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dubins_core.m
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dubins_core.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This function will find the shortest dubins curve between two points
% Input:
% p1/p2: Initial and ending 2-D pose
% In row vectors, e.g. [x, y, theta]
% r: turning radius of the curve
% Output:
% param: a struct that includes 4 field:
% p_init: Initial pose, equals to input p1
% type: One of the 6 types of the dubins curve
% r: Turning radius, same as input r, also the scaling factor for
% the dubins paramaters
% seg_param: angle or normalized length, in row vector [pr1, pr2, pr3]
% Reference:
% https://github.com/AndrewWalker/Dubins-Curves#shkel01
% Shkel, A. M. and Lumelsky, V. (2001). "Classification of the Dubins
% set". Robotics and Autonomous Systems 34 (2001) 179¡V202
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Original Source (in C): Andrew Walker
% MATLAB-lization: Ewing Kang
% Date: 2016.2.28
% contact: f039281310 [at] yahoo.com.tw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Copyright (c) 2018 Ewing Kang %
% Released under GPLv3 license %
% This function is a MATLAB re-written from Andrew Walker's work, which %
% was originally distributed under MIT license in C language %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function param = dubins_core(p1, p2, r)
%%%%%%%%%%%%%%%%%% DEFINE %%%%%%%%%%%%%%%%%
% Here are some usefuldefine headers for better implementation
% there are 6 types of dubin's curve, only one will have minimum cost
% LSL = 1;
% LSR = 2;
% RSL = 3;
% RSR = 4;
% RLR = 5;
% LRL = 6;
% The three segment types a path can be made up of
% L_SEG = 1;
% S_SEG = 2;
% R_SEG = 3;
% The segment types for each of the Path types
%{
DIRDATA = [ L_SEG, S_SEG, L_SEG ;...
L_SEG, S_SEG, R_SEG ;...
R_SEG, S_SEG, L_SEG ;...
R_SEG, S_SEG, R_SEG ;...
R_SEG, L_SEG, R_SEG ;...
L_SEG, R_SEG, L_SEG ];
%}
%%%%%%%%%%%%%%%% END DEFINE %%%%%%%%%%%%%%%%
% the return parameter
param.p_init = p1; % the initial configuration
param.seg_param = [0, 0, 0]; % the lengths of the three segments
param.r = r; % model forward velocity / model angular velocity turning radius
param.type = -1; % path type. one of LSL, LSR, ...
param.flag = 0;
%%%%%%%%%%%%%%%%%%%%%%%%% START %%%%%%%%%%%%%%%%%%%%%%%%%
% First, basic properties and normalization of the problem
dx = p2(1) - p1(1);
dy = p2(2) - p1(2);
D = sqrt( dx^2 + dy^2 );
d = D / r; % distance is shrunk by r, this make lengh calculation very easy
if( r <= 0 )
param.flag = -1;
return;
end
theta = mod(atan2( dy, dx ), 2*pi);
alpha = mod((p1(3) - theta), 2*pi);
beta = mod((p2(3) - theta), 2*pi);
% Second, we find all possible curves
best_word = -1;
best_cost = -1;
test_param(1,:) = dubins_LSL(alpha, beta, d);
test_param(2,:) = dubins_LSR(alpha, beta, d);
test_param(3,:) = dubins_RSL(alpha, beta, d);
test_param(4,:) = dubins_RSR(alpha, beta, d);
test_param(5,:) = dubins_RLR(alpha, beta, d);
test_param(6,:) = dubins_LRL(alpha, beta, d);
for i = 1:1:6
if(test_param(i,1) ~= -1)
cost = sum(test_param(i,:));
if(cost < best_cost) || (best_cost == -1)
best_word = i;
best_cost = cost;
param.seg_param = test_param(i,:);
param.type = i;
end
end
end
if(best_word == -1)
param.flag = -2; % NO PATH
return;
else
return;
end
end
function param = dubins_LSL(alpha, beta, d)
tmp0 = d + sin(alpha) - sin(beta);
p_squared = 2 + (d*d) -(2*cos(alpha - beta)) + (2*d*(sin(alpha) - sin(beta)));
if( p_squared < 0 )
param = [-1, -1, -1];
return;
else
tmp1 = atan2( (cos(beta)-cos(alpha)), tmp0 );
t = mod((-alpha + tmp1 ), 2*pi);
p = sqrt( p_squared );
q = mod((beta - tmp1 ), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return ;
end
end
function param = dubins_LSR(alpha, beta, d)
p_squared = -2 + (d*d) + (2*cos(alpha - beta)) + (2*d*(sin(alpha)+sin(beta)));
if( p_squared < 0 )
param = [-1, -1, -1];
return;
else
p = sqrt( p_squared );
tmp2 = atan2( (-cos(alpha)-cos(beta)), (d+sin(alpha)+sin(beta)) ) - atan2(-2.0, p);
t = mod((-alpha + tmp2), 2*pi);
q = mod(( -mod((beta), 2*pi) + tmp2 ), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return ;
end
end
function param = dubins_RSL(alpha, beta, d)
p_squared = (d*d) -2 + (2*cos(alpha - beta)) - (2*d*(sin(alpha)+sin(beta)));
if( p_squared< 0 )
param = [-1, -1, -1];
return;
else
p = sqrt( p_squared );
tmp2 = atan2( (cos(alpha)+cos(beta)), (d-sin(alpha)-sin(beta)) ) - atan2(2.0, p);
t = mod((alpha - tmp2), 2*pi);
q = mod((beta - tmp2), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return ;
end
end
function param = dubins_RSR(alpha, beta, d)
tmp0 = d-sin(alpha)+sin(beta);
p_squared = 2 + (d*d) -(2*cos(alpha - beta)) + (2*d*(sin(beta)-sin(alpha)));
if( p_squared < 0 )
param = [-1, -1, -1];
return;
else
tmp1 = atan2( (cos(alpha)-cos(beta)), tmp0 );
t = mod(( alpha - tmp1 ), 2*pi);
p = sqrt( p_squared );
q = mod(( -beta + tmp1 ), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return;
end
end
function param = dubins_RLR(alpha, beta, d)
tmp_rlr = (6. - d*d + 2*cos(alpha - beta) + 2*d*(sin(alpha)-sin(beta))) / 8.;
if( abs(tmp_rlr) > 1)
param = [-1, -1, -1];
return;
else
p = mod(( 2*pi - acos( tmp_rlr ) ), 2*pi);
t = mod((alpha - atan2( cos(alpha)-cos(beta), d-sin(alpha)+sin(beta) ) + mod(p/2, 2*pi)), 2*pi);
q = mod((alpha - beta - t + mod(p, 2*pi)), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return;
end
end
function param = dubins_LRL(alpha, beta, d)
tmp_lrl = (6. - d*d + 2*cos(alpha - beta) + 2*d*(- sin(alpha) + sin(beta))) / 8.;
if( abs(tmp_lrl) > 1)
param = [-1, -1, -1]; return;
else
p = mod(( 2*pi - acos( tmp_lrl ) ), 2*pi);
t = mod((-alpha - atan2( cos(alpha)-cos(beta), d+sin(alpha)-sin(beta) ) + p/2), 2*pi);
q = mod((mod(beta, 2*pi) - alpha -t + mod(p, 2*pi)), 2*pi);
param(1) = t;
param(2) = p;
param(3) = q;
return;
end
end