Cam Phase Actuator Control Systems and Methods

Abstract

The systems and methods described herein provide an approach for cam phase angle control where an axial or rotational position of an actuator of a cam phaser has a direct relationship to the phase angle of the cam shaft, allowing for accurate cam phasing without the need for cam shaft or crank shaft position sensors. Providing phase angle adjustability without the need for crank shaft or cam shaft position sensors enables control of phase angle solely by sensing the axial or rotational position of the actuator of the cam phaser.

Description

BACKGROUND

In general, cam phasing systems include a rotary actuator, or phaser, that is configured to adjust a rotational position of a cam shaft relative to a crank shaft of an internal combustion engine.

BRIEF SUMMARY OF THE INVENTION

According to one aspect, the present disclosure provides a cam phasing control system for varying a rotational relationship between a crank shaft and a cam shaft. The cam phasing system comprises a cam phaser including a first component configured to be coupled to a cam shaft and a second component configured to be coupled to a crank shaft, an actuator configured to adjust a rotational position of the first component relative to the second component, an actuator position sensor configured to detect an actuation position of the actuator, and a controller including a processor and a memory. The processor is configured to receive a phase angle command, determine a required actuation position of the actuator based on the phase angle command and a predetermined relationship between an actuation position of the actuator and cam phase angle. The processor is further configured to command the actuator to displace from a first fixed position to a second fixed position, where a magnitude of the displacement between the first fixed position and the second fixed position corresponds with a proportional rotational displacement between the first component and the second component.

According to some aspects, a determination of a required actuation position is done without a cam shaft position sensor and a crank shaft position sensor.

According to another aspect, the present disclosure provides a method of open-loop control of a cam phasing system for varying a rotational relationship between a crank shaft and a cam shaft. The method comprises receiving a phase angle command, determining a required actuation position of a cam phaser actuator based on the phase angle command and a predetermined relationship between actuation positions of the cam phaser actuator and cam phase angle, and commanding the actuator to the required actuation position.

According to another aspect, the present disclosure provides a method of calibrating a cam phasing control system. The method comprises commanding a cam phaser actuator to an end position, detecting the cam shaft position and the crank shaft position, determining the phase angle of the cam shaft relative to the crank shaft based on the cam shaft position and the crank shaft position, and defining a proportional relationship between actuation positions of the cam phaser actuator to phase angles of the cam shaft based on the determined phase angle and a predetermined relationship between actuation positions of the cam phaser actuator and resulting phase angles.

According to another aspect, the present disclosure provides a method of controlling a cam phasing system for varying a rotational relationship between a crank shaft and a cam shaft. The method comprises detecting an error between a commanded actuator position and a sensed actuator position of a cam phaser actuator and determining if the error is within a predetermined range. When the error is outside of the predetermined range, the cam phasing system is operated in an open loop mode. When the error is within the predetermined range, determining if a phase angle reading sensed by a cam shaft position sensor is accurate. When the phase angle reading is determined to be accurate, the cam phasing system is operated in a closed loop mode.

According to another aspect, the present disclosure provides a cam phasing control system operable in an open loop mode and a closed loop mode. The cam phasing control system comprises a cam phaser including a first component configured to be coupled to a cam shaft and a second component configured to be coupled to a crank shaft, an actuator configured to adjust a rotational position of the first component relative to the second component, and a controller in communication with an actuator position sensor configured to detect an actuation position of the actuator, a crank shaft position sensor configured to detect a crank shaft position, and a cam shaft position sensor configured to detect cam shaft position. When the controller is in an open loop mode, the controller is configured to receive a phase angle command, determine a required actuation position of the actuator based on the phase angle command and a predetermined relationship between an actuation position of the actuator and cam phase angle, and command the actuator to displace to the required actuation position. When the controller is in a closed loop mode, the controller is configured to receive the phase angle command, determine an estimated actuation position of the actuator based on the phase angle command and the predetermined relationship between the actuation position of the actuator and the cam phase angle, determine an error between the commanded phase angle and an actual cam phase angle detected by the cam shaft position sensor and the crank shaft position sensor, and command the actuator to displace to an actuator position based on the error and the estimated actuation position.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic illustration of a cam phasing control system according to one aspect of the present disclosure;

FIG. 2 is a diagram of an open loop control method according to one aspect of the present disclosure;

FIG. 3 is a graphical illustration of a proportional relationship between an actuator position and a cam phase angle according to one aspect of the present disclosure;

FIG. 4 is a diagram of a method of calibrating the look up table of FIG. 3;

FIG. 5 is a diagram of a modified closed loop control method according to one aspect of the present disclosure;

FIG. 6 is a diagram of an algorithm for determining an operation mode of a cam phasing control system according to one aspect of the present disclosure;

FIG. 7 is a diagram of an algorithm for determining the accuracy of a cam shaft or crank shaft position sensor reading according to one aspect of the present disclosure;

FIG. 8 is a plot of measured actuator angle, measured phase angle, and engine speed over time;

FIG. 9 illustrates one non-limiting example of a cam phasing system utilizing an axial displacement actuator; and

FIG. 10 illustrates one non-limiting example of a cam phasing system utilizing a rotational displacement actuator.

DETAILED DESCRIPTION OF THE INVENTION

Before any embodiments of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of components set forth in the following description or illustrated in the following drawings. The invention is capable of other embodiments and of being practiced or of being carried out in various ways. Also, it is to be understood that the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. Unless specified or limited otherwise, the terms “mounted,” “connected,” “supported,” and “coupled” and variations thereof are used broadly and encompass both direct and indirect mountings, connections, supports, and couplings. Further, “connected” and “coupled” are not restricted to physical or mechanical connections or couplings.

The following discussion is presented to enable a person skilled in the art to make and use embodiments of the invention. Various modifications to the illustrated embodiments will be readily apparent to those skilled in the art, and the generic principles herein can be applied to other embodiments and applications without departing from embodiments of the invention. Thus, embodiments of the invention are not intended to be limited to embodiments shown, but are to be accorded the widest scope consistent with the principles and features disclosed herein. The following detailed description is to be read with reference to the figures, in which like elements in different figures have like reference numerals. The figures, which are not necessarily to scale, depict selected embodiments and are not intended to limit the scope of embodiments of the invention. Skilled artisans will recognize the examples provided herein have many useful alternatives and fall within the scope of embodiments of the invention.

Currently, cam phasers can be hydraulically actuated, electronically actuated, or mechanically actuated. Typically, mechanically actuated phasers harvest cam torque pulses to enable the rotation of the phaser. In most cases, the system can only control whether or not the system is allowed to rotate in the desired direction, with little control over the speed or final position. The speed of the rotation of the phaser and a stop position of the phaser after the cam torque pulse has ended are functions of a magnitude/direction of the cam torque pulses and a speed of the engine, among other things. Since the cam torque pulses can be large relative to the dampening of the mechanical cam phasing system, the phaser can easily overshoot or undershoot the desired rotation amount. For effective control, these systems rely on cam shaft and crank shaft position sensors read by the engine controller (“ECU”) and require very fast control or continuous cycling on and off. That is, in mechanical systems, a component may lock or unlock rotation between two components. However, the two components being in a locked or unlocked state does not relate to phase angle. Rather, the components being in a locked or unlocked state merely determines if the phaser is allowed to advance or retard the cam shaft relative to the crank shaft. Therefore, the actuator alone cannot command the phaser to drive to a predetermined, predictable position.

Hydraulically actuated phasers typically utilize an oil control solenoid to control oil pressure to enable rotation of the phaser. While this operation can allow bidirectional control of the phaser, hydraulically actuated phasers rely on cam shaft and crank shaft position sensors read by the engine controller (“ECU”) for effective control. That is, the position of a spool in the oil control solenoid does not relate to phase angle. Rather, the position of the spool merely determines if the phaser is being driven to advance or retard the cam shaft relative to the crank shaft. Therefore, the actuator alone cannot command the phaser to drive to a predetermined, predictable position. Hydraulic phasers are also sensitive to the oil pressure, viscosity, and quality of the oil inside the internal combustion engine, which is prone to degradation over time.

Electronically actuated phasers (also known as “e-phasers”) typically utilize an electric rotary actuator to enable rotation of the phaser. In this operation, the rotary actuator must spin faster or slower than the phaser (e.g., faster or slower than cam speed) to actuate the phaser. Similar to hydraulically actuated phasers, electronically actuated phasers rely on cam shaft and crank shaft position sensors read by the engine controller (“ECU”) for effective control. That is, the rotational position of the electric rotary actuator in these conventional e-phasers do not relate to phase angle. Rather, the speed of the rotary actuator merely determines if the phaser is being driven to advance or retard the cam shaft relative to the crank shaft. Therefore, the actuator alone cannot command the phaser to drive to a predetermined, predictable position. Further, these electronically actuated phasers typically require the system to return to a “home position” on engine shut down to learn the position of the phaser.

Due to the deficiencies in these cam phasing systems, it would be desirable to have a cam phasing system capable of altering the relationship between the cam shaft and the crank shaft on an internal combustion engine independently of a magnitude and direction of cam torque pulses and engine speed.

The systems and methods described herein are capable of altering a rotational relationship between a cam shaft and a crank shaft on an internal combustion engine (i.e., cam phasing) independent of engine speed and a magnitude of cam torque pulses, where the position of the actuator of the cam phaser has a direct relationship to the phase angle of the cam shaft relative to the crank shaft. As will be described, the systems and methods provide an approach where an axial or rotational position of an actuator of the cam phaser alone has a direct relationship to the phase angle of the cam shaft, allowing for accurate cam phasing without the need for cam shaft or crank shaft position sensors. Providing phase angle adjustability without the need for crank shaft or cam shaft position sensors enables control of phase angle solely by sensing the axial or rotational position of the cam phaser actuator.

As used herein, cam shaft position sensors refer to sensors that detect the actual rotational position of the cam shaft. This is typically done by the cam shaft position sensor detecting a geometric/structural feature that designates a zero position for the cam shaft (e.g., a feature that designates the beginning of a new revolution). Similarly, crank shaft position sensors refer to sensors that detect the actual rotational position of the crank shaft. This is also typically done by the crank shaft position sensor detecting a geometric/structural feature that designates a zero position for the crank shaft. In conventional cam phasing systems, as noted above, the signals from both the cam shaft and crank shaft position sensors are used to determine the phase angle of the cam shaft relative to the crank shaft in order to determine how to control the cam phaser or the actuator thereof.

FIG. 1 shows a cam phasing system 10 configured to control the phase angle of a cam shaft 14 relative to a crank shaft 16 in both an open loop and a closed loop mode. The cam phasing system 10 can include a cam phaser 12 configured to be coupled between a cam shaft 14 and a crank shaft 16 of an internal combustion engine (not shown). The cam phaser 12 can include a first component 18 (e.g., a cradle rotor) coupled to the cam shaft 14 and a second component 20 (e.g., a sprocket hub) coupled to the crank shaft 16. The first component 18 can drive the cam shaft 14 via its coupling to the cam shaft 14, for example, via one or more fasteners. The second component 20 can be driven by the crank shaft 16, for example, via a belt, chain, or gear train assembly. This can drive the second component 20 to rotate at a speed proportional to the speed of the crank shaft (e.g., half the speed of the crank shaft). It would be known by one of ordinary skill in the art that alternative configurations for the relative coupling of the first component 18, the second component 20, the cam shaft 14, and the crank shaft 16 are possible. For example, in one embodiment, the crank shaft 16 may be coupled to the first component 18 and the cam shaft 14 may be coupled to the second component 20.

The cam phasing system 10 can include an actuator 22 configured to engage the cam phaser 12 to adjust the rotational position of the first component 18 relative to the second component 20. As will be described herein, in some non-limiting examples, the actuator 22 can be configured to directly or indirectly engage an intermediate component (e.g., spider rotor, see FIGS. 9 and 10) of the cam phaser 12 for accurately controlling a rotary position of the intermediate component with a mechanism causing the first component 18 to follow the rotary position of the intermediate component to alter a rotational relationship between the cam shaft 14 and the crank shaft 16 on an internal combustion engine.

The actuator 22 can be configured to provide an axial or rotational input to the cam phaser 12. For example, the actuator 22 can be a linear actuator or solenoid configured to axially displace in response to electrical current. The actuator 22 can also be a mechanical linkage, a hydraulically actuated actuation element, or other mechanism capable of providing an axial force and/or displacement to the cam phaser 12. According to another example, the actuator 22 can be a rotary actuator and may include a stator and a rotor that is electromagnetically coupled to the stator. A current may be applied to the rotary actuator that may result in a rotary output being provided by the rotary actuator in a desired direction at a desired force. In some non-limiting examples, the rotary actuator may be in the form of a brushless DC (BLDC) motor.

The cam phasing system can include a controller 24 including a processor 26 and a memory 28. The memory 28 can be a non-transitory computer readable medium or other form of storage, such as flash or other type of memory, containing programs, software, or instructions executable by the processor 26. According to some non-limiting examples, the controller 24 can be integrated in the engine control unit of the internal combustion engine. In other non-limiting examples, the controller 24 can be separate from the engine control unit. For example, the controller 24 can be integrated into a body of the actuator 22.

In the illustrated non-limiting example, the controller 24 can be in electrical communication with the actuator 22 to supply actuation command signals to the actuator 22. The controller 24 can also be in electrical communication with an actuator position sensor 30 configured to measure/sense an actuation position of the actuator 22. According to some non-limiting examples, the controller 24 can also be in electrical communication with a cam shaft position sensor 32 and a crank shaft position sensor 34 configured to detect the rotational position of the cam shaft 14 and the crank shaft 16, respectively. It is to be understood that cam shaft and crank shaft speeds and accelerations can also be derived from the cam shaft position sensor 32 and the crank shaft position sensor 34.

Open Loop Mode

The cam phasing system 10 of FIG. 1 can be operated in both an open loop and a modified closed loop mode. FIG. 2 shows a non-limiting example of a method of open loop operation 100. Referring to FIGS. 1 and 2, the process can begin by receiving or generating a phase angle command at block 102. The phase angle command can be received by the controller 24, for example, from the engine control unit. According to one non-limiting example, the phase angle command can be generated by the controller 24 based on operating parameters of the internal combustion engine (e.g., engine speed, engine load, etc.). The controller 24 can then determine a required actuation position of the actuator 22 at block 104 based on the phase angle command and a predetermined relationship between actuation positions of the actuator 22 and resulting cam phase angles (see FIG. 3).

Upon determining the required actuation position, the controller 24 can command the actuator 22 (e.g., via signals or current supplied to the actuator 22) to the required actuation position at block 106. That is, the controller 24 can command the actuator 22 to axially or rotationally displace an actuation element to engage or otherwise displace an intermediate component, such as a spider rotor (see FIGS. 9 and 10), from a first fixed position (e.g., a stationary actuation element position) correlating to a first phase angle towards a second fixed position correlating to a second phase angle, where the first and second fixed positions are different positions and the first and second phase angles are different phase angles. The actuator 22 will continue to progress towards the second fixed position until the actuation position sensor 30 detects that the actuator 22 or the actuation element controlled by the actuator 22, is in the second fixed position. As will be described, the magnitude of the displacement of the actuation element by the actuator 22 between the first and second fixed positions corresponds with a proportional rotational displacement between the first component 18 and the second component 20 of the cam phaser 12, thereby proportionally adjusting the phase angle of the cam shaft 14 relative to the crank shaft 16 based on the position of the actuation element of the actuator 22.

During open loop operation, the controller 24 utilizes only the predetermined relationship between actuation positions of the actuator 22 and resulting cam phase angles and the actuation position sensor 30 to control the cam phaser 12 to achieve a desired phase angle. That is, the cam shaft position sensor 32 and the crank shaft position sensor 34 are not needed or utilized during open loop operation. The utilization of the actuation position sensor 30 and the predetermined relationship correlating actuation positions to cam phase angles can allow for rapid and large-magnitude phase angle changes. This open loop operation can also be more robust than closed loop operation as it is independent of the cam shaft and crank shaft trigger wheels (i.e., encoders), which can be susceptible to encoder malfunctions, such as false readings that can take place during large, rapid phase angle changes (such as the detection of a “false zero” or “false missing tooth”). Open loop operation can also reduce the settling time without increasing overshoot by enabling large, rapid phase angle changes via the utilization of the predetermined relationship between actuation positions and resulting cam phase angles, rather than operating in a closed-loop feedback mode.

Referring now to FIGS. 1 and 3, the actuator 22 is configured to provide an axial or rotational input to the cam phaser 12 which corresponds with a known desired rotational displacement between the first component 18 and the second component 20 of the cam phaser 12. The result of this is that displacement of the actuation element of the actuator 22 by a known amount can cause the cam shaft 14 to rotate clockwise or counterclockwise a known amount relative to the crank shaft 16 (also known as phase angle), depending on whether it is desired to advance or retard the timing of the valve opening/closing events controlled by the cam shaft 14.

FIG. 3 illustrates one example of a predetermined, proportional relationship 206 between the fixed actuator positions 202 and the resulting phase angles 204. In the illustrated non-limiting example, each distinct position of the actuator 22, for example, an axial/linear or rotational position of an actuation element thereof, results in a distinct phase angle 204. The resulting phase angle 204 is proportional to the actuator position 202. In one specific non-limiting example of operation, the actuator position 202 can be at a first fixed position, correlating to a first phase angle 204. The controller 24 can command the actuator 22 to displace from the first fixed position towards a second fixed position until the actuation position sensor 30 detects that the actuator 22 is in the second fixed position. With the actuator 22 being in the second fixed position, the resulting phase angle is a second phase angle. In this non-limiting example, each of the first and second fixed positions, and the first and second phase angles, fall along the proportional relationship 206 illustrated in FIG. 3.

In the illustrated non-limiting example, the resulting phase angle is linearly proportional to the actuator position, although other proportional relationships are also possible. For example, the relationship can be substantially linear, where the relationship between the phase angle and actuator position may slightly deviate from being perfectly linear at one or more actuator positions. According to another example, the relationship between the phase angle and actuator position may be inversely proportional. As will be described, the proportionality between the actuator position and the phase angle can be defined by the geometry or configuration of the components of the cam phaser. According to some non-limiting examples, the proportionality is defined by a helical feature internal to the cam phaser (see, e.g., FIG. 9). According to other non-limiting examples, the proportionality is defined by a gear ratio of a gear train internal to the cam phaser (see, e.g., FIG. 10).

Calibration

Referring now to FIGS. 1, 3, and 4, the controller 24 can execute a calibration process 300 configured to define the predetermined relationship previously described. According to some non-limiting examples, the calibration process can utilize the predetermined relationship between actuator positions and phase angles to generate a two-dimensional look up table for use during open loop cam phaser operation. The equation below illustrates one non-limiting example of an equation representative of the predetermined relationship between actuator positions (a) and resulting phase angles (θ).

θ=β(a−a₁)+θ₁  (1)

In the above equation, β is a coefficient representative of the slope of the relationship illustrated in FIG. 3, and (a₁, θ₁) are coefficients representative of a known operating point. As will be described, the β coefficient can be defined by a known geometric feature or configuration of the cam phaser and the known operating point can be determined during a calibration procedure.

The calibration process 300 can begin at block 302 by commanding the actuator 22, by the controller 24, to either a first end position 208 (e.g., a maximum position in a first direction) or a second end position 210 (e.g., a maximum position in a second direction) of the operating range of the actuator 22 (see FIG. 3). For example, for axial actuators, the actuator can be commanded to a first or second end position. In another example, for rotary actuators, the actuator can be commanded to a maximum clockwise or counter-clockwise position.

The controller 24 can then detect the actuator position via the actuation position sensor 30. The controller 24 can then sense or measure the cam shaft position via the cam shaft position sensor 32 and the crank shaft position via the crank shaft position sensor 34 at block 304. Next, the controller 24 can determine the cam phase angle based on the sensed cam shaft and crank shaft positions at block 306 to learn the phase angle at a maximum retard position 212, or a maximum advance position 214 of the cam shaft 14 at the first or second end position 208, 210, respectively.

Once at least one operating point is known (e.g., one cam phase angle and corresponding actuator position), the controller 24 can define the relationship between actuator positions and resulting cam phase angles for the entire actuation range of the actuator 22 using equation (1) above, along with the known operating point, and the predetermined proportional relationship between actuator position and the phase angle (i.e., the slope 206, β) defined by the geometry or configuration of the components of the cam phaser 12. This predetermined relationship (e.g., slope, or linear function) can be known by the controller 24, for example, by being stored within the memory 28 (see FIG. 1).

According to some non-limiting examples, the controller 24 can command the actuator 22 to ramp between the first and second end positions 208, 210 and at a plurality of distinct intermediate positions, and determine the phase angle at each of the plurality of intermediate positions using the cam shaft position sensor 32 and the crank shaft position sensor 34, to generate the relationship illustrated in FIG. 3. In this specific non-limiting example, the controller 24 can interpolate between the plurality of distinct positions. The proportional relationship between actuator position and the phase angle (e.g., the slope 206) can then be calculated or derived from the plurality of data points. According to another non-limiting example, the controller 24 can command the actuator 22 to one of the first and second end position 208, 210, determine the phase angle at that position using the cam shaft and crank shaft position sensors 32, 34, then command the actuator 22 to the other of the first and second end position 208, 210, and again determine the phase angle. The controller 24 can then generate the relationship illustrated in FIG. 3 using these two known positions and interpolate a proportional relationship (e.g., linear or non-linear) between the two known positions.

Calibration instructions and information can be stored within the memory 28 of the controller 24. According to one non-limiting example, the calibration process 300 can be executed at a factory when the vehicle is on or leaving the assembly line. According to other non-limiting examples, the calibration process 300 can be executed at engine start up.

With the predetermined relationship defined, the controller 24 can utilize the predetermined relationship (e.g., FIG. 3/equation 1), along with the actuation position sensor 30 during open and closed loop control of the cam phasing system 10. For example, the controller 24 can implement the predetermined relationship into a two-dimensional look up table at block 308. According to some non-limiting examples, the controller 24 can continually update the look up table based on measured data during closed loop operation, e.g., by utilizing the actuation position sensor 30, the cam shaft position sensor 32, and the crank shaft position sensor 34. This measured data may result in the look up table having portions that deviate from being perfectly proportional (e.g., perfectly linear or not falling directly on the line defined by equation 1) owing to differences between manufactured cam phasers, tolerances, and friction between components, among other factors.

Modified Closed Loop Mode

The cam phasing system 10 of FIG. 1 can also be operated a modified closed loop mode. During closed loop operation, the controller can utilize the readings from the cam shaft position sensor 32 and the crank shaft position sensor 34 to determine the actual cam phase angle relative to a commanded phase angle to determine a phase angle error. This phase angle error can then be used in a feedback loop to command the actuator 22 to adjust the cam phaser 12 to correct the error between the actual cam phase angle relative to the commanded phase angle. In addition, this modified closed loop mode utilizes the predetermined relationship illustrated in FIG. 3 as a feed-forward mechanism. That is, as illustrated in FIG. 5, the controller 24 can utilize the methods described above with respect to FIG. 2 for integration into a modified closed loop control algorithm 350.

The algorithm 350 can begin by receiving or generating a phase angle command at block 352. The controller 24 can then determine an estimated actuation position of the actuator 22 at block 354 based on the phase angle command and the predetermined relationship between actuation positions of the actuator 22 and resulting cam phase angles (see FIG. 3 and equation 1). The controller 24 can then sense or measure the cam shaft position via the cam shaft position sensor 32 and the crank shaft position via the crank shaft position sensor 34 to determine the actual cam phase angle based on the sensed cam shaft and crank shaft position. The controller 24 can then determine a phase angle error at block 356 by comparing the actual cam phase angle relative to the commanded phase angle. The controller 24 can then command the actuator 22 to an actuator position based on the estimated actuation position and the phase angle error at block 358. According to some non-limiting examples, the phase angle error is delivered through a PID controller.

Selecting Open or Closed Loop Modes

In general, open loop control can be particularly beneficial when large magnitude phase angle changes are required, as the cam phasing system can respond more rapidly. For example, conventional closed loop control can lead to error wind up, resulting in a slower response time. Once the current phase angle is near the commanded phase angle, or for small phase angle changes, the modified closed loop mode described herein can be beneficial as closed loop control allows for accurate fine adjustments. Further, the cam phasing system 10 described herein allows for cam phasing operation to continue in open loop mode if the cam shaft position sensor 32 or the crank shaft position sensor 34 were to fail, or provide inaccurate readings, providing a more robust and adaptable system.

Referring now to FIGS. 1 and 6, the controller 24 can execute an algorithm 400 configured to determine if the cam phasing system 10 should be operated in the open or modified closed loop mode. The algorithm 400 can begin at block 402, where the controller 24 can detect an actuator error and determine if the actuator error is within a predetermined range. The actuator error can be defined by a difference between the current phase angle, as derived from the cam shaft and crank shaft position sensors 32, 34 or as derived from the actuation position sensor 30 and the predetermined relationship between actuator positions and phase angles, and the commanded phase angle. In cases where the controller 24 is maintaining a phase angle at a given engine condition (e.g., at idle, or while maintaining some engine speed while cruising), the actuator error may be small as the current phase angle closely matches the commanded phase angle. Conversely, in cases where a phase angle change has just been commanded, the actuator error may be large as the new commanded phase angle is different from the current phase angle.

According to some non-limiting examples, the predetermined range can be defined by a percentage. For example, the predetermined range can be defined by a percentage relative to the current phase angle (e.g., within 10%, 15%, 25%, etc., of the current phase angle). According to other non-limiting examples, the predetermined range can be defined by a phase angle. For example, the predetermined range can be defined by a phase angle relative to the current phase angle (e.g., within 10°, 15°, 25°, etc., of the current phase angle).

If the controller 24 determines that the actuator error is outside of (i.e., not within) the predetermined range at block 402, the controller 24 uses the open loop control mode at block 404 described with respect to FIG. 2. In some non-limiting examples, if the controller 24 determines that the actuator error is within the predetermined range at block 402, the controller 24 may proceed to block 408 to utilize the modified closed loop control mode. Optionally, if the controller 24 determined that the actuator error is within the predetermined range at block 402, the controller 24 may proceed to block 406, where the controller can determine if the phase angle measurement, as derived from the cam shaft position sensor 32 and the crank shaft position sensor 34, is accurate. If the controller 24 determines that the phase angle measurement is not accurate at block 406, the controller 24 uses the open loop control mode at block 404 described with respect to FIG. 2. If the controller 24 determines that the phase angle measurement is accurate, the controller 24 uses the modified closed loop control mode at block 408. It is to be understood that the algorithm 400 can be repeated continuously during operation of the internal combustion engine, and the controller 24 can switch between open and closed loop operation many times depending on the actuator error at any given time.

FIG. 7 illustrates one specific and non-limiting example of a method 500 of determining the accuracy of the phase angle measurement. As previously described, the cam shaft and crank shaft position sensors 32, 34 can sense or detect a geometric feature that designates a zero position for the cam shaft and crank shaft (e.g., a feature that designates the beginning of a new revolution). According to some non-limiting examples, the geometric feature can be a gap on a trigger wheel. According to another non-limiting example, the geometric feature can be a protrusion on a trigger wheel. According to one non-limiting example, the geometric feature can be the “teeth” of a trigger wheel (i.e., an encoder wheel), which can have one missing “tooth.” The detection of this missing tooth can define the geometric feature designating a “zero” position and can be used to determine crank angles relative of the missing tooth. That is, the missing tooth is taken as the “zero” or start of the crank shaft or cam shaft rotation (e.g., the 0° point). The signals from the cam shaft and crank shaft position sensors 32, 34 can thus resemble a sinuous signal, where the falling edge of the sinuous signal can correlate to the passing of a tooth. A large enough gap between the detection of adjacent teeth can be indicative of the zero mark of the cam shaft or crank shaft. It is to be understood that the following description can be used with respect to the crank shaft or the cam shaft to determine if their respective position measurements are accurate. If either position measurement is inaccurate, the resulting phase angle calculation is also inaccurate, and the controller 24 can switch to open loop operation. It is also to be understood that the following description describes one specific and non-limiting example of determining the accuracy of a measured phase angle as sensed by crank and cam position sensors, and that this description could be applied to various other geometric features or trigger wheels, such as those described above.

The method 500 may begin at block 502 when the controller 24 detects the falling edge of a tooth. The controller 24 may then determine if the geometric feature designating a “zero” is detected at block 504, for example, if a large enough gap is detected in the signal from the previous peak caused by the previous tooth. If the controller 24 does not detect a signal condition indicative of a zero, the controller increases the tooth count at block 506 and returns to start. For example, if the current tooth count, as detected by the controller 24 via the crank/cam position sensors, is some number “n”, the controller 24 will increase the tooth count to “n+1”.

If the controller 24 detects a signal condition indicative of a zero, the controller proceeds to block 508 and determines if the current tooth count “n” is equal to the predetermined total number of teeth of the crank or cam trigger wheel (e.g., 50), which can be stored in the memory 28 of the controller 24. If the controller 24 determines that the current tooth count “n” is equal to the predetermined number of teeth, then the controller 24 detected an accurate zero of the trigger wheel (i.e., the missing tooth), and the position of the cam or crank shaft 14, 16 is known (block 510). The controller 24 may then proceed to bock 512 by resetting the current tooth count “n” equal to one and the process can return to the start. Upon the determination that the crank shaft and cam shaft positions are accurate, the controller 24 can switch into the modified closed loop operation.

If the controller 24 determined that the current tooth count “n” is not equal to the predetermined number of teeth, then the controller 24 detected a “false” zero of the trigger wheel (i.e., a gap between teeth instead of the missing tooth), and the position of the cam or crank shaft 14, 16 is unknown (block 514). The controller 24 may then proceed to block 512 by resetting the current tooth count “n” equal to one and the process can return to the start. Upon the determination that the crank shaft and cam shaft positions are inaccurate, the controller 24 can switch into open loop operation. According to some non-limiting examples, the controller 24 defaults to open loop operation.

A “false” zero, as previously noted above, can be caused by a rapid change in phase angle or an error in the signals from the cam shaft or crank shaft position sensors 32, 34. For example, during phasing, the cam shaft speed can vary (e.g., increase or decrease) depending on the direction of phasing. In some specific and non-limiting examples, the cam shaft speed can slow to such an extent that the cam shaft position sensor 32 detects an abnormally large gap between teeth of the trigger wheel, which can cause a signal condition indicative of a zero. That is, the gap between detecting adjacent teeth during phasing operations can be large enough to resemble the geometric feature designating the zero position (i.e., a missing tooth), causing a false reading.

FIG. 8 illustrates one non-limiting example of cam phasing on startup utilizing the cam phasing system 10 described herein. As illustrated, during engine start up the controller 24 can control phase angle in an open loop operation without a proper or accurate phase angle measurement. That is, the cam or crank shaft may have not yet rotated enough times during start up to determine if the signals from the cam shaft or crank shaft position sensors 32, 34 are accurate (see FIG. 1).

Cam Phaser Examples

As previously described herein, the proportionality between the actuator position and the phase angle can be defined by the geometry or configuration of the components of the cam phaser. For example, a geometric feature or component of the cam phaser can be arranged between an input shaft and one of the first and second components of the cam phaser to be coupled to the cam shaft and crank shaft, respectively. According to some non-limiting examples, the proportionality is defined by a helical feature internal to the cam phaser, such as the cam phasers described in U.S. Pat. No. 10,072,537 to Schmitt et al. entitled “Mechanical Cam Phasing Systems and Methods,” the content of which is incorporated herein by reference in its entirety. According to other non-limiting examples, the proportionality is defined by a gear ratio of a gear train internal to the cam phaser, such as the cam phasers described in United States Patent Application No. 2020/031346 to Van Weelden et al. entitled “Systems and Methods for Controlled Relative Rotational Motion,” the content of which is also incorporated herein by reference in its entirety.

As illustrated in FIG. 9, a helical feature internal to the cam phaser can define a relationship between actuation position and rotational positions between rotary components. FIG. 9 shows a cam phasing system 1000 configured to be coupled to a cam shaft (not shown) of an internal combustion engine (not shown). As shown in FIG. 9, the cam phasing system 1000 can include a cradle rotor 1018 (e.g., a first component) configured to be coupled to a cam shaft, a sprocket hub 1020 (e.g., a second component) configured to be coupled to a crank shaft, a spider rotor 1006, an input shaft configured as a helix rod 1008, and an end plate 1010. The sprocket hub 1020, the cradle rotor 1018, the spider rotor 1006, the helix rod 1008, and the end plate 1010 can each share a common central axis 1011, when assembled. The sprocket hub 1020 can include a gear 1012 and the gear 1012 can be connected to an outer diameter of the sprocket hub 1020 and the gear 1012 can be coupled to a crank shaft (not shown) of the internal combustion engine. This can drive the sprocket hub 1020 to rotate at a speed proportional to the speed of the crank shaft.

An actuator 1022 can be configured to engage the helix rod 1008. The actuator 1022 can be configured to apply an axial force to the helix rod 1008 in a direction parallel to, or along, the central axis 1011. The actuator 1022 may be a linear actuator, a mechanical linkage, a hydraulically actuated actuation element, or any other mechanism capable of providing an axial force and/or displacement to the helix rod 1008. That is, the actuator 1022 can be configured to axially displace the helix rod 1008 to a known position, which corresponds with a desired rotational displacement of the spider rotor 1006. The actuator 1022 can be controlled and powered by a controller (e.g., controller 24, FIG. 1).

The helix rod 1008 includes a helical portion 1082 configured to be received within helical features 1056 of the spider rotor 1006. An interaction between the helical portion 1082 of the helix rod 1008 and the helical features 1056 of the spider rotor 1006 can enable the spider rotor 1006 to rotate relative to the sprocket hub 1020 in response to an axial displacement applied by the actuator 1022 on the helix rod 1008. When assembled, as shown in FIG. 9, the spider rotor 1006 can be constrained such that it cannot displace axially. Thus, in response to an axial displacement applied on the helix rod 1008 by the actuator 1022, the spider rotor 1006 is forced to rotate clockwise or counterclockwise a known amount, depending on whether it is desired to advance or retard the valve events controlled by the cam shaft. That is, the spider rotor 1006 will rotate relative to the sprocket hub 1020 due to the interaction between the helical portion 1082 of the helix rod 1008 and the helical features 1056 of the spider rotor 1006.

In operation, when the rotational relationship between the cam shaft, which is fastened to the cradle rotor 1018, and the crank shaft, which is coupled to the sprocket hub 1020, is desired to be altered, a controller (e.g., controller 24 of FIG. 1) can command the actuator 1022 to provide an axial displacement to the helix rod 1008 from a first fixed axial position to a second fixed axial position. When the signal is sent to axially displace the helix rod 1008, the cam phasing system 1000 can transition from a locked state, where the rotational relationship between the cradle rotor 1018 and the sprocket hub 1020 is locked, to an actuation state. In response to the axial displacement applied to the helix rod 1008, the spider rotor 1006 can rotate, either clockwise or counterclockwise depending of the direction of the axial displacement, due to the interaction between the helical portion 1082 of the helix rod 1008 and the helical features 1056 of the spider rotor 1006. The rotation of the spider rotor 1006 can cause the spider rotor 1006 to engage locking features (not separately numbered) to place the cam phasing system 1000 in the actuation state. With the cam phasing system 1000 in the actuation state, the cradle rotor 1018 rotationally follows the spider rotor 1006 (e.g., by harvesting cam torque pulses applied to the cradle rotor 1018) in the same direction that the spider rotor 1006 was rotated. The cradle rotor 1018 will continue to rotate until the cradle rotor 1018 rotationally displaces to a rotational position correlating to the magnitude of the axial displacement of the helix rod 1008 and the angle of the helical features 1056.

In general, the design of the cam phasing system 1000 only requires an input force provided to the helix rod 1008 from the actuator 1022 when relative rotation is desired (e.g., the actuator 1022 displaces between fixed positions, and those fixed positions correlate to a known phase angle between the cam shaft and the crank shaft).

As illustrated in FIG. 10, a gear ratio of a planetary gear train of a cam phaser can define a relationship between actuation position and rotational positions between rotary components. FIG. 10 illustrates a non-limiting example of a cam phasing system 2000 including a planetary actuator 2001. In the illustrated non-limiting example, the mechanical cam phasing system 2000 includes a cradle rotor 2018 (e.g., a first component) configured to be coupled to a cam shaft, a sprocket hub 2020 (e.g., a second component) configured to be coupled to a crank shaft, a cradle rotor 2018, a bearing cage, or spider rotor, 2008, a plurality of locking assemblies 2010, and the planetary actuator 2001. The planetary actuator 2001, the sprocket hub 2020, the cradle rotor 2018, and the bearing cage 2008 can each share a common central axis 2111, when assembled.

In the illustrated non-limiting example, the mechanical cam phasing system 2000 includes an actuator 2022 configured as a rotary actuator. In some non-limiting examples, the rotary actuator 2022 may include a stator and a rotor that is electromagnetically coupled to the stator. A current may be applied to the rotary actuator 2022 that may result in a rotary output being provided by the rotary actuator 2022 in a desired direction at a desired force. In some non-limiting examples, the rotary actuator 2022 may be in the form of a brushless DC (BLDC) motor.

The planetary actuator 2001 includes a first ring gear 2200, a first sun gear 2202, a carrier assembly 2204, a second ring gear 2206, a second sun gear 2208, and an input shaft 2021. The carrier assembly 2204 includes a first set of planet gears 2222, a second set of planet gears 2224, and a carrier plate 2226. The first set of planet gears 2222 and the second set of planet gears 2224 may be arranged on axially opposing sides of the carrier plate 2226. In the illustrated non-limiting example, the first set of planet gears 2222 mesh with the first sun gear 2202 and the second set of planet gears 2224 mesh with the second sun gear 2208.

The first ring gear 2200 may be selectively rotated relative to the second ring gear 2206 in a desired direction. To facilitate the rotation of the first ring gear 2200 relative to the second ring gear 2206, the input shaft 2021, which is rotationally coupled to the rotary actuator 2022, may be rotated in a first direction. The rotation of the input shaft 2021 in the first direction results in rotation of the first sun gear 2202 in the first direction. Rotation of the first sun gear 2202 in the first direction results in rotation of the planet gears of the first set of planet gears 2222 in a second direction opposite the first direction, which rotates the first ring gear 2200 in the second direction. With the second sun gear 2208 being rotationally fixed, this selective rotation of the first sun gear 2202, and thereby the first ring gear 2200, allows the first ring gear 2200 to rotate relative to the second ring gear 2206 in the second direction. The opposite is also true if the input shaft is rotated in the second direction.

The sprocket hub 2020 can include a gear 2011 arranged on an outer diameter thereof, which can be coupled to a crank shaft (not shown) of an internal combustion engine (not shown), for example, via a belt, chain, or gear train assembly. The cradle rotor 2018 may be configured to be attached to the cam shaft (not shown) of the internal combustion engine via a bolt 2034. In general, the cradle rotor 2018 may be in engagement with the locking assemblies 2010.

In the illustrated non-limiting example, the input shaft 2021 may be coupled to the rotary actuator 2022, such that the rotary output provided by the rotary actuator 2022 is rotationally transferred to the input shaft 2021. The second sun gear 2208 is rotationally fixed to the rotary actuator 2022 and prevented from rotating. The rotary actuator 2022 is rotationally coupled to the first sun gear 2202 to control the rotation thereof. In general, the second ring gear 2206 may be configured to be rotationally coupled to the sprocket hub 2020, such that the second ring gear 2206 rotates with the sprocket hub 2020.

In operation, the rotary actuator 2022 may be configured to apply the rotary displacement/torque to the first sun gear 2202 to achieve a known rotary displacement of the first ring gear 2200 based on the gear ratio of the planetary actuator 2001, which corresponds with a known desired rotational displacement of the bearing cage 2008. The rotary actuator 2022 can be controlled and powered by a controller (e.g., controller 24 of FIG. 1).

During operation, the sprocket hub 2020 can be coupled to the crank shaft of the internal combustion engine. The cam shaft of the internal combustion engine can be fastened to the cradle rotor 2018. Thus, the cam shaft and the crank shaft can be coupled to rotate together, with the cam shaft rotating half as fast as the crank shaft, via the mechanical cam phasing system 2000. When the engine is operating and no rotational adjustment of the cam shaft is desired, the mechanical cam phasing system 2000 can be in a locked state to lock the rotational relationship between the sprocket hub 2020 and the cradle rotor 2018, thereby locking the rotational relationship between the cam shaft and the crank shaft. In this locked state, the rotary actuator 2022 does not supply a rotary output to the input shaft 2021 of the planetary actuator 2001, and the first ring gear 2200 and the second ring gear 2206 rotate in unison with the sprocket hub 2020. Therefore, the bearing cage 2008 is not rotated relative to the sprocket hub 2020 and locking assemblies 2010 lock relative rotation between the cradle rotor 2018 and the sprocket hub 2020. Therefore, the rotational relationship between the cam shaft and the crank shaft is unaltered, when the mechanical cam phasing system 2000 is in the locked state.

If it is desired to advance or retard the cam shaft relative to the crank shaft, the rotary actuator 2022 can be commanded by the controller 24 to provide a rotary displacement/torque to the input shaft 2021 of the planetary actuator 2001. That is, the controller 24 can command the actuator 2022 to provide a rotational displacement to the input shaft 2021 from a first fixed rotational position to a second fixed rotational position. The direction and magnitude of the rotation of the input shaft 2021 can be correlated to a known rotation of the first ring gear 2200 relative to the second ring gear 2206. Since the second ring gear 2206 is rotationally coupled to the sprocket hub 2020, the first ring gear 2200 may be rotated relative to the sprocket hub 2020. The desired magnitude and direction of the relative rotation applied to the first ring gear 2200 may be rotationally transferred to the bearing cage 2008 via a coupling therebetween. The coupling is configured to maintain the force applied to the bearing cage 2008 until the cradle rotor 2018 reaches the desired rotational position relative to the sprocket hub 2020, which is determined by the rotary input displacement/force provided by the rotary actuator 2022 and the gear ratio of the planetary actuator 2001. The rotation of the bearing cage 2008 can engage the locking assemblies 2010 and place the cam phasing system 2000 into an actuation state.

In the actuation state, the cradle rotor 2018 rotates in the same rotational direction in which the bearing cage 2008 was rotated. For example, in the non-limiting example where the first ring gear 2200 rotationally biases the bearing cage 2008 clockwise, the cradle rotor 2018 can rotationally displace in a clockwise direction. In general, in response to a given rotary input displacement/force applied to the bearing cage 2008 through the planetary actuator 2001, the cradle rotor 2018 rotationally follows the bearing cage 2008 and eventually reaches a predefined final rotary position of the bearing cage 2008 based on a magnitude of rotational input to the input shaft 2021 and the gear ratio of the planetary actuator 2001.

The rotation of the cradle rotor 2018 with respect to the sprocket hub 2020 that occurs during this phasing process can vary the rotational relationship between the cam shaft and the sprocket hub 2020, which simultaneously alters the rotational relationship between the cam shaft and the crank shaft. As described above, the amount of rotation achieved by the bearing cage 2008 for a given rotary input displacement/torque provided by the rotary actuator 2022 can be known based on the gearing between the first sun gear 2202 and the first ring gear 2200 and the resultant gear ratio defined therebetween. Furthermore, the design of the mechanical cam phasing system 2000 can enable the cradle rotor 2018 to only be allowed to rotate in the same direction as the bearing cage 2008. Thus, during engine operation the mechanical cam phasing system 2000 can alter the rotational relationship between the cam shaft and the crank shaft.

In general, the design of the cam phasing system 2000 only requires an input torque/displacement provided to the input shaft 2021 from the rotary actuator 2022 when relative rotation is desired (e.g., the actuator 2022 rotates between fixed positions, and those fixed positions correlate to a known phase angle between the cam shaft and the crank shaft).

It will be appreciated by those skilled in the art that while the invention has been described above in connection with particular embodiments and examples, the invention is not necessarily so limited, and that numerous other embodiments, examples, uses, modifications and departures from the embodiments, examples and uses are intended to be encompassed by the claims attached hereto. The entire disclosure of each patent and publication cited herein is incorporated by reference, as if each such patent or publication were individually incorporated by reference herein.

Various features and advantages of the invention are set forth in the following claims.

Claims

1. A cam phasing control system for varying a rotational relationship between a crank shaft and a cam shaft, the cam phasing control system comprising: a cam phaser including a first component configured to be coupled to a cam shaft and a second component configured to be coupled to a crank shaft;an actuator configured to adjust a rotational position of the first component relative to the second component;an actuator position sensor configured to detect an actuation position of the actuator; anda controller including a processor and a memory, the processor configured to: receive a phase angle command;determine a required actuation position of the actuator based on the phase angle command and a predetermined relationship between an actuation position of the actuator and cam phase angle; andcommand the actuator to displace from a first fixed position to a second fixed position, wherein a magnitude of the displacement between the first fixed position and the second fixed position corresponds with a proportional rotational displacement between the first component and the second component.

2. The system of claim 1, wherein the predetermined relationship between the actuation position of the actuator and the cam phase angle is linear.

3. The system of claim 1, wherein the determination of the required actuation position is done without a cam shaft position sensor and a crank shaft position sensor.

4. The system of claim 1, wherein the predetermined relationship between the actuation position of the actuator and the cam phase angle is defined by a helical feature arranged between an input shaft of the cam phaser and one of the first component or the second component.

5. The system of claim 4, wherein the actuator is configured to axially displace the input shaft of the cam phaser.

6. The system of claim 1, wherein the predetermined relationship between the actuation position of the actuator and the cam phase angle is defined by a gear ratio of a planetary gear train arranged between an input shaft of the cam phaser and one of the first component or the second component.

7. The system of claim 6, wherein the actuator is configured to rotationally displace the input shaft of the cam phaser.

8. The system of claim 1, wherein the predetermined relationship is governed by the following equation: θ=β(a−a1)+θ1

9. The system of claim 1, further comprising a crank shaft position sensor configured to detect a crank shaft position and a cam shaft position sensor configured to detect a cam shaft position; wherein the processor is configured to measure a cam phase angle based on the crank shaft position and the cam shaft position.

10. The system of claim 9, wherein the predetermined relationship is stored in the memory of the controller as a two-dimensional look up table; wherein the processor is configured to update the two-dimensional look up table based on the measured cam phase angle and the actuation position of the actuator.

11. The system of claim 10, wherein the processor is further configured to execute a calibration procedure, the processor configured to: command the actuator to an end position;determine the cam phase angle based on the crank shaft position and the cam shaft position; andgenerate the two-dimensional look up table based on the determined cam phase angle and a coefficient defined by one of a helical feature or a gear ratio of a planetary gear train arranged between an input shaft of the cam phaser and one of the first component or the second component.

12. The system of claim 9, wherein the processor is further configured to operate in an open loop mode and a closed loop mode, the processor configured to: detect an actuator error between a commanded actuator position and a sensed actuator position of the actuator; anddetermine if the actuator error is within a predetermined range;wherein when the actuator error is outside of the predetermined range, the processor is configured to operate the cam phasing control system in an open loop mode;wherein when the actuator error is within the predetermined range, the processor is configured to determine if a measured cam phase angle detected by the cam shaft position sensor and the crank shaft position sensor is accurate; andwherein when the phase angle reading is determined to be accurate, the processor is configured to operate the cam phasing control system in a closed loop mode.

13. The system of claim 12, wherein when the cam phasing control system is in the closed loop mode, the processor is configured to: receive a phase angle command;determine an estimated actuation position of the actuator based on the phase angle command and the predetermined relationship between the actuation position of the actuator and the cam phase angle;determine a phase angle error between the commanded phase angle and an actual cam phase angle; andcommand the actuator to an actuator position based on the phase angle error and the estimated actuation position.

14. A method of open loop control of a cam phasing system for varying a rotational relationship between a crank shaft and a cam shaft, the method comprising: receiving a phase angle command;determining a required actuation position of a cam phaser actuator based on the phase angle command and a predetermined relationship between actuation positions of the cam phaser actuator and cam phase angles; andcommanding the cam phaser actuator to the required actuation position.

15. The method of claim 14, wherein the predetermined relationship between the actuation positions of the cam phaser actuator and the cam phase angles is linear.

16. The method of claim 14, wherein the determination of the required actuation position is done without a cam shaft position sensor and a crank shaft position sensor.

17. The method of claim 14, wherein commanding the cam phaser actuator to the required actuation position includes displacing the cam phaser actuator from a first fixed position corresponding to a first phase angle to a second fixed position corresponding to a second phase angle.

18. The method of claim 17, wherein a magnitude of the displacement between the first fixed position and the second fixed position corresponds with a proportional rotational displacement between a first component and a second component of a cam phaser.

19. The method of claim 17, wherein the first fixed position and the second fixed position are a first axial position and a second axial position for the cam phaser actuator, the second axial position being distinct from the first axial position; or wherein the first fixed position and the second fixed position are a first rotational position and a second rotational position for the cam phaser actuator, the second rotational position being distinct from the first rotational position.

20. The method of claim 14, wherein determining the required actuation position of the cam phaser actuator includes executing the following equation: θ=β(a−a1)+θ1 wherein a is the required actuation position, θ is the commanded phase angle, β is a coefficient defined by one of a helical feature or a gear ratio of a planetary gear train internal to a cam phaser, and a1, θ1 are coefficients representative of a known operating point for a known actuation position and a corresponding known cam phase angle, respectively.