Fig 1.
(A) The participants rotate a haptic joystick that maps 1:1 to the position θ of the red virtual motor link on the screen. This link is coupled to the first link l1 of the virtual compliant double pendulum by a spring k1. Moving the joystick, and thus the virtual motor link, causes a deflection of the spring, which in turn induces the pendulum motion. The spring torque τ is reflected to the user as force feedback, rendered in green. The task is to hit two colored target balls with the second link endpoint (indicated by the star shape) as often as possible in 40 s. When the target is hit, the participant is rewarded with a point. If the pendulum does not reach the target or swings through it, no point is added to the score. (B) Participants can freely arrange themselves in front of the haptic joystick and position their arm to their preference. The joystick and motor link are both arranged to point to the left. (C) Three pendulum configurations are tested in the experiments, where the equilibrium position of the spring between the first and second link differs as indicated by the colored arrows: P0: qeq = (0, 0)°, P90: qeq = (0, 90)°, P45: qeq = (0, 45)°.
Table 1.
Overview of the evaluated metrics for the P0 configuration.
Table 2.
Overview of the evaluated metrics for the P90 configuration.
Fig 2.
Differently excited pendulum motions.
Compared are the pendulum motions excited by the baseline strategies BL1–3 with the system’s conservative NNM (solid black) and the averaged participant data (dashed black) for the (A) P0 and (B) P90 pendulum configuration. The resulting motions are shown in Cartesian space (left) and joint space, comparing positions (middle) and velocities (right).
Fig 3.
Overlay of excited pendulum trajectories in Exp. 1.
Each light-colored line corresponds to the trial average of an individual participant with (A) P0 and (B) P90. The left plot visualizes the applied joystick motion, i.e., motor link θ (red) and first pendulum link q1 (blue) over one period (target to target) comparing the averaged participant data (dashed) to BL1 (solid). The middle and right plots show the respective trajectories in position and velocity space compared to the NNM trajectory.
Fig 4.
Trend comparison of individual performance metrics.
The individually applied phase lag between the input motor link and the output pendulum link per participant are correlated with (A) the obtained hit score and (B) mode metric values. (C) Sorting the phase lag values per participant from smallest to largest reveals that participants remained consistent with their chosen strategy for the two tested pendula.
Fig 5.
Pendulum trajectories of Exp. 2 with decreased target radius (↓ rt).
Light-colored lines show the averaged trials per individual participant and the overall averages are dashed for (A) P0 and (B) P90. The left plots compare the red motor link motion θ to the first pendulum link q1 (blue) per period. Middle and right plots compare the respective trajectories in position and velocity space with the NNM.
Fig 6.
Pendulum trajectories of Exp. 2 with increased link mass (↑ m2).
Light-colored lines show the averaged trials per individual participant and dashed are the overall averages for (A) P0 and (B) P90. The left plots compare the red motor link motion θ to the first pendulum link q1 (blue) per period. Middle and right plots compare the respective trajectories in position and velocity space with the NNM.
Table 3.
Overview of the evaluated metrics for the P45 configuration.
Fig 7.
Excited pendulum trajectories of Exp. 3 with the P45 configuration.
Light-colored lines show the average data over all trials per individual participant. The left plots compare the red motor link motion θ to the upper pendulum link coordinate q1 (blue) cut by period (target to target). The middle and right plots show the respective trajectories in position and velocity space, compared to the NNM of the original pendulum system (black solid) and a system where the upper spring stiffness was altered to k1 = 6 N m rad−1 (yellow).
Fig 8.
Overview of pendulum kinematics and NNMs.
(A) Schematic simple pendulum with the link-centered mass m1 and a rotational spring with stiffness k1 in its origin. The spring equilibrium is set to the origin (dotted line) of the link angle q1 (qeq = 0). Adding a second link with mass m2 through another spring of stiffness k2 leads to a double pendulum, where the second link angle q2 is defined relative to the first link. While the spring equilibrium at the first link always remains zero, different pendulum configurations can be defined by changing the equilibrium position of the second spring to either describe (B) a fully extended pendulum P0 with qeq = (0, 0)° or (C) a flexed pendulum P90 with . (D) Computing the nonlinear normal modes (NNMs) for the double pendulum configuration P90 reveals two modes, M1 and M2. A generator collects the turning points for the brake orbit oscillation on each energy level. The specific trajectory of the first mode for an energy level of E = 2.5 J is highlighted as an example. (E) Increasing energy changes the period times for both modes of the considered double pendulum system.
Table 4.
Parameter values of experimental double pendulum configurations.
Only for a variation of Experiment 2, the second link mass m2 was increased to 0.625 kg.
Fig 9.
Computed brake orbit trajectories for an energy level of 2.5 J.
The orbits are shown in Cartesian space (left) and joint space (right) for double pendulum configurations (A) P0) and (B) P90. These brake orbits were used as a reference in the human user study.
Fig 10.
Control scheme of the experimental setup.
Participants command the virtual double pendulum implemented in Gazebo through a joystick mapping 1:1 to the position θ of a motor link (red). Connected by a spring, this moves the first pendulum link (q1), reflecting the spring forces τ to users as haptic feedback through the joystick.
Fig 11.
Characterization of pendulum responses and reachable space.
(A,B) To characterize the system responses to different input frequencies, the motor link position θ was commanded sine waves with different frequency values ω in simulation. (C,D) To outline the reachable space of the pendulum systems, the motor link was commanded a sine wave, where the amplitude and frequency were pseudo-randomly changed every 0.1 s. (A,C) show the P0 and (B,D) the P90 configuration. Respective plots on the left depict the system responses in Cartesian space, while the right plots show the corresponding joint space trajectories with the NNM in black (E = 2.5 J).
Table 5.
Applied sine wave frequencies ω to characterize the P0 and P90 pendulum response with a sweep.
For each frequency, the sine amplitude A of the handle motion was manually tuned to reach a set deflection of the first link (q1 ≈ 1 rad ≈ 60°).
Fig 12.
Computed NNM and target placement for the P45 configuration.
(A) Computed NNMs of P45 with the initial parameters from Table 4 in position joint (top) and Cartesian space (bottom). (B) P45 target locations (yellow) arranged between the targets of P0 (blue) and P90 (orange). (C) Comparison of the computed NNM for the original P45 system (cyan) and the same pendulum with an adjusted first spring stiffness k1 (yellow).