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。

Two-link Planar Arm

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

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

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

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。

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。

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

Two-link Planar Arm: Dynamic Model L{1} = link([ 0 1 0 0 0], 'standard'); % D-H param L{2} = link([ 0 1 0 0 0], 'standard'); L{1}.m = 50; %link mass L{2}.m = 50; L{1}.r = [ -0.5 0 0]; % center of mass referred to the link frame L{2}.r = [ -0.5 0 0]; L{1}.I = [ 0 0 10 0 0 0]; % inertia tensor L{2}.I = [ 0 0 10 0 0 0]; 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=[0 9.81 0];

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