Aerospace Modeling Tutorial Lecture 1 – Rigid Body Dynamics Greg and Mario January 21, 2015.

Modeling Lecture 1

Aerospace Modeling TutorialLecture 1 Rigid Body DynamicsGreg and MarioJanuary 21, 2015Reference frames

Translational DynamicsRotational Dynamics(1 dimensional)(3 dimensional)Rotational sim

Rotational sim

We know the angular velocity, but not the anglesuper easy 1 dimensionHow to model rotations 1 dimensionHow to model rotations 3 dimensional

First Idea:Euler angles(yaw, pitch, roll)

Euler angles how do we express the rotation?

How do we use this with our rigid body equations?

?How do we use this with our rigid body equations?

(Sorry for different time scales)

How do we use this with our rigid body equations?


Euler angles why dont we use them?

These are very nonlinear

Major problem: Harry Ball Theorem(Every cow must have at least one cowlick)(You cant comb the hair on a coconut)

A: Quaternions or Rotation Matrices!Q: What do we use instead of Euler Angles?Quaternions in 15 seconds

Very compact and elegant representation of attitudewhich we will not discuss todayRotation Matrices a.k.a. Direction Cosine MatricesHow to rotate vectors from one frame to another?Word to the wise:Everyone uses different conventions. Stick to one. Always check other peoples convention.Rotation Matrices a.k.a. Direction Cosine MatricesDerivative with respect to DCM sim

DCM sim

What makes DCMs hard?Hard to visualizeSolution: convert to Euler angles before plottingMust be initialized right-handed and orthonormalEasiest solution: initialize to identityEasy solution: initialize as Euler angles then convert to DCM.If initial angle is free/unknown, use hard solution: enforce orthonormality as a constraintMatching conditions have 9 equations and 3 degrees of freedomSolution: Enforce small relative rotation = 0Only enforce these threeOnly enforce this at one time point!Doing this more than once destroys LICQ!IMPORTANT:Also enforce diagonal elements positiveHomework 1:Reproduce these plots


Homework 2:Match twosimulationswith differentstatesHomework 3:minimum torque satellite de-tumbleMultiple shooting with 200 timesteps, rk4 integratorHomework 4: OPTIONAL BONUS QUESTIONsame problem as homework 3, but in minimal timeObjective is now end timeAdd bounds on the control:

Aerospace Modeling Tutorial Lecture 1 – Rigid Body Dynamics Greg and Mario January 21, 2015.

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