Multi-contact Walking Pattern Generation based on Model Preview Control of 3D COM Accelerations

We present a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass (COM). A key point in the design of our algorithm is the calculation of contact-stability constraints. Thanks to a mathematical observation on the algebraic nature of the frictional wrench cone, we show that the 3D volume of feasible COM accelerations is a always a downward-pointing cone. We reduce its computation to a convex hull of (dual) 2D points, for which optimal O(n log n) algorithms are readily available. This reformulation brings a significant speedup compared to previous methods, which allows us to compute time-varying contact-stability criteria fast enough for the control loop. Next, we propose a conservative trajectory-wide contact-stability criterion, which can be derived from COM-acceleration volumes at marginal cost and directly applied in a model-predictive controller. We finally implement this pipeline and exemplify it with the HRP-4 humanoid model in multi-contact dynamically walking scenarios

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

ZMP Support Areas for Multicontact Mobility Under Frictional Constraints

International audienceWe propose a method for checking and enforcing multi-contact stability based on the Zero-tilting Moment Point (ZMP). The key to our development is the generalization of ZMP support areas to take into account (a) frictional constraints and (b) multiple non-coplanar contacts. We introduce and investigate two kinds of ZMP support areas. First, we characterize and provide a fast geometric construction for the support area generated by valid contact forces, with no other constraint on the robot motion. We call this set the full support area. Next, we consider the control of humanoid robots using the Linear Pendulum Mode (LPM). We observe that the constraints stemming from the LPM induce a shrinking of the support area, even for walking on horizontal floors. We propose an algorithm to compute the new area, which we call pendular support area. We show that, in the LPM, having the ZMP in the pendular support area is a necessary and sufficient condition for contact stability. Based on these developments, we implement a whole-body controller and generate feasible multi-contact motions where an HRP-4 humanoid locomotes in challenging multi-contact scenarios

Stability of Surface Contacts for Humanoid Robots: Closed-Form Formulae of the Contact Wrench Cone for Rectangular Support Areas

Humanoid robots locomote by making and breaking contacts with their environment. A crucial problem is therefore to find precise criteria for a given contact to remain stable or to break. For rigid surface contacts, the most general criterion is the Contact Wrench Condition (CWC). To check whether a motion satisfies the CWC, existing approaches take into account a large number of individual contact forces (for instance, one at each vertex of the support polygon), which is computationally costly and prevents the use of efficient inverse-dynamics methods. Here we argue that the CWC can be explicitly computed without reference to individual contact forces, and give closed-form formulae in the case of rectangular surfaces -- which is of practical importance. It turns out that these formulae simply and naturally express three conditions: (i) Coulomb friction on the resultant force, (ii) ZMP inside the support area, and (iii) bounds on the yaw torque. Conditions (i) and (ii) are already known, but condition (iii) is, to the best of our knowledge, novel. It is also of particular interest for biped locomotion, where undesired foot yaw rotations are a known issue. We also show that our formulae yield simpler and faster computations than existing approaches for humanoid motions in single support, and demonstrate their consistency in the OpenHRP simulator.Comment: 14 pages, 4 figure

Actuation-Aware Simplified Dynamic Models for Robotic Legged Locomotion

In recent years, we witnessed an ever increasing number of successful hardware implementations of motion planners for legged robots. If one common property is to be identified among these real-world applications, that is the ability of online planning. Online planning is forgiving, in the sense that it allows to relentlessly compensate for external disturbances of whatever form they might be, ranging from unmodeled dynamics to external pushes or unexpected obstacles and, at the same time, follow user commands. Initially replanning was restricted only to heuristic-based planners that exploit the low computational effort of simplified dynamic models. Such models deliberately only capture the main dynamics of the system, thus leaving to the controllers the issue of anchoring the desired trajectory to the whole body model of the robot. In recent years, however, we have seen a number of new approaches attempting to increase the accuracy of the dynamic formulation without trading-off the computational efficiency of simplified models. In this dissertation, as an example of successful hardware implementation of heuristics and simplified model-based locomotion, I describe the framework that I developed for the generation of an omni-directional bounding gait for the HyQ quadruped robot. By analyzing the stable limit cycles for the sagittal dynamics and the Center of Pressure (CoP) for the lateral stabilization, the described locomotion framework is able to achieve a stable bounding while adapting to terrains of mild roughness and to sudden changes of the user desired linear and angular velocities. The next topic reported and second contribution of this dissertation is my effort to formulate more descriptive simplified dynamic models, without trading off their computational efficiency, in order to extend the navigation capabilities of legged robots to complex geometry environments. With this in mind, I investigated the possibility of incorporating feasibility constraints in these template models and, in particular, I focused on the joint torques limits which are usually neglected at the planning stage. In this direction, the third contribution discussed in this thesis is the formulation of the so called actuation wrench polytope (AWP), defined as the set of feasible wrenches that an articulated robot can perform given its actuation limits. Interesected with the contact wrench cone (CWC), this yields a new 6D polytope that we name feasible wrench polytope (FWP), defined as the set of all wrenches that a legged robot can realize given its actuation capabilities and the friction constraints. Results are reported where, thanks to efficient computational geometry algorithms and to appropriate approximations, the FWP is employed for a one-step receding horizon optimization of center of mass trajectory and phase durations given a predefined step sequence on rough terrains. For the sake of reachable workspace augmentation, I then decided to trade off the generality of the FWP formulation for a suboptimal scenario in which a quasi-static motion is assumed. This led to the definition of the, so called, local/instantaneous actuation region and of the global actuation/feasible region. They both can be seen as different variants of 2D linear subspaces orthogonal to gravity where the robot is guaranteed to place its own center of mass while being able to carry its own body weight given its actuation capabilities. These areas can be intersected with the well known frictional support region, resulting in a 2D linear feasible region, thus providing an intuitive tool that enables the concurrent online optimization of actuation consistent CoM trajectories and target foothold locations on rough terrains