1. Two-link Planar Arm
让学生做练习。推导时计算bij对q的偏导数。最终动力学模型保留在黑板上。

2. Two-link Planar Arm

3. Joint Space Dynamic Model
Viscous friction torques
Actuation torques
Coulomb friction torques
Coriolis/ centripetal torque
Force and moment exerted on the environment
Multi-input-multi-output; Strong coupling; Nonlinearity

4. Direct Dynamics and Inverse Dynamics
Given joint torques and initial joint position and velocity, determine joint acceleration
Useful for simulation
Inverse dynamics:
Given joint position, velocity and acceleration, determine joint torques
Useful for trajectory planning and control algorithm implementation

5. Two-link Planar Arm: Inverse Dynamics (Example 4.2)
Matlab Toolbox

6. Two-link Planar Arm: Inverse Dynamics (Example 4.2)
注意惯性项和手臂的姿态有关系，在本例中手臂在慢慢展开，因此对关节1来说惯性项会越来 越大，且由于数值大是主要因素。在0 .25 秒速度达到最大，这时候科氏力和离心力大，注意在0 .25 秒附近对关节力矩的影响比较大。重力项和手臂的姿势有关，注意运动停止后的关节力矩就只有重力项了。

7. Two-link Planar Arm: Direct Dynamics (Example 4.2)
Robot toolbox
Case 1: no actuating torques
Case 2: actuating only the first joint
Case 3: simulate puma560 by yourself
可以课堂上用SIMULINK建立PUMA560模型并仿真；可以先用DRIVEBOT属性一下此机器人的特性。

8. Matlab Toolbox Useful functions Simulink library Roblocks.mdl
Accel (pp 20): Compute manipulator forward dynamics
Coriolis(pp 22): Compute the manipulator Coriolis/centripetal torque components
Gravload(pp 33):Compute the manipulator gravity torque components
Itorque(pp40): Compute the manipulator inertia torque component
Rne(57): Compute inverse dynamics via recursive Newton-Euler formulation
Simulink library
Roblocks.mdl

9. Two-link Planar Arm: Dynamic Model
L{1} = link([ ], 'standard'); % D-H param
L{2} = link([ ], 'standard');
L{1}.m = 50; %link mass
L{2}.m = 50;
L{1}.r = [ ]; % center of mass referred to the link frame
L{2}.r = [ ];
L{1}.I = [ ]; % inertia tensor
L{2}.I = [ ];
L{1}.Jm = 0.01; %inertia of motor
L{2}.Jm = 0.01;
L{1}.G = 100; %gear reduction ratio
L{2}.G = 100;
global mytl;
mytl = robot(L);
mytl.gravity=[ ];

10. Joint Space Dynamic Model
Viscous friction torques
Actuation torques
Coulomb friction torques
Coriolis/ centripetal torque
Force and moment exerted on the environment
Multi-input-multi-output; Strong coupling; Nonlinearity