# US 10,478,690 B2 · Golf clubs and golf club heads having a plurality of sensors for detecting one or more swing parameters

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- Number: US 10,478,690 B2
- Document ID: US10478690B2
- Status: Granted
- Kind: B2
- Application number: 14/951,865
- Filed: 2015-11-25
- Published: 2017-05-25
- Granted: 2019-11-19
- Earliest claimed: 2015-11-25
- Assignees: NIKE, Inc.
- Applicants: NIKE, Inc.
- Inventors: Bradley C. Glenn; Douglas A. Thornton; Jeffrey A. Hadden
- Web page: https://patentgrep.com/patent/US10478690B2

## CPC

- A61B 5/6895: {Sport equipment}
- A63B 60/46: Measurement devices associated with golf clubs, bats, rackets or the like for measuring physical parameters relating to sporting activity, e.g. baseball bats with impact indicators or bracelets for measuring the golf swing
- A61B 5/1114: {Tracking parts of the body}
- A63B 53/04: Heads
- A61B 2503/10: Athletes
- G06F 2218/08: Feature extraction
- A61B: DIAGNOSIS; SURGERY; IDENTIFICATION (usefulness limited to only animals A61D)
- A63B: APPARATUS FOR PHYSICAL TRAINING, GYMNASTICS, SWIMMING, CLIMBING, OR FENCING; BALL GAMES; TRAINING EQUIPMENT (apparatus for passive exercising, massage A61H)
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A golf club head having accelerometer and gyroscope sensors measuring swing motion acceleration and angular rate values. Inputs of force and torque to the golf club head during the swing motion may be unknown, such that a system of motion equations may be used to calculate functions for the unknown input force and torque, as well as roll angle and pitch angle for the golf club head, and compensate for gyroscope bias, without using static leveling processes.

## Claims

### Claim 1

An inertial measurement unit, comprising:
an output device;
an accelerometer;
a gyroscope; and
an application-specific integrated circuit comprising discrete hardware components configured to perform at least:
receive sensor data from the accelerometer and gyroscope coupled to an object in motion between a first time instant and a second time instant;
extract an acceleration measurement and an angular rate measurement from the sensor data;
identify, from the angular rate measurement, a gyroscope saturation event corresponding to an output of the gyroscope reaching a physical limitation of hardware of the gyroscope;
calculate an estimated output error using the received sensor data, wherein the estimated output error includes a saturation function that takes account of the gyroscope saturation event;
input the estimated output error into a system of motion equations for the object, wherein the system of motion equations includes an estimate of a gyroscope bias estimated without using static leveling;
solve the system of motion equations;
output functions for an input force, an input torque, a roll angle, and a pitch angle from the solved system of motion equations; and
calculate and output to the output device, a position of the object in motion between the first time instant and the second time instant using the functions.

### Claim 2 (depends on 1)

The inertial measurement unit of claim 1, wherein the acceleration measurement comprises acceleration values resolved along three mutually-perpendicular axes and the angular rate measurement comprises angular rate values about the three mutually-perpendicular axes.

### Claim 3 (depends on 2)

The inertial measurement unit of claim 2, wherein the estimated output error (ē) is modelled as a 6×1 vector of the form:
e(1)=sat(ωx+Bx)−sat({circumflex over (ω)}x+{circumflex over (B)}x);
e(2)=sat(ωy+By)−sat({circumflex over (ω)}y+{circumflex over (B)}y);
e(3)=sat(ωz+Bz)−sat({circumflex over (ω)}z+{circumflex over (B)}z);
e(4)=ax−âx;
e(5)=ay−ây; and
e(6)=az−âz,
wherein ωx, ωy, and ωz are measured angular rates along an x-axis, y-axis, and z-axis, respectively, {circumflex over (ω)}x, {circumflex over (ω)}y, and {circumflex over (ω)}z are angular rate estimates along the x-axis, y-axis, and z-axis, respectively; ax, ay, and az are measured linear accelerations along the x-axis, y-axis, and z-axis, respectively; âx, ây, and âz are estimated linear accelerations along the x-axis, y-axis, and z-axis, respectively, Bx, By, and Bz are measured angular rates biases along the x-axis, y-axis, and z-axis, respectively, and {circumflex over (B)}x, {circumflex over (B)}y, and {circumflex over (B)}z are angular rate bias estimates along the x-axis, y-axis, and z-axis, respectively.

### Claim 4 (depends on 2)

The inertial measurement unit of claim 2, wherein the system of motion equations for the object comprises twenty-nine coupled differential equations of the form:
⁢
d
⁢
ω
^
x
dt
=
ζ
^
x
J
x
-
(
J
z
-
J
y
)
J
x
⁢
ω
^
y
⁢
ω
^
z
+
L
1
⁢
e
_
;
(
Equation
⁢
⁢
1
)
⁢
d
⁢
ω
^
y
dt
=
ζ
^
y
J
y
-
(
J
x
-
J
z
)
J
y
⁢
ω
^
x
⁢
ω
^
z
+
L
2
⁢
e
_
;
(
Equation
⁢
⁢
2
)
⁢
d
⁢
ω
^
z
dt
=
ζ
^
z
J
z
-
(
J
y
-
J
x
)
J
z
⁢
ω
^
x
⁢
ω
^
y
+
L
3
⁢
e
_
;
(
Equation
⁢
⁢
3
)
⁢
d
⁢
ζ
^
x
1
dt
=
ζ
^
x
2
+
L
4
⁢
e
_
;
(
Equation
⁢
⁢
4
)
⁢
d
⁢
ζ
^
x
2
dt
=
m
^
ζ
x
⁢
ζ
^
x
1
+
L
5
⁢
e
_
;
(
Equation
⁢
⁢
5
)
⁢
d
⁢
m
^
ζ
x
dt
=
L
6
⁢
ζ
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
6
)
⁢
d
⁢
ζ
^
y
1
dt
=
ζ
^
y
2
+
L
7
⁢
e
_
;
(
Equation
⁢
⁢
7
)
⁢
d
⁢
ζ
^
y
2
dt
=
m
^
ζ
y
⁢
ζ
^
y
1
+
L
8
⁢
e
_
;
(
Equation
⁢
⁢
8
)
⁢
d
⁢
m
^
ζ
y
dt
=
L
9
⁢
ζ
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
9
)
⁢
d
⁢
ζ
^
z
1
dt
=
ζ
^
z
2
+
L
10
⁢
e
_
;
(
Equation
⁢
⁢
10
)
⁢
d
⁢
ζ
^
z
2
dt
=
m
^
ζ
z
⁢
ζ
^
z
1
+
L
11
⁢
e
_
;
(
Equation
⁢
⁢
11
)
⁢
d
⁢
m
^
ζ
z
dt
⁢
L
12
⁢
ζ
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
12
)
⁢
d
⁢
B
^
x
dt
=
L
13
⁢
e
_
;
(
Equation
⁢
⁢
13
)
⁢
d
⁢
B
^
y
dt
=
L
14
⁢
e
_
;
(
Equation
⁢
⁢
14
)
⁢
d
⁢
B
^
z
dt
=
L
15
⁢
e
_
;
(
Equation
⁢
⁢
15
)
d
⁢
x
^
2
dt
=
ω
^
z
⁢
y
^
2
-
ω
^
y
⁢
z
^
2
-
sin
⁡
(
θ
^
)
⁢
g
+
f
^
x
1
(
mass
)
+
L
16
⁢
e
_
;
(
Equation
⁢
⁢
16
)
d
⁢
y
^
2
dt
=
-
ω
^
z
⁢
x
^
2
+
ω
^
x
⁢
z
^
2
+
sin
⁡
(
ϕ
^
)
⁢
cos
⁡
(
θ
^
)
⁢
g
+
f
^
y
1
(
mass
)
+
L
17
⁢
e
_
;
(
Equation
⁢
⁢
17
)
d
⁢
z
^
2
dt
=
ω
^
y
⁢
x
^
2
-
ω
^
x
⁢
y
^
2
+
cos
⁡
(
θ
^
)
⁢
cos
⁡
(
ϕ
^
)
⁢
g
+
(
f
^
z
1
(
mass
)
-
g
)
+
L
18
⁢
e
_
;
(
Equation
⁢
⁢
18
)
⁢
d
⁢
f
^
x
1
dt
=
f
^
x
2
+
L
19
⁢
e
_
;
(
Equation
⁢
⁢
19
)
⁢
d
⁢
f
^
x
2
dt
=
-
m
^
f
y
⁢
f
^
x
1
+
L
20
⁢
e
_
;
(
Equation
⁢
⁢
20
)
⁢
d
⁢
m
^
f
x
dt
=
L
21
⁢
f
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
21
)
⁢
d
⁢
f
^
y
1
dt
=
f
^
y
2
+
L
22
⁢
e
_
;
(
Equation
⁢
⁢
22
)
⁢
d
⁢
f
^
y
2
dt
=
-
m
^
f
y
⁢
f
^
y
1
+
L
23
⁢
e
_
;
(
Equation
⁢
⁢
23
)
⁢
d
⁢
m
^
f
y
dt
=
L
24
⁢
f
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
24
)
⁢
d
⁢
f
^
z
1
dt
=
f
^
z
2
+
L
25
⁢
e
_
;
(
Equation
⁢
⁢
25
)
⁢
d
⁢
f
^
z
2
dt
=
-
m
^
f
z
⁢
f
^
z
1
+
L
26
⁢
e
_
;
(
Equation
⁢
⁢
26
)
⁢
d
⁢
m
^
f
z
dt
=
L
27
⁢
f
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
27
)
d
⁢
ϕ
^
dt
=
ω
^
x
+
ω
^
y
⁢
sin
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
ω
^
z
⁢
cos
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
L
28
⁢
e
_
;
⁢
⁢
⁢
and
(
Equation
⁢
⁢
28
)
⁢
d
⁢
θ
^
dt
=
ω
^
y
⁢
cos
⁡
(
ϕ
^
)
+
ω
^
z
⁢
sin
⁡
(
ϕ
^
)
+
L
29
⁢
e
_
,
(
Equation
⁢
⁢
29
)
wherein subscripts x, y, and z correspond to the three mutually-perpendicular axes, subscripts x1, y1, and z1 correspond to displacements along the x, y, and z axes, subscripts x2, y2, and z1 correspond to derivatives with respect to the x, y, and z axes, ω is an angular rate, {circumflex over (ω)} is an angular rate estimate, J is a moment of inertia, {circumflex over (ζ)} is a torque estimate, {circumflex over (m)}ζ is a torque frequency estimate ({circumflex over (m)}ζx is a torque frequency estimate about the x-axis, {circumflex over (m)}ζy is a torque frequency estimate about the y-axis, {circumflex over (m)}ζz is a torque frequency estimate about the z-axis), B is an angular rate bias, {circumflex over (B)} is an angular rate bias estimate, {circumflex over (x)}2 is a velocity estimate along x-axis, ŷ2 is a velocity estimate along y-axis, {circumflex over (z)}2 is a velocity estimate along z-axis, {circumflex over (f)} is a force estimate, {circumflex over (m)}f is a force frequency estimate, {tilde over (ϕ)} is a pitch estimate, {circumflex over (θ)} is a roll estimate, g is acceleration due to gravity, and L1-L29 are twenty-nine unknown design gains.

### Claim 5 (depends on 4)

The inertial measurement unit of claim 4, wherein the twenty-nine equations utilize a full-order unknown input observer for the object in motion.

### Claim 6 (depends on 1)

The inertial measurement unit of claim 1, wherein the system of motion equations utilize a reduced-order unknown input observer.

### Claim 7 (depends on 1)

The inertial measurement unit of claim 1, wherein the object in motion is a golf club.

### Claim 8 (depends on 7)

The inertial measurement unit of claim 7, wherein the inertial measurement unit is positioned within a club head of the golf club.

### Claim 9 (depends on 1)

The inertial measurement unit of claim 1, wherein the input force and the input torque are modelled as sinusoidal functions with unknown frequencies.

### Claim 10 (depends on 4)

The inertial measurement unit of claim 4, wherein the twenty-nine equations are solved simultaneously using a solver selected from the group consisting of Runge-Kutta, Adams, Rosenbrock, Trapezoidal Rule, Dorman-Prince, and Euler.

### Claim 11

A golf club head comprising:
an accelerometer sensor;
a gyroscope sensor; and
an application-specific integrated circuit comprising discrete hardware components configured to:
receive sensor data from the accelerometer sensor and the gyroscope sensor responsive to a motion of the golf club head between a first time instant and a second time instant;
extract an acceleration measurement and an angular rate measurement from the sensor data;
identify, from the angular rate measurement, a gyroscope saturation event corresponding to an output of the gyroscope sensor reaching a physical limitation of hardware of the gyroscope;
calculate an estimated output error using the received sensor data, wherein the estimated output error includes a saturation function that takes account of the gyroscope saturation event;
input the estimated output error into a system of motion equations for the golf club, wherein the system of motion equations includes an estimate of a gyroscope bias estimated without using static leveling;
solve the system of motion equations; and
output functions describing a position of the golf club head between the first time instant and the second time instant.

### Claim 12 (depends on 11)

The golf club head of claim 11, wherein the application-specific integrated circuit is further configured to:
calculate and output a position of the golf club head between the first time instant and the second time instant using functions describing an input force, an input torque, a roll angle and a pitch angle.

### Claim 13 (depends on 11)

The golf club head of claim 11, wherein the accelerometer sensor measures linear acceleration along three mutually-perpendicular axes and the gyroscope sensor measures angular velocities about the three mutually-perpendicular axes.

### Claim 14 (depends on 11)

The golf club head of claim 11, wherein the estimated output error (ē) is modelled as a 6×1 vector of the form:
e(1)=sat(ωx+Bx)−sat({circumflex over (ω)}x+{circumflex over (B)}x);
e(2)=sat(ωy+By)−sat({circumflex over (ω)}y+{circumflex over (B)}y);
e(3)=sat(ωz+Bz)−sat({circumflex over (ω)}z+{circumflex over (B)}z);
e(4)=ax−âx;
e(5)=ay−ây; and
e(6)=az−âz,
wherein {circumflex over (ω)}x, {circumflex over (ω)}y, and {circumflex over (ω)}z are measured angular rates along an x-axis, y-axis, and z-axis, respectively, {circumflex over (ω)}x, {circumflex over (ω)}y, and {circumflex over (ω)}z are angular rate estimates along the x-axis, y-axis, and z-axis, respectively; ax, ay, and az are measured linear accelerations along the x-axis, y-axis, and z-axis, respectively; âx, ây, and âz are estimated linear accelerations along the x-axis, y-axis, and z-axis, respectively, Bx, By, and Bz are measured angular rates biases along the x-axis, y-axis, and z-axis, respectively, and {circumflex over (B)}x, {circumflex over (B)}y, and {circumflex over (B)}z are angular rate bias estimates along the x-axis, y-axis, and z-axis, respectively.

### Claim 15 (depends on 12)

The golf club head of claim 12, wherein the system of motion equations comprises twenty-nine coupled differential equations of the form:
⁢
d
⁢
ω
^
x
dt
=
ζ
^
x
J
x
-
(
J
z
-
J
y
)
J
x
⁢
ω
^
y
⁢
ω
^
z
+
L
1
⁢
e
_
;
(
Equation
⁢
⁢
1
)
⁢
d
⁢
ω
^
y
dt
=
ζ
^
y
J
y
-
(
J
x
-
J
z
)
J
y
⁢
ω
^
x
⁢
ω
^
z
+
L
2
⁢
e
_
;
(
Equation
⁢
⁢
2
)
⁢
d
⁢
ω
^
z
dt
=
ζ
^
z
J
z
-
(
J
y
-
J
x
)
J
z
⁢
ω
^
x
⁢
ω
^
y
+
L
3
⁢
e
_
;
(
Equation
⁢
⁢
3
)
⁢
d
⁢
ζ
^
x
1
dt
=
ζ
^
x
2
+
L
4
⁢
e
_
;
(
Equation
⁢
⁢
4
)
⁢
d
⁢
ζ
^
x
2
dt
=
m
^
ζ
x
⁢
ζ
^
x
1
+
L
5
⁢
e
_
;
(
Equation
⁢
⁢
5
)
⁢
d
⁢
m
^
ζ
x
dt
=
L
6
⁢
ζ
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
6
)
⁢
d
⁢
ζ
^
y
1
dt
=
ζ
^
y
2
+
L
7
⁢
e
_
;
(
Equation
⁢
⁢
7
)
⁢
d
⁢
ζ
^
y
2
dt
=
m
^
ζ
y
⁢
ζ
^
y
1
+
L
8
⁢
e
_
;
(
Equation
⁢
⁢
8
)
⁢
d
⁢
m
^
ζ
y
dt
=
L
9
⁢
ζ
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
9
)
⁢
d
⁢
ζ
^
z
1
dt
=
ζ
^
z
2
+
L
10
⁢
e
_
;
(
Equation
⁢
⁢
10
)
⁢
d
⁢
ζ
^
z
2
dt
=
m
^
ζ
z
⁢
ζ
^
z
1
+
L
11
⁢
e
_
;
(
Equation
⁢
⁢
11
)
⁢
d
⁢
m
^
ζ
z
dt
⁢
L
12
⁢
ζ
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
12
)
⁢
d
⁢
B
^
x
dt
=
L
13
⁢
e
_
;
(
Equation
⁢
⁢
13
)
⁢
d
⁢
B
^
y
dt
=
L
14
⁢
e
_
;
(
Equation
⁢
⁢
14
)
⁢
d
⁢
B
^
z
dt
=
L
15
⁢
e
_
;
(
Equation
⁢
⁢
15
)
d
⁢
x
^
2
dt
=
ω
^
z
⁢
y
^
2
-
ω
^
y
⁢
z
^
2
-
sin
⁡
(
θ
^
)
⁢
g
+
f
^
x
1
(
mass
)
+
L
16
⁢
e
_
;
(
Equation
⁢
⁢
16
)
d
⁢
y
^
2
dt
=
-
ω
^
z
⁢
x
^
2
+
ω
^
x
⁢
z
^
2
+
sin
⁡
(
ϕ
^
)
⁢
cos
⁡
(
θ
^
)
⁢
g
+
f
^
y
1
(
mass
)
+
L
17
⁢
e
_
;
(
Equation
⁢
⁢
17
)
d
⁢
z
^
2
dt
=
ω
^
y
⁢
x
^
2
-
ω
^
x
⁢
y
^
2
+
cos
⁡
(
θ
^
)
⁢
cos
⁡
(
ϕ
^
)
⁢
g
+
(
f
^
z
1
(
mass
)
-
g
)
+
L
18
⁢
e
_
;
(
Equation
⁢
⁢
18
)
⁢
d
⁢
f
^
x
1
dt
=
f
^
x
2
+
L
19
⁢
e
_
;
(
Equation
⁢
⁢
19
)
⁢
d
⁢
f
^
x
2
dt
=
-
m
^
f
y
⁢
f
^
x
1
+
L
20
⁢
e
_
;
(
Equation
⁢
⁢
20
)
⁢
d
⁢
m
^
f
x
dt
=
L
21
⁢
f
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
21
)
⁢
d
⁢
f
^
y
1
dt
=
f
^
y
2
+
L
22
⁢
e
_
;
(
Equation
⁢
⁢
22
)
⁢
d
⁢
f
^
y
2
dt
=
-
m
^
f
y
⁢
f
^
y
1
+
L
23
⁢
e
_
;
(
Equation
⁢
⁢
23
)
⁢
d
⁢
m
^
f
y
dt
=
L
24
⁢
f
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
24
)
⁢
d
⁢
f
^
z
1
dt
=
f
^
z
2
+
L
25
⁢
e
_
;
(
Equation
⁢
⁢
25
)
⁢
d
⁢
f
^
z
2
dt
=
-
m
^
f
z
⁢
f
^
z
1
+
L
26
⁢
e
_
;
(
Equation
⁢
⁢
26
)
⁢
d
⁢
m
^
f
z
dt
=
L
27
⁢
f
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
27
)
d
⁢
ϕ
^
dt
=
ω
^
x
+
ω
^
y
⁢
sin
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
ω
^
z
⁢
cos
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
L
28
⁢
e
_
;
⁢
⁢
⁢
and
(
Equation
⁢
⁢
28
)
⁢
d
⁢
θ
^
dt
=
ω
^
y
⁢
cos
⁡
(
ϕ
^
)
+
ω
^
z
⁢
sin
⁡
(
ϕ
^
)
+
L
29
⁢
e
_
.
(
Equation
⁢
⁢
29
)
wherein subscripts x, y, and z correspond to the three mutually-perpendicular axes, subscripts x1, y1, and z1 correspond to displacements along the x, y, and z axes, subscripts x2, y2, and z2 correspond to derivatives with respect to the x, y, and z axes, ω is an angular rate, {circumflex over (ω)} is an angular rate estimate, J is a moment of inertia, {circumflex over (ζ)} is a torque estimate, {circumflex over (m)}ζ is a torque frequency estimate ({circumflex over (m)}ζx is a torque frequency estimate about the x-axis, {circumflex over (m)}ζy is a torque frequency estimate about the y-axis, {circumflex over (m)}ζz is a torque frequency estimate about the z-axis), B is an angular rate bias, {circumflex over (B)} is an angular rate bias estimate, {circumflex over (x)}2 is a velocity estimate along x-axis, ŷ2 is a velocity estimate along y-axis, {circumflex over (z)}2 is a velocity, estimate along z-axis, {circumflex over (f)} is a force estimate, {circumflex over (m)}f is a force frequency estimate, {circumflex over (ϕ)} is a pitch estimate, {circumflex over (θ)} is a roll estimate, g is acceleration due to gravity, L1-L29 and are twenty-nine unknown design gains.

### Claim 16 (depends on 15)

The golf club head of claim 15, wherein the twenty-nine equations utilize a full-order unknown input observer.

### Claim 17 (depends on 11)

The golf club head of claim 11, wherein the system of motion equations utilizes a reduced-order unknown input observer.

### Claim 18 (depends on 11)

The golf club head of claim 11, further comprising a transceiver, wherein the computer-readable instructions, when executed, further cause the processor to:
transmit the output functions, using the transceiver, to a remote computing device.

### Claim 19 (depends on 11)

The golf club head of claim 11, wherein the accelerometer, the gyroscope, and the an application-specific integrated circuit are integrated into an inertial measurement unit (IMU).

### Claim 20 (depends on 19)

The golf club head of claim 19, wherein the IMU is configured to be removably positioned within the golf club head.

### Claim 21 (depends on 20)

The golf club head of claim 20, wherein the golf club head has a moment of inertia having a standard deviation of no more than 200 g·cm2 regardless of whether the IMU is positioned within the golf club head.

### Claim 22

A method comprising:
receiving, by an application-specific integrated circuit comprising discrete hardware components, sensor data from an inertial measurement unit (IMU) coupled to an object in motion between a first time instant and a second time instant;
identifying, by the application-specific integrated circuit, from the received sensor data, a gyroscope saturation event corresponding to an output of a gyroscope reaching a physical limitation of hardware of the gyroscope;
calculating, by the application-specific integrated circuit, an estimated output error using the received sensor data, wherein the estimated output error includes a saturation function that takes account of the gyroscope saturation event;
inputting, by the application-specific integrated circuit, the estimated output error into a system of motion equations for the object;
solving, by the application-specific integrated circuit, the system of motion equations; and
outputting, by the application-specific integrated circuit, functions describing a position of the object in motion between the first time instant and the second time instant.

### Claim 23 (depends on 2)

The method of claim 2, wherein the estimated output error (ē) is modelled as a 6×1 vector of the form:
e(1)=sat(ωx+Bx)−sat({circumflex over (ω)}x+{circumflex over (B)}x);
e(2)=sat(ωy+By)−sat({circumflex over (ω)}y+{circumflex over (B)}y);
e(3)=sat(ωz+Bz)−sat({circumflex over (ω)}z+{circumflex over (B)}z);
e(4)=ax−âx;
e(5)=ay−ây; and
e(6)=az−âz,
wherein ωx, ωy, and ωz are measured angular rates along an x-axis, y-axis, and z-axis, respectively, {circumflex over (ω)}x, {circumflex over (ω)}y, and {circumflex over (ω)}z are angular rate estimates along the x-axis, y-axis, and z-axis, respectively; ax, ay, and az are measured linear accelerations along the x-axis, y-axis, and z-axis, respectively; âx, ây, and âz are estimated linear accelerations along the x-axis, y-axis, and z-axis, respectively, Bx, By, and Bz are measured angular rates biases along the x-axis, y-axis, and z-axis, respectively, and {circumflex over (B)}x, {circumflex over (B)}y, and {circumflex over (B)}z are angular rate bias estimates along the x-axis, y-axis, and z-axis, respectively.

### Claim 24 (depends on 22)

The method of claim 22, wherein the system of motion equations for the object comprises twenty-nine coupled differential equations of the form:
⁢
d
⁢
ω
^
x
dt
=
ζ
^
x
J
x
-
(
J
z
-
J
y
)
J
x
⁢
ω
^
y
⁢
ω
^
z
+
L
1
⁢
e
_
;
(
Equation
⁢
⁢
1
)
⁢
d
⁢
ω
^
y
dt
=
ζ
^
y
J
y
-
(
J
x
-
J
z
)
J
y
⁢
ω
^
x
⁢
ω
^
z
+
L
2
⁢
e
_
;
(
Equation
⁢
⁢
2
)
⁢
d
⁢
ω
^
z
dt
=
ζ
^
z
J
z
-
(
J
y
-
J
x
)
J
z
⁢
ω
^
x
⁢
ω
^
y
+
L
3
⁢
e
_
;
(
Equation
⁢
⁢
3
)
⁢
d
⁢
ζ
^
x
1
dt
=
ζ
^
x
2
+
L
4
⁢
e
_
;
(
Equation
⁢
⁢
4
)
⁢
d
⁢
ζ
^
x
2
dt
=
m
^
ζ
x
⁢
ζ
^
x
1
+
L
5
⁢
e
_
;
(
Equation
⁢
⁢
5
)
⁢
d
⁢
m
^
ζ
x
dt
=
L
6
⁢
ζ
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
6
)
⁢
d
⁢
ζ
^
y
1
dt
=
ζ
^
y
2
+
L
7
⁢
e
_
;
(
Equation
⁢
⁢
7
)
⁢
d
⁢
ζ
^
y
2
dt
=
m
^
ζ
y
⁢
ζ
^
y
1
+
L
8
⁢
e
_
;
(
Equation
⁢
⁢
8
)
⁢
d
⁢
m
^
ζ
y
dt
=
L
9
⁢
ζ
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
9
)
⁢
d
⁢
ζ
^
z
1
dt
=
ζ
^
z
2
+
L
10
⁢
e
_
;
(
Equation
⁢
⁢
10
)
⁢
d
⁢
ζ
^
z
2
dt
=
m
^
ζ
z
⁢
ζ
^
z
1
+
L
11
⁢
e
_
;
(
Equation
⁢
⁢
11
)
⁢
d
⁢
m
^
ζ
z
dt
⁢
L
12
⁢
ζ
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
12
)
⁢
d
⁢
B
^
x
dt
=
L
13
⁢
e
_
;
(
Equation
⁢
⁢
13
)
⁢
d
⁢
B
^
y
dt
=
L
14
⁢
e
_
;
(
Equation
⁢
⁢
14
)
⁢
d
⁢
B
^
z
dt
=
L
15
⁢
e
_
;
(
Equation
⁢
⁢
15
)
d
⁢
x
^
2
dt
=
ω
^
z
⁢
y
^
2
-
ω
^
y
⁢
z
^
2
-
sin
⁡
(
θ
^
)
⁢
g
+
f
^
x
1
(
mass
)
+
L
16
⁢
e
_
;
(
Equation
⁢
⁢
16
)
d
⁢
y
^
2
dt
=
-
ω
^
z
⁢
x
^
2
+
ω
^
x
⁢
z
^
2
+
sin
⁡
(
ϕ
^
)
⁢
cos
⁡
(
θ
^
)
⁢
g
+
f
^
y
1
(
mass
)
+
L
17
⁢
e
_
;
(
Equation
⁢
⁢
17
)
d
⁢
z
^
2
dt
=
ω
^
y
⁢
x
^
2
-
ω
^
x
⁢
y
^
2
+
cos
⁡
(
θ
^
)
⁢
cos
⁡
(
ϕ
^
)
⁢
g
+
(
f
^
z
1
(
mass
)
-
g
)
+
L
18
⁢
e
_
;
(
Equation
⁢
⁢
18
)
⁢
d
⁢
f
^
x
1
dt
=
f
^
x
2
+
L
19
⁢
e
_
;
(
Equation
⁢
⁢
19
)
⁢
d
⁢
f
^
x
2
dt
=
-
m
^
f
y
⁢
f
^
x
1
+
L
20
⁢
e
_
;
(
Equation
⁢
⁢
20
)
⁢
d
⁢
m
^
f
x
dt
=
L
21
⁢
f
^
x
1
⁢
e
_
;
(
Equation
⁢
⁢
21
)
⁢
d
⁢
f
^
y
1
dt
=
f
^
y
2
+
L
22
⁢
e
_
;
(
Equation
⁢
⁢
22
)
⁢
d
⁢
f
^
y
2
dt
=
-
m
^
f
y
⁢
f
^
y
1
+
L
23
⁢
e
_
;
(
Equation
⁢
⁢
23
)
⁢
d
⁢
m
^
f
y
dt
=
L
24
⁢
f
^
y
1
⁢
e
_
;
(
Equation
⁢
⁢
24
)
⁢
d
⁢
f
^
z
1
dt
=
f
^
z
2
+
L
25
⁢
e
_
;
(
Equation
⁢
⁢
25
)
⁢
d
⁢
f
^
z
2
dt
=
-
m
^
f
z
⁢
f
^
z
1
+
L
26
⁢
e
_
;
(
Equation
⁢
⁢
26
)
⁢
d
⁢
m
^
f
z
dt
=
L
27
⁢
f
^
z
1
⁢
e
_
;
(
Equation
⁢
⁢
27
)
d
⁢
ϕ
^
dt
=
ω
^
x
+
ω
^
y
⁢
sin
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
ω
^
z
⁢
cos
⁡
(
ϕ
^
)
⁢
tan
⁡
(
θ
^
)
+
L
28
⁢
e
_
;
⁢
⁢
⁢
and
(
Equation
⁢
⁢
28
)
⁢
d
⁢
θ
^
dt
=
ω
^
y
⁢
cos
⁡
(
ϕ
^
)
+
ω
^
z
⁢
sin
⁡
(
ϕ
^
)
+
L
29
⁢
e
_
,
(
Equation
⁢
⁢
29
)
wherein subscripts x, y, and z correspond to the three mutually-perpendicular axes, subscripts x1, y1, and z1 correspond to displacements along the x, y, and z axes, subscripts x2, y2, and z2 correspond to derivatives with respect to the x, y, and z axes, ω is an angular rate, {circumflex over (ω)} is an angular rate estimate, J is a moment of inertia, {circumflex over (ζ)} is a torque estimate, {circumflex over (m)}ζ is a torque frequency estimate ({circumflex over (m)}ζx is a torque frequency estimate about the x-axis, {circumflex over (m)}ζy is a torque frequency estimate about the y-axis, {circumflex over (m)}ζz is a torque frequency estimate about the z-axis), B is an angular rate bias, {circumflex over (B)} is an angular rate bias estimate, {circumflex over (x)}2 is a velocity estimate along x-axis, ŷ2 is a velocity estimate along y-axis, {circumflex over (z)}2 is a velocity, estimate along z-axis, {circumflex over (f)} is a force estimate, {circumflex over (m)}f is a force frequency estimate, {circumflex over (ϕ)} is a pitch estimate, {circumflex over (θ)} is a roll estimate, g is acceleration due to gravity, L1-L29 and are twenty-nine unknown design gains.