Tensegrity robots correspond to a lightweight, modular, and resilient class of mobile platforms that have been proposed for rugged operations, such as the exploration of other planets and moons. While the many actuated cables of these robots enable adaptation to highly unstructured environments, they also produce high dimensional and strongly coupled nonlinear system dynamics that are unintuitive and difficult to control. This project aims to provide new algorithmic, mathematical, and software tools that address these challenges for tensegrity locomotion in uneven terrain. The ultimate vision is to employ a curated library of learned feedback motion primitives within a high-level planner that matches them appropriately to terrain features in order to efficiently plan effective long-horizon traversal.
Publications:
2024 |
Chen, N; Wang, K; Johnson, W; Kramer-Bottiglio, R; Bekris, K; Aanjaneya, M Learning Differentiable Tensegrity Dynamics Using Graph Neural Networks Inproceedings Conference on Robot Learning (CoRL), Munich, Germany, 2024. @inproceedings{Chen:2024aa, title = {Learning Differentiable Tensegrity Dynamics Using Graph Neural Networks}, author = {N Chen and K Wang and W Johnson and R Kramer-Bottiglio and K Bekris and M Aanjaneya}, url = {https://openreview.net/pdf?id=5Awumz1VKU}, year = {2024}, date = {2024-11-01}, booktitle = {Conference on Robot Learning (CoRL)}, address = {Munich, Germany}, abstract = {Tensegrity robots are composed of rigid struts and flexible cables and constitute an emerging class of hybrid rigid-soft robotic systems. They are promising systems for a wide-array of applications, ranging from locomotion to assembly. They are difficult to control and model accurately, however, due to their high number of degrees of freedom and compliance. To address this issue, prior works have introduced a differentiable physics engine designed for tensegrity robots based on first-principles. In contrast, this work proposes the use of graph neural networks to model contact dynamics over a graph representation of tensegrity robots, which leverages the natural graph-like cable connectivity between the rod end caps. This learned simulator can accurately model 3-bar and 6-bar tensegrity robot dynamics in simulation-to-simulation experiments where MuJoCo is used as the ground truth. It can also achieve higher accuracy than the previous differentiable engine for a real 3-bar tensegrity robot, where the robot state is only being partially observable. When compared against direct applications of recent graph neural network simulators, the proposed approach is computationally more efficient both for training and inference, while achieving higher accuracy.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } Tensegrity robots are composed of rigid struts and flexible cables and constitute an emerging class of hybrid rigid-soft robotic systems. They are promising systems for a wide-array of applications, ranging from locomotion to assembly. They are difficult to control and model accurately, however, due to their high number of degrees of freedom and compliance. To address this issue, prior works have introduced a differentiable physics engine designed for tensegrity robots based on first-principles. In contrast, this work proposes the use of graph neural networks to model contact dynamics over a graph representation of tensegrity robots, which leverages the natural graph-like cable connectivity between the rod end caps. This learned simulator can accurately model 3-bar and 6-bar tensegrity robot dynamics in simulation-to-simulation experiments where MuJoCo is used as the ground truth. It can also achieve higher accuracy than the previous differentiable engine for a real 3-bar tensegrity robot, where the robot state is only being partially observable. When compared against direct applications of recent graph neural network simulators, the proposed approach is computationally more efficient both for training and inference, while achieving higher accuracy. |
2023 |
Wang, K; Johnson, W; Lu, S; Huang, X; Booth, J; Kramer-Bottiglio, R; Aanjaneya, M; Bekris, K Real2sim2real Transfer for Control of Cable-Driven Robots Via a Differentiable Physics Engine Inproceedings IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Detroit, MI, 2023. @inproceedings{Wang:2023aa, title = {Real2sim2real Transfer for Control of Cable-Driven Robots Via a Differentiable Physics Engine}, author = {K Wang and W Johnson and S Lu and X Huang and J Booth and R Kramer-Bottiglio and M Aanjaneya and K Bekris}, url = {https://arxiv.org/abs/2209.06261}, year = {2023}, date = {2023-10-01}, booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, address = {Detroit, MI}, abstract = {Tensegrity robots, composed of rigid rods and flexible cables, exhibit high strength-to-weight ratios and significant deformations, which enable them to navigate unstructured terrains and survive harsh impacts. They are hard to control, however, due to high dimensionality, complex dynamics, and a coupled architecture. Physics-based simulation is a promising avenue for developing locomotion policies that can be transferred to real robots. Nevertheless, modeling tensegrity robots is a complex task due to a substantial sim2real gap. To address this issue, this paper describes a Real2Sim2Real (R2S2R) strategy for tensegrity robots. This strategy is based on a differentiable physics engine that can be trained given limited data from a real robot. These data include offline measurements of physical properties, such as mass and geometry for various robot components, and the observation of a trajectory using a random control policy. With the data from the real robot, the engine can be iteratively refined and used to discover locomotion policies that are directly transferable to the real robot. Beyond the R2S2R pipeline, key contributions of this work include computing non-zero gradients at contact points, a loss function for matching tensegrity locomotion gaits, and a trajectory segmentation technique that avoids conflicts in gradient evaluation during training. Multiple iterations of the R2S2R process are demonstrated and evaluated on a real 3-bar tensegrity robot.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } Tensegrity robots, composed of rigid rods and flexible cables, exhibit high strength-to-weight ratios and significant deformations, which enable them to navigate unstructured terrains and survive harsh impacts. They are hard to control, however, due to high dimensionality, complex dynamics, and a coupled architecture. Physics-based simulation is a promising avenue for developing locomotion policies that can be transferred to real robots. Nevertheless, modeling tensegrity robots is a complex task due to a substantial sim2real gap. To address this issue, this paper describes a Real2Sim2Real (R2S2R) strategy for tensegrity robots. This strategy is based on a differentiable physics engine that can be trained given limited data from a real robot. These data include offline measurements of physical properties, such as mass and geometry for various robot components, and the observation of a trajectory using a random control policy. With the data from the real robot, the engine can be iteratively refined and used to discover locomotion policies that are directly transferable to the real robot. Beyond the R2S2R pipeline, key contributions of this work include computing non-zero gradients at contact points, a loss function for matching tensegrity locomotion gaits, and a trajectory segmentation technique that avoids conflicts in gradient evaluation during training. Multiple iterations of the R2S2R process are demonstrated and evaluated on a real 3-bar tensegrity robot. |
Zhao, L; Wu, Y; Yan, W; Zhan, W; Huang, X; Booth, J; Mehta, A; Bekris, K; Kramer-Bottiglio, R; Balkcom, D Starblocks: Soft Actuated Self-Connecting Blocks for Building Deformable Lattice Structures Journal Article IEEE Robotics and Automation Letters, 8 (8), pp. 4521–4528, 2023. @article{Zhao:2023aa, title = {Starblocks: Soft Actuated Self-Connecting Blocks for Building Deformable Lattice Structures}, author = {L Zhao and Y Wu and W Yan and W Zhan and X Huang and J Booth and A Mehta and K Bekris and R Kramer-Bottiglio and D Balkcom}, url = {https://ieeexplore.ieee.org/document/10146508}, year = {2023}, date = {2023-08-01}, journal = {IEEE Robotics and Automation Letters}, volume = {8}, number = {8}, pages = {4521--4528}, abstract = {In this paper, we present a soft modular block inspired by tensegrity structures that can form load-bearing structures through self-assembly. The block comprises a stellated compliant skeleton, shape memory alloy muscles, and permanent magnet connectors. We classify five deformation primitives for individual blocks: bend, compress, stretch, stand, and shrink, which can be combined across modules to reason about full-lattice deformation. Hierarchical function is abundant in nature and in human-designed systems. Using multiple self-assembled lattices, we demonstrate the formation and actuation of 3-dimensional shapes, including a load-bearing pop-up tent, a self-assembled wheel, a quadruped, a block-based robotic arm with gripper, and non-prehensile manipulation. To our knowledge, this is the first example of active deformable modules (blocks) that can reconfigure into different load-bearing structures on-demand.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we present a soft modular block inspired by tensegrity structures that can form load-bearing structures through self-assembly. The block comprises a stellated compliant skeleton, shape memory alloy muscles, and permanent magnet connectors. We classify five deformation primitives for individual blocks: bend, compress, stretch, stand, and shrink, which can be combined across modules to reason about full-lattice deformation. Hierarchical function is abundant in nature and in human-designed systems. Using multiple self-assembled lattices, we demonstrate the formation and actuation of 3-dimensional shapes, including a load-bearing pop-up tent, a self-assembled wheel, a quadruped, a block-based robotic arm with gripper, and non-prehensile manipulation. To our knowledge, this is the first example of active deformable modules (blocks) that can reconfigure into different load-bearing structures on-demand. |
2022 |
Lu, S; Johnson, W; Wang, K; Huang, X; Booth, J; Kramer-Bottiglio, R; Bekris, K 6n-Dof Pose Tracking for Tensegrity Robots Inproceedings International Symposium on Robotics Research (ISRR), 2022. @inproceedings{Lu:2022aa, title = {6n-Dof Pose Tracking for Tensegrity Robots}, author = {S Lu and W Johnson and K Wang and X Huang and J Booth and R Kramer-Bottiglio and K Bekris}, year = {2022}, date = {2022-10-01}, booktitle = {International Symposium on Robotics Research (ISRR)}, abstract = {Tensegrity robots, which are composed of rigid compressive elements (rods) and flexible tensile elements (e.g., cables), have a variety of advantages, including flexibility, light weight, and resistance to mechanical impact. Nevertheless, the hybrid soft-rigid nature of these robots also complicates the ability to localize and track their state. This work aims to address what has been recognized as a grand challenge in this domain, i.e., the pose tracking of tensegrity robots through a markerless, vision-based method, as well as novel, onboard sensors that can measure the length of the robot's cables. In particular, an iterative optimization process is proposed to estimate the 6-DoF poses of each rigid element of a tensegrity robot from an RGB-D video as well as endcap distance measurements from the cable sensors. To ensure the pose estimates of rigid elements are physically feasible, i.e., they are not resulting in collisions between rods or with the environment, physical constraints are introduced during the optimization. Real-world experiments are performed with a 3-bar tensegrity robot, which performs locomotion gaits. Given ground truth data from a motion capture system, the proposed method achieves less than 1 cm translation error and 3 degrees rotation error, which significantly outperforms alternatives. At the same time, the approach can provide pose estimates throughout the robot's motion, while motion capture often fails due to occlusions.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } Tensegrity robots, which are composed of rigid compressive elements (rods) and flexible tensile elements (e.g., cables), have a variety of advantages, including flexibility, light weight, and resistance to mechanical impact. Nevertheless, the hybrid soft-rigid nature of these robots also complicates the ability to localize and track their state. This work aims to address what has been recognized as a grand challenge in this domain, i.e., the pose tracking of tensegrity robots through a markerless, vision-based method, as well as novel, onboard sensors that can measure the length of the robot's cables. In particular, an iterative optimization process is proposed to estimate the 6-DoF poses of each rigid element of a tensegrity robot from an RGB-D video as well as endcap distance measurements from the cable sensors. To ensure the pose estimates of rigid elements are physically feasible, i.e., they are not resulting in collisions between rods or with the environment, physical constraints are introduced during the optimization. Real-world experiments are performed with a 3-bar tensegrity robot, which performs locomotion gaits. Given ground truth data from a motion capture system, the proposed method achieves less than 1 cm translation error and 3 degrees rotation error, which significantly outperforms alternatives. At the same time, the approach can provide pose estimates throughout the robot's motion, while motion capture often fails due to occlusions. |
Wang, K; Aanjaneya, M; Bekris, K A Recurrent Differentiable Engine for Modeling Tensegrity Robots Trainable with Low-Frequency Data Inproceedings IEEE International Conference on Robotics and Automation (ICRA), 2022. @inproceedings{Wang:2022aa, title = {A Recurrent Differentiable Engine for Modeling Tensegrity Robots Trainable with Low-Frequency Data}, author = {K Wang and M Aanjaneya and K Bekris}, year = {2022}, date = {2022-07-01}, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, abstract = {Tensegrity robots, composed of rigid rods and flexible cables, are difficult to accurately model and control given the presence of complex dynamics and high number of DoFs. Differentiable physics engines have been recently proposed as a data-driven approach for model identification of such complex robotic systems. These engines are often executed at a high-frequency to achieve accurate simulation. Ground truth trajectories for training differentiable engines, however, are not typically available at such high frequencies due to limitations of real-world sensors. The present work focuses on this frequency mismatch, which impacts the modeling accuracy. We proposed a recurrent structure for a differentiable physics engine of tensegrity robots, which can be trained effectively even with low-frequency trajectories. To train this new recurrent engine in a robust way, this work introduces relative to prior work: (i) a new implicit integration scheme, (ii) a progressive training pipeline, and (iii) a differentiable collision checker. A model of NASA's icosahedron SUPERballBot on MuJoCo is used as the ground truth system to collect training data. Simulated experiments show that once the recurrent differentiable engine has been trained given the low-frequency trajectories from MuJoCo, it is able to match the behavior of MuJoCo's system. The criterion for success is whether a locomotion strategy learned using the differentiable engine can be transferred back to the ground-truth system and result in a similar motion. Notably, the amount of ground truth data needed to train the differentiable engine, such that the policy is transferable to the ground truth system, is 1% of the data needed to train the policy directly on the ground-truth system.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } Tensegrity robots, composed of rigid rods and flexible cables, are difficult to accurately model and control given the presence of complex dynamics and high number of DoFs. Differentiable physics engines have been recently proposed as a data-driven approach for model identification of such complex robotic systems. These engines are often executed at a high-frequency to achieve accurate simulation. Ground truth trajectories for training differentiable engines, however, are not typically available at such high frequencies due to limitations of real-world sensors. The present work focuses on this frequency mismatch, which impacts the modeling accuracy. We proposed a recurrent structure for a differentiable physics engine of tensegrity robots, which can be trained effectively even with low-frequency trajectories. To train this new recurrent engine in a robust way, this work introduces relative to prior work: (i) a new implicit integration scheme, (ii) a progressive training pipeline, and (iii) a differentiable collision checker. A model of NASA's icosahedron SUPERballBot on MuJoCo is used as the ground truth system to collect training data. Simulated experiments show that once the recurrent differentiable engine has been trained given the low-frequency trajectories from MuJoCo, it is able to match the behavior of MuJoCo's system. The criterion for success is whether a locomotion strategy learned using the differentiable engine can be transferred back to the ground-truth system and result in a similar motion. Notably, the amount of ground truth data needed to train the differentiable engine, such that the policy is transferable to the ground truth system, is 1% of the data needed to train the policy directly on the ground-truth system. |
2021 |
Shah, D; Booth, J; Baines, R; Wang, K; Vespignani, M; Bekris, K; Kramer-Bottiglio, R Tensegrity Robotics Journal Article Soft Robotics, 2021. @article{Shah:2021aa, title = {Tensegrity Robotics}, author = {D Shah and J Booth and R Baines and K Wang and M Vespignani and K Bekris and R Kramer-Bottiglio}, doi = {10.1089/soro.2020.0170}, year = {2021}, date = {2021-12-01}, journal = {Soft Robotics}, abstract = {Numerous recent advances in robotics have been inspired by the biological principle of tensile integrity --- or ``tensegrity''--- to achieve remarkable feats of dexterity and resilience. Tensegrity robots contain compliant networks of rigid struts and soft cables, allowing them to change their shape by adjusting their internal tension. Local rigidity along the struts provides support to carry electronics and scientific payloads, while global compliance enabled by the flexible interconnections of struts and ca- bles allows a tensegrity to distribute impacts and prevent damage. Numerous techniques have been proposed for designing and simulating tensegrity robots, giving rise to a wide range of locomotion modes including rolling, vibrating, hopping, and crawling. Here, we review progress in the burgeoning field of tensegrity robotics, highlighting several emerging challenges, including automated design, state sensing, and kinodynamic motion planning.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Numerous recent advances in robotics have been inspired by the biological principle of tensile integrity --- or ``tensegrity''--- to achieve remarkable feats of dexterity and resilience. Tensegrity robots contain compliant networks of rigid struts and soft cables, allowing them to change their shape by adjusting their internal tension. Local rigidity along the struts provides support to carry electronics and scientific payloads, while global compliance enabled by the flexible interconnections of struts and ca- bles allows a tensegrity to distribute impacts and prevent damage. Numerous techniques have been proposed for designing and simulating tensegrity robots, giving rise to a wide range of locomotion modes including rolling, vibrating, hopping, and crawling. Here, we review progress in the burgeoning field of tensegrity robotics, highlighting several emerging challenges, including automated design, state sensing, and kinodynamic motion planning. |
Meng, P; Wang, W; Balkcom, D; Bekris, K ASCE Earth and Space Conference 2021, Seattle, WA, 2021. @conference{Meng:2021aa, title = {Proof-Of-Concept Designs for the Assembly of Modular, Dynamic Tensegrities into Easily Deployable Structures}, author = {P Meng and W Wang and D Balkcom and K Bekris}, url = {https://par.nsf.gov/servlets/purl/10294210}, year = {2021}, date = {2021-10-01}, booktitle = {ASCE Earth and Space Conference 2021}, address = {Seattle, WA}, abstract = {Dynamic tensegrity robots are inspired by tensegrity structures in architecture; arrangements of rigid rods and flexible elements allow the robots to deform. This work proposes the use of multiple, modular, tensegrity robots that can move and compliantly connect to assemble larger, compliant, lightweight, strong structures and scaffolding. The focus is on proof-of-concept designs for the modular robots themselves and their docking mechanisms, which can allow the easy deployment of structures in unstructured environments. These mechanisms include (electro)magnets to allow each individual robot to connect and disconnect on cue. An exciting direction is the design of specific module and structure designs to fit the mission at hand. For example, this work highlights how the considered three bar structures could stack to form a column or deform on one side to create an arch. A critical component of future work will involve the development of algorithms for automatic design and layout of modules in structures.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Dynamic tensegrity robots are inspired by tensegrity structures in architecture; arrangements of rigid rods and flexible elements allow the robots to deform. This work proposes the use of multiple, modular, tensegrity robots that can move and compliantly connect to assemble larger, compliant, lightweight, strong structures and scaffolding. The focus is on proof-of-concept designs for the modular robots themselves and their docking mechanisms, which can allow the easy deployment of structures in unstructured environments. These mechanisms include (electro)magnets to allow each individual robot to connect and disconnect on cue. An exciting direction is the design of specific module and structure designs to fit the mission at hand. For example, this work highlights how the considered three bar structures could stack to form a column or deform on one side to create an arch. A critical component of future work will involve the development of algorithms for automatic design and layout of modules in structures. |
Wang, K; Aanjaneya, M; Bekris, K Sim2Sim Evaluation of a Novel Data-Efficient Differentiable Physics Engine for Tensegrity Robots Inproceedings IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2021. @inproceedings{Wang:2021ab, title = {Sim2Sim Evaluation of a Novel Data-Efficient Differentiable Physics Engine for Tensegrity Robots}, author = {K Wang and M Aanjaneya and K Bekris}, url = {https://arxiv.org/abs/2011.04929}, year = {2021}, date = {2021-09-01}, booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, abstract = {Learning policies in simulation is promising for reducing human effort when training robot controllers. This is especially true for soft robots that are more adaptive and safe but also more difficult to accurately model and control. The sim2real gap is the main barrier to successfully transfer policies from simulation to a real robot. System identification can be applied to reduce this gap but traditional identification methods require a lot of manual tuning. Data-driven alternatives can tune dynamical models directly from data but are often data hungry, which also incorporates human effort in collecting data. This work proposes a data-driven, end-to-end differentiable simulator focused on the exciting but challenging domain of tensegrity robots. To the best of the authors' knowledge, this is the first differentiable physics engine for tensegrity robots that supports cable, contact, and actuation modeling. The aim is to develop a reasonably simplified, data-driven simulation, which can learn approximate dynamics with limited ground truth data. The dynamics must be accurate enough to generate policies that can be transferred back to the ground-truth system. As a first step in this direction, the current work demonstrates sim2sim transfer, where the unknown physical model of MuJoCo acts as a ground truth system. Two different tensegrity robots are used for evaluation and learning of locomotion policies, a 6-bar and a 3-bar tensegrity. The results indicate that only 0.25% of ground truth data are needed to train a policy that works on the ground truth system when the differentiable engine is used for training against training the policy directly on the ground truth system.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } Learning policies in simulation is promising for reducing human effort when training robot controllers. This is especially true for soft robots that are more adaptive and safe but also more difficult to accurately model and control. The sim2real gap is the main barrier to successfully transfer policies from simulation to a real robot. System identification can be applied to reduce this gap but traditional identification methods require a lot of manual tuning. Data-driven alternatives can tune dynamical models directly from data but are often data hungry, which also incorporates human effort in collecting data. This work proposes a data-driven, end-to-end differentiable simulator focused on the exciting but challenging domain of tensegrity robots. To the best of the authors' knowledge, this is the first differentiable physics engine for tensegrity robots that supports cable, contact, and actuation modeling. The aim is to develop a reasonably simplified, data-driven simulation, which can learn approximate dynamics with limited ground truth data. The dynamics must be accurate enough to generate policies that can be transferred back to the ground-truth system. As a first step in this direction, the current work demonstrates sim2sim transfer, where the unknown physical model of MuJoCo acts as a ground truth system. Two different tensegrity robots are used for evaluation and learning of locomotion policies, a 6-bar and a 3-bar tensegrity. The results indicate that only 0.25% of ground truth data are needed to train a policy that works on the ground truth system when the differentiable engine is used for training against training the policy directly on the ground truth system. |
Surovik, D; Wang, K; Vespignani, M; Bruce, J; Bekris, K Adaptive Tensegrity Locomotion: Controlling a Compliant Icosahedron with Symmetry-Reduced Reinforcement Learning Journal Article International Journal of Robotics Research (IJRR), 2021. @article{Surovik:2021aa, title = {Adaptive Tensegrity Locomotion: Controlling a Compliant Icosahedron with Symmetry-Reduced Reinforcement Learning}, author = {D Surovik and K Wang and M Vespignani and J Bruce and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/reinf_learning_tensegrity_locomotion.pdf}, year = {2021}, date = {2021-01-01}, journal = {International Journal of Robotics Research (IJRR)}, abstract = {Tensegrity robots, which are prototypical examples of hybrid soft-rigid robots, exhibit dynamical properties that provide ruggedness and adaptability. They also bring about, however, major challenges for locomotion control. Due to high dimensionality and the complex evolution of contact states, data-driven approaches are appropriate for producing viable feedback policies for tensegrities. Guided Policy Search (GPS), a sample-efficient hybrid framework for optimization and reinforcement learning, has previously been applied to generate periodic, axis-constrained locomotion by an icosahedral tensegrity on flat ground. Varying environments and tasks, however, create a need for more adaptive and general locomotion control that actively utilizes an expanded space of robot states. This implies significantly higher needs in terms of sample data and setup effort. This work mitigates such requirements by proposing a new GPS- based reinforcement learning pipeline, which exploits the vehicle's high degree of symmetry and appropriately learns contextual behaviors that are sustainable without periodicity. Newly achieved capabilities include axially-unconstrained rolling, rough terrain traversal, and rough incline ascent. These tasks are evaluated for a small variety of key model parameters in simulation and tested on the NASA hardware prototype, SUPERball. Results confirm the utility of symmetry exploitation and the adaptability of the vehicle. They also shed light on numerous strengths and limitations of the GPS framework for policy design and transfer to real hybrid soft-rigid robots.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Tensegrity robots, which are prototypical examples of hybrid soft-rigid robots, exhibit dynamical properties that provide ruggedness and adaptability. They also bring about, however, major challenges for locomotion control. Due to high dimensionality and the complex evolution of contact states, data-driven approaches are appropriate for producing viable feedback policies for tensegrities. Guided Policy Search (GPS), a sample-efficient hybrid framework for optimization and reinforcement learning, has previously been applied to generate periodic, axis-constrained locomotion by an icosahedral tensegrity on flat ground. Varying environments and tasks, however, create a need for more adaptive and general locomotion control that actively utilizes an expanded space of robot states. This implies significantly higher needs in terms of sample data and setup effort. This work mitigates such requirements by proposing a new GPS- based reinforcement learning pipeline, which exploits the vehicle's high degree of symmetry and appropriately learns contextual behaviors that are sustainable without periodicity. Newly achieved capabilities include axially-unconstrained rolling, rough terrain traversal, and rough incline ascent. These tasks are evaluated for a small variety of key model parameters in simulation and tested on the NASA hardware prototype, SUPERball. Results confirm the utility of symmetry exploitation and the adaptability of the vehicle. They also shed light on numerous strengths and limitations of the GPS framework for policy design and transfer to real hybrid soft-rigid robots. |
2020 |
Wang, K; Aanjaneya, M; Bekris, K A First Principles Approach for Data-Efficient System Identification of Spring-Rod Systems Via Differentiable Physics Engines Conference Learning for Dynamics and Control (L4DC), Berkeley, CA, 2020. @conference{Wang:2020aa, title = {A First Principles Approach for Data-Efficient System Identification of Spring-Rod Systems Via Differentiable Physics Engines}, author = {K Wang and M Aanjaneya and K Bekris}, year = {2020}, date = {2020-06-01}, booktitle = {Learning for Dynamics and Control (L4DC)}, address = {Berkeley, CA}, abstract = {We propose a novel differentiable physics engine for system identification of complex spring-rod assemblies. Unlike black-box data-driven methods for learning the evolution of a dynamical system and its parameters, we modularize the design of our engine using a discrete form of the governing equations of motion, similar to a traditional physics engine. We further reduce the dimension from 3D to 1D for each module, which allows efficient learning of system parameters using linear regression. As a side benefit, the regression parameters correspond to physical quantities, such as spring stiffness or the mass of the rod, making the pipeline explainable. The approach significantly reduces the amount of training data required, and also avoids iterative identification of data sampling and model training. We compare the performance of the proposed engine with previous solutions, and demonstrate its efficacy on tensegrity systems, such as NASA's icosahedron.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } We propose a novel differentiable physics engine for system identification of complex spring-rod assemblies. Unlike black-box data-driven methods for learning the evolution of a dynamical system and its parameters, we modularize the design of our engine using a discrete form of the governing equations of motion, similar to a traditional physics engine. We further reduce the dimension from 3D to 1D for each module, which allows efficient learning of system parameters using linear regression. As a side benefit, the regression parameters correspond to physical quantities, such as spring stiffness or the mass of the rod, making the pipeline explainable. The approach significantly reduces the amount of training data required, and also avoids iterative identification of data sampling and model training. We compare the performance of the proposed engine with previous solutions, and demonstrate its efficacy on tensegrity systems, such as NASA's icosahedron. |
Littlefield, Z Efficient and Asymptotically Optimal Kinodynamic Motion Planning PhD Thesis Rutgers, the State University of New Jersey, 2020. @phdthesis{Littlefield:2020aa, title = {Efficient and Asymptotically Optimal Kinodynamic Motion Planning}, author = {Z Littlefield}, url = {http://www.cs.rutgers.edu/~kb572/pubs/LittlefieldThesisMay2020.pdf}, year = {2020}, date = {2020-05-01}, volume = {PhD}, school = {Rutgers, the State University of New Jersey}, abstract = {This dissertation explores properties of motion planners that build tree data structures in a robottextquoterights state space. Sampling-based tree planners are especially useful for planning for systems with significant dynamics, due to the inherent forward search that is per- formed. This is in contrast to roadmap planners that require a steering local planner in order to make a graph containing multiple possible paths. This dissertation explores a family of motion planners for systems with significant dynamics, where a steering local planner may be computationally expensive or may not exist. These planners focus on providing practical path quality guarantees without prohibitive computational costs. These planners can be considered successors of each other, in that each sub- sequent algorithm addresses some drawback of its predecessor. The first algorithm, Sparse-RRT, addresses a drawback of the RRT method by considering path quality during the tree construction process. Sparse-RRT is proven to be probabilistically complete under mild conditions for the first time here, albeit with a poor convergence rate. The second algorithm presented, SST, provides probabilistic completeness and asymptotic near-optimality properties that are provable, but at the cost of additional algorithmic overhead. SST is shown to improve the convergence rate compared to Sparse-RRT. The third algorithm, DIRT, incorporates learned lessons from these two algorithms and their shortcomings, incorporates task space heuristics to further improve runtime performance, and simplifies the parameters to more user-friendly ones. DIRT is also shown to be probabilistically complete and asymptotically near-optimal. Application areas explored using this family of algorithms include evaluation of distance functions for planning in belief space, manipulation in cluttered environments, and locomotion planning for an icosahedral tensegrity-based rover prototype that requires a physics engine to simulate its motions.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } This dissertation explores properties of motion planners that build tree data structures in a robottextquoterights state space. Sampling-based tree planners are especially useful for planning for systems with significant dynamics, due to the inherent forward search that is per- formed. This is in contrast to roadmap planners that require a steering local planner in order to make a graph containing multiple possible paths. This dissertation explores a family of motion planners for systems with significant dynamics, where a steering local planner may be computationally expensive or may not exist. These planners focus on providing practical path quality guarantees without prohibitive computational costs. These planners can be considered successors of each other, in that each sub- sequent algorithm addresses some drawback of its predecessor. The first algorithm, Sparse-RRT, addresses a drawback of the RRT method by considering path quality during the tree construction process. Sparse-RRT is proven to be probabilistically complete under mild conditions for the first time here, albeit with a poor convergence rate. The second algorithm presented, SST, provides probabilistic completeness and asymptotic near-optimality properties that are provable, but at the cost of additional algorithmic overhead. SST is shown to improve the convergence rate compared to Sparse-RRT. The third algorithm, DIRT, incorporates learned lessons from these two algorithms and their shortcomings, incorporates task space heuristics to further improve runtime performance, and simplifies the parameters to more user-friendly ones. DIRT is also shown to be probabilistically complete and asymptotically near-optimal. Application areas explored using this family of algorithms include evaluation of distance functions for planning in belief space, manipulation in cluttered environments, and locomotion planning for an icosahedral tensegrity-based rover prototype that requires a physics engine to simulate its motions. |
2019 |
Littlefield, Z; Surovik, D; Vespignani, M; Bruce, J; Wang, W; Bekris, K Kinodynamic Planning for Spherical Tensegrity Locomotion with Effective Gait Primitives Journal Article International Journal of Robotics Research (IJRR), 2019. @article{Littlefield:2019aa, title = {Kinodynamic Planning for Spherical Tensegrity Locomotion with Effective Gait Primitives}, author = {Z Littlefield and D Surovik and M Vespignani and J Bruce and W Wang and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/kinodynamic_tensegrity.pdf}, year = {2019}, date = {2019-10-01}, journal = {International Journal of Robotics Research (IJRR)}, abstract = {Tensegrity-based robots can achieve locomotion through shape deformation and compliance. They are highly adaptable to their surroundings, have light weight, low cost and are physically robust. Their high dimensionality and strongly dynamic nature, however, complicate motion planning. Efforts to-date have primarily considered quasi-static reconfiguration and short-term dynamic motion of tensegrity robots, which do not fully exploit the underlying system dynamics in the long term. Longer-horizon planning has previously required costly search over the full space of valid control inputs. This work synthesizes new and existing approaches to produce dynamic long-term motion while balancing the computational demand. A numerical process based upon quasi-static assumptions is first applied to deform the system into an unstable configuration, causing forward motion. The dynamical characteristics of the result are then altered via a few simple parameters to produce a small but diverse set of useful behaviors. The proposed approach takes advantage of identified symmetries on the prototypical spherical tensegrity robot, which reduce the number of needed gaits but allow motion along different directions. These gaits are first combined with a standard search method to achieve long term planning in environments where the developed gaits are effective. For more complex environments, the various motion primitives are paired with the fall-back option of random valid actions and are used by an informed sampling-based kinodynamic motion planner with anytime properties. Evaluations using a physics-based model for the prototypical robot demonstrate that modest but efficiently-applied search effort can unlock the utility of dynamic tensegrity motion to produce high-quality solutions.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Tensegrity-based robots can achieve locomotion through shape deformation and compliance. They are highly adaptable to their surroundings, have light weight, low cost and are physically robust. Their high dimensionality and strongly dynamic nature, however, complicate motion planning. Efforts to-date have primarily considered quasi-static reconfiguration and short-term dynamic motion of tensegrity robots, which do not fully exploit the underlying system dynamics in the long term. Longer-horizon planning has previously required costly search over the full space of valid control inputs. This work synthesizes new and existing approaches to produce dynamic long-term motion while balancing the computational demand. A numerical process based upon quasi-static assumptions is first applied to deform the system into an unstable configuration, causing forward motion. The dynamical characteristics of the result are then altered via a few simple parameters to produce a small but diverse set of useful behaviors. The proposed approach takes advantage of identified symmetries on the prototypical spherical tensegrity robot, which reduce the number of needed gaits but allow motion along different directions. These gaits are first combined with a standard search method to achieve long term planning in environments where the developed gaits are effective. For more complex environments, the various motion primitives are paired with the fall-back option of random valid actions and are used by an informed sampling-based kinodynamic motion planner with anytime properties. Evaluations using a physics-based model for the prototypical robot demonstrate that modest but efficiently-applied search effort can unlock the utility of dynamic tensegrity motion to produce high-quality solutions. |
2018 |
Surovik, D; Bruce, J; Wang, K; Vespignani, M; Bekris, K Any-Axis Tensegrity Rolling Via Bootstrapped Learning and Symmetry Reduction Conference International Symposium on Experimental Robotics (ISER), Buenos Aires, Argentina, 2018. @conference{Surovik:2018aa, title = {Any-Axis Tensegrity Rolling Via Bootstrapped Learning and Symmetry Reduction}, author = {D Surovik and J Bruce and K Wang and M Vespignani and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/any_axis_tensegrity_rolling.pdf}, year = {2018}, date = {2018-11-01}, booktitle = {International Symposium on Experimental Robotics (ISER)}, address = {Buenos Aires, Argentina}, abstract = {Tensegrity rovers incorporate design principles that give rise to many desirable properties, such as adaptability and robustness, while also creating challenges in terms of locomotion control. A recent milestone in this area combined reinforcement learning and optimal control to effect fixed-axis rolling of NASA's 6-bar spherical tensegrity rover prototype, SUPERball, with use of 12 actuators. The new 24-actuator version of SUPERball presents the potential for greatly increased locomotive abilities, but at a drastic nominal increase in the size of the data-driven control problem. This paper is focused upon unlocking those abilities while crucially moderating data requirements by incorporating symmetry reduction into the controller design pipeline, along with other new considerations. Experiments in simulation and on the hardware prototype demonstrate the resulting capability for any-axis rolling on the 24-actuator version of SUPERball, such that it may utilize diverse ground-contact patterns to smoothly locomote in arbitrary directions.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Tensegrity rovers incorporate design principles that give rise to many desirable properties, such as adaptability and robustness, while also creating challenges in terms of locomotion control. A recent milestone in this area combined reinforcement learning and optimal control to effect fixed-axis rolling of NASA's 6-bar spherical tensegrity rover prototype, SUPERball, with use of 12 actuators. The new 24-actuator version of SUPERball presents the potential for greatly increased locomotive abilities, but at a drastic nominal increase in the size of the data-driven control problem. This paper is focused upon unlocking those abilities while crucially moderating data requirements by incorporating symmetry reduction into the controller design pipeline, along with other new considerations. Experiments in simulation and on the hardware prototype demonstrate the resulting capability for any-axis rolling on the 24-actuator version of SUPERball, such that it may utilize diverse ground-contact patterns to smoothly locomote in arbitrary directions. |
Zhu, S; Surovik, D; Bekris, K; Boularias, A Efficient Model Identification for Tensegrity Locomotion Conference IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain, 2018. @conference{196, title = {Efficient Model Identification for Tensegrity Locomotion}, author = {S Zhu and D Surovik and K Bekris and A Boularias}, url = {https://www.cs.rutgers.edu/~kb572/pubs/model_identification_tensegrity.pdf}, year = {2018}, date = {2018-10-01}, booktitle = {IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, address = {Madrid, Spain}, abstract = {This paper aims to identify in a practical manner unknown physical parameters, such as mechanical models of actuated robot links, which are critical in dynamical robotic tasks. Key features include the use of an off-the-shelf physics engine and the Bayesian optimization framework. The task being considered is locomotion with a high-dimensional, compliant Tensegrity robot. A key insight, in this case, is the need to project the space of models into an appropriate lower dimensional space for time efficiency. Comparisons with alternatives indicate that the proposed method can identify the parameters more accurately within the given time budget, which also results in more precise locomotion control.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } This paper aims to identify in a practical manner unknown physical parameters, such as mechanical models of actuated robot links, which are critical in dynamical robotic tasks. Key features include the use of an off-the-shelf physics engine and the Bayesian optimization framework. The task being considered is locomotion with a high-dimensional, compliant Tensegrity robot. A key insight, in this case, is the need to project the space of models into an appropriate lower dimensional space for time efficiency. Comparisons with alternatives indicate that the proposed method can identify the parameters more accurately within the given time budget, which also results in more precise locomotion control. |
Rennie, C; Bekris, K Discovering a Library of Rhythmic Gaits for Spherical Tensegrity Locomotion Conference IEEE International Conference on Robotics and Automation (ICRA), Brisbane, Australia, 2018. @conference{Rennie:2018aa, title = {Discovering a Library of Rhythmic Gaits for Spherical Tensegrity Locomotion}, author = {C Rennie and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/gps_bo_svm_tensegrity.pdf}, year = {2018}, date = {2018-05-01}, booktitle = {IEEE International Conference on Robotics and Automation (ICRA)}, address = {Brisbane, Australia}, abstract = {Tensegrity robots, which combine both rigid and soft elements, provide exciting new locomotion capabilities but introduce significant control challenges given their high-dimensionality and non-linear nature. This work first defines an effective parameterization of a spherical tensegrity for generating rhythmic gaits based on Central Pattern Generators (CPG). This allows the definition of periodic and rhythmic control signals, while exposing only five gait parameters. Then, this work proposes a framework for optimizing such gaits by exploring the parameter space through Bayesian Optimization on an underlying Gaussian Process regression model. The objective is to provide gaits that allow the platform to move along different directions with high velocity. Additionally, kNN binary classifiers are trained to estimate whether a parameter sample will result in an effective gait. The classification biases the sampling toward subspaces likely to yield effective gaits. An asynchronous communication layer is defined between the optimization and classification processes. The proposed gait discovery process is shown to efficiently optimize the parameters of gaits defined given the novel CPG architecture and outperforms less holistic approaches and Monte Carlo sampling}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Tensegrity robots, which combine both rigid and soft elements, provide exciting new locomotion capabilities but introduce significant control challenges given their high-dimensionality and non-linear nature. This work first defines an effective parameterization of a spherical tensegrity for generating rhythmic gaits based on Central Pattern Generators (CPG). This allows the definition of periodic and rhythmic control signals, while exposing only five gait parameters. Then, this work proposes a framework for optimizing such gaits by exploring the parameter space through Bayesian Optimization on an underlying Gaussian Process regression model. The objective is to provide gaits that allow the platform to move along different directions with high velocity. Additionally, kNN binary classifiers are trained to estimate whether a parameter sample will result in an effective gait. The classification biases the sampling toward subspaces likely to yield effective gaits. An asynchronous communication layer is defined between the optimization and classification processes. The proposed gait discovery process is shown to efficiently optimize the parameters of gaits defined given the novel CPG architecture and outperforms less holistic approaches and Monte Carlo sampling |
Surovik, D; Bekris, K Symmetric Reduction of Tensegrity Rover Dynamics for Efficient Data-Driven Control Conference ASCE Earth and Space Conference, Symposium on "Tensegrity - Structural Concept and Applications", Cleveland, Ohio, 2018. @conference{Surovik:2018ab, title = {Symmetric Reduction of Tensegrity Rover Dynamics for Efficient Data-Driven Control}, author = {D Surovik and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/asce_sym.pdf}, year = {2018}, date = {2018-04-01}, booktitle = {ASCE Earth and Space Conference, Symposium on "Tensegrity - Structural Concept and Applications"}, address = {Cleveland, Ohio}, abstract = {Tensegrity robots consist of disconnected rods suspended within a network of length-actuated cables, which gives them a high degree of compliance and adaptability suitable for traversing rugged terrain. These vehicles, however, undergo complex contact dynamics that prevent the use of traditional control techniques based on mathematical analyses of equations of motion. Data-driven approaches are thus an appropriate choice for controller design, but are themselves hindered by the high number of degrees of freedom and correspondingly large state spaces. This paper presents a scheme for exploiting the 24th-order symmetry of an icosahedral tensegrity robot to vastly reduce the breadth of the controller input space without loss of information. Symmetric properties and state reduction operations are detailed and placed in the context of a data-driven control pipeline. Results are illustrated by comparing the input and output of a locomotive controller in both raw and symmetry-reduced dynamical spaces. The findings suggest a strong relief of the data requirements for training locomotive controllers.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Tensegrity robots consist of disconnected rods suspended within a network of length-actuated cables, which gives them a high degree of compliance and adaptability suitable for traversing rugged terrain. These vehicles, however, undergo complex contact dynamics that prevent the use of traditional control techniques based on mathematical analyses of equations of motion. Data-driven approaches are thus an appropriate choice for controller design, but are themselves hindered by the high number of degrees of freedom and correspondingly large state spaces. This paper presents a scheme for exploiting the 24th-order symmetry of an icosahedral tensegrity robot to vastly reduce the breadth of the controller input space without loss of information. Symmetric properties and state reduction operations are detailed and placed in the context of a data-driven control pipeline. Results are illustrated by comparing the input and output of a locomotive controller in both raw and symmetry-reduced dynamical spaces. The findings suggest a strong relief of the data requirements for training locomotive controllers. |
2017 |
Littlefield, Z; Surovik, D; Wang, W; Bekris, K From Quasi-Static to Kinodynamic Planning for Spherical Tensegrity Locomotion Conference International Symosium on Robotics Research (ISRR), Puerto Varas, Chile, 2017. @conference{Littlefield:2017aa, title = {From Quasi-Static to Kinodynamic Planning for Spherical Tensegrity Locomotion}, author = {Z Littlefield and D Surovik and W Wang and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/isrr17_quasistatic_kinodynamic.pdf}, year = {2017}, date = {2017-12-01}, booktitle = {International Symosium on Robotics Research (ISRR)}, address = {Puerto Varas, Chile}, abstract = {Tensegrity-based robots can achieve locomotion through shape deformation and compliance. They are highly adaptable to surroundings, have light weight, low cost and high endurance. Their high dimensionality and highly dynamic nature, however, complicate motion planning. So far, only rudimentary quasi-static solutions have been achieved, which do not utilize tensegrity dynamics. This work explores a spectrum of planning methods that increasingly allow dynamic motion for such platforms. Symmetries are first identified for a prototypical spherical tensegrity robot, which reduce the number of needed gaits. Then, a numerical process is proposed for generating quasi-static gaits that move forward the systemtextquoterights center of mass in different directions. These gaits are combined with a search method to achieve a quasi-static solution. In complex environments, however, this approach is not able to fully explore the space and utilize dynamics. This motivates the application of sampling-based, kinodynamic planners. This paper proposes such a method for tensegrity locomotion that is informed and has anytime properties. The proposed solution allows the generation of dynamic motion and provides good quality solutions. Evaluation using a physics-based model for the prototypical robot highlight the benefits of the proposed scheme and the limits of quasi-static solutions.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Tensegrity-based robots can achieve locomotion through shape deformation and compliance. They are highly adaptable to surroundings, have light weight, low cost and high endurance. Their high dimensionality and highly dynamic nature, however, complicate motion planning. So far, only rudimentary quasi-static solutions have been achieved, which do not utilize tensegrity dynamics. This work explores a spectrum of planning methods that increasingly allow dynamic motion for such platforms. Symmetries are first identified for a prototypical spherical tensegrity robot, which reduce the number of needed gaits. Then, a numerical process is proposed for generating quasi-static gaits that move forward the systemtextquoterights center of mass in different directions. These gaits are combined with a search method to achieve a quasi-static solution. In complex environments, however, this approach is not able to fully explore the space and utilize dynamics. This motivates the application of sampling-based, kinodynamic planners. This paper proposes such a method for tensegrity locomotion that is informed and has anytime properties. The proposed solution allows the generation of dynamic motion and provides good quality solutions. Evaluation using a physics-based model for the prototypical robot highlight the benefits of the proposed scheme and the limits of quasi-static solutions. |
2016 |
Littlefield, Z; Caluwaerts, K; Bruce, J; SunSpiral, V; Bekris, K Integrating Simulated Tensegrity Models with Efficient Motion Planning for Planetary Navigation Conference International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-SAIRAS 2016), Beijing, China, 2016. @conference{Littlefield:2016ab, title = {Integrating Simulated Tensegrity Models with Efficient Motion Planning for Planetary Navigation}, author = {Z Littlefield and K Caluwaerts and J Bruce and V SunSpiral and K Bekris}, url = {https://www.cs.rutgers.edu/~kb572/pubs/isairas_littlefield.pdf}, year = {2016}, date = {2016-06-01}, booktitle = {International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-SAIRAS 2016)}, address = {Beijing, China}, abstract = {Tensegrity-based robots use compression elements and tension cables to create lightweight structures that can reconfigure their shape. These characteristics are especially suited for planetary exploration, including for hard to traverse areas, such as lava tubes. While these capabilities are desirable for transporting these robots beyond Earth as well as reducing material costs, they complicate the control process. With such dynamic and reconfiguring parts, both simulating the motions of the tensegrity robot and planning future motions become challenging. New simulation tools for tensegrity rovers and state-of-the-art planning algorithms have been recently developed which can help to address these challenges, but have yet to used in tandem. This work integrates a recent sampling-based motion planner, which has been shown to converge to asymptotically optimal solutions even for systems with dynamics, with a novel tensegrity rover simulation tool, which has been verified in terms of accuracy against a hardware prototype. This paper shows that it is possible to get complex, long-duration trajectories for planetary navigation through this integration. At the same time, this integration is computationally demanding which motivated a parallel implementation of the proposed integration. With the parallel implementation, it is possible to observe improving path quality as computation time increases. This framework allows the consideration of planning under uncertainty to compute robust solutions, which is even more computationally demanding.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Tensegrity-based robots use compression elements and tension cables to create lightweight structures that can reconfigure their shape. These characteristics are especially suited for planetary exploration, including for hard to traverse areas, such as lava tubes. While these capabilities are desirable for transporting these robots beyond Earth as well as reducing material costs, they complicate the control process. With such dynamic and reconfiguring parts, both simulating the motions of the tensegrity robot and planning future motions become challenging. New simulation tools for tensegrity rovers and state-of-the-art planning algorithms have been recently developed which can help to address these challenges, but have yet to used in tandem. This work integrates a recent sampling-based motion planner, which has been shown to converge to asymptotically optimal solutions even for systems with dynamics, with a novel tensegrity rover simulation tool, which has been verified in terms of accuracy against a hardware prototype. This paper shows that it is possible to get complex, long-duration trajectories for planetary navigation through this integration. At the same time, this integration is computationally demanding which motivated a parallel implementation of the proposed integration. With the parallel implementation, it is possible to observe improving path quality as computation time increases. This framework allows the consideration of planning under uncertainty to compute robust solutions, which is even more computationally demanding. |