Balancing on tightropes and slacklines

Abstract

Balancing on a tightrope or a slackline is an example of a neuromechanical task where the whole body both drives and responds to the dynamics of the external environment, often on multiple timescales. Motivated by a range of neurophysiological observations, here we formulate a minimal model for this system and use optimal control theory to design a strategy for maintaining an upright position. Our analysis of the open and closed-loop dynamics shows the existence of an optimal rope sag where balancing requires minimal effort, consistent with qualitative observations and suggestive of strategies for optimizing balancing performance while standing and walking. Our consideration of the effects of nonlinearities, potential parameter coupling and delays on the overall performance shows that although these factors change the results quantitatively, the existence of an optimal strategy persists.

Figures

Figure 1.

(a) Schematic of a man balancing on a rope and (b) associated mechanical model describing the transverse motion on the plane . The linearized equations of motion for this system are reported in (3.1) and (3.2). (c) Spectrum of the linearized system defined in (3.1) and (3.2) as a function of R (top). The circular markers correspond to . The markers for the imaginary eigenvalues for R = 0 is missing because as R → 0. (Online version in colour.)

Figure 2.

(a) Schematics of the balancing control loop. (b) Condition number of the controllability matrix (solid) and observability matrix (dashed) as a function of normalized radius R. (c) Influence of the normalized radius R on time to steady state (i), maximum control torque (ii), stabilization S (iii) and energy spent for control U (iv). Data obtained by simulating the full nonlinear model (A 3) and (A 4) with and LQG controller based on measurement of (see §3.2). (Online version in colour.)

Figure 3.

Total control energy spent for stabilization as a function of system parameters m and R. Data obtained by simulating the full nonlinear model (A 3) and (A 4) and (a) LQG controller based on measurement of (solid line, m = 0.1; dotted line, m = 1; dashed line, m = 30; see §3.2) and (b) LQ controller with full state information (appendix B). For comparison, we also plot the total control energy spent for stabilization as a function of parameters R and γ. Data obtained by simulating the full nonlinear model (A 3) and (A 4) with and (c) LQG controller based on measurement of (solid line, ; dotted line, ; dashed line, ; see §3.2) and (d) LQ controller with full state information (appendix C). (Online version in colour.)

Figure 4.

(a) Mechanical model for the motion on a tightrope with large pretension. The linearized equation of motion are reported in (5.2) and (5.3). (b) Total energy spent for stabilization as a function of spring stiffness k. Data obtained by simulating the full nonlinear model with LQG controller based on measurement of . (Online version in colour.)

Figure 5.

(a) LQG control system structure in presence of feedback delay . (b) Stability region and (c) controllability matrix condition number as a function of R and . Data obtained by simulating the full nonlinear model (A 3) and (A 4) with m = 30 and LQG controller based on measurement of . The white region corresponds to unstable dynamics, whereas the grey levels in the stability region code for the total amount of energy U spent for stabilizing the system. (d) Stability region as a function of R and with as in §5.1.

Figure 6.

Influence of the normalized radius R on time to (a) steady-state , (b) maximum control torque , (c) stabilization success and (d) energy spent for control U. Data obtained by simulating the full nonlinear model (A 3) and (A 4) with m = 30 and LQG controller (solid) or LQ controller (dashed). (Online version in colour.)