Master of Science

thesisThis research studies the passive dynamics of an under-actuated trotting quadruped. The goal of this project is to perform three-dimensional (3D) dynamic simulations of a trotting quadruped robot to find proper leg configurations and stiffness range, in order to achieve stable trotting gait. First, a 3D simulation framework that includes all the six degrees of freedom of the body is introduced. Directionally compliant legs together with different leg configurations are employed to achieve passive stability. Compliant legs passively support the body during stance phase and during flight phase a motor is used to retract the legs. Leg configurations in the robot's sagittal and frontal plane are introduced. Numerical experiments are conducted to search the design space of the leg, focusing on increasing the passive stability of the robot. Increased stability is defined as decreased pitching, rolling, and yawing motion of the robot. The results indicate that optimized leg parameters can guarantee passive stable trotting with reduced roll, pitch, and yaw. Studies suggest that a quadruped robot with compliant legs is dynamically stable while trotting. Results indicate that the robot based on a biological model (i.e., caudal inclination of humeri and cranial inclination of femora) has the best performance. Stiff springs at hips and shoulders, soft spring at knees and elbows, and stiff springs at ankles and wrists are recommended. The results of this project provide a conceptual framework for understanding the movements of a trotting quadruped

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

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

Co-Designing Robots by Differentiating Motion Solvers

We present a novel algorithm for the computational co-design of legged robots and dynamic maneuvers. Current state-of-the-art approaches are based on random sampling or concurrent optimization. A few recently proposed methods explore the relationship between the gradient of the optimal motion and robot design. Inspired by these approaches, we propose a bilevel optimization approach that exploits the derivatives of the motion planning sub-problem (the inner level) without simplifying assumptions on its structure. Our approach can quickly optimize the robot's morphology while considering its full dynamics, joint limits and physical constraints such as friction cones. It has a faster convergence rate and greater scalability for larger design problems than state-of-the-art approaches based on sampling methods. It also allows us to handle constraints such as the actuation limits, which are important for co-designing dynamic maneuvers. We demonstrate these capabilities by studying jumping and trotting gaits under different design metrics and verify our results in a physics simulator. For these cases, our algorithm converges in less than a third of the number of iterations needed for sampling approaches, and the computation time scales linearly.Comment: 8 pages, 7 figures, submitted to IROS 202

Validation of a laboratory method for evaluating dynamic properties of reconstructed equine racetrack surfaces.

BackgroundRacetrack surface is a risk factor for racehorse injuries and fatalities. Current research indicates that race surface mechanical properties may be influenced by material composition, moisture content, temperature, and maintenance. Race surface mechanical testing in a controlled laboratory setting would allow for objective evaluation of dynamic properties of surface and factors that affect surface behavior.ObjectiveTo develop a method for reconstruction of race surfaces in the laboratory and validate the method by comparison with racetrack measurements of dynamic surface properties.MethodsTrack-testing device (TTD) impact tests were conducted to simulate equine hoof impact on dirt and synthetic race surfaces; tests were performed both in situ (racetrack) and using laboratory reconstructions of harvested surface materials. Clegg Hammer in situ measurements were used to guide surface reconstruction in the laboratory. Dynamic surface properties were compared between in situ and laboratory settings. Relationships between racetrack TTD and Clegg Hammer measurements were analyzed using stepwise multiple linear regression.ResultsMost dynamic surface property setting differences (racetrack-laboratory) were small relative to surface material type differences (dirt-synthetic). Clegg Hammer measurements were more strongly correlated with TTD measurements on the synthetic surface than the dirt surface. On the dirt surface, Clegg Hammer decelerations were negatively correlated with TTD forces.ConclusionsLaboratory reconstruction of racetrack surfaces guided by Clegg Hammer measurements yielded TTD impact measurements similar to in situ values. The negative correlation between TTD and Clegg Hammer measurements confirms the importance of instrument mass when drawing conclusions from testing results. Lighter impact devices may be less appropriate for assessing dynamic surface properties compared to testing equipment designed to simulate hoof impact (TTD).Potential relevanceDynamic impact properties of race surfaces can be evaluated in a laboratory setting, allowing for further study of factors affecting surface behavior under controlled conditions

Understanding and Improving Locomotion: The Simultaneous Optimization of Motion and Morphology in Legged Robots

There exist many open design questions in the field of legged robotics. Should leg extension and retraction occur with a knee or a prismatic joint? Will adding a compliant ankle lead to improved energetics compared to a point foot? Should quadrupeds have a flexible or a rigid spine? Should elastic elements in the actuation be placed in parallel or in series with the motors? Though these questions may seem basic, they are fundamentally difficult to approach. A robot with either discrete choice will likely need very different components and use very different motion to perform at its best. To make a fair comparison between two design variations, roboticists need to ask, is the best version of a robot with a discrete morphological variation better than the best version of a robot with the other variation? In this dissertation, I propose to answer these type of questions using an optimization based approach. Using numerical algorithms, I let a computer determine the best possible motion and best set of parameters for each design variation in order to be able to compare the best instance of each variation against each other. I developed and implemented that methodology to explore three primary robotic design questions. In the first, I asked if parallel or series elastic actuation is the more energetically economical choice for a legged robot. Looking at a variety of force and energy based cost functions, I mapped the optimal motion cost landscape as a function of configurable parameters in the hoppers. In the best case, the series configuration was more economical for an energy based cost function, and the parallel configuration was better for a force based cost function. I then took this work a step further and included the configurable parameters directly within the optimization on a model with gear friction. I found, for the most realistic cost function, the electrical work, that series was the better choice when the majority of the transmission was handled by a low-friction rotary-to-linear transmission. In the second design question, I extended this analysis to a two-dimensional monoped moving at a forward velocity with either parallel or series elastic actuation at the hip and leg. In general it was best to have a parallel elastic actuator at the hip, and a series elastic actuator at the leg. In the third design question, I asked if there is an energetic benefit to having an articulated spinal joint instead of a rigid spinal joint in a quadrupedal legged robot. I found that the answer was gait dependent. For symmetrical gaits, such as walking and trotting, the rigid and articulated spine models have similar energetic economy. For asymmetrical gaits, such as bounding and galloping, the articulated spine led to significant energy savings at high speeds. The combination of the above studies readily presents a methodology for simultaneously optimizing for motion and morphology in legged robots. Aside from giving insight into these specific design questions, the technique can also be extended to a variety of other design questions. The explorations in turn inform future hardware development by roboticists and help explain why animals in nature move in the ways that they do.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144074/1/yevyes_1.pd

Energetics and Passive Dynamics of Quadruped Robot Planar Running Gaits

Quadruped robots find application in military for load carrying over uneven terrain, humanitarian de-mining, and search and rescue missions. The energy required for quadruped robot locomotion needs to be supplied from on-board energy source which can be either electrical batteries or fuels such as gasolene/diesel. The range and duration of missions very much depend on the amount of energy carried, which is highly limited. Hence, energy efficiency is of paramount importance in building quadruped robots. Study of energy efficiency in quadruped robots not only helps in efficient design of quadruped robots, but also helps understand the biomechanics of quadrupedal animals. This thesis focuses on the energy efficiency of planar running gaits and presents: (a) derivation of cost of transport expressions for trot and bounding gaits, (b) advantages of articulated torso over rigid torso for quadruped robot, (c) symmetry based control laws for passive dynamic bounding and design for inherent stability, and (d) effect of asymmetry in zero-energy bounding gaits

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

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