IEEE Robotics & Automation Magazine - December 2018 - 40
derivative, and integral gains respectively; m is the vehicle
mass; and g is the gravity vector.
Attitude Control
The attitude controller consists of two cascade controllers: an
outer loop for the attitude control that generates desired body
rates and a rate controller that creates desired moments. The
attitude controller is based on quaternion error, which is
given by
q err = q des, I B 7 qt I* B = c
q w, err
m.
q v, err
(13)
The desired body rate ~ des is generated from the vector
part of the quaternion error q v,err as follows:
~ des = k q sign(q w, err) q v, err ,
(14)
where k q is a tuning parameter. Note that the sign of the real
part of the quaternion error is used to avoid the unwinding
phenomenon.
The desired moments M des are computed as
t,
M des = k r (~ des - ~t ) - roff # B Fdes + ~t # J~
(15)
where k r is the rate controller gain, ~t is the estimated angular velocity, roff is the center of mass offset, and J is the vehicle's inertia matrix.
Control Allocation
The control allocation problem deals with finding rotor
speeds n 1, f, n 6 and rotor angular positions a to satisfy (10).
This control allocation problem has been studied in detail in
[22] for marine vessels, while a general survey about control
allocation can be found in [23]. Solving this equation is challenging for two reasons. First, any solution is not unique
because of the overactuated nature of the platform. Second,
the rotor angular positions appear in a nonlinear fashion in
the allocation matrix (11).
One approach to solve this problem would be to perform a
nonlinear least-squares optimization. However, the computation time required and the available computation resources
on board the platform inhibit such a solution. Another approach is to use a nonlinear model- predictive attitude controller [24] that generates near-optimal rotor-speed and
tilting-angle commands. This method is appealing because it
can easily account for tilting-angle dynamics, but it can be
computationally expensive. Note that this allocation needs to
be solved at a high rate to allow the system to perform in an
agile manner.
Here, we propose a means to transform the nonlinear allocation problem into a linear problem through variable transformation. We show that this solution also satisfies the
constraint of a positive rotor speed. To this end, we decompose the force generated by each rotor, as shown in (4), along
the vertical and lateral axes of the rotor. We define the vertical
and lateral forces of the ith rotor as
40
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
december 2018
Fv, i = nn 2i cos a i ,
2
i
Fl, i = nn sin a i .
(16)
(17)
We also define the vector of dimension 12 # 1 of all of the
vertical and lateral forces as
Fdec = (Fv, 1, Fl, 1, f, Fv, 6, Fl, 6) < .
(18)
By rearranging (10), we obtain
c
F
m = A static Fdec ,
M
(19)
where A static is a static allocation matrix of dimensions
6 # 12 that does not depend on the rotor orientation a. At
this point, we can easily invert (19) to calculate Fdec from a
desired wrench ^F <, M
IEEE Robotics & Automation Magazine - December 2018
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2018
Contents
IEEE Robotics & Automation Magazine - December 2018 - Cover1
IEEE Robotics & Automation Magazine - December 2018 - Cover2
IEEE Robotics & Automation Magazine - December 2018 - Contents
IEEE Robotics & Automation Magazine - December 2018 - 2
IEEE Robotics & Automation Magazine - December 2018 - 3
IEEE Robotics & Automation Magazine - December 2018 - 4
IEEE Robotics & Automation Magazine - December 2018 - 5
IEEE Robotics & Automation Magazine - December 2018 - 6
IEEE Robotics & Automation Magazine - December 2018 - 7
IEEE Robotics & Automation Magazine - December 2018 - 8
IEEE Robotics & Automation Magazine - December 2018 - 9
IEEE Robotics & Automation Magazine - December 2018 - 10
IEEE Robotics & Automation Magazine - December 2018 - 11
IEEE Robotics & Automation Magazine - December 2018 - 12
IEEE Robotics & Automation Magazine - December 2018 - 13
IEEE Robotics & Automation Magazine - December 2018 - 14
IEEE Robotics & Automation Magazine - December 2018 - 15
IEEE Robotics & Automation Magazine - December 2018 - 16
IEEE Robotics & Automation Magazine - December 2018 - 17
IEEE Robotics & Automation Magazine - December 2018 - 18
IEEE Robotics & Automation Magazine - December 2018 - 19
IEEE Robotics & Automation Magazine - December 2018 - 20
IEEE Robotics & Automation Magazine - December 2018 - 21
IEEE Robotics & Automation Magazine - December 2018 - 22
IEEE Robotics & Automation Magazine - December 2018 - 23
IEEE Robotics & Automation Magazine - December 2018 - 24
IEEE Robotics & Automation Magazine - December 2018 - 25
IEEE Robotics & Automation Magazine - December 2018 - 26
IEEE Robotics & Automation Magazine - December 2018 - 27
IEEE Robotics & Automation Magazine - December 2018 - 28
IEEE Robotics & Automation Magazine - December 2018 - 29
IEEE Robotics & Automation Magazine - December 2018 - 30
IEEE Robotics & Automation Magazine - December 2018 - 31
IEEE Robotics & Automation Magazine - December 2018 - 32
IEEE Robotics & Automation Magazine - December 2018 - 33
IEEE Robotics & Automation Magazine - December 2018 - 34
IEEE Robotics & Automation Magazine - December 2018 - 35
IEEE Robotics & Automation Magazine - December 2018 - 36
IEEE Robotics & Automation Magazine - December 2018 - 37
IEEE Robotics & Automation Magazine - December 2018 - 38
IEEE Robotics & Automation Magazine - December 2018 - 39
IEEE Robotics & Automation Magazine - December 2018 - 40
IEEE Robotics & Automation Magazine - December 2018 - 41
IEEE Robotics & Automation Magazine - December 2018 - 42
IEEE Robotics & Automation Magazine - December 2018 - 43
IEEE Robotics & Automation Magazine - December 2018 - 44
IEEE Robotics & Automation Magazine - December 2018 - 45
IEEE Robotics & Automation Magazine - December 2018 - 46
IEEE Robotics & Automation Magazine - December 2018 - 47
IEEE Robotics & Automation Magazine - December 2018 - 48
IEEE Robotics & Automation Magazine - December 2018 - 49
IEEE Robotics & Automation Magazine - December 2018 - 50
IEEE Robotics & Automation Magazine - December 2018 - 51
IEEE Robotics & Automation Magazine - December 2018 - 52
IEEE Robotics & Automation Magazine - December 2018 - 53
IEEE Robotics & Automation Magazine - December 2018 - 54
IEEE Robotics & Automation Magazine - December 2018 - 55
IEEE Robotics & Automation Magazine - December 2018 - 56
IEEE Robotics & Automation Magazine - December 2018 - 57
IEEE Robotics & Automation Magazine - December 2018 - 58
IEEE Robotics & Automation Magazine - December 2018 - 59
IEEE Robotics & Automation Magazine - December 2018 - 60
IEEE Robotics & Automation Magazine - December 2018 - 61
IEEE Robotics & Automation Magazine - December 2018 - 62
IEEE Robotics & Automation Magazine - December 2018 - 63
IEEE Robotics & Automation Magazine - December 2018 - 64
IEEE Robotics & Automation Magazine - December 2018 - 65
IEEE Robotics & Automation Magazine - December 2018 - 66
IEEE Robotics & Automation Magazine - December 2018 - 67
IEEE Robotics & Automation Magazine - December 2018 - 68
IEEE Robotics & Automation Magazine - December 2018 - 69
IEEE Robotics & Automation Magazine - December 2018 - 70
IEEE Robotics & Automation Magazine - December 2018 - 71
IEEE Robotics & Automation Magazine - December 2018 - 72
IEEE Robotics & Automation Magazine - December 2018 - 73
IEEE Robotics & Automation Magazine - December 2018 - 74
IEEE Robotics & Automation Magazine - December 2018 - 75
IEEE Robotics & Automation Magazine - December 2018 - 76
IEEE Robotics & Automation Magazine - December 2018 - 77
IEEE Robotics & Automation Magazine - December 2018 - 78
IEEE Robotics & Automation Magazine - December 2018 - 79
IEEE Robotics & Automation Magazine - December 2018 - 80
IEEE Robotics & Automation Magazine - December 2018 - 81
IEEE Robotics & Automation Magazine - December 2018 - 82
IEEE Robotics & Automation Magazine - December 2018 - 83
IEEE Robotics & Automation Magazine - December 2018 - 84
IEEE Robotics & Automation Magazine - December 2018 - 85
IEEE Robotics & Automation Magazine - December 2018 - 86
IEEE Robotics & Automation Magazine - December 2018 - 87
IEEE Robotics & Automation Magazine - December 2018 - 88
IEEE Robotics & Automation Magazine - December 2018 - 89
IEEE Robotics & Automation Magazine - December 2018 - 90
IEEE Robotics & Automation Magazine - December 2018 - 91
IEEE Robotics & Automation Magazine - December 2018 - 92
IEEE Robotics & Automation Magazine - December 2018 - 93
IEEE Robotics & Automation Magazine - December 2018 - 94
IEEE Robotics & Automation Magazine - December 2018 - 95
IEEE Robotics & Automation Magazine - December 2018 - 96
IEEE Robotics & Automation Magazine - December 2018 - 97
IEEE Robotics & Automation Magazine - December 2018 - 98
IEEE Robotics & Automation Magazine - December 2018 - 99
IEEE Robotics & Automation Magazine - December 2018 - 100
IEEE Robotics & Automation Magazine - December 2018 - 101
IEEE Robotics & Automation Magazine - December 2018 - 102
IEEE Robotics & Automation Magazine - December 2018 - 103
IEEE Robotics & Automation Magazine - December 2018 - 104
IEEE Robotics & Automation Magazine - December 2018 - 105
IEEE Robotics & Automation Magazine - December 2018 - 106
IEEE Robotics & Automation Magazine - December 2018 - 107
IEEE Robotics & Automation Magazine - December 2018 - 108
IEEE Robotics & Automation Magazine - December 2018 - 109
IEEE Robotics & Automation Magazine - December 2018 - 110
IEEE Robotics & Automation Magazine - December 2018 - 111
IEEE Robotics & Automation Magazine - December 2018 - 112
IEEE Robotics & Automation Magazine - December 2018 - 113
IEEE Robotics & Automation Magazine - December 2018 - 114
IEEE Robotics & Automation Magazine - December 2018 - 115
IEEE Robotics & Automation Magazine - December 2018 - 116
IEEE Robotics & Automation Magazine - December 2018 - 117
IEEE Robotics & Automation Magazine - December 2018 - 118
IEEE Robotics & Automation Magazine - December 2018 - 119
IEEE Robotics & Automation Magazine - December 2018 - 120
IEEE Robotics & Automation Magazine - December 2018 - 121
IEEE Robotics & Automation Magazine - December 2018 - 122
IEEE Robotics & Automation Magazine - December 2018 - 123
IEEE Robotics & Automation Magazine - December 2018 - 124
IEEE Robotics & Automation Magazine - December 2018 - 125
IEEE Robotics & Automation Magazine - December 2018 - 126
IEEE Robotics & Automation Magazine - December 2018 - 127
IEEE Robotics & Automation Magazine - December 2018 - 128
IEEE Robotics & Automation Magazine - December 2018 - 129
IEEE Robotics & Automation Magazine - December 2018 - 130
IEEE Robotics & Automation Magazine - December 2018 - 131
IEEE Robotics & Automation Magazine - December 2018 - 132
IEEE Robotics & Automation Magazine - December 2018 - 133
IEEE Robotics & Automation Magazine - December 2018 - 134
IEEE Robotics & Automation Magazine - December 2018 - 135
IEEE Robotics & Automation Magazine - December 2018 - 136
IEEE Robotics & Automation Magazine - December 2018 - Cover3
IEEE Robotics & Automation Magazine - December 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2023
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2023
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2023
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2023
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2022
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2022
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2022
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2022
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2021
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2021
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2021
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2021
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2020
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2020
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2020
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2020
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2019
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2019
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2019
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2019
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2018
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2018
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2018
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2018
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2017
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2017
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2017
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2017
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2016
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2016
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2016
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2016
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2015
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2015
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2015
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2015
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2014
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2014
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2014
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2014
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2013
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2013
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2013
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2013
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2012
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2012
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2012
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2012
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2011
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2011
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2011
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_march2011
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2010
https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_september2010
https://www.nxtbookmedia.com