Rible.jl

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Quaternion Coordinates — RibleQCF ​

RibleQCF is an extension package that adds quaternion-based coordinates (QC) as an alternative to the default Natural Coordinate Formulation (NCF). A single rigid body is described by 7 generalized coordinates — 3 for translation and 4 for a unit quaternion — yielding 6 degrees of freedom after the unit-norm constraint is enforced. The inertia representation and its numerical influence follow the framework established in [2] and [3].

Tip

QCF produces a non-constant mass matrix, which requires recomputations during updates and has involved jacobian formulations.

Coordinate Type QC ​

The core type QCF.QC stores mass and a diagonalized inertia tensor along with several precomputed scalars derived from them:

julia
struct QC{T,JT} <: AbstractCoordinates{3,T}
    m      # mass
    m⁻¹    # 1/mass
    γ      # max diagonal of inertia (regularization parameter)
    Jγ     # J - γ·I
    γ⁻¹    # 1/γ
    J⁻¹γ   # J⁻¹ - (1/γ)·I
end

Construct a QC instance by passing mass and a 3×3 inertia matrix:

julia
m = 0.5
J = Diagonal(SA[0.001, 0.001, 0.002])

qc = QCF.QC(m, J)

Only the diagonal of the inertia matrix is used; off-diagonal terms are discarded internally. The optional keyword γ defaults to maximum(diag(J)) and acts as a regularization parameter that keeps the rotational mass matrix well-conditioned.

Checking the coordinate trait

You can query whether a body uses constant-mass coordinates at any level:

julia
has_constant_mass_matrix(body.coords)   # Val(false) for QC
has_constant_mass_matrix(body)          # propagates from coords
has_constant_mass_matrix(structure)     # propagates from bodies

Nonconstant Mass Matrix ​

Unlike NCF (where has_constant_mass_matrix returns Val(true)), the QCF generalized mass matrix depends on the current quaternion state:

julia
has_constant_mass_matrix(::QCF.QC) = Val(false)

The 7×7 mass matrix is block-diagonal:

where is the 4×4 left-quaternion-multiplication matrix. Because changes every step, the solver must recompute and refactorize it at each iteration. Rible handles this automatically through its solver dispatch: when it detects Val(false), it selects the Zhong06_Nonconstant_Mass solver family instead of the constant-mass family.

Practical implications ​

  • Cost per step is higher than NCF due to per-step mass-matrix assembly and factorization. For single-body simulations the difference is negligible; for large multi-body systems with many QCF bodies, consider whether NCF suffices.

  • Tight tolerances help. The quaternion constraint   drifts if the solver tolerances are loose. Using ftol = 1e-14 and maxiters = 50 is a good default.

Intrinsic Constraint ​

QC carries one intrinsic algebraic constraint — the unit quaternion norm:

This constraint is enforced automatically by the solver alongside any joint constraints. The framework provides the constraint function, its Jacobian, and the force-Jacobian in constraints.jl.

Complete Example: Spinning Top with QCF ​

The bundled spinning-top model supports both NCF and QCF via the cT argument. Below we construct a QCF top, verify its nonconstant-mass trait, and run a full contact simulation.

Construction ​

julia
include(joinpath(pathof(Rible), "../../examples/robots/spinningtop.jl"))

origin_position = [0.0, 0.0, 0.5]
R = RotX(0.0)
origin_velocity = [1.0, 0.0, 0.0]
Ω = [0.0, 0.0, 200.0]

top = make_top(origin_position, R, origin_velocity, Ω, :QCF; μ = 0.95, e = 0.5, loadmesh = true)
Robot{Structure{TypeSortedCollections.TypeSortedCollection{Tuple{Vector{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}}}, 1}, Vector{Int64}, Connectivity{Vector{Vector{Int64}}}, Rible.StructureState{CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}, Vector{CoordinatesState{Float64, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, Vector{Float64}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}}}}, @NamedTuple{activated_bits::BitVector, persistent_bits::BitVector, friction_coefficients::Vector{Float64}, restitution_coefficients::Vector{Float64}, gaps::Vector{Float64}}, Rible.StructureCache{InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseVector{Float64, Int64}}}}, ControlHub{Vector{Int64}, TypeSortedCollections.TypeSortedCollection{Tuple{Vector{ErrorGauge{Signifier{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}}, PositionCaptum, Vector{Float64}}}}, 1}, TypeSortedCollections.TypeSortedCollection{Tuple{Vector{ExternalForceActuator{Signifier{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}}, NaiveOperator, Vector{Float64}, Float64}}}, 1}, Coalition, @NamedTuple{c::Vector{Float64}, e::Vector{Float64}, u::Vector{Float64}}}, StructArrays.StructVector{CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}, @NamedTuple{t::Vector{Float64}, q::Vector{Vector{Float64}}, q̇::Vector{Vector{Float64}}, q̈::Vector{Vector{Float64}}, F::Vector{Vector{Float64}}, λ::Vector{Vector{Float64}}, s::Vector{Vector{Float64}}, p::Vector{Vector{Float64}}, c::Vector{Vector{Float64}}}, Int64}, Vector{Vector{Rible.Contact{Float64}}}, Vector{@NamedTuple{na::Int64, bodyid2act_idx::Vector{Vector{Int64}}, persistent_idx::Vector{Int64}, activated_bits::BitVector, H::LinearAlgebra.Diagonal{Float64, Vector{Float64}}, activated_restitution_coefficients::Vector{Float64}, D::Matrix{Float64}, Dper::Matrix{Float64}, Dimp::Matrix{Float64}, ∂Dq̇∂q::Matrix{Float64}, ∂DᵀΛ∂q::Matrix{Float64}, ŕ::Vector{Float64}, L::Matrix{Float64}, Lv::Matrix{Float64}, Λ::Vector{Float64}, Γ::Vector{Float64}}}, StructArrays.StructVector{@NamedTuple{c::Vector{Float64}, e::Vector{Float64}, u::Vector{Float64}}, @NamedTuple{c::Vector{Vector{Float64}}, e::Vector{Vector{Float64}}, u::Vector{Vector{Float64}}}, Int64}}(Structure{TypeSortedCollections.TypeSortedCollection{Tuple{Vector{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, 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Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5)]), RigidBodyState{3, 3, Float64, 9}(Rible.CartesianFrame{3, 3, Float64, 9}([0.0, 0.0, 0.5], Rible.Axes{3, Float64, 9}([1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]), Rible.LocusState{3, 3, Float64, 9}(Rible.CartesianFrame{3, 3, Float64, 9}([0.0, 0.0, 0.5], Rible.Axes{3, Float64, 9}([1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0], Rible.ContactState{3, Float64, 9}(false, true, Inf, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]), [1.0, 0.0, 0.0]), Rible.LocusState{3, 3, Float64, 9}[Rible.LocusState{3, 3, Float64, 9}(Rible.CartesianFrame{3, 3, Float64, 9}([0.03111269837220809, 0.03111269837220809, 0.5], Rible.Axes{3, Float64, 9}([0.0 0.0 1.0; 1.0 0.0 0.0; 0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]), [0.0, 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Signifier{Int64}(1, 2), Signifier{Int64}(1, 3), Signifier{Int64}(1, 4), Signifier{Int64}(1, 5), Signifier{Int64}(1, 6)], [[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12], [13, 14, 15], [16, 17, 18]], [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], 7, 18, 18, 0, 18, Vector{Int64}[], Vector{Int64}[]), Rible.StructureState{CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}, Vector{CoordinatesState{Float64, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, Vector{Float64}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}}}}(CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}(0.0, [0.0, 0.0, 0.5, 1.0, 0.0, 0.0, 0.0], [1.0, 0.0, 0.0, -0.0, 0.0, 0.0, 100.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0], Float64[], [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 100.0], [0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.0, 0.0, -0.03795882, 0.0, 0.0, 0.0]), CoordinatesState{Float64, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, Vector{Float64}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}}[CoordinatesState{Float64, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}, Vector{Float64}, SubArray{Float64, 1, Vector{Float64}, Tuple{Vector{Int64}}, false}}(0.0, [0.0, 0.0, 0.5, 1.0, 0.0, 0.0, 0.0], [1.0, 0.0, 0.0, -0.0, 0.0, 0.0, 100.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0], Float64[], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.0, 0.0, -0.03795882, 0.0, 0.0, 0.0])]), (activated_bits = Bool[0, 0, 0, 0, 0, 0], persistent_bits = Bool[0, 0, 0, 0, 0, 0], friction_coefficients = [0.95, 0.95, 0.95, 0.95, 0.95, 0.95], restitution_coefficients = [0.5, 0.5, 0.5, 0.5, 0.5, 0.5], gaps = [Inf, Inf, Inf, Inf, Inf, Inf]), Rible.StructureCache{InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseVector{Float64, Int64}}}(InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseVector{Float64, Int64}}(sparse([1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7], [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], 7, 7), sparse([1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7], [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], 7, 7), sparse(Int64[], Int64[], Float64[], 7, 7), sparse(Int64[], Int64[], Float64[], 7, 7), sparsevec(Int64[], Float64[], 7), sparsevec(Int64[], Float64[], 7), true))), ControlHub{Vector{Int64}, TypeSortedCollections.TypeSortedCollection{Tuple{Vector{ErrorGauge{Signifier{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}}, PositionCaptum, Vector{Float64}}}}, 1}, TypeSortedCollections.TypeSortedCollection{Tuple{Vector{ExternalForceActuator{Signifier{RigidBody{3, 3, Float64, 9, RibleQCF.QC{Float64, LinearAlgebra.Diagonal{Float64, StaticArraysCore.SVector{3, Float64}}}, RigidBodyCache{StaticArraysCore.MMatrix{3, 7, Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}}, NaiveOperator, Vector{Float64}, Float64}}}, 1}, Coalition, @NamedTuple{c::Vector{Float64}, e::Vector{Float64}, u::Vector{Float64}}}(Int64[], TypeSortedCollections.TypeSortedCollection{Tuple{Vector{ErrorGauge{Signifier{RigidBody{3, 3, Float64, 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Float64, 21}, InertiaCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, StaticArraysCore.MVector{7, Float64}}, Nothing}, GeometryBasics.Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}, (:position, :normal, :color), Tuple{Vector{GeometryBasics.Point{3, Float64}}, Vector{GeometryBasics.Vec{3, Float32}}, Vector{ColorTypes.RGB{Float32}}}, Vector{GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}}}(RigidBodyProperty{3, Float64, 9}(true, true, 1, :generic, 0.5838707, [0.00022129 0.0 0.0; 0.0 0.00022129 0.0; 0.0 0.0 0.00030207], Locus{3, Float64, 9}([0.0, 0.0, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.0, 0.0), Locus{3, Float64, 9}[Locus{3, Float64, 9}([0.03111269837220809, 0.03111269837220809, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5), Locus{3, Float64, 9}([0.03111269837220809, 0.03111269837220809, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5), Locus{3, Float64, 9}([0.03111269837220809, 0.03111269837220809, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5), Locus{3, Float64, 9}([0.03111269837220809, 0.03111269837220809, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5), Locus{3, Float64, 9}([0.0, 0.0, -0.03795882], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5), Locus{3, Float64, 9}([0.0, 0.0, 0.0], Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), 0.95, 0.5)]), RigidBodyState{3, 3, Float64, 9}(Rible.CartesianFrame{3, 3, Float64, 9}([0.0, 0.0, 0.5], Rible.Axes{3, Float64, 9}([1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]), Rible.LocusState{3, 3, Float64, 9}(Rible.CartesianFrame{3, 3, Float64, 9}([0.0, 0.0, 0.5], Rible.Axes{3, Float64, 9}([1.0 0.0 0.0; 0.0 1.0 0.0; 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4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7], [1.7127079677058636, 1.7127079677058636, 1.7127079677058636, 827.6227364518159, 1129.7392561796737, 1129.7392561796737, 827.6227364518159], 7, 7), sparse([4, 5, 6, 7, 4, 5, 6, 7, 4, 5, 6, 7, 4, 5, 6, 7], [4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7], [-0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0], 7, 7), sparse([4, 5, 6, 7, 4, 5, 6, 7, 4, 5, 6, 7, 4, 5, 6, 7], [4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7], [-0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0], 7, 7), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, -0.0, 0.0], true), nothing), Mesh{3, Float64, GeometryBasics.NgonFace{3, GeometryBasics.OffsetInteger{-1, UInt32}}}(...)), 5), NaiveOperator(1), [0.0, 1.0, 0.0], [0.0])],), ([1],)), Coalition(1, 0, 1, 0, 1, Vector{Int64}[], [[1]], 1, [[1]]), (c = Float64[], e = [0.16999999999999998], u = [0.0])), CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}[CoordinatesState{Float64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}(0.0, [0.0, 0.0, 0.5, 1.0, 0.0, 0.0, 0.0], [1.0, 0.0, 0.0, -0.0, 0.0, 0.0, 100.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0], Float64[], [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 100.0], [0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.03111269837220809, 0.03111269837220809, 0.0, 0.0, 0.0, -0.03795882, 0.0, 0.0, 0.0])], Vector{Rible.Contact{Float64}}[[Rible.Contact{Float64}(1, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0])), Rible.Contact{Float64}(2, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0])), Rible.Contact{Float64}(3, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0])), Rible.Contact{Float64}(4, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0])), Rible.Contact{Float64}(5, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0])), Rible.Contact{Float64}(6, 0.95, 0.5, Rible.ContactState{3, Float64, 9}(false, true, 1.0, Rible.Axes{3, Float64, 9}([-0.0 -0.0 1.0; 1.0 -0.0 0.0; -0.0 1.0 0.0]), [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]))]], @NamedTuple{na::Int64, bodyid2act_idx::Vector{Vector{Int64}}, persistent_idx::Vector{Int64}, activated_bits::BitVector, H::LinearAlgebra.Diagonal{Float64, Vector{Float64}}, activated_restitution_coefficients::Vector{Float64}, D::Matrix{Float64}, Dper::Matrix{Float64}, Dimp::Matrix{Float64}, ∂Dq̇∂q::Matrix{Float64}, ∂DᵀΛ∂q::Matrix{Float64}, ŕ::Vector{Float64}, L::Matrix{Float64}, Lv::Matrix{Float64}, Λ::Vector{Float64}, Γ::Vector{Float64}}[(na = 0, bodyid2act_idx = [[0, 0, 0, 0, 0, 0]], persistent_idx = [], activated_bits = [0, 0, 0, 0, 0, 0], H = Diagonal(Float64[]), activated_restitution_coefficients = [], D = Matrix{Float64}(undef, 0, 7), Dper = Matrix{Float64}(undef, 0, 7), Dimp = Matrix{Float64}(undef, 0, 7), ∂Dq̇∂q = Matrix{Float64}(undef, 0, 7), ∂DᵀΛ∂q = [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0], ŕ = [], L = Matrix{Float64}(undef, 0, 0), Lv = Matrix{Float64}(undef, 0, 0), Λ = [], Γ = [])], [(c = Float64[], e = [0.16999999999999998], u = [0.0])])

Verify the nonconstant-mass trait ​

julia
body1 = get_bodies(top.structure)[1]
has_constant_mass_matrix(body1.coords) === Val(false)
true

Contact setup and simulation ​

The solver object is the same as for NCF — a Zhong06 integrator with an inner contact solver. Internally, the framework routes QCF bodies to the nonconstant-mass code path.

julia
planes = StaticContactSurfaces([
    HalfSpace([0.0, 0.0, 1.0], [0.0, 0.0, 0.0]),
])

contact_model = RestitutionFrictionCombined(
    NewtonRestitution(),
    CoulombFriction(),
)

prob = DynamicsProblem(top; env=planes, contact_model)

solver = DynamicsSolver(
    Zhong06(),
    InnerLayerContactSolver(InteriorPointMethod()),
)

sim_result = solve!(
    prob,
    solver;
    tspan = (0.0, 0.2),
    dt = 1e-3,
    ftol = 1e-14,
    maxiters = 50,
    exception = false,
)

tip_pos = get_trajectory!(top, 1, 1).u[end]
tip_vel = get_velocity!(top, 1, 1).u[end]
tip_pos, tip_vel
([0.15600299208285928, -0.0005131221481740021, 0.3038000000000203], [1.1026244296346408, -8.79940158342844, -1.9619999999997975])

Visualization ​

Because the top was constructed with loadmesh = true, vis! renders the STL geometry directly:

julia
key_steps = round.(Int, range(1, length(top.traj.t), length = 4))

fig_traj = plot_traj!(
    top;
    do_slide = false,
    show_info = false,
    show_loci = false,
    show_background = false,
    show_ground = false,
    gridsize = (2, 2),
    at_steps = key_steps,
)

Comparing QCF and NCF results ​

You can reuse the same initial conditions with :NCF to compare:

julia
top_ncf = make_top(origin_position, R, origin_velocity, Ω, :NCF; μ = 0.95, e = 0.5, loadmesh = false)

has_constant_mass_matrix(get_bodies(top_ncf.structure)[1].coords) === Val(true)
true

References ​

  1. X. Xu and W. Zhong. On the Numerical Influences of Inertia Representation for Rigid Body Dynamics in Terms of Unit Quaternion. Journal of Applied Mechanics-Transactions of the ASME 83, 061006 (2016). Accessed on Oct 30, 2022.

  2. X. Xu, J. Luo and Z. Wu. The Numerical Influence of Additional Parameters of Inertia Representations for Quaternion-Based Rigid Body Dynamics. Multibody System Dynamics 49, 237–270 (2020). Accessed on Oct 13, 2022.