Michael Posa

A key challenge in contact-rich dexterous manipulation is the need to jointly reason over global geometry and nonsmooth contact dynamics. End-to-end policies bypass this complexity, but often require large amounts of data and transfer poorly from simulation to reality. We address the limitations with a simple insight: dexterous manipulation is inherently hierarchical--at a high level, a robot decides where to touch (geometry); at a low level it determines how to move the object through contact dynamics. Building on this insight, we propose a hierarchical RL--MPC framework in which a high-level reinforcement learning (RL) policy predicts a contact intention, a novel object-centric interface that specifies (i) an object-surface contact location and (ii) a post-contact object subgoal pose. Conditioned on the contact intention, a low-level contact-implicit model predictive control (MPC) optimizes local contact modes and real-time (re)plans through contact dynamics to generate robot actions that robustly move the object toward each subgoal. We evaluate the framework on non-prehensile tasks, including geometry-generalized pushing across diverse object shapes, pivoting/flipping-based object reorientation, and environment-assisted object repositioning. It achieves high success rate with substantially reduced data (10 times less than end-to-end baselines), highly robust performance, and zero-shot sim-to-real transfer.

A key challenge in contact-rich dexterous manipulation is the need to jointly reason over global geometry and nonsmooth contact dynamics. End-to-end policies bypass this complexity, but often require large amounts of data and transfer poorly from simulation to reality. We address the limitations with a simple insight: dexterous manipulation is inherently hierarchical--at a high level, a robot decides where to touch (geometry); at a low level it determines how to move the object through contact dynamics. Building on this insight, we propose a hierarchical RL--MPC framework in which a high-level reinforcement learning (RL) policy predicts a contact intention, a novel object-centric interface that specifies (i) an object-surface contact location and (ii) a post-contact object subgoal pose. Conditioned on the contact intention, a low-level contact-implicit model predictive control (MPC) optimizes local contact modes and real-time (re)plans through contact dynamics to generate robot actions that robustly move the object toward each subgoal. We evaluate the framework on non-prehensile tasks, including geometry-generalized pushing across diverse object shapes, pivoting/flipping-based object reorientation, and environment-assisted object repositioning. It achieves high success rate with substantially reduced data (10 times less than end-to-end baselines), highly robust performance, and zero-shot sim-to-real transfer.

Object geometry is key information for robot manipulation. Yet, object reconstruction is a challenging task because cameras only capture partial observations of objects, especially when occlusion occurs. In this paper, we leverage two extra sources of information to reduce the ambiguity of vision signals. First, generative models learn priors of the shapes of commonly seen objects, allowing us to make reasonable guesses of the unseen part of geometry. Second, contact information, which can be obtained from videos and physical interactions, provides sparse constraints on the boundary of the geometry. We combine the two sources of information through contact-guided 3D generation. The guidance formulation is inspired by drag-based editing in generative models. Experiments on synthetic and real-world data show that our approach improves the reconstruction compared to pure 3D generation and contact-based optimization.

Abstract When legged robots impact their environment executing dynamic motions, 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, 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. Code and video of the experiments are available at https://impact-invariant-control.github.io/ .

Non-prehensile manipulation of diverse objects remains a core challenge in robotics, driven by unknown physical properties and the complexity of contact-rich interactions. Recent advances in contact-implicit model predictive control (CI-MPC), with contact reasoning embedded directly in the trajectory optimization, have shown promise in tackling the task efficiently and robustly. However, demonstrations have been limited to narrowly curated examples. In this work, we showcase the broader capabilities of CI-MPC through precise planar pushing tasks over a wide range of object geometries, including multi-object domains. These scenarios demand reasoning over numerous inter-object and object-environment contacts to strategically manipulate and de-clutter the environment, challenges that were intractable for prior CI-MPC methods. To achieve this, we introduce Consensus Complementarity Control Plus (C3+), an enhanced CI-MPC algorithm integrated into a complete pipeline spanning object scanning, mesh reconstruction, and hardware execution. Compared to its predecessor C3, C3+ achieves substantially faster solve times, enabling real-time performance even in multi-object pushing tasks. On hardware, our system achieves overall 98% success rate across 33 objects, reaching pose goals within tight tolerances. The average time-to-goal is approximately 0.5, 1.6, 3.2, and 5.3 minutes for 1-, 2-, 3-, and 4-object tasks, respectively. Project page: https://dairlab.github.io/push-anything.

General robot manipulation requires the handling of previously unseen objects. Learning a physically accurate model at test time can provide significant benefits in data efficiency, predictability, and reuse between tasks. Tactile sensing can compliment vision with its robustness to occlusion, but its temporal sparsity necessitates careful online exploration to maintain data efficiency. Direct contact can also cause an unrestrained object to move, requiring both shape and location estimation. In this work, we propose a learning and exploration framework that uses only tactile data to simultaneously determine the shape and location of rigid objects with minimal robot motion. We build on recent advances in contact-rich system identification to formulate a loss function that penalizes physical constraint violation without introducing the numerical stiffness inherent in rigid-body contact. Optimizing this loss, we can learn cuboid and convex polyhedral geometries with less than 10s of randomly collected data after first contact. Our exploration scheme seeks to maximize Expected Information Gain and results in significantly faster learning in both simulated and real-robot experiments. More information can be found at https://dairlab.github.io/activetactile

Bipedal robots demonstrate potential in navigating challenging terrains through dynamic ground contact. However, current frameworks often depend solely on proprioception or use manually designed visual pipelines, which are fragile in real-world settings and complicate real-time footstep planning in unstructured environments. To address this problem, we present a vision-based hierarchical control framework that integrates a reinforcement learning high-level footstep planner, which generates footstep commands based on a local elevation map, with a low-level Operational Space Controller that tracks the generated trajectories. We utilize the Angular Momentum Linear Inverted Pendulum model to construct a low-dimensional state representation to capture an informative encoding of the dynamics while reducing complexity. We evaluate our method across different terrain conditions using the underactuated bipedal robot Cassie and investigate the capabilities and challenges of our approach through simulation and hardware experiments.

To achieve general-purpose dexterous manipulation, robots must rapidly devise and execute contact-rich behaviors. Existing model-based controllers cannot globally optimize in real time over the exponential number of possible contact sequences. Instead, progress in contact-implicit control leverages simpler models that, while still hybrid, make local approximations. Locality limits the controller to exploit only nearby interactions, requiring intervention to richly explore contacts more broadly. Our approach leverages the strengths of local complementarity-based control combined with low-dimensional, but global, sampling of possible end effector locations. Our key insight is to consider a contact-free stage preceding a contact-rich stage at every control loop. Our algorithm, in parallel, samples end effector locations to which the contact-free stage can move the robot, then considers the cost predicted by contact-rich MPC local to each sampled location. The result is a globally-informed, contact-implicit controller capable of real-time dexterous manipulation. We demonstrate our controller on precise, non-prehensile manipulation of non-convex objects with a Franka arm. Project webpage: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://approximating-global-ci-mpc.github.io</uri>.

To achieve general-purpose dexterous manipulation, robots must rapidly devise and execute contact-rich behaviors. Existing model-based controllers are incapable of globally optimizing in real-time over the exponential number of possible contact sequences. Instead, recent progress in contact-implicit control has leveraged simpler models that, while still hybrid, make local approximations. However, the use of local models inherently limits the controller to only exploit nearby interactions, potentially requiring intervention to richly explore the space of possible contacts. We present a novel approach which leverages the strengths of local complementarity-based control in combination with low-dimensional, but global, sampling of possible end-effector locations. Our key insight is to consider a contact-free stage preceding a contact-rich stage at every control loop. Our algorithm, in parallel, samples end effector locations to which the contact-free stage can move the robot, then considers the cost predicted by contact-rich MPC local to each sampled location. The result is a globally-informed, contact-implicit controller capable of real-time dexterous manipulation. We demonstrate our controller on precise, non-prehensile manipulation of non-convex objects using a Franka Panda arm. Project page: https://approximating-global-ci-mpc.github.io

Traversing rough terrain requires dynamic bipeds to stabilize themselves through foot placement without stepping in unsafe areas. Planning these footsteps online is challenging given non-convexity of the safe terrain, and imperfect perception and state estimation. This paper addresses these challenges with a full-stack perception and control system for achieving underactuated walking on discontinuous terrain. First, we develop model-predictive footstep control (MPFC), a single mixed-integer quadratic program which assumes a convex polygon terrain decomposition to optimize over discrete foothold choice, footstep position, ankle torque, template dynamics, and footstep timing at over 100 Hz. We then propose a novel approach for generating convex polygon terrain decompositions online. Our perception stack decouples safe-terrain classification from fitting planar polygons, generating a temporally consistent terrain segmentation in real time using a single CPU thread. We demonstrate the performance of our perception and control stack through outdoor experiments with the underactuated biped Cassie, achieving state of the art perceptive bipedal walking on discontinuous terrain. Supplemental Video: https://youtu.be/JK16KJXJxi4

Robotic manipulation and locomotion often entail nearly-simultaneous collisions—such as heel and toe strikes during a foot step—with outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators and state estimation commonly lack the fidelity and 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 outcomes to simultaneous frictional impacts with any impact ordering. 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 an algorithm for tight approximation of the post-impact velocity set with probabilistic guarantees. We demonstrate our approach on several examples drawn from manipulation and legged locomotion, and compare the predictions to other models of rigid and compliant collisions.

Non-prehensile manipulation enables fast interactions with objects by circumventing the need to grasp and ungrasp as well as handling objects that cannot be grasped through force closure. Current approaches to non-prehensile manipulation focus on static contacts, avoiding the underactuation that comes with sliding. However, the ability to control sliding contact, essentially removing the no-slip constraint, opens up new possibilities in dynamic manipulation. In this paper, we explore a challenging dynamic non-prehensile manipulation task that requires the consideration of the full spectrum of hybrid contact modes. We leverage recent methods in contact-implicit MPC to handle the multi-modal planning aspect of the task. We demonstrate, with careful consideration of integration between the simple model used for MPC and the low-level tracking controller, how contact-implicit MPC can be adapted to dynamic tasks. Surprisingly, despite the known inaccuracies of frictional rigid contact models, our method is able to react to these inaccuracies while still quickly performing the task. Moreover, we do not use common aids such as reference trajectories or motion primitives, highlighting the generality of our approach. To the best of our knowledge, this is the first application of contact-implicit MPC to a dynamic manipulation task in three dimensions.

Model-based approaches for planning and control for bipedal locomotion have a long history of success. It can provide stability and safety guarantees while being effective in accomplishing many locomotion tasks. Model-free reinforcement learning, on the other hand, has gained much popularity in recent years due to computational advancements. It can achieve high performance in specific tasks, but it lacks physical interpretability and flexibility in re-purposing the policy for a different set of tasks. For instance, we can initially train a neural network (NN) policy using velocity commands as inputs. However, to handle new task commands like desired hand or footstep locations at a desired walking velocity, we must retrain a new NN policy. In this work, we attempt to bridge the gap between these two bodies of work on a bipedal platform. We formulate a model-based reinforcement learning problem to learn a reduced-order model (ROM) within a model predictive control (MPC). Results show a 49% improvement in viable task region size and a 21% reduction in motor torque cost. All videos and code are available at https://sites.google.com/view/ymchen/research/rl-for-roms.

The hybrid nature of multi-contact robotic systems, due to making and breaking contact with the environment, creates significant challenges for high-quality control. Existing model-based methods typically rely on either good prior knowledge of the multi-contact model or require significant offline model tuning effort, thus resulting in low adaptability and robustness. In this paper, we propose a real-time adaptive multi-contact model predictive control framework, which enables online adaption of the hybrid multi-contact model and continuous improvement of the control performance for contact-rich tasks. This framework includes an adaption module, which continuously learns a residual of the hybrid model to minimize the gap between the prior model and reality, and a real-time multi-contact MPC controller. We demonstrated the effectiveness of the framework in synthetic examples, and applied it on hardware to solve contact-rich manipulation tasks, where a robot uses its end-effector to roll different unknown objects on a table to track given paths. The hardware experiments show that with a rough prior model, the multi-contact MPC controller adapts itself on-the-fly with an adaption rate around 20 Hz and successfully manipulates previously unknown objects with non-smooth surface geometries. Accompanying media can be found at: https://sites.google.com/view/adaptive-contact-implicit-mpc/home

In contact-rich tasks, the hybrid, multi-modal nature of contact dynamics poses great challenges in model representation, planning, and control. Recent efforts have attempted to address these challenges via data-driven methods, learning dynamical models in combination with model predictive control. Those methods, while effective, rely solely on minimizing forward prediction errors to hope for better task performance with MPC controllers. This weak correlation can result in data inefficiency as well as limitations to overall performance. In response, we propose a novel strategy: using a policy gradient algorithm to find a simplified dynamics model that explicitly maximizes task performance. Specifically, we parameterize the stochastic policy as the perturbed output of the MPC controller, thus, the learned model representation can directly associate with the policy or task performance. We apply the proposed method to contact-rich tasks where a three-fingered robotic hand manipulates previously unknown objects. Our method significantly enhances task success rate by up to 15% in manipulating diverse objects compared to the existing method while sustaining data efficiency. Our method can solve some tasks with success rates of 70% or higher using under 30 minutes of data. All videos and codes are available at https://sites.google.com/view/lcs-rl.

We propose a hybrid model predictive control algorithm, consensus complementarity control, 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 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 multicontact system. We further demonstrate the effectiveness of our method on a physical experiment accomplishing a high-dimensional, multicontact manipulation task with a robot arm.

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 within a few minutes of online learning, by collecting only a few thousand environment samples.

Reduced-order models (ROM) are popular in online motion planning due to their simplicity. A good ROM for control captures critical task-relevant aspects of the full dynamics while remaining low dimensional. 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 a user-specified distribution of tasks and corresponding cost functions. To demonstrate our work, we optimized models for a bipedal robot Cassie. We show in simulation that the optimal ROM reduces the cost of Cassie's joint torques by up to 23% and increases its walking speed by up to 54%. We also show hardware result that the real robot walks on flat ground with 10% lower torque cost. All videos and code can be found at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://sites.google.com/view/ymchen/research/optimal-rom</uri> .

Bipedal robots promise the ability to traverse rough terrain quickly and efficiently, and indeed, humanoid robots can now use strong ankles and careful foot placement to traverse discontinuous terrain. However, more agile underactuated bipeds have small feet and weak ankles, and must maintain balance primarily through constant adjustment of their next footstep position. We introduce a new model-predictive foot placement controller for underactuated bipedal robots which jointly optimizes over the robot's discrete choice of stepping surface, impending footstep position sequence, ankle torque in the sagittal plane, and center of mass trajectory, to track a velocity command. The controller is formulated as a single Mixed Integer Quadratic Program (MIQP) which is solved at 50–200 Hz, depending on terrain complexity. We implement a state of the art real-time elevation mapping and convex terrain decomposition framework to inform the controller of its surroundings in the form on convex polygons representing steppable terrain. We investigate the capabilities and challenges of our approach through hardware experiments on the underactuated biped Cassie.

Robotic manipulation can greatly benefit from the data efficiency, robustness, and predictability of model-based methods if robots can quickly generate models of novel objects they encounter. This is especially difficult when effects like complex joint friction lack clear first-principles models and are usually ignored by physics simulators. Further, numerically-stiff contact dynamics can make common model-building approaches struggle. We propose a method to simultaneously learn contact and continuous dynamics of a novel, possibly multi-link object by observing its motion through contact-rich trajectories. We formulate a system identification process with a loss that infers unmeasured contact forces, penalizing their violation of physical constraints and laws of motion given current model parameters. Our loss is unlike prediction-based losses used in differentiable simulation. Using a new dataset of real articulated object trajectories and an existing cube toss dataset, our method outperforms differentiable simulation and end-to-end alternatives with more data efficiency. See our project page for code, datasets, and media: https://sites.google.com/view/continuous-contact-nets/home

In a world designed for legs, quadrupeds, bipeds, and humanoids have the opportunity to impact emerging robotics applications from logistics, to agriculture, to home assistance. The goal of this survey is to cover the recent progress toward these applications that have been driven by model-based optimization for the real-time generation and control of movement. The majority of the research community has converged on the idea of generating locomotion control laws by solving an optimal control problem (OCP) in either a model-based or data-driven manner. However, solving the most general of these problems online remains intractable due to complexities from intermittent unidirectional contacts with the environment, and from the many degrees of freedom of legged robots. This survey covers methods that have been pursued to make these OCPs computationally tractable, with a specific focus on how environmental contacts are treated, how the model can be simplified, and how these choices affect the numerical solution methods employed. The survey focuses on model-based optimization while paving its way for broader combination with learning-based formulations to accelerate progress in this growing field.

In contact-rich tasks, the hybrid, multi-modal nature of contact dynamics poses great challenges in model representation, planning, and control. Recent efforts have attempted to address these challenges via data-driven methods, learning dynamical models in combination with model predictive control. Those methods, while effective, rely solely on minimizing forward prediction errors to hope for better task performance with MPC controllers. This weak correlation can result in data inefficiency as well as limitations to overall performance. In response, we propose a novel strategy: using a policy gradient algorithm to find a simplified dynamics model that explicitly maximizes task performance. Specifically, we parameterize the stochastic policy as the perturbed output of the MPC controller, thus, the learned model representation can directly associate with the policy or task performance. We apply the proposed method to contact-rich tasks where a three-fingered robotic hand manipulates previously unknown objects. Our method significantly enhances task success rate by up to 15% in manipulating diverse objects compared to the existing method while sustaining data efficiency. Our method can solve some tasks with success rates of 70% or higher using under 30 minutes of data. All videos and codes are available at https://sites.google.com/view/lcs-rl.

Model-based approaches for planning and control for bipedal locomotion have a long history of success. It can provide stability and safety guarantees while being effective in accomplishing many locomotion tasks. Model-free reinforcement learning, on the other hand, has gained much popularity in recent years due to computational advancements. It can achieve high performance in specific tasks, but it lacks physical interpretability and flexibility in re-purposing the policy for a different set of tasks. For instance, we can initially train a neural network (NN) policy using velocity commands as inputs. However, to handle new task commands like desired hand or footstep locations at a desired walking velocity, we must retrain a new NN policy. In this work, we attempt to bridge the gap between these two bodies of work on a bipedal platform. We formulate a model-based reinforcement learning problem to learn a reduced-order model (ROM) within a model predictive control (MPC). Results show a 49% improvement in viable task region size and a 21% reduction in motor torque cost. All videos and code are available at https://sites.google.com/view/ymchen/research/rl-for-roms.

The hybrid nature of multi-contact robotic systems, due to making and breaking contact with the environment, creates significant challenges for high-quality control. Existing model-based methods typically rely on either good prior knowledge of the multi-contact model or require significant offline model tuning effort, thus resulting in low adaptability and robustness. In this paper, we propose a real-time adaptive multi-contact model predictive control framework, which enables online adaption of the hybrid multi-contact model and continuous improvement of the control performance for contact-rich tasks. This framework includes an adaption module, which continuously learns a residual of the hybrid model to minimize the gap between the prior model and reality, and a real-time multi-contact MPC controller. We demonstrated the effectiveness of the framework in synthetic examples, and applied it on hardware to solve contact-rich manipulation tasks, where a robot uses its end-effector to roll different unknown objects on a table to track given paths. The hardware experiments show that with a rough prior model, the multi-contact MPC controller adapts itself on-the-fly with an adaption rate around 20 Hz and successfully manipulates previously unknown objects with non-smooth surface geometries.

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 being 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.

Bipedal robots promise the ability to traverse rough terrain quickly and efficiently, and indeed, humanoid robots can now use strong ankles and careful foot placement to traverse discontinuous terrain. However, more agile underactuated bipeds have small feet and weak ankles, and must constantly adjust their planned footstep position to maintain balance. We introduce a new model-predictive footstep controller which jointly optimizes over the robot's discrete choice of stepping surface, impending footstep position sequence, ankle torque in the sagittal plane, and center of mass trajectory, to track a velocity command. The controller is formulated as a single Mixed Integer Quadratic Program (MIQP) which is solved at 50-200 Hz, depending on terrain complexity. We implement a state of the art real-time elevation mapping and convex terrain decomposition framework to inform the controller of its surroundings in the form on convex polygons representing steppable terrain. We investigate the capabilities and challenges of our approach through hardware experiments on the underactuated biped Cassie.

This work presents an instance-agnostic learning framework that fuses vision with dynamics to simultaneously learn shape, pose trajectories, and physical properties via the use of geometry as a shared representation. Unlike many contact learning approaches that assume motion capture input and a known shape prior for the collision model, our proposed framework learns an object's geometric and dynamic properties from RGBD video, without requiring either category-level or instance-level shape priors. We integrate a vision system, BundleSDF, with a dynamics system, ContactNets, and propose a cyclic training pipeline to use the output from the dynamics module to refine the poses and the geometry from the vision module, using perspective reprojection. Experiments demonstrate our framework's ability to learn the geometry and dynamics of rigid and convex objects and improve upon the current tracking framework.

When legged robots impact their environment executing dynamic motions, 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, 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. Code and video of the experiments are available at https://impact-invariant-control.github.io/.

Reduced-order models (ROM) are popular in online motion planning due to their simplicity. A good ROM for control captures critical task-relevant aspects of the full dynamics while remaining low dimensional. 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 a user-specified distribution of tasks and corresponding cost functions. To demonstrate our work, we optimized models for a bipedal robot Cassie. We show in simulation that the optimal ROM reduces the cost of Cassie's joint torques by up to 23% and increases its walking speed by up to 54%. We also show hardware result that the real robot walks on flat ground with 10% lower torque cost. All videos and code can be found at https://sites.google.com/view/ymchen/research/optimal-rom.