Multilayer Hierarchical Control for Team-of-Teams Escort Formation
This example demonstrates multi-agent escort formation control comparing a 3-Layer Pushforward Control Architecture ($f_\star F$) against a Direct 2-Layer Solve on the full sheaf $F$ using the generic Layered architecture.
Mathematical & Categorical Overview
- Generic Topology Specification (
LayeredEscortSpec):- Ring 1 (6 agents): Escorts Target 1 (node 16) at radius $r_1 = 0.3\,\text{m}$.
- Ring 2 (6 agents): Escorts Target 2 (node 17) at radius $r_2 = 0.36\,\text{m}$.
- Support Pool (3 agents): Bridges Ring 1 and Ring 2, observing Target 1 and Target 2 directly.
- Pushforward Sheaf $f_\star F$ (3 coarse nodes):
- Coarsens Fiber 1 $\to$ Node 1, Fiber 2 $\to$ Node 2, Support Pool $\to$ Node 3.
- Fiber bases are structured and managed via
LayeredFiberBases. - Pseudoinverse lifting $T^+$ projects high-level midpoint references back onto the subspace of exact subteam consensus.
using CellularSheaves
using CellularSheaves.Formations
using CellularSheaves.AgentControllers
using CellularSheaves.ControlSheaves.Layered
using CellularSheaves.ControlSheaves.NestedSystems
using CellularSheaves.Pushforwards
using CellularSheaves.GraphHomomorphisms
using LinearAlgebra
using Statistics
using Plots
using Printf— Topology Specification —
const r_ring = 0.3
rings = [
RingSpec(1, 6, r_ring),
RingSpec(2, 6, r_ring * 1.2)
]
supports = [
SupportSpec(1, 2, 3)
]
spec = LayeredEscortSpec(rings, supports)CellularSheaves.ControlSheaves.Layered.LayeredEscortSpec(CellularSheaves.ControlSheaves.Layered.RingSpec[CellularSheaves.ControlSheaves.Layered.RingSpec(1, 6, 0.3, [1]), CellularSheaves.ControlSheaves.Layered.RingSpec(2, 6, 0.36, [1])], CellularSheaves.ControlSheaves.Layered.SupportSpec[CellularSheaves.ControlSheaves.Layered.SupportSpec(1, 2, 3, [1], [1])], 4, true, 2, 1, 2, 15, 17, UnitRange{Int64}[1:6, 7:12], UnitRange{Int64}[13:15], [16, 17])— Sheaf Topology & Homomorphism —
F = build_layered_escort_sheaf(spec)
f = build_layered_homomorphism(spec)
PfF = pushforward_sheaf(f, F)
bases = build_layered_fiber_bases(f, F, spec)CellularSheaves.ControlSheaves.Layered.LayeredFiberBases(GraphHomomorphism: 17 source vertices → 3 target vertices
vertex_map: [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2]
fiber sizes: [7, 7, 3]
, [[1.0 0.5999999999999996 0.0 6.857834927532851e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17; 0.9999999999999999 0.44999999999999973 -0.0 4.9960036108131996e-17; -7.703719777548943e-34 0.2598076211353315 -0.0 0.9999999999999997; 0.0 0.0 1.0 0.0; 0.0 1.0 0.0 0.0; 1.0 0.1499999999999999 0.0 4.163336342344337e-17; -1.461837931635933e-17 0.25980762113533146 -0.0 0.9999999999999998; 0.0 -0.0 1.0 -0.0; -2.7755575615628914e-17 0.9999999999999994 -0.0 7.753736490788077e-17; 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0; -0.0 -0.0 1.0 -0.0; -5.551115123125783e-17 0.9999999999999991 -0.0 1.6653345369377336e-16; 1.0 0.14999999999999972 0.0 2.0816681711721685e-17; 1.0816665110907539e-17 -0.2598076211353313 0.0 0.9999999999999997; 0.0 0.0 0.9999999999999998 0.0; -2.7755575615628914e-17 0.999999999999999 0.0 1.914243483398994e-16; 1.0 0.4499999999999996 0.0 6.938893903907228e-17; 6.154798468514088e-18 -0.2598076211353314 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999998 0.0; 0.0 0.9999999999999991 0.0 1.4832681464315452e-16; 1.0 0.29999999999999977 0.0 5.010788220678779e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17], [0.9999999999999994 0.0 0.72 5.730145749088649e-18; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17; 0.9999999999999994 -0.0 0.54 -6.938893903907228e-18; 1.0151139188489342e-17 -0.0 0.3117691453623979 0.999999999999999; -0.0 1.0 -0.0 0.0; 8.326672684688674e-17 -0.0 1.0 5.440544929897799e-17; 0.9999999999999996 0.0 0.18000000000000005 0.0; 2.030227837697869e-17 -0.0 0.3117691453623979 0.9999999999999993; -0.0 1.0 -0.0 -0.0; 8.326672684688674e-17 -0.0 0.9999999999999999 0.0; 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.9999999999999998 0.0 0.17999999999999977 -3.469446951953614e-17; -2.757226071654248e-17 0.0 -0.31176914536239786 0.9999999999999993; 0.0 0.9999999999999998 0.0 0.0; 0.0 0.0 1.0 6.1039660892657e-17; 0.9999999999999996 0.0 0.5399999999999999 -6.938893903907228e-18; -4.384165187475787e-17 0.0 -0.3117691453623978 0.999999999999999; 0.0 0.9999999999999998 0.0 0.0; 2.7755575615628914e-17 0.0 0.9999999999999999 5.440544929897799e-17; 0.9999999999999994 0.0 0.36 -2.482474975886393e-17; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17], [1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 -0.0; -0.0 -0.0 -0.0 1.0; 1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 -0.0; -0.0 -0.0 -0.0 1.0; 1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0]], Dict(2 => [0.9999999999999994 0.0 0.36 -2.482474975886393e-17; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17], 1 => [1.0 0.29999999999999977 0.0 5.010788220678779e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17]), Dict(2 => [1.0000000000000004 -2.549771467806916e-17 0.0 -0.36000000000000015; -7.3337345365904196e-34 -2.87623209734096e-17 1.0 -4.071040469505696e-50; -3.375966153774958e-17 1.3978462343592527e-16 -3.10384214854243e-32 0.9999999999999999; 2.5497714678069177e-17 1.0000000000000013 -8.225998148910653e-17 -1.3978462343592505e-16], 1 => [1.0 -1.3148774930209175e-16 0.0 -0.3; -1.7582056375572068e-17 6.869771928148037e-17 0.0 1.0000000000000007; 0.0 0.0 1.0000000000000002 0.0; -1.4929318261206478e-18 1.0000000000000004 0.0 3.9404731836805973e-16]))— Simulation Parameters —
DT = 0.05
STEPS = 200200Setup LQR gain for tests
dyn_test = QuadrotorDynamics()
Ad, Bd = CellularSheaves.AgentControllers.discrete_matrices(dyn_test, DT)([0.9999999999999999 0.0 0.0 0.0 0.012262499999999997 0.049999999999999996 0.0 0.0 0.0 0.00020437499999999994; 0.0 0.9999999999999999 0.0 -0.012262499999999997 0.0 0.0 0.049999999999999996 0.0 -0.00020437499999999994 0.0; 0.0 0.0 0.9999999999999999 0.0 0.0 0.0 0.0 0.049999999999999996 0.0 0.0; 0.0 0.0 0.0 0.9999999999999999 0.0 0.0 0.0 0.0 0.049999999999999996 0.0; 0.0 0.0 0.0 0.0 0.9999999999999999 0.0 0.0 0.0 0.0 0.049999999999999996; 0.0 0.0 0.0 0.0 0.49049999999999994 0.9999999999999999 0.0 0.0 0.0 0.012262499999999999; 0.0 0.0 0.0 -0.49049999999999994 0.0 0.0 0.9999999999999999 0.0 -0.012262499999999999 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9999999999999999 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9999999999999999 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.9999999999999999], [0.0 0.0 0.00025546874999999973; 0.0 -0.00025546874999999973 0.0; 0.0024999999999999996 0.0 0.0; 0.0 0.12499999999999999 0.0; 0.0 0.0 0.12499999999999999; 0.0 0.0 0.02043749999999999; 0.0 -0.02043749999999999 0.0; 0.09999999999999999 0.0 0.0; 0.0 5.0 0.0; 0.0 0.0 5.0])Standard LQR Cost Matrices (moderate velocity gain)
Q_diag = [500.0, 500.0, 500.0, 150.0, 150.0, 100.0, 100.0, 100.0, 5.0, 5.0]
Q_lqr = Matrix(Diagonal(Q_diag))
R_lqr = Matrix(Diagonal([0.005, 0.005, 0.005]))
K_test = CellularSheaves.AgentControllers.solve_dare(Ad, Bd, 10*Q_lqr, R_lqr)3×10 Matrix{Float64}:
0.0 0.0 21.1663 0.0 0.0 0.0 0.0 10.5247 0.0 0.0
0.0 -1.46672 0.0 2.39234 0.0 0.0 -1.07034 0.0 0.255508 0.0
1.46672 0.0 0.0 0.0 2.39234 1.07034 0.0 0.0 0.0 0.255508Target position, velocity, and acceleration trajectories
target1_pos(t) = [1.0 + 0.5 * cos(0.5 * t), 0.5 * sin(0.5 * t), 1.5 + 0.01 * sin(1.0 * t), 1.0]
target2_pos(t) = [-1.0 - 0.5 * cos(0.5 * t), -0.5 * sin(0.5 * t), 1.5 + 0.01 * cos(1.0 * t), 1.0]
target1_vel(t) = [-0.25 * sin(0.5 * t), 0.25 * cos(0.5 * t), 0.01 * cos(1.0 * t), 0.0]
target2_vel(t) = [0.25 * sin(0.5 * t), -0.25 * cos(0.5 * t), -0.01 * sin(1.0 * t), 0.0]
target1_acc(t) = [-0.125 * cos(0.5 * t), -0.125 * sin(0.5 * t), -0.01 * sin(1.0 * t), 0.0]
target2_acc(t) = [0.125 * cos(0.5 * t), 0.125 * sin(0.5 * t), -0.01 * cos(1.0 * t), 0.0]
target_trajs = [target1_pos, target2_pos]
target_vels = [target1_vel, target2_vel]
target_accels = [target1_acc, target2_acc]
prob_pf = LayeredEscortProblem(
spec, F, f, PfF, bases,
SystemBinding(dynamics=dyn_test, K_lqr=K_test),
target_trajs, target_vels, target_accels,
DT, STEPS, nothing
)CellularSheaves.ControlSheaves.Layered.LayeredEscortProblem(CellularSheaves.ControlSheaves.Layered.LayeredEscortSpec(CellularSheaves.ControlSheaves.Layered.RingSpec[CellularSheaves.ControlSheaves.Layered.RingSpec(1, 6, 0.3, [1]), CellularSheaves.ControlSheaves.Layered.RingSpec(2, 6, 0.36, [1])], CellularSheaves.ControlSheaves.Layered.SupportSpec[CellularSheaves.ControlSheaves.Layered.SupportSpec(1, 2, 3, [1], [1])], 4, true, 2, 1, 2, 15, 17, UnitRange{Int64}[1:6, 7:12], UnitRange{Int64}[13:15], [16, 17]), A network sheaf with 17 vertex stalks and 19 edge stalks.
, GraphHomomorphism: 17 source vertices → 3 target vertices
vertex_map: [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2]
fiber sizes: [7, 7, 3]
, A network sheaf with 3 vertex stalks and 2 edge stalks.
, CellularSheaves.ControlSheaves.Layered.LayeredFiberBases(GraphHomomorphism: 17 source vertices → 3 target vertices
vertex_map: [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2]
fiber sizes: [7, 7, 3]
, [[1.0 0.5999999999999996 0.0 6.857834927532851e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17; 0.9999999999999999 0.44999999999999973 -0.0 4.9960036108131996e-17; -7.703719777548943e-34 0.2598076211353315 -0.0 0.9999999999999997; 0.0 0.0 1.0 0.0; 0.0 1.0 0.0 0.0; 1.0 0.1499999999999999 0.0 4.163336342344337e-17; -1.461837931635933e-17 0.25980762113533146 -0.0 0.9999999999999998; 0.0 -0.0 1.0 -0.0; -2.7755575615628914e-17 0.9999999999999994 -0.0 7.753736490788077e-17; 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0; -0.0 -0.0 1.0 -0.0; -5.551115123125783e-17 0.9999999999999991 -0.0 1.6653345369377336e-16; 1.0 0.14999999999999972 0.0 2.0816681711721685e-17; 1.0816665110907539e-17 -0.2598076211353313 0.0 0.9999999999999997; 0.0 0.0 0.9999999999999998 0.0; -2.7755575615628914e-17 0.999999999999999 0.0 1.914243483398994e-16; 1.0 0.4499999999999996 0.0 6.938893903907228e-17; 6.154798468514088e-18 -0.2598076211353314 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999998 0.0; 0.0 0.9999999999999991 0.0 1.4832681464315452e-16; 1.0 0.29999999999999977 0.0 5.010788220678779e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17], [0.9999999999999994 0.0 0.72 5.730145749088649e-18; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17; 0.9999999999999994 -0.0 0.54 -6.938893903907228e-18; 1.0151139188489342e-17 -0.0 0.3117691453623979 0.999999999999999; -0.0 1.0 -0.0 0.0; 8.326672684688674e-17 -0.0 1.0 5.440544929897799e-17; 0.9999999999999996 0.0 0.18000000000000005 0.0; 2.030227837697869e-17 -0.0 0.3117691453623979 0.9999999999999993; -0.0 1.0 -0.0 -0.0; 8.326672684688674e-17 -0.0 0.9999999999999999 0.0; 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.9999999999999998 0.0 0.17999999999999977 -3.469446951953614e-17; -2.757226071654248e-17 0.0 -0.31176914536239786 0.9999999999999993; 0.0 0.9999999999999998 0.0 0.0; 0.0 0.0 1.0 6.1039660892657e-17; 0.9999999999999996 0.0 0.5399999999999999 -6.938893903907228e-18; -4.384165187475787e-17 0.0 -0.3117691453623978 0.999999999999999; 0.0 0.9999999999999998 0.0 0.0; 2.7755575615628914e-17 0.0 0.9999999999999999 5.440544929897799e-17; 0.9999999999999994 0.0 0.36 -2.482474975886393e-17; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17], [1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 -0.0; -0.0 -0.0 -0.0 1.0; 1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 -0.0; -0.0 -0.0 -0.0 1.0; 1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0]], Dict(2 => [0.9999999999999994 0.0 0.36 -2.482474975886393e-17; -2.5497714678069125e-17 0.0 6.62750913262382e-17 0.9999999999999988; 0.0 0.9999999999999999 0.0 0.0; 1.1102230246251565e-16 0.0 1.0 8.487470974431272e-17], 1 => [1.0 0.29999999999999977 0.0 5.010788220678779e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17]), Dict(2 => [1.0000000000000004 -2.549771467806916e-17 0.0 -0.36000000000000015; -7.3337345365904196e-34 -2.87623209734096e-17 1.0 -4.071040469505696e-50; -3.375966153774958e-17 1.3978462343592527e-16 -3.10384214854243e-32 0.9999999999999999; 2.5497714678069177e-17 1.0000000000000013 -8.225998148910653e-17 -1.3978462343592505e-16], 1 => [1.0 -1.3148774930209175e-16 0.0 -0.3; -1.7582056375572068e-17 6.869771928148037e-17 0.0 1.0000000000000007; 0.0 0.0 1.0000000000000002 0.0; -1.4929318261206478e-18 1.0000000000000004 0.0 3.9404731836805973e-16])), CellularSheaves.ControlSheaves.NestedSystems.SystemBinding(QuadrotorDynamics(9.81, 0.5, 0.01, 0.01), [0.0 0.0 21.166309247170133 0.0 0.0 0.0 0.0 10.52467760470746 0.0 0.0; 0.0 -1.4667218880511859 0.0 2.392340797060175 0.0 0.0 -1.0703393879965994 0.0 0.25550801774491644 0.0; 1.4667218880511854 0.0 0.0 0.0 2.3923407970601733 1.070339387996599 0.0 0.0 0.0 0.2555080177449164], Dict{Symbol, CellularSheaves.ControlSheaves.NestedSystems.SystemBinding}(), Dict{Int64, CellularSheaves.ControlSheaves.NestedSystems.AgentBinding}()), Any[Main.var"##2388".target1_pos, Main.var"##2388".target2_pos], Any[Main.var"##2388".target1_vel, Main.var"##2388".target2_vel], Any[Main.var"##2388".target1_acc, Main.var"##2388".target2_acc], 0.05, 200, nothing)Simulation 1: 3-Layer Pushforward Architecture (f_★F) —
@time res_pf = run_layered_escort_simulation(prob_pf; use_feedforward=true)
@time res_pf = run_layered_escort_simulation(prob_pf; use_feedforward=true)CellularSheaves.ControlSheaves.Layered.LayeredEscortResult(CellularSheaves.ControlSheaves.Layered.LayeredEscortProblem(CellularSheaves.ControlSheaves.Layered.LayeredEscortSpec(CellularSheaves.ControlSheaves.Layered.RingSpec[CellularSheaves.ControlSheaves.Layered.RingSpec(1, 6, 0.3, [1]), CellularSheaves.ControlSheaves.Layered.RingSpec(2, 6, 0.36, [1])], CellularSheaves.ControlSheaves.Layered.SupportSpec[CellularSheaves.ControlSheaves.Layered.SupportSpec(1, 2, 3, [1], [1])], 4, true, 2, 1, 2, 15, 17, UnitRange{Int64}[1:6, 7:12], UnitRange{Int64}[13:15], [16, 17]), A network sheaf with 17 vertex stalks and 19 edge stalks.
, GraphHomomorphism: 17 source vertices → 3 target vertices
vertex_map: [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2]
fiber sizes: [7, 7, 3]
, A network sheaf with 3 vertex stalks and 2 edge stalks.
, CellularSheaves.ControlSheaves.Layered.LayeredFiberBases(GraphHomomorphism: 17 source vertices → 3 target vertices
vertex_map: [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2]
fiber sizes: [7, 7, 3]
, [[1.0 0.5999999999999996 0.0 6.857834927532851e-17; 1.4929318261206358e-18 -2.786040707021194e-17 0.0 0.9999999999999996; 0.0 0.0 0.9999999999999999 0.0; -1.5937503311477594e-36 0.9999999999999996 0.0 6.156822356180239e-17; 0.9999999999999999 0.44999999999999973 -0.0 4.9960036108131996e-17; -7.703719777548943e-34 0.2598076211353315 -0.0 0.9999999999999997; 0.0 0.0 1.0 0.0; 0.0 1.0 0.0 0.0; 1.0 0.1499999999999999 0.0 4.163336342344337e-17; -1.461837931635933e-17 0.25980762113533146 -0.0 0.9999999999999998; 0.0 -0.0 1.0 -0.0; -2.7755575615628914e-17 0.9999999999999994 -0.0 7.753736490788077e-17; 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0; -0.0 -0.0 1.0 -0.0; -5.551115123125783e-17 0.9999999999999991 -0.0 1.6653345369377336e-16; 1.0 0.14999999999999972 0.0 2.0816681711721685e-17; 1.0816665110907539e-17 -0.2598076211353313 0.0 0.9999999999999997; 0.0 0.0 0.9999999999999998 0.0; -2.7755575615628914e-17 0.999999999999999 0.0 1.914243483398994e-16; 1.0 0.4499999999999996 0.0 6.938893903907228e-17; 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[[0.6546744517197463, -0.36159406204325606, 1.5099894134183978, 1.0], [-0.6546744517197463, 0.36159406204325606, 1.4995399787436046, 1.0], [-2.4472255495510295e-17, 5.67582268698009e-17, 1.5047646960810013, 1.0]], [[0.6638212702624722, -0.3701133092338915, 1.5099539377725761, 1.0], [-0.6638212702624722, 0.3701133092338915, 1.499041291066235, 1.0], [-2.457799958533101e-17, 5.155618145886673e-17, 1.5044976144194055, 1.0]], [[0.6731781895681941, -0.3784012476539641, 1.5098935824662338, 1.0], [-0.6731781895681941, 0.3784012476539641, 1.4985449996619138, 1.0], [-2.4680872583978596e-17, 7.425095404114362e-17, 1.504219291064074, 1.0]], [[0.6827393618669269, -0.38645269761174544, 1.509808498356204, 1.0], [-0.6827393618669269, 0.38645269761174544, 1.4980523450006884, 1.0], [-2.4780810199177565e-17, 6.933455622253327e-17, 1.5039304216784461, 1.0]], [[0.6924988117372128, -0.3942626272130975, 1.509698898108451, 1.0], [-0.6924988117372128, 0.3942626272130975, 1.497564558462642, 1.0], [2.8778256424573934e-18, 6.456563620726583e-17, 1.5036317282855465, 1.0]], [[0.7024504398405675, -0.4018261555062442, 1.509565055666515, 1.0], [-0.7024504398405675, 0.4018261555062442, 1.4970828592600858, 1.0], [2.2767958737124483e-17, 8.770275003074432e-17, 1.5033239574633004, 1.0]], [[0.7125880267333654, -0.40913855553220513, 1.5094073055667978, 1.0], [-0.7125880267333654, 0.40913855553220513, 1.4966084513901616, 1.0], [7.818834571799171e-17, 8.323763284999403e-17, 1.5030078784784797, 1.0]], [[0.7229052367537916, -0.41619525727899365, 1.5092260421023933, 1.0], [-0.7229052367537916, 0.41619525727899365, 1.4961425206254777, 1.0], [2.2589604059240406e-17, 7.892865083353558e-17, 1.5026842813639356, 1.0]], [[0.7333956219814228, -0.42299185053772326, 1.509021718337563, 1.0], [-0.7333956219814228, 0.42299185053772326, 1.495686231550294, 1.0], [2.2505242195810034e-17, 1.0253407257049495e-16, 1.5023539749439285, 1.0]], [[0.7440526262669681, -0.42952408765884564, 1.5087948449753086, 1.0], [-0.7440526262669681, 0.42952408765884564, 1.4952407246496668, 1.0], [7.793531285111915e-17, 9.854534054069527e-17, 1.5020177848124878, 1.0]], [[0.7548695893296503, -0.4357878862067941, 1.5085459890808828, 1.0], [-0.7548695893296503, 0.4357878862067941, 1.4948071134588332, 1.0], [7.785756423537277e-17, 9.472052318744498e-17, 1.501676551269858, 1.0]], [[0.7658397509196617, -0.4417793315113711, 1.5082757726644198, 1.0], [-0.7658397509196617, 0.4417793315113711, 1.4943864817799493, 1.0], [2.227204493892492e-17, 9.106201089708692e-17, 1.5013311272221845, 1.0]], [[0.7769562550431036, -0.44749467911429175, 1.507984871126235, 1.0], [-0.7769562550431036, 0.44749467911429175, 1.4939798809731517, 1.0], [7.771225513316904e-17, 8.757209012071263e-17, 1.5009823760496934, 1.0]], [[0.7882121542467562, -0.4529303571093466, 1.5076740115686749, 1.0], [-0.7882121542467562, 0.4529303571093466, 1.493588327328704, 1.0], [2.2133634228912708e-17, 8.425294194520563e-17, 1.5006311694486896, 1.0]], [[0.7996004139600122, -0.45808296837472745, 1.507343970978741, 1.0], [-0.7996004139600122, 0.45808296837472745, 1.4932127995267999, 1.0], [1.0533640493316556e-16, 8.110664073013279e-17, 1.5002783852527706, 1.0]], [[0.8111139168912556, -0.4629492926961186, 1.5069955742860266, 1.0], [-0.8111139168912556, 0.4629492926961186, 1.492854236191373, 1.0], [4.976485106014572e-17, 7.813515281133677e-17, 1.4999249052386998, 1.0]], [[0.822745467475934, -0.46752628877922453, 1.5066296923008218, 1.0], [-0.822745467475934, 0.46752628877922453, 1.492513533544026, 1.0], [4.9708039668940525e-17, 7.534033527204094e-17, 1.499571612922424, 1.0]], [[0.8344877963735563, -0.4718110961504812, 1.506247239537542, 1.0], [-0.8344877963735563, 0.4718110961504812, 1.4921915431639425, 1.0], [1.0516600626482015e-16, 7.272393478223191e-17, 1.4992193913507423, 1.0]], [[0.8463335650107904, -0.47580103694475806, 1.5058491719289175, 1.0], [-0.8463335650107904, 0.47580103694475806, 1.4918890698593834, 1.0], [4.960533039267703e-17, 7.028758650704837e-17, 1.4988691208941505, 1.0]], [[0.8582753701678338, -0.47949361757893655, 1.5054364844366608, 1.0], [-0.8582753701678338, 0.47949361757893655, 1.4916068696560851, 1.0], [4.9559496697573075e-17, 9.57883887004848e-17, 1.498521677046373, 1.0]], [[0.870305748605187, -0.4828865303103194, 1.505010208564579, 1.0], [-0.870305748605187, 0.4828865303103194, 1.4913456479075888, 1.0], [1.0502853382407578e-16, 9.3716599291276e-17, 1.4981779282360839, 1.0]], [[0.8824171817279313, -0.4859776546788933, 1.5045714097803515, 1.0], [-0.8824171817279313, 0.4859776546788933, 1.4911060575322241, 1.0], [-6.032136832901562e-18, 9.182908869599215e-17, 1.497838733656288, 1.0]], [[0.8946021002846102, -0.4887650588325485, 1.5041211848524176, 1.0], [-0.8946021002846102, 0.4887650588325485, 1.4908886973811533, 1.0], [2.1688840477431698e-17, 9.012703654731407e-17, 1.4975049411167856, 1.0]], [[0.9068528890977614, -0.49124700073442285, 1.5036606591086241, 1.0], [-0.9068528890977614, 0.49124700073442285, 1.4906941107415548, 1.0], [7.716918491537449e-17, 8.86115065724305e-17, 1.4971773849250896, 1.0]], [[0.9191618918231568, -0.49342192925161826, 1.5031909836234936, 1.0], [-0.9191618918231568, 0.49342192925161826, 1.490522783978689, 1.0], [7.714218888728309e-17, 8.728344592824319e-17, 1.4968568838010914, 1.0]], [[0.9315214157347691, -0.49528848512460866, 1.5027133323411364, 1.0], [-0.9315214157347691, 0.49528848512460866, 1.4903751453202376, 1.0], [7.711902049605413e-17, 8.61436846094244e-17, 1.496544238830687, 1.0]], [[0.9439237365324725, -0.4968455018167322, 1.5022288991410024, 1.0], [-0.9439237365324725, 0.4968455018167322, 1.4902515637859584, 1.0], [2.1588542989920133e-17, 8.519293492969771e-17, 1.4962402314634804, 1.0]], [[0.9563611031694825, -0.4980920062432396, 1.5017388948538044, 1.0], [-0.9563611031694825, 0.4980920062432396, 1.4901523482653267, 1.0], [7.708422214094728e-17, 1.121873666922935e-16, 1.4959456215595657, 1.0]], [[0.9688257426965041, -0.4990272193794397, 1.5012445442350706, 1.0], [-0.9688257426965041, 0.4990272193794397, 1.490077746745474, 1.0], [7.707261392490859e-17, 1.1161630435608578e-16, 1.4956611454902724, 1.0]], [[0.9813098651195742, -0.4996505567475648, 1.5007470829038954, 1.0], [-0.9813098651195742, 0.4996505567475648, 1.4900279456913479, 1.0], [7.706487682781907e-17, 1.1123568043207484e-16, 1.4953875142976216, 1.0]], [[0.9938056682685548, -0.4999616287820504, 1.5002477542545336, 1.0], [-0.9938056682685548, 0.4999616287820504, 1.490003069579648, 1.0], [1.0481659130074144e-16, 1.1104573279782325e-16, 1.495125411917091, 1.0]], [[1.0063053426732327, -0.49996024107300086, 1.4997478063485634, 1.0], [-1.0063053426732327, 0.49996024107300086, 1.490003180587698, 1.0], [1.0481660852550638e-16, 1.1104658016441932e-16, 1.4948754934681308, 1.0]], [[1.0188010764439883, -0.499646394487689, 1.4992484887953819, 1.0], [-1.0188010764439883, 0.499646394487689, 1.4900282784380363, 1.0], [1.048205041069779e-16, 1.1123822200228659e-16, 1.494638383616709, 1.0]], [[1.0312850601539678, -0.4990202851700145, 1.4987510496288325, 1.0], [-1.0312850601539678, 0.4990202851700145, 1.490078300399107, 1.0], [1.0482827561054439e-16, 1.116205385415146e-16, 1.4944146750139697, 1.0]], [[1.0437494917197232, -0.49808230441792034, 1.4982567321877702, 1.0], [-1.0437494917197232, 0.49808230441792034, 1.4901531214420587, 1.0], [1.0483991817926907e-16, 1.1219329084671119e-16, 1.4942049268149145, 1.0]], [[1.0561865812772617, -0.4968330384388438, 1.497766772008362, 1.0], [-1.0561865812772617, 0.4968330384388438, 1.4902525545532501, 1.0], [1.0485542453692549e-16, 8.520054535070068e-17, 1.494009663280806, 1.0]], [[1.0685885560504536, -0.49527326798335664, 1.4972823937358906, 1.0], [-1.0685885560504536, 0.49527326798335664, 1.490376351201687, 1.0], [1.0487478499254486e-16, 8.615297654074723e-17, 1.4938293724687888, 1.0]], [[1.0809476652087684, -0.49340396785722057, 1.4968048080637772, 1.0], [-1.0809476652087684, 0.49340396785722057, 1.49052420196022, 1.0], [1.0489798744647263e-16, 8.729441356273127e-17, 1.4936645050119985, 1.0]], [[1.0932561847112878, -0.49122630631216624, 1.4963352087074808, 1.0], [-1.0932561847112878, 0.49122630631216624, 1.4906957372789524, 1.0], [1.3268059301355928e-16, 8.862414305566932e-17, 1.4935154729932165, 1.0]], [[1.1055064221339823, -0.4887416443157734, 1.495874769420829, 1.0], [-1.1055064221339823, 0.4887416443157734, 1.4908905284089211, 1.0], [1.3271143356970718e-16, 9.014133398191288e-17, 1.493382648914875, 1.0]], [[1.1176907214772256, -0.4859515347009104, 1.4954246410622467, 1.0], [-1.1176907214772256, 0.4859515347009104, 1.4911080884737464, 1.0], [1.3274606545620157e-16, 9.184503814651978e-17, 1.4932663647679965, 1.0]], [[1.1298014679505437, -0.4828577211952647, 1.494985948718208, 1.0], [-1.1298014679505437, 0.4828577211952647, 1.4913478736865693, 1.0], [1.327844670292407e-16, 9.373419078984504e-17, 1.4931669112023886, 1.0]]])Simulation 2: Direct 2-Layer Solve on Sheaf F —
function direct_simulation()
sim_data_dir = Vector{Vector{Vector{Float64}}}()
qstar_history_dir = Vector{Matrix{Float64}}()
target_history_dir = Vector{Vector{Vector{Float64}}}()
time_grid = 0:DT:(STEPS*DT)
# Initialize agent states for direct simulation using AgentState and JointTikhonovFilter
agent_states_dir = [AgentState(zeros(10), dyn_test, DT, K_test, 0.02; use_velocity=true) for _ in 1:spec.n_agents]
for state in agent_states_dir
state.x[3] = 1.5
end
for step in 1:STEPS
t = time_grid[step]
p1 = target1_pos(t)
p2 = target2_pos(t)
p3_center = (p1 + p2) ./ 2.0
push!(target_history_dir, [p1, p2, p3_center])
# Direct Harmonic Solves on F for position, velocity, and acceleration
qstar_agents = solve_direct_harmonic(F, spec.target_nodes, [p1, p2])
qstar_dot_agents = solve_direct_harmonic(F, spec.target_nodes, [target1_vel(t), target2_vel(t)])
qstar_ddot_agents = solve_direct_harmonic(F, spec.target_nodes, [target1_acc(t), target2_acc(t)])
push!(qstar_history_dir, qstar_agents)
# Agent Dynamics Integration with Feedforward Velocity & Acceleration Support
step_states = Vector{Vector{Float64}}()
for i in 1:spec.n_agents
q_ref_i = qstar_agents[i, 1:3]
q_dot_i = qstar_dot_agents[i, 1:3]
q_ddot_i = qstar_ddot_agents[i, 1:3]
step_agent!(agent_states_dir[i], q_ref_i, q_dot_i, q_ddot_i, DT)
push!(step_states, copy(agent_states_dir[i].x))
end
push!(sim_data_dir, step_states)
end
return (sim_data_dir, qstar_history_dir, target_history_dir, time_grid)
end
@time direct_simulation()
@time sim_data_dir, qstar_history_dir, target_history_dir, time_grid = direct_simulation()([[[0.0006119512823327152, 6.274827146943245e-5, 1.4937261322410988, -0.030702518150181046, 0.2994257038936835, 0.04895610258661724, 0.0050198617175545995, -0.25095471035603817, -1.228100726007242, 11.977028155747343], [0.0005603596483704115, 0.00015210760273763794, 1.4937261322410988, -0.07442573834257521, 0.27418209094576734, 0.044828771869632945, 0.012168608219011043, -0.2509547103560372, -2.977029533703009, 10.967283637830697], [0.000457176380445805, 0.00015210760273763783, 1.4937261322410988, -0.07442573834257515, 0.22369486504993538, 0.03657411043566442, 0.012168608219011034, -0.2509547103560377, -2.9770295337030066, 8.947794601997415], [0.0004055847464835018, 6.274827146943244e-5, 1.4937261322410988, -0.030702518150181043, 0.19845125210201942, 0.032446779718680166, 0.005019861717554599, -0.2509547103560372, -1.2281007260072418, 7.9380500840807775], [0.000457176380445805, 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1.0], [-0.8946021002846102, 0.4887650588325485, 1.4908886973811533, 1.0], [0.0, 0.0, 1.4975049411167856, 1.0]], [[0.9068528890977614, -0.49124700073442285, 1.5036606591086241, 1.0], [-0.9068528890977614, 0.49124700073442285, 1.4906941107415548, 1.0], [0.0, 0.0, 1.4971773849250893, 1.0]], [[0.9191618918231568, -0.49342192925161826, 1.5031909836234936, 1.0], [-0.9191618918231568, 0.49342192925161826, 1.490522783978689, 1.0], [0.0, 0.0, 1.4968568838010912, 1.0]], [[0.9315214157347691, -0.49528848512460866, 1.5027133323411364, 1.0], [-0.9315214157347691, 0.49528848512460866, 1.4903751453202376, 1.0], [0.0, 0.0, 1.4965442388306869, 1.0]], [[0.9439237365324725, -0.4968455018167322, 1.5022288991410024, 1.0], [-0.9439237365324725, 0.4968455018167322, 1.4902515637859584, 1.0], [0.0, 0.0, 1.4962402314634804, 1.0]], [[0.9563611031694825, -0.4980920062432396, 1.5017388948538044, 1.0], [-0.9563611031694825, 0.4980920062432396, 1.4901523482653267, 1.0], [0.0, 0.0, 1.4959456215595655, 1.0]], [[0.9688257426965041, -0.4990272193794397, 1.5012445442350706, 1.0], [-0.9688257426965041, 0.4990272193794397, 1.490077746745474, 1.0], [0.0, 0.0, 1.4956611454902724, 1.0]], [[0.9813098651195742, -0.4996505567475648, 1.5007470829038954, 1.0], [-0.9813098651195742, 0.4996505567475648, 1.4900279456913479, 1.0], [0.0, 0.0, 1.4953875142976216, 1.0]], [[0.9938056682685548, -0.4999616287820504, 1.5002477542545336, 1.0], [-0.9938056682685548, 0.4999616287820504, 1.490003069579648, 1.0], [0.0, 0.0, 1.495125411917091, 1.0]], [[1.0063053426732327, -0.49996024107300086, 1.4997478063485634, 1.0], [-1.0063053426732327, 0.49996024107300086, 1.490003180587698, 1.0], [0.0, 0.0, 1.4948754934681308, 1.0]], [[1.0188010764439883, -0.499646394487689, 1.4992484887953819, 1.0], [-1.0188010764439883, 0.499646394487689, 1.4900282784380363, 1.0], [0.0, 0.0, 1.4946383836167092, 1.0]], [[1.0312850601539678, -0.4990202851700145, 1.4987510496288325, 1.0], [-1.0312850601539678, 0.4990202851700145, 1.490078300399107, 1.0], [0.0, 0.0, 1.4944146750139697, 1.0]], [[1.0437494917197232, -0.49808230441792034, 1.4982567321877702, 1.0], [-1.0437494917197232, 0.49808230441792034, 1.4901531214420587, 1.0], [0.0, 0.0, 1.4942049268149145, 1.0]], [[1.0561865812772617, -0.4968330384388438, 1.497766772008362, 1.0], [-1.0561865812772617, 0.4968330384388438, 1.4902525545532501, 1.0], [0.0, 0.0, 1.494009663280806, 1.0]], [[1.0685885560504536, -0.49527326798335664, 1.4972823937358906, 1.0], [-1.0685885560504536, 0.49527326798335664, 1.490376351201687, 1.0], [0.0, 0.0, 1.4938293724687888, 1.0]], [[1.0809476652087684, -0.49340396785722057, 1.4968048080637772, 1.0], [-1.0809476652087684, 0.49340396785722057, 1.49052420196022, 1.0], [0.0, 0.0, 1.4936645050119988, 1.0]], [[1.0932561847112878, -0.49122630631216624, 1.4963352087074808, 1.0], [-1.0932561847112878, 0.49122630631216624, 1.4906957372789524, 1.0], [0.0, 0.0, 1.4935154729932165, 1.0]], [[1.1055064221339823, -0.4887416443157734, 1.495874769420829, 1.0], [-1.1055064221339823, 0.4887416443157734, 1.4908905284089211, 1.0], [0.0, 0.0, 1.4933826489148752, 1.0]], [[1.1176907214772256, -0.4859515347009104, 1.4954246410622467, 1.0], [-1.1176907214772256, 0.4859515347009104, 1.4911080884737464, 1.0], [0.0, 0.0, 1.4932663647679965, 1.0]], [[1.1298014679505437, -0.4828577211952647, 1.494985948718208, 1.0], [-1.1298014679505437, 0.4828577211952647, 1.4913478736865693, 1.0], [0.0, 0.0, 1.4931669112023886, 1.0]]], 0.0:0.05:10.0)— 3. Render Animations & Visualizations —
r1, r2, r3 = r_ring, r_ring * 1.2, r_ring * 0.8(0.3, 0.36, 0.24)Render 3-Layer Pushforward Animation
animate_comprehensive_escort(res_pf.sim_data, res_pf.qstar_history, res_pf.target_history, time_grid, r1, r2, r3, "multilayer_comprehensive_animation.gif"; frame_step=4, fps=10, title_prefix="3-Layer Pushforward Architecture (f_★F)")Plots.Animation("/tmp/jl_3e9mbZ", ["000001.png", "000002.png", "000003.png", "000004.png", "000005.png", "000006.png", "000007.png", "000008.png", "000009.png", "000010.png", "000011.png", "000012.png", "000013.png", "000014.png", "000015.png", "000016.png", "000017.png", "000018.png", "000019.png", "000020.png", "000021.png", "000022.png", "000023.png", "000024.png", "000025.png", "000026.png", "000027.png", "000028.png", "000029.png", "000030.png", "000031.png", "000032.png", "000033.png", "000034.png", "000035.png", "000036.png", "000037.png", "000038.png", "000039.png", "000040.png", "000041.png", "000042.png", "000043.png", "000044.png", "000045.png", "000046.png", "000047.png", "000048.png", "000049.png", "000050.png"])Render Direct 2-Layer Solve Animation
animate_comprehensive_escort(sim_data_dir, qstar_history_dir, target_history_dir, time_grid, r1, r2, r3, "direct_sheaf_comprehensive_animation.gif"; frame_step=4, fps=10, title_prefix="Direct 2-Layer Solve (F)")
println("Multilayer Escort simulation with generic architecture complete.")[ Info: Saved animation to /home/runner/work/CellularSheaves.jl/CellularSheaves.jl/docs/src/generated/layered/direct_sheaf_comprehensive_animation.gif
Multilayer Escort simulation with generic architecture complete.
Visual Comparison Animations
