@article{winkler17b,
                  author    = {Winkler, Alexander W and 
                              Farshidian, Farbod and 
                              Pardo, Diego and 
                              Neunert, Michael and 
                              Buchli, Jonas},
                  title     = {Fast Trajectory Optimization for Legged Robots 
                              using Vertex-based ZMP Constraints},
                  journal   = {IEEE Robotics and Automation Letters (RA-L)},
                  year      = {2017},
                  month     = {oct},
                  pages     = {2201-2208},
                  doi       = {10.1109/LRA.2017.2723931},
                  volume    = {2},
                  keywords  = {legged locomotion, trajectory optimization}
                }
                

@inproceedings{farshidian17b,
  author    = {Farshidian, Farbod and 
               Jelavic, Edo and
               Winkler, Alexander W and
               Buchli, Jonas},
  title     = {Robust Whole-Body Motion Control of Legged Robots},
  booktitle = {IEEE/RSJ International Conference on 
               Intelligent Robots and Systems (IROS)},
  year      = {2017},
  abstract  = {We introduce a robust control architecture for
               the whole-body motion control of torque controlled robots
               with arms and legs. The method is based on the robust
               control of contact forces in order to track a planned Center
               of Mass trajectory. Its appeal lies in the ability to guarantee
               robust stability and performance despite rigid body model
               mismatch, actuator dynamics, delays, contact surface stiffness,
               and unobserved ground profiles. Furthermore, we introduce a
               task space decomposition approach which removes the coupling
               effects between contact force controller and the other noncontact
               controllers. Finally, we verify our control performance
               on a quadruped robot and compare its performance to a
               standard inverse dynamics approach on hardware.}            
}

We introduce a robust control architecture for the whole-body motion control of torque controlled robots with arms and legs. The method is based on the robust control of contact forces in order to track a planned Center of Mass trajectory. Its appeal lies in the ability to guarantee robust stability and performance despite rigid body model mismatch, actuator dynamics, delays, contact surface stiffness, and unobserved ground profiles. Furthermore, we introduce a task space decomposition approach which removes the coupling effects between contact force controller and the other noncontact controllers. Finally, we verify our control performance on a quadruped robot and compare its performance to a standard inverse dynamics approach on hardware.

@inproceedings{pardo17,
  author    = {Pardo, Diego and
               Neunert, Michael and 
               Winkler, Alexander W and
               Grandia, Ruben and
               Buchli, Jonas},
  title     = {Hybrid direct collocation and control in the constraint-
               consistent subspace for dynamic legged robot locomotion},
  booktitle = {Robotics, Science and Systems (RSS)},
  year      = {2017},
  abstract  = {In this paper, we present an algorithm for optimal planning and
               control of legged robot locomotion. Given the desired contact 
               sequence, this method generates gaits and dynamic motions for 
               legged robots without resorting to simplified stability criteria. 
               The method uses direct collocation for searching for solutions 
               within the constraint-consistent subspace defined by the robot’s 
               contact configuration. For the differential equation constraints 
               of the collocation algorithm, we use the so-called direct 
               dynamics of a constrained multibody system. The dynamics of a 
               legged robot is different for each contact configuration. Our 
               method deals with such a hybrid nature, and it allows for 
               velocity discontinuities when contacts are made. We introduce 
               the projected impact dynamics constraint to enforce consistency 
               during mode switching. We stabilize the plan using an inverse 
               dynamics controller consistent with the constraints and 
               compatible with the optimal feed-forward control of the motion 
               plan. As a whole, this approach reduces the complexity associated 
               with specifying dynamic motions of a floating-base robot under 
               the constant influence of contact forces. We apply this method 
               on a hydraulically actuated quadruped robot. We show two type of 
               gaits on the physical system (walking and trotting), and other 
               dynamic motions in simulation (jumping and leaping). The results 
               presented here are one of the few examples of an optimal control 
               problem satisfactorily solved and transferred to a real 
               torque-controlled legged robot.},       
}

In this paper, we present an algorithm for optimal planning and control of legged robot locomotion. Given the desired contact sequence, this method generates gaits and dynamic motions for legged robots without resorting to simplified stability criteria. The method uses direct collocation for searching for solutions within the constraint-consistent subspace defined by the robot’s contact configuration. For the differential equation constraints of the collocation algorithm, we use the so-called direct dynamics of a constrained multibody system. The dynamics of a legged robot is different for each contact configuration. Our method deals with such a hybrid nature, and it allows for velocity discontinuities when contacts are made. We introduce the projected impact dynamics constraint to enforce consistency during mode switching. We stabilize the plan using an inverse dynamics controller consistent with the constraints and compatible with the optimal feed-forward control of the motion plan. As a whole, this approach reduces the complexity associated with specifying dynamic motions of a floating-base robot under the constant influence of contact forces. We apply this method on a hydraulically actuated quadruped robot. We show two type of gaits on the physical system (walking and trotting), and other dynamic motions in simulation (jumping and leaping). The results presented here are one of the few examples of an optimal control problem satisfactorily solved and transferred to a real torque-controlled legged robot.

@inproceedings{winkler17a,
  author    = {Winkler, Alexander W and 
               Farshidian, Farbod and 
               Neunert, Michael and 
               Pardo, Diego and 
               Buchli, Jonas},
  title     = {Online Walking Motion and Foothold Optimization 
               for Quadruped Locomotion},
  booktitle = {IEEE International Conference on Robotics and Automation (ICRA)},
  year      = {2017},
  pages     = {5308-5313},
  doi       = {10.1109/ICRA.2017.7989624},
  abstract  = {We present an algorithm that generates walking
               motions for quadruped robots without the use of an explicit
               footstep planner by simultaneously optimizing over both the
               Center of Mass (CoM) trajectory and the footholds. Feasibility
               is achieved by imposing stability constraints on the CoM
               related to the Zero Moment Point and explicitly enforcing
               kinematic constraints between the footholds and the CoM
               position. Given a desired goal state, the problem is solved 
               online by a Nonlinear Programming solver to generate the walking
               motion. Experimental trials show that the algorithm is able to
               generate walking gaits for multiple steps in milliseconds that
               can be executed on a real quadruped robot.} 
}

We present an algorithm that generates walking motions for quadruped robots without the use of an explicit footstep planner by simultaneously optimizing over both the Center of Mass (CoM) trajectory and the footholds. Feasibility is achieved by imposing stability constraints on the CoM related to the Zero Moment Point and explicitly enforcing kinematic constraints between the footholds and the CoM position. Given a desired goal state, the problem is solved online by a Nonlinear Programming solver to generate the walking motion. Experimental trials show that the algorithm is able to generate walking gaits for multiple steps in milliseconds that can be executed on a real quadruped robot.

@inproceedings{farshidian17a,
  author    = {Farshidian, Farbod and 
              Neunert, Michael and 
              Winkler, Alexander W and 
              Rey, Gonzalo and 
              Buchli, Jonas},
  title     = {An Efficient Optimal Planning and Control Framework 
              For Quadrupedal Locomotion},
  booktitle = {IEEE International Conference on Robotics and Automation (ICRA)},
  year      = {2017},
  pages     = {93-100},
  doi       = {10.1109/ICRA.2017.7989016},
  abstract  = {In this paper, we present an efficient Dynamic
               Programing framework for optimal planning and control of
               legged robots. First we formulate this problem as an optimal
               control problem for switched systems. Then we propose a
               multi–level optimization approach to find the optimal switching
               times and the optimal continuous control inputs. Through
               this scheme, the decomposed optimization can potentially be
               done more efficiently than the combined approach. Finally,
               we present a continuous-time constrained LQR algorithm
               which simultaneously optimizes the feedforward and feedback
               controller with O(n) time-complexity. In order to validate our
               approach, we show the performance of our framework on a
               quadrupedal robot. We choose the Center of Mass dynamics
               and the full kinematic formulation as the switched system model
               where the switching times as well as the contact forces and the
               joint velocities are optimized for different locomotion tasks 
               such as gap crossing, walking and trotting.}
}

In this paper, we present an efficient Dynamic Programing framework for optimal planning and control of legged robots. First we formulate this problem as an optimal control problem for switched systems. Then we propose a multi–level optimization approach to find the optimal switching times and the optimal continuous control inputs. Through this scheme, the decomposed optimization can potentially be done more efficiently than the combined approach. Finally, we present a continuous-time constrained LQR algorithm which simultaneously optimizes the feedforward and feedback controller with O(n) time-complexity. In order to validate our approach, we show the performance of our framework on a quadrupedal robot. We choose the Center of Mass dynamics and the full kinematic formulation as the switched system model where the switching times as well as the contact forces and the joint velocities are optimized for different locomotion tasks such as gap crossing, walking and trotting.

@article{neunert2017,
  author    = {Michael Neunert and 
               Farbod Farshidian and 
               Alexander W. Winkler and 
               Jonas Buchli},
  title     = {Trajectory Optimization Through Contacts and 
               Automatic Gait Discovery for Quadrupeds},
  journal   = {IEEE Robotics and Automation Letters (RA-L)},
  year      = {2017},
  pages     = {1502-1509},
  volume    = {2},
  doi       = {10.1109/LRA.2017.2665685},
  abstract  = {In this work we present a Trajectory Optimization
               framework for whole-body motion planning through contacts.
               We demonstrate how the proposed approach can be applied to
               automatically discover different gaits and dynamic motions on a
               quadruped robot. In contrast to most previous methods, we do
               not pre-specify contact-switches, -timings, -points or gait 
               patterns, but they are a direct outcome of the optimization. 
               Furthermore, we optimize over the entire dynamics of the robot, 
               which enables the optimizer to fully leverage the capabilities of
               the robot. To illustrate the spectrum of achievable motions, we 
               show eight different tasks, which would require very different 
               control structures when solved with state-of-the-art methods. 
               Using our Trajectory Optimization approach, we are solving each 
               task with a simple, high level cost function and without any 
               changes in the control structure. Furthermore, we fully integrate
               our approach with the robot’s control and estimation framework 
               such that we are able to run the optimization online. Through 
               several hardware experiments we show that the optimized 
               trajectories and control inputs can be directly applied to 
               physical systems.},
  keywords  = {Multilegged Robots, Motion and Path Planning,
               Optimization and Optimal Control}
}

In this work we present a Trajectory Optimization framework for whole-body motion planning through contacts. We demonstrate how the proposed approach can be applied to automatically discover different gaits and dynamic motions on a quadruped robot. In contrast to most previous methods, we do not pre-specify contact-switches, -timings, -points or gait patterns, but they are a direct outcome of the optimization. Furthermore, we optimize over the entire dynamics of the robot, which enables the optimizer to fully leverage the capabilities of the robot. To illustrate the spectrum of achievable motions, we show eight different tasks, which would require very different control structures when solved with state-of-the-art methods. Using our Trajectory Optimization approach, we are solving each task with a simple, high level cost function and without any changes in the control structure. Furthermore, we fully integrate our approach with the robot’s control and estimation framework such that we are able to run the optimization online. Through several hardware experiments we show that the optimized trajectories and control inputs can be directly applied to physical systems.

@inproceedings{buchli2017,
  author    = {Jonas Buchli and 
               Farbod Farshidian and 
               Alexander W. Winkler and 
               Timothy Sandy and
               Markus Gifthaler},
  title     = {Optimal and Learning Control for Autonomous Robots},
  booktitle = {arXiv},
  year      = {2017},
  abstract  = {Optimal and Learning Control for Autonomous Robots has been taught 
               in the Robotics, Systems and Controls Masters at ETH Zurich with the 
               aim to teach optimal control and reinforcement learning for closed loop 
               control problems from a unified point of view. The starting point is the 
               formulation of of an optimal control problem and deriving the different 
               types of solutions and algorithms from there. These lecture notes aim at 
               supporting this unified view with a unified notation wherever possible, 
               and a bit of a translation help to compare the terminology and notation 
               in the different fields. The course assumes basic knowledge of Control 
               Theory, Linear Algebra and Stochastic Calculus.}
}

Optimal and Learning Control for Autonomous Robots has been taught in the Robotics, Systems and Controls Masters at ETH Zurich with the aim to teach optimal control and reinforcement learning for closed loop control problems from a unified point of view. The starting point is the formulation of of an optimal control problem and deriving the different types of solutions and algorithms from there. These lecture notes aim at supporting this unified view with a unified notation wherever possible, and a bit of a translation help to compare the terminology and notation in the different fields. The course assumes basic knowledge of Control Theory, Linear Algebra and Stochastic Calculus.