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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
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• System Model
• Control of total body force and moments
L
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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
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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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