# Matlab代写｜ESE 4481 ASSIGNMENT #3: DYNAMICS

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This is not a partner assignment. Please work individually. Do not plagiarise.

## 3.Assignment

This assignment is to help you grasp dynamics. It walks through the derivation of conservation of translational and angular momentum for the UAV. The dynamics are derived i a frame centered at the c.g. of the UAV and rotating with the aircraft’s principle axes, the body frame.

The derivation in this homework assignment makes practical assumptions. A simplified dynamics for the quadcopter may satisfy the following 5 assumptions [1]:

• The body has a fixed mass distribution and constant mass;
• The air is at rest relative to the earth;
• The earth is fixed in inertial space;
• Flight in the earth atmosphere is close to the surface of earth, so the earth surface can be approximated as flat;
• Gravity is uniform so that the aircraft c.g. and center of mass are coinci dent, and gravitational forces do not change with altitude.

## 4.Notation

X is a force in the x body axis. Y is a force in the y body axis. Z is a force in the z body axis. L is a moment about the body x axis. M is a moment about body y axis. N is a moment about the body z axis.

Assume the standard aerospace notation discussed in class holds with (Z-Y-X) euler angles. ϕ is the roll euler angle between the earth (NED) frame and the body frame. θ is the pitch euler angle between the earth frame and the body frame. ψ is the yaw euler angle between the earth frame and the body frame.

Recall that a vector in the body frame pb can represented as follows:

(4.1) pb = RbEpE = R(ϕ)R(θ)R(ψ)pE

where pE is the vector as measured in the earth frame.

Assume that V b is the airspeed vector as measured in the body frame with components as follows:

V b = uvw

## 5.Translational and Angular Moment Balance in the Body Frame

The linear momentum of the UAV is P,

P = mV,

and the angular momentum of the UAV is

H = Jω.

For the time being, assume that the rotors do not spin and are also fixed so that they do not contribute to angular momentum.

Newton’s 2nd law applied to a quadcopter can be written in the earth frame (inertial frame) as follows:

(5.2)  FE = d dt (P) E,

where m is the mass of the UAV, V is the translational velocity of the body, ωb/E is the angular velocity of the body, and F and M represent applied moments and forces,

(5.3) ME = d dt (H) E,

Rewrite (5.2) in a frame that rotates relative to the earth frame, the body frame.

Assume that the centers of the body frame and the earth frame are coincident, but the body frame rotates relative to the earth frame by ωb/E. These laws now take the form under the assumption of constant mass and mass distribution:

(5.4)˙V b = 1/m （Fb ωb/E × mV b）,

(5.5)˙ωb/E = J1 （Mb ωb/E × b/E）

Create a simulink diagram that models these equations with F, M as inputs and V, ω as outputs. Assume that m, the mass of the quadcopter with prop guards (68g), and J, the moment of inertia with prop guards, are constants (reusing values from your previous homework).

## 6.Rotational Kinmeatics

Recall that the rate of change of the euler angles˙ϕ, θ˙,˙ψ is not the same as the body axis rotational rate that would be measured in a gyroscope p, q, and r.

The rate of change of euler angles can be thought of as the magnitude of angular velocity vectors for the Earth axis [2].

Changing the sum of the vectors rotated to the body axis frame, form the familiar roll rate, pitch rate, and yaw rate. This is done as follows (recalling the Z-Y-X order):

(6.1)

Let the vector Θ represent the euler angles,

Θ = (ϕ, θ, ψ)T .

This yields the following expression that relates the rate of change of euler angles x to the angular velocities roll rate, pitch rate, and yaw rate ωb/E = (p, q, r)T .

(6.2)

where H is the matrix in (6.2). To calculate the rate of change of euler angles in terms of angular rates the equation is inverted as follows

(6.4)˙Θ =H(Θ)1ωb/E

Construct a simulink diagram that takes an input ω and outputs Θ. These equations will have singularities so please note when these equations are invalid.

## 7.Model Gravity

Transform the gravity vector from the earth frame to the body frame. Assume that gravity points downwards and has a constant acceleration 9.81 m/s. Write out the gravity vector in body coordinates as a function of ϕ and θ.

(7.1)Fgb = RbEFgE

Create a simulink diagram that models gravity in the body frame. Construct a simulink diagram that takes an input Θ and outputs Fgb.

## 8.Propulsive Motor Forces and Moments

The speed of the Parrot’s four propellers is n1, n2, n3 , and n4. Use units of Hz for motor speeds.Assume that the x distance from the UAV center of mass to a rotor’s center:

W = 0.047625;

Assume that the y distance from the UAV center of mass to a rotor’s center:

L = 0.047625;

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