The document summarizes a paper that uses Kane's method to model the dynamics of a small helicopter. It reviews the paper's motivation for dynamic modeling, provides an overview of Kane's method and popular modeling techniques, and walks through the paper's application of Kane's method to the helicopter system. This includes deriving the helicopter's kinematics, defining generalized coordinates and speeds, calculating partial velocities and forces, and determining the final equations of motion.

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Paper Review: Sandino et al
Tutorial for the application of Kane’s Method to model a
small-size helicopter
Daniel Kuntz
April 20, 2015
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
1 Motivation
Dynamics of Multi-body Systems
Popular Methods
2 Paper Overview
3 Paper Walkthrough
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
4 Paper Analysis
5 Conclusion
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Dynamics of Multi-body Systems
Popular Methods
The Need for Dynamic Models
We need dynamic models to:
Gain a deeper understanding of how your machine behaves
Learn how best to control your machine
We have seen modelling in class from the perspective of the
Laplace transforms, however, these can only be used to model
Linear Time-Invariant (LTI) systems.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Dynamics of Multi-body Systems
Popular Methods
More Complicated Cases
Systems where not all parts are best described in an inertial or
”world” reference frame.
These are referred to as multi-body systems (MBSs). These are
common in:
Robotics
Aerospace
Aviation
Industrial Automation
Non-trival systems of this type are inherently non-linear.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Dynamics of Multi-body Systems
Popular Methods
Example
0
Video courtesy of SpaceX
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Dynamics of Multi-body Systems
Popular Methods
Popular Methods for Modelling Multi-body Systems
According to [1] there are two classes of methods to model these
systems.
Vector Methods
Newton-Euler
Scalar Methods
Lagrangian Dynamics
Kane’s method borrows concepts from both, but is classified as a
Vector Method.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Dynamics of Multi-body Systems
Popular Methods
Why Kane’s Method?
Kane’s method is touted as a superior approach by it proponents
because it:
Encapsulates holonomic (position) constraints by the use of
generalized coordinates (as in the Lagrangian method).
Also encapsulates non-holonomic (velocity) velocity
constraints through the use of generalized speeds. (Which
requires Lagrange’s Method of Undetermined Multipliers)
Results in a compact, first order representation of the
equations of motion.
Is more systematic and therefore easier to learn [2].
Is becoming the industry standard where complex systems
need to be modeled.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Objective
This paper has the stated purpose of describing the process of
using Kane’s Method to model a familiar system. While describing
key points of the method, they walk through the process of setting
up the problem and describe it’s solution.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Kane’s Method Breakdown
1 Find a set up generalized coordinates that describe the system
including reference frames for different bodies.
2 Create coordinate frame rotation matrices that describe the
rotation of each frame in terms of the systems generalized
coordinates.
3 Pick generalized speeds that compactly represent the system’s
kinematic equations.
4 Find the partial velocities and partial angular velocities of
point of interest in the system.
5 Find the generalized active forces and the generalized inertia
forces.
6 Use Kane’s equation to find the equations of motion.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Key Body Points
Figure 1 : Points on helicopter body
0
Image courtesy [1]
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Forces
Figure 2 : Forces on helicopter body
0
Image courtesy [1]
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
System Kinematics
The kinematics of the system describe how motion must happen in
a system when certain parts are travelling at certain speeds. To
define the kinematics of the system it is required to define:
generalized coordinates
generalized speeds
Effectively, these encapsulate the motion constraints of the system
so we don’t have to worry about them later.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Coordinates
Helicopter has 7 DOF, and two rigid bodies. The generalized
coordinates q1, · · · , q7 then correlate to:
Table 1 : Generalized coordinates
Coordinate Num. Description
1, · · · , 3 position of COM in inertial frame
4, · · · , 6 Euler Angles of helicopter body
w/r to inertial frame
7 Rotation of main rotor
w/r to helicopter body
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Coordinate Frame Rotation Matrices
Table 2 : Frame rotation matricies
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Kinematic Equations
The kinematic equation are the first 6 of 12 needed equations,
these are defined as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Speeds
Generalized speeds must be in the form:
A u1 · · · u7
T
+ B = 0
Where A, B are functions of the generalized coordinates q1, · · · , q7
And dependent generalized speeds (u7) must be written as:
u7 = C u1 · · · u6
T
+ D
Where C, D are constants
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Speeds (cont.)
Under the constraints from the last slide we carefully pick
generalized speeds in order to make the kinematic equations more
compact. The author chooses:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Final Kinematic Equations
Solving the equation for the generalized velocities leads to the final
kinematic differential equations:
Where ˙q7, the dependent speed, can be written as ˙q7 = u7 = ωMR
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Dynamics
The dynamics of the system, as opposed to the kinematics of the
describe how velocities of the part change when subjected to an
outside force. To calculate these requires:
partial velocities
partial angular velocities
generalized active forces
generalized ineria forces
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Partial Velocities
For points of interest Pj (j = 1, · · · , µ) we want to find their partial
velocities, these are given by:
N
vPj
= ∂N v
Pj
∂u1
· · · ∂N v
Pj
∂u6
u1 · · · u6
T
+
∂NvPj
∂t
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Partial Angular Velocities
Similarly, the partial angular velocities are given by:
N
ωBk
= ∂N ωBk
∂u1
· · · ∂N ωBk
∂u6
u1 · · · u6
T
+
∂NωBk
∂t
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Summary of Partial Velocities
The partial velocities and angular velocities for the helicopter
system are thus:
Table 3 : Partial velocities and angular velocities
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Active Forces
The generalized active forces are defined by Kane to be:
(Fr )Pj
∂vPj
∂ur
· RPj
(Fr )Bk
∂ωBk
∂ur
· TBk
Fr =
µ
j=1
(Fr )Pj
+
ν
k=1
(Fr )Bk
Where RPj
is the resultant of active forces acting at point Pj , TBk
is the resultant of all torques acting on body Bk, µ is the number
of points and ν is the number of bodies
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Active Forces (cont.)
The author calculates the generalized forces for the first and fourth
partial velocities as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Active Forces (cont.)
And gives the 2,3,5,6th as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Inertia Forces
The inertial force and inertial torque are defined as:
R∗
Pj
−mN
Pj
NdNvPj
dt
T∗
Bk
−IBk /BO
k ·
NdNωBk
dt
−N
ωBk
× IBk /BO
k ·N
ωBk
where mPj
is the mass of point j and IBk /BO
k is the inertia dyadic
body k about it’s center of mass.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Inertia Forces (cont.)
Similar to the generalized active forces, the generalized inertia
forces are given by:
(Fr )∗
Pj
∂vPj
∂ur
· R∗
Pj
(Fr )∗
Bk
∂ωBk
∂ur
· T∗
Bk
F∗
r =
µ
j=1
(Fr )∗
Pj
+
ν
k=1
(Fr )∗
Bk
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Inertia Forces (cont.)
The author calculates the generalized forces for the first and fourth
partial angular velocities as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Generalized Inertia Forces (cont.)
And gives the 2,3,5,6th as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Constants
The Kxxx constants are given by:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Kane’s Equation
Once Fr and F∗
r have been calculated, the dynamic equations are
given by Kane’s equation, which is simply:
Fr + F∗
r = 0
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Dynamical Differential Equations
Once Kane’s equation has been applied, the author calculates the
dynamic differential equations as:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Full Equations of Motion (EOM)
The author’s result can be put into non-linear state space form:
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Helicopter Configuration
Derivation of Kinematics
Derivation of Dynamics
Equations of Motion
Computation
Computation of EOM
The author generated MATLAB code for system dynamics using
Autolev and MotionGenesis software. Scripts were generated for
both Newton-Euler and Kane’s method. He then executed the
script and timed it for both cases. The following results were
obtained:
Newton-Euler: 2.35 s (24 kB MATLAB code)
Kane’s Method: 1.58 s (16 kB MATLAB code)
Showing empirically that Kane’s method is superior to
Newton-Euler in computation. It was not compared to Lagrange’s
Method however.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
General Observations
Importance
Paper is not important in the popular sense, it does not add
anything novel to the field of research; nor does it attempt to.
However, it is the best paper that I found for actually learning
Kane’s method. This is important to practitioners because learning
it is a very difficult undertaking.
Quality
I found the paper to be of very high quality. It is descriptive, does
not skip steps, covers the derivation in detail and because I am
actually implementing the given model in Simulink for a controls
design class, I can attest to the accuracy of the equations.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
The Good
High quality, accurate derivations
Very well written
One of the best resources for learning Kane’s method
Model complex but not too complex to exceed a newcomer’s
level of understanding.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
The Bad
Model is not realistic
Main rotor speed is modelled as constant, while the force from
the rotor is variable.
Mass and inertia of the rear rotor is considered to be negligible.
Forces and torques from main and tail rotors are used as
inputs, however, these are not the actual inputs to a real life
helicopter plant.
Theory behind Kane’s Method is not explained, paper is
purely utilitarian.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
In the End
This is a great paper for people new to learning Kane’s method and
serves as a ”gentle as possible” introduction to the method. People
who know Kane’s method will not derive a better understanding
from this paper. Overall the paper achieves its intended goal.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
References
[1] Sandino, L; Bejar, M; Ollero, A. Tutorial for the application of
Kane’s Method to model a small-size helicopter. 2011. Proc. of
the 1st Workshop on Research, Development and education on
Unmanned Aerial Systems (RED-UAS 2011).
[2] Kane, Thomas R. Dynamics, theory and applications.
McGraw-Hill series in mechanical engineering. 1985. ISBN
0-07-037846-0.
Daniel Kuntz Paper Review: Sandino et al

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Motivation
Paper Overview
Paper Walkthrough
Paper Analysis
Conclusion
Questions
Daniel Kuntz Paper Review: Sandino et al