Sensorless position detection for electric motor

1. An apparatus comprising:

an electric motor including a stator and a translator;

a three-phase inverter electrically coupled to the electric motor;

a power source electrically coupled to the three-phase inverter; and

a controller communicatively coupled to the three-phase inverter and programmed to

determine at least three measurements at different times of flux linkage from the electric motor;

represent the measurements in Clarke coordinates;

determine Clarke coordinates of a center of a circle defined by the Clarke coordinates of the measurements; and

determine a position of the translator relative to the stator based on the Clarke coordinates of the center of the circle.

2. The apparatus of claim 1 , wherein the electric motor is a linear electric motor.

3. The apparatus of claim 1 , wherein the power source is a supercapacitor.

4. The apparatus of claim 1 , wherein the electric motor lacks a position sensor.

5. The apparatus of claim 1 , further comprising a position sensor coupled to the electric motor to detect a position of the translator relative to the stator, wherein the controller is further programmed to receive data indicating the position of the translator relative to the stator from the position sensor, and provide a warning in response to a difference between the position of the translator from the position sensor and the position of the translator based on the center of the circle exceeding a threshold.

6. The apparatus of claim 1 , wherein the controller is further programmed to charge the power source with power generated by movement of the electric motor.

7. The apparatus of claim 1 , further comprising a vehicle frame and a wheel assembly movable relative to the vehicle frame, wherein the electric motor is coupled to the vehicle frame and to the wheel assembly so that as the wheel assembly moves relative to the vehicle frame, the translator moves relative to the stator.

8. A method for controlling an electric motor including a stator and a translator, the method comprising:

determining at least three flux-linkage measurements at different times from the electric motor;

representing the flux-linkage measurements in Clarke coordinates;

determining Clarke coordinates of a center of a circle defined by the Clarke coordinates of the flux-linkage measurements; and

determining a position of the translator relative to the stator based on the Clarke coordinates of the center of the circle.

9. The method of claim 8 , further comprising providing proportional compensation to the center of the circle.

10. The method of claim 9 , further comprising providing integral compensation to the center of the circle.

11. The method of claim 9 , wherein the proportional compensation is further based on a weighting factor, wherein the weighting factor is inversely related to a condition number of a matrix of the measurements of flux linkage.

12. The method of claim 11 , wherein the weighting factor is given by an equation of this form:

K

R

=

1

κ

⁡

(

A

T

⁢

A

)

x

wherein K R is the weighting factor, K(.) is the condition number of a matrix, x is a positive number, and the matrix A is given by the following equation:

A

=

[

ψ

α

,

1

-

ψ

α

,

2

ψ

β

,

1

-

ψ

β

,

2

⋮

⋮

ψ

α

,

N

-

1

-

ψ

α

,

N

ψ

β

,

N

-

1

-

ψ

β

,

N

]

wherein Ψα, i is the ith flux-linkage measurement in the alpha dimension in Clarke coordinates, Ψβ, i is the ith flux-linkage measurement in the beta dimension in Clarke coordinates, and N is the number of flux-linkage measurements.

13. The method of claim 9 , wherein the proportional compensation forms a feedback loop in which a measured back-emf is proportionally compensated and then, after integration, becomes one of the flux-linkage measurements used to determine the Clarke coordinates of the center of the circle.

14. The method of claim 8 , further comprising actuating the electric motor based on the position of the translator relative to the stator.

15. The method of claim 8 , wherein collecting at least three flux-linkage measurements includes determining at least four flux-linkage measurements, and determining the Clarke coordinates of the center of the circle includes applying least-squares estimation to the Clarke coordinates of the flux-linkage measurements.

16. The method of claim 15 , wherein applying the least-squares estimation includes applying this equation:

Ψ c =A † B

wherein Ψ c is the center of the circle in Clarke coordinates, A and B are matrices given by the following equations, and the dagger symbol represents the pseudoinverse of a matrix:

A

=

[

ψ

α

,

1

-

ψ

α

,

2

ψ

β

,

1

-

ψ

β

,

2

⋮

⋮

ψ

α

,

N

-

1

-

ψ

α

,

N

ψ

β

,

N

-

1

-

ψ

β

,

N

]

B

=

1

2

⁡

[

ψ

α

,

1

2

-

ψ

α

,

2

2

+

ψ

β

,

1

2

-

ψ

β

,

2

2

⋮

ψ

α

,

N

-

1

2

-

ψ

α

,

N

2

+

ψ

β

,

N

-

1

2

-

ψ

β

,

N

2

]

wherein Ψα, i is the ith flux-linkage measurement in the alpha dimension in Clarke coordinates, Ψβ, i is the ith flux-linkage measurement in the beta dimension in Clarke coordinates, and N is the number of flux-linkage measurements.

17. The method of claim 8 , further comprising charging a power source with power generated by movement of the electric motor.

18. A controller comprising a processor and a memory storing instructions executable by the processor to:

determine at least three flux-linkage measurements at different times from an electric motor, the electric motor including a translator and a stator;

represent the flux-linkage measurements in Clarke coordinates;

determine Clarke coordinates of a center of a circle defined by the Clarke coordinates of the flux-linkage measurements; and

determine a position of the translator relative to the stator based on the Clarke coordinates of the center of the circle.