机器人学回炉重造（4）：动力学仿真（附牛顿-欧拉递归逆动力学算法matlab代码）

写在前面

动力学

另外，在高自由度的情况下，牛顿-欧拉算法更容易通过编程实现，核心如下：

平面三连杆机械臂瞬态运动的牛顿-欧拉递归逆动力学求解Matlab

问题描述

自写Matlab实现代码

% 三连杆机械臂瞬态运动的牛顿-欧拉递归逆动力学求解：
% 参数：（运动指令）各关节运动角度， 关节速度， 关节加速度（3*1矩阵）
% 返回值：各关节力矩（3*1矩阵）
function tau = three_dof_dynamics(theta, theta_d, theta_dd)
% 改进D-H参数
th(1) = theta(1)*pi/180; d(1) = 0; a(1) = 0; alp(1) = 0;
th(2) = theta(2)*pi/180; d(2) = 0; a(2) = 4; alp(2) = 0;
th(3) = theta(3)*pi/180; d(3) = 0; a(3) = 3; alp(3) = 0;
% base_link的各项初始值
w00 = [0; 0; 0]; v00 = [0; 0; 0]; w00d = [0; 0; 0]; v00d = [0; 9.8; 0];
% 各关节p及各link质心pc的距离(假设质心在几何中心)
p10 = [0; 0; 0]; p21 = [4; 0; 0];
p32 = [3; 0; 0]; p43 = [2; 0; 0];
pc11 = [2; 0; 0]; pc22 = [1.5; 0; 0];
pc33 = [1; 0; 0];
z = [0; 0; 1];
% 各连杆质量
m1 = 20; m2 = 15; m3 = 10;
% 惯性张量
I1 = [0 0 0; 0 0 0; 0 0 0.5]; I2 = [0 0 0; 0 0 0; 0 0 0.2]; I3 = [0 0 0; 0 0 0; 0 0 0.1];
% 各齐次矩阵function T = MDHTrans(alpha, a, d, theta)
% 旋转矩阵
T01 = MDHTrans(alp(1), a(1), d(1), th(1));
T12 = MDHTrans(alp(2), a(2), d(2), th(2));
T23 = MDHTrans(alp(3), a(3), d(3), th(3));
R01 = T01(1:3, 1:3); R12 = T12(1:3, 1:3); R23 = T23(1:3, 1:3);
R10 = R01'; R21 = R12'; R32 = R23';
R34 = [1 0 0; 0 1 0; 0 0 1];
%% Outward iterations: i: 0->2
% i = 0 连杆1
w11 = R10*w00 + theta_d(1)*z;
w11d = R10*w00d + cross(R10*w00, z*theta_d(1)) + theta_dd(1)*z;
v11d = R10*(cross(w00d, p10) + cross(w00, cross(w00, p10)) + v00d);
vc11d = cross(w11d, pc11) + cross(w11, cross(w11, pc11)) + v11d;
F11 = m1*vc11d;
N11 = I1*w11d + cross(w11, I1*w11);
% i = 1 连杆2
w22 = R21*w11 + theta_d(2)*z;
w22d = R21*w11d + cross(R21*w11, z*theta_d(2)) + theta_dd(2)*z;
v22d = R21*(cross(w11d, p21) + cross(w11, cross(w11, p21)) + v11d);
vc22d = cross(w22d, pc22) + cross(w22, cross(w22, pc22)) + v22d;
F22 = m2*vc22d;
N22 = I2*w22d + cross(w22, I2*w22);
% i = 2 连杆3
w33 = R32*w22 + theta_d(3)*z;
w33d = R32*w22d + cross(R32*w22, z*theta_d(3)) + theta_dd(3)*z;
v33d = R32*(cross(w22d, p32) + cross(w22, cross(w22, p32)) + v22d);
vc33d = cross(w33d, pc33) + cross(w33, cross(w33, pc33)) + v33d;
F33 = m3*vc33d;
N33 = I3*w33d + cross(w33, I3*w33);%% Inward iterations: i: 3->1
f44 = [0; 0; 0]; n44 = [0; 0; 0];
% i = 3
f33 = R34*f44 + F33;
n33 = N33 + R34*n44 + cross(pc33, F33) + cross(p43, R34*f44);
tau(3) = n33'*z;
% i = 2
f22 = R23*f33 + F22;
n22 = N22 + R23*n33 + cross(pc22, F22) + cross(p32, R23*f33);
tau(2) = n22'*z;
% i =1
f11 = R12*f22 + F11;
n11 = N11 + R12*n22 + cross(pc11, F11) + cross(p21, R12*f22);
tau(1) = n11'*z;end

运行结果：

验证代码

clear;
clc;
% 改进D-H
th(1) = 0; d(1) = 0; a(1) = 0; alp(1) = 0;
th(2) = 0; d(2) = 0; a(2) = 4; alp(2) = 0;
th(3) = 0; d(3) = 0; a(3) = 3; alp(3) = 0;% DH parameters  th     d    a    alpha
L1 = Link([th(1), d(1), a(1), alp(1)], 'modified');
L2 = Link([th(2), d(2), a(2), alp(2)], 'modified');
L3 = Link([th(3), d(3), a(3), alp(3)], 'modified');L1.m = 20; L2.m = 15; L3.m = 10;% 连杆质量
% 连杆质心位置
L1.r = [2 0 0];
L2.r = [1.5 0 0];
L3.r = [1 0 0];
% 连杆惯性张量
L1.I = [0 0 0; 0 0 0; 0 0 0.5];
L2.I = [0 0 0; 0 0 0; 0 0 0.2];
L3.I = [0 0 0; 0 0 0; 0 0 0.1];
robot = SerialLink([L1, L2, L3]);
robot.name = 'Plan3R';
robot.comment = 'xuuyann';
robot.display()
% Forward Pose Kinematics
theta=[10, 20, 30]*pi/180;
robot.teach()
robot.plot(theta);
t0=robot.fkine(theta)    %末端执行器位姿
% TAU = R.rne(Q, QD, QDD, GRAV)
tau = robot.rne(theta, [1, 2, 3], [0.5, 1, 1.5], [0 9.8 0])

运算结果：

验证完毕。

六自由度机械臂瞬态运动的牛顿-欧拉递归逆动力学求解Matlab（未验证）（2019.11.29已通过函数验证）

前情提要：机器人学回炉重造（1）：正运动学、标准D-H法与改进D-H法的区别与应用（附ABB机械臂运动学建模matlab代码）

自写代码

% 六自由度机械臂瞬态运动的牛顿-欧拉递归逆动力学求解：
% 参数：（运动指令）各关节运动角度， 关节速度， 关节加速度（6*1矩阵）
% 返回值：各关节力矩（6*1矩阵）
% 由于我不清楚机械臂实际质量参数，因此下面的惯性张量及质心位置均为假设杜撰
function tau = myNewtonEuler(theta, theta_d, theta_dd)
%% 初始化
% 改进D-H参数
th(1) = theta(1)*pi/180; d(1) = 0; a(1) = 0; alp(1) = 0;
th(2) = theta(2)*pi/180; d(2) = 0; a(2) = 0.320; alp(2) = pi/2;
th(3) = theta(3)*pi/180; d(3) = 0; a(3) = 0.975; alp(3) = 0;
th(4) = theta(4)*pi/180; d(4) = 0.887; a(4) = 0.2; alp(4) = pi/2;
th(5) = theta(5)*pi/180; d(5) = 0; a(5) = 0; alp(5) = -pi/2;
th(6) = theta(6)*pi/180; d(6) = 0; a(6) = 0; alp(6) = pi/2;
w00 = [0; 0; 0]; v00 = [0; 0; 0]; w00d = [0; 0; 0]; v00d = [0; 0; 9.8];%base_link的各项初始值z = [0; 0; 1];
% 各连杆质量L1.m = 7.36; L2.m = 36.26; L3.m = 12.44;
% L4.m = 1.21; L5.m = 1.21; L6.m = 1;
m1 = 7.36; m2 = 36.26; m3 = 12.44; m4 = 1.21; m5 = 1.21; m6 = 1;
% 惯性张量
I1 = [0 0 0; 0 0 0; 0 0 0.6]; I2 = [0 0 0; 0 0 0; 0 0 0.5]; I3 = [0 0 0; 0 0 0; 0 0 0.4];
I4 = [0 0 0; 0 0 0; 0 0 0.3]; I5 = [0 0 0; 0 0 0; 0 0 0.2]; I6 = [0 0 0; 0 0 0; 0 0 0.1];% 各齐次矩阵function T = MDHTrans(alpha, a, d, theta)
T01 = MDHTrans(alp(1), a(1), d(1), th(1));
T12 = MDHTrans(alp(2), a(2), d(2), th(2));
T23 = MDHTrans(alp(3), a(3), d(3), th(3));
T34 = MDHTrans(alp(4), a(4), d(4), th(4));
T45 = MDHTrans(alp(5), a(5), d(5), th(5));
T56 = MDHTrans(alp(6), a(6), d(6), th(6));% 各关节p及各link质心pc的距离(假设质心在几何中心)
p10 = T01(1: 3, 4); p21 = T12(1: 3, 4); p32 = T23(1: 3, 4);
p43 = T34(1: 3, 4); p54 = T45(1: 3, 4); p65 = T56(1: 3, 4); p76 = [0, 0, 0]';
pc11 = 0.5 * p21; pc22 = 0.5 * p32; pc33 = 0.5 * p43;
pc44 = 0.5 * p54; pc55 = 0.5 * p65; pc66 = 0.5 * p76;% 旋转矩阵
R01 = T01(1:3, 1:3); R12 = T12(1:3, 1:3); R23 = T23(1:3, 1:3);
R34 = T34(1:3, 1:3); R45 = T45(1:3, 1:3); R56 = T56(1:3, 1:3);
R10 = R01'; R21 = R12'; R32 = R23';
R43 = R34'; R54 = R45'; R65 = R56';
R67 = [1 0 0; 0 1 0; 0 0 1]; R76 = R67';%% Outward iterations: i: 0->5
% 连杆1到连杆6向外迭代
% i = 0
w11 = R10*w00 + theta_d(1)*z;
w11d = R10*w00d + cross(R10*w00, z*theta_d(1)) + theta_dd(1)*z;
v11d = R10*(cross(w00d, p10) + cross(w00, cross(w00, p10)) + v00d);
vc11d = cross(w11d, pc11) + cross(w11, cross(w11, pc11)) + v11d;
F11 = m1*vc11d;
N11 = I1*w11d + cross(w11, I1*w11);
% i = 1
w22 = R21*w11 + theta_d(2)*z;
w22d = R21*w11d + cross(R21*w11, z*theta_d(2)) + theta_dd(2)*z;
v22d = R21*(cross(w11d, p21) + cross(w11, cross(w11, p21)) + v11d);
vc22d = cross(w22d, pc22) + cross(w22, cross(w22, pc22)) + v22d;
F22 = m2*vc22d;
N22 = I2*w22d + cross(w22, I2*w22);
% i = 2
w33 = R32*w22 + theta_d(3)*z;
w33d = R32*w22d + cross(R32*w22, z*theta_d(3)) + theta_dd(3)*z;
v33d = R32*(cross(w22d, p32) + cross(w22, cross(w22, p32)) + v22d);
vc33d = cross(w33d, pc33) + cross(w33, cross(w33, pc33)) + v33d;
F33 = m3*vc33d;
N33 = I3*w33d + cross(w33, I3*w33);
% i= 3
w44 = R43*w33 + theta_d(4)*z;
w44d = R43*w33d + cross(R43*w33, z*theta_d(4)) + theta_dd(4)*z;
v44d = R43*(cross(w33d, p43) + cross(w33, cross(w33, p43)) + v33d);
vc44d = cross(w44d, pc44) + cross(w44, cross(w44, pc44)) + v44d;
F44 = m4*vc44d;
N44 = I4*w44d + cross(w44, I4*w44);
% i = 4
w55 = R54*w44 + theta_d(5)*z;
w55d = R54*w44d + cross(R54*w44, z*theta_d(5)) + theta_dd(5)*z;
v55d = R54*(cross(w44d, p54) + cross(w44, cross(w44, p54)) + v44d);
vc55d = cross(w55d, pc55) + cross(w55, cross(w55, pc55)) + v55d;
F55 = m5*vc55d;
N55 = I5*w55d + cross(w55, I5*w55);
% i = 5
w66 = R65*w55 + theta_d(6)*z;
w66d = R65*w55d + cross(R65*w55, z*theta_d(6)) + theta_dd(6)*z;
v66d = R65*(cross(w55d, p65) + cross(w55, cross(w55, p65)) + v55d);
vc66d = cross(w66d, pc66) + cross(w66, cross(w66, pc66)) + v66d;
F66 = m6*vc66d;
N66 = I6*w66d + cross(w66, I6*w66);%% Inward iterations: i: 6->1
% 连杆6到连杆1向内迭代
f77 = [0; 0; 0]; n77 = [0; 0; 0];
% i = 6
f66 = R67*f77 + F66;
n66 = N66 + R67*n77 + cross(pc66, F66) + cross(p76, R67*f77);
tau(6) = n66'*z;
% i = 5
f55 = R56*f66 + F55;
n55 = N55 + R56*n66 + cross(pc55, F55) + cross(p65, R56*f66);
tau(5) = n55'*z;
% i = 4
f44 = R45*f55 + F44;
n44 = N44 + R45*n55 + cross(pc44, F44) + cross(p54, R45*f55);
tau(4) = n44'*z;
% i = 3
f33 = R34*f44 + F33;
n33 = N33 + R34*n44 + cross(pc33, F33) + cross(p43, R34*f44);
tau(3) = n33'*z;
% i = 2
f22 = R23*f33 + F22;
n22 = N22 + R23*n33 + cross(pc22, F22) + cross(p32, R23*f33);
tau(2) = n22'*z;
% i =1
f11 = R12*f22 + F11;
n11 = N11 + R12*n22 + cross(pc11, F11) + cross(p21, R12*f22);
tau(1) = n11'*z;end

运行结果：

验证代码：

% Modified DH
% ABB robot
clear;
clc;
th(1) = 0; d(1) = 0; a(1) = 0; alp(1) = 0;
th(2) = 0; d(2) = 0; a(2) = 0.320; alp(2) = pi/2;
th(3) = 0; d(3) = 0; a(3) = 0.975; alp(3) = 0;
th(4) = 0; d(4) = 0.887; a(4) = 0.2; alp(4) = pi/2;
th(5) = 0; d(5) = 0; a(5) = 0; alp(5) = -pi/2;
th(6) = 0; d(6) = 0; a(6) = 0; alp(6) = pi/2;
% DH parameters  th     d    a    alpha  sigma
L1 = Link([th(1), d(1), a(1), alp(1), 0], 'modified');
L2 = Link([th(2), d(2), a(2), alp(2), 0], 'modified');
L3 = Link([th(3), d(3), a(3), alp(3), 0], 'modified');
L4 = Link([th(4), d(4), a(4), alp(4), 0], 'modified');
L5 = Link([th(5), d(5), a(5), alp(5), 0], 'modified');
L6 = Link([th(6), d(6), a(6), alp(6), 0], 'modified');L1.m = 7.36; L2.m = 36.26; L3.m = 12.44;
L4.m = 1.21; L5.m = 1.21; L6.m = 1;
% L1.r =
L1.I = [0 0 0; 0 0 0; 0 0 0.6];
L2.I = [0 0 0; 0 0 0; 0 0 0.5];
L3.I = [0 0 0; 0 0 0; 0 0 0.4];
L4.I = [0 0 0; 0 0 0; 0 0 0.3];
L5.I = [0 0 0; 0 0 0; 0 0 0.2];
L6.I = [0 0 0; 0 0 0; 0 0 0.1];T01 = MDHTrans(alp(1), a(1), d(1), th(1));
T12 = MDHTrans(alp(2), a(2), d(2), th(2));
T23 = MDHTrans(alp(3), a(3), d(3), th(3));
T34 = MDHTrans(alp(4), a(4), d(4), th(4));
T45 = MDHTrans(alp(5), a(5), d(5), th(5));
T56 = MDHTrans(alp(6), a(6), d(6), th(6));
% 各关节p及各link质心pc的距离(假设质心在几何中心)
p10 = T01(1: 3, 4); p21 = T12(1: 3, 4); p32 = T23(1: 3, 4);
p43 = T34(1: 3, 4); p54 = T45(1: 3, 4); p65 = T56(1: 3, 4); p76 = [0, 0, 0]';
pc11 = 0.5 * p21
pc22 = 0.5 * p32
pc33 = 0.5 * p43
pc44 = 0.5 * p54
pc55 = 0.5 * p65
pc66 = 0.5 * p76
% 连杆质心位置
L1.r = pc11; L2.r = pc22; L3.r = pc33;
L4.r = pc44; L5.r = pc55; L6.r = pc66;robot = SerialLink([L1, L2, L3, L4, L5, L6]);
robot.name='ABBRobot-6-dof';
robot.display() theta = [-20, 120, -15, 30, 20, 10]*pi/180;
robot.teach();
robot.plot(theta);
% t = robot.fkine(theta)    %末端执行器位姿
% [~, T] = My_Forward_Kinematics_MDH(theta, d, a, alp)
qd = [6 5 4 3 2 1]; qdd = [6 5 4 3 2 1];
tau = robot.rne(theta, qd, qdd, [0, 0, 9.8])
TAU = myNewtonEuler(theta*180/pi, qd', qdd')

运行结果：

qd=[1 1 1 1 1 1]’; qdd = [6 5 4 3 2 1]’;

qd=[1 1 1 1 1 1]’; qdd=[1 1 1 1 1 1]’;

2019.11.29
感谢vaultboy2020指出的错误，正文程序已改正并得到正确的验证结果

2021.7.30
原文程序是基于9.10工具箱运行的，10.3.1以上版本的工具箱运行则会出现如下错误：
解决该错误的办法：
在建立机器人模型后，加上电机惯量，例如
L1.Jm = 0; L2.Jm = 0; L3.Jm = 0; L4.Jm = 0; L5.Jm = 0; L6.Jm = 0; % 新版的rne需要加电机惯量
即可运行成功。

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