We define a theoretical framework for the experimental study of neuromechanical control in animals, based on mathematical concepts from dynamical systems theory. This approach allows experiments, results and theoretical models to be shared among biologists, engineers and mathematicians, and is applicable to the study of control in any system, biological, artificial or simulated, provided the system exhibits stable rhythmic solutions. The basis of this framework is the notion that rhythmic systems are best expressed as periodic functions of their phase. Using phase as a predictor, an extrapolated prediction of future animal motions can be compared with the motions that occur when a perturbation is applied. Phase also serves as a succinct summary of the kinematic state, allowing the difference between the expected state as summarized by phase and the phase found in the perturbed animal -- a “residual phase”. In the first chapter we introduce the key concepts and describe how the residual phase may be used to identify the neuromechanical control architecture of an animal. In the following two chapters we use residual phase to analyze running arthropods subjected to perturbations. In the final chapter, we extend the kinematic phase based models to the construction of a linearized approximation of animal dynamics based on Floquet theory. The Floquet model allows us to directly test the “Templates and Anchors Hypothesis” of motor control and to characterize a “template” -- a low dimensional model of the dynamics of the animal.

In chapter 2, our residual phase results from running cockroaches over a hurdle show that kinematic phase was reset, while running frequency was closely maintained to within ±5%. Kinematic phase changes were distributed bi-modally with modes one step (half a cycle) apart, which corresponds to a left-right reflection of the kinematic state of the body. The results decrease the plausibility of feedforward control and support the use of neural feedback for this task. Based on the results, we propose a controller that expresses the timing of the two leg tripods of the animals as two coupled phase oscillators, which in turn, may also be coupled to a master clock.

In chapter 3, we analyze cockroaches which ran onto a movable cart that translated laterally. The specific impulse imposed on animals was 50±4 cm/s (mean,SD), nearly twice their forward speed 25±6 cm/s. Animals corrected for these perturbations by decreasing stride frequency, thereby demonstrating neural feedback. Trials fell into two classes, one class slowing down after a step (50 ms), the other after nearly three steps (130 ms). Classes were predicted by the kinematic phase of the animal at onset of perturbation. We hypothesize that the differences in response time is a consequence of the mechanical posture of the animal during perturbation, as expressed by the phase, and the coupling of neural and mechanical control.

In chapter 4 we attempted to use kinematic phase methods to reconstruct the linearized (Floquet) structure of running cockroaches when viewed as nonlinear oscillators. The development of this approach required several innovations in applied mathematics and statistics. We analyzed foot and body positions of 34 animals running on a treadmill. Results showed that cockroaches running at preferred speed possess a six dimensional template with each dimension recovering by less than 50% in a stride (P<0.05, 11 animals, 24 trials, 532 strides).

Bioinspired robotics has experienced unprecedented advancements. Robots can now run, swim and fly. Despite these advancements, the animals that inspired these robots remain substantially more versatile, using the same set of multifunctional appendages to perform many different tasks or behaviors. Understanding how biomechanics and behavior contribute to these capabilities will offer new insight into the science of animal movement and potentially inspire new multifunctional robotic technologies. Here, the ghost crab, Ocypode quadrata, was examined. These crabs, which are among the fastest land invertebrates, use relatively simple, unspecialized appendages to run, climb, burrow and dexterously capture prey. This dissertation focuses on ghost crabs’ burrowing and climbing behaviors. Both of these involve complex behavioral suites that involve the walking legs, the chelae and the body. Crabs demonstrated specialized postures, locomotion in confined spaces and goal-directed manipulation of both the environment and themselves. Both burrowing and climbing strategies involved compensatory strategies that allowed the crabs to maintain performance across a wide range of environmental conditions. Crabs do not rely on specialized appendage features but rather on the ability to use all appendages together. The findings presented in this dissertation represent new insight into the biomechanics of multifunctionality and offer inspiration for new bio-inspired robots with multi-use parts that permit, not simply obstacle negotiation, but also modification of the environment.

The single greatest difference between biological organisms and human technologies today is perhaps robustness. Robustness is broadly defined as a system's ability to maintain performance despite disturbances. Qualitatively, a complex system may be called robust if it exhibits some or all of the following properties - multi-functionality, fault tolerance, damage resistance, modularity and redundancy. However, even the best engineering approaches that have attempted to consider some of these aspects find themselves 'fragile', slow and computationally intensive. On the other hand, I contend that biological systems are truly robust and capable of a plethora of complex activities such as locomotion, reproduction, respiration etc. In addition, animals constantly overcome challenges of growth and perform self-repair and learning, still a challenge for even the best engineered systems today. Using the cockroach as my model organism, I have investigated the role of exoskeletons in enabling robust high-speed locomotor behavior in cockroaches. Specifically, I have discovered that the cockroach exoskeleton

(1) is effective at dealing with external loads/impulses through body reconfiguration,

(2) facilitates (or enhances) rapid horizontal to vertical transitions during high-speed running, and

(3) compensates for damage (loss of appendages (or its parts)) undergoing a limited decrement in high-speed running performance.

Based on the impressive locomotion performance by cockroaches despite perturbations, internal and external to the animals, we propose that robustness is a crucial measure of the effectiveness of a system's performance.

Life above ground requires that animals navigate a highly three-dimensional world. Both cursorial and arboreal animals must rapidly negotiate a myriad of complex and often unpredictable substrata within their habitats (e.g. dense vegetation; forest canopy) in which discontinuous supports challenge secure footholds. Traveling rapidly through such demanding terrain necessitates a behavioral repertoire that consists of both terrestrial and aerial modes of locomotion. Lizards represent an important model system in the study of general principles of how animals move. Moreover, developing capabilities of ambulation in cluttered environments to assist in search and rescue operations is in demand in robotics. Within the scope of this dissertation, I investigate whether multiple coordinated tail reflexes and responses are necessary for the successful navigation of a highly three-dimensional environment that challenges animals' locomotor systems with numerous obstacles, discontinuous supports and slippery surfaces.

In Chapter One, I present data on challenging single footholds in wall-running geckos, which lead to the discovery of a control structure the significance of which had not been previously recognized. Although the remarkable climbing performance of geckos has traditionally been attributed to specialized feet, I showed that a gecko's tail functions as an emergency fifth leg to prevent falling during rapid climbing. A response initiated by slipping causes the tail tip to push against the vertical surface, thereby preventing pitch-back of the head and upper body. When confronted with insurmountable gaps the lizards exhibited tail movement as they recovered from free fall. Lizards could also control body pitch and induce turning during simulated aerial descent. These experiments suggested that the secret to the gecko's arboreal acrobatics includes an active tail.

In the context of measuring locomotor performance as a function of foot-substrate interaction, I perturbed geckos even further and found that when a gecko falls with its back to the ground, a swing of its tail induces the most rapid air-righting response yet measured. Chapter Two investigates the tail as an effective torque source first attempting to generatesimple, low parameter models of thesystem, then developing thee-dimensional analytical models of multi-body systems to investigate righting performance in two species of lizard, and how it is affected by variations in tail length, mass distribution, tail placement and orientation. These results suggest that large, active tails can function as effective control appendages. Lizards gliding in a vertical wind tunnel could use appendage inertia to induce turns in yaw whereby tails twice the torso length have better yield. Robots can also serve as physical models to test our understanding of animal locomotion. In this spirit, a physical model and robot prototype tests the model's predictive capacity further, while also demonstrating feasibility of tail use in robots.

In Chapter Three, I present data from field research conducted in Southeast Asian lowland tropical rainforest, the natural habitat of my model system H. platyurus. A species heretofore not known to glide exhibits considerable horizontal transit of over 4m. Geckos with tails that glided to the tree trunk were able to remain attached to it upon landing in the majority of trials, whereas geckos without tails generally became dislodged upon impact. Moreover, I present how they carry out landing on vertical tree trunks despite approaching them at very high speeds. I propose a mechanically-mediated solution to how landing on a wall could be stabilized by the caudal appendage.

In Chapter Four, I explore whether representatives of lizard taxa other than Gekkonidae night utilize their tails for improving the stability, maneuverability and overall robustness of a mode of terrestrial locomotion: Rapid climbing on tree bark. Observations from the field in Malaysian lowland tropical rainforest, where I observed these animals' behavior initiated this study. I discovered that they have specialized subcaudal scales which are keeled such as to engage with a rough substrate. I present materials testing measurements using an Instron machine, where we determined that each scale can support one to several times body weight, depending on species. Acanthosaurus crucigera, Gonocephalus grandis and Iguana iguana were sampled. When the scales are prevented from engaging the animals' performance decreases dramatically.

The interaction between cognitive control and biomechanics has enabled animal locomotion capabilities that far surpass state of the art mobile robot performance. But there has been little research focusing on the interaction of the cognitive processes, like learning and decision making, with the biomechanics of the animal coupled to the mechanical properties of the environment. Understanding the cognitive biomechanical strategies, specifically how cognitive processes may work synergistically with the unique mechanical properties of the organism in its environment, is likely to inspire future generations of bioinspired search-and-rescue and environmental monitoring robots. In this dissertation, two model organisms were used to study the cognitive biomechanics of arboreal locomotion. First, high speed branch locomotion was examined using the American cockroach, Periplaneta americana. This cockroach is one of the fastest runners relative to its body length. This hexapedal animal uses an alternating tripod to run up to 1.5 meters per second, becoming bipedal at the fastest speeds. Second, the Fox squirrel, Sciurus niger was studied. This arboreal specialist exhibits high performance climbing, branch navigation, and gap crossing via targeted leaping. Fox squirrels demonstrated learning to improve targeted leaping performance, a novel parkour-like wall jumping behavior, and decisions on when to leap that was dependent on the mechanical behavior of the branch and geometric properties of the gap to cross. These discoveries give new insights into the cognitive biomechanics of arboreal locomotion, and represent a promising path towards a principled understanding of canopy navigation.

Maneuverability in animals is unparalleled when compared to the most maneuverable human-engineered mobile robot. Maneuverability arises in part from animals' ability to integrate multimodal sensory information with an ongoing motor program while interacting within a spatiotemporally complex world. Complicating this integration, actions from the nervous system must operate through the mechanics of the body. Since sensors and muscles are fused to a mechanical frame, mechanical processing occurs at both at the sensory (input) and motor (output) levels. To reveal the basic organization of the neural and mechanical parts of organisms during locomotion, I studied high-speed sensorimotor tasks in a remarkably maneuverable insect, the cockroach, which integrates sensory information to navigate through irregular, unpredictable environments.

Animals can expend energy to acquire information by emitting signals or moving sensory structures. However, it is not clear if the energy from locomotion, itself, could permit a different form of sensing, in which animals transfer energy from movement to reconfigure a passive sensor. In the first chapter, I demonstrate that cockroaches can transfer the self-generated energy from locomotion to actively control the state of the antenna via passive mechanical elements, with important effects on body control. This chapter advances our current understanding of sensorimotor integration during rapid running by showing how the whole body, not just the sensor, can participate in sensory acquisition.

Information flow from individual sensory units operating on locomotion-driven appendages to the generation of motor patterns is not well understood. The nervous system must rapidly integrate sensory information from noisy channels while constrained by neural conduction delays. When executing high-speed wall following using their antennae, cockroaches presumably integrate information between self and obstacles to generate appropriate turns, preventing collisions. Previous work on modeling high-speed wall following within a control theoretic framework predicted that a sensory controller for antenna tactile sensing of wall position (P) and the derivative of position (D) was sufficient for control of the body. I hypothesized that individual mechanoreceptive units along the antenna were tuned to enable stable running. Extracellular multi-unit recordings revealed P and D sensitivity and variable-latency responses, suggesting the antenna may function as a delay line. In the second chapter, I show how individual sensor units distributed on the antenna precondition neural signals for the control of high-speed turning.

Since sensors of animals are embedded within the body, they must function through the mechanics of the body. In Chapter 3, I studied mechanical properties of the primary tactile sensors of cockroaches, the antennae, using experimental and engineering approaches. I revealed how both the static and dynamic properties of the antenna may influence sensory acquisition during quasi-static and dynamic sensorimotor tasks. Further elucidation of antennal mechanical tuning will lead to new hypotheses, integrating distributed mechanosensory inputs from a dynamic sensory appendage operating on a moving body.

During rapid escape from predators, the neuromechanical system of animals is pushed to operate closer to its limits. When operating at such extremes, small animals are true escape artists benefiting from enhanced maneuverability, in part due to scaling. In Chapter 4, I show a novel neuromechanical strategy used by the cockroach P. americana and the gecko H. platyrus which may facilitate their escape when encountering a gap. Both species ran rapidly at 12-15 body lengths-per-second toward a ledge without braking, dove off the ledge, attached their feet by claws like a grappling hook, and used a pendulum-like motion that can exceed one meter-per-second to swing around to an inverted position under the ledge, out of sight. In cockroaches, I show that the behavior is mediated by a rapid claw-engagement reflex initiated during the fall. Finally, I show how the novel behavior has inspired the design of a small, hexapedal robot that can assist rescuers during natural and human-made disasters.

Animals must navigate a complex and ever-changing world to find mates, forage for food, and escape from predators. Consequently, the robustness of an animal’s locomotor system (i.e., the capacity for its locomotor system to function despite experiencing perturbations) can have major implications for the animal’s survival and fitness. Perturbations can take a variety of forms, and they may be internal or external to the locomotor system. An internal perturbation may be an injury or the total loss of a body part involved in movement, whereas an external perturbation may be an obstacle that an animal must overcome. Considering the frequency with which they encounter perturbations, terrestrial arthropods appear to have highly robust locomotor systems. They often lose legs, and because they are generally small in stature, they must frequently alter the mechanics of their movements during running to navigate obstacles and terrain variability. Still, terrestrial arthropods include animals with a variety of body plans and locomotor strategies, and the robustness of these animals relative to one another has not been well studied. Furthermore, the relative impacts of different types of internal and external perturbations on individual arthropod locomotor systems are not well studied. A better understanding of the factors that determine how capably locomotor systems overcome perturbations could lead to reveal deeper insights about the diversity of morphologies and behaviors that we observe in arthropods today. At the same time, features of arthropod locomotor systems that confer robustness to particular perturbations could inspire engineers seeking to design robust robots.

My dissertation explores the costs and limits of robustness in the locomotor systems of cockroaches. Many species of cockroaches are remarkable runners, and they’ve been used extensively in studies about obstacle negotiation and leg loss. I use oxygen consumption to measure the energetic cost of locomotion, and I measure the cost of running before and after leg loss and with and without obstacles present. Initially, I focus on one species of cockroach as I compare the added costs associated with different types of leg loss and the interactive effects of leg loss and obstacle encounters. In the final chapter, I compare the consequences of leg loss among species of cockroach that display distinct morphologies and behaviors. This work emphasizes the context-dependent nature of robustness. Not all perturbations affect the same locomotor system in the same way, and not all locomotor systems experience the same perturbations in the same way.

Chapter 1 explores the energetic cost associated with leg loss in the cockroach Blaberus discoidalis, a species which has been used frequently to study the energetics and mechanics of locomotion and has shown itself adept at overcoming perturbations. It seeks to determine if different types of internal perturbations have different consequences for the cockroach. I measured the energetic cost of running and endurance of these animals as they ran on a treadmill across a range of speeds. I looked at how these measurements changed for individuals after losing different pairs of legs, and I sought to relate any changes to observable differences in their stride kinematics. I found energetic costs associated with front and middle leg loss, but not hind leg loss. At the same time, I found reductions in endurance capacity after middle and hind leg loss, but not after front leg loss. Observed changes in the animals’ movements matched with expectations based on the role each pair of legs plays during locomotion. Animals missing front legs were destabilized in pitch, animals missing middle legs were destabilized in roll, and animals missing both middle and hind legs struggled to generate propulsive forces that would have allowed them to move forward in the chamber and extend their runs through intermediate locomotion. These results show that the costs of leg loss depend on the leg lost.

Whereas Chapter 1 focuses on the internal perturbation of leg loss, Chapter 2 considers the costs associated with an external perturbation: regular encounters with a hurdle. It seeks to determine if internal perturbations can affect a locomotor system’s response to external perturbations and vice versa. Again, I focused exclusively on the B. discoidalis cockroach. As in Chapter 1, I measured the energetic cost of running on a treadmill, and I compared that measurement for an individual before and after losing legs from a particular pair of legs. However, trials in Chapter 2 were limited to one speed. In addition, for each condition of leg loss (i.e., intact, five-legged, and four-legged), I measured the cost of running on a flat treadmill and on a treadmill with a hurdle present. Similarly to Chapter 1, I found that, during hurdle negotiation, the costs associated with leg loss also depended on the leg that was lost. Animals missing front legs often appeared out of control when climbing up and down hurdles, but there were no energetic costs associated with front leg loss. In contrast, animals missing middle legs rolled excessively as they clambered over hurdles, and middle leg loss exacerbated the cost of encountering hurdles. Finally, animals missing hind legs did not display an increased cost of encountering hurdles, but they showed they were incapable of overcoming the hurdle at the set speed of the treadmill. These results provide further evidence of the importance of considering a leg’s role in the locomotor system when predicting the impact of its loss. At the same time, the results are specific to the particularities of the experimental set up, and Chapter 2 raises questions about how results would if the speed of the animals varied and if the type and frequency of obstacle encounters varied. A true understanding of the robustness of a biological system requires not just an understanding of the system itself, but also a strong understanding of the ecological and behavioral context in which it operates.

Chapter 3 examines how differences in the design of the locomotor system itself influences its robustness. It seeks to determine if an analogous internal perturbation has the same impact on different locomotor systems. It explores the costs of leg loss in three distinct species of cockroach: the gracile sprinter Periplaneta americana, the powerful runner B. discoidalis, and the lumbering burrower Gromphadorhina portentosa. These three species represent a continuum of running performance capabilities, with P. americana cockroaches being the fastest runners and G. portentosa cockroaches being the slowest. There is often a tradeoff between robustness and performance, so it was hypothesized that the species would also represent a continuum of robustness to leg loss, with G. portentosa cockroaches being the most robust and P. americana cockroaches being the least robust. As before, I measured the energetic cost of running on a treadmill, and I compared that measurement for an individual before and after leg loss from a particular pair of legs. P. americana and B. discoidalis cockroaches displayed similar response patterns to leg loss, with front leg loss causing pitch instability and middle leg loss causing roll instability. However, P. americana cockroaches exhibited more extreme responses, with higher energetic costs associated with front and middle leg loss. In addition, whereas B. discoidalis cockroaches maintained the capacity to run at all speeds included in the study, regardless of which legs they lost, most P. americana cockroaches missing their middle and hind legs failed to run at the highest speed in the study. Interestingly, although G. portentosa cockroaches showed a similar response to front leg loss as the other two species, they did not display consistently higher running costs after middle and hind leg loss. In fact, they readily switched to a new gait that minimized roll instability after the loss of two middle legs. These results suggest that, even if locomotor systems consist of superficially similar components that appear to operate in similar ways, the function and relative contribution of those components may vary from system to system. Understanding the impact of a perturbation on a locomotor system requires an understanding of the particularities of that system. The diversity of life represents both an enduring challenge for biologists who wish to understand all of its facets, and a limitless source of opportunity for engineers who seek new inspiration for their inventions.

Dynamic locomotion behaviors in both animals and machines emerge from high-dimensional, nonlinear, and coupled interactions between neuromuscular or actuator feedback control and the mechanical interactions between the body and the surrounding environment. The complexity of the hierarchical control systems and mechanical behaviors of multiple distributed appendages and body segments necessary to achieve robust locomotion behavior in challenging environments has made elucidation of general principles underlying the remarkable adaptive and dynamic locomotion behaviors observed in animals, and consequently the achievement of such behaviors in engineered systems, a significant challenge. In this dissertation, we take the first steps towards developing a principled theory and practice for understanding mechanisms of whole body dynamic locomotion control through the hierarchical composition of distributed embodied control modules. Specifically, we introduce a data-driven approach for discovery of hierarchical neuromechanical control systems via the targeting of a low dimensional posture principle, demonstrate how the embodiment of multifunctional actuation affordances and recruitment of multiple body segments enables robust, high performance dynamic ground righting maneuvers, and show that the sequential composition of substrate manipulation modes can enable deep, rapid burrowing in small organisms.

In Chapter 1, we present an approach for discovering task-specific attracting invariant postural submanifolds directly from ensemble behavioral measurement data. Specifically, our approach is grounded in the templates and anchors hypothesis for dynamic locomotion control, which hypothesizes that the preferred postural degrees of freedom used for a specific task are anchored - rendered stable and robust to perturbations - by fast neuromechanical feedback control forces. We present both a theory and practice for identifying these dynamic posture principles that depends only on a local model of the hypothesized template-anchor dynamics, and therefore is generalizable to a wide range of locomotion phenomena including both periodic and transient behavior. We introduce an expressive model of periodic locomotion behavior, the generalized anchored Hopf model (GAHM), that affords direct numerical tests of the efficacy of our approach. We find that our data-driven method is capable of accurately detecting template submanifolds with a surprisingly wide range of geometric and dynamical properties, and is robust to measurement noise and parameter errors.

In Chapter 2, we explore how the composition of multifunctional actuation affordances can enable robust dynamic transitional maneuvers. Specifically, we use the ground righting behavior of the house gecko H. platyurus to explore how the composition of mechanical affordances embodied in the tail and spine may enable robust performance in response to different environmental contact conditions and tail autotomy. We develop a parsimonious axial plane ground righting template model capable of predicting righting times and trajectories for the rolling behavior of bodies with different effective axial plane shapes, allowing us to test the hypothesis that control modules beyond tail driven actuation alone contribute to ground righting performance. To reconstruct the axial plane ground righting kinematics of multiple body segments, we measure the full 3D spatial kinematics of landmarks on the tail, torso, and legs of geckos ground righting on substrate conditions either allowing or preventing tail-ground contact. We discover comparable total ground righting times across distinct tail actuation modes, with performance benefits afforded when the tail is employed in contact with the substrate. Further, we find that in individuals with the tail autotomized, ground righting can still be achieved using dynamic spine motions with comparable performance to tail-intact individuals. Finally, we take the first step towards exploring dynamic-spine tail compositions by showing that measured ground righting performance exceeds the predictions of the passive template model, and that a shared axial plane spine twisting control mode is present for all tail actuation strategies.

Finally, in Chapter 3, we explore how adaptive appendage control can enable rapid, deep burrowing in complex multiphase substrates for small, force-limited organisms. Specifically, we use the Pacific mole crab, E. analoga, to study how leg kinematics adjust in response to depth-dependent substrate resistive forces encountered during burrowing. To test the hypothesis that appendage control is adjusted in response to increasing substrate resistive forces, we use optically transparent substrates with significantly different resistive properties to precisely measure the effects of body depth and substrate resistance on appendage kinematic control. We find that the stride periods of the anterior excavating appendages increase as substrate resistive forces increase. We further find evidence suggesting the presence of an additional substrate manipulation mode, legged cavity expansion, that may be sequentially composed with excavation strides to enable burrowing to increased depths. Finally, we show that the control program of the anterior excavating appendage pairs is robust to loss of function of a posterior appendage pair, and take the first step towards identifying the adaptive, sequentially composable substrate manipulation modules that enable burrowing using particle image velocimetry (PIV) measurements of the substrate mechanical response.

Rapid terrestrial locomotion is a fascinating but difficult problem that roboticists face. Considerations must be made towards dynamics, controls, estimation, the environment, electronics, and mechanical design. Wheeled and legged robots have been heavily researched and have gained much success, however they still have their limitations, especially when ground contact is lost.

Inspired by the high performance of lizards actively using their tails to stabilize perturbations and rapidly reorient their body, we propose to add an active inertial appendage to terrestrial robots that will greatly improve their stability and maneuverability.

In this dissertation we present three main focus areas. We first develop the capabilities of a novel tailed robot with an active single degree-of-freedom (DOF) tail. We then discuss the single degree of freedom orientation sensing developed for this robot. Finally, we construct a nonlinear controller to enable reorientation of the body of a 2-link robot in three-dimensions while being constrained to a tail that only has two DOF of actuation.

The first 177 (g) active tailed robot, Tailbot, has a single DOF of actuation and contains attitude inertial sensors. By utilizing both contact forces and zero net angular momentum maneuvering, this tailed robot can rapidly right itself in a fall, avoid flipping over after a large perturbation, and smoothly transition between surfaces of different slopes. We used a modeling approach to show that a tail-like design offers significant advantages to other alternatives, including reaction wheels, when the speed of response is important.

In order to provide the reliable orientation sensing for the single DOF robot we design a novel time varying complementary filter. Complementary filtering (CF) is a signal processing method

that is commonly used for the fusion of gyroscope and accelerometer measurements in order to robustly estimate the attitude of a rigid body. The traditional CF uses linear time invariant filters to combine the two different measurements of the same physical quantity. Here we present an extension to the CF method with time-varying filters. A fuzzy logic method is developed to adjust the parameters. Stability analysis as well as experimental results are presented to verify the proposed

method.

Finally, we propose a nonlinear control scheme for attitude control of a falling, two link active tailed robot with only two DOF of actuation. We derive a simplified expression for the robot's angular momentum and invert this expression to solve for the shape velocities that drive the body's

angular momentum to a desired value. By choosing a body angular velocity vector parallel to the axis of error rotation, the controller steers the robot towards its desired orientation. The proposed scheme is accomplished through feedback laws as opposed to feedforward trajectory generation, is fairly robust to model uncertainties, and is simple enough to implement on a low power microcontroller.

We verify our approach by implementing the controller on a small 175 (g) robot platform, enabling rapid maneuvers approaching the spectacular capability of animals.