In biomechanics and robotics, elasticity plays a crucial role in enhancing locomotion efficiency and stability. Traditional approaches in legged robots often employ series elastic actuators (SEA) with discrete rigid components, which, while effective, add weight and complexity. This paper presents an innovative alternative by integrating continuously compliant structures into the lower legs of a bipedal robot, fundamentally transforming the SEA concept. Our approach replaces traditional rigid segments with lightweight, deformable materials, reducing overall mass and simplifying the actuation design. This novel design introduces unique challenges in modeling, sensing, and control, due to the infinite dimensionality of continuously compliant elements. We address these challenges through effective approximations and control strategies. The paper details the design and modeling of the compliant leg structure, presents low-level force and kinematics controllers, and introduces a high-level posture controller with a gait scheduler. Experimental results demonstrate successful bipedal walking using this new design.

This paper presents a novel robotic leg design and an associated control approach, which aims at providing an extension to the classical series elastic actuation concept. We propose to directly integrate the series compliance into the structure of the robotic leg itself, as opposed to co-locating spring and motor as done in traditional series elastic actuators. Our approach will eliminate mechanical design complexity and lead to a reduction of mass in the legs. This will, as a secondary benefit, improve the energy efficiency of locomotion. The primary contribution of this work is a model-based controller that can stably and precisely regulate the ground contact forces during stance. This control approach is demonstrated in a set of test-bench experiments, in which we control the contact forces of a modified version of the robotic leg ScarlETH. Here, the rigid shank is replaced by a continuously compliant element made of spring steel. This work presents the first step towards a new generation of robotic legs with structural compliance.

This work introduces a new design paradigm for robotic legs. Our concept extends upon classical series elastic actuation and directly integrates the series compliance into the structure of the leg. This integration reduces the mechanical design complexity and can potentially reduce the overall weight of the leg. In this paper, we introduce a prototype leg with a continuously compliant shank and derive and analyze a non-linear beam model that is used to predict its contact forces. The model is validated in two static experiments: one on the isolated shank and one on the full leg. It shows good agreement with measurements. In our validation, we also studied the influence of the model discretization, showing that about 10 nodes are sufficent. Furthermore, we introduce the concept of a force workspace: the range of forces that can be created by controlling the joint angles of the leg. Due to the coupling of nonlinear deformations and nonlinearities in the kinematics, this workspace is non-trivial. In particular, we demonstrate that it is bounded by a force-singularity in which the force/joint-angle relationship cannot be inverted. The results presented in this work can be applied in the development of state-estimators, set-point filters and controllers, and they can inform the future design of suitable geometries of compliant elements.

Upright locomotion has many health benefits for patients with spinal cord injury. Passive gait orthoses, such as an isocentric reciprocal gait orthosis (IRGO), allow patients to walk by pushing themselves forward with forearm supports. To move the legs, the IRGO physically couples the motion of stance and swing leg through a linkage. Unfortunately, this locomotion is associated with high metabolic effort. To reduce the metabolic demand while maintaining the advantages of simplicity and low weight of an IRGO, we propose the extension of an IRGO with a single actuator directly integrated into the rocking mechanism and investigate the use of a Hybrid Zero Dynamics based controller to support the user. This is a time-invariant feedback controller, in which a stable reciprocal gait is designed by means of numerically optimized virtual constraints. We evaluate the feasibility of this approach in a sagittal plane model of the dynamic system. For a range of walking speeds and step lengths, we are able to show that the chosen approach has the potential to safely support impaired users walking with IRGOs. The obtained gaits were periodic, stable, maintained a minimal ground clearance and could be implemented with a driving torque of 50 Nm.

For a class of biped robots with impulsive dynamics and a non-empty set of passive gaits (unactuated, periodic motions of the biped model), we present a method for computing continuous families of locally optimal gaits with respect to a class of commonly used energetic cost functions (e.g., the integral of torque-squared). We compute these families using only the passive gaits of the biped, which are globally optimal gaits with respect to these cost functions. Our approach fills in an important gap in the literature when computing a library of locally optimal gaits, which often do not make use of these globally optimal solutions as seed values. We demonstrate our approach on a well-studied two-link biped model.

We present a homotopic approach to generating energetically optimal gaits for legged robots that maps passive (i.e., unactuated) gaits of an energetically conservative model of the robot to a model with user-defined target dynamics with dissipation and actuation (i.e., the more “realistic” legged model). Our core contribution is advancing the state-of-the-art towards a turn-key approach where the seed values are known by design and do not rely on domain-specific knowledge to generate or randomly guess across a range of energetic cost functions and desired gait properties (e.g., walking speed, hopping height, etc.), which can limit the usefulness of the typical optimization-based approach. We demonstrate this methodology on a parallel elastic actuated planar monoped with five degrees of freedom. Our work also demonstrates an explicit connection between passive gaits and optimally actuated motions, which has long been an area of interest in the fields of robotics and biome-chanics.

Controlling soft robots with precision is a challenge due to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman operator theory offers a way to construct explicit dynamical models of soft robots and to control them using established model-based control methods. This approach is data driven, yet yields an explicit control-oriented model rather than just a “black-box” input-output mapping. This work describes a Koopman-based system identification method and its application to model predictive control (MPC) design for soft robots. Three MPC controllers are developed for a pneumatic soft robot arm via the Koopman-based approach, and their performances are evaluated with respect to several real-world trajectory following tasks. In terms of average tracking error, these Koopman-based controllers are more than three times more accurate than a benchmark MPC controller based on a linear state-space model of the same system, demonstrating the utility of the Koopman approach in controlling real soft robots.

Soft robots are challenging to model due in large part to the nonlinear properties of soft materials. Fortunately, this softness makes it possible to safely observe their behavior under random control inputs, making them amenable to large-scale data collection and system identification. This paper implements and evaluates a system identification method based on Koopman operator theory in which models of nonlinear dynamical systems are constructed via linear regression of observed data by exploiting the fact that every nonlinear system has a linear representation in the infinite-dimensional space of real-valued functions called observables. The approach does not suffer from some of the shortcomings of other nonlinear system identification methods, which typically require the manual tuning of training parameters and have limited convergence guarantees. A dynamic model of a pneumatic soft robot arm is constructed via this method, and used to predict the behavior of the real system. The total normalized-root-mean-square error (NRMSE) of its predictions is lower than that of several other identified models including a neural network, NLARX, nonlinear Hammerstein-Wiener, and linear state space model.

In this paper, we investigated how the stiffness and damping properties of soft robotic feet affect the stability and energetic economy of bipedal robotic walking. To this end, we manufactured four different spherical feet from the following materials: hollow rubber, Sorbothane, Norsorex, and Neoprene. The materials were specifically chosen to cover a wide range of stiffness and damping values. The impact response of each design was first characterized in a drop test rig. We then evaluated the performance of each foot in an extensive series of walking experiments on the planar bipedal robot RAM one. Our results showed that, at low speeds, the feet with lower damping had a smaller energy cost of walking, possibly due to greater return of mechanical energy at lift-off. However, at speeds above 0.5m\s, the feet with lower damping started to exhibit a bouncing behaviour which led to higher walking instability and increased the energy cost of walking. Additionally, we found the feet with lower stiffness to be more economical across all walking speeds. Our results provide insight into the role of foot properties in bipedal walking and may help with the design of walking robots.

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.

This paper introduces RAMone, a series-elastic planar biped with knees, built to study the energetics of bipedal locomotion. RAMone is designed to achieve versatile, economical motion using lightweight legs with highly compliant joints. To demonstrate the hardware, we present a virtual-model controller that regulates body height and forward speed while walking. This controller achieves stable walking for a range of body heights, both in hardware and simulation. For these gaits, we verify that the energetic cost of transport decreases as walking height increases, in agreement with previous results for robot models and humans.

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.

Bellows-like actuators are popular in soft robotic systems. Sensing the movement of these actuators with traditional sensors is challenging. This work proposes and tests a sensing system based on the changing mutual inductance of wire coils on bellows. A method for modeling the changes in mutual inductance between coils of tightly-packed wires is introduced. Changes in mutual inductance are measured through the self-inductance of a circuit made up of the coils in series. The self-inductance of the circuit measures the bellows bend-angle. The experiments show an approximately quadratic relationship between the bend angle and the measured inductance. From a bend angle of 121 ° to −67 ° the inductance of the circuit increases by 19 %. The bias-inducing effects of shear strain, torsional strain, non-uniform bending, and nearby metal are also explored.

In this paper we use optimal control on a geared electric DC motor to compare the energetic efficiency of a simulation of conceptual monoped hoppers with either parallel elastic actuation (PEA) or series elastic actuation (SEA). The energy is measured using three cost functions: positive actuator work, electrical losses, and positive electrical work. For PEA, the presence of the motor inertia in the collision losses leads to increased collision losses at large transmission ratios, which lead to energetically costly compensatory strategies where the SEA is at its most efficient. At small transmission ratios, the motor force increases for both cases, leading to increased thermal losses. In agreement with those theoretical predictions, our work shows that for positive actuator work and positive electrical work the optimal parameter choice for SEA is significantly more energetically efficient than the optimal choice for PEA. For electrical losses, a suitable choice of the transmission ratio can lead to negligible cost values for both actuator concepts.

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.

In this paper, we introduce a novel method to measure the contraction length and force output of Pneumatic Artificial Muscles. Our key innovation is to make the braid of these fiber-reinforced actuators out of insulated wires that form a single electric circuit. Changes in the inductance and resistance of this circuit can be related to muscle contraction length and muscle force. We implemented the proposed approach in two McKibben muscles that were evaluated under a variety of air pressures (0-200 kPa) and loads (0-45 N). The experiments confirmed our theoretical prediction of a roughly linear relationship between the measured inductance and the contraction length, as well as a strong correlation between the measured resistance and the actuator force. We were able to determine force with a resolution of 5N and contraction length with a resolution of 0.5 mm. This technique can be used to create flexible, precise, and robust self-sensing actuators that benefit a multitude of robotic applications.

Building on our previous work on passive dynamic walking with quadrupeds, we show that a large variety of gaits can be created completely passively by a quadrupedal model with elastic legs. Similar to the well-known Spring Loaded Inverted Pendulum model for bipeds, we created a conceptual quadrupedal model with elastic massless legs. To obtain a well-defined sequence of ground contact, we defined three distinct phases for each leg: stance, swing, and wait for touch down. Since a leg cannot make contact during swing, modifying the duration of this phase allows us to prevent feet from striking the ground prematurely. Gaits were identified in a single shooting implementation, such that the contact sequence was only influenced by the starting values of the numerical integration. By varying these values, we were able to identify trotting, pacing, walking, toelting, bounding, and galloping within a single model. For each of the identified gaits, we report the footfall pattern, ground contact forces, speed, and first order limit cycle stability.

In this paper, we explore the potential of trajectory optimization for unspecified contact sequences as a tool to identify optimal gaits and motions for legged robots. This work is based on a recently proposed method that states the mechanical dynamics in a floating base description, makes the ground contact forces part of the free variable vector, and implements the requirement that a foot is either on the ground or that the corresponding contact force is zero via a set of complimentary conditions. We introduce an algorithmic improvement that uses higher order integration for states that are continuous through collisions and thus increases the accuracy of the obtained solutions. The benefits of the proposed changes are evaluated with the models of a 1D hopper and a 2D bipedal robot, and we additionally compare our results with analytic solutions and an established multiple shooting implementation. The proposed method was able to automatically discover walking and running as the most energetically economic ways of locomotion for a conceptual biped that is moving at different speed. It additionally discovers an elastic walking gait that is used at intermediate velocities.

This paper presents a method for state estimation on a ballbot; i.e., a robot balancing on a single sphere. Within the framework of an extended Kalman filter and by utilizing a complete kinematic model of the robot, sensory information from different sources is combined and fused to obtain accurate estimates of the robot's attitude, velocity, and position. This information is to be used for state feedback control of the dynamically unstable system. Three incremental encoders (attached to the omniwheels that drive the ball of the robot) as well as three rate gyroscopes and accelerometers (attached to the robot's main body) are used as sensors. For the presented method, observability is proven analytically for all essential states in the system, and the algorithm is experimentally evaluated on the Ballbot Rezero.

In this work we apply optimal control to create running gaits for the model of an electrically driven one legged hopper, and compare the results obtained for five different objective functions. By using high compliant series elastic actuators, the motions of joint and motor are decoupled, which allows the exploitation of natural dynamics. Depending on the cost function, this exploitation varies. Energy is injected at different points of time, the amplitude of actuator action changes significantly, and the optimal gear ratios differ by a factor of two. Variations are, however, comparable over a wide range of hopping heights and running velocities. Purely force-based cost functions prove to be ill-suited for such non-conservative systems, and it is shown that thermal electrical losses, in contrast to common belief, do not dominate energy expenditure. The numerical results are corroborated by detailed analytical considerations which give general insights into optimal excitation with electric actuators.

This paper introduces the mechanical design and the control concept of the Series Compliant Articulated Robotic Leg ScarlETH which was developed at ETH Zurich for fast, efficient, and versatile locomotion. Inspired by biological systems, we seek to achieve this through large compliances in the joints which enable natural dynamics, allow temporary energy storage, and improve the passive adaptability. A sophisticated chain and cable pulley design minimizes the segment masses, places the overall CoG close to the hip joint, and maximizes the range of motion. Nonlinearities in the damping and an appropriate low-level controller allow for precise torque control during stance and for fast task space position control during swing. This paved the road for the combined application of a virtual model controller for ground contact and a modified Raibert style controller for flight phase which was successfully tested in planar running.

This work introduces a framework for the creation and analysis of efficient gaits for legged systems based on the exploitation of natural dynamics. It summarizes the theory behind hybrid dynamic modeling, the identification of optimal periodic motions with single shooting and direct collocation, and the analysis of first order stability. Three examples introduce various aspects of gait creation and analysis: a stability study of a passive dynamic walker determines the ideal position of the leg's center of mass, the cost of transportation is minimized for a prismatic monopod hopper based on series elastic actuators, and a basic controller is created for the model of a bounding robot. The presented tools and examples are freely available at www.asl.ethz.ch/people/cremy/personal/GaitCreation.

In this paper, we are presenting a method to estimate terrain properties (such as small-scale geometry or surface friction) to improve the assessment of stability and the guiding of foot placement of legged robots in rough terrain. Haptic feedback, expressed through joint motor currents and ground contact force measurements that arises when prescribing a predefined motion was collected for a variety of ground samples (four different shapes and four different surface properties). Features were extracted from this data and used for training and classification by a multiclass AdaBoost machine learning algorithm. In a single leg testbed, the algorithm could correctly classify about 94% of the terrain shapes, and about 73% of the surface samples.

In the present study, we applied the principles of passive dynamic walking onto the three dimensional motion of a simplified quadrupedal model. We extended the simulation framework of a planar system to include a rolling degree of freedom and searched for limit cycles that represent periodic gaits. Among the eight different gaits that we identified, were three kinds of trots and paces, as well as a lateral and diagonal single foot sequence. We could show that a distinct relation exists between the lateral spacing of the legs and the relative phase of the front and the back legs, and a certain trade-off between efficiency and dynamic stability. In agreement with established bipedal models, our results showed that the lateral rolling motion is invariably unstable.

SLIP models are generally known as one of the best and simplest abstractions describing the spring-like leg behavior found in human and animal running, and have thus been subject to exhaustive investigation. To exploit these findings in real robots, we utilize an operational space controller that projects the behavior of the SLIP model onto the dynamics of an actual segmented robotic leg. Additionally, we introduce a method to compensate for the energetic losses at the impact collisions, which are not accounted for in the simplified SLIP assumptions. This allows the direct application of existing dead-beat control strategies to arbitrary robotic legs, for which we can show that the collision and compensation effects in the actual leg enlarge the regions of stable running and reduce the minimally required locomotion speed. The necessary joint torque profiles can be generated in large part passively, for example by using high compliance series elastic actuators.

In the present study, we investigate a control strategy for hopping motions of an articulated leg that is driven by series elastic actuation. A highly compliant spring in the knee joint allows the exploitation of periodic energy storage but creates a major control challenge by severely limiting the bandwidth of closed-loop position or force control. This handicap is intensified by slow actuators, substantial delays, and the kinematic coupling of the articulated design. With classic closed-loop control strategies failing, an adaptive open-loop control algorithm is presented, that, over a series of jumps, estimates the compression of the actuator springs, and gradually modifies the motor inputs in order to minimize slipping and create a purely vertical motion.