The transformation between the global and body coordinate frames is described below. Therefore, it is necessary to define a rotation matrix to transform variables between the coordinate systems. The state variables for velocity are in the body frame but the state variables for position are in the global frame. For more advanced navigation including waypoint trajectory following, the waypoints are given in the global coordinate system. GPS, for example, measures the quadrotor’s position and ground speed in the global frame while accelerometers and rate gyros produce measurements in the body frame. It is important to note that different common sensors produce measurements in both coordinate systems. The position of the quadrotor is given in the global frame while the velocity and angular velocity are defined in the quadrotor body frame. This notation uses notation taken from the aeronautics literature, specifically the North, East, Down (NED) coordinate system. The complete list of state variables is shown below. Surrounding fluid velocities (wind) are negligible.The Earth is flat and non-rotating (difference of gravity by altitude or the spin of the earth is negligible).The propellers are considered to be rigid and therefore blade flapping is negligible (deformation of propeller blades due to high velocities and flexible material).The thrust and drag of each motor is proportional to the square of the motor velocity.The quadrotor structure is rigid and symmetrical with a center of mass aligned with the center of the body frame of the vehicle.This model relies on several assumptions: The basic quadrotor structure used for the model development is shown in the figure above depicting the Euler angles of roll, pitch, and yaw, a body coordinate frame. As will be shown below, the rotational and translational dynamics are coupled which presents an interesting control problem. While the quadrotor can move in 6 degrees of freedom (3 translational and 3 rotational), there are only 4 inputs that can be controlled (the speeds of the 4 motors). The quadrotor is classified as an under-actuated system. The simple design results in small platforms which are battery operated, able to perform a stable hover, and safe to use in indoor environments. To move in the lateral directions, the relative speeds of each motor in the lateral pair are varied to create a desired lateral thrust offset. Altitude is controlled by varying the thrust from each motor by equal amounts to provide a net thrust vector and without a rotational moment. Yaw is controlled by varying the speeds of the pairs of motors to create a non-zero net counter torque. The elimination of the rotating moment allows the vehicle to maintain a constant heading while hovering. This design results in the reaction torques from the pairs of motors being exactly opposed by each other if they are all spinning at the same speed. The motors are arranged in pairs along the horizontal and vertical axes with the forward pair rotating clockwise and the horizontal pair rotating counter-clockwise. Each rotor consists of a brushless DC motor and rotor with a fixed pitch. The quadrotor vehicle operates on the concept of variable torques and thrusts. Variable and Coordinate System Definition The goal is to make a model that is simple while still being as realistic as possible. Once all the components of the equations are developed, it is possible to simplify the equations of motion after making several assumptions about the method which the quadrotor will be operated. The quadrotor is defined by a set of non-linear equations which make accurate simulation as well as control difficult. In order to develop accurate control systems for the quadrotor platform, it is necessary to develop and analyze the equations of motion that define the quadrotor system. System Modeling ■ Model Verification ■ Controller Design ■ Simulation EnvironmentĪutopilot Implementation ■ System Set-up ■ Experiments
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