The Penn Jerboa: A Platform for Exploring Parallel Composition of Templates

We have built a 12DOF, passive-compliant legged, tailed biped actuated by four brushless DC motors. We anticipate that this machine will achieve varied modes of quasistatic and dynamic balance, enabling a broad range of locomotion tasks including sitting, standing, walking, hopping, running, turning, leaping, and more. Achieving this diversity of behavior with a single under-actuated body, requires a correspondingly diverse array of controllers, motivating our interest in compositional techniques that promote mixing and reuse of a relatively few base constituents to achieve a combinatorially growing array of available choices. Here we report on the development of one important example of such a behavioral programming method, the construction of a novel monopedal sagittal plane hopping gait through parallel composition of four decoupled 1DOF base controllers. For this example behavior, the legs are locked in phase and the body is fastened to a boom to restrict motion to the sagittal plane. The platform's locomotion is powered by the hip motor that adjusts leg touchdown angle in flight and balance in stance, along with a tail motor that adjusts body shape in flight and drives energy into the passive leg shank spring during stance. The motor control signals arise from the application in parallel of four simple, completely decoupled 1DOF feedback laws that provably stabilize in isolation four corresponding 1DOF abstract reference plants. Each of these abstract 1DOF closed loop dynamics represents some simple but crucial specific component of the locomotion task at hand. We present a partial proof of correctness for this parallel composition of template reference systems along with data from the physical platform suggesting these templates are anchored as evidenced by the correspondence of their characteristic motions with a suitably transformed image of traces from the physical platform.Comment: Technical Report to Accompany: A. De and D. Koditschek, "Parallel composition of templates for tail-energized planar hopping," in 2015 IEEE International Conference on Robotics and Automation (ICRA), May 2015. v2: Used plain latex article, correct gap radius and specific force/torque number

Parallel Composition of Templates for Tail-Energized Planar Hopping

We have built a 4DOF tailed monoped that hops along a boom permitting free sagittal plane motion. This underactuated platform is powered by a hip motor that adjusts leg touchdown angle in flight and balance in stance, along with a tail motor that adjusts body shape in flight and drives energy into the passive leg shank spring during stance. The motor control signals arise from the application in parallel of four simple, completely decoupled 1DOF feedback laws that provably stabilize in isolation four corresponding 1DOF abstract reference plants. Each of these abstract 1DOF closedloopdynamicsrepresentssomesimplebutcrucialspecific component of the locomotion task at hand. We present a partial proof of correctness for this parallel composition of “template” reference systems along with data from the physical platform suggesting these templates are “anchored” as evidenced by the correspondence of their characteristic motions with a suitably transformed image of traces from the physical platform. For more information: http://kodlab.seas.upenn.edu/Avik/ICRA201

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

Toward Dynamical Sensor Management for Reactive Wall-following

We propose a new paradigm for reactive wallfollowing by a planar robot taking the form of an actively steered sensor model that augments the robot’s motion dynamics. We postulate a foveated sensor capable of delivering third-order infinitesimal (range, tangent, and curvature) data at a point along a wall (modeled as an unknown smooth plane curve) specified by the angle of the ray from the robot’s body that first intersects it. We develop feedback policies for the coupled (point or unicycle) sensorimotor system that drive the sensor’s foveal angle as a function of the instantaneous infinitesimal data, in accord with the trade-off between a desired standoff and progress-rate as the wall’s curvature varies unpredictably in the manner of an unmodeled noise signal. We prove that in any neighborhood within which the thirdorder infinitesimal data accurately predicts the local “shape” of the wall, neither robot will ever hit it. We empirically demonstrate with comparative physical studies that the new active sensor management strategy yields superior average tracking performance and avoids catastrophic collisions or wall losses relative to the passive sensor variant. This work was supported by AFOSR MURI FA9550–10–1−0567. For further information, visit Kod*lab

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

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

Frontal plane stabilization and hopping with a 2DOF tail

The Jerboa, a tailed bipedal robot with two hip-actuated, passive-compliant legs and a doubly actuated tail, has been shown both formally and empirically to exhibit a variety of stable hopping and running gaits in the sagittal plane. In this paper we take the first steps toward operating Jerboa as a fully spatial machine by addressing the predominant mode of destabilization away from the sagittal plane: body roll. We develop a provably stable controller for underactuated aerial stabilization of the coupled body roll and tail angles, that uses just the tail torques. We show that this controller is successful at reliably reorienting the Jerboa body in roughly 150 ms of freefall from a large set of initial conditions. This controller also enables (and appears intuitively to be crucial for) sustained empirically stable hopping in the frontal plane by virtue of its substantial robustness against destabilizing perturbations and calibration errors. The controller as well as the analysis methods developed here are applicable to any robotic platform with a similar doubly-actuated spherical tail joint

Design Principles for a Family of Direct-Drive Legged Robots

This letter introduces Minitaur, a dynamically running and leaping quadruped, which represents a novel class of direct-drive (DD) legged robots. We present a methodology that achieves the well-known benefits of DD robot design (transparency, mechanical robustness/efficiency, high-actuation bandwidth, and increased specific power), affording highly energetic behaviors across our family of machines despite severe limitations in specific force. We quantify DD drivetrain benefits using a variety of metrics, compare our machines\u27 performance to previously reported legged platforms, and speculate on the potential broad-reaching value of “transparency” for legged locomotion. For more information: Kod*lab

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