Fig. 1 LRA motors and schematic: ( a ) LRA with/without housing and ( b ) LRA cross section illustration More

Fig. 2 Measured oscillation frequencies of an LRA at different spring elongation positions More

Fig. 3 LRA schematic of coil (black rectangle), spring, and mass (gray rectangle) More

Fig. 4 Electromagnetic model: magnetic force imparted to the mass as a function of normalized spring elongation ( δ ) for different normalized mass offset positions ( ρ ) More

Fig. 5 LRA in force fields with various directions and gradients: ( a ) homogeneous field, ( b )field with positive gradient, ( c ) field with negative gradient, ( d ) field with positive gradient, negative direction, and ( e ) field with negative gradient, negative direction More

Fig. 6 Normalized natural frequency for ρ = 0.2 as a function of δ ¯ : this plot shows how the actual natural frequency of the LRA can vary with equilibrium position. The six black dots correspond to the six example LRA configurations illustrated in Fig. 7 . More

Fig. 7 Example LRA configurations: black bar represents the coil. Gray bar represents the mass. Dotted line represents the position of the mass when the spring is unstretched. More

Fig. 8 Normalized natural frequency for ρ = 0.2 as a function of V ¯ : This plot shows how the actual natural frequency of the LRA can vary with steady-state voltage More

Fig. 9 Excitation profiles used in experiments More

Fig. 10 Hardware used for experiments More

Fig. 11 Hardware connection diagram More

Fig. 12 Acceleration response from LRA. Response over entire 2-s duration shown in ( a ). Response between t  = 0.38 and t  = 0.46 shown in ( b ). ( a ) LRA acceleration response. ( b ) LRA acceleration response to voltage step from 66% to 100% of maximum voltage. More

Fig. 13 Experimental and model LRA frequencies across a range of equilibrium points: ( a ) LRA effective frequency—experimental results and ( b ) normalized natural frequency for ρ = 0.2 as a function of V ¯ ; zoomed-in portion showing portion of segments 4 and 5 More

Fig. 1 Schematic diagram of a two-link robot manipulator More

Fig. 2 Block diagram for simple discrete-time UDE-based controller More

Fig. 3 Block diagram for discrete-time UDE-based controller–observer structure More