Floquet multipliers (orbital stability of a cyclic movement)

Quantifies how quickly perturbations to a limit-cycle movement (e.g. gait) decay from one stride to the next. At each phase of the cycle a linear Poincare (return) map is fitted between the SAME phase in consecutive strides (dS_{n+1} = J dS_n); the Floquet multipliers are its eigenvalues. The maximum modulus < 1 means the movement is orbitally stable (perturbations shrink stride to stride).

Usage

floquetMultipliers(states)

Arguments

states: Either a [stride, phase, state] array, or a list of phase x state matrices (one per stride). Strides must be consecutive and sampled at the same phases; the state is a vector (e.g. joint angles and velocities).

Value

a floquet_result: max_multiplier (largest modulus over phases), multipliers (eigenvalues of the section return map), per_phase_max (the Dingwell per-phase maximum multiplier) and orbitally_stable.

References

Hurmuzlu Y, Basdogan C (1994) J Biomech Eng 116:30-36; Dingwell JB, Kang HG (2007) J Biomech Eng 129:586-593.

Examples

set.seed(1)
A <- matrix(c(0.5, 0.1, 0, 0.3), 2, 2)             # stride-to-stride return map
P <- 4; nS <- 80; strides <- array(0, c(nS, P, 2)); d <- rnorm(2)
cyc <- cbind(sin(1:P), cos(1:P))
for (n in 1:nS) { for (k in 1:P) strides[n, k, ] <- cyc[k, ] + d
  d <- as.numeric(A %*% d) + rnorm(2, 0, 0.01) }
floquetMultipliers(strides)$max_multiplier          # ~ 0.5 (< 1, stable)
#> [1] 0.4317407