uact_manipulator.rst

Underactuated Manipulator

In this section, the model of an underactuated manipulator is treated. The system consists of two bars with the mass M₁ and M₂ which are connected to each other via the joint G₂. The angle between them is designated by θ₂. The joint G₁ connects the first rod with the inertial system, the angle to the x-axis is labeled θ₁. In the joint G₁ the actuating torque Q is applied. The bars have the moments of inertia I₁ and I₂. The distances between the centers of mass to the joints are r₁ and r₂.

The modeling was taken from the thesis of Carsten Knoll (April, 2009) where in addition the inertia parameter η was introduced.

$$\begin{equation*} \eta = \frac{m_2 l_1 r_2}{I_2 + m_2 r_2^2} \end{equation*}$$

For the example shown here, strong inertia coupling was assumed with η = 0.9. By partial linearization to the output y = θ₁ one obtains the state representation with the states x = [θ₁, θ̇₁, θ₂, θ̇₂]^T and the new input $\tilde{u} = \ddot{\theta}_1$.

$$\begin{aligned} \begin{eqnarray*} \dot{x}_1 & = & x_2 \\\ \dot{x}_2 & = & \tilde{u} \\\ \dot{x}_3 & = & x_4 \\\ \dot{x}_4 & = & -\eta x_2^2 \sin(x_3) - (1 + \eta \cos(x_3))\tilde{u} \end{eqnarray*} \end{aligned}$$

For the system, a trajectory is to be determined for the transfer between two equilibrium positions within an operating time of T = 1.8[s].

$$\begin{aligned} \begin{equation*} x(0) = \begin{bmatrix} 0 \\ 0 \\ 0.4 \pi \\ 0 \end{bmatrix} \rightarrow x(T) = \begin{bmatrix} 0.2 \pi \\ 0 \\ 0.2 \pi \\ 0 \end{bmatrix} \end{equation*} \end{aligned}$$

The trajectory of the inputs should be without cracks in the transition to the equilibrium positions (ũ(0) = ũ(T) = 0).

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Source Code

/../../examples/ex4_UnderactuatedManipulator.py