Time-projection control to recover inter-sample disturbances, application to bipedal walking control

We present a new walking 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 3LP, the 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. The projection controller reacts to disturbances immediately and compared to the discrete LQR controller, it provides superior performance in recovering intermittent external pushes. Further analysis of closed-loop eigenvalues and disturbance rejection strength show strong stabilization properties for this architecture. An analysis of viable regions also show that the proposed controller covers most of the maximal viable set of states. It is computationally much faster than Model Predictive Controllers (MPC) and yet optimal over an infinite horizon

Push recovery with stepping strategy based on time-projection control

In this paper, we present a simple control framework for on-line push recovery with dynamic stepping properties. Due to relatively heavy legs in our robot, we need to take swing dynamics into account and thus use a linear model called 3LP which is composed of three pendulums to simulate swing and torso dynamics. Based on 3LP equations, we formulate discrete LQR controllers and use a particular time-projection method to adjust the next footstep location on-line during the motion continuously. This adjustment, which is found based on both pelvis and swing foot tracking errors, naturally takes the swing dynamics into account. Suggested adjustments are added to the Cartesian 3LP gaits and converted to joint-space trajectories through inverse kinematics. Fixed and adaptive foot lift strategies also ensure enough ground clearance in perturbed walking conditions. The proposed structure is robust, yet uses very simple state estimation and basic position tracking. We rely on the physical series elastic actuators to absorb impacts while introducing simple laws to compensate their tracking bias. Extensive experiments demonstrate the functionality of different control blocks and prove the effectiveness of time-projection in extreme push recovery scenarios. We also show self-produced and emergent walking gaits when the robot is subject to continuous dragging forces. These gaits feature dynamic walking robustness due to relatively soft springs in the ankles and avoiding any Zero Moment Point (ZMP) control in our proposed architecture.Comment: 20 pages journal pape

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

Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.Comment: 15 pages. This work has been accepted to IEEE Transactions on Robotics (2019

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

Towards understanding human locomotion

Die zentrale Frage, die in der vorliegenden Arbeit untersucht wurde, ist, wie man die komplizierte Dynamik des menschlichen Laufens besser verstehen kann. In der wissenschaftlichen Literatur werden zur Beschreibung von Laufbewegungen (Gehen und Rennen) oftmals minimalistische "Template"-Modelle verwendet. Diese sehr einfachen Modelle beschreiben nur einen ausgewählten Teil der Dynamik, meistens die Schwerpunktsbahn. In dieser Arbeit wird nun versucht, mittels Template-Modellen das Verständnis des Laufens voranzubringen. Die Analyse der Schwerpunktsbewegung durch Template-Modelle setzt eine präzise Bestimmung der Schwerpunktsbahn im Experiment voraus. Hierfür wird in Kapitel 2.3 eine neue Methode vorgestellt, welche besonders robust gegen die typischen Messfehler bei Laufexperimenten ist. Die am häfigsten verwendeten Template-Modelle sind das Masse-Feder-Modell und das inverse Pendel, welche zur Beschreibung der Körperschwerpunktsbewegung gedacht sind und das Drehmoment um den Schwerpunkt vernachlässigen. Zur Beschreibung der Stabilisierung der Körperhaltung (und damit der Drehimpulsbilanz) wird in Abschnitt 3.3 das Template-Modell "virtuelles Pendel" für das menschliche Gehen eingeführt und mit experimentellen Daten verglichen. Die Diskussion möglicher Realisierungsmechanismen legt dabei nahe, dass die Aufrichtung des menschlichen Gangs im Laufe der Evolution keine große mechanische Hürde war. In der Literatur wird oft versucht, Eigenschaften der Bewegung wie Stabilität durch Eigenschaften der Template-Modelle zu erklären. Dies wird in modifizierter Form auch in der vorliegen Arbeit getan. Hierzu wird zunächst eine experimentell bestimmte Schwerpunktsbewegung auf das Masse-Feder-Modell übertragen. Anschließend wird die Kontrollvorschrift der Schritt-zu-Schritt-Anpassung der Modellparameter identifiziert sowie eine geeignete Näherung angegeben, um die Stabilität des Modells, welches mit dieser Kontrollvorschrift ausgestattet wird, zu analysieren. Der Vergleich mit einer direkten Bestimmung der Stabilität aus einem Floquet-Modell zeigt qualitativ gute Übereinstimmung. Beide Ansätze führen auf das Ergebnis, dass beim langsamen menschlichen Rennen Störungen innerhalb von zwei Schritten weitgehend abgebaut werden. Zusammenfassend wurde gezeigt, wie Template-Modelle zum Verständnis der Laufbewegung beitragen können. Gerade die Identifikation der individuellen Kontrollvorschrift auf der Abstraktionsebene des Masse-Feder-Modells erlaubt zukünftig neue Wege, aktive Prothesen oder Orthesen in menschenähnlicher Weise zu steuern und ebnet den Weg, menschliches Rennen auf Roboter zu übertragen.Human locomotion is part of our everyday life, however the mechanisms are not well enough understood to be transferred into technical devices like orthoses, protheses or humanoid robots. In biomechanics often minimalistic so-called template models are used to describe locomotion. While these abstract models in principle offer a language to describe both human behavior and technical control input, the relation between human locomotion and locomotion of these templates was long unclear. This thesis focusses on how human locomotion can be described and analyzed using template models. Often, human running is described using the SLIP template. Here, it is shown that SLIP (possibly equipped with any controller) cannot show human-like disturbance reactions, and an appropriate extension of SLIP is proposed. Further, a new template to describe postural stabilization is proposed. Summarizing, this theses shows how simple template models can be used to enhance the understanding of human gait

Towards Robust Bipedal Locomotion:From Simple Models To Full-Body Compliance

Thanks to better actuator technologies and control algorithms, humanoid robots to date can perform a wide range of locomotion activities outside lab environments. These robots face various control challenges like high dimensionality, contact switches during locomotion and a floating-base nature which makes them fall all the time. A rich set of sensory inputs and a high-bandwidth actuation are often needed to ensure fast and effective reactions to unforeseen conditions, e.g., terrain variations, external pushes, slippages, unknown payloads, etc. State of the art technologies today seem to provide such valuable hardware components. However, regarding software, there is plenty of room for improvement. Locomotion planning and control problems are often treated separately in conventional humanoid control algorithms. The control challenges mentioned above are probably the main reason for such separation. Here, planning refers to the process of finding consistent open-loop trajectories, which may take arbitrarily long computations off-line. Control, on the other hand, should be done very fast online to ensure stability. In this thesis, we want to link planning and control problems again and enable for online trajectory modification in a meaningful way. First, we propose a new way of describing robot geometries like molecules which breaks the complexity of conventional models. We use this technique and derive a planning algorithm that is fast enough to be used online for multi-contact motion planning. Similarly, we derive 3LP, a simplified linear three-mass model for bipedal walking, which offers orders of magnitude faster computations than full mechanical models. Next, we focus more on walking and use the 3LP model to formulate online control algorithms based on the foot-stepping strategy. The method is based on model predictive control, however, we also propose a faster controller with time-projection that demonstrates a close performance without numerical optimizations. We also deploy an efficient implementation of inverse dynamics together with advanced sensor fusion and actuator control algorithms to ensure a precise and compliant tracking of the simplified 3LP trajectories. Extensive simulations and hardware experiments on COMAN robot demonstrate effectiveness and strengths of our method. This thesis goes beyond humanoid walking applications. We further use the developed modeling tools to analyze and understand principles of human locomotion. Our 3LP model can describe the exchange of energy between human limbs in walking to some extent. We use this property to propose a metabolic-cost model of human walking which successfully describes trends in various conditions. The intrinsic power of the 3LP model to generate walking gaits in all these conditions makes it a handy solution for walking control and gait analysis, despite being yet a simplified model. To fill the reality gap, finally, we propose a kinematic conversion method that takes 3LP trajectories as input and generates more human-like postures. Using this method, the 3LP model, and the time-projecting controller, we introduce a graphical user interface in the end to simulate periodic and transient human-like walking conditions. We hope to use this combination in future to produce faster and more human-like walking gaits, possibly with more capable humanoid robots

A Foot Placement Strategy for Robust Bipedal Gait Control

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance

Analytic and Learned Footstep Control for Robust Bipedal Walking

Bipedal walking is a complex, balance-critical whole-body motion with inherently unstable inverted pendulum-like dynamics. Strong disturbances must be quickly responded to by altering the walking motion and placing the next step in the right place at the right time. Unfortunately, the high number of degrees of freedom of the humanoid body makes the fast computation of well-placed steps a particularly challenging task. Sensor noise, imprecise actuation, and latency in the sensomotoric feedback loop impose further challenges when controlling real hardware. This dissertation addresses these challenges and describes a method of generating a robust walking motion for bipedal robots. Fast modification of footstep placement and timing allows agile control of the walking velocity and the absorption of strong disturbances. In a divide and conquer manner, the concepts of motion and balance are solved separately from each other, and consolidated in a way that a low-dimensional balance controller controls the timing and the footstep locations of a high-dimensional motion generator. Central pattern generated oscillatory motion signals are used for the synthesis of an open-loop stable walk on flat ground, which lacks the ability to respond to disturbances due to the absence of feedback. The Central Pattern Generator exhibits a low-dimensional parameter set to influence the timing and the landing coordinates of the swing foot. For balance control, a simple inverted pendulum-based physical model is used to represent the principal dynamics of walking. The model is robust to disturbances in a way that it returns to an ideal trajectory from a wide range of initial conditions by employing a combination of Zero Moment Point control, step timing, and foot placement strategies. The simulation of the model and its controller output are computed efficiently in closed form, supporting high-frequency balance control at the cost of an insignificant computational load. Additionally, the sagittal step size produced by the controller can be trained online during walking with a novel, gradient descent-based machine learning method. While the analytic controller forms the core of reliable walking, the trained sagittal step size complements the analytic controller in order to improve the overall walking performance. The balanced whole-body walking motion arises by using the footstep coordinates and the step timing predicted by the low-dimensional model as control input for the Central Pattern Generator. Real robot experiments are presented as evidence for disturbance-resistant, omnidirectional gait control, with arguably the strongest push-recovery capabilities to date

SAFETY-GUARANTEED TASK PLANNING FOR BIPEDAL NAVIGATION IN PARTIALLY OBSERVABLE ENVIRONMENTS

Bipedal robots are becoming more capable as basic hardware and control challenges are being overcome, however reasoning about safety at the task and motion planning levels has been largely underexplored. I would like to make key steps towards guaranteeing safe locomotion in cluttered environments in the presence of humans or other dynamic obstacles by designing a hierarchical task planning framework that incorporates safety guarantees at each level. This layered planning framework is composed of a coarse high-level symbolic navigation planner and a lower-level local action planner. A belief abstraction at the global navigation planning level enables belief estimation of non-visible dynamic obstacle states and guarantees navigation safety with collision avoidance. Both planning layers employ linear temporal logic for a reactive game synthesis between the robot and its environment while incorporating lower level safe locomotion keyframe policies into formal task specification design. The high-level symbolic navigation planner has been extended to leverage the capabilities of a heterogeneous multi-agent team to resolve environment assumption violations that appear at runtime. Modifications in the navigation planner in conjunction with a coordination layer allow each agent to guarantee immediate safety and eventual task completion in the presence of an assumption violation if another agent exists that can resolve said violation, e.g. a door is closed that another dexterous agent can open. The planning framework leverages the expressive nature and formal guarantees of LTL to generate provably correct controllers for complex robotic systems. The use of belief space planning for dynamic obstacle belief tracking and heterogeneous robot capabilities to assist one another when environment assumptions are violated allows the planning framework to reduce the conservativeness traditionally associated with using formal methods for robot planning.M.S