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A.3.2.1 Equations of Motion
A.3.2.1 Equations of Motion
The first step in controlling the ascent of the launch vehicle is understanding its dynamics. We develop a mathematical model that predicts the launch vehicle’s actual behavior. We plan to use this model as a simulator in our SIMULINK code. The Trajectory group designs a nominal trajectory assuming that the launch vehicle is a point mass. It is our task to make the actual launch vehicle follow this trajectory. While the Trajectory group only worries about the path of the center of mass of the launch vehicle, we concern ourselves with the orientation of the launch vehicle. Theoretically the launch vehicle’s center of mass could follow the trajectory while body tumbled. In the real world, however, this motion would be disastrous.
We model the launch vehicle as an axisymmetric object. We design the controller in such a way to keep the orientation of the launch vehicle so that the axis of rotation is tangent to the designed trajectory. We vector the thrust to make up for variations in wind and other aerodynamic forces to keep the launch vehicle in the correct orientation. At this point in the design process, we were not concern with how the thrust would be vectored, just that it could be changed with time. We also assumed that the difference between the center of mass and the center of gravity were negligible because there are much larger perturbing forces that act on the launch vehicle. Below is a schematic of the forces acting on the launch vehicle during its ascent.
Fig. A.3.2.1.1: Forces acting on the launch vehicle during ascent.
(Jeffery Stuart)
The forces acting on the launch vehicle are weight, thrust, and the aerodynamic force which act through the center of mass (cm), center of thrust (ct), and the center of pressure (cp) respectively. One early assumption made was that the cp, cm, and ct were all located along the axis of symmetry. The spin moment would be applied by another sub system for spin stabilization. Spin stabilization calculations were done for every stage, but were only taken advantage of in the third and final stage. The angles δ and κ orient the direction of the thrust as shown in Fig. A.3.2.1.1. These angles are what the controller manipulated to keep the rocket on the desired path.
The trajectory group modeled the launch vehicle as a point mass located at the center of mass of the launch vehicle in their mathematical model. Below Figure A.3.2.1.2 shows the unit vectors used.
Fig. A.3.2.1.2: Unit vectors describing orientation of the center of mass of the launch vehicle with respect to the center of the Earth
(Albert Chaney)
The e unit vectors are inertially fixed. The a unit vectors and the b unit vectors are rotations away from the inertial as shown in Fig. A.3.2.1.2. These definitions were the basis from which we were able to derive the equations of motion for the launch vehicle as it ascended.
The equations for the forces are below:
/ (A.3.2.1.1)where m is the mass of the launch vehicle, g is the gravitational acceleration of the earth.
/ (A.3.2.1.2)where the components A1, A2, and A3 are calculated with Eq. A.3.2.1.3.
/ (A.3.2.1.3)where ρ is the density of air, CDi is the coefficient of drag, Si is the reference area, and v∞i is the velocity of the freestream in that direction.
/ (A.3.2.1.4)where T is the magnitude of the thrust, and the angles δ and κ are defined as before.
/ (A.3.2.1.5)where S is the magnitude of the spin moment.
To get the moments about the center of mass, we computed the cross product of the forces and their displacement vectors. The displacement vectors are listed below.
/ (A.3.2.1.6)/ (A.3.2.1.7)
where dt1, dt2, and dt3 are the components of the displacement vector from the center of mass to the center of thrust and dp1, dp2, dp3 are the components of the displacement vector from the center of pressure to the center of mass.
Furthermore, since we assumed that the center of mass, center of pressure, and the center of thrust all lie along the axis of symmetry, , which reduces Eq. A.3.2.1.6 and Eq. A.3.2.1.7 to the following.
/ (A.3.2.1.8)/ (A.3.2.1.9)
The moments about the center of mass in the body frame can then be calculated by crossing the forces with their displacement vectors. The moments are shown below.
/ (A.3.2.1.10)where Mg is the moment due to weight which acts through the center of mass.
/ (A.3.2.1.11)where MS is the spin moment.
/ (A.3.2.1.12)where MA is the moment due to the aerodynamic forces. It is calculated by finding the cross product of Eq. A.3.2.1.7 and Eq. A.3.2.1.2.
/ (A.3.2.1.13)/ (A.3.2.1.14)
Next the moment from the thrust will be derived by calculating the cross product of Eq. A.3.2.1.8 and Eq. A.3.2.1.4.
/ (A.3.2.1.15)/ (A.3.2.1.16)
/ (A.3.2.1.17)
An inertial measurement unit in the launch vehicle will be able to sense the angular accelerations in the vehicle. A central processing unit will then calculate the angular velocity vector between the b frame and the r frame. All of the moments have been found in the r frame. They can be related to the angular velocity vector through Euler’s equation of motion, which is shown below.
/ (A.3.2.1.18)/ (A.3.2.1.19)
Where is the sum of the moments in the r frame, is the angular momentum from the b frame to the r frame, I is the moment of inertia matrix, and is the angular velocity from the b frame to the r frame. The inertia matrix is shown below.
/ (A.3.2.1.20)where Iij is the component of the inertia matrix.
We placed the r frame on the launch vehicle such that it was aligned with the principle axis. This allows us to eliminate all of the products of inertias and reduces the inertia matrix to the one below.
/ (A.3.2.1.21)The values for the inertia matrix were provided by the Structures group.
The kinematic equations are developed by first taking the time derivative of Eq. A.3.2.1.19.
/ (A.3.2.1.22)/ (A.3.2.1.23)
Equation A.3.2.1.22 can then be combined with Eq. A.3.2.1.18. Also notice that I22 = I11.
/ (A.3.2.1.24)/ (A.3.2.1.25)
/ (A.3.2.1.26)
/ (A.3.2.1.27)
The left hand side of Eq. A.3.2.1.23 is the sum of the moments acting on the launch vehicle calculated in Eq. A.3.2.1.11, Eq. A.3.2.1.14, and Eq. A.3.2.1.17. We then combined Eq. A.3.2.1.11, Eq. A.3.2.1.14, Eq. A.3.2.1.17, and the equations above to get the simplified equations describing the orientation of the launch vehicle during ascent which are shown below. Also recall that, since we assume that the center of pressure, and center of thrust act on the axis of rotation of the launch vehicle.
/ (A.3.2.1.28)/ (A.3.2.1.29)
/ (A.3.2.1.30)
The desired orientation could be compared to the actual orientation found through these equations. The angles δ and κ could then be found to correct the launch vehicle to the desired orientation. These kinematic equations would have to be combined with equations of motion derived from Newton’s laws to give a complete picture of the dynamics of the launch vehicle. We also discussed using Euler angles to describe the orientation of the vehicles so the vehicle’s orientation was easily conceptualized. The problem with Euler angles is that they have singularities. This path of determining the EOMs and coding them into the simulator was scrapped when a pre developed six degree of freedom box was found in SIMULINK. Below is the built in subsystem that was used in the SIMULINK code.
Fig. A.3.2.1.3: Six degree of freedom simulator.
(Mike Walker and Alfred Lynam)
This built in subsystem is perfectly suited for our application. It takes in an inertia matrix and its time rate of change, the vehicle’s mass and its time rate of change, the moments influencing the launch vehicle, and the forces acting on the launch vehicle. It then uses quaternions to calculate the orientation, and behavior of the launch vehicle. This system will be discussed in detail in subsequent chapters of this appendix.
We decided that the built in function was superior to us coding in our own subsystem. Using the built in function would save us both time and effort while being more reliable. Also this built in function takes advantage of quaternions, which eliminates the possibility of a singularity in the analysis of the launch vehicle’s ascent. What was important from this analysis was the moment equations, as they are inputs to the built in subsystem.
One difficulty that we ran into when using the built in block was it uses a North-East-Down reference frame to describe the body fixed vectors. The body fixed vectors, b frame, that the Trajectory group defines are South-East-Down. This required us to convert all of their calculations before we input them into the simulator. Below is a diagram what is built into MATLAB’s aerospace blockset.
Fig. A.3.2.1.4: SIMULINK built in Euler angle rotation
(MATLAB Aerospace block set help)
MATLAB uses a 1-2-3, roll, pitch, yaw (φ,θ,ψ) Euler angle rotation. We used a similar rotation shown below.
Fig. A.3.2.1.5: Rotation sequence used in our launch vehicle model
(Jeffery Stuart)
The r3 direction is the axis of symmetry. Its direction was swapped in the equations when the difference between what we had derived and what is defined in MATLAB was noticed. We also use a 1-2-3 (χ,ξ,η) Euler angle rotation. We call the first rotation (χ) pitch or the steering angle. The second rotation (ξ) is referred to as yaw or the pointing angle. The third and final rotation (η) is called spin or roll.
Further work in this section of the analysis would be completing the equations of motion and creating a code to integrate them. The code could be used to verify or contradict the findings from the built in SIMULINK model.
Author: Albert Chaney