A hybrid dynamical extension of averaging and its application to the analysis of legged gait stability

We extend a smooth dynamical systems averaging technique to a class of hybrid systems with a limit cycle that is particularly relevant to the synthesis of stable legged gaits. After introducing a definition of hybrid averageability sufficient to recover the classical result, we illustrate its applicability by analysis of first a one-legged and then a two-legged hopping model. These abstract systems prepare the ground for the analysis of a significantly more complicated two legged model—a new template for quadrupedal running to be analyzed and implemented on a physical robot in a companion paper. We conclude with some rather more speculative remarks concerning the prospects for further extension and generalization of these ideas

Modular Hopping and Running via Parallel Composition

Though multi-functional robot hardware has been created, the complexity in its functionality has been constrained by a lack of algorithms that appropriately manage flexible and autonomous reconfiguration of interconnections to physical and behavioral components. Raibert pioneered a paradigm for the synthesis of planar hopping using a composition of ``parts\u27\u27: controlled vertical hopping, controlled forward speed, and controlled body attitude. Such reduced degree-of-freedom compositions also seem to appear in running animals across several orders of magnitude of scale. Dynamical systems theory can offer a formal representation of such reductions in terms of ``anchored templates,\u27\u27 respecting which Raibert\u27s empirical synthesis (and the animals\u27 empirical performance) can be posed as a parallel composition. However, the orthodox notion (attracting invariant submanifold with restriction dynamics conjugate to a template system) has only been formally synthesized in a few isolated instances in engineering (juggling, brachiating, hexapedal running robots, etc.) and formally observed in biology only in similarly limited contexts. In order to bring Raibert\u27s 1980\u27s work into the 21st century and out of the laboratory, we design a new family of one-, two-, and four-legged robots with high power density, transparency, and control bandwidth. On these platforms, we demonstrate a growing collection of $\{$body, behavior$\}$ pairs that successfully embody dynamical running / hopping ``gaits\u27\u27 specified using compositions of a few templates, with few parameters and a great deal of empirical robustness. We aim for and report substantial advances toward a formal notion of parallel composition---embodied behaviors that are correct by design even in the presence of nefarious coupling and perturbation---using a new analytical tool (hybrid dynamical averaging). With ideas of verifiable behavioral modularity and a firm understanding of the hardware tools required to implement them, we are closer to identifying the components required to flexibly program the exchange of work between machines and their environment. Knowing how to combine and sequence stable basins to solve arbitrarily complex tasks will result in improved foundations for robotics as it goes from ad-hoc practice to science (with predictive theories) in the next few decades

Averaged Anchoring of Decoupled Templates in a Tail-Energized Monoped

We refine and advance a notion of parallel composition to achieve for the first time a stability proof and empirical demonstration of a steady-state gait on a highly coupled 3DOF legged platform controlled by two simple (decoupled) feedback laws that provably stabilize in isolation two simple 1DOF mechanical subsystems. Specifically, we stabilize a limit cycle on a tailed monoped to excite sustained sagittal plane translational hopping energized by tail-pumping during stance. The constituent subsystems for which the controllers are nominally designed are: (i) a purely vertical bouncing mass (controlled by injecting energy into its springy shaft); and (ii) a purely tangential rimless wheel (controlled by adjusting the inter-spoke stepping angle).We introduce the use of averaging methods in legged locomotion to prove that this “parallel composition” of independent 1DOF controllers achieves an asymptotically stable closed-loop hybrid limit cycle for a dynamical system that approximates the 3DOF stance mechanics of our physical tailed monoped.We present experimental data demonstrating stability and close agreement between the motion of the physical hopping machine and numerical simulations of the (mathematically tractable) approximating model. More information: http://kodlab.seas.upenn.edu/Avik/AveragingTSLI

Vertical hopper compositions for preflexive and feedback-stabilized quadrupedal bounding, pacing, pronking, and trotting

This paper applies an extension of classical averaging methods to hybrid dynamical systems, thereby achieving formally specified, physically effective and robust instances of all virtual bipedal gaits on a quadrupedal robot. Gait specification takes the form of a three parameter family of coupling rules mathematically shown to stabilize limit cycles in a low degree of freedom template: an abstracted pair of vertical hoppers whose relative phase locking encodes the desired physical leg patterns. These coupling rules produce the desired gaits when appropriately applied to the physical robot. The formal analysis reveals a distinct set of morphological regimes determined by the distribution of the body’s inertia within which particular phase relationships are naturally locked with no need for feedback stabilization (or, if undesired, must be countermanded by the appropriate feedback), and these regimes are shown empirically to analogously govern the physical machine as well. In addition to the mathematical stability analysis and data from physical experiments we summarize a number of extensive numerical studies that explore the relationship between the simple template and its more complicated anchoring body models. For more information: Kod*la

Analytically-Guided Design of a Tailed Bipedal Hopping Robot

We present the first fully spatial hopping gait of a 12 DoF tailed biped driven by only 4 actuators. The control of this physical machine is built up from parallel compositions of controllers for progressively higher DoF extensions of a simple 2 DoF, 1 actuator template. These template dynamics are still not themselves integrable, but a new hybrid averaging analysis yields a conjectured closed form representation of the approximate hopping limit cycle as a function of its physical and control parameters. The resulting insight into the role of the machine\u27s kinematic and dynamical design choices affords a redesign leading to the newly achieved behavior

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

In silico case studies of compliant robots: AMARSI deliverable 3.3

In the deliverable 3.2 we presented how the morphological computing ap- proach can significantly facilitate the control strategy in several scenarios, e.g. quadruped locomotion, bipedal locomotion and reaching. In particular, the Kitty experimental platform is an example of the use of morphological computation to allow quadruped locomotion. In this deliverable we continue with the simulation studies on the application of the different morphological computation strategies to control a robotic system

Simulation and Control of Running Models

This work focuses on the locomotion of one-legged robots, with focus on approaches that stabilize passive limit cycles. Locomotion based on the socalled passive gaits promises to greatly reduce the actuation effort required for legged robots to move. In this work, the passive gaits of robots of varying complexity are characterized and stabilizing controllers are reviewed from the literature and newly formulated. The robots are modelled as hybrid dynamical systems and numerically simulated, thereby allowing to validate the proposed control strategies. Firstly, the vertical control through energy regulation of a one-dimensional hopper is considered. Secondly, the SLIP model is reviewed and then extended to the “pitchingSLIP”, with the aim of characterizing its passive gaits with somersaults. Two controllers based on energy and angular momentum regulation are then formulated to stabilize passive gaits with somersaults, making the control effort converge to zero. A further extension of the SLIP template, denominated “bodySLIP”, is then used to test the control approach on a more realistic model. The controllers shall be later extended to more complex cases, in which the somersaults are not necessarily present in the passive gaits. Thirdly, the locomotion of a one-legged robot with a body link is studied. Raibert’s control approach based on the foot placement algorithm is reviewed and compared to the non-dissipative touchdown controller of Hyon and Emura. The latter is then extended to be used with continuous torque profiles and to perform velocity tracking. Moreover, damping is added to the joints in order to study its effect on the controller, which was then modified to achieve stable running even in such conditions. The results obtained shall lay the foundations for a later test on hardware on DLR’s quadruped Bert

Dynamical Systems Theory

The quest to ensure perfect dynamical properties and the control of different systems is currently the goal of numerous research all over the world. The aim of this book is to provide the reader with a selection of methods in the field of mathematical modeling, simulation, and control of different dynamical systems. The chapters in this book focus on recent developments and current perspectives in this important and interesting area of mechanical engineering. We hope that readers will be attracted by the topics covered in the content, which are aimed at increasing their academic knowledge with competences related to selected new mathematical theoretical approaches and original numerical tools related to a few problems in dynamical systems theory