Quadrupedal locomotion can be described as a constrained optimization problem that is very hard to solve due to the high dimensional, nonlinear and non-smooth system dynamics. In this paper, we propose a formulation that can be solved within few seconds using sequential quadratic programming. This method considers only a simplified model that just sufficiently represents the system dynamics. The output is a very coarse plan, which can be accurately and robustly followed on a real system using hierarchical whole-body control combined with inverted pendulum-based reactive stepping. Using the fully torque controllable quadrupedal robot ANYmal, we present successful experiments for walking, trotting, and gait transitions even under substantial external disturbances.

This paper presents a framework which allows a quadrupedal robot to execute dynamic gaits including trot, pace and dynamic lateral walk, as well as a smooth transition between them. Our method relies on an online ZMP based motion planner which continuously updates the reference motion trajectory as a function of the contact schedule and the state of the robot. The planner is coupled with a hierarchical whole-body controller which optimizes the whole-body motion and contact forces by solving a cascade of prioritized tasks. We tested our framework on ANYmal, a fully torque controllable quadrupedal robot which is actuated by series-elastic actuators.

This paper introduces ANYmal, a quadrupedal robot that features outstanding mobility and dynamic motion capability. Thanks to novel, compliant joint modules with integrated electronics, the 30 kg, 0.5 m tall robotic dog is torque controllable and very robust against impulsive loads during running or jumping. The presented machine was designed with a focus on outdoor suitability, simple maintenance, and user-friendly handling to enable future operation in real world scenarios. Performance tests with the joint actuators indicated a torque control bandwidth of more than 70 Hz, high disturbance rejection capability, as well as impact robustness when moving with maximal velocity. It is demonstrated in a series of experiments that ANYmal can execute walking gaits, dynamically trot at moderate speed, and is able to perform special maneuvers to stand up or crawl very steep stairs. Detailed measurements unveil that even full-speed running requires less than 280 W, resulting in an autonomy of more than 2 h.

Quadrupedal locomotion on sloped terrains poses different challenges than walking in a mostly flat environment. The robot's configuration needs to be explicitly controlled in order to avoid slipping and kinematic limits. To this end, information about the terrain's inclination is required for carefully planning footholds, the pose of the main body, and modulation of the ground reaction forces. This is even more important for dynamic trotting, as only two support legs are available to compensate for gravity and drive a desired motion. We propose a reliable method for estimating the parameters of the terrain quadrupedal robots move on, in the face of limited perception capabilities and drifting robot pose estimates. By fusing inertial measurements, kinematic data from joint encoders and contact information from force sensors, the local inclination can be robustly estimated and used to optimize the contact forces to reduce slippage. The estimated terrain information, namely the pitch and roll angles of the ground plane, is exploited in an extended version of our previous model-based control approach. Our improved control framework enabled StarlETH, a medium-sized, fully autonomous, torque-controllable quadrupedal robot, to trot on slopes of up to 21°.

Methods of optimizing a single trajectory are mature enough for planning in many applications. Yet such optimization methods applied to high Degree-Of-Freedom robots either consume too much time to be real-time or approximate the dynamics such that they lack physical consistency. In this paper, we present a method of precomputing optimized trajectories and compressing the information to get a compact representation of the optimal policy function. By varying the initial configuration of a robot and optimizing multiple trajectories, the controller gains knowledge about the optimal policy function. Such computation can be performed on a powerful workstation or even supercomputers instead of an onboard computer of the robot. The precomputed optimal trajectories are stored in a Single-hidden Layer Feedforward neural Network (SLFN) using Optimally Pruned Extreme Learning Machine (OP-ELM). This ensures minimal representation of the model and fast evaluation of the SLFN. We first explain our method using a simple time-optimal control problem with an analytical solution. We then demonstrate how this method can work even for high dimensional state by optimizing a foothold strategy of a full quadruped robot in simulation.

Developing control methods that allow legged robots to move with skill and agility remains one of the grand challenges in robotics. In order to achieve this ambitious goal, legged robots must possess a wide repertoire of motor skills. A scalable control architecture that can represent a variety of gaits in a unified manner is therefore desirable. Inspired by the motor learning principles observed in nature, we use an optimization approach to automatically discover and fine-tune parameters for agile gaits. The success of our approach is due to the controller parameterization we employ, which is compact yet flexible, therefore lending itself well to learning through repetition. We use our method to implement a flying trot, a bound and a pronking gait for StarlETH, a fully autonomous quadrupedal robot.

This paper describes a hierarchical operational space control (OSC) method based on least square optimization and outlines different ways to reduce the dimensionality of the optimization vector. The framework allows to emulate various behaviors by prioritized task-space motion, joint torque, and contact force optimization. Moreover, a methodology is introduced to partially excite the natural dynamics of the robot by open-loop motor regulation while the entire behavior is stabilized by hierarchical OSC. As a major contribution, the presented control strategies are tested and validated in real hardware walking, trotting, and pronking experiments using a fully torque controllable quadrupedal robot.

In this paper we present a method for fusing optical flow and inertial measurements. To this end, we derive a novel visual error term which is better suited than the standard continuous epipolar constraint for extracting the information contained in the optical flow measurements. By means of an unscented Kalman filter (UKF), this information is then tightly coupled with inertial measurements in order to estimate the egomotion of the sensor setup. The individual visual landmark positions are not part of the filter state anymore. Thus, the dimensionality of the state space is significantly reduced, allowing for a fast online implementation. A nonlinear observability analysis is provided and supports the proposed method from a theoretical side. The filter is evaluated on real data together with ground truth from a motion capture system.

This paper provides an overview about StarlETH: a compliant quadrupedal robot that is designed to study fast, efficient, versatile, and robust locomotion. The platform is driven by highly compliant series elastic actuation, which makes the system fully torque controllable, energetically efficient, and well suited for dynamic maneuvers. Using model-based control strategies, this medium dog-sized machine is capable of various gaits ranging from static walking to dynamic running over challenging terrain. StarlETH is equipped with an onboard PC, batteries, and various sensor equipment that enables enduring autonomous operation. In this paper, we provide an overview about the underlying locomotion control algorithms, outline a real-time control and simulation environment, and conclude the work with a number of experiments to demonstrate the performance of the presented hardware and controllers.

This paper introduces a novel batch optimization based calibration framework for legged robots. Given a non-degenerate calibration dataset and considering the stochastic models of the sensors, the task is formulated as a maximum likelihood problem. In order to facilitate the derivation of consistent measurement equations, the trajectory of the robot and other auxiliary variables are included into the optimization problem. This formulation can be transformed into a nonlinear least squares problem which can be readily solved. Applied to our legged robot StarIETH, the framework estimates kinematic parameters (segment lengths, body dimensions, angular offsets), accelerometer and gyroscope biases, as well as full inter-sensor calibrations. The generic structure easily allows the inclusion of additional sensor modalities. Based on datasets obtained on the real robot the consistency and performance of the presented approach are successfully evaluated.

Quadrupedal animals move through their environments with unmatched agility and grace. An important part of this is the ability to choose between different gaits in order to travel optimally at a certain speed or to robustly deal with unanticipated perturbations. In this paper, we present a control framework for a quadrupedal robot that is capable of locomoting using several gaits. We demonstrate the flexibility of the algorithm by performing experiments on StarlETH, a recently-developed quadrupedal robot. We implement controllers for a static walk, a walking trot, and a running trot, and show that smooth transitions between them can be performed. Using this control strategy, StarlETH is able to trot unassisted in 3D space with speeds of up to 0.7m/s, it can dynamically navigate over unperceived 5-cm high obstacles and it can recover from significant external pushes.

This paper addresses the problem of evaluating and estimating the mechanical robustness of footholds for legged robots in unstructured terrain. In contrast to approaches that rely on human expert knowledge or human defined criteria to identify appropriate footholds, our method uses the robot itself to assess whether a certain foothold is adequate or not. To this end, one of the robot's legs is employed to haptically explore an unknown foothold. The robustness of the foothold is defined by a simple metric as a function of the achievable ground reaction forces. This haptic feedback is associated with the foothold shape to estimate the robustness of untouched footholds. The underlying shape clustering principles are tested on synthetic data and in hardware experiments using a single-leg testbed.

This paper presents a state estimation approach for legged robots based on stochastic filtering. The key idea is to extract information from the kinematic constraints given through the intermittent contacts with the ground and to fuse this information with inertial measurements. To this end, we design an unscented Kalman filter based on a consistent formulation of the underlying stochastic model. To increase the robustness of the filter, an outliers rejection methodology is included into the update step. Furthermore, we present the nonlinear observability analysis of the system, where, by considering the special nature of 3D rotations, we obtain a relatively simple form of the corresponding observability matrix. This yields, that, except for the global position and the yaw angle, all states are in general observable. This also holds if only one foot is in contact with the ground. The presented filter is evaluated on a real quadruped robot trotting over an uneven and slippery terrain.

This paper presents the application of reinforcement learning to improve the performance of highly dynamic single legged locomotion with compliant series elastic actuators. The goal is to optimally exploit the capabilities of the hardware in terms of maximum jump height, jump distance, and energy efficiency of periodic hopping. These challenges are tackled with the reinforcement learning method Policy Improvement with Path Integrals (PI2) in a model-free approach to learn parameterized motor velocity trajectories as well as highlevel control parameters. The combination of simulation and hardware-based optimization allows to efficiently obtain optimal control policies in an up to 10-dimensional parameter space. The robotic leg learns to temporarily store energy in the elastic elements of the joints in order to improve the jump height and distance. In addition, we present a method to learn time-independent control policies and apply it to improve the energetic efficiency of periodic hopping.