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Cross-coupling is the influence between the two axes. The cross-coupling effect arises due to internal rotating parts of the system that have a net angular momentum about a direction. The cross-coupling effect, in this case, produces moment proportional to the angular yawing velocity and yawing moment proportional angular rolling velocity. Cross-coupling can be minimized by using gimbal momentum wheels. Wheels reduce the cross-coupling effect by filtering big changes in momentum or by control moment gyros and reaction wheels. On Im11. Are shown plots of the angular velocity.

Im11. Angular velocity

On the image Im12. We can see the Simulink model with linear PD control. Derivative control improves reaction time, derivative jump instantly, and suppress the disturbance at the static working regime. Linear Derivative control should suppress linear disturbance around the working point of a nonlinear system.

Im12. Simulink model of the object and linear PD control system

Plots are shown on images Im13, Im14, and Im15.

The system is destabilized. We get out of the working point of the linear control system (caused by object non-linearity). We can try to tune control gains again but it is not guaranteed that we can succeed. If we have to tune control gains again, we have to check system properties (equilibrium states, stability and reality of those states, etc…). In this case, performance can not be improved.

After implementing the non-linear control, the system was stable. After changing the desired points, we can easily re-tune control gains. The Nonlinear Simulink model is shown on image Im19. Below the model image, we can see output comparation.

Im19. System model

Simulink model of the closed-loop control for UAV is shown on the image Im22 and on Im23 is the position control system.

Im22. UAV linear control Simulink model

Im23. UAV position control system On images below are shown plots from the task.

I did not have success in tuning control parameters for this purpose. UAV is trying to follow the eight-shape but it is not good enough. What is important, the system is stable but it is not giving a nominal working regime. The Simulink model is shown on image 24, on the images below are shown plots from the task.

I'm. 24

### Robotics: Analysis and Control

The car subsystem of the Simulink model is shown on Im1.

Im1. Car kinematic subsystem

Closed-loop control of the car Simulink model is shown on the Im2. We have to tune control gains to avoid gain destabilization. Gains have to be < 1, in different proportions we can balance between static error and reaction time.

Im2. Closed-loop control system for car

Results of the simulation are shown on images Im3 and Im4 (states and inputs respectively).

in (b). Include plots of the states and inputs, and explain any differences in performance.

Plots of the states and inputs are shown on images Im5 and Im6. After adding the non- linearity saturation type, the system is unstable and can not reach the reference position. In reality, every system has saturation, and its part of his physical description. Saturation can affect system stability, control winding up, etc. To calculate proper control, we have to include all saturation of the object and control system.

#### b)Using the model from (c) and the initial conditions from (b), set up a control to follow a line

specified as aX + bY + c = 0 with a = 2, b = 1 and c = −25. Provide a plot of the states and input γ and list the control gains you used. Show that (x, y) converges to the specified line. The Simulink model is shown on Im7.

Plots are shown on Im8, Im9, and Im10.