Integration of vertical COM motion and angular momentum in an extended Capture Point tracking controller for bipedal walking

In this paper, we demonstrate methods for bipedal walking control based on the Capture Point (CP) methodology. In particular, we introduce a method to intuitively derive a CP reference trajectory from the next three steps and extend the linear inverted pendulum (LIP) based CP tracking controller introduced in [1], generalizing it to a model that contains vertical CoM motions and changes in angular momentum. Respecting the dynamics of general multibody systems, we propose a measurement-based compensation of multi-body effects, which leads to a stable closed-loop dynamics of bipedal walking robots. In addition we propose a ZMP projection method, which prevents the robots feet from tilting and ensures the best feasible CP tracking. The extended CP controller’s performance is validated in OpenHRP3 [2] simulations and compared to the controller proposed in [1]

On Time Optimization of Centroidal Momentum Dynamics

Recently, the centroidal momentum dynamics has received substantial attention to plan dynamically consistent motions for robots with arms and legs in multi-contact scenarios. However, it is also non convex which renders any optimization approach difficult and timing is usually kept fixed in most trajectory optimization techniques to not introduce additional non convexities to the problem. But this can limit the versatility of the algorithms. In our previous work, we proposed a convex relaxation of the problem that allowed to efficiently compute momentum trajectories and contact forces. However, our approach could not minimize a desired angular momentum objective which seriously limited its applicability. Noticing that the non-convexity introduced by the time variables is of similar nature as the centroidal dynamics one, we propose two convex relaxations to the problem based on trust regions and soft constraints. The resulting approaches can compute time-optimized dynamically consistent trajectories sufficiently fast to make the approach realtime capable. The performance of the algorithm is demonstrated in several multi-contact scenarios for a humanoid robot. In particular, we show that the proposed convex relaxation of the original problem finds solutions that are consistent with the original non-convex problem and illustrate how timing optimization allows to find motion plans that would be difficult to plan with fixed timing.Comment: 7 pages, 4 figures, ICRA 201

Bipedal Walking Analysis, Control, and Applications Towards Human-Like Behavior

Realizing the essentials of bipedal walking balance is one of the core studies in both robotics and biomechanics. Although the recent developments of walking control on bipedal robots have brought the humanoid automation to a different level, the walking performance is still limited compared to human walking, which also restricts the related applications in biomechanics and rehabilitation. To mitigate the discrepancy between robotic walking and human walking, this dissertation is broken into three parts to develop the control methods to improve three important perspectives: predictive walking behavior, gait optimization, and stepping strategy. To improve the predictive walking behavior captured by the model predictive control (MPC) which is transitionally applied with the nonlinear tracking control in sequence, a quadratic program (QP)-based controller is proposed to unify center of mass (COM) planning using MPC and a nonlinear torque control with control Lyapunov function (CLF). For the gait optimization, we focus on the algorithms of trajectory optimization with direct collocation framework. We propose a robust trajectory optimization using step-time sampling for a simple walker under terrain uncertainties. Towards generating human-like walking gait with multi-domain (phases), we improve the optimization through contact with more accurate transcription method for level walking, and generalize the hybrid zero dynamics (HZD) gait optimization with modified contact conditions for walking on various terrains. The results are compared with human walking gaits, where the similar trends and the sources of discrepancies are identified. In the third part for stepping strategy, we perform step estimation based on capture point (CP) for different human movements, including single-step (balance) recovery, walking and walking with slip. The analysis provides the insights of the efficacy and limitation of CP-based step estimation for human gait

Online receding horizon planning of multi-contact locomotion

Legged robots can traverse uneven terrain by using multiple contacts between their limbs and the environment. Nevertheless, to enable reliable operation in the real world, legged robots necessarily require the capability to online re-plan their motions in response to changing conditions, such as environment changes, or state deviations due to external force perturbations. To approach this goal, Receding Horizon Planning (RHP) can be a promising solution. RHP refers to the planning framework that can constantly update the motion plan for immediate execution. To achieve successful RHP, we typically need to consider an extended planning horizon, which consists of an execution horizon that plans the motion to be executed, and a prediction horizon that foresees the future. Although the prediction horizon is never executed, it is important to the success of RHP. This is because the prediction horizon serves as a value function approximation that evaluates the feasibility and the future effort required for accomplishing the given task starting from a chosen robot state. Having such value information can guide the execution horizon toward the states that are beneficial for the future. Nevertheless, computing such multi-contact motions for a legged robot to traverse uneven terrain can be time-consuming, especially when considering a long planning horizon. The computation complexity typically comes from the simultaneous resolution of the following two sub-problems: 1) selecting a gait pattern that specifies the sequence in which the limbs break and make contact with the environment; 2)synthesizing the contact and motion plan that determines the robot state trajectory along with the contact plan, i.e., contact locations and contact timings. The issue of gait pattern selection introduces combinatorial complexity into the planning problem,while the computation of the contact and motion plan brings high-dimensionality and non-convexity due to the consideration of complex non-linear dynamics constraints. To facilitate online RHP of multi-contact motions, in this thesis, we focus on exploring novel methods to address these two sub-problems efficiently. To give more detail, we firstly consider the problem of planning contact and motion plans in an online receding horizon fashion. In this case, we pre-specifying the gait pattern as a priori. Although this helps us to avoid the combinatorial complexity, the resulting planning problem is still high-dimensional and non-convex, which can hinder online computation. To improve the computation speed, we propose to simplify the modeling of the value function approximation that is required for guiding the RHP. This leads to 1) Receding Horizon Planning with Multiple Levels of Model Fidelity, where we compute the prediction horizon with a convex relaxed model; 2) Locally- Guided Receding Horizon Planning—where we propose to learn an oracle to predict local objectives (intermediate goals) for completing a given task, and then we use these local objectives to construct local value functions to guide a short-horizon RHP. We evaluate our methods for planning centroidal trajectories of a humanoid robot walking on moderate slopes as well as large slopes where static stability cannot be maintained.The result of multi-fidelity RHP demonstrates that we can accelerate the computation speed by relaxing the model accuracy in the prediction horizon. However, the relaxation cannot be arbitrary. Furthermore, owing to the shortened planning horizon, we find that locally-guided RHP demonstrates the best computation efficiency (95%-98.6%cycles converge online). This computation advantage enables us to demonstrate online RHP for our real-world humanoid robot Talos walking in dynamic environments that change on-the-fly. To handle the combinatorial complexity that arises from the gait pattern selection issue, we propose the idea of constructing a map from the task specifications to the gait pattern selections for a given environment model and performance objective(cost). We show that for a 2D half-cheetah model and a quadruped robot, a direct mapping between a given task and an optimal gait pattern can be established. We use supervised learning to capture the structure of this map in the form of gait regions.Furthermore, we also find that the trajectories in each gait region are qualitatively similar. We utilize this property to construct a warm-starting trajectory for each gait region, i.e., the mean of the trajectories discovered in each region. We empirically show that these warm-starting trajectories can improve the computation speed of our trajectory optimization problem up to 60 times when compared with random initial guesses. Moreover, we also conduct experimental trials on the ANYmal robot to validate our method

CYCLOIDAL GAIT WITH DOUBLE SUPPORT PHASE FOR THE NAO HUMANOID ROBOT

The commercial Nao humanoid robot has 11 DOF in legs. Even if these legs include 12 revolute joints, only 11 actuators are employed to control the walking of the robot. Under such conditions, the mobility of the pelvis and that of the oscillating foot are mutually constrained at each step. Besides, the original gait provided by the manufacturer company of the Nao employs only single support phases during the walking. Because of both issues, the reduced mobility in legs and the use of only single support phases, the stability of the walking is affected. To contribute to improving such stability, in this paper an approach is proposed that incorporates a double support phase and a gait based on cycloidal time functions for motions of the pelvis and those of the oscillating foot. To assess the stability of the walking an index is applied, which is based on the notion of zero-moment point (ZMP) of the static foot at each step. Results of experimental tests show that the proposed gait enhances the stability of the robot during the walking

CYCLOIDAL GAIT WITH DOUBLE SUPPORT PHASE FOR THE NAO HUMANOID ROBOT

The commercial Nao humanoid robot has 11 DOF in legs. Even if these legs include 12 revolute joints, only 11 actuators are employed to control the walking of the robot. Under such conditions, the mobility of the pelvis and that of the oscillating foot are mutually constrained at each step. Besides, the original gait provided by the manufacturer company of the Nao employs only single support phases during the walking. Because of both issues, the reduced mobility in legs and the use of only single support phases, the stability of the walking is affected. To contribute to improving such stability, in this paper an approach is proposed that incorporates a double support phase and a gait based on cycloidal time functions for motions of the pelvis and those of the oscillating foot. To assess the stability of the walking an index is applied, which is based on the notion of zero-moment point (ZMP) of the static foot at each step. Results of experimental tests show that the proposed gait enhances the stability of the robot during the walking

Bipedal Walking Analysis, Control, and Applications Towards Human-Like Behavior

Realizing the essentials of bipedal walking balance is one of the core studies in both robotics and biomechanics. Although the recent developments of walking control on bipedal robots have brought the humanoid automation to a different level, the walking performance is still limited compared to human walking, which also restricts the related applications in biomechanics and rehabilitation. To mitigate the discrepancy between robotic walking and human walking, this dissertation is broken into three parts to develop the control methods to improve three important perspectives: predictive walking behavior, gait optimization, and stepping strategy. To improve the predictive walking behavior captured by the model predictive control (MPC) which is transitionally applied with the nonlinear tracking control in sequence, a quadratic program (QP)-based controller is proposed to unify center of mass (COM) planning using MPC and a nonlinear torque control with control Lyapunov function (CLF). For the gait optimization, we focus on the algorithms of trajectory optimization with direct collocation framework. We propose a robust trajectory optimization using step-time sampling for a simple walker under terrain uncertainties. Towards generating human-like walking gait with multi-domain (phases), we improve the optimization through contact with more accurate transcription method for level walking, and generalize the hybrid zero dynamics (HZD) gait optimization with modified contact conditions for walking on various terrains. The results are compared with human walking gaits, where the similar trends and the sources of discrepancies are identified. In the third part for stepping strategy, we perform step estimation based on capture point (CP) for different human movements, including single-step (balance) recovery, walking and walking with slip. The analysis provides the insights of the efficacy and limitation of CP-based step estimation for human gait