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.

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.

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.