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Natural dynamics, nonlinear optimization, and, more recently, convex optimization are available methods for stiffness design of energy-efficient series elastic actuators. Natural dynamics and general nonlinear optimization only work for a limited set of load kinetics and kinematics, cannot guarantee convergence to a global optimum, or depend on initial conditions to the numerical solver. Convex programs alleviate these limitations and allow a global solution in polynomial time, which is useful when the space of optimization variables grows (e.g., when designing optimal nonlinear springs or co-designing spring, controller, and reference trajectories). Our previous work introduced the stiffness design of series elastic actuators via convex optimization when the transmission dynamics are negligible, which is an assumption that applies mostly in theory or when the actuator uses a direct or quasi-direct drive. In this work, we extend our analysis to include friction at the transmission. Coulomb friction at the transmission results in a non-convex expression for the energy dissipated as heat, but we illustrate a convex approximation for stiffness design. We experimentally validated our framework using a series elastic actuator with specifications similar to the knee joint of the Open Source Leg, an open-source robotic knee-ankle prosthesis. 

This paper presents a new model and phase-variable controller for sit-to-stand motion in above-knee amputees. The model captures the effect of work done by the sound side and residual limb on the prosthesis, while modeling only the prosthetic knee and ankle with a healthy hip joint that connects the thigh to the torso. The controller is parametrized by a biomechanical phase variable rather than time and is analyzed in simulation using the model. We show that this controller performs well with minimal tuning, under a range of realistic initial conditions and biological parameters such as height and body mass. The controller generates kinematic trajectories that are comparable to experimentally observed trajectories in non-amputees. Furthermore, the torques commanded by the controller are consistent with torque profiles and peak values of normative human sit-to-stand motion. Rise times measured in simulation and in non-amputee experiments are also similar. Finally, we compare the presented controller with a baseline proportional-derivative controller demonstrating the advantages of the phase-based design over a set-point based design. 

Recently, it has been shown that light-weight, passive, ankle exoskeletons with spring-based energy store-and-release mechanisms can reduce the muscular effort of human walking. The stiffness of the spring in such a device must be properly tuned in order to minimize the muscular effort. However, this muscular effort changes for different locomotion conditions (e.g., walking speed), causing the optimal spring stiffness to vary as well. Existing passive exoskeletons have a fixed stiffness during operation, preventing it from responding to changes in walking conditions. Thus, there is a need of a device and auto-tuning algorithm that minimizes the muscular effort across different walking conditions, while preserving the advantages of passive exoskeletons. In this letter, we developed a quasi-passive ankle exoskeleton with a variable stiffness mechanism capable of self-tuning. As the relationship between the muscular effort and the optimal spring stiffness across different walking speeds is not known a priori, a model-free, discrete-time extremum seeking control (ESC) algorithm was implemented for real-time optimization of spring stiffness. Experiments with an able-bodied subject demonstrate that as the walking speed of the user changes, ESC automatically tunes the torsional stiffness about the ankle joint. The average RMS EMG readings of tibialis anterior and soleus muscles at slow walking speed decreased by 26.48% and 7.42%, respectively. 

Compared to rigid actuators, Series Elastic Actuators (SEAs) offer a potential reduction of motor energy consumption and peak power, though these benefits are highly dependent on the design of the torque-elongation profile of the elastic element. In the case of linear springs, natural dynamics is a traditional method for this design, but it has two major limitations: arbitrary load trajectories are difficult or impossible to analyze and it does not consider actuator constraints. Parametric optimization is also a popular design method that addresses these limitations, but solutions are only optimal within the space of the parameters. To overcome these limitations, we propose a non-parametric convex optimization program for the design of the nonlinear elastic element that minimizes energy consumption and peak power for an arbitrary periodic reference trajectory. To obtain convexity, we introduce a convex approximation to the expression of peak power; energy consumption is shown to be convex without approximation. The combination of peak power and energy consumption in the cost function leads to a multiobjective convex optimization framework that comprises the main contribution of this paper. As a case study, we recover the elongation-torque profile of a cubic spring, given its natural oscillation as the reference load. We then design nonlinear SEAs for an ankle prosthesis that minimize energy consumption and peak power for different trajectories and extend the range of achievable tasks when subject to actuator constraints.