Morphing UAV Intelligent Shape and Flight Control

Morphing UAV Intelligent Shape and Flight Control - I
Kumar Digvijay Mishra
Control of Morphing Aerostructures
kumardigvijaymishra@yahoo.co.uk
January 11, 2021
Kumar Digvijay Mishra January 11, 2021 1 / 69

Overview
1 Introduction
2 A-RLC Architecture Functionality
3 Learning Air Vehicle Shape Changes
Overview of RL
Implementation of Shape Change Learning Agent
4 Mathematical Modeling of Morphing Air Vehicle
Aerodynamic Modeling
Constitutive Equations
Model Grid
Dynamical Modeling
Reference Trajectory
Shape Memory Alloy Actuator Dynamics
Control Effectors on Morphing Wing
5 Morphing Control Law
6 Numerical Examples
7 Conclusion
Kumar Digvijay Mishra January 11, 2021 2 / 69

Introduction
Definition
Morphing refers to both small-scale and large-scale shape changes or
transconfigurations
Two classes of morphing in context of flight vehicles:
1 Morphing for Control
A small-scale or component level, in-flight physical or virtual
shape change, used to achieve multiple control objectives such as
noise suppression, flutter suppression, load alleviation and active flow
separation control
Kumar Digvijay Mishra 1. Introduction January 11, 2021 3 / 69

Introduction
Definition
Morphing refers to both small-scale and large-scale shape changes or
transconfigurations
Two classes of morphing in context of flight vehicles:
1 Morphing for Control
A small-scale or component level, in-flight physical or virtual
shape change, used to achieve multiple control objectives such as
noise suppression, flutter suppression, load alleviation and active flow
separation control
2 Morphing for Mission Adaptation (MMA)
A large-scale, relatively slow, in-flight shape change to enable a
single vehicle to perform multiple diverse mission profiles

Introduction
Essential functionalities of practical MMA capability are:
1 When to reconfigure: Driven by mission priorities/tasks, and leads to
optimal shape being a system parameter

Introduction
2 How to reconfigure: Problem of sensing, actuation, and control

Introduction
2 How to reconfigure: Problem of sensing, actuation, and control
3 Learning to reconfigure: Actuation scheme(s) to produce most
optimal shape

Introduction
Previous control design research1:
◦ Vortex lattice method used to calculate aerodynamics for the
case of variable wing dihedral2
◦ Desired dynamics chosen for each mission phase
◦ Developed H∞ model-following controllers
1
Boothe et al. (2005) and Abdulrahim and Lind(2005)
2
inclination of aircraft’s wing from horizontal, esp upwards, away from fuselage

Introduction
Previous control design research1:
◦ Vortex lattice method used to calculate aerodynamics for the
case of variable wing dihedral2
◦ Desired dynamics chosen for each mission phase
◦ Developed H∞ model-following controllers
Current control design approach:
◦ High-level description of functionality of A-RLC architecture
◦ Development of learning agent, air vehicle model and simulation,
smart actuators, and adaptive controller
1
Boothe et al. (2005) and Abdulrahim and Lind(2005)
2
inclination of aircraft’s wing from horizontal, esp upwards, away from fuselage

A-RLC Architecture Functionality
Adaptive-Reinforcement Learning Control composed of:
1 Reinforcement Learning (RL)
Kumar Digvijay Mishra 2. A-RLC Architecture Functionality January 11, 2021 6 / 69

2 Structured Adaptive Model Inversion (SAMI)

Both sub-systems interact significantly during:

1 Episodic learning stage: when optimal shape change policy is learned

2 Operational stage: when the plant morphs, and tracks a trajectory

2 Operational stage: when the plant morphs, and tracks a trajectory
A-RLC addresses optimal shape changing of air vehicle, rather than
isolated components or individual actuation.

RL Module of A-RLC Architecture
1 Initially commands an arbitrary action from the set of admissible
actions

actions
2 This action is sent to the plant, which produces a shape change

actions
3 Cost associated with resultant shape change in terms of system
states, parameters, and user-defined performance measures is
evaluated with the cost function and then passed to the agent

actions
4 Agent modifies its action-value function based on Q-learning
algorithm

actions
algorithm
5 For the next episode, the agent chooses a new action based on the
current policy and its updated action-value function, and the
sequence repeats itself

actions
algorithm
5 For the next episode, the agent chooses a new action based on the
current policy and its updated action-value function, and the
sequence repeats itself
6 RL sub-system improved by applying Sequential Function
Approximation to generalize learning from previously experienced
quantized states and actions to the continuous state-action space.

SAMI Module of A-RLC Architecture
1 Shape changes in the plant due to actions generated by RL agent
cause plant dynamics to change

2 By calculating commanded moments, SAMI controller tracks a
reference trajectory irrespective of the changing dynamics of the
vehicle caused by these shape changes

3 SAMI is a non-linear technique based on concepts of Feedback
Linearization, Dynamic Inversion, and Structured Model Reference
Adaptive Control

Adaptive Control
4 Dynamic Inversion is approx since system parameters are inaccurately
modeled

Adaptive Control
modeled
5 Adaptive control structure is wrapped around the dynamic inverter to
account for uncertainties in system parameters

Adaptive Control
modeled
5 Adaptive control structure is wrapped around the dynamic inverter to
account for uncertainties in system parameters
6 Applications: tracking reference trajectories of spacecraft, planetary
entry vehicles, and aircraft; handling of actuator failures; facilitation
of correct adaptation in presence of actuator saturation

3.1 Overview of Reinforcement Learning
Definition
RL is a method of learning from interaction between an agent and its
environment to achieve a specified goal.
Kumar Digvijay Mishra 3. Learning Air Vehicle Shape Changes January 11, 2021 10 / 69

Definition
◦ Agent is the learner and decision-maker with the goal of maximizing
total reward it receives over the long run.

Definition
◦ Environment is everything outside the agent, that it interacts with.

Definition
◦ Environment is everything outside the agent, that it interacts with.
◦ Agent’s Policy is the mapping from states to probabilities of
selecting a specified possible action at a time step

Overview of Reinforcement Learning
◦ Agent interacts with its environment at discrete time steps, t = 0,
1, 2, 3....
Kumar Digvijay Mishra 3.1 Learning Air Vehicle Shape Changes January 11, 2021 11 / 69

1, 2, 3....
◦ At each time step t, agent receives some representation of
environment’s state, st ∈ S, where S is a set of possible states, and
on that basis it selects an action, at ∈ A(st), where A(st) is a set of
actions available in state st.

1, 2, 3....
◦ One time step later, as a result of agent’s action, it receives a
numerical reward, rt+1 = R, and finds itself in a new state, st+1.

1, 2, 3....
◦ One time step later, as a result of agent’s action, it receives a
numerical reward, rt+1 = R, and finds itself in a new state, st+1.
◦ Here agent’s policy πt(s, a) indicates the probability that agent
chose action at = a for the environment in given state of st = s at
time t.

RL algos are based on estimating value functions for a policy π:
State-value function V π(s) estimates how good it is, under policy π,
for the agent to be in state s.

◦ Defined as the expected return starting from s and thereafter
following π
V π
(st) ≡ E[
∞
X
k=0
γk
rt+k] (1)
where γ is discount factor, rt+k is sequence of rewards

following π
V π
(st) ≡ E[
∞
X
k=0
γk
rt+k] (1)
Action-value function Qπ(s, a) estimates how good it is, under policy
π, for the agent to perform action a in state s.

following π
V π
(st) ≡ E[
∞
X
k=0
γk
rt+k] (1)
Action-value function Qπ(s, a) estimates how good it is, under policy
π, for the agent to perform action a in state s.
◦ Defined as the expected return starting from s, taking action a,
and thereafter following policy π
Qπ
(s, a) ≡ r(s, a) + γV ∗
(δ(s, a)) (2)
where δ(s, a) denotes the state resulting from applying action a
to state s & V ∗(δ(s, a)) is state-value function for optimal policy π∗

◦ Policy Evaluation: Process of computing V π(s) or Qπ(s, a)

◦ Policy Improvement: For a state s, an action can be chosen from all
possible actions to improve π to π‘ s.t. V π(s) or Qπ(s, a) are
maximized

maximized
Evaluate new V π‘
(s) or Qπ‘
(s, a) for improved policy π‘

maximized
(s) or Qπ‘
Policy π‘ can be further improved to π“

maximized
(s) or Qπ‘
◦ Optimal Policy: A policy π∗ that has :

maximized
(s) or Qπ‘
Either an optimal state-value function V ∗
(s) = maxπV π
(s)

maximized
(s) or Qπ‘
Either an optimal state-value function V ∗
(s) = maxπV π
(s)
Or an optimal action-value function Q∗
(s, a) = maxπQπ
(s, a)

Goal of RL is to find optimal policy π∗ by policy iteration
◦ Optimal action-value function Q∗ in terms of optimal state-value
function V ∗ :
Q∗
(s, a) ≡ E[r(s, a) + γV ∗
(δ(s, a))]
≡ E[r(s, a)] + E[γV ∗
(δ(s, a))]
≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a)V ∗
(s
0
)
(3)
where P(s‘|s, a) is probability of taking action a in state s to produce
next state s‘

Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a)V ∗
(s
0
)

Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a)V ∗
(s
0
)
Optimal state-value function V ∗(s‘) is action-value function Q(s‘, a‘)
maximized for action a‘
Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a) max
a0
Q(s
0
, a
0
) (4)

Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a)V ∗
(s
0
)
Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a) max
a0
Q(s
0
, a
0
) (4)
Iterative update of each action-value function Q(s,a) to converge to
optimal Q∗(s, a) gives following training rule
Qn(s, a) ← (1 − α)Qn−1(s, a) + α[r + γ max
a0
Qn−1(s
0
, a
0
)] (5)

Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a)V ∗
(s
0
)
Q∗
(s, a) ≡ E[r(s, a)] + γ
X
s0
P(s
0
|s, a) max
a0
Q(s
0
, a
0
) (4)
Iterative update of each action-value function Q(s,a) to converge to
optimal Q∗(s, a) gives following training rule
Qn(s, a) ← (1 − α)Qn−1(s, a) + α[r + γ max
a0
Qn−1(s
0
, a
0
)] (5)
For Q-learning algorithm,
Q(s, a) ← Q(s, a) + α[r + γ max
a0
Q(s
0
, a
0
) − Q(s, a)] (6)

Policy Iteration
Major methods for PI:
1 Dynamic Programming
Kumar Digvijay Mishra 3.1 Overview of Reinforcement Learning contd... January 11, 2021 16 / 69

Policy Iteration
2 Monte Carlo Method

Policy Iteration
3 Temporal-Difference Learning

Policy Iteration
3 Temporal-Difference Learning
4 Q-Learning Technique

Major Methods of Policy Iteration
1. Dynamic Programming
Refers to a collection of algos to compute optimal policies given a
perfect model of the environment as a Markov decision process

Use value functions to organize and structure the search for good
policies

policies
Limited utility in RL because of

policies
1 Requires model of environment’s dynamics to learn optimal behavior

policies
2 their assumption of a perfect model, and

policies
2 their assumption of a perfect model, and
3 computational expense

2. Monte Carlo Methods
Estimates functions using an iterative, incremental procedure

Solution based on averaging sample returns

To ensure availability of well-defined returns, returns are defined only
for episodic tasks

for episodic tasks
Upon completion of an episode, value estimates and policies are
changed

for episodic tasks
changed
Advantages:

for episodic tasks
changed
Advantages:
1 Contrary to DP, MC methods can be used to learn optimal behavior
directly from interaction with the environment with no model of
environment’s dynamics

for episodic tasks
changed
Advantages:
2 Can be used with simulation

for episodic tasks
changed
Advantages:
3 Easy and efficient to focus MC methods on a small subset of states

for episodic tasks
changed
Advantages:
Limitation:

for episodic tasks
changed
Advantages:
Limitation:
Convergence properties not well-understood for RL

3. Temporal-Difference Methods
Sutton’s3method of TD is a form of policy evaluation method in DP
which attempts to choose a control policy π0
3
Sutton (1988)
4
Williams and Baird 1993

Expected discounted rewards, V π(i), for each state i in S are
predicted using policy π0 to improve to new policy π1
3
Sutton (1988)
4

Advantages:
3
Sutton (1988)
4

Advantages:
1 Less computation
3
Sutton (1988)
4

Advantages:
1 Less computation
2 No assumption of perfect model of environment
3
Sutton (1988)
4

Advantages:
1 Less computation
3 Howard’s4
algo converges to some policy
3
Sutton (1988)
4

Advantages:
1 Less computation
3 Howard’s4
Limitations:
3
Sutton (1988)
4

Advantages:
1 Less computation
3 Howard’s4
Limitations:
1 Can’t learn optimal functions directly. Why ?
3
Sutton (1988)
4

Advantages:
1 Less computation
3 Howard’s4
Limitations:
1 Can’t learn optimal functions directly. Why ?
Because it fixes the policy and then determines corresponding value
functions
3
Sutton (1988)
4

4. Q-Learning Technique
A form of successive approximation technique of DP
5
Watkins and Davan 1989
Kumar Digvijay Mishra 3.1 Overview of Reinforcement Learning January 11, 2021 20 / 69

Proposed and developed by Watkins5
5

Learns optimal value functions directly, as opposed to fixing a policy
and determining corresponding value functions, like TD
5

Advantages:
5

Advantages:
1 Learns optimal value functions directly
5

Advantages:
2 Avoids need to sweep over the state-action space
5

Advantages:
3 First provably convergent direct adaptive optimal control algo
5

Advantages:
3 First provably convergent direct adaptive optimal control algo
4 Less computations
5

Reinforcement Learning
◦ Has been applied to wide variety of physical control tasks, both real
and simulated

and simulated
◦ When state-space is too large to enumerate value functions,
function approximators are used including,

and simulated
1 Neural networks

and simulated
1 Neural networks
2 Clustering

and simulated
1 Neural networks
2 Clustering
3 Nearest-neighbor methods

and simulated
1 Neural networks
2 Clustering
4 Tile coding, and

and simulated
1 Neural networks
2 Clustering
4 Tile coding, and
5 Cerebellar model articulator controller

3.2 Implementation of Shape Change Learning Agent
For morphing wing problem:
RL agent independently maneuvers wing from some initial state to
final state, characterized by aerodynamic properties of the wing

Agent learns from its interaction with the environment to command a
series of actions that changes the states of morphing wing’s thickness
or camber toward an optimal shape - optimal policy

Environment is the resulting aerodynamics to which the wing is
subjected to

subjected to
RL agent has no prior knowledge of relationship between actions and
thickness and camber of morphing wing

subjected to
RL agent knows all possible actions that can be applied

subjected to
RL agent has accurate, real-time information of morphing wing shape,
present aerodynamics, and current reward provided by environment

subjected to
RL agent has accurate, real-time information of morphing wing shape,
present aerodynamics, and current reward provided by environment
◦ Sensory info from SMA

RL agent uses a 1-step Q-learning method - an off-policy TD control algo
Q(s, a) ← Q(s, a) + α[r + γ max
a0
Q(s
0
, a
0
) − Q(s, a)]
Initialize Q(s,a) arbitrarily

Q(s, a) ← Q(s, a) + α[r + γ max
a0
Q(s
0
, a
0
) − Q(s, a)]
Repeat (for each episode)
- Initialize s
- Repeat (for each step of the episode)
* Choose a from s using policy derived from Q(s,a) (e.g.
-Greedy Policy)
* Take action a, observe reward r, state s’
* Q(s, a) ← Q(s, a) + α[r + γ maxa0 Q(s
0
, a
0
) − Q(s, a)]
* s ← s’
- until s is terminal

Q(s, a) ← Q(s, a) + α[r + γ max
a0
Q(s
0
, a
0
) − Q(s, a)]
Repeat (for each episode)
- Initialize s
- Repeat (for each step of the episode)
* Choose a from s using policy derived from Q(s,a) (e.g.
-Greedy Policy)
* Take action a, observe reward r, state s’
* Q(s, a) ← Q(s, a) + α[r + γ maxa0 Q(s
0
, a
0
) − Q(s, a)]
* s ← s’
- until s is terminal
return Q(s,a)

Q(s,a) can be updated using:
1 a random policy, with each action having the same probability of
selection

selection
2 an -greedy policy, where is a small value

selection
Action a with the maximum Q(s,a) is selected with probability 1 − ,
otherwise a random action is selected

selection
Action a with the maximum Q(s,a) is selected with probability 1 − ,
otherwise a random action is selected
RL agent learns -greedy policy defined as:
if (probability 1 − )
a = argmaxaQ(s, a)
else
a = rand(ai )
(7)

As # of learning episodes increase, learned action-value function
Q(s,a) converges asymptotically to optimal action-value function
Q∗(s, a)

As # of learning episodes increase, learned action-value function
Q(s,a) converges asymptotically to optimal action-value function
Q∗(s, a)
Q-Learning method is an off-policy one because it evaluates target
policy (the greedy policy) while following another policy

With fewer # of states actions of RL problem, action-value
function Q(s,a) can be tabulated, where an entity in the table is
action value for each state-action pair

States in RL problem for morphing vehicle are shape of wings

States in morphing vehicle are continuous there are an infinite # of
state-action pairs

state-action pairs
To reduce # of state-action pairs, artificially quantize states into
discrete sets while maintaining integrity of learned action-value
function

state-action pairs
function
Depth of quantization is determined through experiments

state-action pairs
function
Depth of quantization is determined through experiments
# of state-action pairs must be kept manageable while adding more
morphing state variables to existing wing states of thickness camber

4.1 Aerodynamic Modeling of Morphing Air Vehicle
Constant strength doublet-source panel6model used to calculate
aerodynamic properties of various wing configurations
6
Niksch 2009
Kumar Digvijay Mishra 4. Mathematical Modeling January 11, 2021 27 / 69

Better than CFD methods
6
Niksch 2009

Assumption: Flow is incompressible and inviscid
6
Niksch 2009

=⇒ Model is valid for linear range of angle-of-attack
6
Niksch 2009

Limitations of the model:
6
Niksch 2009

1 Sensitive to the grid size, location of panels, and # of panels
6
Niksch 2009

because model uses a panel method to determine aerodynamics
6
Niksch 2009

because model uses a panel method to determine aerodynamics
2 Only NACA 4-Digit Series airfoils considered
◦ Have explicit equations that describe upper and lower
geometries of airfoil sections as a function of camber and thickness
◦ Easier to examine and optimize to achieve best possible wing
shape
◦ Have blunt leading edges for thick/thin airfoil sections
◦ Are ideal for subsonic flight conditions
6
Niksch 2009

Versatile Aerodynamic Modeling of Morphing Air Vehicle
Morphing DOF and flight condition parameters in model are:
1 Wing thickness, t
2 Maximum camber
3 Location of maximum camber
4 Root chord, cr
5 Tip chord, ct
6 Sweep angle, Λ
7 Dihedral angle, Γ
8 Wing span, b
9 Wing angle-of-attack, α
Kumar Digvijay Mishra 4.1 Mathematical Modeling January 11, 2021 28 / 69

4.2 Constitutive Equations
Constant strength doublet-source panel method is used for modeling
wing
7
Katz and Plotkin 2001

wing
Basic potential flow theory7used to obtain equations for aerodynamic
forces on wing
52
φ = 0 (8)
Laplace equation where 52 is the Laplace operator and φ is the
velocity potential
7

wing
Basic potential flow theory7used to obtain equations for aerodynamic
forces on wing
52
φ = 0 (8)
Laplace equation where 52 is the Laplace operator and φ is the
velocity potential
Assumption: Incompressible and inviscid flow
7

Using Green’s identity, solution to Laplace equation is formed with a
sum of source σ and doublet µ distributions along the boundary SB

φ = −
1
4π
Z
SB
[σ(
1
r
) − µn · 5(
1
r
)]dS + φ∞ (9)

φ = −
1
4π
Z
SB
[σ(
1
r
) − µn · 5(
1
r
)]dS + φ∞ (9)
Assuming wake convects at trailing edge of wing as a set of thin
doublets, (9) becomes

φ = −
1
4π
Z
SB
[σ(
1
r
) − µn · 5(
1
r
)]dS + φ∞ (9)
Assuming wake convects at trailing edge of wing as a set of thin
doublets, (9) becomes
φ =
1
4π
Z
Body+Wake
µn · 5(
1
r
)dS −
1
4π
Z
Body
σ(
1
r
) + φ∞ (10)

Boundary Conditions
BC is:
No-penetration condition - requires normal velocity of flow at surface
to be zero

Boundary Conditions
BC is:
to be zero
Specified by:

Boundary Conditions
BC is:
to be zero
Specified by:
1 Direct formulation/Neumann condition: forces normal velocity
component of flow at surface to be zero

Boundary Conditions
BC is:
to be zero
Specified by:
1 Direct formulation/Neumann condition: forces normal velocity
component of flow at surface to be zero
2 Indirect formulation/Dirichlet condition: specifies a value for the
potential function, and indirectly satisfies zero normal flow condition on
boundary

φ =
1
4π
Z
Body+Wake
µn · 5(
1
r
)dS −
1
4π
Z
Body
σ(
1
r
) + φ∞
Using Dirichlet BC, potential is specified at all points on the boundary

φ =
1
4π
Z
Body+Wake
µn · 5(
1
r
)dS −
1
4π
Z
Body
σ(
1
r
) + φ∞
If a point is placed inside the surface, inner potential, φi , is defined by
the singularity distributions along the surface
φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞ (11)

φ =
1
4π
Z
Body+Wake
µn · 5(
1
r
)dS −
1
4π
Z
Body
σ(
1
r
) + φ∞
φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞ (11)
Do you see any problem in the integral ?

φ =
1
4π
Z
Body+Wake
µn · 5(
1
r
)dS −
1
4π
Z
Body
σ(
1
r
) + φ∞
φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞ (11)
Do you see any problem in the integral ?
Singularity as r → 0

φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞
To evaluate integrals near singular points, no-penetration BC is
enforced by setting φi to a constant value

φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞
By choosing φi to be φ∞, we get
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

= 0 (12)

φi =
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

+ φ∞
By choosing φi to be φ∞, we get
1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

= 0 (12)
Integral equations are reduced to a set of linear algebraic equations

System divided into N panels for surface Nw panels for wake

System divided into N panels for surface Nw panels for wake
BC specified at a collocation point, which for the Dirichlet BC is
inside body and at center of panel

1
4π
Z
Body+Wake
µ
∂
∂n

1
r

dS −
1
4π
Z
Body
σ

1
r

= 0
For N collocation points for N panels, above equation becomes,
N
X
k=1
1
4π
Z
BodyPanel
µn · 5

1
r

dS +
Nw
X
l=1
1
4π
Z
WakePanel
µn · 5

1
r

dS
−
N
X
k=1
1
4π
Z
BodyPanel
σ

1
r

dS = 0
(13)

N
X
k=1
1
4π
Z
BodyPanel
µn · 5

1
r

dS +
Nw
X
l=1
1
4π
Z
WakePanel
µn · 5

1
r

dS
−
N
X
k=1
1
4π
Z
BodyPanel
σ

1
r

dS = 0
For each collocation point inside body,

N
X
k=1
1
4π
Z
BodyPanel
µn · 5

1
r

dS +
Nw
X
l=1
1
4π
Z
WakePanel
µn · 5

1
r

dS
−
N
X
k=1
1
4π
Z
BodyPanel
σ

1
r

dS = 0
N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0 (14)

N
X
k=1
1
4π
Z
BodyPanel
µn · 5

1
r

dS +
Nw
X
l=1
1
4π
Z
WakePanel
µn · 5

1
r

dS
−
N
X
k=1
1
4π
Z
BodyPanel
σ

1
r

dS = 0
N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0 (14)
A is doublet influence coefficient, and

N
X
k=1
1
4π
Z
BodyPanel
µn · 5

1
r

dS +
Nw
X
l=1
1
4π
Z
WakePanel
µn · 5

1
r

dS
−
N
X
k=1
1
4π
Z
BodyPanel
σ

1
r

dS = 0
N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0 (14)
A is doublet influence coefficient, and
B is source influence coefficient

N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0
Kutta condition(KC): No circulation at the trailing edge of a wing
section

N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0
section
Using KC, doublet strength of wake panel ≡ difference between
trailing edge panels on upper surface and lower surface

N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0
section
KC eliminates wake contribution in above equation,

N
X
k=1
Akµk +
Nw
X
l=1
Al µl −
N
X
k=1
Bkσk = 0
section
KC eliminates wake contribution in above equation,



a11 . . . a1N
.
.
.
...
.
.
.
aN1 . . . aNN






µ1
.
.
.
µN


 = −



b11 . . . b1N
.
.
.
...
.
.
.
bN1 . . . bNN






σ1
.
.
.
σN


 (15)




a11 . . . a1N
.
.
.
...
.
.
.
aN1 . . . aNN






µ1
.
.
.
µN


 = −



b11 . . . b1N
.
.
.
...
.
.
.
bN1 . . . bNN






σ1
.
.
.
σN



Set of N equations with N unknown doublet strengths (µ’s)




a11 . . . a1N
.
.
.
...
.
.
.
aN1 . . . aNN






µ1
.
.
.
µN


 = −



b11 . . . b1N
.
.
.
...
.
.
.
bN1 . . . bNN






σ1
.
.
.
σN



Set of N equations with N unknown doublet strengths (µ’s)
Solvable by matrix inversion



µ1
.
.
.
µN


 = −



a11 . . . a1N
.
.
.
...
.
.
.
aN1 . . . aNN



−1 


b11 . . . b1N
.
.
.
...
.
.
.
bN1 . . . bNN






σ1
.
.
.
σN




Determination of Aerodynamic Forces
After computing doublet strengths, aerodynamic forces acting on
each panel are found

1 Determine tangential (l,m) and normal (n) perturbation velocity
components for each of the panels

ql = −
∂µ
∂l
, qm = −
∂µ
∂m
, qn = −σ (16)

ql = −
∂µ
∂l
, qm = −
∂µ
∂m
, qn = −σ (16)
2 Total velocity of each panel k is computed with
Qk = (Q∞l , Q∞m, Q∞n)k + (ql , qm, qn)k (17)

Qk = (Q∞l , Q∞m, Q∞n)k + (ql , qm, qn)k
These velocities are expressed in local panel coordinate system

3 Pressure coefficient at each panel is found using modified form of
Bernoulli’s equation

Cpk
= 1 −
Q2
k
Q2
∞
(18)

Cpk
= 1 −
Q2
k
Q2
∞
(18)
4 Summation of aerodynamic forces from each panel gives total
aerodynamic force

Cpk
= 1 −
Q2
k
Q2
∞
(18)
4 Summation of aerodynamic forces from each panel gives total
aerodynamic force
X
CFk
= −
Cpk
4S
S
· nk (19)

5 Parasitic drag coefficient CD0 is calculated based on wing area S and
equivalent parasite area f

CD0 =
f
S
(20)

CD0 =
f
S
(20)
where equivalent parasite area is related to wetted area by
log10f = a + b log10Swet (21)
a, b - correlation coefficients related to equivalent skin friction of
aircraft

CD0 =
f
S
(20)
where equivalent parasite area is related to wetted area by
log10f = a + b log10Swet (21)
a, b - correlation coefficients related to equivalent skin friction of
aircraft
6 Oswald efficiency factor is,
e = 1.78(1 − 0.045 AR0.68
) − 0.64 (22)
valid if sweep angle of leading edge of wing 300

4.3 Model Grid
Cosine spacing used to generate grid for both chordwise and spanwise
paneling8
xvi =
c(y)
2

1 − cos

iπ
N + 1

; i = 1 : N (23)
7
Moran 1984
Kumar Digvijay Mishra 4. Mathematical Modeling of Morphing Air Vehicle January 11, 2021 42 / 69

4.3 Model Grid
Cosine spacing used to generate grid for both chordwise and spanwise
paneling8
xvi =
c(y)
2

1 − cos

iπ
N + 1

; i = 1 : N (23)
Cosine spacing gives more panels near leading and trailing edges as
well as root and tip of the wing where many aerodynamic changes
occur
7
Moran 1984

4.3 Model Grid
Straight tapered wing configuration with cosine spacing

4.3 Model Grid
Swept wing configuration with cosine spacing
More panels placed where many aerodynamic changes occur:
1 Leading and trailing edges as well as root and tip of the wing
2 Four corner points of the wing

4.3 Model Grid
Delta wing configuration with cosine spacing
More panels placed where many aerodynamic changes occur:
1 Leading and trailing edges as well as root and tip of the wing
2 Four corner points of the wing

Model Grid Accuracy
As # of panels decrease, accuracy of model also decreases
Kumar Digvijay Mishra 4.3 Mathematical Modeling of Morphing Air Vehicle January 11, 2021 46 / 69

Model Grid Accuracy
Balance between accuracy and computational time is achieved by
defining set # of panels beyond which accuracy does not increase
appreciably

Model Grid Accuracy
appreciably
Ex: Doubling no. of panels is acceptable if accuracy of the
model increased by 10%

Model Grid Accuracy
appreciably
Ex: Doubling no. of panels is acceptable if accuracy of the
model increased by 10%
50% increase in the set of panels is unnecessary, if accuracy is
increased by less than 1%

4.4 Dynamical Modeling

Dynamical Modeling
Velocity vector v with body-axis components is
v =


u
v
w

 (24)

Dynamical Modeling
Velocity vector v with body-axis components is
v =


u
v
w

 (24)
Angular velocity vector ω with roll(p), pitch(q), and yaw(r)
components in body-axis coordinate system is
ω =


p
q
r

 (25)

Dynamical Modeling
Derivative of velocity vector v with respect to time yields
v̇N
= v̇B
+ ω × v =


u̇ + qw − rv
v̇ + ru − pw
ẇ + pv − qu

 (26)

Dynamical Modeling
v̇N
= v̇B
+ ω × v =


u̇ + qw − rv
v̇ + ru − pw
ẇ + pv − qu

 (26)
u̇, v̇, and ẇ: acceleration of body along forward, lateral, and
down/up-ward direction

Dynamical Modeling
v̇N
= v̇B
+ ω × v =


u̇ + qw − rv
v̇ + ru − pw
ẇ + pv − qu

 (26)
There is always forward acceleration, but no lateral acceleration for
longitudinal maneuvers

Dynamical Modeling
v̇N
= v̇B
+ ω × v =


u̇ + qw − rv
v̇ + ru − pw
ẇ + pv − qu

 (26)
There is always forward acceleration, but no lateral acceleration for
longitudinal maneuvers
Notice association of {roll(p), pitch(q), yaw(r)} components of
angular velocity with translational velocity components add to
acceleration components (u̇, v̇, ẇ)

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=


−mg sin(θ) − D cos(α)cos(β) + L sin(α)cos(β) + T
mg sin(φ)cos(θ) + D cos(α)sin(β) − L sin(α)sin(β)
mg cos(φ)cos(θ) − D sin(α) − L cos(α)


(27)

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=




(27)
G: Gravitational force; T: Propulsive(thrust) force

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=




(27)
D: Drag force; L: Lift force

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=




(27)
(X, Y, Z): Aerodynamic forces calculated using constant strength
source-doublet panel method

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=




(27)
ψ, θ, φ: Yaw, pitch roll angles

Dynamical Modeling
Force vector is,
F =


X + Tx + Gx
Y + Ty + Gy
Z + Tz + Gz


=




(27)
ψ, θ, φ: Yaw, pitch roll angles
(α, β): Aerodynamic angles (AoA, sideslip)

Dynamical Modeling
From Newton’s 2nd law, translational equations of motion are:
m(u̇ +qw −rv) = −mgsin(θ)−Dcos(α)cos(β)+Lsin(α)cos(β)+T (28)
m(v̇ +ru−pw) = −mgsin(φ)cos(θ)−Dcos(α)sin(β)−Lsin(α)sin(β) (29)
m(ẇ + pv − qu) = −mgcos(φ)cos(θ) − Dsin(α) − Lcos(α) (30)
p, q, r: roll, pitch, yaw components of angular velocity vector, ω
u, v, w: components of body-axis translational velocity vector, v
ψ, θ, φ: yaw(unused), pitch, roll angles
α, β: aerodynamic angles
T: Thrust; D: Drag ; L: Lift
Kumar Digvijay Mishra 4.4 Modeling of Morphing Air Vehicle January 11, 2021 51 / 69

Dynamical Modeling
Euler’s equation L = ḣ
N
gives relationship between vector sum of
moments L and angular momentum vector h

Dynamical Modeling
N
Angular momentum vector related to inertia matrix I and body-axis
angular velocity vector ω by
h = Iω (31)

Dynamical Modeling
N
h = Iω (31)
Inertia matrix is,
I =
Ixx −Ixy −Ixz
−Ixy Iyy −Iyz
−Ixz −Iyz Izz

(32)

Dynamical Modeling
N
h = Iω (31)
Inertia matrix is,
I =
Ixx −Ixy −Ixz
−Ixy Iyy −Iyz
−Ixz −Iyz Izz

(32)
1 These moments-of-inertia are time-varying because aircraft wing
morphs during flight

Dynamical Modeling
N
h = Iω (31)
Inertia matrix is,
I =
Ixx −Ixy −Ixz
−Ixy Iyy −Iyz
−Ixz −Iyz Izz

(32)
1 These moments-of-inertia are time-varying because aircraft wing
morphs during flight
2 Should be computed for each state of morphing wing aircraft as well as
transitions between them. How to compute them in real-time?

Dynamical Modeling
Time derivative of angular momentum vector h is,
ḣ
N
= Iω̇B
+ İ
B
ω + ω × Iω
=








Ixx ṗ − Ixy q̇ − Ixz ˙
r + ˙
Ixx p − ˙
Ixy q − ˙
Ixzr+p(−Ixzq + Ixy r)
+ q(−Iyzq − Iyy r) + r(Izzq + Iyzr)
−Ixy ṗ − Iyy q̇ − Iyz ˙
r − ˙
Ixy p + ˙
Iyy q − ˙
Iyzr+p(Ixx r + Ixzp)
+ q(−Ixy r + Iyzp) + r(−Ixzr − Izzp)
−Ixzṗ − Iyzq̇ + Izz ˙
r − ˙
Ixzp − ˙
Iyzq + ˙
Izzr+p(−Ixy p − Ixx q)
+q(Iyy p + Ixy q) + r(−Iyzp + Ixzq)








(33)

Dynamical Modeling
ḣ
N
= Iω̇B
+ İ
B
ω + ω × Iω
=








r + ˙
Ixx p − ˙
Ixy q − ˙
r − ˙
Ixy p + ˙
Iyy q − ˙
r − ˙
Ixzp − ˙
Iyzq + ˙








(33)
9 inertia terms that vary in temporal dimension

Dynamical Modeling
ḣ
N
= Iω̇B
+ İ
B
ω + ω × Iω
=








r + ˙
Ixx p − ˙
Ixy q − ˙
r − ˙
Ixy p + ˙
Iyy q − ˙
r − ˙
Ixzp − ˙
Iyzq + ˙








(33)
9 inertia terms that vary in temporal dimension
Is it prudent to neglect time-varying inertia due to morphing?

Dynamical Modeling
Moments are due to ?

Dynamical Modeling
Aerodynamic forces

Dynamical Modeling
Aerodynamic forces
where aerodynamic forces are calculated by ?

Dynamical Modeling
Aerodynamic forces
Constant strength source-doublet panel method

Dynamical Modeling
Aerodynamic forces
Aerodynamic moment vector is,
L =


LA
MA
NA

 (34)

Dynamical Modeling
Aerodynamic forces
L =


LA
MA
NA

 (34)
LA: Roll moment MA: Pitch moment NA: Yaw moment

Dynamical Modeling
Aerodynamic forces
L =


LA
MA
NA

 (34)
Why not account for moment produced by propulsive forces?

Dynamical Modeling
Aerodynamic forces
L =


LA
MA
NA

 (34)
Why not account for moment produced by propulsive forces?
Because thrust force is assumed to act along the line parallel to
body’s x-axis at C-of-G location

Rotational Equation of Motion for Morphing Wing
Using Euler’s equation L = ḣ
N
,
LA = Ixx ṗ − Ixy q̇ − Ixz ˙
r + ˙
Ixx p − ˙
Ixy q − ˙
(35)

N
,
r + ˙
Ixx p − ˙
Ixy q − ˙
(35)
MA = −Ixy ṗ + Iyy q̇ − Iyz ˙
r − ˙
Ixy p + ˙
Iyy q − ˙
(36)

N
,
r + ˙
Ixx p − ˙
Ixy q − ˙
(35)
r − ˙
Ixy p + ˙
Iyy q − ˙
(36)
NA = −Ixzṗ − Iyzq̇ + Izz ˙
r − ˙
Ixzp − ˙
Iyzq + ˙
+ q(Iyy p + Ixy q) + r(−Iyzp + Ixzq)
(37)

N
,
r + ˙
Ixx p − ˙
Ixy q − ˙
(35)
r − ˙
Ixy p + ˙
Iyy q − ˙
(36)
NA = −Ixzṗ − Iyzq̇ + Izz ˙
r − ˙
Ixzp − ˙
Iyzq + ˙
+ q(Iyy p + Ixy q) + r(−Iyzp + Ixzq)
(37)
Time-varying inertias are present in roll, pitch and yaw moments.
Why neglect them?

Exercise - Inertia of an aircraft
Ixx = N/m2 Ixy = N/m2 Ixz = N/m2

Iyx = N/m2 Iyy = N/m2 Iyz = N/m2

Izz = N/m2 Izy = N/m2 Izz = N/m2

Izz = N/m2 Izy = N/m2 Izz = N/m2
Rudder Fault =⇒ Transience in Inertia - Is ˙
Iij negligible?

Dynamical Modeling

Dynamical Modeling
Aerodynamic angles α and β are,
tan(α) =
w
VT
(38)
tan(β) =
v
VT
(39)

Dynamical Modeling
tan(α) =
w
VT
(38)
tan(β) =
v
VT
(39)
Body axis system

Dynamical Modeling
tan(α) =
w
VT
(38)
tan(β) =
v
VT
(39)
Body axis system
w: velocity along body z-axis
v: velocity along body y-axis
VT : total body velocity

Dynamical Modeling

Dynamical Modeling
Aerodynamic angles change at a rate
of,
α̇ =
1
VT
(ẇ cos α − u̇ sin α) (40)
β̇ =
1
VT
(v̇ cos β − u̇ sin β) (41)

Dynamical Modeling
Aerodynamic angles change at a rate
of,
α̇ =
1
VT
(ẇ cos α − u̇ sin α) (40)
β̇ =
1
VT
(v̇ cos β − u̇ sin β) (41)
Body axis system

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