Feedforward Layered Control for 6-Agent Escort Formation ($SE(3)$ Homogeneous Sheaf)

This example demonstrates eliminating tracking lag in the layered control architecture by combining joint Tikhonov reference-and-velocity filtering with a feedforward LQR control law.

Theoretical & Mathematical Background

1. Harmonic Reference & Velocity Solves

High-level spatial coordination computes the harmonic extension $q^*(t)$ by solving the restricted Laplacian linear system:

\[H q^*(t) = -L_{AB} p(t)\]

where $p(t)$ is the vector of target positions. For time-invariant sheaf restrictions $H = L_{AA}$, differentiating this identity with respect to time yields the exact harmonic reference velocity $\dot{q}^*(t)$:

\[H \dot{q}^*(t) = -L_{AB} \dot{p}(t)\]

A second distributed tree solve on worker processes computes $\dot{q}^*(t)$ using the pre-factorized restricted Laplacian $H$.

2. Joint Tikhonov Filtering

On each agent's flight computer, a JointTikhonovFilter smooths both $q^*(t)$ and $\dot{q}^*(t)$:

\[\epsilon \dot{x}_{\text{ref}} = -x_{\text{ref}} + q^*(t), \quad \epsilon \dot{v}_{\text{ref}} = -v_{\text{ref}} + \dot{q}^*(t)\]

3. Feedforward + Feedback Control Law

Standard LQR uses feedback $u_{\text{fb}} = -K(x - x_{\text{ref}})$, which incurs tracking lag when following moving references. The feedforward controller adds $u_{\text{ff}} = B^\dagger (v_{\text{ref}} - A_c x)$ where $B^\dagger = (B_c^\top B_c)^{-1} B_c^\top$ is the pseudoinverse of the control input matrix, yielding total control input:

\[u = -K (x - x_{\text{ref}}) + B^\dagger (v_{\text{ref}} - A_c x)\]

using CellularSheaves
using CellularSheaves.Formations
using CellularSheaves.AgentControllers
using CellularSheaves.DistributedLayeredControl
using LinearAlgebra
using Statistics
using Distributed
using Plots
using Printf

A single house style for every figure below

default(framestyle = :box, grid = true, gridalpha = 0.18, gridstyle = :dot,
    titlefontsize = 10, guidefontsize = 9, legendfontsize = 8, tickfontsize = 8,
    markerstrokewidth = 0, size = (720, 380))

const RING_COLOR = :steelblue
const TARGET_COLOR = :black

:black

Setup 10D Quadrotor Dynamics & Controllers

dyn = QuadrotorDynamics()
DT = 0.05
nx = 10

10

Standard LQR Cost Matrices (moderate velocity gain)

Q_diag_std = [500.0, 500.0, 500.0, 150.0, 150.0, 100.0, 100.0, 100.0, 5.0, 5.0]
Q_std = Matrix(Diagonal(Q_diag_std))
R_lqr = Matrix(Diagonal([0.005, 0.005, 0.005]))

3×3 Matrix{Float64}:
 0.005  0.0    0.0
 0.0    0.005  0.0
 0.0    0.0    0.005

High-Velocity-Gain LQR Cost Matrices (increased velocity state weights: 100.0 -> 1000.0)

Q_diag_highvel = [500.0, 500.0, 500.0, 150.0, 150.0, 1000.0, 1000.0, 1000.0, 5.0, 5.0]
Q_highvel = Matrix(Diagonal(Q_diag_highvel))

10×10 Matrix{Float64}:
 500.0    0.0    0.0    0.0    0.0     0.0     0.0     0.0  0.0  0.0
   0.0  500.0    0.0    0.0    0.0     0.0     0.0     0.0  0.0  0.0
   0.0    0.0  500.0    0.0    0.0     0.0     0.0     0.0  0.0  0.0
   0.0    0.0    0.0  150.0    0.0     0.0     0.0     0.0  0.0  0.0
   0.0    0.0    0.0    0.0  150.0     0.0     0.0     0.0  0.0  0.0
   0.0    0.0    0.0    0.0    0.0  1000.0     0.0     0.0  0.0  0.0
   0.0    0.0    0.0    0.0    0.0     0.0  1000.0     0.0  0.0  0.0
   0.0    0.0    0.0    0.0    0.0     0.0     0.0  1000.0  0.0  0.0
   0.0    0.0    0.0    0.0    0.0     0.0     0.0     0.0  5.0  0.0
   0.0    0.0    0.0    0.0    0.0     0.0     0.0     0.0  0.0  5.0

SE(3) Homogeneous Coordination Sheaf Construction (D = 4)

const NA = 6
const NT = 1
const TV1 = NA + 1
r_ring = 0.3

0.3

Build escort ring with Agent 1 as observer

sheaf = build_escort_ring(NA, TV1, r_ring; observers=[1])

A network sheaf with 7 vertex stalks and 7 edge stalks.

Target position p(t), velocity pdot(t), and acceleration pddot(t)

target1_pos(node, t)   = [0.5cos(0.5*t), 0.5sin(0.5*t), 1.5 + 0.1sin(1.0*t), 1.0]
target1_vel(node, t)   = [-0.25sin(0.5*t), 0.25cos(0.5*t), 0.1cos(1.0*t), 0.0]
target1_accel(node, t) = [-0.125cos(0.5*t), -0.125sin(0.5*t), -0.1sin(1.0*t), 0.0]

target1_accel (generic function with 1 method)

Provision Worker Processes

Workers start with stderr detached: a worker inherits the parent's stderr, which while these docs are built is the pipe capturing the page's output, and holding it open stalls the build. Each worker still reports through its stdout, which the parent relays.

workers_pids = redirect_stderr(devnull) do
    addprocs(NA; exeflags = ["--project=$(Base.active_project())",
        "--sysimage=$(unsafe_string(Base.JLOptions().image_file))"])
end

@everywhere workers_pids begin
    using CellularSheaves
end

Simulation Framework

STEPS = 200
epsilon = 0.02
init_states = [[r_ring*i/NA, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] for i in 1:NA]
dyns = [QuadrotorDynamics(m=0.5 + 0.05*i, Ixx=0.01 + 0.002*i, Iyy=0.01 + 0.002*i) for i in 1:NA]

lqr_configs = [(init_states[i], dyns[i], LQRController(dyns[i], DT, Q_std, R_lqr).K) for i in 1:NA]

prob_pos = LayeredControlProblem(;
    sheaf=sheaf,
    target_nodes=[TV1],
    target_trajectory_func=target1_pos,
    agent_configs=lqr_configs,
    dt=DT,
    steps=STEPS,
    r_ring=r_ring
)

prob_vel = LayeredControlProblem(;
    sheaf=sheaf,
    target_nodes=[TV1],
    target_trajectory_func=target1_pos,
    target_velocity_func=target1_vel,
    agent_configs=lqr_configs,
    dt=DT,
    steps=STEPS,
    r_ring=r_ring
)

prob_accel = LayeredControlProblem(;
    sheaf=sheaf,
    target_nodes=[TV1],
    target_trajectory_func=target1_pos,
    target_velocity_func=target1_vel,
    target_acceleration_func=target1_accel,
    agent_configs=lqr_configs,
    dt=DT,
    steps=STEPS,
    r_ring=r_ring
)

CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredControlProblem(A network sheaf with 7 vertex stalks and 7 edge stalks.
, [7], Main.var"##1959".target1_pos, Main.var"##1959".target1_vel, Main.var"##1959".target1_accel, Tuple{Vector{Float64}, CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics, Matrix{Float64}}[([0.049999999999999996, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.55, 0.012, 0.012), [0.0 0.0 23.155681190378882 0.0 0.0 0.0 0.0 11.519911616866297 0.0 0.0; 0.0 -1.7599735965077932 0.0 2.870686340109049 0.0 0.0 -1.2843436264323052 0.0 0.30659989356367856 0.0; 1.7599735965077927 0.0 0.0 0.0 2.8706863401090486 1.2843436264323052 0.0 0.0 0.0 0.30659989356367856]), ([0.09999999999999999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.6, 0.014, 0.014), [0.0 0.0 25.232354358509784 0.0 0.0 0.0 0.0 12.554408254133824 0.0 0.0; 0.0 -2.0532605811338844 0.0 3.349078558831746 0.0 0.0 -1.4983720900987731 0.0 0.3576954723740814 0.0; 2.0532605811338835 0.0 0.0 0.0 3.3490785588317467 1.498372090098773 0.0 0.0 0.0 0.3576954723740814]), ([0.15, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.65, 0.016, 0.016), [0.0 0.0 27.301777873345102 0.0 0.0 0.0 0.0 13.585644127011617 0.0 0.0; 0.0 -2.3465282100371168 0.0 3.8274451665066107 0.0 0.0 -1.712387261425324 0.0 0.40878901933445405 0.0; 2.3465282100371163 0.0 0.0 0.0 3.827445166506611 1.7123872614253237 0.0 0.0 0.0 0.4087890193344541]), ([0.19999999999999998, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.7, 0.018000000000000002, 0.018000000000000002), [0.0 0.0 29.36341085933043 0.0 0.0 0.0 0.0 14.613375885779773 0.0 0.0; 0.0 -2.6397737216488646 0.0 4.30578250903451 0.0 0.0 -1.92638724392375 0.0 0.459880244545805 0.0; 2.6397737216488633 0.0 0.0 0.0 4.3057825090345085 1.9263872439237484 0.0 0.0 0.0 0.45988024454580495]), ([0.25, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.75, 0.02, 0.02), [0.0 0.0 31.41672665850292 0.0 0.0 0.0 0.0 15.637366559271296 0.0 0.0; 0.0 -2.932994355813146 0.0 4.784086934163531 0.0 0.0 -2.140370142072015 0.0 0.5109688582551377 0.0; 2.932994355813146 0.0 0.0 0.0 4.784086934163528 2.140370142072015 0.0 0.0 0.0 0.5109688582551376]), ([0.3, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.8, 0.022, 0.022), [0.0 0.0 33.46121357508947 0.0 0.0 0.0 0.0 16.657385889028156 0.0 0.0; 0.0 -3.2261873539624526 0.0 5.262354791718899 0.0 0.0 -2.354334061434512 0.0 0.5620545708736221 0.0; 3.2261873539624535 0.0 0.0 0.0 5.2623547917189 2.3543340614345123 0.0 0.0 0.0 0.5620545708736221])], 0.05, 200, 0.3, 3)

Case 1: Position-Only Tracking (1-stage solve)

init_distributed_agents!(workers_pids, lqr_configs, DT, epsilon; use_velocity=false)
sim_pos_res = run_layered_simulation(prob_pos, workers_pids; mode=:distributed)

CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredSimulationResult(CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredControlProblem(A network sheaf with 7 vertex stalks and 7 edge stalks.
, [7], Main.var"##1959".target1_pos, nothing, nothing, Tuple{Vector{Float64}, CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics, Matrix{Float64}}[([0.049999999999999996, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.55, 0.012, 0.012), [0.0 0.0 23.155681190378882 0.0 0.0 0.0 0.0 11.519911616866297 0.0 0.0; 0.0 -1.7599735965077932 0.0 2.870686340109049 0.0 0.0 -1.2843436264323052 0.0 0.30659989356367856 0.0; 1.7599735965077927 0.0 0.0 0.0 2.8706863401090486 1.2843436264323052 0.0 0.0 0.0 0.30659989356367856]), ([0.09999999999999999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.6, 0.014, 0.014), [0.0 0.0 25.232354358509784 0.0 0.0 0.0 0.0 12.554408254133824 0.0 0.0; 0.0 -2.0532605811338844 0.0 3.349078558831746 0.0 0.0 -1.4983720900987731 0.0 0.3576954723740814 0.0; 2.0532605811338835 0.0 0.0 0.0 3.3490785588317467 1.498372090098773 0.0 0.0 0.0 0.3576954723740814]), ([0.15, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.65, 0.016, 0.016), [0.0 0.0 27.301777873345102 0.0 0.0 0.0 0.0 13.585644127011617 0.0 0.0; 0.0 -2.3465282100371168 0.0 3.8274451665066107 0.0 0.0 -1.712387261425324 0.0 0.40878901933445405 0.0; 2.3465282100371163 0.0 0.0 0.0 3.827445166506611 1.7123872614253237 0.0 0.0 0.0 0.4087890193344541]), ([0.19999999999999998, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.7, 0.018000000000000002, 0.018000000000000002), [0.0 0.0 29.36341085933043 0.0 0.0 0.0 0.0 14.613375885779773 0.0 0.0; 0.0 -2.6397737216488646 0.0 4.30578250903451 0.0 0.0 -1.92638724392375 0.0 0.459880244545805 0.0; 2.6397737216488633 0.0 0.0 0.0 4.3057825090345085 1.9263872439237484 0.0 0.0 0.0 0.45988024454580495]), ([0.25, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.75, 0.02, 0.02), [0.0 0.0 31.41672665850292 0.0 0.0 0.0 0.0 15.637366559271296 0.0 0.0; 0.0 -2.932994355813146 0.0 4.784086934163531 0.0 0.0 -2.140370142072015 0.0 0.5109688582551377 0.0; 2.932994355813146 0.0 0.0 0.0 4.784086934163528 2.140370142072015 0.0 0.0 0.0 0.5109688582551376]), ([0.3, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.8, 0.022, 0.022), [0.0 0.0 33.46121357508947 0.0 0.0 0.0 0.0 16.657385889028156 0.0 0.0; 0.0 -3.2261873539624526 0.0 5.262354791718899 0.0 0.0 -2.354334061434512 0.0 0.5620545708736221 0.0; 3.2261873539624535 0.0 0.0 0.0 5.2623547917189 2.3543340614345123 0.0 0.0 0.0 0.5620545708736221])], 0.05, 200, 0.3, 3), [0.05025635306433698 0.1001860262522584 0.15006411530239802 0.1999937955507906 0.25002664725578755 0.3001110788407213; 0.05371421673452679 0.10269719358362983 0.15093305651525 0.19991612815962548 0.2503934701640751 0.30161786861965034; 0.06458851596781354 0.11061408873657079 0.15370800166202703 0.1997338677958589 0.25162318331435335 0.30644397773297777; 0.0846534104807252 0.12525464828853594 0.15889742312011676 0.19949919221045814 0.25401811893349047 0.31549517573242; 0.11361929270851286 0.14641933254672743 0.1664501386219327 0.19925093181150513 0.2575905180130497 0.3286988609860483; 0.15001728980810117 0.17303618820440936 0.17598353731598895 0.19900336616851472 0.26216669310352164 0.34540097612776605; 0.19185595420181195 0.20364437402564964 0.18696301945071675 0.19875249101022208 0.2674819610513003 0.3646805318550805; 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Case 2: Joint Position & Velocity Tracking (2-stage solve)

init_distributed_agents!(workers_pids, lqr_configs, DT, epsilon; use_velocity=true)
sim_vel_res = run_layered_simulation(prob_vel, workers_pids; mode=:distributed)

CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredSimulationResult(CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredControlProblem(A network sheaf with 7 vertex stalks and 7 edge stalks.
, [7], Main.var"##1959".target1_pos, Main.var"##1959".target1_vel, nothing, Tuple{Vector{Float64}, CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics, Matrix{Float64}}[([0.049999999999999996, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.55, 0.012, 0.012), [0.0 0.0 23.155681190378882 0.0 0.0 0.0 0.0 11.519911616866297 0.0 0.0; 0.0 -1.7599735965077932 0.0 2.870686340109049 0.0 0.0 -1.2843436264323052 0.0 0.30659989356367856 0.0; 1.7599735965077927 0.0 0.0 0.0 2.8706863401090486 1.2843436264323052 0.0 0.0 0.0 0.30659989356367856]), ([0.09999999999999999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.6, 0.014, 0.014), [0.0 0.0 25.232354358509784 0.0 0.0 0.0 0.0 12.554408254133824 0.0 0.0; 0.0 -2.0532605811338844 0.0 3.349078558831746 0.0 0.0 -1.4983720900987731 0.0 0.3576954723740814 0.0; 2.0532605811338835 0.0 0.0 0.0 3.3490785588317467 1.498372090098773 0.0 0.0 0.0 0.3576954723740814]), ([0.15, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.65, 0.016, 0.016), [0.0 0.0 27.301777873345102 0.0 0.0 0.0 0.0 13.585644127011617 0.0 0.0; 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Case 3: Joint Position, Velocity & Acceleration Tracking (3-stage solve — Differential Flatness)

init_distributed_agents!(workers_pids, lqr_configs, DT, epsilon; use_velocity=true)
sim_accel_res = run_layered_simulation(prob_accel, workers_pids; mode=:distributed)

CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredSimulationResult(CellularSheaves.ControlSheaves.DistributedLayeredControl.LayeredControlProblem(A network sheaf with 7 vertex stalks and 7 edge stalks.
, [7], Main.var"##1959".target1_pos, Main.var"##1959".target1_vel, Main.var"##1959".target1_accel, Tuple{Vector{Float64}, CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics, Matrix{Float64}}[([0.049999999999999996, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.55, 0.012, 0.012), [0.0 0.0 23.155681190378882 0.0 0.0 0.0 0.0 11.519911616866297 0.0 0.0; 0.0 -1.7599735965077932 0.0 2.870686340109049 0.0 0.0 -1.2843436264323052 0.0 0.30659989356367856 0.0; 1.7599735965077927 0.0 0.0 0.0 2.8706863401090486 1.2843436264323052 0.0 0.0 0.0 0.30659989356367856]), ([0.09999999999999999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.6, 0.014, 0.014), [0.0 0.0 25.232354358509784 0.0 0.0 0.0 0.0 12.554408254133824 0.0 0.0; 0.0 -2.0532605811338844 0.0 3.349078558831746 0.0 0.0 -1.4983720900987731 0.0 0.3576954723740814 0.0; 2.0532605811338835 0.0 0.0 0.0 3.3490785588317467 1.498372090098773 0.0 0.0 0.0 0.3576954723740814]), ([0.15, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.65, 0.016, 0.016), [0.0 0.0 27.301777873345102 0.0 0.0 0.0 0.0 13.585644127011617 0.0 0.0; 0.0 -2.3465282100371168 0.0 3.8274451665066107 0.0 0.0 -1.712387261425324 0.0 0.40878901933445405 0.0; 2.3465282100371163 0.0 0.0 0.0 3.827445166506611 1.7123872614253237 0.0 0.0 0.0 0.4087890193344541]), ([0.19999999999999998, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.7, 0.018000000000000002, 0.018000000000000002), [0.0 0.0 29.36341085933043 0.0 0.0 0.0 0.0 14.613375885779773 0.0 0.0; 0.0 -2.6397737216488646 0.0 4.30578250903451 0.0 0.0 -1.92638724392375 0.0 0.459880244545805 0.0; 2.6397737216488633 0.0 0.0 0.0 4.3057825090345085 1.9263872439237484 0.0 0.0 0.0 0.45988024454580495]), ([0.25, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.75, 0.02, 0.02), [0.0 0.0 31.41672665850292 0.0 0.0 0.0 0.0 15.637366559271296 0.0 0.0; 0.0 -2.932994355813146 0.0 4.784086934163531 0.0 0.0 -2.140370142072015 0.0 0.5109688582551377 0.0; 2.932994355813146 0.0 0.0 0.0 4.784086934163528 2.140370142072015 0.0 0.0 0.0 0.5109688582551376]), ([0.3, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], CellularSheaves.ControlSheaves.AgentControllers.QuadrotorDynamics(9.81, 0.8, 0.022, 0.022), [0.0 0.0 33.46121357508947 0.0 0.0 0.0 0.0 16.657385889028156 0.0 0.0; 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Clean up worker processes

rmprocs(workers_pids)

Task (done) @0x00007fe2b4d23ee0

Visualizations

1. Joint Position, Velocity & Acceleration Tracking Escort Animation

animate_layered_escort(sim_accel_res; frame_step = 4, filename = "layered_escort_feedforward.gif", fps = 10)
nothing # hide

[ Info: Saved animation to /home/runner/work/CellularSheaves.jl/CellularSheaves.jl/docs/src/generated/layered/layered_escort_feedforward.gif

2. Tracking Lag Comparison

We compare the per-agent tracking lag $||x_i[1:3] - q^*_i[1:3]||$ over time across three reference formulations:

1. Position-Only Reference: LQR feedback only, constant tracking lag (~15 cm).
2. Joint Pos & Vel Reference: Solves $H \dot{q}^* = -L_{AB} \dot{p}(t)$, significantly reduces velocity lag (~5 mm).
3. Joint Pos, Vel & Accel Reference (Differential Flatness): Solves $H \ddot{q}^* = -L_{AB} \ddot{p}(t)$, pre-tilts quadrotor attitude and boosts thrust, eliminating tracking lag (< 0.1 mm).

ts = (1:STEPS) .* DT

err_pos   = zeros(STEPS, NA)
err_vel   = zeros(STEPS, NA)
err_accel = zeros(STEPS, NA)

for step in 1:STEPS
    for i in 1:NA
        qstar_3d = sim_pos_res.qstar_history[step, i, 1:3]
        err_pos[step, i]   = norm(sim_pos_res.sim_data[step, i, 1:3] - qstar_3d)
        err_vel[step, i]   = norm(sim_vel_res.sim_data[step, i, 1:3] - qstar_3d)
        err_accel[step, i] = norm(sim_accel_res.sim_data[step, i, 1:3] - qstar_3d)
    end
end

mean_err_pos   = mean(err_pos, dims=2)[:]
mean_err_vel   = mean(err_vel, dims=2)[:]
mean_err_accel = mean(err_accel, dims=2)[:]

p_comp = plot(;
    title="Tracking Lag Elimination (Mean Agent Distance to Harmonic Extension)",
    xlabel="time [s]",
    ylabel="tracking lag [m]",
    yscale=:log10,
    xlims=(0, ts[end]),
    ylims=(1e-5, 1.0),
    legend=:topright,
    bottom_margin=5Plots.mm,
    left_margin=5Plots.mm
)

plot!(p_comp, ts, mean_err_pos,   lw=2.0, color=:crimson,     label="Position-Only Reference")
plot!(p_comp, ts, mean_err_vel,   lw=2.0, color=:darkorange,  label="Joint Pos & Vel Reference")
plot!(p_comp, ts, mean_err_accel, lw=2.0, color=:forestgreen, label="Joint Pos, Vel & Accel (Differential Flatness)")

savefig(p_comp, "tracking_lag_comparison.png")

"/home/runner/work/CellularSheaves.jl/CellularSheaves.jl/docs/src/generated/layered/tracking_lag_comparison.png"