Прямая задача: ошибка отображения «углы →\to→ поза» на тестовой выборке и время счёта

Список литературы

параметры

Определение модели

Численное интегрирование

Вычисление правой части

Приведение к системе первого порядка

Упругие опоры. Матрицы демпфирования C и жёсткости K

Матрица масс M

Введение

параметры

При равных дисбаланса

Суммарный кинетический момент двух роторов из

ля роторов с тройной симметрией лопастей тензоры Jup и Jdown не зависят от угла поворота.

Матрица масс зависит от времени только через фазу вращения роторов α(t), п

Для учёта различного положения центров масс тел относительно точки, к которой приводится тензор, используется теорема Гюйгенса–Штейнера.

блочную структуру

Для роторов rrot,i=pi+ci, где pi — положение системы координат ротора, а ci — положение его центра масс в этой системе.

Динамика системы описывается линейной матричной моделью второго порядка

бщая для обоих роторов угловая скорость вращения по модулю в момент времени t,

величина дисбалансирующей массы

диагональными по осям X, Y, Z:

упруго-вязких

еремещения центра вращения qt и малые угловые отклонения системы qr

φ, θ, ψ

Координаты вектора x, y, z

𝐂

where

The scheme investigated here is single-stage and does not require inner iterations, so the cost of a step is fixed and predictable, which makes it especially interesting for hardware-in-the-loop (HIL) applications where execution time is more important than accuracy.

A- and L-stability. The L-stability of the scheme means that the infinitely stiff decaying modes are suppressed entirely, so that the stiffness of the constraints does not restrict the step size from the standpoint of stability.

The physical meaning of the multipliers was checked statically: at a constant prescribed crank angle the mechanism must be at rest, and the actuator multiplier must equal the derivative of the potential energy with respect to the crank angle, dU/dθ, taken along the constraint manifold (gravity plus the energy of the spring). The configuration for each angle is taken from the exact analytical solution of the loop geometry, and dU/dθ is computed by automatic differentiation of the same potential energy along that solution; the states are indeed static — the maximal acceleration in them does not exceed 2.8⋅10−14 m/s2. The relative discrepancy between the actuator multiplier and dU/dθ is 2.9⋅10−15 over the range θ∈[0.4,2.2] rad.

This reference shares no code with the CROS core: it uses analytically eliminated coordinates rather than numerical projection onto the null space of the constraint Jacobian.

Constraint residual. The closure error of a hinge — the distance between the hinge point computed from one link and from the other — is evaluated from the raw state without using the multipliers, over t∈[0,20] s. The attachment to ground is satisfied to machine precision: its constraint rows contain only identity blocks, so the solution of the linear system reproduces them exactly. On halving the step the closure error of the hinges decreases by factors of 3.44, 3.83, 3.91 and 3.99, that is, the ratios converge to 4 — the signature of second order. The closure error is not equal to zero, so the scheme does not enforce the constraint exactly; it enforces it to the accuracy of the discretization, and the achievable geometric tolerance is determined by the step size.

and the resulting index-2 or index-1 system is discretized directly.

In the figures these variants are denoted as index 1, index 1 + stabilization and index 1 + projection, and the direct form as index 3.

using StaticArrays.jl; the figures were produced with Makie.jl.

index 3: {Mq¨=Q(q,q˙,t)+CT(q)λ,Φ(q,t)=0, where

The scheme is A- and L-stable, so it tolerates the stiff constraint dynamics that force explicit index-reduced integrators to very small steps.