This study introduces an innovative actuator that resembles a motor with a non-uniform permanent magnetic field. We have developed a prototype of the actuator by combining a standard motor, characterized by a uniform magnetic field, with a custom rotary magnetic spring exhibiting a non-uniform magnetic field. We have also presented a systematic computational approach to customize the magnetic field to minimize the energy consumption of the actuator when used for a user-defined oscillatory task. Experiments demonstrate that this optimized actuator significantly lowers energy consumption in a typical oscillatory task, such as pick-and-place or oscillatory limb motion during locomotion, compared to conventional motors. Our findings imply that incorporating task-optimized non-uniform permanent magnetic fields into conventional motors and direct-drive actuators could enhance the energy efficiency of robotic systems.

Springs can provide force at zero net energy cost by recycling negative mechanical work to benefit motor-driven robots or spring-augmented humans. However, humans have limited force and range of motion, and motors have a limited ability to produce force. These limits constrain how much energy a conventional spring can store and, consequently, how much assistance a spring can provide. In this paper, we introduce an approach to accumulating negative work in assistive springs over several motion cycles. We show that, by utilizing a novel floating spring mechanism, the weight of a human or robot can be used to iteratively increase spring compression, irrespective of the potential energy stored by the spring. Decoupling the force required to compress a spring from the energy stored by a spring advances prior works, and could enable spring-driven robots and humans to perform physically demanding tasks without the use of large actuators.

Springs are efficient in storing and returning elastic potential energy but are unable to hold the energy they store in the absence of an external load. Lockable springs use clutches to hold elastic potential energy in the absence of an external load, but have not yet been widely adopted in applications, partly because clutches introduce design complexity, reduce energy efficiency, and typically do not afford high fidelity control over the energy stored by the spring. Here, we present the design of a novel lockable compression spring that uses a small capstan clutch to passively lock a mechanical spring. The capstan clutch can lock over 1000 N force at any arbitrary deflection, unlock the spring in less than 10 ms with a control force less than 1% of the maximal spring force, and provide an 80% energy storage and return efficiency (comparable to a highly efficient electric motor operated at constant nominal speed). By retaining the form factor of a regular spring while providing high-fidelity locking capability even under large spring forces, the proposed design could facilitate the development of energy-efficient spring-based actuators and robots.

Springs are commonly used in wearable robotic devices to provide assistive joint torque without the need for motors and batteries. However, different tasks (such as walking or running) and different users (such as athletes with strong legs or the elderly with weak legs) necessitate different assistive joint torques, and therefore, springs with different stiffness. Variable stiffness springs are a special class of springs which can exert more or less torque upon the same deflection, provided that the user is able to change the stiffness of the spring. In this paper, we present a novel variable stiffness spring design in which the user can select a preferred spring stiffness similar to switching gears on a bicycle. Using a leg-swing experiment, we demonstrate that the user can increment and decrement spring stiffness in a large range to effectively assist the hip joint during leg oscillations. Variable stiffness springs with human-selectable stiffness could be key components of wearable devices which augment locomotion tasks, such as walking, running, and swimming.

The Spring-Loaded Inverted Pendulum (SLIP) is one of the simplest models of robot locomotion. SLIP is commonly used to predict the center of mass motion and derive simple control laws for stable locomotion. However, the SLIP model is not integrable, which means that no closed-form relation can be derived to understand how the design and control parameters of the SLIP model affect stable locomotion. There exist a number of different analytical approximations to the SLIP model when considering small step lengths and symmetric steps. In this paper, we present a novel approximation to the SLIP model without relying on the small step length and the symmetric step assumption. The model was found to accurately predict the stability of the SLIP model for large and asymmetric steps and was used to design a controller to stabilize the SLIP model in a couple of steps.

Direct-drive motors (DDMs) have been increasingly used for robot actuation because they provide high-fidelity torque control, but they typically have low torque density. Gearing can be used to increase the torque density of motors, but gearing decreases the power density of the actuator. Parallel elastic actuators (PEAs), composed of a spring attached in parallel to a motor, can increase both the torque and power density of the actuator without jeopardizing torque control fidelity. PEAs can also generate efficient oscillatory motion by applying the torque of the motor through resonant oscillations. However, conventional fixed stiffness springs used in PEAs only enable efficient oscillatory motion at a fixed resonant frequency defined by the stiffness of the spring. In this article, we present a parallel variable stiffness actuator (PVSA) consisting of a DDM connected in parallel to a variable stiffness spring. PVSAs retain the torque control bandwidth of DDMs and PEAs and can be used to amplify the torque and power of the motor over a range of oscillation frequencies. We present a compact design of a PVSA where a direct-drive motor, a high energy density composite spring, and a variable stiffness mechanism are arranged in a conventional cylindrical geometry, similar to a motor-gearbox assembly. We foresee the use of PVSAs in mobile robots and wearable devices, where energy efficient oscillatory motion at different frequencies, along with high torque and power density is indispensable.

Energetically-passive robot exoskeletons, mimicking the function of the bicycle, could enable humans to reach previously unprecedented mobility. However, energetically-passive robot exoskeletons require a sophisticated mechanism to enable the human to supply energy, similar to what is enabled by the variable gear transmission mechanism of the bicycle. In this work, we present a new type of human-driven compliant transmission mechanism that could enable humans to supply energy when the leg is in the air, store the supplied energy, and release the stored energy when the leg is on the ground, in order to amplify the leg force and power. The compliant transmission mechanism presented in this paper is the first prototype and key component of a future human-driven artificial limb that aims to augment human mobility without using external energy.

Theory suggests a linear relation between stiffness and the energy stored by a linear helical spring at constant deformation. This relation implies that increasing the stiffness of a helical spring upon deformation requires more energy at larger deformations. State-of-the-art variable stiffness spring actuators, used to drive robots and human assistive and augmentation devices, are characterized by a similar relation: increasing stiffness as the spring is deformed costs more energy as more energy is stored by the spring. This feature imposes an apparently fundamental limitation on variable stiffness spring actuation in demanding tasks, such as lifting more, jumping higher, or running faster, because, in all these tasks, the variable stiffness spring should store a considerable amount of energy and provide different stiffness to accommodate different weights in lifting, heights in jumping, and speeds in running. Here, we present an innovative variable stiffness spring design, where the energy cost of changing stiffness is independent of the energy stored by the spring. The key element of the new design is a novel floating spring which changes stiffness without changing the energy stored by the spring. Springs possessing the aforementioned feature could pave the way towards variable stiffness robot actuation and human augmentation using smaller motors and smaller battery packs.

In this paper, we introduce a new type of compliant actuator named the Parallel Variable Stiffness Actuator (PVSA) which consists of a variable stiffness spring placed in parallel with a direct-drive motor. Parallel variable stiffness actuators provide (i) high-fidelity force control and (ii) controllable energy storage, as they inherit the benefits of direct-drive motors and variable stiffness springs. We present a compact design of the PVSA using a flat motor connected to an adjustable mechanical advantage torsional spring. We show that this PVSA is (1) not subject to the fundamental force control bandwidth limitation of series elastic and variable stiffness actuators, and most notably, (2) enables resonant energy accumulation despite the limited deformation of the spring and the constrained motion of the load attached to the actuator. The latter differentiates parallel variable stiffness actuators from fixed-stiffness parallel elastic actuators. PVSAs may be used with smaller direct-drive motors to match the peak power of larger motors without compromising force control fidelity. PVSAs may be used to implement resonant forcing under joint angle limitations in walking, jumping, running, swimming robots, or robotic exoskeletons used to augmented human motion in the aforementioned tasks.

Theory suggests an inverse relation between the stiffness and the energy storage capacity for linear helical springs: reducing the active length of the spring by 50% increases its stiffness by 100%, but reduces its energy storage capacity by 50%. State-of-the-art variable stiffness actuators used to drive robots are characterized by a similar inverse relation, implying reduced energy storage capacity for increased spring stiffness. This relation limits the potential of the variable stiffness actuation technology when it comes to human performance augmentation in natural tasks, e.g., jumping, weight-bearing and running, which may necessitate a spring exoskeleton with large stiffness range and high energy storage capacity. In this paper, we theoretically show that the trade-off between stiffness range and energy storage capacity is not fundamental; it is possible to develop variable stiffness springs with simultaneously increasing stiffness and energy storage capacity. Consistent with the theory, we experimentally show that a controllable volume air spring, has a direct relation between its stiffness range and energy storage capacity. The mathematical conditions presented in this paper may be used to develop actuators that could bypass the limited energy storage capacity of current variable stiffness spring technology.

Operational space feedback controllers can be used for tracking, motion coordination, stabilization, and a variety of other practical tasks. However, classical operational space controllers are only applicable to fully actuated robots and do not take into account fundamental physical limitations affecting the actuators in practical application. Here, we present an online computable operational space controller that extends to underactuated and overactuated systems, and which takes actuator limitations rigorously into account. In the proposed formulation, the control inputs are computed using constrained quadratic programs that have the minimal number of decision variables, and are strictly convex by construction. The resulting feedback controller is efficiently computable, and is applicable to fully actuated, underactuated, and overactuated systems. This is demonstrated via numerical simulations and experiments using two torque controlled robots.

Prioritized multi-objective optimization has been widely used within the operational space inverse dynamics control framework. In this paper, we present a constrained prioritized multi-objective optimization-base control formulation that extends to impedance control, including the `simple' impedance controller, which does not require the dynamic model. The main contribution of this paper is the dynamic-model-free prioritized feedback control formulation which encompasses arbitrary number of priority levels and takes the saturation constraints on the control inputs rigorously into account. The utility of the proposed formulation is demonstrated by a combined inverse dynamics impedance controller used to simulate stable locomotion of a planar anthropometric biped robot.

We present a differential-algebraic formulation with switching constraints to model the nonsmooth dynamics of robotic systems subject to changing constraints and multiple impacts. The formulation combines a single structurally simple governing equation, a set of switching kinematic constraints, and the plastic impact law, to represent the dynamics of robots that interact with their environment. The main contribution of this formulation is a novel algorithmic impact resolution method which provides an explicit solution to the classical plastic impact law in the case of multiple simultaneous impacts. This method serves as an alternative to prior linear-complementarity-based formulations which offer an implicit impact resolution through iterative calculation. We demonstrate the utility of the proposed method by simulating the locomotion of a planar anthropometric biped.

Theoretical studies suggest and experimental evidence confirms that maintaining and changing human joint stiffness by coactivated antagonistic muscles are metabolically expensive, even if muscles do not perform net mechanical work. Based on this observation, we posit that effective human augmentation can be achieved by actuators operated in parallel to human joints, even if these actuators only supplement joint stiffness without doing net mechanical work. In this article, we present a prototype variable-length leaf-spring actuator capable of large-range stiffness modulation. The key feature of the actuator is that it provides intrinsically low-energy-cost stiffness modulation even for large output deflection, by keeping the force on the driving motor low. Variable stiffness actuators use two motors to provide both stiffness and equilibrium position modulation as they are designed to do net mechanical work. The proposed actuator conceptually differs from variable stiffness actuators because first, it uses a single motor to only provide stiffness modulation, second, it does not provide equilibrium position modulation, and third, unless externally loaded, it cannot do net mechanical work. Using this actuator, we demonstrate stiffness augmentation during human-machine collaboration in challenging postural stabilization and weight-bearing tasks. Our results indicate that the proposed actuator can be used to complement a biological system by restoring or extending its functionality with low energy cost, and that variable stiffness spring actuators could effectively augment humans by doing no or a limited amount of mechanical work.

Compliant actuators are typically designed to possess a tunable positive stiffness characteristic in order to generate restoring force upon displacement. These actuators either require two independent motor units or closed-loop control to change both their equilibrium position and output stiffness. The introduction of negative stiffness, in combination with tunable positive stiffness, may reduce the complexity and extend the capability of these actuators in unexpected ways. In this paper, we present a compliant actuator that employs a passive negative stiffness mechanism in conjunction with an effectively tunable positive stiffness mechanism. We show that such actuator enables open-loop stiffness modulation and equilibrium position control using a single motor unit, as opposed to more conventional variable stiffness and series elastic actuators. The paper presents the theoretical foundation of positive-negative stiffness actuators and demonstrates low-power stiffness modulation and equilibrium position control achievable with a prototype positive-negative stiffness actuator.

Optimal control provides a systematic approach to control robots. However, computing optimal controllers for hardware-in-the-loop control is sensitively affected by modeling assumptions, computationally expensive in online implementation, and time-consuming in practical application. This makes the theoretical appeal of optimization challenging to exploit in real-world implementation. In this paper, we present a novel online optimal control formulation that aims to address the above-mentioned limitations. The formulation combines a model with measured state information to efficiently find near-optimal feedback controllers. The idea to combine a model with measurements from the actual motion is similar to what is used in model predictive control formulations, with the difference that here the model is not used for future prediction, the optimization is performed along the measured trajectory of the system, and the online computation is reduced to a minimum; it requires a small-scale, one time step, static optimization, instead of a large-scale, finite time horizon, dynamic optimization. The formulation can be used to solve optimal control problems defined with nonlinear cost, nonlinear dynamics, and box-constrained control inputs. Numerical simulations and hardware-in-the-loop experiments demonstrate the effectiveness of the proposed hardware-in-the-loop optimal control approach.

Stiffness modulators are devices that promote a novel means of actuation; they provide stiffness modulation without deliberately doing mechanical work. These type of compliant actuators may be used for human augmentation to complement co-contracted antagonistic muscles and as such reduce muscle activity and metabolic energy cost. Despite the theoretical appeal of this concept, its implementation remains elusive in practical applications. This is particularly true for human augmentation which requires a portable stiffness modulator. In this paper, we present a compact, lightweight, and self-contained stiffness modulator. Using this device, we demonstrate stiffness augmentation of the human knee joint in a sit to stand task. The experimental results indicate that the proposed device is able to assist a human by reducing muscle activity while drawing minimal battery power.

This paper introduces an analytical approach for the design of variable stiffness mechanisms. The basis of this approach is a general model - representing the potential energy function and the physical constraints - covering the design space of variable stiffness mechanisms. Using this model, we present a systematic procedure to analytically define classes of variable stiffness mechanisms from first principles. Consequently, we identify mechanisms capable of infinite range stiffness modulation using bounded motor forces, and define the simplest mathematical model representing mechanisms in this class. A prototype mechanism consistent with this canonical model is designed, fabricated and experimentally tested. The experimental data are consistent with our theoretical predictions showing constant motor force independent of the output deflection and output stiffness when the mechanism is subject to external load.

Compliant actuators have found their place in areas of prosthetics, rehabilitation and robot locomotion because they enable safe human-robot and stable robot-environment interaction, both non-trivial to achieve using conventional rigid actuation. These actuators are capable of varying their equilibrium position and apparent output stiffness in a way humans change the resting position and compliance of their limbs. Just like antagonistically actuated human joints, these actuators require two motor units to provide control over the equilibrium position and the positive joint stiffness. Here we present a novel compliant actuation concept which affords control over the equilibrium position and joint stiffness using a single motor unit. In order to achieve this unconventional functionality, the actuator combines a passive positive feedback (negative stiffness) mechanism with an efficiently tunable negative feedback (positive stiffness) mechanism. This provides a novel design with two distinct operation modes, one leading to unprecedented stiffness tunability, while the other enabling equilibrium point controllability. We present the first practical implementation of this actuator using a prototype prosthetic limb design along with experimental data testifying the range of tunability, covering compliant to rigid behaviour, without paying much on the power input.

Designing robotic systems capable of low-power operation, inherent to their compliant actuation, has been elusive in practical application. In this paper, we propose a physical measure to mathematically define mechanical designs that are suitable to realize stiffness modulation with low power cost. Using this measure, we present a mathematical formulation of an ideal variable stiffness mechanism unaffected by the external load during its operation. We then analyze several existing mechanisms from the literature to relate design features with analytical conditions inherent to low power stiffness modulation in practical designs. Through this analysis, we identify an approximate practical realization of an ideal actuator capable of stiffness modulation with inherently low power cost. Similar to a number of existing efficient variable stiffness mechanisms, this mechanism is able to hold a given stiffness setting with zero input force under no external load. However, unlike many other previously designed mechanisms, it enables infinite range stiffness modulation using finite control forces. A practical variable stiffness mechanism that is capable of infinite range stiffness modulation using finite control forces leads to lower power cost and reduced energy consumption.

Compliant actuators enabling low-power stiffness adaptation are missing ingredients and key enablers of next generation robotic systems. One of the key components of these actuators is the mechanism implementing stiffness adaptation that requires sophisticated control and nontrivial mechanical design. However, despite recent advances in controlling these systems, their design remains experience based and not well understood. In this paper, we present an optimization-based computational framework for the design of intrinsically low-power compliant variable stiffness mechanisms. The core ingredient of this framework is the mathematical formulation of the design problem-provided by a constrained nonlinear parameter optimization-which is computationally solved here to identify optimal variable stiffness designs. We show the basic capability of this formulation in finding parameters for variable stiffness mechanisms that require the least power by design. Further, we demonstrate the generality of this method in cross-comparing mechanisms with different kinematic topology to identify the one that requires the least power by design.

Energy efficient compliant actuation is the missing ingredient and key enabler of next-generation autonomous systems, domestic robots, prosthetic devices, orthotic devices, and wearable exoskeletons, to name a few. For all these devices, one would wish to develop actuators enabling wide range impedance modulation with low energy cost. Using conventional and biologically-inspired compliant actuation, previous research led to functional devices but with high energy cost. Here we introduce a minimalistic compliant actuator to realize impedance modulation with low energy cost. Using this actuator we demonstrate stiffness augmentation in human-machine collaboration. We argue that the non-biologically-inspired actuation concept presented here may effectively complement a biological system, by restoring or extending its functionality, with negligible energy cost.

Anthropomorphic robots that aim to approach human performance agility and efficiency are typically highly redundant not only in their kinematics but also in actuation. Variable-impedance actuators, used to drive many of these devices, are capable of modulating torque and impedance (stiffness and/or damping) simultaneously, continuously, and independently. These actuators are, however, nonlinear and assert numerous constraints, e.g., range, rate, and effort limits on the dynamics. Finding a control strategy that makes use of the intrinsic dynamics and capacity of compliant actuators for such redundant, nonlinear, and constrained systems is nontrivial. In this study, we propose a framework for optimization of torque and impedance profiles in order to maximize task performance, which is tuned to the complex hardware and incorporating real-world actuation constraints. Simulation study and hardware experiments 1) demonstrate the effects of actuation constraints during impedance control, 2) show applicability of the present framework to simultaneous torque and temporal stiffness optimization under constraints that are imposed by real-world actuators, and 3) validate the benefits of the proposed approach under experimental conditions.

This paper presents a comparative study of approaches to control robots with variable impedance actuators (VIAs) in ways that imitate the behavior of humans. We focus on problems where impedance modulation strategies are recorded from human demonstrators for transfer to robotic systems with differing levels of heterogeneity, both in terms of the dynamics and actuation. We categorize three classes of approach that may be applied to this problem, namely, 1) direct, 2) feature-based, and 3) inverse optimal approaches to transfer. While the first is restricted to highly biomorphic plants, the latter two are shown to be sufficiently general to be applied to various VIAs in a way that is independent of the mechanical design. As instantiations of such transfer schemes, 1) a constraint-based method and 2) an apprenticeship learning framework are proposed, and their suitability to different problems in robotic imitation, in terms of efficiency, ease of use, and task performance, is characterized. The approaches are compared in simulation on systems of varying complexity, and robotic experiments are reported for transfer of behavior from human electromyographic data to two different variable passive compliance robotic devices.

Anthropomorphic robots that aim to approach human performance agility and efficiency are typically highly redundant not only in their kinematics but also in actuation. Variable-impedance actuators, used to drive many of these devices, are capable of modulating torque and passive impedance (stiffness and/or damping) simultaneously and independently. Here, we propose a framework for simultaneous optimisation of torque and impedance (stiffness) profiles in order to optimise task performance, tuned to the complex hardware and incorporating real-world constraints. Simulation and hardware experiments validate the viability of this approach to complex, state dependent constraints and demonstrate task performance benefits of optimal temporal impedance modulation.

Considerable research effort has gone into the design of variable passive stiffness actuators (VSAs). A number of different mechanical designs have been proposed, aimed at either a biomorphic (i.e., antagonistic) design, compactness, or simplified modelling and control. In this paper, we propose a (model-based) unified control methodology that is able to exploit the benefits of variable stiffness independent of the specifics of the mechanical design. Our approach is based on forming constraints on commands sent to the VSA to ensure that the equilibrium position and stiffness of the VSA are tracked to the desired values. We outline how our approach can be used for tracking stiffness and equilibrium position both in joint and task space, and how it may be used in the context of constrained local optimal control. In our experiments we illustrate the utility of our approach in the context of online teleoperation, to transfer compliant human behaviour to a variable stiffness device.

This paper presents an approach for the closed-loop control of actuated biped that allows natural looking and energy efficient walking. Rather than prescribe kinematic trajectories or kinematic constraints, the approach is based on the prescription of state dependent torques that ¿encourage¿ patterned movement. Some of the prescribed torques are referenced to the inertial reference frame, which largely decouples the angular dynamics of the robot, and as such greatly simplifies the selection of control parameters. Implementation of torques from the inertial coordinate frames is enabled by a joint torque computation which is motivated by Gauss's principle of least constraint. The proposed approach is implemented in simulation on an anthropomorphic biped, and is shown to quickly converge to a natural looking gait limit cycle. Simulations are conducted with various control parameters and different initial conditions. The authors also show that walking speed can be altered in a simple manner by varying two intuitive controller parameters. The mechanical cost of transport computed on a representative dynamic walk is used to validate energy efficiency of the proposed control approach.

This paper presents an approach for the closed-loop control of a fully actuated biped robot that leverages its natural dynamics when walking. Rather than prescribing kinematic trajectories, the approach proposes a set of state-dependent torques, each of which can be constructed from a combination of low-gain spring-damper couples. Accordingly, the limb motion is determined by interaction of the passive control elements and the natural dynamics of the biped, rather than being dictated by a reference trajectory. In order to implement the proposed approach, the authors develop a model-based transformation from the control torques that are defined in a mixed reference frame to the actuator joint torques. The proposed approach is implemented in simulation on an anthropomorphic biped. The simulated biped is shown to converge to a stable, natural-looking walk from a variety of initial configurations. Based on these simulations, the mechanical cost of transport is computed and shown to be significantly lower than that of trajectory-tracking approaches to biped control, thus validating the ability of the proposed idea to provide efficient dynamic walking. Simulations further demonstrate walking at varying speeds and on varying ground slopes. Finally, controller robustness is demonstrated with respect to forward and backward push-type disturbances and with respect to uncertainty in model parameters.