Partial Feedback Linearization (PFL)

\[\newcommand{\vect}[1]{\boldsymbol{#1}} \newcommand{\dvect}[1]{\dot{\boldsymbol{#1}}} \newcommand{\ddvect}[1]{\ddot{\boldsymbol{#1}}} \newcommand{\mat}[1]{\boldsymbol{#1}}\]

Partial Feedback Linearization (PFL) [1-3] is a classical method from control theory. With PFL it is possible to provoke a linear response in both joints of the double pendulum even if operated as a pendubot or acrobot. For an intuition of its functionality consider the manipulator equation for the acrobot (\(u_1 \equiv 0\))

\[\begin{split}\left[ {\begin{array}{cc} M_{11} & M_{12} \\ M_{21} & M_{22} \\ \end{array} } \right] \left[ {\begin{array}{c} \ddot{q}_1 \\ \ddot{q}_2 \\ \end{array} } \right] + \left[ {\begin{array}{cc} C_{11} & C_{12} \\ C_{21} & C_{22} \\ \end{array} } \right] \left[ {\begin{array}{c} \dot{q}_1 \\ \dot{q}_2 \\ \end{array} } \right] + \left[ {\begin{array}{c} G_1 \\ G_2 \\ \end{array} } \right] + \left[ {\begin{array}{c} F_1 \\ F_2 \\ \end{array} } \right] - \left[ {\begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} } \right] \left[ {\begin{array}{c} 0 \\ u_2 \\ \end{array} } \right] = 0\end{split}\]

The unactuated upper part of the vector equation can be solved for the acceleration \(\ddot{q}_1\) and then plugged into the lower part of the equation. The control input \(u_2\) can now be designed as PD control with an energy term

\[u_2(\vect{x}) = -k_p(q_2 - q_2^{d}) - k_d\dot{q_2} + k_e(E - E^{d})\dot{q}_1 \label{eq:pfl_acro_col}\]

with the desired configuration \(q_2^{d}\) of the second link, the total energy \(E\), the desired total energy \(E^{d}\) and the gain parameters \(k_p, k_d\) and \(k_e\). The above described method is called collocated PFL. Similarly, it is also possible to eliminate \(\ddot{q}_2\) instead of \(\ddot{q}_1\) from the equations which is than called non-collocated PFL. Partial feedback linearization for the pendubot can be done on the same way. The collocated control law in this case reads

\[u_1(\vect{x}) = -k_p(q_1 - q_1^{d}) - k_d\dot{q}_1 + k_e(E - E^{d})\dot{q}_2. \label{eq:_pfl_pendu_col}\]

References