Determining gravity toolface and inclination in a rotating downhole tool

1. A system for controlling a drilling operation by determining gravity toolface and inclination, comprising:

a downhole tool comprising an internal bore through which a drilling fluid passes during a drilling operation in a borehole;

a steering assembly;

a sensor assembly disposed in a radially offset location within the downhole tool, wherein the radially offset location is relative to a longitudinal axis of the downhole tool, wherein the sensor assembly consists of a three-axis accelerometer package and an angular rate sensing device, wherein the three-axis accelerometer package comprises three accelerometers, and wherein the angular rate sensing device senses an angular velocity of the downhole tool; and

a processor in communication with the sensor assembly and the steering assembly, wherein the processor is coupled to at least one memory device containing a set of instruction that, when executed by the processor, cause the processor to

receive an output from the sensor assembly;

determine a centripetal acceleration of the downhole tool based, at least in part, on the output;

determine a tangential acceleration of the downhole tool based, at least in part, on a first output at a first time from the angular rate sensing device and a second output at a second time from the angular rate sensing device, wherein a digital filter is implemented to determine at least one of the centripetal acceleration and the tangential acceleration;

determine at least one of the gravity toolface and the inclination of the downhole tool using at least one of the centripetal acceleration and a tangential acceleration; and

alter at least one of a direction and a rotation of the steering assembly based, at least in part, on the at least one of the gravity toolface and inclination.

2. The system of claim 1 , wherein the three accelerometers comprise:

a first accelerometer oriented to sense a first component in a first direction within a plane;

a second accelerometer oriented to sense a second component in a second direction within the plane, wherein the second direction is perpendicular to the first direction; and

a third accelerometer oriented to sense a third component in a third direction perpendicular to the plane.

3. The system of claim 2 , wherein the angular rate sensing device comprises a gyroscope.

4. The system of claim 2 , wherein the gravity toolface Θ is determined using at least one of the following equations:

x =( g *sin Θ)+ a;

y =(− g *cos Θ)− r;

with x corresponding to the sensed first component from the first accelerometer, y corresponding to the sensed second component from the second accelerometer; g corresponding to the force of gravity, a corresponding to the tangential acceleration, and r corresponding to the centripetal acceleration.

5. The system of claim 2 , wherein the output comprises:

the sensed first component from the first accelerometer;

the sensed second component from the second accelerometer;

the sensed third component from the third accelerometer; and

an angular speed from the angular rate sensing device.

6. The system of claim 1 , wherein:

the centripetal acceleration is determined using the following equation:

r=ω 2 *radius,

where r corresponds to the centripetal acceleration, ω corresponds to an angular speed output of the angular rate sensing device, and radius corresponds to a radial distance of the angular rate sensing device from a longitudinal axis of the downhole tool; and

the tangential acceleration is determined using the following equation:

a =((ω 2 −ω 1 )/( t 2 −t 1 ))*radius

where a corresponds to the tangential acceleration, ω 2 corresponds to an angular speed output of the angular rate sensing device at time t 2 , ω 1 corresponds to an angular speed output of the angular rate sensing device at time t 1 , and radius corresponds to a radius of the downhole tool.

7. The system of claim 1 , wherein the sensor assembly is implemented on a single printed circuit board (PCB).

8. The system of claim 7 , wherein the angular rate sensing device comprises a gyroscope implemented in a single integrated circuit chip coupled to the PCB.

9. A method for controlling a drilling operation by determining gravity toolface and inclination, comprising:

positioning a downhole tool comprising an internal bore through which a drilling fluid passes during a drilling operation within a borehole, wherein:

the downhole tool comprises a sensor assembly disposed, wherein the sensor assembly is in a radially offset location within the downhole tool, wherein the radially offset location is relative to a longitudinal axis of the downhole tool;

the downhole tool comprises a steering assembly; and

the sensor assembly consisting of a three-axis accelerometer package and an angular rate sensing device, wherein the three-axis accelerometer package comprises three accelerometers, wherein the angular rate sensing device senses an angular velocity of the downhole tool;

determining a centripetal acceleration of the downhole tool based, at least in part, on an output of the sensor assembly;

determining a tangential acceleration of the downhole tool based, at least in part, on a first output at a first time from the angular rate sensing device and a second output at a second time from the angular rate sensing device, wherein a digital filter is implemented to determine at least one of the centripetal acceleration and the tangential acceleration;

determining at least one of the gravity toolface and the inclination of the downhole tool using at least one of the centripetal acceleration and a tangential acceleration;

transmitting the at least one of the gravity toolface and the inclination to a steering control, wherein the steering control alters at least one of a direction and a rotation of the steering assembly based, at least in part, on the at least one of the gravity toolface and the inclination.

10. The method of claim 9 , wherein the three accelerometers comprise:

a first accelerometer oriented to sense a first component in a first direction within a plane; and

a second accelerometer oriented to sense a second component in a second direction within the plane, wherein the second direction is perpendicular to the first direction.

11. The method of claim 10 , further comprising a third accelerometer of the three accelerometers oriented to sense a third component in a third direction perpendicular to the plane.

12. The method of claim 10 , wherein:

the centripetal acceleration is determined using the following equation:

r=ω 2 *radius,

where r corresponds to the centripetal acceleration, ω corresponds to an angular speed output of the angular rate sensing device, and radius corresponds to a radial distance of the angular rate sensing device from a longitudinal axis of the downhole tool;

the tangential acceleration is determined using the following equation:

a =((ω 2 −ω 1 )/( t 2 −t 1 ))*radius

where a corresponds to the tangential acceleration, ω 2 corresponds to an angular speed output of the angular rate sensing device at time t 2 , ω 1 corresponds to an angular speed output of the angular rate sensing device at time t 1 , and radius corresponds to a radius of the downhole tool; and

the gravity toolface Θ is determined using at least one of the following equations:

x =( g *sin Θ)+ a;

y =(− g *cos Θ)− r;

with x corresponding to the sensed first component from the first accelerometer, y corresponding to the sensed second component from the second accelerometer; g corresponding to the force of gravity, a corresponding to the tangential acceleration, and r corresponding to the centripetal acceleration.

13. A system for controlling a drilling operation by determining gravity toolface and inclination, comprising:

a downhole tool comprising an internal bore through which a drilling fluid passes during a drilling operation in a borehole;

a steering assembly;

a first sensor assembly disposed in a first radially offset location within the downhole tool, wherein the first radially offset location is relative to a longitudinal axis of the downhole tool, wherein the first sensor assembly is implemented on a first printed circuit board (PCB), wherein the first sensor assembly consists of a first accelerometer and a second accelerometer;

a second sensor assembly disposed in a second radially offset location within the downhole tool, wherein the second radially offset location is relative to a longitudinal axis of the downhole tool, wherein the second sensor assembly is implemented on a second PCB, wherein the second sensor assembly consists of a third accelerometer and a fourth accelerometer; and

a processor in communication with the steering assembly, the first sensor assembly, and the second sensor assembly, wherein the processor is coupled to at least one memory device containing a set of instruction that, when executed by the processor, cause the processor to:

receive a first output from the first sensor assembly;

receive a second output from the second sensor assembly;

determine a centripetal acceleration of the downhole tool based, at least in part, on the first output and the second output;

determine a first tangential acceleration of the downhole tool based, at least in part, on the first output, wherein a digital filter is implemented to determine at least one of the centripetal acceleration and the first tangential acceleration;

determine at least one of the gravity toolface and the inclination of the downhole tool using at least one of the centripetal acceleration and a tangential acceleration; and

alter at least one of a direction and a rotation of the steering assembly based, at least in part, on the least one of the gravity toolface and inclination.

14. The system of claim 13 , wherein:

the first accelerometer is oriented to sense a first component in a first direction within a plane;

the second accelerometer is oriented to sense a second component in a second direction within the plane, wherein the second direction is perpendicular to the first direction;

the third accelerometer is oriented to sense a third component in a third direction within the plane, wherein the third direction is opposite the first direction;

the fourth accelerometer is oriented to sense a fourth component in a fourth direction within the plane, wherein the fourth direction is perpendicular to the third direction and opposite the second direction.

15. The system of claim 14 , wherein:

the centripetal acceleration is determined using the following equation:

r =−( y+y 2)/2

where r corresponds to the centripetal acceleration, y corresponds to a second sensed component from the second accelerometer, and y2 corresponds to the fourth sensed component from the fourth accelerometer; and

a second tangential acceleration is determined using the following equation:

a =( x+x 2)/2

where a corresponds to the tangential acceleration, x corresponds to a first sensed component from the first accelerometer, and x2 corresponds to the third sensed component from the third accelerometer.

16. The system of claim 15 , wherein the gravity toolface Θ is determined using at least one of the following equations:

x =( g *sin Θ)+ a;

x 2=(− g *sin Θ)+ a;

y =(− g *cos Θ)− r;

y 2=( g *cos Θ)− r

with x corresponding to a first sensed component from the first accelerometer, x2 corresponding to the third sensed component from the third accelerometer, y corresponding to a second sensed component from the second accelerometer, y2 corresponding to the fourth sensed component from the fourth accelerometer; g corresponding to the force of gravity, a corresponding to the tangential acceleration, and r corresponding to the centripetal acceleration.

17. The system of claim 14 wherein the output comprises:

the sensed first component from the first accelerometer;

the sensed second component from the second accelerometer;

the sensed third component from the third accelerometer; and

the sensed fourth component from the fourth accelerometer.

18. The system of claim 13 , wherein the first sensor assembly is implemented on a first printed circuit board (PCB).