Avik De

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 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.

Haptic feedback enhances rhythmic motor control by reducing variability, not improving convergence rate

Stability and performance during rhythmic motor behaviors such as locomotion are critical for survival across taxa: falling down would bode well for neither cheetah nor gazelle. Little is known about how haptic feedback, particularly during discrete events such as the heel-strike event during walking, enhances rhythmic behavior. To determine the effect of haptic cues on rhythmic motor performance, we investigated a virtual paddle juggling behavior, analogous to bouncing a table tennis ball on a paddle. Here, we show that a force impulse to the hand at the moment of ball-paddle collision categorically improves performance over visual feedback alone, not by regulating the rate of convergence to steady state (e.g., via higher gain feedback or modifying the steady-state hand motion), but rather by reducing cycle-to-cycle variability. This suggests that the timing and state cues afforded by haptic feedback decrease the nervous systems uncertainty of the state of the ball to enable more accurate control but that the feedback gain itself is unaltered. This decrease in variability leads to a substantial increase in the mean first passage time, a measure of the long-term metastability of a stochastic dynamical system. Rhythmic tasks such as locomotion and juggling involve intermittent contact with the environment (i.e., hybrid transitions), and the timing of such transitions is generally easy to sense via haptic feedback. This timing information may improve metastability, equating to less frequent falls or other failures depending on the task.

Toward dynamical sensor management for reactive wall-following

We propose a new paradigm for reactive wall-following by a planar robot taking the form of an actively steered sensor model that augments the robots 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 robots body that first intersects it. We develop feedback policies for the coupled (point or unicycle) sensorimotor system that drive the sensors 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 walls curvature varies unpredictably in the manner of an unmodeled noise signal. We prove that in any neighborhood within which the third-order 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.

Motor sizing for legged robots using dynamic task specification

We explore an approach to incorporating task and motor thermal dynamics in the selection of actuators for legged robots, using both analytical and simulation methods. We develop a motor model with a thermal component and apply it to a vertical climbing task; in the process, we optimally choose gear ratio and therefore eliminate it as a design parameter. This approach permits an analytical proof that continuous operation yields superior thermal performance to intermittent operation. We compare the results of motor sizing using our proposed method with more conventional techniques such as using the continuously permissible current specification. Our simulations are run across a database of commercially available motors, and we envision that our results might be of immediate use to robot designers for motor as well as gearbox selection.

Toward SLAM on Graphs

We present an algorithm for SLAM on planar graphs. We assume that a robot moves from node to node on the graph using odometry to measure the distance between consecutive landmark observations. At each node, the robot follows a branch chosen at random, without reporting which branch it follows. A low-level process detects (with some uncertainty) the presence of landmarks, such as corners, branches, and bumps, but only triggers a binary flag for landmark detection (i.e., the robot is oblivious to the details or “appearance” of the landmark). Under uncertainties of the robot’s odometry, landmark detection, and the current landmark position of the robot, we present an E-M-based SLAM algorithm for two cases: (1) known, arbitrary topology with unknown edge lengths and (2) unknown topology, but restricted to “elementary” 1- and 2-cycle graphs. In the latter case, the algorithm (flexibly and reversibly) closes loops and allows for dynamic environments (adding and deleting nodes).

Active Sensing for Dynamic, Non-holonomic, Robust Visual Servoing

We consider the problem of visually servoing a legged vehicle with unicycle-like nonholonomic constraints subject to second-order fore-aft dynamics in its horizontal-plane. We target applications to rugged environments characterized by complex terrain likely to significantly perturb the robots nominal dynamics. At the same time, it is crucial that the camera avoid “obstacle” poses where absolute localization would be compromised by even partial loss of landmark visibility. Hence, we seek a controller whose robustness against disturbances and obstacle avoidance capabilities can be assured by a strict global Lyapunov function. Since the nonholonomic constraints preclude smooth point stabilizability we introduce an extra degree of sensory freedom, affixing the camera to an actuated panning axis on the robots back. Smooth stabilizability to the robot-orientation-indifferent goal cycle no longer precluded, we construct a controller and strict global Lyapunov function with the desired properties. We implement several versions of the scheme on a RHex robot maneuvering over slippery ground and document its successful empirical performance.

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.

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.

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.

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.