GJK算法 matlab

GJK算法是一种用于计算两个凸多边形之间最小距离的算法。在Matlab中，可以使用以下代码实现GJK算法：

function [distance, simplex] = gjk(A, B, support)
    % 初始化
    d = [1; -1];
    v = support(A, B, d(1) * (A(:, 1) - B(:, 1)));
    simplex = v;
    d = -v;
    
    % 迭代
    while true
        v = support(A, B, d);
        if dot(v, d) < 0
            distance = norm(simplex(end, :));
            break;
        end
        simplex(end + 1, :) = v;
        [simplex, d] = doSimplex(simplex);
    end
end

function [simplex, d] = doSimplex(simplex)
    switch size(simplex, 1)
        case 2
            [simplex, d] = doLine(simplex);
        case 3
            [simplex, d] = doTriangle(simplex);
        case 4
            [simplex, d] = doTetrahedron(simplex);
    end
end

function [simplex, d] = doLine(simplex)
    AB = simplex(2, :) - simplex(1, :);
    AO = -simplex(1, :);
    if dot(AB, AO) > 0
        d = cross(cross(AB, AO), AB);
    else
        simplex(1, :) = [];
        d = -AO;
    end
end

function [simplex, d] = doTriangle(simplex)
    AB = simplex(2, :) - simplex(1, :);
    AC = simplex(3, :) - simplex(1, :);
    AO = -simplex(1, :);
    ABC = cross(AB, AC);
    if dot(cross(ABC, AC), AO) > 0
        if dot(AC, AO) > 0
            simplex(2, :) = [];
            d = cross(cross(AC, AO), AC);
        else
            if dot(AB, AO) > 0
                simplex(3, :) = [];
                d = cross(cross(AB, AO), AB);
            else
                simplex([1 2]) = simplex([2 1]);
                d = -AO;
            end
        end
    else
        if dot(cross(AB, ABC), AO) > 0
            if dot(AB, AO) > 0
                simplex(3, :) = [];
                d = cross(cross(AB, AO), AB);
            else
                simplex([1 2]) = simplex([2 1]);
                d = -AO;
            end
        else
            distance = dot(ABC, AO) / norm(ABC);
            simplex = simplex(abs(dot(simplex, ABC)) == max(abs(dot(simplex, ABC))), :);
            d = cross(cross(simplex(end, :), -AO), simplex(end, :));
        end
    end
end

function [simplex, d] = doTetrahedron(simplex)
    AB = simplex(2, :) - simplex(1, :);
    AC = simplex(3, :) - simplex(1, :);
    AD = simplex(4, :) - simplex(1, :);
    AO = -simplex(1, :);
    ABC = cross(AB, AC);
    ACD = cross(AC, AD);
    ADB = cross(AD, AB);
    if dot(cross(ABC, AB), AO) > 0
        simplex([1 2 3]) = simplex([2 1 3]);
        d = doTriangle(simplex(1:3, :));
    elseif dot(cross(ACD, AC), AO) > 0
        simplex([1 2 3]) = simplex([3 1 2]);
        d = doTriangle(simplex(1:3, :));
    elseif dot(cross(ADB, AD), AO) > 0
        simplex([1 2 3]) = simplex([4 1 2]);
        d = doTriangle(simplex(1:3, :));
    else
        distance = dot(ABC, AO) / norm(ABC);
        simplex = simplex(abs(dot(simplex, ABC)) == max(abs(dot(simplex, ABC))), :);
        d = cross(cross(simplex(end, :), -AO), simplex(end, :));
    end
end

function v = support(A, B, d)
    v = supportPoint(A, d) - supportPoint(B, -d);
end

function v = supportPoint(V, d)
    [~, i] = max(dot(V, repmat(d', size(V, 1), 1), 2));
    v = V(i, :);
end

其中，`A`和`B`分别是两个凸多边形的顶点坐标，`support`是一个函数，用于计算一个方向上的支撑点。函数返回值`distance`是两个凸多边形之间的最小距离，`simplex`是GJK算法中的单纯形。