# 2. Dynamics, Control and Trajectory Following

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1 2. Dynamics, Control and Trajectory Following

2 This module Flying vehicles: how do they work? Quick refresher on aircraft dynamics with reference to the magical flying space potato How I learned to stop worrying and love control Attitude control from a robotics perspective Zen and the art of trajectory following Position control and hierarchical architecture

3 Early morning stimulus question 4 What do all aircraft have in common? What differentiates them from non-aircraft? Form groups of 2-3 people and yell at them until you all agree

4 It s ok, don t be afraid.

5 A quick refresher An aircraft can be thought of as a magical flying space potato* magic *or a flying brick, if you prefer

6 Basic potato dynamics To maneuver, the potato applies magic (which we ll call forces and torques ) It s not important how this magic is produced magic

7 Basic potato dynamics Like all magic potatoes, when not accelerating, forces on the aircraft must be in equilibrium magic mg

8 Basic potato dynamics Unbalanced forces result in an acceleration proportional to the mass/inertia of the potato: magic Resultant acceleration Net force mg

9 Basic potato dynamics We represent the location of the potato in space by a coordinate vector, defined in an fixed inertial reference frame

10 Basic potato dynamics There are lots of ways of representing rotation. eg. rotation matrices, Euler angles, quaternions

11 Basic potato dynamics Given a rigid space potato (a firm assumption), internal forces and torques are generated due to body-intrinsic rotations*: *You may know of these as gyroscopic and Coriolis forces

12 Basic potato dynamics Now we know enough to write the dynamics of magical and non-magical forces:

13 Basic potato dynamics But we also care about the angle and trajectory of potato flight the integral of spud velocities:

14 Basic potato dynamics We have the complete equations of motion but where does the magic come from? Aerodynamics!

15 Basic potato dynamics There are lots of different ways of making magic

16 Basic potato dynamics Different magic systems create different couplings between forces in each direction Few aircraft are truly decoupled in all axes It is this cross-coupling that makes control interesting

17 Early morning stimulus question 5 What is this control thing, anyway? How is a controlled system different from a robotic system? Form groups of 2-3 people and yell at them until you all agree

18 position The control thing: what is it? Control is the process of making current system states converge to desired final states This can be orientation, position, as well as velocities and accelerations This is done by using state measurements to compute commands leading to convergence goal time

19 Example: a simple rotorcraft Consider a super-simple dynamic model of a rotary-wing vehicle (a quadrotor in particular) Fixed direction of thrust in the body frame No flapping torques, forces, etc Rotor pairs generate pure torques Assume for now that rotor thrust is constant and exactly equals gravity

20 Underactuation This system is underactuated we have fewer control inputs than states to control* Motion in a desired direction must arise as a result of movement in another direction So what do we do? * This is where smart people start talking about stuff like group theory

21 Attitude control Fortunately, we don t care about every state all at once* instead, let s focus on attitude Helicopters are inherently unstable, so attitude control is important Rotation is approximately decoupled in hover, so we can treat pitch and roll separately *If we did, we would have a deep conversation about controllability

22 Example: PD control Let s consider only pitch things are vastly simplified when cross-coupling is ignored!, This is a second-order system we can stabilise it with Proportional-Derivative control* *Proofs usually follow statements of this kind

23 Example: PD control We can also use to drive the system to a particular reference angle, We can think of this feedback process as a series of interconnected dynamic systems: PD system dynamics -1

24 Position control But wait a minute pitch angle feeds into the longitudinal position dynamics! We can use a simple linearised model to build a controller for position around hover:, This is also a second order system PD ahoy!

25 Example: PD control We can use as a virtual control term:* The closed-loop system will converge to and *Assuming time-scale separation of rotational and translational dynamics

26 Cascaded system This builds a nested control structure PD PD attitude dynamics longitudinal dynamics -1 We can use a sequence of desired positions to make the craft follow a desired trajectory, guiding the aircraft through space -1

27 Early morning stimulus question 6 Where does state knowledge come from? Where do reference trajectories come from? Form groups of 2-3 people and yell at them until you all agree

28 Tip of the iceberg There are more methods of controlling dynamic systems than you can shake a stick at: PD, PID, PD 2 LQR, LQG, backstepping, sliding mode Robust control, adaptive control, etc just to name a few! Each and every one has been applied to UAVs somehow and they ve been done for a variety of coordinate systems: Euler angles, Tait-Bryan angles Quaternions Rotation matrices Axis-angle

29 Questions?

30 Zwischenspiel.

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