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🔥 内容介绍
一、研究背景与仿真框架设计
欠驱动四旋翼无人机是典型的四输入六自由度欠驱动系统，凭借结构简单、机动灵活的优势，在巡检、测绘、竞速等场景得到广泛应用。在激进轨迹跟踪、传感器测量噪声、外部强风扰等复杂工况下，不同控制算法的性能差异极大，传统基于欧拉角的PID控制在高动态场景下极易出现姿态失稳、轨迹跟踪发散的问题，完全无法适配高复杂度作业需求。
本研究搭建统一的MATLAB/Simulink仿真框架，针对欠驱动四旋翼的全非线性动力学模型，在完全一致的硬件参数、仿真步长、测试工况下，系统性对比几何控制、微分平坦度控制、INDI（增量非线性动态逆）、NMPC（非线性模型预测控制）四类主流先进控制算法的性能差异，覆盖激进轨迹跟踪、测量噪声干扰、外部风扰三大核心测试场景，量化不同算法的适用边界，为工程场景下的控制器选型提供精准的量化依据。
仿真框架采用统一的四旋翼物理参数配置：整机质量1.2kg，旋翼臂长0.25m，转动惯量对角矩阵为diag(0.0023, 0.0023, 0.0042) kg·m²，所有算法的仿真步长统一设置为1ms，传感器采样频率统一为200Hz，从根源上消除仿真条件差异带来的性能对比偏差，保证测试结果的公平性与可复现性。
二、四类先进控制算法核心原理实现
1. 几何控制算法
几何控制直接在特殊欧氏群SE(3)上推导控制律，完全不使用欧拉角或四元数的局部参数化表示，从根源上规避了传统欧拉角控制的奇异点问题，非常适合大角度机动场景。控制律直接基于四旋翼的旋转矩阵、位置、速度误差设计，将位置环输出的期望合力映射到机体坐标系，通过姿态误差的动态补偿直接生成期望力矩，实现全姿态的稳定跟踪。该算法无需对动力学模型做线性化近似，在大角度机动场景下依然可以保证全局渐近稳定性，代码实现简洁，计算量极低，非常适合机载嵌入式平台部署。
2. 微分平坦度控制
四旋翼系统已被严格证明是微分平坦系统，选择位置坐标与偏航角作为平坦输出，系统所有的状态量与输入量都可以表示为平坦输出及其各阶导数的代数函数，无需求解复杂的微分方程即可完成轨迹规划与控制律设计。在轨迹跟踪场景下，直接通过期望轨迹的位置、速度、加速度、加加速度、加加加速度的高阶微分，即可直接解算出四旋翼所需的总推力与期望姿态角，大幅简化轨迹跟踪的控制逻辑，对平滑连续的轨迹具备天然的优秀跟踪性能。
3. INDI增量非线性动态逆控制
INDI算法是在传统非线性动态逆的基础上改进得到的增量式控制方法，不直接依赖精确的系统动力学模型，而是利用传感器实时测量得到的当前状态量，计算控制量的增量，仅需要非常粗略的模型先验信息即可实现高精度控制。该算法对模型参数不确定性的鲁棒性极强，即使四旋翼的升力系数、转动惯量等参数出现较大偏差，依然可以维持稳定的控制性能，非常适合模型参数难以精准标定的工程场景。
4. NMPC非线性模型预测控制
NMPC算法在每个控制周期内，基于四旋翼的全非线性动力学模型，在满足所有物理约束的前提下，在线求解有限时间域内的最优控制序列，直接将无人机的最大角速度、最大加速度、电机转速饱和等物理约束显式嵌入优化问题，从根源上避免控制量超出执行机构的饱和边界，在多约束场景下可以生成最优的控制输出，是目前理论上性能上限最高的先进控制算法之一。
⛳️ 运行结果
📣 部分代码
%% plot_paper_figures.m
% Paper-style figure entry point. This file reads saved benchmark results and
% exports editable PDF plus high-resolution PNG without rerunning simulations.
%
% Edit the settings in Sections 1-5 for paper layout, colors, line widths,
% labels, legends, and output size.
clear; clc; close all;
%% ========================================================================
%% 1. What to export
figuresToMake = ["disturbance_monte_carlo", "trajectory_sweep_3d"];
outputDir = fullfile(pwd, "results", "paper_figures");
savePdf = true;
savePng = true;
%% ========================================================================
%% 2. Global paper style
P.fontName = "Times New Roman";
P.fontSize = 8;
P.axisLineWidth = 0.5;
P.gridAlpha = 0.18;
P.titleFontSize = 9;
P.labelFontSize = 8;
P.legendFontSize = 7;
P.export.pdfResolution = 1200;
P.export.pngResolution = 600;
P.export.pdfContentType = "vector"; % set "image" if vector PDF is slow
P.export.pngContentType = "image";
P.export.padding = "tight";
%% ========================================================================
%% 3. Controller display style
P.controllerOrder = ["lee", "johnson", "lu", ...
"sun_dfbc", "sun_dfbc_indi", "sun_nmpc", "sun_nmpc_indi", ...
"tal", "geometric_indi"];
P.controllerLabel.lee = "Geometric Control";
P.controllerLabel.johnson = "Log-Geometric Control";
P.controllerLabel.lu = "On-Manifold MPC";
P.controllerLabel.sun_dfbc = "DFBC";
P.controllerLabel.sun_dfbc_indi = "DFBC+INDI";
P.controllerLabel.sun_nmpc = "NMPC";
P.controllerLabel.sun_nmpc_indi = "NMPC+INDI";
P.controllerLabel.tal = "DF-based INDI";
P.controllerLabel.geometric_indi = "Geometric INDI";
P.controllerColor.lee = [0.000, 0.447, 0.741];
P.controllerColor.johnson = [0.850, 0.325, 0.098];
P.controllerColor.lu = [0.929, 0.694, 0.125];
P.controllerColor.sun_dfbc = [0.494, 0.184, 0.556];
P.controllerColor.sun_dfbc_indi = [0.494, 0.184, 0.556];
P.controllerColor.sun_nmpc = [0.466, 0.674, 0.188];
P.controllerColor.sun_nmpc_indi = [0.466, 0.674, 0.188];
P.controllerColor.tal = [0.301, 0.745, 0.933];
P.controllerColor.geometric_indi = [0.635, 0.078, 0.184];
P.controllerLineStyle.lee = "-";
P.controllerLineStyle.johnson = "-";
P.controllerLineStyle.lu = "-";
P.controllerLineStyle.sun_dfbc = "--";
P.controllerLineStyle.sun_dfbc_indi = "-";
P.controllerLineStyle.sun_nmpc = "--";
P.controllerLineStyle.sun_nmpc_indi = "-";
P.controllerLineStyle.tal = "-";
P.controllerLineStyle.geometric_indi = "-";
%% ========================================================================
%% 4. Disturbance Monte Carlo boxchart figure
P.mc.resultDir = fullfile(pwd, "results", "disturbance_monte_carlo", ...
"random_gust");
P.mc.trajNames = ["figure8_horizontal", "helix_flip"];
P.mc.controllerNames = strings(0,1); % empty means all saved controllers
P.mc.levelNames = ["low", "medium", "high"];
P.mc.levelLabel.low = "Low disturbance";
P.mc.levelLabel.medium = "Medium disturbance";
P.mc.levelLabel.high = "High disturbance";
P.mc.failureRmseThreshold = 5; % [m], set inf to keep all finite runs
P.mc.figureWidthCm = 16;
P.mc.figureHeightCm = 9;
P.mc.boxWidth = 0.10;
P.mc.boxLineWidth = 0.55;
P.mc.boxFaceAlpha = 0.25;
P.mc.medianColor = [0.80, 0.10, 0.10];
P.mc.outlierMarker = "o";
P.mc.outlierMarkerSize = 3.5;
P.mc.legendLocation = "northwest";
P.mc.legendNumColumns = 1;
P.mc.showLegend = true;
P.mc.yLabel = "RMS position tracking error (m)";
P.mc.xLabel = "";
P.mc.title.figure8_horizontal = "Horizontal figure-eight";
P.mc.title.helix_flip = "Helix flip";
P.mc.yLimit.figure8_horizontal = [];
P.mc.yLimit.helix_flip = [];
P.mc.disturbanceLabelFontSize = 6.5;
%% ========================================================================
%% 5. Trajectory sweep 3-D figure
P.sweep.resultFile = "latest"; % "latest" or a .mat file path
P.sweep.trajNames = ["figure8_horizontal", "helix_flip"];
P.sweep.figureWidthCm = 16;
P.sweep.figureHeightCm = 7;
P.sweep.tileSpacing = "compact";
P.sweep.padding = "compact";
P.sweep.actualColor = [0.000, 0.447, 0.741];
P.sweep.refColor = [0.850, 0.325, 0.098];
P.sweep.actualLineStyle = "-";
P.sweep.refLineStyle = "--";
P.sweep.actualLineWidth = 0.9;
P.sweep.refLineWidth = 0.8;
P.sweep.viewAzEl = [35, 25];
P.sweep.axisEqual = true;
P.sweep.legendLocation = "northeast";
P.sweep.showLegend = true;
P.sweep.xLabel = "x (m)";
P.sweep.yLabel = "y (m)";
P.sweep.zLabel = "z (m)";
P.sweep.title.figure8_horizontal = "Horizontal figure-eight";
P.sweep.title.helix_flip = "Helix flip";
%% ========================================================================
%% 6. Run selected paper plots
P.outputDir = outputDir;
P.savePdf = savePdf;
P.savePng = savePng;
applyPaperDefaults(P);
if any(figuresToMake == "disturbance_monte_carlo")
makePaperDisturbanceBoxFigures(P);
end
if any(figuresToMake == "trajectory_sweep_3d")
makePaperTrajectorySweepFigure(P);
end
fprintf('Paper figures saved to:\n %s\n', outputDir);
%% ========================================================================
%% Local plotting functions
function applyPaperDefaults(P)
set(groot, ...
'defaultAxesFontSize', P.fontSize, ...
'defaultAxesFontName', char(P.fontName), ...
'defaultAxesLineWidth', P.axisLineWidth, ...
'defaultAxesLabelFontSizeMultiplier', 1, ...
'defaultAxesTitleFontSizeMultiplier', 1, ...
'defaultTextFontName', char(P.fontName), ...
'defaultLegendFontName', char(P.fontName));
end
function makePaperDisturbanceBoxFigures(P)
resultFile = fullfile(P.mc.resultDir, "disturbance_benchmark_results.mat");
if ~isfile(resultFile)
error("Monte Carlo result not found: %s.", resultFile);
end
S = load(resultFile, 'results');
results = S.results;
trajNames = selectedNames(P.mc.trajNames, unique(results.Trajectory, ...
'stable'));
controllerNames = selectedNames(P.mc.controllerNames, ...
unique(results.Controller, 'stable'));
controllerOrder = appendMissing(P.controllerOrder, controllerNames);
levelNames = selectedNames(P.mc.levelNames, ...
unique(results.DisturbanceLevel, 'stable'));
levelLabels = disturbanceLevelLabels(levelNames, P.mc.levelLabel);
outDir = fullfile(P.outputDir, "disturbance_monte_carlo");
ensureDir(outDir);
for iTraj = 1:numel(trajNames)
trajName = trajNames(iTraj);
mask = string(results.Trajectory) == trajName ...
& ismember(string(results.Controller), controllerNames) ...
& ismember(string(results.DisturbanceLevel), levelNames) ...
& results.IsFinite;
if isfinite(P.mc.failureRmseThreshold)
mask = mask & isfinite(results.RMSE) ...
& results.RMSE <= P.mc.failureRmseThreshold;
end
if ~any(mask)
warning("No Monte Carlo rows selected for trajectory %s.", trajName);
continue;
end
fig = paperFigure(P.mc.figureWidthCm, P.mc.figureHeightCm);
ax = axes(fig);
hold(ax, 'on');
grid(ax, 'on');
ax.GridAlpha = P.gridAlpha;
legendHandles = gobjects(0);
legendLabels = strings(0,1);
nLevels = numel(levelNames);
nCtrl = numel(controllerOrder);
offsets = groupedOffsets(nCtrl, 0.70);
for iLevel = 1:nLevels
for iCtrl = 1:nCtrl
ctrlName = controllerOrder(iCtrl);
rowMask = mask ...
& string(results.DisturbanceLevel) == levelNames(iLevel) ...
& string(results.Controller) == ctrlName;
y = results.RMSE(rowMask);
y = y(isfinite(y));
if isempty(y)
continue;
end
x = ones(numel(y),1)*(iLevel + offsets(iCtrl));
color = controllerColor(P, ctrlName, iCtrl);
h = boxchart(ax, x, y, ...
'BoxFaceColor', color, ...
'BoxFaceAlpha', P.mc.boxFaceAlpha, ...
'BoxEdgeColor', color, ...
'BoxMedianLineColor', P.mc.medianColor, ...
'WhiskerLineColor', [0.15 0.15 0.15], ...
'MarkerStyle', char(P.mc.outlierMarker), ...
'MarkerColor', color, ...
'MarkerSize', P.mc.outlierMarkerSize, ...
'BoxWidth', P.mc.boxWidth, ...
'LineWidth', P.mc.boxLineWidth);
if iLevel == 1
legendHandles(end+1,1) = h; %#ok<AGROW>
legendLabels(end+1,1) = controllerLabel(P, ctrlName); %#ok<AGROW>
end
end
end
xlim(ax, [0.5, nLevels + 0.5]);
xticks(ax, 1:nLevels);
ylabel(ax, P.mc.yLabel, 'FontSize', P.labelFontSize);
xlabel(ax, P.mc.xLabel, 'FontSize', P.labelFontSize);
title(ax, titleFor(P.mc.title, trajName), ...
'FontSize', P.titleFontSize, 'Interpreter', 'none');
formatAxes(ax, P);
applyOptionalYLimit(ax, P.mc.yLimit, trajName);
addDisturbanceLevelLabels(ax, levelLabels, P.mc.disturbanceLabelFontSize);
xticklabels(ax, repmat({''}, 1, nLevels));
if P.mc.showLegend && ~isempty(legendHandles)
legend(ax, legendHandles, cellstr(legendLabels), ...
'Location', char(P.mc.legendLocation), ...
'NumColumns', P.mc.legendNumColumns, ...
'FontSize', P.legendFontSize, ...
'Interpreter', 'none');
end
savePaperFigure(fig, fullfile(outDir, "mc_" + trajName), ...
P.mc.figureWidthCm, P.mc.figureHeightCm, P);
end
end
function makePaperTrajectorySweepFigure(P)
resultFile = string(P.sweep.resultFile);
if resultFile == "latest"
resultFile = latestSweepResultsFile();
end
if ~isfile(resultFile)
error("Trajectory sweep result not found: %s.", resultFile);
end
S = load(resultFile, 'results');
results = S.results;
trajNames = selectedNames(P.sweep.trajNames, unique(results.Trajectory, ...
'stable'));
data = loadSweepData(results, trajNames);
if isempty(data)
error("No saved trajectory run data found for the selected sweep.");
end
outDir = fullfile(P.outputDir, "trajectory_sweep");
ensureDir(outDir);
fig = paperFigure(P.sweep.figureWidthCm, P.sweep.figureHeightCm);
tl = tiledlayout(fig, 1, numel(data), ...
'TileSpacing', char(P.sweep.tileSpacing), ...
'Padding', char(P.sweep.padding));
for i = 1:numel(data)
ax = nexttile(tl);
p = data(i).log.p;
pd = data(i).log.pd;
plot3(ax, p(1,:), p(2,:), p(3,:), ...
'LineStyle', char(P.sweep.actualLineStyle), ...
'Color', P.sweep.actualColor, ...
'LineWidth', P.sweep.actualLineWidth, ...
'DisplayName', 'actual');
hold(ax, 'on');
plot3(ax, pd(1,:), pd(2,:), pd(3,:), ...
'LineStyle', char(P.sweep.refLineStyle), ...
'Color', P.sweep.refColor, ...
'LineWidth', P.sweep.refLineWidth, ...
'DisplayName', 'reference');
grid(ax, 'on');
ax.GridAlpha = P.gridAlpha;
if P.sweep.axisEqual
axis(ax, 'equal');
end
view(ax, P.sweep.viewAzEl(1), P.sweep.viewAzEl(2));
set(ax, 'ZDir', 'reverse');
xlabel(ax, P.sweep.xLabel, 'FontSize', P.labelFontSize);
ylabel(ax, P.sweep.yLabel, 'FontSize', P.labelFontSize);
zlabel(ax, P.sweep.zLabel, 'FontSize', P.labelFontSize);
title(ax, titleFor(P.sweep.title, data(i).name), ...
'FontSize', P.titleFontSize, 'Interpreter', 'none');
setTrajectoryLimits(ax, [p, pd]);
formatAxes(ax, P);
if P.sweep.showLegend && i == numel(data)
legend(ax, 'Location', char(P.sweep.legendLocation), ...
'FontSize', P.legendFontSize, 'Interpreter', 'none');
end
end
savePaperFigure(fig, fullfile(outDir, "trajectory_sweep_3d"), ...
P.sweep.figureWidthCm, P.sweep.figureHeightCm, P);
end
function fig = paperFigure(widthCm, heightCm)
fig = figure('Color', 'w', 'Units', 'centimeters', ...
'Position', [0, 0, widthCm, heightCm], ...
'PaperPositionMode', 'auto', ...
'InvertHardcopy', 'off');
end
function savePaperFigure(fig, stem, widthCm, heightCm, P)
[folder, ~, ~] = fileparts(stem);
ensureDir(folder);
set(fig, 'Units', 'centimeters');
set(fig, 'Position', [0, 0, widthCm, heightCm]);
drawnow;
if P.savePdf
exportgraphics(fig, stem + ".pdf", ...
'ContentType', P.export.pdfContentType, ...
'BackgroundColor', 'white', ...
'Resolution', P.export.pdfResolution, ...
'Width', widthCm, ...
'Height', heightCm, ...
'Padding', P.export.padding, ...
'Units', 'centimeters');
end
if P.savePng
exportgraphics(fig, stem + ".png", ...
'ContentType', P.export.pngContentType, ...
'BackgroundColor', 'white', ...
'Resolution', P.export.pngResolution, ...
'Width', widthCm, ...
'Height', heightCm, ...
'Padding', P.export.padding, ...
'Units', 'centimeters');
end
end
function data = loadSweepData(results, trajNames)
data = struct('name', {}, 'time', {}, 'log', {}, 'par', {});
for i = 1:numel(trajNames)
idx = find(string(results.Trajectory) == trajNames(i), 1);
if isempty(idx) || strlength(string(results.ErrorMessage(idx))) > 0
continue;
end
matFile = string(results.MatFile(idx));
if ~isfile(matFile)
continue;
end
S = load(matFile, 'time', 'log', 'par');
S.name = trajNames(i);
data(end+1) = S; %#ok<AGROW>
end
end
function resultFile = latestSweepResultsFile()
rootDir = fullfile(pwd, "results", "main_trajectory_sweep");
files = dir(fullfile(rootDir, "*", "main_trajectory_sweep_results.mat"));
if isempty(files)
error("No trajectory sweep result found under %s.", rootDir);
end
[~, idx] = max([files.datenum]);
resultFile = string(fullfile(files(idx).folder, files(idx).name));
end
function names = selectedNames(value, defaultNames)
names = string(value);
names = names(strlength(names) > 0);
if isempty(names)
names = string(defaultNames);
end
names = names(:).';
end
function values = appendMissing(values, observed)
values = string(values(:)).';
observed = string(observed(:)).';
values = [values, observed(~ismember(observed, values))];
values = values(ismember(values, observed));
end
function offsets = groupedOffsets(n, width)
if n <= 1
offsets = 0;
else
offsets = linspace(-width/2, width/2, n);
end
end
function color = controllerColor(P, ctrlName, idx)
field = matlab.lang.makeValidName(char(ctrlName));
if isfield(P.controllerColor, field)
color = P.controllerColor.(field);
else
color = lines(max(idx, 1));
color = color(end,:);
end
end
function label = controllerLabel(P, ctrlName)
field = matlab.lang.makeValidName(char(ctrlName));
if isfield(P.controllerLabel, field)
label = string(P.controllerLabel.(field));
else
label = string(ctrlName);
end
end
function txt = titleFor(titleStruct, name)
field = matlab.lang.makeValidName(char(name));
if isfield(titleStruct, field)
txt = string(titleStruct.(field));
else
txt = string(name);
end
end
function applyOptionalYLimit(ax, yLimits, trajName)
field = matlab.lang.makeValidName(char(trajName));
if isstruct(yLimits) && isfield(yLimits, field) ...
&& ~isempty(yLimits.(field))
ylim(ax, yLimits.(field));
end
end
function labels = disturbanceLevelLabels(levelNames, labelStruct)
labels = string(levelNames);
for i = 1:numel(labels)
field = matlab.lang.makeValidName(char(labels(i)));
if isstruct(labelStruct) && isfield(labelStruct, field)
labels(i) = string(labelStruct.(field));
end
end
end
function addDisturbanceLevelLabels(ax, labels, fontSize)
fig = ancestor(ax, 'figure');
n = numel(labels);
if n == 0
return;
end
ax.Units = 'normalized';
pos = ax.Position;
bottomPad = 0.10;
ax.Position = [pos(1), pos(2) + bottomPad, ...
pos(3), max(0.1, pos(4) - bottomPad)];
boxW = 0.92*pos(3)/n;
boxH = 0.05;
boxY = max(0.01, pos(2) + 0.025);
for i = 1:n
xCenter = pos(1) + pos(3)*(i - 0.5)/n;
annotation(fig, 'textbox', ...
[xCenter - boxW/2, boxY, boxW, boxH], ...
'String', char(labels(i)), ...
'HorizontalAlignment', 'center', ...
'VerticalAlignment', 'top', ...
'Interpreter', 'none', ...
'EdgeColor', 'none', ...
'FitBoxToText', 'off', ...
'FontSize', fontSize);
end
end
function setTrajectoryLimits(ax, pAll)
span = max(max(pAll, [], 2) - min(pAll, [], 2));
margin = 0.1*max(span, 1);
xlim(ax, [min(pAll(1,:))-margin, max(pAll(1,:))+margin]);
ylim(ax, [min(pAll(2,:))-margin, max(pAll(2,:))+margin]);
zlim(ax, [min(pAll(3,:))-margin, max(pAll(3,:))+margin]);
end
function formatAxes(ax, P)
ax.FontName = char(P.fontName);
ax.FontSize = P.fontSize;
ax.LineWidth = P.axisLineWidth;
ax.TickLabelInterpreter = 'none';
box(ax, 'on');
end
function ensureDir(folder)
if strlength(string(folder)) == 0
return;
end
if ~exist(folder, 'dir')
mkdir(folder);
end
end
🔗 参考文献
[1] T. Lee, M. Leok, and N. H. McClamroch, “Geometric Tracking Control of a Quadrotor UAV on SE(3),” Mar. 10, 2010, arXiv: arXiv:1003.2005. doi: 10.48550/arXiv.1003.2005.
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