Imprecise dynamic walking with time-projection control

We present a new walking foot-placement controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of the 3LP model, our proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. This control paradigm reacts to disturbances immediately and includes rules to account for swing dynamics and leg-retraction. We apply it to a simulated Atlas robot in position-control, always commanded to perform in-place walking. The stance hip joint in our robot keeps the torso upright to let the robot naturally fall, and the swing hip joint tracks the desired footstep location. Combined with simple Center of Pressure (CoP) damping rules in the low-level controller, our foot-placement enables the robot to recover from strong pushes and produce periodic walking gaits when subject to persistent sources of disturbance, externally or internally. These gaits are imprecise, i.e., emergent from asymmetry sources rather than precisely imposing a desired velocity to the robot. Also in extreme conditions, restricting linearity assumptions of the 3LP model are often violated, but the system remains robust in our simulations. An extensive analysis of closed-loop eigenvalues, viable regions and sensitivity to push timings further demonstrate the strengths of our simple controller

3LP: a linear 3D-walking model including torso and swing dynamics

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.Comment: Journal paper under revie

Evolutionary robotics and neuroscience

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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

Simulation of Mechanical Systems With Multiple Frictional Contacts

There are several applications in robotics and manufacturing in which nominally rigid objects are subject to multiple frictional contacts with other objects. In most previous work, rigid body models have been used to analyze such systems. There are two fundamental problems with such an approach. Firstly, the use of frictional laws, such as Coulomb\u27s law, introduce inconsistencies and ambiguities when used in conjunction with the principles of rigid body dynamics. Secondly, hypotheses traditionally used to model frictional impacts can lead to solutions which violate principles of energy conservation. In this paper these problems are explained with the help of examples. A new approach to the simulation of mechanical systems with multiple, frictional constraints is proposed which is free of inconsistencies

Impact-Aware Task-Space Quadratic-Programming Control

Generating on-purpose impacts with rigid robots is challenging as they may lead to severe hardware failures due to abrupt changes in the velocities and torques. Without dedicated hardware and controllers, robots typically operate at a near-zero velocity in the vicinity of contacts. We assume knowing how much of impact the hardware can absorb and focus solely on the controller aspects. The novelty of our approach is twofold: (i) it uses the task-space inverse dynamics formalism that we extend by seamlessly integrating impact tasks; (ii) it does not require separate models with switches or a reset map to operate the robot undergoing impact tasks. Our main idea lies in integrating post-impact states prediction and impact-aware inequality constraints as part of our existing general-purpose whole-body controller. To achieve such prediction, we formulate task-space impacts and its spreading along the kinematic tree of a floating-base robot with subsequent joint velocity and torque jumps. As a result, the feasible solution set accounts for various constraints due to expected impacts. In a multi-contact situation of under-actuated legged robots subject to multiple impacts, we also enforce standing stability margins. By design, our controller does not require precise knowledge of impact location and timing. We assessed our formalism with the humanoid robot HRP-4, generating maximum contact velocities, neither breaking established contacts nor damaging the hardware