical roots and handles noise in inputs and outputs alike, instead of prediction losses which consider output noise only. Our insights into the generalizability and physical relevance of violation implicit formulations match evidence from prior works and are validated through a toy problem, inspired by rigid-contact models and referenced throughout our theoretical analysis.
@inproceedings{Bianchini2022, title = {Generalization Bounded Implicit Learning of Nearly Discontinuous Functions}, author = {Bianchini, Bibit and Halm, Mathew and Matni, Nikolai and Posa, Michael}, booktitle = {Proceedings of The 4th Annual Learning for Dynamics and Control Conference (L4DC)}, pages = {1112--1124}, year = {2022}, editor = {Firoozi, Roya and Mehr, Negar and Yel, Esen and Antonova, Rika and Bohg, Jeannette and Schwager, Mac and Kochenderfer, Mykel}, volume = {168}, series = {Proceedings of Machine Learning Research}, month = {23--24 Jun}, publisher = {PMLR}, url = {https://proceedings.mlr.press/v168/bianchini22a.html}, arxiv = {2112.06881} }
Learning Linear Complementarity Systems

Wanxin Jin, Alp Aydinoglu, Mathew Halm, and Michael Posa

In Proceedings of The 4th Annual Learning for Dynamics and Control Conference (L4DC), 2022

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This paper investigates the learning, or system identification, of a class of piecewise-affine dynamical systems known as linear complementarity systems (LCSs). We propose a violation-based loss which enables efficient learning of the LCS parameterization, without prior knowledge of the hybrid mode boundaries, using gradient-based methods. The proposed violation-based loss incorporates both dynamics prediction loss and a novel complementarity - violation loss. We show several properties attained by this loss formulation, including its differentiability, the efficient computation of first- and second-order derivatives, and its relationship to the traditional prediction loss, which strictly enforces complementarity. We apply this violation-based loss formulation to learn LCSs with tens of thousands of (potentially stiff) hybrid modes. The results demonstrate a state-of-the-art ability to identify piecewise-affine dynamics, outperforming methods which must differentiate through non-smooth linear complementarity problems.
@inproceedings{Jin2022, title = {Learning Linear Complementarity Systems}, author = {Jin, Wanxin and Aydinoglu, Alp and Halm, Mathew and Posa, Michael}, booktitle = {Proceedings of The 4th Annual Learning for Dynamics and Control Conference (L4DC)}, pages = {1137--1149}, year = {2022}, editor = {Firoozi, Roya and Mehr, Negar and Yel, Esen and Antonova, Rika and Bohg, Jeannette and Schwager, Mac and Kochenderfer, Mykel}, volume = {168}, series = {Proceedings of Machine Learning Research}, month = {23--24 Jun}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v168/jin22a/jin22a.pdf}, url = {https://proceedings.mlr.press/v168/jin22a.html}, arxiv = {2112.13284}, code = {https://github.com/DAIRLab/Learning-LCS} }
Stabilization of Complementarity Systems via Contact-Aware Controllers

Alp Aydinoglu, Philip Sieg, Victor Preciado, and Michael Posa

IEEE Transactions on Robotics (TRO), 2022

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We propose a framework for provably stable local control of multi-contact robotic systems, directly utilizing force measurements and exploiting the complementarity structure of contact dynamics. Since 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, we present a control framework which can close the loop on tactile sensors. Critically, this framework is non-combinatoric, enabling optimization algorithms to automatically synthesize provably stable control policies. We demonstrate this approach on multiple examples, including underactuated multi-contact problems, quasi-static friction problems and a high-dimensional problem with ten contacts.
@article{Aydinoglu2021b, title = {Stabilization of Complementarity Systems via Contact-Aware Controllers}, author = {Aydinoglu, Alp and Sieg, Philip and Preciado, Victor and Posa, Michael}, journal = {IEEE Transactions on Robotics (TRO)}, year = {2022}, youtube = {fZiJh7coMXc}, arxiv = {2008.02104}, doi = {10.1109/TRO.2021.3120931}, url = {https://ieeexplore.ieee.org/document/9614168}, volume = {38}, number = {3}, pages = {1735-1754} }
Validating Robotics Simulators on Real World Impacts

Brian Acosta*, William Yang*, and Michael Posa

IEEE Robotics and Automation Letters (RA-L), 2022

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A realistic simulation environment is an essential tool in every roboticist’s toolkit, with uses ranging from planning and control to training policies with reinforcement learning. Despite the centrality of simulation in modern robotics, little work has been done to compare the performance of robotics simulators against real-world data, especially for scenarios involving dynamic motions with high speed impact events. Handling dynamic contact is the computational bottleneck for most simulations, and thus the modeling and algorithmic choices surrounding impacts and friction form the largest distinctions between popular tools. Here, we evaluate the ability of several simulators to reproduce real-world trajectories involving impacts. Using experimental data, we identify system-specific contact parameters of popular simulators Drake, MuJoCo, and Bullet, analyzing the effects of modeling choices around these parameters. For the simple example of a cube tossed onto a table, simulators capture inelastic impacts well while failing to capture elastic impacts. For the higher-dimensional case of a Cassie biped landing from a jump, the simulators capture the bulk motion well but the accuracy is limited by numerous model differences between the real robot and the simulators.
@article{Acosta2022, title = {Validating Robotics Simulators on Real World Impacts}, author = {Acosta*, Brian and Yang*, William and Posa, Michael}, journal = {IEEE Robotics and Automation Letters (RA-L)}, arxiv = {2110.00541}, year = {2022}, youtube = {Ao6xJt4TpgU}, url = {https://ieeexplore.ieee.org/document/9772943}, volume = {7}, number = {3}, pages = {6471-6478}, doi = {10.1109/LRA.2022.3174367} }
Fundamental Challenges in Deep Learning for Stiff Contact Dynamics

Mihir Parmar*, Mathew Halm*, and Michael Posa

In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2021

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Frictional contact has been extensively studied as the core underlying behavior of legged locomotion and manipulation, and its nearly-discontinuous nature makes planning and control difficult even when an accurate model of the robot is available. Here, we present empirical evidence that learning an accurate model in the first place can be confounded by contact, as modern deep learning approaches are not designed to capture this non-smoothness. We isolate the effects of contact’s non-smoothness by varying the mechanical stiffness of a compliant contact simulator. Even for a simple system, we find that stiffness alone dramatically degrades training processes, generalization, and data-efficiency. Our results raise serious questions about simulated testing environments which do not accurately reflect the stiffness of rigid robotic hardware. Significant additional investigation will be necessary to fully understand and mitigate these effects, and we suggest several avenues for future study.
@inproceedings{Parmar2021, arxiv = {2103.15406}, author = {Parmar*, Mihir and Halm*, Mathew and Posa, Michael}, booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, month = mar, title = {{Fundamental Challenges in Deep Learning for Stiff Contact Dynamics}}, year = {2021}, youtube = {G0V3_oQCGTk}, code = {https://github.com/DAIRLab/ContactLearningBias}, website = {https://sites.google.com/view/contact-learning-bias}, url = {https://ieeexplore.ieee.org/document/9636383} }
Impact Invariant Control with Applications to Bipedal Locomotion

William Yang and Michael Posa

In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2021

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When legged robots impact their environment, they undergo large changes in their velocities in a small amount of time. Measuring and applying feedback to these velocities is challenging, and is further complicated due to uncertainty in the impact model and impact timing. This work proposes a general framework for adapting feedback control during impact by projecting the control objectives to a subspace that is invariant to the impact event. The resultant controller is robust to uncertainties in the impact event while maintaining maximum control authority over the impact invariant subspace. We demonstrate the utility of the projection on a walking controller for a planar five-link-biped and on a jumping controller for a compliant 3D bipedal robot, Cassie. The effectiveness of our method is shown to translate well on hardware.
@inproceedings{Yang2021, booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, author = {Yang, William and Posa, Michael}, arxiv = {2103.06907}, file = {::}, month = mar, title = {{Impact Invariant Control with Applications to Bipedal Locomotion}}, year = {2021}, youtube = {EF_tWrT-xeQ}, url = {https://ieeexplore.ieee.org/document/9636094} }
Stability analysis of complementarity systems with neural network controllers

Alp Aydinoglu, Fazlyab Mahyar, Manfred Morari, and Michael Posa

In Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control (HSCC), 2021

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Complementarity problems, a class of mathematical optimization problems with orthogonality constraints, are widely used in many robotics tasks, such as locomotion and manipulation, due to their ability to model non-smooth phenomena (e.g., contact dynamics). In this paper, we propose a method to analyze the stability of complementarity systems with neural network controllers. First, we introduce a method to represent neural networks with rectified linear unit (ReLU) activations as the solution to a linear complementarity problem. Then, we show that systems with ReLU network controllers have an equivalent linear complementarity system (LCS) description. Using the LCS representation, we turn the stability verification problem into a linear matrix inequality (LMI) feasibility problem. We demonstrate the approach on several examples, including multi-contact problems and friction models with non-unique solutions.
@inproceedings{Aydinoglu2021, author = {Aydinoglu, Alp and Mahyar, Fazlyab and Morari, Manfred and Posa, Michael}, booktitle = {Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control (HSCC)}, title = {Stability analysis of complementarity systems with neural network controllers}, arxiv = {2011.07626}, year = {2021}, url = {https://dl.acm.org/doi/abs/10.1145/3447928.3456651} }
Contact-Aware Controller Design for Complementarity Systems

Alp Aydinoglu, V.M. Victor Preciado, and Michael Posa

In IEEE International Conference on Robotics and Automation (ICRA), 2020

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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.
@inproceedings{Aydinoglu2020, address = {Paris, France}, author = {Aydinoglu, Alp and Preciado, V.M. Victor and Posa, Michael}, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, doi = {10.1109/ICRA40945.2020.9197568}, isbn = {9781728173955}, issn = {10504729}, title = {{Contact-Aware Controller Design for Complementarity Systems}}, year = {2020}, arxiv = {1909.11221}, url = {https://ieeexplore.ieee.org/abstract/document/9197568/} }
ContactNets: Learning of Discontinuous Contact Dynamics with Smooth, Implicit Representations

Samuel Pfrommer*, Mathew Halm*, and Michael Posa

In The Conference on Robot Learning (CoRL), 2020

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Common methods for learning robot dynamics assume motion is continuous, causing unrealistic model predictions for systems undergoing discontinuous impact and stiction behavior. In this work, we resolve this conflict with a smooth, implicit encoding of the structure inherent to contact-induced discontinuities. Our method, ContactNets, learns parameterizations of inter-body signed distance and contact-frame Jacobians, a representation that is compatible with many simulation, control, and planning environments for robotics. We furthermore circumvent the need to differentiate through stiff or non-smooth dynamics with a novel loss function inspired by the principles of complementarity and maximum dissipation. Our method can predict realistic impact, non-penetration, and stiction when trained on 60 seconds of real-world data.
@inproceedings{Pfrommer2020, arxiv = {2009.11193}, author = {Pfrommer*, Samuel and Halm*, Mathew and Posa, Michael}, booktitle = {The Conference on Robot Learning (CoRL)}, title = {{ContactNets: Learning of Discontinuous Contact Dynamics with Smooth, Implicit Representations}}, year = {2020}, code = {https://github.com/DAIRLab/contact-nets}, youtube = {I6p8JrIp1Es}, url = {https://proceedings.mlr.press/v155/pfrommer21a.html} }
Optimal Reduced-order Modeling of Bipedal Locomotion

Yu-Ming Chen and Michael Posa

In IEEE International Conference on Robotics and Automation (ICRA), 2020

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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{Chen2020, address = {Paris, France}, author = {Chen, Yu-Ming and Posa, Michael}, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, doi = {10.1109/ICRA40945.2020.9197004}, isbn = {9781728173955}, issn = {10504729}, title = {{Optimal Reduced-order Modeling of Bipedal Locomotion}}, year = {2020}, arxiv = {1909.10111}, youtube = {RQfBb8jRDGk}, url = {https://ieeexplore.ieee.org/abstract/document/9197004/} }
Modeling and Analysis of Non-unique Behaviors in Multiple Frictional Impacts

Mathew Halm and Michael Posa

In Robotics: Science and Systems (RSS), 2019

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Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. Models that address frictional contact must be inherently non-smooth; rigid-body 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 present an approach to capturing simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh’s graphical method, naturally capture the set of potential post-impact velocities. We prove that, under modest conditions, the presented approach is guaranteed to terminate. This is, to the best of our knowledge, the first such guarantee for simultaneous frictional impacts.
@inproceedings{Halm2019, address = {Freiburg im Breisgau, Germany}, arxiv = {1902.01462}, author = {Halm, Mathew and Posa, Michael}, booktitle = {Robotics: Science and Systems (RSS)}, title = {{Modeling and Analysis of Non-unique Behaviors in Multiple Frictional Impacts}}, year = {2019}, url = {http://roboticsproceedings.org/rss15/p22.pdf} }
A Quasi-static Model and Simulation Approach for Pushing, Grasping, and Jamming

Mathew Halm and Michael Posa

In The Workshop on the Algorithmic Foundations of Robotics (WAFR), 2018

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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).
@inproceedings{Halm2018, address = {Merida, Mexico}, author = {Halm, Mathew and Posa, Michael}, booktitle = {The Workshop on the Algorithmic Foundations of Robotics (WAFR)}, keywords = {dynamics,linear complementarity problems,manipulation and grasping,quasi-static motion,rigid body motion,simulation}, title = {{A Quasi-static Model and Simulation Approach for Pushing, Grasping, and Jamming}}, year = {2018}, arxiv = {1902.03487}, url = {https://link.springer.com/chapter/10.1007/978-3-030-44051-0_29} }
Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

Michael Posa, Twan Koolen, and Russ Tedrake

In Robotics: Science and Systems (RSS), 2017

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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{Posa2017, author = {Posa, Michael and Koolen, Twan and Tedrake, Russ}, booktitle = {Robotics: Science and Systems (RSS)}, doi = {10.15607/rss.2017.xiii.032}, isbn = {9780992374730}, issn = {2330765X}, title = {{Balancing and Step Recovery Capturability via Sums-of-Squares Optimization}}, volume = {13}, year = {2017}, url = {http://www.roboticsproceedings.org/rss13/p32.pdf} }
Stability analysis and control of rigid-body systems with impacts and friction

Michael Posa, Mark Tobenkin, and Russ Tedrake

IEEE Transactions on Automatic Control (TAC), 2016

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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.
@article{Posa2016, author = {Posa, Michael and Tobenkin, Mark and Tedrake, Russ}, doi = {10.1109/TAC.2015.2459151}, issn = {00189286}, journal = {IEEE Transactions on Automatic Control (TAC)}, keywords = {Control design,Lyapunov analysis and stability verification,rigid-body dynamics with impacts and friction,sumsof-squares (sos)}, month = jun, number = {6}, pages = {1423--1437}, title = {{Stability analysis and control of rigid-body systems with impacts and friction}}, volume = {61}, year = {2016}, url = {https://ieeexplore.ieee.org/document/7163521} }
Optimization and stabilization of trajectories for constrained dynamical systems

Michael Posa, Scott Kuindersma, and Russ Tedrake

In IEEE International Conference on Robotics and Automation (ICRA), 2016

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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\97 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.
@inproceedings{Posa2016a, address = {Stockholm, Sweden}, author = {Posa, Michael and Kuindersma, Scott and Tedrake, Russ}, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, doi = {10.1109/ICRA.2016.7487270}, isbn = {9781467380263}, issn = {10504729}, month = may, pages = {1366--1373}, title = {{Optimization and stabilization of trajectories for constrained dynamical systems}}, volume = {2016-June}, year = {2016}, youtube = {iTDtMTJ1Z14}, url = {https://ieeexplore.ieee.org/abstract/document/7487270/} }
Balance control using center of mass height variation: limitations imposed by unilateral contact

Twan Koolen, Michael Posa, and Russ Tedrake

In IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids), 2016

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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.
@inproceedings{Koolen2016, author = {Koolen, Twan and Posa, Michael and Tedrake, Russ}, booktitle = {IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids)}, doi = {10.1109/HUMANOIDS.2016.7803247}, isbn = {9781509047185}, issn = {21640580}, pages = {8--15}, publisher = {IEEE}, title = {{Balance control using center of mass height variation: limitations imposed by unilateral contact}}, year = {2016}, url = {https://ieeexplore.ieee.org/document/7803247} }
An Architecture for Online Affordance-based Perception and Whole-body Planning

Maurice Fallon, Scott Kuindersma, Sisir Karumanchi, Matthew Antone, Toby Schneider, Hongkai Dai, Claudia Pérez D’Arpino, Robin Deits, Matt DiCicco, Dehann Fourie, Twan Koolen, Pat Marion, Michael Posa, Andrés Valenzuela, Kuan-Ting Yu, Julie Shah, Karl Iagnemma, Russ Tedrake, and Seth Teller

Journal of Field Robotics (JFR), 2015

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@article{Fallon2015, author = {Fallon, Maurice and Kuindersma, Scott and Karumanchi, Sisir and Antone, Matthew and Schneider, Toby and Dai, Hongkai and D'Arpino, Claudia P{\'{e}}rez and Deits, Robin and DiCicco, Matt and Fourie, Dehann and Koolen, Twan and Marion, Pat and Posa, Michael and Valenzuela, Andr{\'{e}}s and Yu, Kuan-Ting and Shah, Julie and Iagnemma, Karl and Tedrake, Russ and Teller, Seth}, doi = {10.1002/rob.21546}, journal = {Journal of Field Robotics (JFR)}, month = mar, number = {2}, pages = {229--254}, title = {{An Architecture for Online Affordance-based Perception and Whole-body Planning}}, url = {http://doi.wiley.com/10.1002/rob.21546}, volume = {32}, year = {2015} }
A Direct Method for Trajectory Optimization of Rigid Bodies Through Contact

Michael Posa, Cecilia Cantu, and Russ Tedrake

International Journal of Robotics Research (IJRR), 2014

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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 environ- ment. 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 dy- namics as a Linear Complementarity Problem (LCP) for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints (MPCC). We leverage Sequential Quadratic Programming (SQP) to naturally resolve contact constraint forces while simul- taneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high dimensional systems with large numbers of possi- ble 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.
@article{Posa2014, author = {Posa, Michael and Cantu, Cecilia and Tedrake, Russ}, doi = {10.1177/0278364913506757}, journal = {International Journal of Robotics Research (IJRR)}, keywords = {Trajectory optimization,locomotion planning,optimal control,planning with contacts,planning with impacts and friction}, month = jan, number = {1}, pages = {69--81}, title = {{A Direct Method for Trajectory Optimization of Rigid Bodies Through Contact}}, url = {http://ijr.sagepub.com/content/33/1/69.short}, volume = {33}, year = {2014} }
Lyapunov Analysis of Rigid Body Systems with Impacts and Friction via Sums-of-Squares

Michael Posa, Mark Tobenkin, and Russ Tedrake

In Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control (HSCC), 2013

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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.
@inproceedings{Posa2013, author = {Posa, Michael and Tobenkin, Mark and Tedrake, Russ}, booktitle = {Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control (HSCC)}, doi = {10.1145/2461328.2461340}, isbn = {9781450315678}, keywords = {Lyapunov analysis and stability verification,Rigid body dynamics with impacts and friction,Sums-of-squares}, month = apr, pages = {63--72}, publisher = {ACM}, title = {{Lyapunov Analysis of Rigid Body Systems with Impacts and Friction via Sums-of-Squares}}, volume = {1}, year = {2013}, url = {https://dl.acm.org/doi/10.1145/2461328.2461340} }
Direct Trajectory Optimization of Rigid Body Dynamical Systems Through Contact

Michael Posa and Russ Tedrake

In The Workshop on the Algorithmic Foundations of Robotics (WAFR), 2012

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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 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 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.
@inproceedings{Posa2012, address = {Cambridge, MA}, author = {Posa, Michael and Tedrake, Russ}, booktitle = {The Workshop on the Algorithmic Foundations of Robotics (WAFR)}, month = jun, pages = {527--542}, title = {{Direct Trajectory Optimization of Rigid Body Dynamical Systems Through Contact}}, year = {2012}, youtube = {pH1pDXnCBx4}, url = {https://link.springer.com/chapter/10.1007/978-3-642-36279-8_32} }

Theses

Optimization for control and planning of multi-contact dynamic motion

Michael Antonio Posa

Massachusetts Institute of Technology, 2017

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The fundamental promise of robotics centers on the ability to productively interact with a complex and changing world. Yet, current robots are largely limited to basic tasks in structured environments and act slowly and cautiously, afraid of incidental contact. In this thesis, we consider a class of control and planning problems for robots dynamically interacting with their environment. We address challenges that arise from non-smooth motions induced by contact, where discontinuities result from impact events and frictional forces. First, we examine the problem of trajectory optimization in contact-rich environments, and present two algorithms for synthesizing motions which make and break contact. The novel contactimplicit trajectory optimization algorithm lifts the problem and reasons over the set of possible contacts forces. In doing so, we eliminate the requirement for an a priori sequencing of the active contacts, and avoid explicit combinatorial complexity. We also introduce a direct collocation algorithm for optimizing high-accuracy trajectories, given an arbitrary contact schedule. This approach eliminates drift in the numerical integration of contact constraints, even when constraints result in closed kinematic chains and require non-minimal coordinates. Second, this thesis concerns questions of control synthesis and provable stability verification of a robot making and breaking contact. To verify stability, we introduce an algorithm for discovering polynomial Lyapunov functions, where the system dynamics include impacts and friction. We leverage the measure differential inclusion representation of non-smooth contact mechanics to efficiently optimize over Lyapunov functions in multi-contact settings. Since avoiding hazardous falls is a primary necessity for bipedal walking robots, we use similar tools to characterize the capabilities of multiple simple models used for balancing and push recovery. Using the notions of barrier functions and occupation measures, we explicitly bound the set of disturbances from which a robot can recover by balancing or stepping. The primary contributions of this thesis are computational in nature, and we heavily leverage modern approaches to both general nonlinear programming and convex optimization. Sums-of-squares, an approach to polynomial optimization utilizing semidefinite programming, plays a central role in our methods for formal stability analysis.
@phdthesis{Posa_thesis, title = {Optimization for control and planning of multi-contact dynamic motion}, author = {Posa, Michael Antonio}, year = {2017}, school = {Massachusetts Institute of Technology}, url = {https://dspace.mit.edu/handle/1721.1/111860} }