This work presents Quad-SDK, an open source ROS-based full stack software framework for agile quadrupedal locomotion. The design of Quad-SDK is focused on the vertical integration of planning, control, estimation, communication, and development tools which enable agile quadrupedal locomotion in simulation and hardware with minimal user changes for multiple platforms. Furthermore, the modular software architecture allows researchers to experiment with their own implementations of different components while leveraging the existing framework. Quad-SDK also offers Gazebo simulation support and a suite of visualization and data-processing tools for rapid development. This work presents the high-level architecture of the software, its core features, and demonstrations of the agile locomotion it enables.

This abstract presents methods that reduce the computational demand of including second-order dynamics sensitivity information into optimization algorithms for robots in contact with the environment. The work extends a previous study on unconstrained whole-body motion planning. A full second-order Differential Dynamic Programming (DDP) algorithm is presented where all the necessary dynamics partial derivatives are computed with lower computational complexity than conventional DDP methods. Compared to using a firstorder approximation, the second-order partials more accurately represent the dynamics locally, but since they are tensorial and expensive to compute, they are often ignored. This work illustrates how to avoid the need for computing the derivative tensor by instead leveraging reverse-mode accumulation of derivatives. We also exploit the structure of the contact-constrained dynamics to further accelerate these computations. The relative performance of the proposed approach is benchmarked on a simulated model of the MIT Mini Cheetah executing a bounding gait

Impact-invariant control is a general framework for adapting model-based controllers for robots undergoing impacts. The framework projects the tracking objectives into a subspace that is invariant to impact forces, thus resulting in a controller that is robust to uncertainties in the impact event while minimally sacrificing control authority. In this work, we apply impact-invariant control to a SLIP-inspired running controller for the bipedal robot Cassie. This, to our knowledge is the first example of a model-based running controller demonstrated on hardware for Cassie. We detail our controller framework and the effect of the impact-invariant projection on the stability of the controller.

This work addresses the problem of multi-contact motion planning for quadrupedal legged robots on non-flat terrain. A common challenge in this domain is the selection of a contact schedule (i.e. contact configurations and contact switching event times) and respective sequences of foothold positions. Our approach is centered on formulating Markov decision processes using the evaluation of kinematic and dynamic feasibility criteria in the form of linear programs and are used in place of physical simulation. The resulting MDPs are solved using a policy- gradient-based reinforcement learning algorithm. Specifically, we train neural-network policies which generate reference foothold positions, Center-of-Mass poses and velocities as well contact switching timings. We evaluate our method in unstructured environments using proprioceptive and exteroceptive sensory input and on a suite of relevant terrain scenarios such as stairs, gaps and stepping stones.

This paper presents the design and experimental evaluations of an articulated robotic leg. The main scope of this new design is to combine high payload capabilities with agile locomotion. While both high payloads and agility have been achieved individually by different walking robots, designs intended for both have not been proposed yet. We optimized pseudo-direct-drive actuators to provide high torques density while simultaneously having good efficiency, low heat generation, and sufficient velocity to address the challenge. A two-DOF leg prototype has been designed and manufactured. The experimental test setup for the leg prototype is explained, and the results of the first hopping experiments are reported.

Terrain geometry is, in general, non-smooth, nonlinear, non-convex, and, if perceived through a robot-centric visual unit, appears partially occluded and noisy. This work presents the complete control pipeline capable of handling the aforementioned problems in real-time. We formulate a trajectory optimization problem that jointly optimizes over the base pose and footholds, subject to a heightmap. To avoid converging into undesirable local optima, we deploy a graduated optimization technique. We embed a compact, contact-force free stability criterion that is compatible with the non-flat ground formulation. Our experiments demonstrate stair climbing, walking on stepping stones, and over gaps, utilizing various dynamic gaits.

The Whole-Body Locomotion Framework (WoLF) is an end-to-end software suite devoted to the loco-manipulation of quadruped robots. WoLF abstracts the complexity of planning and control of quadrupedal robot hardware into a simple to use and robust software that can be connected through multiple tele-operation devices to different quadruped robot models. Furthermore, WoLF allows controlling mounted devices, such as arms or pan-tilt cameras, jointly with the quadrupedal platform. In this short paper, we introduce the main features of WoLF and its overall software architecture.

Navigation planning for legged robots has distinct challenges compared to wheeled and tracked systems due to the ability to lift legs off the ground and step over obstacles. While most navigation planners assume a fixed traversability value for a single terrain patch, we had previously proposed a reachability-based navigation planner for legged robots, which approximates the robot morphology by a set of reachability and body volumes. It uses a learned convolutional neural network to restrict valid footholds to feasible regions and incrementally builds a probabilistic roadmap graph for fast planning queries. We extend this work using an additional neural network trained in simulation, which predicts locomotion cost and risk over a given terrain. This produces safe and low-cost paths and increases planning speed by leveraging batched processing of the cost prediction network on GPU to rapidly validate navigation graph edges through locomotion risk thresholding. We deployed this navigation planner in team CERBERUS’ winning entry to the DARPA Subterranean Challenge finals, where the proposed planner powered the local navigation of four ANYmal quadrupeds during all three competition runs without a single navigation failure.

For legged robots to perform agile, highly dy- namic and contact-rich motions, whole-body trajectories com- putation of under-actuated complex systems subject to non- linear dynamics is required. In this work, we present hands- on applications of Horizon, a novel open-source framework for trajectory optimization tailored to robotic systems, that provides a collection of tools to simplify dynamic motion generation. Horizon was tested on a broad range of behaviours involving several robotic platforms: we introduce its building blocks and describe the complete procedure to generate three complex motions using its intuitive and straightforward API.

This paper proposes a model predictive control (MPC) framework for realizing dynamic walking gaits on the MIT Humanoid. In addition to adapting footstep location and timing online, the proposed method can reason about varying height, contact wrench, torso rotation, kinematic limit and negotiating uneven terrains. Specifically, a linear MPC (LMPC) optimizes for the desired footstep location by linearizing the single rigid body dynamics with respect to the current footstep location. A low-level task-space controller tracks the predicted state and control trajectories from the LMPC to leverage the full-body dynamics. Finally, an adaptive gait frequency scheme is employed to modify the step frequency and enhance the robustness of the walking controller. Both LMPC and taskspace control can be efficiently solved as quadratic programs (QP), and thus amenable for real-time applications. Simulation studies where the MIT Humanoid traverses a wave field and recovers from impulsive disturbances validated the proposed approach.

Legged robots that can operate autonomously in remote and hazardous environments will greatly increase opportunities for exploration into under-explored areas. Exteroceptive perception is crucial for fast and energy-efficient locomotion: perceiving the terrain before making contact with it enables planning and adaptation of the gait ahead of time to maintain speed and stability. However, utilizing exteroceptive perception robustly for locomotion has remained a grand challenge in robotics. Snow, vegetation, and water visually appear as obstacles on which the robot cannot step – or are missing altogether due to high reflectance. Additionally, depth perception can degrade due to difficult lighting, dust, fog, reflective or transparent surfaces, sensor occlusion, and more. For this reason, the most robust and general solutions to legged locomotion to date rely solely on proprioception. This severely limits locomotion speed, because the robot has to physically feel out the terrain before adapting its gait accordingly. Here we present a robust and general solution to integrating exteroceptive and proprioceptive perception for legged locomotion. We leverage an attention-based recurrent encoder that integrates proprioceptive and exteroceptive input. The encoder is trained end-to-end and learns to seamlessly combine the different perception modalities without resorting to heuristics. The result is a legged locomotion controller with high robustness and speed.

The deployment of robots within realistic environments requires the capability to plan and refine the locomanipulation trajectories on the fly to avoid unexpected interactions with a dynamic environment. This extended abstract provides a pipeline to offline plan a configuration space global trajectory based on a randomized strategy, and to online locally refine it depending on any change of the dynamic environment and the robot state. The offline planner directly plans in the contact space, and additionally seeks for whole-body feasible configurations compliant with the sampled contact states. The planned trajectory, made by a discrete set of contacts and configurations, can be seen as a graph and it can be online refined during the execution of the global trajectory. The online refinement is carried out by a graph optimization planner exploiting visual information. It locally acts on the global initial plan to account for possible changes in the environment. While the offline planner is a concluded work, tested on the humanoid COMAN+, the online local planner is still a work-in-progress which has been tested on a reduced model of the CENTAURO robot to avoid dynamic and static obstacles interfering with a wheeled motion task. Both the COMAN+ and the CENTAURO robots have been designed at the Italian Institute of Technology (IIT).

In this paper, we introduce a two-step calibration routine to identify the planar center of mass (CoM) position and the effective centroidal dynamics parameters of any quadruped using only joint sensors and an inertial measurement unit (IMU). Our proposed calibration routine consists of two steps: A bisection search method is used to locate the position of the planar CoM, and a sinusoidal excitation method is used to extract moments of inertia about each body axis. We verifu the inertial parameter identification method in simulation, and we implemented the center of mass finding algorithm in both simulation and hardware. The results of hardware CoM finding experiments verified in a balancing controller that requires 5mm CoM position accuracy.

Rigid body model has been widely adopted to plan the centroidal motion of legged robot. However, the rigid body motion evolves on the SE(3) manifold, which is nonlinear and not trivial to parameterize. The existing Model Predictive Control (MPC) either uses local parameterization that suffers from singularities, or geometric based liearization that makes the problem state dependent or hard to vectorize. In this work, we focus on the MPC on matrix Lie group. We linearize the configuration error and design a quadratic cost function in its Lie algebra. Given an initial condition, the linearized configuration error dynamics is globally valid and evolve independently of the system trajectory. The quadratic cost function could also ensure exponential convergence rate. The proposed MPC has been experimentally validated in MIT mini Cheetah locomotion and pose control.

We present a versatile nonlinear model predictive control (NMPC) approach for quadrupedal locomotion. Our formulation jointly optimizes a base trajectory and a set of footholds over a finite time horizon based on simplified dynamics models. We leverage second-order sensitivity analysis and a sparse Gauss-Newton (SGN) method to solve the resulting optimal control problems. We further describe our ongoing effort to verify our approach through simulation and hardware experiments. Finally, we extend our locomotion framework to deal with challenging tasks that comprise gap crossing, movement on stepping stones, and multi-robot control.

We propose a new neck design of legged robots to achieve robust visual-inertial state estimation in dynamic locomotion. Our neck design absorbs the impact from the legs and reduces the periodic vertical movement of the head using the dynamic vibration absorber (DVA), which can readily adapt to the frequency of oscillation of the body. Due to this smooth movement of the head, sensors in the head can measure data stably, which is impossible when sensors are directly attached to the body. We present the mechanical design of the neck as a combination of suspension mechanism with dynamic vibration absorber, which provide an adjustable notch filter for the vertical motion of the robot. Simulation and experimental results are performed to verify the effect of the proposed neck on the head of robots, manifesting that the states estimated from the visual-inertial sensors on the head can be precise even during aggressive motion by rendering both the inertial information and the feature tracking more stable and robust. The presented design can overcome the disadvantage that the localization of robots with legs can only operate well in gentle locomotion.

In this paper, we introduce a Model Predictive Control (MPC) formulation that calculates the ground reaction forces and torques for a quadruped equipped with two reaction wheels. We augment the centroidal dynamics with two additional angular momentum states from the wheels, and we further simplify the dynamics to formulate the problem as a convex quadratic program. With the simplified dynamics, we are able to run a hardware implementation of the MPC with a ten-time-step horizon and close the control loop at 250 Hertz. Experiment and simulation results demonstrate improved attitude stabilization and up to 40 percent reduction in angular error. A video demonstration of some of the experiments is available1 . Our approach and formulation provide a framework for more involved acrobatic maneuvers for future work.

Humanoid bipedal platforms are finally moving from controlled lab environments to real-world applications in unstructured scenarios. This has been recently achieved with substantial improvements in hardware and software architectures. In particular, the hardware design of bipedal humanoid robots has changed to be robust and resilient to impacts, as well as lightweight and efficient for dynamic movements. At the same time, planning and control architectures permit now to generate and control high dynamic, contact rich, motions employing the dynamics and the full body of the system. In this short paper, we present Kangaroo, a new humanoid bipedal robot designed by PAL Robotics for research on agile and dynamic locomotion. Kangaroo exploits novel mechanical solutions based on linear actuators and closed/parallel kinematics chains. We present the mechanical design and the whole planning and control pipeline developed to make Kangaroo jump on real experiments to demonstrate the superior robustness and resilience of the platform.

This work presents a simplified model-based tra- jectory optimization (TO) formulation for planning motions on quadruped mobile manipulators that locomote while carrying heavy payload. The formulation considers both robot and payload dynamics and simultaneously plans locomotion and payload manipulation trajectories. Thanks to the heavy payload manipulation planning the resulted payload-aware planner is less prone to leg singular configurations in kinematically demanding motions compared to its locomotion-only counter- part. The method is validated on the quadruped bi-manual CENTAURO robot carrying a payload that consists 85 % of the arms’ payload capacity and exceeds 15 % of the robot’s mass.

This abstract presents methods that reduce the computational demand of including second-order dynamics sensitivity information into optimization algorithms for robots in contact with the environment. The work extends a previous study on unconstrained whole-body motion planning. A full second-order Differential Dynamic Programming (DDP) algorithm is presented where all the necessary dynamics partial derivatives are computed with lower computational complexity than conventional DDP methods. Compared to using a firstorder approximation, the second-order partials more accurately represent the dynamics locally, but since they are tensorial and expensive to compute, they are often ignored. This work illustrates how to avoid the need for computing the derivative tensor by instead leveraging reverse-mode accumulation of derivatives. We also exploit the structure of the contact-constrained dynamics to further accelerate these computations. The relative performance of the proposed approach is benchmarked on a simulated model of the MIT Mini Cheetah executing a bounding gait

Impact-invariant control is a general framework for adapting model-based controllers for robots undergoing impacts. The framework projects the tracking objectives into a subspace that is invariant to impact forces, thus resulting in a controller that is robust to uncertainties in the impact event while minimally sacrificing control authority. In this work, we apply impact-invariant control to a SLIP-inspired running controller for the bipedal robot Cassie. This, to our knowledge is the first example of a model-based running controller demonstrated on hardware for Cassie. We detail our controller framework and the effect of the impact-invariant projection on the stability of the controller.

Online planning of whole-body motions for legged robots is challenging due to the inherent nonlinearity in the robot dynamics. To address this limitation, we propose a nonlinear MPC framework, the BiConMP, which can generate whole body trajectories online by efficiently exploiting the structure of the robot dynamics. BiConMP is used to generate various cyclic gaits on a real quadruped robot and its performance is evaluated on different terrain, countering unforeseen pushes and transitioning online between different gaits. Further, we demonstrate the ability of BiConMP to generate non-trivial acyclic whole-body dynamic motions on the robot. Finally, an extensive empirical analysis on the effects of planning horizon and frequency on the nonlinear MPC framework is reported and discussed.

This work presents Quad-SDK, an open source ROS-based full stack software framework for agile quadrupedal locomotion. The design of Quad-SDK is focused on the vertical integration of planning, control, estimation, communication, and development tools which enable agile quadrupedal locomotion in simulation and hardware with minimal user changes for multiple platforms. Furthermore, the modular software architecture allows researchers to experiment with their own implementations of different components while leveraging the existing framework. Quad-SDK also offers Gazebo simulation support and a suite of visualization and data-processing tools for rapid development. This work presents the high-level architecture of the software, its core features, and demonstrations of the agile locomotion it enables.

In this abstract, we examine the problem of push recovery for bipedal robot locomotion and present a reactive decision-making and robust planning framework for locomotion resilient to external perturbations. Rejecting perturbations is an essential capability of bipedal robots and has been widely studied in the locomotion literature. However, adversarial disturbances and aggressive turning can lead to negative lateral step width (i.e., crossed-leg scenarios) with unstable motions and self-collision risks. These motion planning problems are computationally difficult and have not been explored under a hierarchically integrated task and motion planning method. We explore a planning and decision-making framework that closely ties linear-temporal-logic-based reactive synthesis with trajectory optimization incorporating the robot’s full-body dynamics, kinematics, and leg collision avoidance constraints. Between the high-level discrete symbolic decision-making and the lowlevel continuous motion planning, behavior trees serve as a reactive interface to handle perturbations occurring at any time of the locomotion process. Our experimental results show the efficacy of our method in generating resilient recovery behaviors in response to diverse perturbations from any direction with bounded magnitudes.

Mathematical optimization is the workhorse be- hind several aspects of modern robotics and control. In these applications, the focus is on constrained optimization, and the ability to work on manifolds (such as the classical matrix Lie groups), along with a specific requirement for robustness and speed. In recent years, augmented Lagrangian methods have seen a resurgence due to their robustness and flexibility, their connections to (inexact) proximal-point methods, and their interoperability with Newton or semismooth Newton methods. In the sequel, we present primal-dual augmented Lagrangian method for inequality-constrained problems on manifolds, which we introduced in our recent work, as well as an efficient C++ implementation suitable for use in robotics applications and beyond.

Legged robotic systems leverage ground contact and the reaction forces they provide to achieve agile locomotion. However, uncertainty coupled with the discontinuous nature of contact can lead to failure in real-world environments with unexpected height variations, such as rocky hills or curbs. To enable dynamic traversal of extreme terrain, this work introduces the utilization of proprioception to estimate and react to unknown hybrid events and elevation changes and a two-degree-of-freedom tail to improve control independent of contact. Simulation results over unforeseen elevation changes show that our method can stabilize locomotion over height changes of up to 1.5 times the leg length.

This paper tackles the problem of nonprehensile object transportation through a legged manipulator. A wholebody control architecture is devised to prevent sliding of the object placed on a tray-like end-effector and retain the legged robot balance during walking. The controller solves a quadratic optimization problem to realize the sought transportation task while maintaining the contact forces between the tray and the object and between the legs and the ground within their respective friction cones.

Agile maneuvers such as sprinting and high-speed turning in the wild are challenging for legged robots. Reinforcement learning provides a promising framework for addressing this challenge. We present an end-to-end learned controller that achieves record agility for the MIT Mini Cheetah, sustaining speeds up to 3.9 m/s. This system runs and turns fast on natural terrains like grass, ice, and gravel and responds robustly to disturbances. Our controller is a neural network trained in simulation via reinforcement learning and transferred to the real world. The two key components are: (i) an adaptive curriculum on velocity commands; and (ii) an online system identification strategy for sim-to-real transfer leveraged from prior work. Videos of the robot’s behaviors are available at: https://agility.csail.mit.edu/.

In this work, we show the implementation of a control framework for non-gaited legged system control. The presented approach decouples the gait optimization from the motion stabilization treating the contact phase as a decision variable. The system has been tested against the state-of-theart Mixed Integer Quadratic Programming solver and verified in various simulation environments.

In this extended abstract we give a brief introduction to our work [1]. We propose a foothold evaluation criterion that considers the transition feasibility for both linear and angular dynamics to overcome complex scenarios. We present convex and nonlinear formulations as a direct extension of [2] in a receding-horizon fashion to take into account also angular dynamics. The criterion is integrated with a Vision-based Foothold Adaptation (VFA) strategy that considers robot kinematics, leg collisions and terrain morphology. We verify the validity of the selected footholds and of the generated trajectories in simulation and experiments with the 90kg quadruped robot HyQ.

Robotic exploration of celestial bodies has been performed solely with wheeled systems until today, making access to highly unstructured, compressible, and sloped areas very challenging. On the other hand, walking robots have advanced rapidly over the last decade and have reached a maturity level where commercial applications have become viable. Thus, it is only a matter of time until walking robots will be used to explore so far unreachable regions of our solar system. In this work, we showcase the recent advancements of our dynamically walking legged systems for future planetary exploration missions. In detail, we summarize our work performed on walking on steep, granular slopes, low-gravity locomotion, robot-payload integration and mission scenario execution. Finally, we showcase several real-world deployments and our lessons learned from the experiments.

This work addresses the problem of multi-contact motion planning for quadrupedal legged robots on non-flat terrain. A common challenge in this domain is the selection of a contact schedule (i.e. contact configurations and contact switching event times) and respective sequences of foothold positions. Our approach is centered on formulating Markov decision processes using the evaluation of kinematic and dynamic feasibility criteria in the form of linear programs and are used in place of physical simulation. The resulting MDPs are solved using a policy- gradient-based reinforcement learning algorithm. Specifically, we train neural-network policies which generate reference foothold positions, Center-of-Mass poses and velocities as well contact switching timings. We evaluate our method in unstructured environments using proprioceptive and exteroceptive sensory input and on a suite of relevant terrain scenarios such as stairs, gaps and stepping stones.