Equations of motion of a pr2/aerial robot are given by,
$$ \dot{x} = V cos(\theta)$$$$ \dot{y} = V sin(\theta) $$$$ \dot{\theta} = \omega $$with \(x \), \(y \) and \(\theta \) being the position and heading of the robot, and \( V \) and \( \omega \) being the control inputs.
Discretize and linearize the equation above about a set point given by \( \hat{x} \), \( \hat{y} \), \( \hat{\theta} \), \( \hat{V} \) and \( \hat{\omega} \), and express the resulting equation in a state space form. Assume uncertainities only in the heading velocity and heading angles.
Further, assume that the measurement of position along X and Y directions is available. Assuming that the robot is moving away from the origin at 45-degrees, sensor covariance noise of \( 10^{-3} \) for each of the sensors. Assume process noise covariance of \( 10^{-4} \) for velocity and \( 10^{-2} \) for heading.
Action of an acrobat can be described using a 3-link serial robotic system, where the first joint that is attached to a fixed point is unactuated, and the other 2 joints are actuated. Derive the equations of motion that describe the movement dynamics of the acrobat model thus proposed.
Acrobat problem is a classical control theory problem where the desired task is to go from a position where the acrobat is hanging such that all the angles are zero, to a position where the acrobat is upright (actuated angles zero, and unactuated angle 180 degrees). Use GPOPS to compute the desired trajectory between the two positions. Impose the constaint that the actuated angles cannot be more than 120 degrees and less than -40 degrees.
Derive the equations of motion for the system you are interested in. Formulate a precise problem that you will solve, and a formal scheme of how you will test if you solved this problem.