The Furuta pedulum, or rotational inverted pendulum, consists of a driven arm which rotates in the horizontal plane and a pendulum attached to that arm which is free to rotate in the vertical plane (Fig 1).
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The Furuta pendulum was first developed at the Tokyo Institute of Technology by Katsuhisa Furuta ^{[1]}^{[2]}^{[3]}^{[4]} and his colleagues. The pendulum is underactutated and extremely nonlinear due to the gravitational forces and the coupling arising from the Coriolis and centripetal forces. Since then, dozens, possibly hundreds of papers and theses have used the system to demonstrate linear and nonlinear control laws ^{[5]}^{[6]}. The system has also been the subject of two texts ^{[7]}^{[8]}.
Despite the great deal of attention the system has received, very few publications successfully derive (or use) the full dynamics. Many authors ^{[3]}^{[7]} have only considered the rotational inertia of the pendulum for a single principal axis (or neglected it altogether ^{[8]}). In other words, the inertia tensor only has a single nonzero element (or none), and the remaining two diagonal terms are zero. It is possible to find a pendulum system where the moment of inertia in one the three principal axes is approximately zero, but not two.
A few authors ^{[2]}^{[4]}^{[5]}^{[9]}^{[10]}^{[11]} have considered slender symmetric pendulums where the moments of inertia for two of the principal axes are equal and the remaining moment of inertia is zero. Of the dozens of publications surveyed for this wiki only a single conference paper ^{[12]} and journal paper ^{[13]} were found to include all three principal inertial terms of the pendulum. Both papers used a Lagrangian formulation but each contained minor errors (presumably typographical).
The equations of motion presented here are an extract from a paper^{[14]} on the Furuta pendulum dynamics derived at the University of Adelaide.
Consider the rotational inverted pendulum mounted to a DC motor as shown in Fig. 1. The DC motor is used to apply a torque τ_{1} to Arm 1. The link between Arm 1 and Arm 2 is not actuated but free to rotate. The two arms have lengths L_{1} and L_{2}. The arms have masses m_{1} and m_{2} which are located at l_{1} and l_{2} respectively, which are the lengths from the point of rotation of the arm to its center of mass. The arms have inertia tensors and (about the centre of mass of the arm). Each rotational joint is viscously damped with damping coefficients b_{1} and b_{2}, where b_{1} is the damping provided by the motor bearings and b_{2} is the damping arising from the pin coupling between Arm 1 and Arm 2.
A right hand coordinate system has been used to define the inputs, states and the Cartesian coordinate systems 1 and 2. The coordinate axes of Arm 1 and Arm 2 are the principal axes such that the inertia tensors are diagonal.
The angular rotation of Arm 1, θ_{1}, is measured in the horizontal plane where a counterclockwise direction (when viewed from above) is positive. The angular rotation of Arm 2, θ_{2} , is measured in the vertical plane where a counterclockwise direction (when viewed from the front) is positive. When the Arm is hanging down in the stable equilibrium position θ_{2} = 0 .
The torque the servomotor applies to Arm 1, τ_{1}, is positive in a counterclockwise direction (when viewed from above). A disturbance torque, τ_{2}, is experienced by Arm 2, where a counterclockwise direction (when viewed from the front) is positive.
Before deriving the dynamics of the system a number of assumptions must be made. These are:
The nonlinear equations of motion are given by ^{[14]}
and
Most Furuta pendulums tend to have long slender arms, such that the moment of inertia along the axis of the arms is negligible. In addition, most arms have rotational symmetry such that the moments of inertia in two of the principal axes are equal. Thus, the inertia tensors may be approximated as follows:
Further simplifications are obtained by making the following substitutions. The total moment of inertia of Arm 1 about the pivot point (using the parallel axis theorem) is . The total moment of inertia of Arm 2 about its pivot point is . Finally, define the total moment of inertia the motor rotor experiences when the pendulum (Arm 2) is in its equilibrium position (hanging vertically down), .
Substituting the previous definitions into the governing DEs gives the more compact form
and
