2-DoF Manipulator (realistic)

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This article is about the dynamics of 2-DoF manipulator, with a bit of twist. In textbooks, you will usually find a manipulator with magical actuators that can apply infinitely large torque and do not have any inertial properties. In reality, however, actuators can only apply limited torques and have velocity limits. Moreover, motors' (actuators') inertial properties may affect the dynamics greatly (as discussed here).

Let's derive a more realistic dynamics of a manipulator in following steps:
1. analyze motion (kinematics)
2. get the energy of the robot
3. derive dynamics using Euler-Lagrangian

1. Kinematics



What we care, in fact, is simply the kinetic energy and the potential energy of the system. Meaning, we need to know how fast the masses are moving (velocity) and how high they are located (position).


Let's define the coordinate system first,
\begin{equation} \displaylines{ \hat{x} = \begin{bmatrix} 1\\0 \end{bmatrix}, \quad \hat{y} = \begin{bmatrix} 0\\1 \end{bmatrix} \\ \hat{e}_1 = c_1 \hat{x} + s_1\hat{y}, \quad \hat{e}_2 = c_{12} \hat{x} + s_{12} \hat{y} } \end{equation}

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