How to put a spline through points representing a path

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Amritz Ansara
Amritz Ansara 2021-3-23
评论： darova 2021-3-24
I have this path here. The waypoints are marked in red. The line going through each point (linearly) is in blue. The path was made programmatically - if you'd like the code to make the path it is:
%% Create Path
%have it come from the south going north
r = 15;
start_pos = [-r * 7 0 0];
%initial approach to curve
waypoints = [start_pos;
-r*5 1.9 -0;
-r*3 1.7 0;
-r*2.5 1.8 0;
-r*2 2.4 0;
-r*1.5 3.4 0;
-r*1 3.7 0;
-r*0.5 3.4 0];
%Set constant radius curve
angle_req = 90;
a = linspace(0, angle_req*pi/180, 10);
round_x = r * sin(a) + waypoints(end, 1);
round_y = r * cos(a) - r + waypoints(end, 2);
for i = 1:length(a)
waypoints = [waypoints; round_x(i) round_y(i) 0];
end
%Set exit
last_point = waypoints(end, :);
waypoints = [waypoints;
last_point(1) + r*1*cos(a(end)) last_point(2) - r*1*sin(a(end)) 0;
last_point(1) + r*2.5*cos(a(end)) last_point(2) - r*2.5*sin(a(end)) 0;
last_point(1) + max(r*5, 100)*cos(a(end)) last_point(2) - max(r*5, 100)*sin(a(end)), 0];
%last_point(1) + r*1*cos(a(end))- 0.2*r*sin(a(end)) last_point(2) - r*1*sin(a(end)) 0;
%last_point(1) + r*2.5*cos(a(end)) - 0.5*r*sin(a(end)) last_point(2) - r*2.5*sin(a(end)) 0;
%last_point(1) + max(r*5, 100)*cos(a(end)) - 0.5*r*sin(a(end)) last_point(2) - max(r*5, 100)*sin(a(end)), 0];
%plot
plot(waypoints(:, 2), waypoints(:, 1), 'LineWidth', 5)
hold on;
plot(waypoints(:, 2), waypoints(:, 1), 'ro', 'LineWidth', 3)
hold off;
st = sprintf("A %d degree turn (anticlockwise) of %d metre radius", angle_req, r);
grid on;
title(st)
ylabel('X Distance (m)'); xlabel('Y Distance(m)')
I am worried about the fact that there could be discontinuous curvature, when I put the waypoints into the driving scenario and draw up the vehicle's trajectory.
I would like to put a smooth spline through these points (which will guarantee a curve with continuous curvature) and then resample my waypoints from there. How do I do this, considering most spline fitting tools rely on a one-to-one correspondence between x-values and y-values? I've tried the curve fitting toolbox but it sends me an error relating to this correspondence.
  2 个评论
Bruno Luong
Bruno Luong 2021-3-24
Cubic spline provides a C^2 interpolation result, meaning the second derivatve is continuous (in fact continuity is enrured for all derivative of order <= 2). So the curvature is continuous, it might change quickly but it is contnuous.

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采纳的回答

Bruno Luong
Bruno Luong 2021-3-24
编辑：Bruno Luong 2021-3-24
Put this next-below to your code
waypoints=unique(waypoints,'rows','stable'); % <- this is required because your example as duplicated points
% Spline interpolation
t=cumsum([0;sqrt(sum(diff(waypoints,1,1).^2,2))]);
ti=linspace(t(1),t(end),512);
xyzi=interp1(t,waypoints,ti,'spline');
%plot
figure
plot(waypoints(:,2),waypoints(:,1),'or',xyzi(:,2),xyzi(:,1),'b');
axis equal
You don't have to think what kind of angle to turn and it can work for multiple winding path.

更多回答（1 个）

darova
darova 2021-3-24
What about rotation?
R = @(a)[cosd(a) -sind(a);sind(a) cosd(a)]; % rotation maitrx
v1 = R(45)*[x(:) y(:)]'; % rotate data to make exclusive X coordinate
x1 = linspace(v1(1,1),v1(1,end),100); % denser mesh
y1 = spline(v1(1,:),v1(2,:),x1); % interpolation with spline
v2 = R(-45)*[x1(:) y1(:)]'; % rotate to original position

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