Energy efficient control of a 1D hopper through tunable damping Gorkem Secer 1 , Uluc Saranli 2 1,2 Dept. of Computer Engineering, Middle East Technical University, Turkey 1 gorkem.secer@ceng.metu.edu.tr Introduction In this paper, we report on accurate and efficient deadbeat control of a serially-actuated vertical hopper platform. Our control approach is based on virtually tuning the damping co- efficient of the elastic leg by controlling the ground reaction force through the position of the series actuator. In previous work, we demonstrated on a simulated vertical hopper that modulating leg damping performs considerably better than changing leg stiffness during stance, a commonly used method for controlling energy for the Spring-Loaded In- verted Pendulum (SLIP) model [3]. In this context, our find- ings indicated that tuning the damping coefficient instead of the stiffness i) substantially reduces actuation power require- ments, ii) achieves more than four-fold increase in positive actuator work, and iii) achieves more accurate and agile dead- beat control performance since it preserves the accuracy of analytic approximations to the passive SLIP dynamics during stance due to less aggressive actuator usage. In subsequent work, we have extended these controllers based on the virtual modulation of leg damping to planar hopping and proposed a hierarchical template/anchor framework to realize them on platforms with more complex dynamics [4]. In this paper, our focus is mainly on the experimental valida- tion of our simulation results summarized above for damping- based deadbeat control of vertically constrained hopping. To this end, we conducted extensive experiments, yielding re- sults that were in agreement with our simulation results. We have also extended our proposed control approach, maximiz- ing performance by eliminating negative work altogether and achieving more effective embedding of SLIP dynamics. Control of Platform Our hopper platform consists of a vertically constrained mass, connected serially to a pair of helical linear springs through a ball screw actuated with a brushless DC motor as shown in Fig. 2. Kinematic and dynamic parameters of the plat- form were found through system identification experiments, yielding the spring rest length l 0 = 0.2m, unsprung distance of COM to the toe r 0 = 0.375m, the body mass m b = 3.81kg, the actuator mass m a = 1.01kg, the toe mass m t = 0.7kg, the spring stiffness k p = 6200N/m, radial damping coefficient of the leg d p = 3.75Ns/m, and the vertical damping of the linear guide d f = 1.5Ns/m. As suggested in our earlier work [4], we consider an extended version of the SLIP model (SLIP+) shown in Fig. 2 as a tem- plate model for hopping. SLIP+ has three adjustable leg pa- rameters, the spring constant k, the damping coefficient d and Figure 1: Vertically constrained hopper platform with series elastic actiation. a constant force f in parallel with the spring. We model the control problem as a once-per-step selection of these parame- ters for SLIP+. This approach can be formalized as the single- step deadbeat control problem [k i , d i , f i ]= arg min [k,d, f ] ||z ∗ − P(z k , k, d , f )|| 2 , where z ∗ denotes the desired height of the hopper at the apex of a jump, and P(z k , k, d , f ) denotes the apex Poincar´ e return map for SLIP+ consisting of the composition of flight and stance maps. Analytic approximations to this return map al- low effective implementation of this controller [2]. We re- strict k = k p for tunable damping control by following our previous approach. On the other hand, traditional variable stiffness control can be obtained by restricting d = d p and f = 0. After choosing virtual leg parameters, we map the de- sired template to the physical anchor, and realize its vector field on the Center-Of-Mass coordinates for the platform by controlling the position of the series elastic actuator to follow u ∗ = m b + m a m b  d p − d k p ˙ r + k p − k k p (r − r 0 ) − f k p  . (1) Unfortunately, traditional linear controllers cannot achieve this position control task with sufficient accuracy due to non- linearities and low mechanical transparency of our actuator. To this end, we use a robust 2DOF controller based on [1], consisting of three components: a feedforward friction and spring force compensator, a disturbance observer based on velocity estimations, and a PD position feedback controller.