In this work, the Divergent Component of Motion (DCM) method is expanded to include angular coordinates for the first time. This work introduces the idea of spatial DCM, which adds an angular objective to the existing linear DCM theory. To incorporate the angular component into the framework, a discussion is provided on extending beyond the linear motion of the Linear Inverted Pendulum model (LIPM) towards the Single Rigid Body model (SRBM) for DCM. This work presents the angular DCM theory for a 1D rotation, simplifying the SRBM rotational dynamics to a flywheel to satisfy necessary linearity constraints. The 1D angular DCM is mathematically identical to the linear DCM and defined as an angle which is ahead of the current body rotation based on the angular velocity. This theory is combined into a 3D linear and 1D angular DCM framework, with discussion on the feasibility of simultaneously achieving both sets of objectives. A simulation in MATLAB and hardware results on the TORO humanoid are presented to validate the framework's performance.

One of the essential aspects of humanoid robot running is determining the limb-swinging trajectories. During the flight phases, where the ground reaction forces are not available for regulation, the limb swinging trajectories are significant for the stability of the next stance phase. Due to the conservation of angular momentum, improper leg and arm swinging results in highly tilted and unsustainable body configurations at the next stance phase landing. In such cases, the robotic system fails to maintain locomotion independent of the stability of the center of mass trajectories. This problem is more apparent for fast and high flight time trajectories. This paper proposes a real-time nonlinear limb trajectory optimization problem for humanoid running. The optimization problem is tested on two different humanoid robot models, and the generated trajectories are verified using a running algorithm for both robots in a simulation environment.

This study extends the whole-body control (WBC) formulation for bipedal humanoid robots that include closed (parallel) kinematic chains in their structure. Along with general formulation, we also stress the implementation of this formulation on Kangaroo, which is a highly dynamic humanoid robot developed by PAL Robotics. This 76-DOF robot includes 24 independent closed-kinematic chains in its structure and constitutes a good case study for our approach. We discuss the WBC formulation for various control structures, including inverse dynamics control (IDC) and Modular Passive Tracking Control (MPTC). As a test scenario, we employ a 3D spring-loaded inverted pendulum (SLIP) jumping trajectory with disturbance rejection as the desired CoM trajectory.

Running and jumping are locomotion modes that allow legged robots to rapidly traverse great distances and overcome difficult terrain. In this article, we show that the 3-D divergent component of motion (3D-DCM) framework, which was successfully used for generating walking trajectories in previous works, retains its validity and coherence during flight phases, and, therefore, can be used for planning running and jumping motions. We propose a highly efficient motion planner that generates stable center-of-mass (CoM) trajectories for running and jumping with arbitrary contact sequences and time parametrizations. The proposed planner constructs the complete motion plan as a sequence of motion phases that can be of different types: stance, flight, transition phases, etc. We introduce a unified formulation of the CoM and DCM waypoints at the start and end of each motion phase, which makes the framework extensible and enables the efficient waypoint computation in matrix and algorithmic form. The feasibility of the generated reference trajectories is demonstrated by extensive whole-body simulations with the humanoid robot TORO.

Embedding elastic elements into legged robots through mechanical design enables highly efficient oscillating patterns that resemble natural gaits. However, current trajectory planning techniques miss the opportunity of taking advantage of these natural motions. This work proposes a locomotion planning method that aims to unify traditional trajectory generation with modal oscillations. Our method utilizes task-space linearized modes for generating center of mass trajectories on the sagittal plane. We then use nonlinear optimization to find the gait timings that match these trajectories within the Divergent Component of Motion planning framework. This way, we can robustly translate the modes-aware centroidal motions into joint coordinates. We validate our approach with promising results and insights through experiments on a compliant quadrupedal robot.

Gait generation frameworks for humanoid robots typically assume a constant centroidal angular momentum (CAM) throughout the walking cycle, which induces undesirable contact torques in the feet and results in performance degradation. In this work, we present a novel algorithm to learn the CAM online and include the obtained knowledge within the closed-form solutions of the Divergent Component of Motion (DCM) locomotion framework. To ensure a reduction of the contact torques at the desired center of pressure position, a CAM trajectory is generated and explicitly tracked by a whole-body controller. Experiments with the humanoid robot TORO demonstrate that the proposed method significantly increases the maximum step length and walking speed during locomotion.

In this paper, we present a highly efficient Divergent Component of Motion (DCM) reference trajectory generator capable of adapting online to large perturbations acting on the center-of-mass (push recovery) and on the swing foot (stumble recovery). For push recovery, we propose an analytic solution for a footstep adjustment strategy based on the DCM dynamics. The proposed algorithm considers double support phases explicitly and is active throughout the motion, i.e., during both single and double support phases. For stumble recovery, we introduce a continuous DCM trajectory adaptation based on the instantaneous tracking error of the swing foot. Our method is highly efficient, computing a push recovery solution within 10 microseconds on the robot hardware. Furthermore, it achieves robust locomotion for large external perturbations, which we demonstrate in simulations and experiments with the humanoid robot TORO.

This paper presents methods that facilitate the implementation of dynamic walking on torque-controlled robots in real world experiments. The work uses the Divergent Component of Motion (DCM) for walking trajectory generation and control. The DCM controller is embedded into a whole-body controller (WBC) that produces a full-body walking behavior. While in simulation the combination of DCM and WBC is sufficient for achieving sophisticated walking gaits, during our initial experiments several real-world issues, detailed in this paper, prevented the original control framework from functioning. This work presents the improvements to the original control framework that enabled a breakthrough on the way to achieving torque-based dynamic walking on a real robot.

When walking at high speeds, the swing legs of robots produce a non-negligible angular momentum rate. To accommodate this, we provide a reference trajectory generator for bipedal walking that incorporates predicted centroidal angular momentum at the planning stage. This can be done efficiently as the Centroidal Moment Pivot (CMP), Instantaneous Capture Point (ICP) and the center of mass (CoM) all have closed-form trajectory solutions due to their linear dynamics. This is then used to produce smooth, continuous trajectories. We furthermore provide a lightweight model to estimate angular momentum as induced during leg swing of the gait cycle. Our proposed trajectory generator is tested thoroughly in simulation and has been shown to successfully operate on the real hardware.

This paper presents a new method for planning and controlling dynamic multi-contact motions for humanoid robots. Our motion planner takes a sequence of multi-contact stances and generates closed-form reference trajectories for the robot center of mass (CoM) position, velocity, and acceleration, based on the concept of Divergent Component of Motion (DCM). The timing of the contact transitions and the end-effector trajectories are automatically computed such that the motion is feasible with respect to kinematic and dynamic constraints. We verify the constraints using a simplified model of the robot to achieve a very fast planner that finds a feasible solution within a few seconds. The reference trajectories serve as inputs to a passivity-based whole-body controller which includes a DCM controller for tracking the CoM trajectory. We demonstrate the robustness of our approach in simulation and experiments with the humanoid robot TORO.

This paper presents a novel multi-step closed-form walking trajectory generator based on the concept of Divergent Component of Motion (DCM) that guarantees smoothness of all resulting reference trajectories. Further, we introduce an analytical method for footstep adjustment to recover from strong disturbances. The DCM trajectory is adjusted to guarantee smoothness of control outputs. Additionally, we present a momentum-based disturbance observer that improves robustness w.r.t. strong continuous perturbations. The proposed methods are verified in simulations.

While only continuous motions are possible, the way in which contacts appear and disappear confers to legged locomotion a characteristic discontinuous nature that is traditionally shared by the algorithms used for legged locomotion planning. In this paper, we show that this discontinuous nature can disappear if the notion of collision is well redefined and we efficiently solve two different practical problems of legged locomotion planning with algorithms based on an approach that establishes a bridge between discrete and continuous planning. The first problem consists of reactive footstep planning with a biped robot and the second one consists of nongaited locomotion planning with a hexapod.

This paper works toward bridging the gap between observations and analysis of human-running motions, i.e., motion science and robust humanoid robot control. It is based on the concept of biologically inspired deadbeat (BID) control, which facilitates both 3D running on flat ground and on 3D stepping stones. Further contributions include explicit foot step targeting during running, leg crossover avoidance, and the embedding of BID control into a quadratic-program-based whole-body controller. The controller is based on the encoding of leg forces and center-of-mass (CoM) trajectories during stance as polynomial splines, allowing for intuitive and purely analytical controller design. It allows a real-time implementation, is highly robust against perturbations, and enables versatile running patterns. This paper provides a method for purely analytical foot-step targeting, introduces a new method to increase kinematic feasibility on complex robot models, and presents advanced whole-body running simulations, including high-speed running and push recovery. The paper closes the circle to human motion science by comparing BID-based CoM trajectories and ground reaction forces to data from human-running experiments.

This paper introduces a Biologically Inspired Dead-beat (BID) controller for bipedal running in 3D. The controller runs in real-time, is extremely robust against perturbations and allows for versatile running patterns. It is based on the encoding of leg forces and CoM trajectories during stance as polynomial splines, allowing for intuitive and primarily analytical controller design. The performance of the control framework is tested in various simulations for a bipedal point-mass model.

In this paper, the concept of divergent component of motion (DCM, also called “Capture Point”) is extended to 3-D. We introduce the “Enhanced Centroidal Moment Pivot point” (eCMP) and the “Virtual Repellent Point” (VRP), which allow for the encoding of both direction and magnitude of the external forces and the total force (i.e., external plus gravitational forces) acting on the robot. Based on eCMP, VRP, and DCM, we present methods for real-time planning and tracking control of DCM trajectories in 3-D. The basic DCM trajectory generator is extended to produce continuous leg force profiles and to facilitate the use of toe-off motion during double support. The robustness of the proposed control framework is thoroughly examined, and its capabilities are verified both in simulations and experiments.

This paper works with the concept of Divergent Component of Motion (DCM), also called `(instantaneous) Capture Point'. We present two real-time DCM trajectory generators for uneven (three-dimensional) ground surfaces, which lead to continuous leg (and corresponding ground reaction) force profiles and facilitate the use of toe-off motion during double support. Thus, the resulting DCM trajectories are well suited for real-world robots and allow for increased step length and step height. The performance of the proposed methods was tested in numerous simulations and experiments on IHMC's Atlas robot and DLR's humanoid robot TORO.

In this paper, we extend the Divergent Component of Motion (DCM, also called `Capture Point') to 3D. We introduce the “Enhanced Centroidal Moment Pivot point” (eCMP) and the “Virtual Repellent Point” (VRP), which allow for the encoding of both direction and magnitude of the external (e.g. leg) forces and the total force (i.e. external forces plus gravity) acting on the robot. Based on eCMP, VRP and DCM, we present a method for real-time planning and control of DCM trajectories in 3D. We address the problem of underactuation and propose methods to guarantee feasibility of the finally commanded forces. The capabilities of the proposed control framework are verified in simulations.

This paper builds up on the Capture Point concept and exploits the simple form of the dynamical equations of the Linear Inverted Pendulum model when formulated in terms of the center of mass and the Capture Point. The presented methods include (i) the derivation of a Capture Point (CP) control principle based on the natural dynamics of the linear inverted pendulum (LIP), which stabilizes the walking robot and motivates (ii) the design of a CP tracking and a CP end-of-step controller. The exponential stability of the CP control law is proven. Tilting is avoided by proper projection of the commanded zero moment point. The robustness of the derived control algorithms is analyzed analytically and verified in simulation and experiments.