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Aerial Robotics

Rigid Body Motion

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-491/691

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Rigid Body Motion

We need a mathematical model to describe the relationship between a robot’s motion evolution (translational & rotational velocities) and its pose (position & orientation) over time

Different systems have different constraints and capabilities:

Underactuated systems
Non-Holonomic Constraints
Multi-joint, multi-DoF systems

Start with simple Rigid Body Motion

CS491/691 C. Papachristos

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Rigid Body Kinematics

Rigid Body Kinematics – Euler :

Translational Rates (Linear Velocity)

Rotational Rates (Angular Velocity):

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Rigid Body Kinematics

Evolution of position vector – Euler :

Overall:

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Rigid Body Kinematics

Evolution of SO(3) orientation – Euler :

Significantly more challenging…

Relationship between Body Frame angular rates and�Euler Angle derivatives:

I.e., we can invert:

But how did we obtain the original expression?

And how do we systematically develop a generic Rigid Body Motion formulation?

CS491/691 C. Papachristos

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Rotational Motion

Or

equivalently:

Solution:

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Rotational Motion

Solution:

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Which means:

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Rotational Motion

Note: Rodrigues’ formula:

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1)

2)

3)

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Rotational Motion

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Rodrigues’ Formula we saw before

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Rotational Motion

Rotational Motion of a Frame – Rotation Matrix Derivative

Describe motion of the {B} unit vector endpoints as viewed from {S}

From definition of Rotation Matrix:

We can define its Time-Derivative (rate of change of its elements) as:

We may also express in Body Frame by using:

From the above we also get the relationships:

From Vector�

And we get:

Note: {B},{S} both at the origin, shown at distance only for visualization clarity

Rotation:

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( origin at{S} &� expressed at{S} ) :

Note:�Differential Equation

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Rotational Motion

Note: Derived by expanding

left-hand-side and rearranging

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Rotational Motion

Note: {B},{S} both at the origin, shown at distance only for visualization clarity

CS491/691 C. Papachristos

( origin at{S} &� expressed at{S} ) :

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Rotational Motion

Note: {B},{S} both at the origin, shown at distance only for visualization clarity

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( origin at{S} &� expressed at{S} ) :

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SO(3)

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Traceless Skew-Symmetric matrices:

Algebra

well-

defined

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Remember: Rigid Body Kinematics – Euler

Evolution of orientation – Euler :

Representation:

Differentiate:

Express Body-Frame Angular Rates:

… express also for individual Intermediate Frames:

Obtain:

https://www.ics.uci.edu/~welling/teaching/KernelsICS273B/MatrixCookBook.pdf

Use slide 11

relation:

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Remember: Rigid Body Kinematics – Euler

I.e. for the Evolution of orientation – Euler :

Which we can subsequently invert to get:

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Rigid Body Motion

In Homogeneous

Representation:

In Homogeneous

Representation:

Whose solution is:

In Homogeneous

Representation:

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Rigid Body Motion

Twist

General structure for the motion representation:

Motion of this type is specified by a Twist (“Spatial Velocity”):

whose “Matrix(/Bracket) Representation” is:

I.e. the solution to the Rigid Body motion equation is written as:

Whose solution is:

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SE(3)

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Algebra

well-

defined

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with:

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and:

where:

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where:

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Screw Motion

I.e. :

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Screw Motion

where: and

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Screw Motion

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Screw Motion

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Euclidian Motion Generators

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Screw Motion & Twist

Again from slide 11:

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Screw Motion & Twist

and we have:

Or equivalently:

and:

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Adjoint of Lie Group

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Adjoint of Lie Group

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Remember:

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Time for Questions !

CS-491/691

CS491/691 C. Papachristos