Autonomous Control of a Quadrotor

Mangal Kothari

Assistant Professor

Department of Aerospace Engineering

Indian Institute of Technology, Kanpur

Organization

• How does a quadrotor work ?

• Mathematical modeling

• Control

• An application

Quadrotor UAV

• Mechanical simplicity

• Ease of construction

• Good maneuverability

• Great platform to learn auto-pilot development

• Test new control strategies

Mechanics

• Four propellers generating four thrust forces and torques

• Two possible configuration ``+’’ and ``x’’

• One set of the propellers (1 and 3) rotates in clockwise direction and another set of the propellers in anti-clockwise direction

• For hovering flight, thrust and torque balance are achieved

Hovering Condition

• Thrust balance

• Moment balance

• Torque balance (w1^2+w3^2-w2^2-w4^2)

Autonomous Capability

Coordinated Frames

• Describe relative position and orientation of objects – Aircraft relative to direction of wind

– Camera relative to aircraft

– Aircraft relative to inertial frame

• Some things most easily calculated or described in certain reference frames – Newton’s law

– Aircraft attitude

– Aerodynamic forces/torques

– Accelerometers, rate gyros

– GPS

– Mission requirements

Rotation of Reference Frame

Rotation of Reference Frame

Inertial Frame and Vehicle Frame

• Vehicle frame has same orientation

as inertial frame

• Vehicle frame is fixed at cm of aircraft

• Inertial and vehicle frames are referred

to as NED frames

• Nx, Ey, Dz

Euler Angles

• Need way to describe attitude of aircraft

• Common approach: Euler angles

• Pro: Intuitive

• Con: Mathematical singularity

– Quaternions are alternative for overcoming singularity

Vehicle-1 Frame

Vehicle-2 Frame

Body Frame

Inertial Frame to Body Frame Transformation

Translational Kinematics

Rotational Kinematics

Inverting gives

State Equations

Six of the 12 state equations for the UAV come from the kinematic

equations relating positions and velocities:

The remaining six equations will come from applying Newton’s 2nd law

to the translational and rotational motion of the aircraft.

Translational Dynamics

Newton’s 2nd Law:

What is ?

• is the sum of all external forces

• is the mass of the aircraft

• Time derivative taken wrt inertial frame

Differentiation of a Vector

Translational Dynamics

can be expressed in body frame as

where

Since we have that

Rotational Dynamics

Newton’s 2nd Law:

• is the angular momentum vector

• is the sum of all external moments

• Time derivative taken wrt inertial frame

Expressed in the body frame,

Rotational Dynamics

Because is unchanging in the body frame, and

Rearranging we get

where

Angular Momentum

Angular Momentum

Inertia Components

Moments of inertia

Products of Inertia

The Tensor of Inertia

Inertia matrix (J)

Inertia matrix

Rotational Dynamics

If the aircraft is symmetric about the plane, then and

This symmetry assumption helps simplify the analysis. The inverse of

becomes

Rotational Dynamics

Define

’s are functions of moments and products of inertia

Gravity Force

expressed in vehicle frame

expressed in body frame

Equation of Motion Summary

System of 12 first-order ODE’s

RPM Controlled Quadrotor

PWM Signal

Control Philosophy

• Force balance – Thrust support weight of a quadrotor

• Moment balance – Controlling attitude rate of control

• Quadrotor is controlled by using four motors

Quadrotor Control Design

• PID control design – Inner and outer loops design

• Dynamic inversion approach – Inner and outer loop design

PID: Stabilization

Inner loop stabilization

Demanded pitch, rolling, and yaw moments

PID: Position Hold

Outer loop design

Demanded thrust, roll, and, pitch commands

Dynamic Inversion Approach

• Nonlinear control design: works for the whole flight regime

• Cancel the all nonlinearities in the system through the feedback law

• Requires the complete knowledge of the plant

DI: Stabilization

• Choosing second order error dynamics

Inner loop Design

Position Control

Second order error dynamics

Demanded control commands

DI – Based on Position and Euler Angle Regulation

Performance of DI Design Based on Position and Euler Angle Set Points

Position Attitude

High frequency maneuver

Drawbacks of Neglecting Coordinate Transformation in Attitude

Tracking with assumption Tracking without assumption

DI: Stabilization using Body Rates

• Choosing first order error dynamics

Performance Comparison of PID Design and Proposed DI Design

Commanded trajectory

Initial condition

PID Vs Proposed DI Design

• Case 1 : w = (6.5* ∏/10) rad/s

– PID controller performance degrades

Position Attitude

PID Vs Proposed DI Design

• Case 1 : w = (6.5* ∏/10) rad/s

– PID controller generates strong control jerks

Position Commanded control inputs

PID Vs Proposed DI Design

• Case 2 : w = (7.325* ∏/10) rad/s

– PID controller fails to track commanded trajectory

PID Proposed DI design

PID Vs Proposed DI Design

• Case 2 : w = (7.325* ∏/10) rad/s

– PID controller fails to track commanded trajectory but the proposed DI design tracks

smoothly

PID Proposed DI design

PID Vs Proposed DI Design

• Case 3 : w = (11.5* ∏/10) rad/s

– Performance of proposed DI design degrades

– Control saturation

Position Commanded control inputs

Cooperative Target Capturing

55

Problem Formulation

Consider a pursuit-evasion game

Define

Problem statement

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Holonomic System

The safe-reachable area of the evader can be computed using Voronoi partition approach

as follows

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Safe-Reachable Areas

The rate of change in the area A

Choose

The area can be computed using the following

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Gradient of A

Perturbation along the LOS Perturbation perpendicular to the LOS

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Cooperative Capturing Strategy

Pursuit evasion geometry

Lemma

The proposed strategy points towards

the negative of the gradient vector

Proof

Guidance command

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Evasive Action

Lemma

Under the proposed strategy, the rate of change in the area is less than equal to zero

for any admissible evader control input. The rate of change is equal to zero if and only

if the evader follows the following strategy

Proof

where equality hold if and only if

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Capturability

The rate of change in squared relative distance

Lemma

If the rate of change of the area is zero, then under the proposed strategy, the capture

is guaranteed to occur.

Proof

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Numerical Result

63

Non-holonomic Systems

Safe-reachable set of an evader

D: Dubins distance

Solve:

Difficulties:

• Computing the safe-reachable area is intractable

• The optimization problem is combinatorial in the

nature

Non-holonomic system – subject to motion constraint (e.g. turn radius)

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Solution Approach

Compute approximate area

• Discreatize the workspace

• Employ a fast search method (Dijakstra type algorithm) to compute the

number of cells those have positive cost

Sequentially compute the control command

• Use a look-ahead strategy with control parameterization

• Choose ui from

• Solve the simplified problem

-1000 -800 -600 -400 -200 0 200 400 600 800 1000-1000

-800

-600

-400

-200

0

200

400

600

800

1000

X(m)

Y(m

)

Approximate area: A cell is safe-reachable if the

following cost is positive

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Numerical Results

66

Trajectories of pursuers and evader Safe-reachable areas

Cooperative PPN Law

• Pure Proportional navigation (PPN) law is used to design the capturing strategy

– The rate of rotation of the velocity vector is proportional to the rate of rotation of the LOS

– Easy to implement

– Optimal and efficient

– To capture attain (collision triangle condition) and then follow the collision course that result in a collision

– To avoid capture, apply invers-PN (I-PN) law

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Numerical Result

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Questions ???