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A low-loss hydraulic transmission offers an efficient and lightweight means to route power from a central source to distal actuators, with example applications in legged and wearable robots. For example, one could develop a completely passive hydraulic transmission for an exoskeleton that routes power from a healthy limb to a limb weakened by neurological injury to enable self-assist or rehabilitation. In this article, we introduce and model a variable hydraulic transmission that features rolling diaphragm cylinders and a discretely variable transmission ratio. The variable transmission utilizes digital hydraulics, wherein valves switch redundantly mounted cylinders in and out of the hydraulic circuit. The article discusses the impact of configuration on the available transmission ratios and passivity. Our modeling efforts focus on the parasitic effects that arise with power losses between the driving and driven joints, compliance and energy storage in the transmission, and drift during switching. We carefully validate the model, comparing analytically predicted fine and gross behaviors to experimental observations. Experimental results demonstrate the feasibility and utility of the proposed device and the associated model which serves as an important tool for the design and control of digital hydraulic transmission systems.

In this paper, we develop theoretical foundations for a new class of rehabilitation robot: body-powered devices that route power between a user's joints. By harvesting power from a healthy joint to assist an impaired joint, novel bimanual and self-assist therapies are enabled. This approach complements existing robotic therapies aimed at promoting recovery of motor function after neurological injury. We employ hydraulic transmissions for routing power, or equivalently for coupling the motions of a user's joints. Fluid power routed through flexible tubing imposes constraints within a limb or between homologous joints across the body. Variable transmissions allow constraints to be steered on the fly, and simple valve switching realizes free space and locked motion. We examine two methods for realizing variable hydraulic transmissions: using valves to switch among redundant cylinders (digital hydraulics) or using an intervening electromechanical link. For both methods, we present a rigorous mathematical framework for describing and controlling the resulting constraints. Theoretical developments are supported by experiments using a prototype fluid-power exoskeleton.

Online control design using a high-fidelity, full-order model for a bipedal robot can be challenging due to the size of the state space of the model. A commonly adopted solution to overcome this challenge is to approximate the fullorder model (anchor) with a simplified, reduced-order model (template), while performing control synthesis. Unfortunately it is challenging to make formal guarantees about the safety of an anchor model using a controller designed in an online fashion using a template model. To address this problem, this paper proposes a method to generate safety-preserving controllers for anchor models by performing reachability analysis on template models by relying on functions that bound the difference between the two models. This paper describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the anchor model in an online fashion. The method is illustrated on a 5-link RABBIT model, and is shown to allow the robot to walk safely while utilizing controllers designed in an online fashion.

In this paper, we systematically investigate passive gaits that emerge from the natural mechanical dynamics of a bipedal system. We use an energetically conservative model of a simple spring-leg biped that exhibits well-defined swing leg dynamics. Through a targeted continuation of periodic motions of this model, we systematically identify different gaits that emerge from simple bouncing in place. We show that these gaits arise along one-dimensional manifolds that bifurcate into different branches with distinctly different motions. The branching is associated with repeated breaks in symmetry of the motion. Among others, the resulting passive dynamic gaits include walking, running, hopping, skipping and galloping. Our work establishes that the most common bipedal gaits can be obtained as different oscillatory motions (or nonlinear modes) of a single mechanical system with a single set of parameter values. For each of these gaits, the timing of swing leg motion and vertical motion is matched. This work thus supports the notion that different gaits are primarily a manifestation of the underlying natural mechanical dynamics of a legged system. Our results might explain the prevalence of certain gaits in nature, and may provide a blueprint for the design and control of energetically economical legged robots.

In this paper, we examine the question of whether parallel elastic actuation or series elastic actuation is better suited for hopping robots. To this end, we compare and contrast the two actuation concepts in energy optimal hopping motions. To enable a fair comparison, we employ optimal control to identify motion trajectories, actuator inputs, and system parameters that are optimally suited for each actuator concept. In other words, we compare the best possible hopper with parallel elastic actuation to the best possible hopper with series elastic actuation. The optimizations are conducted for three different cost functions: positive mechanical motor work, thermal electrical losses, and positive electrical work. Furthermore, we look at three representative cases for converting rotary motor motion to linear leg motion in a legged robot. Our model featured an electric DC-motor model, a gearbox with friction, damping in the leg spring, and contact collisions. We find that the optimal actuator choice depends both on the cost function and conversion of motor motion to leg motion. When considering only thermal electrical losses, parallel elastic actuation always performs better. In terms of positive mechanical motor work and positive electrical work, series elastic actuation is better when there is little friction in the gear-train. For higher gear-train friction parallel elastic actuation is more economical for these cost functions as well.

In this letter, we identify passive dynamic bounding gaits of quadrupedal robots and investigate the dynamic similarities between these gaits and those of a bipedal model. For this purpose, we develop a passive dynamic bounding model of a quadruped with an extended main body. This model is based on the established SLIP model, yet has dynamic swing leg motions instead of a predefined angle of attack. We present a gait-identification framework that is based on a numerical continuation approach and that allows us to find all possible passive periodic motions that emerge from pronking and bounding in place. These gaits include forward pronking as well as forward bounding with gathered suspension, extended suspension, and with two suspensions. By conducting a parameter study on the body inertia, we can draw parallels between bipedal and quadrupedal gaits, and can also show how the structure of possible solutions changes with the main body inertia. In particular, we see sudden changes in the gait structure at certain critical values of the main body inertia. This letter shows connections between different types of quadrupedal gaits and may serve as a guideline in the design and control of gaits for energetically economical legged robots. It also sheds some light at the question why animals with different morphology use different gaits at the same Froude numbers.

This paper uses optimal control to simultaneously optimize the motion and morphology of a realistic model of a 2D Monoped. In particular, we compare the energetics of four different actuator configurations: a parallel elastic actuator (PEA) in the hip and a series elastic actuator in the leg (SEA), series hip and parallel leg, series hip and series leg, and parallel hip and parallel leg. We use realistic models with mass in the legs and feet, damping in the springs, and detailed DC electric motor models. The comparison is carried out for the cost of transport of three energetic measures: positive motor work, electrical losses, and positive electrical work, and evaluated as a function of velocity. In our optimization we include motor parameters, stiffness, and spring pre-compression terms as free variables, ensuring that we compare the energetically optimal version of each configuration at each velocity. We show that for the positive motor work and the electrical losses costs of transport (COT), the parallel hip and series leg configuration is energetically optimal. For the electrical work, the optimal configuration is speed dependent, with series hip and parallel leg optimal at low speeds, and both series hip series leg and parallel hip series leg optimal at high speeds.

If the control authority of a running system is insufficient to reach a target state in a single step, i.e. if deadbeat control is not possible, then a stabilizing controller is faced with the decision on how to plan intermediate steps. In this work, we compare the performance of a simple greedy control policy (that computes deadbeat inputs and simply caps them) with the optimal performance found by an exhaustive search through decision space. The performance criterion used in this study is the basin of attraction: the set of all states from where the target state will be reached in a finite number of steps. Using the planar spring-loaded inverted pendulum (SLIP) as a model for a running robot, we compare the two control schemes and a fully passive behavior. To this end, we extended the passive slip model to include a controllable, yet limited variation of the touchdown angle and of the damping in the leg spring. We quantified the number of steps that it would take for the model to fall or converge from arbitrary initial states. The paper highlights how the passive stabilization, that is inherent to the SLIP model, greatly influences the dynamics of the controlled system. Furthermore, it reveals some new insights into the structure of basins of attractions of SLIP-like running models.

This paper presents a novel variable transmission system that is based on the concept of digital hydraulics. In the proposed system, sets of rolling-diaphragm cylinders are mounted via different effective lever arms to an input and output joint. A variable subset of these cylinders is connected via three-way two-position on/off valves to a common hydraulic manifold. This introduces a controllable constraint on the hydraulic flow and creates a programmable hydraulic transmission. With three single-acting cylinders, we could realize 37 different transmission ratios. We investigated the nonholonomic flow constraint analytically, in simulation, and with an experimental prototype. Using water as fluid, we show that a very stiff transmission (124.2 Nm/rad) can be achieved within the range of ±6°. Theoretical transmission ratios are tracked with R-squared values of more than 0.996 and backlash is smaller than 1.4%. Furthermore, we show the applicability of the proposed transmission in the simulation of a body-powered knee-ankle exoskeleton.

This paper presents a simplistic passive dynamic model that is able to create realistic quadrupedal walking, tölting, and trotting motions. The model is inspired by the bipedal spring loaded inverted pendulum (SLIP) model and consists of a distributed mass on four massless legs. Each of the legs is either in ground contact, retracted for swing, or is ready for touch down with a predefined angle of attack. Different gaits, that is, periodic motions differing in interlimb coordination patterns, are generated by choosing different initial model states. Contact patterns and ground reaction forces (GRFs) evolve solely from these initial conditions. By identifying appropriate system parameters in an optimization framework, the model is able to closely match experimentally recorded vertical GRFs of walking and trotting of Warmblood horses, and of tölting of Icelandic horses. In a detailed study, we investigated the sensitivity of the obtained solutions with respect to all states and parameters and quantified the improvement in fitting GRF by including an additional head and neck segment. Our work suggests that quadrupedal gaits are merely different dynamic modes of the same structural system and that we can interpret different gaits as different nonlinear elastic oscillations that propel an animal forward.