Neurobionics Lab

Adjustable-Stiffness Nonlinear Spring Testbed


To accurately mimic the behavior of a biological leg, a prosthetic leg like the OSL needs to match not only the nominal behavior of walking when everything goes according to plan, but also the mechanical impedance with which the leg responds to perturbations. To do this, the device must be able to sense both the joint torque it is applying and achieve a stiffness which is much softer than the typical high-gear-ratio transmission naturally provides. Both of these goals are supported by including a series elastic actuator (SEA) in the prosthetic joints. The design of these SEA introduces many tradeoffs: size, weight, complexity, energy efficiency, force sensing resolution, and closed loop impedance rendering quality. To perfect our SEA designs before putting them on a person’s leg, we’ve built a testbench setup that places the spring—in the same form factor as in the OSL—between two OSL motors with representative 50:1 transmissions.


Our approach extends the Open Source Leg’s Selectable Series-Elasticity design, which mates an involute gear-shaft to a collection of spring disks. Each spring disk features an array of inward-pointing flexible beams that engage with the gear teeth to act as a (nearly linear) rotary spring. The amount of series elasticity can be selected by adding or removing spring disks. Our extension aims to re-design the spring disks to allow selection of nonlinear spring behaviors. This is based on two competing effects: when a beam rolls up on the tip of a tooth, the contact force vector tilts towards the radial direction—producing a softening-type nonlinearity; when a beam is initially disengaged from a tooth, the angle where the beam first engages with the tooth marks an abrupt increase in stiffness—a stiffening-type nonlinearity.

Spring-disk design. Involute gear shaft (left), inward-facing flexure (right). Stiffening non-linearities are introduced by designing some of the gear teeth to engage only after an initial deflection.

The springs are designed to, in part, measure joint torque in a feedback controller designed to hide the effects of transmission friction in prosthetic legs.  This transmission friction is problematic because it prevents a person wearing the powered prosthetic leg from swinging the leg forward under natural pendulum dynamics as they typically do with a passive prosthesis. But thanks to the spring’s ability to measure the output torque, we can use a disturbance estimator—a specialized type of linear feedback controller—to measure and directly compensate for the nonlinear friction effects in the transmission.

Block diagram of a delay compensated disturbance observer.

The performance we can expect using this strategy ties back to the design of the spring profile because the measurement noise in the deflection sensing is related to torque noise by the marginal spring rate at any particular deflection. Using our optimization methods, we can design springs that optimally trade off the goals of disturbance rejection and energy savings while guaranteeing robust satisfaction of constraints over a range of activities of daily living. In particular, we hope to exploit stiffening nonlinearity in the spring to attain high torque fidelity at low torques while still satisfying absolute deflection limits at high torques and remaining close to the energy-optimal performance.

Contributors: Gray C. Thomas, Isaac Harris, Elliott Rouse, Negin Nikafrooz, Brandon Peterson

External Collaborators: Robert Gregg, Luke Mooney, Edgar Bolívar-Nieto