Julián Andres Gómez Gómez
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Camilo E. Moncada Guayazán
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Sebastián Roa Prada
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Hernando Gonzalez Acevedo
Abstract
Gimbals are mechatronic systems well known for their use in the stabilization of cameras which are under the effect of sudden movements. Gimbals help keeping cameras at previously defined fixed orientations, so that the captured images have the highest quality. This paper focuses on the design of a Linear Quadratic Gaussian, LQG, controller, based on the physical modeling of a commercial Gimbal with two degrees of freedom (2DOF), which is used for first-person applications in unmanned aerial vehicle (UAV). This approach is proposed to make a more realistic representation of the system under study, since it guarantees high accuracy in the simulation of the dynamic response, as compared to the prediction of the mathematical model of the same system.
The development of the model starts by sectioning the Gimbal into a series of interconnected links. Subsequently, a fixed reference system is assigned to each link body and the corresponding homogeneous transformation matrices are established, which will allow the calculation of the orientation of each link and the displacement of their centers of mass. Once the total kinetic and potential energy of the mechanical components are obtained, Lagrange’s method is utilized to establish the mathematical model of the mechanical structure of the Gimbal. The equations of motion of the system are then expressed in state space form, with two inputs, two outputs and four states, where the inputs are the torques produced by each one of the motors, the outputs are the orientation of the first two links, and the states are the aforementioned orientations along with their time derivatives. The state space model was implemented in MATLAB’s Simulink environment to compare its prediction of the transient response with the prediction obtained with the representation of the same system using MATLAB’s SimMechanics physical modelling interface.
The mathematical model of each one of the three-phase Brushless DC motors is also expressed in state space form, where the three inputs of each motor model are the voltages of the corresponding motor phases, its two outputs are the angular position and angular velocity, and its four states are the currents in two of the phases, the orientation of the motor shaft and its rate of change. This model is experimentally validated by performing a switching sequence in both the simulation model and the physical system and observing that the transient response of the angular position of the motor shaft is in accordance with the theoretical model.
The control system design process starts with the interconnection of the models of the mechanical components and the models of the Brushless DC Motor, using their corresponding state space representations. The resulting model features six inputs, two outputs and eight states. The inputs are the voltages in each phase of the two motors in the Gimbal, the outputs are the angular positions of the first two links, and the states are the currents in two of the phases for each motor and the orientations of the first two links, along with their corresponding time derivatives.
An optimal LQG control system is designed using MATLAB’s dlqr and Kalman functions, which calculate the gains for the control system and the gains for the states estimated by the observer. The external excitation in each of the phases is carried out by pulse width modulation.
Finally, the transient response of the overall system is evaluated for different reference points. The simulation results show very good agreement with the experimental measurements.
Applications of the local state-space form of constrained mechanical systems in multibody dynamics and robotics
