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Minimum Snap Trajectory Generation for Control of

Quadrotors

(Best Paper ICRA 2011)

Daniel Mellinger and Vijay Kumar

GRASP Lab, UPenn

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(Very) Brief Outline

• Goal is to develop planning and control techniques for control of an autonomous quadrotor

• Experimental setup– Ascending Technologies Hummingbird

• 200g payload• 20min

– Vicon Motion Capture system• IR and visual cameras to capture position and

orientation from markers on the target• 200Hz

• System Model

• Control of total body force and moments

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L

Differential Flatness

• A complicated thing I don’t really understand

1 - Select a set of ‘flat’ outputs

2 - The inputs must be able to be written as a function of the flat outputs and a (limited number of) their derivatives

3 - ?

4 – Planning and control (profit)

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• For the quadrotor, the full state is:

(position, velocity, orientation, angular velocity)

• Flat output selection

• Then allowable trajectories are smooth functions of position and yaw angle

• Then, need to find equations expressing the inputs (engine speeds) as a function of the flat outputs

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Orientation as a function of flat outputs

• Body z-axis unit vector is the acceleration direction

• Further rotation by the yaw angle gives the x unit vector (in the intermediate yaw frame)

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Angular velocity as a function of flat outputs

• Body acceleration

• Differentiate (u1 is the total force from the motors)

• Isolate angular rate components

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Angular accelerations as a function of flat outputs

• Angular acceleration

• x and y components (differantiate and dot-product with the x and y body vectors)

• z-component

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Step 3 - ?

• Then, force is a function of the flat outputs

• Angular velocity and acceleration are functions of the flat outputs

• Then, u = f(x, y, z, ψ)

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Control

• From a defined trajectory

• Control law to find desired force to travel to target trajectory

• Now need desired rotation matrix

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• As before, get the intermediate yaw frame, then the body axis unit vectors

• Orientation error

• Angular velocity error

• Control variables

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Trajectories

• Keyframes (defined trajectory points) with ‘safety corridors’ between each keyframe

• Link keyframes with polynomial paths in the flat output space

• Minimise curvature for smooth achievable paths

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• i.e.

• “The cost function [...] is similar to that used by Flash and Hogan who showed human reaching trajectories appear to minimize the integral of the square of the norm of the jerk (the derivative of acceleration, kr = 3). In our system, since the inputs u2 and u3 appear as functions of the fourth derivatives of the positions, we generate trajectories that minimize the integral of the square of the norm of the snap (the second derivative of acceleration, kr = 4).

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• Components are decoupled which means they can be solved separately

• Re-consider the general problem for w

• Can also consider time scaling to determine the temporal length of the path segments

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Notes on the paper

• At the page limit – efficient use of space– reference previous work and theories

• Nice graphical representations of video data

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