Patent Publication Number: US-8981682-B2

Title: Asymmetric and general vibration waveforms from multiple synchronized vibration actuators

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 13/030,663, filed Feb. 18, 2011, and entitled SYNCHRONIZED VIBRATION DEVICE FOR HAPTIC FEEDBACK, which is a continuation of U.S. application Ser. No. 11/476,436, filed Jun. 27, 2006, issued on Apr. 5, 2011 as U.S. Pat. No. 7,919,945, which claims the benefit of the filing date of U.S. Provisional Patent Application No. 60/694,468 filed Jun. 27, 2005 and entitled SYNCHRONIZED VIBRATION DEVICE FOR HAPTIC FEEDBACK, the entire disclosures of which are hereby expressly incorporated by reference herein. This application also claims the benefit of the filing dates of U.S. Provisional Patent Application No. 61/453,739, filed Mar. 17, 2011 and entitled ASYMMETRIC AND GENERAL VIBRATION WAVEFORMS FROM MULTIPLE SYNCHRONIZED VIBRATION ACTUATORS, and of U.S. Provisional Patent Application No. 61/511,268, filed Jul. 25, 2011 and entitled ASYMMETRIC AND GENERAL VIBRATION WAVEFORMS FROM MULTIPLE SYNCHRONIZED VIBRATION ACTUATORS, the entire disclosures of which are hereby expressly incorporated by reference herein. 
    
    
     BACKGROUND OF THE INVENTION 
     Vibration devices are used in a wide range of applications including haptic displays, haptic interfaces, force feedback devices, vibratory feeders, beauty products, personal hygiene products, personal pleasure products, personal massagers, tree harvesters, and seismic vibrators. Some widely used products that include haptic displays include the DUALSHOCK® 3 wireless controller for Sony Computer Entertainment&#39;s PlayStation® 3; the PlayStation® Move motion controller for motion gaming with Sony Computer Entertainment&#39;s PlayStation® 3; Microsoft Corporation&#39;s Xbox 360 Wireless Speed Wheel; and the Wii Remoter™ Plus controller which is used for motion gaming with the Nintendo Wii. 
     Vibration actuators are typically the smallest and lowest cost method for generating haptic sensations. Therefore, it is advantageous to use vibration actuators to create a wide range of haptic sensations. Common low cost vibration actuators include Eccentric Rotating Mass actuators (ERMs) and Linear Resonant Actuators (LRAs). One of the advantages of both ERMs and LRAs is that they can generate relatively large vibration forces from low power input. Both ERMs and LRAs generally build up kinetic energy during their ramp-up period; an ERM does this as the velocity of its rotating mass increases, and an LRA does this as the amplitude of vibration of its moving mass increases. These low cost actuators are used in many applications, including in consumer electronics products such as smartphones and videogame controllers. 
     Many smartphones today use either a single ERM or a single LRA to produce alerts by vibrating the entire device. This has the advantage that the vibration alert can be felt while the device is inside a person&#39;s pocket. Game controllers (also commonly termed interchangeably as “videogame controllers” or simply “controllers”) often incorporate two ERMs within a two-handed device such as the Xbox 360 Wireless Controller or the Xbox 360 Wireless Speed Wheel (both devices from Microsoft). Sometimes such dual-ERM controllers are configured with one ERM having a large rotating mass and the other ERM having a small rotating mass. A single-handed controller such as the Wii Remote™ Plus (from Nintendo) will typically have a single ERM to provide vibration feedback to the user. 
     A common limitation of most existing vibration devices is the inability to define the directionality of the vibratory forces. ERM actuators generate centripetal forces that rotate in a plane, and generally the direction of vibration (that is to say, the instantaneous direction of the rotating centripetal force vector) cannot be not sensed in haptic applications due in part to the high rate of change of the direction of vibrations. In an ERM a centripetal force is applied onto the eccentric mass by the motor shaft, and an equal and opposite centrifugal force is applied onto the motor shaft. In this document both the terms centripetal and centrifugal are used with the understanding that these are equal but opposite forces. LRAs vibrate back and forth, and thus it may be possible to sense the axis of vibration, but it is not possible to provide more of a sensation in the forward direction relative to the backward direction or vice versa. Since haptic applications are often integrated with audio and video displays such as in computer gaming where directions are an integral component of the game, it is desirable to provide a haptic sensation that also corresponds to a direction. Moreover, it is be useful to generate haptic cues of directionality for applications where a person does not have visual cues, such as to guide a vision-impaired person. Therefore, it is desirable to provide a human-perceptible indication of directionality in vibratory haptic displays and interfaces. In addition, it is advantageous to use vibration actuators to generate a wide range of vibration waveforms including both directional and non-directional waveforms. 
     There have been some haptic vibration devices that provide a sensation of vibration direction, but these prior implementations have disadvantages. Specifically, asymmetric vibrations have been used to generate a haptic sensation that is larger in one direction than the opposite direction. 
     However, existing asymmetric vibrators are complex, costly, or have limited controllability. They tend to be bulky and have low power efficiency. Tappeiner et. al. demonstrated a vibration device that generated asymmetric directional haptic cues (Tappeiner, H. W.; Klatzky, R. L.; Unger, B.; Hollis, R., “Good vibrations: Asymmetric vibrations for directional haptic cues”, World Haptics 2009, Third Joint Euro Haptics Conference and Symposium on Haptic Interfaces for Virtual Environments and Teleoperator Systems), yet this device uses a high power and an expensive 6-DOF magnetic levitation haptic device. Amemiya et. al. (Tomohiro Amemiya; Hideyuki Ando; Taro Maeda; “Kinesthetic Illusion of Being Pulled Sensation Enables Haptic Navigation for Broad Social Applications, Ch. 21, Advances in Haptics, pp. 403-414”) illustrated a device that also generates asymmetric vibrations for haptic applications, yet this device uses a complex and large linkage system with 6 links and it appears that the direction of vibration cannot be modified in real-time. 
     Another limitation of vibration devices that use ERMs is that the amplitude of vibration is dependent on the frequency of vibration, since the vibration forces are generated from centripetal acceleration of an eccentric mass. Some prior approaches have used multiple ERMs to control frequency and amplitude independently, but in the process also generate undesirable torque effects due to the offset between the ERMs. 
     SUMMARY OF THE INVENTION 
     It is desirable to produce not only haptic directional cues but also to be able to render legacy haptic effects, so that a single game controller could generate both new and existing sensations. Therefore, one aspect of this disclosure, and associated embodiments described in this specification include vibration devices and methods of operating those devices employing low cost vibration actuators that use a low amount of electrical power to generate asymmetric vibrations and to control the direction of the vibration forces, as well as other vibration waveforms. The aspects described within this disclosure address these limitations through a design that can generate unique directional haptic cues that can be controlled in all 360 degrees of a plane, as well as other vibration waveforms. To provide a wide range of vibration effects it is desirable to be able to control the amplitude and frequency independently. This disclosure describes a vibration device that uses ERMs and allows independent control of vibration force and frequency, without generating torque vibrations. Since no torque effects are created, the vibration force can be brought all the way down to zero, even when the ERMs are rotating. With this approach the vibration forces can be turned on and off without having to wait for the ERMs to spin up to speed or spin down, and therefore allow vibration forces to be turned on and off more quickly and effectively increase the responsiveness of the vibration device. 
     The present disclosure overcomes the disadvantages and limitations of known vibration devices by providing means of generating asymmetric and general waveform vibration profiles from multiple vibration actuators. These vibrations can apply a force vibration, torque vibration, or a combined force and torque vibration onto an object or onto the ground. Numerous embodiments and alternatives are provided below. 
     In accordance with an embodiment, a vibration device is provided. The vibration device comprises multiple vibration actuators or mechanical oscillators in which the phases and frequencies of the actuators are controlled and synchronized. The amplitude of vibration of each actuator may also be controllable to provide a wider range of waveforms. Through this synchronization and control is it possible in some embodiments to generate arbitrary shaped vibration waveforms, including asymmetric waveforms that have higher peaks in one direction and lower peaks in the opposite direction. With this approach the advantages of asymmetric vibrations can be realized with low cost and low power vibration actuators. Moreover, the direction of vibration can be controlled in multiple degrees of freedom. 
     Many vibration actuators generate sinusoidal waveforms. Common low cost vibration actuators include ERMs and LRAs. Pivoting actuators and rocking actuators are less common, but are described in U.S. Pat. No. 7,994,741, and can replace LRAs in many applications. Other types of actuators that can generate vibrations, that is to say “Vibration Actuators”, include voice coils, linear actuators, linear force actuators, pneumatic actuators, hydraulic actuators, piezoelectric actuators, electrostatic actuators, electoactive polymers (EAPs), solenoids, ultrasonic motors, and motors that drive vibrating linkages. This disclosure combines multiple Vibration Actuators in a manner whereby the vibration waveforms from the Vibration Actuators are superimposed to generate a desired combined waveform. 
     In many embodiments multiple Vibration Actuators are mounted onto a mounting platform (equivalently termed as a “base plate” or a “sub-frame” or a “housing”) so that the vibration forces from each actuator are vectorially added together to generate a combined vibration waveform. A control method is used to control the phase and frequency of each of the Vibration Actuators. In some embodiments the amplitude is also controlled. Furthermore, in some embodiments the method of Fourier synthesis is used to select the desired phase, frequency, and amplitude of each of the members of a set of sinusoidal waveforms to generate a desired combined vibration waveform. Indeed, with enough actuators it is possible to approximate an arbitrary vibration waveform, including both symmetric and asymmetric vibration waveforms. 
     A single Vibration Actuator generates vibration forces, torque, or force and torques. Many Vibration Actuators generate simple vibration waveforms such as sinusoidal waveforms. In this disclosure multiple Vibration Actuators are configured such that they generate a combined vibration waveform. In many embodiments multiple Vibration Actuators are secured to a mounting platform such that force and torques from individual Vibration Actuators are vectorially added to generate a combined vibration force, torque, or force and torque. This mounting platform can be held by a person, worn by a person or otherwise placed in contact with a person, attached to a person, attached to an object, or placed on a surface. In other embodiments the mounting platform is a part of a locomotion device the vibration forces generate propulsion forces. 
     Multiple Vibration Actuators can be synchronized together by controlling their frequency of vibration, and the relative phase of vibration. In some cases, the amplitude of vibrations is also controlled. The frequency, phase, and amplitude of vibration can refer to the characteristics of vibration force, torque, or force and torque waveforms. In many Vibration Actuators, including most LRAs and ERMs, vibration forces coincide temporally with the motion of a moving mass. With LRAs a mass oscillates back and forth. With an ERM, an eccentric mass rotates about an axis. Accordingly synchronization of Vibration Actuators can also refer to the control of the frequency and phase of the motion of multiple moving masses, and in some cases the amplitudes of motion. 
     One category of waveforms that is specifically useful for haptic applications is asymmetric waveforms. These waveforms generate peak forces, peak accelerations, or peak rate of change of force (“jerk”) in one direction that are larger than these peaks in the opposite direction. Asymmetric waveforms can also apply to torque, or torque and force waveforms. These waveforms can provide a haptic sensation that corresponds to a specific direction. In one example, two LRAs are mounted onto a mounting platform such that their force vectors are aligned with each other, which are defined as an LRA Pair. The frequency of one LRA is set to twice the frequency of the other LRA in the pair. 
     The relative phase of vibration between the two actuators is synchronized such that in one direction the peaks of the two vibration forces occur at the same time with the same sign and thus through superposition with constructive (positive) interference combine to increase the magnitude of the overall vibration force. In the opposite direction the peaks of the two actuators occur at the same time but with opposite signs and thus through superposition with destructive (negative) interference partially cancel each other out and thereby reduce the magnitude of the overall vibration force. With this approach, a larger peak force is felt in one direction and a lower peak force is felt in the opposite direction. The relative phase of vibration between the two actuators can be changed so that the larger force of the combined waveform switches sign and occurs in the opposite direction. In haptic applications, a higher peak force can be sensed as a more significant sensory input than a lower magnitude force that has a longer duration. Thereby, by creating larger peak forces in one direction than the opposite direction, a haptic cue can be generated in a desired direction such as forwards and backwards. 
     In another example, a set of LRAs is mounted onto a mounting platform in an orientation such that the LRAs within the set have their force vectors aligned with each other. Fourier waveform synthesis can be is used to select the phase, frequency, and amplitude of each LRA to approximate a desired vibration waveform. One example waveform is a Sawtooth waveform, which creates a more sudden change of force in one direction than the opposite direction. In this manner, the Sawtooth waveform can be used to generate directional haptic cues. When the number of LRAs in a set is three, the Sawtooth waveform would be generated with the first harmonic at relative amplitude of 1; the second harmonic is at relative amplitude of ½; and the third harmonic with a relative amplitude of ⅓. Other waveforms can be generated from a set of LRAs that generate high peak forces in one direction, and lower peak force in the opposite direction. 
     In another example, two LRA pairs are mounted onto a mounting platform. The LRA pairs are oriented such that one of the LRA pairs is aligned with an x-axis of a plane, and the second LRA pair is aligned with a y-axis of the same plane. The phase, frequency, and amplitude of vibration of each LRA is controlled by a microprocessor, FPGA or other controller. In each LRA pair, one LRA is operated at twice the frequency of the other LRA in the pair, so that an asymmetric vibration waveform can be generated along the axis of each LRA pair. The frequencies of the LRA pair aligned with the x-axis are set equal to the frequencies of the LRA pair aligned with the y-axis. 
     By controlling the amplitude and phase of vibration of the LRA pairs it is possible, through vector superposition of the vibration forces, to generate a combined vibration force that is aligned with an arbitrary direction in the plane. In this fashion the direction of asymmetric vibrations can be independently controlled in all 360 degrees of a plane. In a similar embodiment, two LRA pairs are mounted onto a mounting platform with their axes of force in the same geometric plane, but the two LRA pairs are not orthogonal to each other, as the x and y axes are. Rather, the axes of the LRA pairs span the geometric plane in a linear algebra sense. Thus, even without orthogonally aligned pairs, asymmetric vibrations can be independently controlled in all 360 degrees of a plane. 
     In another example, three pairs of LRAs are mounted onto a mounting platform with their axes of force oriented such that the three axes span the three dimensional space. In this fashion, asymmetric vibrations can be arbitrarily generated in any 3D direction. 
     In another example, two LRA pairs are mounted onto a mounting platform with the axes of both pairs aligned to a single axis. The two pairs are spaced a set distance apart to generate a desired torque effect. Both pairs can be controlled so that their forces occur simultaneous in the same direction, thereby generating a net force on the mounting platform, but no net torque about a central point in the mounting platform. With another control approach the force from one pair can be synchronized such that it is in the opposite direction of the other pair, thereby generating a net torque onto the mounting platform. A further variation is to have each pair generate an asymmetric vibration waveform, so that the peak torque applied to the mounting platform is larger in one direction than the other direction. In this fashion it is possible to generate a haptic cue that provides a sense of rotational direction. This is an example of asymmetric torque vibration. 
     According to one aspect of the disclosure, a controller (e.g., a microprocessor or FPGA) is used to synchronize the vibration of multiple Vibration Actuators, and the controller implementation depends on the type of vibration actuator being controlled. LRAs, rocking actuators, and pivoting actuators can use similar controllers and include an actuator driver that controls the voltage or current applied to the actuator. An LRA has a moving mass that translates back and forth and a restoring spring that centers the mass. 
     In a similar fashion, rocking actuators and pivoting actuators have a mass with rotational inertia that rocks back and forth with a restoring spring that centers it. LRAs, rocking actuators, and pivoting actuators typically have a resonant frequency, and the actuator driver typically uses a sinusoidal or square wave profile. When the actuator driver operates near the resonant frequency of the actuator, large vibration forces can be generated. LRAs, rocking actuators, and pivoting actuators can by synchronized with open-loop controllers, with no need for sensors. A controller can generate a driving waveform for each actuator at a desired frequency, phase, and amplitude. 
     The precision of a controller for synchronizing LRAs, rocking actuators, and pivoting actuators can be improved in a number of ways. The physical properties of the LRA, rocking actuator, or pivoting actuators generate a phase lag between the actuator driver waveform and the vibration force waveform generated by the actuator. This phase lag may vary based upon the frequency at which the actuator is being driven. In addition, during actuator ramp-up and changes in frequency, phase, or amplitude there can be further discrepancy between the actuator driver waveform and the vibration force waveform. 
     The precision of the synchronization can be improved by characterizing the actuator dynamics and phase lag and incorporating this information into the controller so that the vibration forces are synchronized in the desired fashion. An additional method for improving the precision of synchronization adds a sensor that measures the motion of the moving mass or vibration force of each actuator, and uses closed-loop feedback control to correct the control signal in real-time. Many consumer products use MEMS sensors such as 3-axis accelerometers for motion sensing. It is possible to periodically characterize each of the actuators through an occasional calibration routine which drives each of the vibration actuators in turn with a known vibration test pattern and utilizes a motion sensing sensor to measure the resulting vibration. 
     Other embodiments use ERM actuators. An ERM employs a motor with an eccentric mass attached to a shaft that is connected to the motors&#39; rotor, which rotates relative to the motor&#39;s stator that is attached to a motor housing. A single ERM generates a rotating centripetal force in a plane that is normal to the axis of rotation of its shaft. Some embodiments use ERM pairs, which consist of two ERMs with similar characteristics that are mounted on a mounting platform with their axes of rotation aligned. In some embodiments, the eccentric rotating masses of the two ERMs are controlled to counter-rotate relative to each other at the same frequency. 
     When the ERMs in a pair are counter-rotating, superposition of the centripetal forces yields a sinusoidal vibrational force along a linear axis, which is defined as the force axis and is normal to the axis of rotation of the ERMs. A controller or mechanical coupling (such as gears or timing belts) can synchronize the speed of each ERM and the relative phase of rotation between the eccentric masses of the two ERMs in the pair. Synchronization of the phase controls the direction of the force axis of force generated by the pair. Accordingly, a synchronized ERM pair can generate a vibrational force along a linear axis, where the direction of the force axis is controlled by the relative phase of the ERMs. Increasing the rotational speed of a single ERM can increase the magnitude of its vibrational force. An ERM pair can also be controlled in a co-rotating mode where both eccentric masses rotate in the same direction, and thereby generate a combined centripetal vibrational force. These centripetal forces can be used to generate legacy effects that emulate existing or historic game controllers. Co-rotating ERMs can also be controlled to modify the magnitude of vibration force independently from the frequency of vibration. 
     One example described herein uses two ERM pairs mounted onto a mounting platform, with the axes of rotation all four ERMs aligned in parallel directions. One ERM pair is operated at twice the frequency of the second ERM pair. The phase of each ERM pair is controlled such that the directions of the force axis of both ERM Pairs are aligned parallel with each other. Furthermore, the timing of vibration is synchronized between the two ERM pairs such that in one direction the peaks of the vibration forces of both ERM pairs occur at the same time, and in the same direction and thus through constructive interference combine to increase the magnitude of the overall vibration force. In the first direction the peaks of the vibration forces of both ERM pairs occur at the same time, but in opposite directions and thus through destructive interference partially cancel each other out and thereby reduced the magnitude of the overall vibration force. With this approach an asymmetric vibration is generated since a larger peak force is generated in one direction and a lower peak force is generated in the opposite direction. With this embodiment the direction of asymmetric vibrations can be independently controlled in all 360 degrees of a plane. 
     ERMs can be synchronized to generate a wide range of vibration waveforms using Fourier synthesis, in a manner similar to the methods used with LRAs. The mass and eccentricity of the rotating mass of each ERM pair can be selected for the appropriate amplitude of force at a desired frequency. Alternatively, co-rotating pairs of ERMs can by synchronized to control the amplitude of vibration force, without the restriction of a fixed eccentricity of rotating masses. Multiple ERM pairs can be mounted on a mounting platform with the axes of rotations aligned. 
     In this configuration arbitrary waveform profiles can be approximated, and the direction of the force axis is controllable in the plane that is normal to the axes of rotation of the ERMs. In other embodiments, ERM pairs can be mounted in orientations where the axis of rotation of one ERM pair is not aligned in parallel with the axis of rotation of a second ERM pair. Furthermore, there are useful embodiments where there are multiple ERMs that are activated where no two ERMs have parallel axes of vibration nor orthogonal axes of vibration. An example of this would be four ERMs each mounted on a face of a tetrahedron. By activating various co-rotating and counter-rotating pairs of these ERMs, a variety of salient haptic effects can be achieved. With this approach 3D vibrations can be generated. Synchronized vibration can also be generated by combining both an ERM and an LRA actuator on a single mounting platform. 
     Asymmetric waveforms can be generated with as few as two Vibration Actuators. By using Fourier synthesis the fidelity of the waveform can be increased to an arbitrary precision by adding additional actuators that generate harmonics of the waveform. Thus, while an ERM pair or an LRA pair may be employed, one could also use ERM triads or ERM quads, LRA triads or LRA quads, and so forth. 
     Co-rotating pairs of ERMs can be controlled so the eccentric masses are 180 degrees out of phase. With this approach, the overall vibration force will be significantly reduced. In this configuration, a gyroscopic effect could be sensed and used to generate haptic effects. Generally a haptic gyroscopic effect requires a large rotational inertia. 
     The typical control method used to synchronize ERM actuators differs from that of LRAs. For synchronization it is necessary to control the frequency and phase of the ERMs. Most existing ERM actuators are controlled in on-off operation or low precision speed control, and are not synchronized in their phase. One method to control the frequency and phase of an ERM is to use a position sensor such as an encoder or potentiometer on the shaft of the motor. Feedback control can be used to achieve the desired synchronization, but the added cost of the position sensor is undesirable. 
     A component employed in accordance with aspects of this disclosure is a low cost frequency and phase sensor, which can be used to synchronize ERMs. In one embodiment, the frequency and phase sensor detects when the rotating mass passes by. The sensor may be an optical reflective sensor, optical pass-through sensor, hall-effect sensor, or other type of sensor. A microprocessor or other controller can track the time at which the rotating mass of each ERM passes by. The interval between successive times the mass rotates by the sensor is used to calculate the frequency of rotation. Multiple sensor measurements can be used to increase the resolution of the sensor. In addition, a state observer can be used to predict the position of the eccentric mass in intervals between sensor measurements. The relative time between sensors on different ERMs is used to measure the relative phase of each ERM. With the information of frequency and relative phase of each ERM, the controller can speed up or slow down each ERM to achieve the desired synchronization. Another method, as previously, mentioned, is to use a MEMS accelerometer that may have a primary use of motion sensing for a consumer device—yet also have a secondary use to characterize and calibrate the ERMs and LRAs. 
     According to another aspect, a motor driver may be used to control the voltage or current applied to each ERM. The motor driver can rotate the motor in a single direction, or could be a bidirectional motor controller such as an H-Bridge. When a bidirectional motor controller is used, reverse voltage can be applied to an ERM to slow it down more quickly and thus allow for faster adjustment to synchronization requirements. In addition, when a bidirectional motor controller is used an ERM Pair can be operated both as in counter-rotating directions and in co-rotating directions. The co-rotating directions can generate a large combined centripetal vibration force, and can be used to generate legacy vibration effects to simulate non-synchronized ERM vibrations. 
     Synchronization of multiple Vibration Actuators can be used to generate a wide range of vibration effects. These include direction effects, gyroscopic effect, amplitude effects, and 2D and 3D effects. 
     According to one aspect of the disclosure, a vibration device comprises a mounting platform and a plurality of actuators. Each of the plurality of actuators is configured to build up an amplitude of that actuator&#39;s force output over successive cycles of operation. Each of the plurality of actuators is attached to the mounting platform so the force outputs of the plurality of actuators are superimposed onto the mounting platform. The plurality of actuators is configured to simultaneously generate force waveforms, corresponding to the force outputs, for at least two different harmonics of a desired force output waveform such that each actuator generates a single harmonic of the desired output waveform. 
     In one example, each of the plurality of actuators is selected from the group consisting of a linear resonant actuator, an eccentric rotating mass actuator, a pivoting actuator, and a rocking actuator. In another example, two of the plurality of actuators comprise interleaved eccentric rotating masses arranged so the two actuators are individually controllable by a controller to simultaneously generate non-zero force outputs such that the superimposed force outputs of the two actuators sum to substantially zero force and substantially zero torque. 
     In one alternative, the at least two different harmonics include a first harmonic of the desired output waveform. In another alternative, the at least two different harmonics include a second harmonic of the desired output waveform. And in a third alternative, the at least two different harmonics include a third harmonic of the desired output waveform. 
     According to one example, the vibration device further comprises a controller coupled to the plurality of actuators to control an amplitude of the desired output waveform. According to another example, the vibration device is configured to generate haptic directional cues. 
     In one alternative, the vibration device is arranged in a handheld electronic device selected from the group consisting of a remote control, a game controller, and a watch, and the vibration device is configured to generate one or more haptic effects for the handheld electronic device. In this case, the game controller may be selected from the group consisting of a driving game controller and a motion game controller. 
     According to another aspect of the disclosure, a vibration device comprises a mounting platform, a plurality of eccentric rotating mass actuators, and a controller. Each of the plurality of eccentric rotating mass actuators is attached to the mounting platform. The controller is coupled to the plurality of eccentric rotating mass actuators to independently control a frequency and a phase of each eccentric rotating mass actuator. 
     In one alternative, the plurality of eccentric rotating mass actuators includes two pairs of eccentric rotating mass actuators. Each pair is aligned and attached to the mounting platform such that: the first pair of eccentric rotating mass actuators is configured to counter-rotate at a first rotational frequency f 1  to produce a first linear vibrating force, and the second pair of eccentric rotating mass actuators is configured to counter-rotate at a second rotational frequency f 2  to produce a second linear vibrating force, the second rotational frequency f 2  being an integer multiple of the first rotational frequency f 1 , so that a combined linear vibration force waveform on the mounting platform generated by operation of the two pairs of eccentric rotating mass actuators is asymmetric. 
     In one example, axes of revolution of each eccentric rotating mass actuator of the two pairs of eccentric rotating mass actuators are substantially parallel. In another example, the two pairs of eccentric rotating mass actuators are controlled by the controller to generate centripetal forces such that the first linear vibrating force, when the first pair is operating at the first rotational frequency f 1 , is substantially twice the second linear vibrating force when the second pair is operating at the second rotational frequency f 2 . 
     In a further example, the axes of revolution of each eccentric rotating mass actuator of the two pairs of eccentric rotating mass actuators are collinear. In yet another example, centripetal force vectors of each eccentric rotating mass actuator of the two pairs of eccentric rotating mass actuators are coplanar. And in another example, the controller is configured to generate haptic directional cues using the two pairs of eccentric rotating mass actuators. 
     According to another alternative, relative phases between the plurality of eccentric rotating mass actuators are controlled by the controller to cancel out centripetal forces generated by each of the eccentric rotating mass actuators; and torques generated by the centripetal forces by each of the eccentric rotating mass actuators cancel each other out. 
     In one example, a frequency, direction and relative phase between each of the eccentric rotating mass actuators are controlled by the controller to produce a combined vibration force along a predetermined axis. In this case, the combined vibration force along the predetermined axis may be asymmetric. 
     In another example, the plurality of eccentric rotating mass actuators includes a first eccentric rotating mass actuator and a second eccentric rotating mass actuator, the first and second eccentric rotating mass actuators having the same eccentricity; and the controller is configured to operate the first and second eccentric rotating mass actuators at the same frequency and the same phase relative to other eccentric rotating mass actuators in the plurality of eccentric rotating mass actuators. 
     In a further alternative, the plurality of eccentric rotating mass actuators comprises a first eccentric rotating mass actuator having a first axis of rotation, and a second eccentric rotating mass actuator having a second axis of rotation, the first and second axes being collinear. The first eccentric rotating mass actuator has a first eccentric mass with a center of eccentricity at a first position projected onto the first axis of rotation. The second eccentric rotating mass actuator has a second eccentric mass with a center of eccentricity at a second position projected onto the second axis of rotation. A distance between the first and second positions is substantially zero. 
     In another alternative, the plurality of eccentric rotating mass actuators comprises a first eccentric rotating mass actuator having a first axis of rotation, a second eccentric rotating mass actuator having a second axis of rotation, and third eccentric rotating mass actuator having a third axis of rotation, where the first, second and third axes being collinear. The first eccentric rotating mass actuator has a first eccentric mass with a center of eccentricity at a first position projected onto the first axis of rotation. The second eccentric rotating mass actuator has a second eccentric mass with a center of eccentricity at a second position projected onto the second axis of rotation. And the third eccentric rotating mass actuator has a third eccentric mass with a center of eccentricity at a third position projected onto the third axis of rotation. A distance between the first position and the second positions times the second eccentricity is equal to a distance between the first position and the third position times the third eccentricity. In this case, the eccentricity of the first eccentric mass may be equal to the second eccentricity plus the third eccentricity. 
     In another aspect of the disclosure, a vibration device comprising a mounting platform, a pair of linear resonant actuators arranged in parallel and attached to the mounting platform, and a controller. Each linear resonant actuator including a moveable mass. The controller is coupled to the pair of linear resonant actuators. The controller is configured to control a first one of the linear resonant actuators to impart a first sinusoidal vibration force of a first frequency f 1  onto the mounting platform, and to control a second one of the linear resonant actuators to impart a second sinusoidal vibration force of a second vibration frequency f 2  onto the mounting platform, the second frequency f 2  being an integer multiple of the first frequency f 1 . The controller is further configured to control amplitudes and phases of the first and second sinusoidal vibration forces to generate a combined vibration waveform that is asymmetric. 
     In one example, the first and second linear resonant actuators are each operable over a range of frequencies including resonant frequencies of the second linear resonant actuators. In this case, the resonant frequency of the second linear resonant actuator may be tuned to be an integer multiple of the resonant frequency of the first linear resonant actuator. 
     In another example, the vibration device is configured to generate haptic directional cues using the first and second linear resonant actuators. In a further example, the vibration device is configured to produce haptic effects that correspond to one or more computer-generated visual events. 
     In yet another example, the vibration device is arranged in a handheld controller, and the vibration device is configured to generate effects for the handheld controller. In a further example, the vibration device is arranged in a device wearable by a user, and the vibration device is configured to generate haptic effects for the wearable device. And in yet another example, n the vibration device is part of a navigation device for navigating a user from waypoint to waypoint. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a chart illustrating a number of different waveform types suitable for use with aspects of the present disclosure. 
         FIG. 2  illustrates a pair of vibration profiles having a phase difference. 
         FIG. 3  illustrates a pair of in-phase vibration profiles. 
         FIG. 4  illustrates a linear motion vibration actuator for use with aspects of the present disclosure. 
         FIGS. 5A-B  illustrate an example of a linear motion vibration actuator in accordance with aspects of the present disclosure. 
         FIGS. 6A-B  illustrate another example of a linear motion vibration actuator in accordance with aspects of the present disclosure. 
         FIGS. 7A-B  illustrate a further example of a linear motion vibration actuator in accordance with aspects of the present disclosure. 
         FIGS. 8A-B  illustrate yet another example of a linear motion vibration actuator in accordance with aspects of the present disclosure. 
         FIG. 9  illustrates a further example of a linear motion vibration actuator in accordance with aspects of the present disclosure. 
         FIG. 10  illustrates a vibration device in accordance with aspects of the present disclosure. 
         FIG. 11  illustrates the vibration device of  FIG. 10  for generating a counterclockwise rotation in accordance with aspects of the present disclosure. 
         FIG. 12  illustrates the vibration device of  FIG. 10  for generating a clockwise rotation in accordance with aspects of the present disclosure. 
         FIG. 13  illustrates the vibration device of  FIG. 10  for generating a change in the direction of force in accordance with aspects of the present disclosure. 
         FIG. 14  illustrates a vibration device employing non-orthogonal linear actuators in accordance with aspects of the present disclosure. 
         FIG. 15  illustrates a vibration device employing a set of linear actuators for generation of a three dimensional force vector in accordance with aspects of the present disclosure. 
         FIG. 16  illustrates a game controller in accordance with aspects of the present disclosure. 
         FIG. 17  illustrates a vibration device in accordance with aspects of the present disclosure. 
         FIG. 18  illustrates another vibration device in accordance with aspects of the present disclosure. 
         FIG. 19  illustrates a vibration device for generating a combined torque in accordance with aspects of the present disclosure. 
         FIG. 20  illustrates another vibration device for generating a combined torque in accordance with aspects of the present disclosure. 
         FIG. 21  illustrates a rotary vibration actuator with eccentric mass in accordance with aspects of the present disclosure. 
         FIG. 22  illustrates a vibration device with a pair of eccentric mass actuators in accordance with aspects of the present disclosure. 
         FIG. 23  illustrates synchronous vibration of eccentric mass actuators in accordance with aspects of the present disclosure. 
         FIGS. 24A-C  illustrate a pivoting actuator in accordance with aspects of the present disclosure. 
         FIGS. 25A-C  illustrate another pivoting actuator in accordance with aspects of the present disclosure. 
         FIG. 26  illustrates a pivoting actuator utilizing a pair of spring devices in accordance with aspects of the present disclosure. 
         FIGS. 27A-F  illustrate a further pivoting actuator in accordance with aspects of the present disclosure. 
         FIG. 28  illustrates a synchronized vibration system employing rotary actuators in accordance with aspects of the present disclosure. 
         FIGS. 29A-B  illustrate game controllers in accordance with aspects of the present disclosure. 
         FIG. 30  illustrates a rocking actuator in accordance with aspects of the present disclosure. 
         FIG. 31  illustrates a vibration system in accordance with aspects of the present disclosure. 
         FIG. 32  illustrates control of a vibration system in accordance with aspects of the present disclosure. 
         FIG. 33  illustrates control of a vibration system in accordance with aspects of the present disclosure. 
         FIG. 34  illustrates control of a vibration system in accordance with aspects of the present disclosure. 
         FIG. 35  illustrates a vibration system in accordance with aspects of the present disclosure. 
         FIGS. 36A-B  illustrate equation parameter and pattern selection processing in accordance with aspects of the present disclosure. 
         FIG. 37  illustrates a haptic interface system in accordance with aspects of the present disclosure. 
         FIG. 38  illustrates another haptic interface system in accordance with aspects of the present disclosure. 
         FIG. 39  illustrates control of vibration profiles in accordance with aspects of the present disclosure. 
         FIG. 40  illustrates a vibration actuator in accordance with aspects of the present disclosure. 
         FIG. 41  illustrates another vibration actuator in accordance with aspects of the present disclosure. 
         FIG. 42  illustrates a vibration device controller in accordance with aspects of the present disclosure. 
         FIG. 43  illustrates a vibration device with two linear resonant actuators for use with aspects of the disclosure. 
         FIG. 44  illustrates superposition of two synchronized sine waves with a phase offset that generates a combined waveform with asymmetry according to aspects of the disclosure. 
         FIG. 45  illustrates time steps within a vibration cycle of two linear resonant actuators generating an asymmetric waveform according to aspects of the disclosure. 
         FIG. 46  illustrates two linear resonant actuators directly attached to one another for use with aspects of the disclosure. 
         FIG. 47  illustrates an alternative example of two linear resonant actuators attached in line with one another for use with aspects of the disclosure. 
         FIG. 48  illustrates a vibration device that uses a slider-crank linkage for use with aspects of the disclosure. 
         FIG. 49  illustrates a vibration device with n LRAs for use with aspects of the disclosure. 
         FIG. 50  illustrates an asymmetric pulse train according to aspects of the disclosure. 
         FIG. 51  illustrates a pulse train with zero DC according to aspects of the disclosure. 
         FIG. 52  is a flow diagram illustrating a process for maximizing asymmetry according to aspects of the disclosure. 
         FIG. 53  illustrates an example of waveform asymmetry according to aspects of the disclosure. 
         FIG. 54  illustrates another example of waveform asymmetry according to aspects of the disclosure. 
         FIG. 55  illustrates a further example of waveform asymmetry according to aspects of the disclosure. 
         FIG. 56  illustrates synchronized triangular waveforms according to aspects of the disclosure. 
         FIG. 57  illustrates a vibration device that can generate asymmetric torques according to aspects of the disclosure. 
         FIG. 58  illustrates a controller for General Synchronized Vibration of a pair of linear force actuators according to aspects of the disclosure. 
         FIG. 59  illustrates a linear force actuator with a sensor that detects when a moving mass passes a midpoint position according to aspects of the disclosure. 
         FIG. 60  illustrates a sensor attached to a mounting platform according to aspects of the disclosure. 
         FIG. 61  illustrates a vibration device controller that uses sensor measurements to update a commanded amplitude, phase and/or frequency according to aspects of the disclosure. 
         FIG. 62  illustrates a vibration device that includes two orthogonal sets of LRAs according to aspects of the disclosure. 
         FIG. 63  illustrates a vibration device that includes two non-orthogonal sets of LRAs according to aspects of the disclosure. 
         FIG. 64  illustrates an ERM for use with aspects of the disclosure. 
         FIG. 65  illustrates a vibration device using an arbitrary number of ERMs according to aspects of the disclosure. 
         FIG. 66  illustrates a vibration device having 4 ERMs for use with aspects of the disclosure. 
         FIG. 67  illustrates time steps within a vibration cycle of ERMs generating an asymmetric waveform according to aspects of the disclosure. 
         FIG. 68  illustrates an example vibration device with a plurality of ERM pairs. 
         FIG. 69  illustrates a vibration device with four vertically stacked ERMs in one example used according to aspects of the disclosure. 
         FIG. 70  illustrates time steps of an asymmetric waveform for a vibration device with four ERMs that are vertically stacked, according to aspects of the disclosure. 
         FIG. 71  illustrates a vibration device with two ERMs that rotate in the same direction. 
         FIG. 72  illustrates a vibration device with four co-rotating pairs of ERMs according to aspects of the disclosure. 
         FIGS. 73A-B  illustrate vibration devices with two ERMs mounted in different arrangements according to aspects of the disclosure. 
         FIG. 74  illustrates an eccentric mass configured for use as a reaction wheel according to aspects of the disclosure. 
         FIG. 75  illustrates an ERM pair with interleaved masses according to aspects of the disclosure. 
         FIGS. 76A-B  illustrate example configurations having three ERMs for use with aspects of the disclosure. 
         FIG. 77  illustrates another configuration with three ERMs arranged in a row. 
         FIG. 78  illustrates an ERM with a sensor for use with aspects of the disclosure. 
         FIG. 79  illustrates an ERM with a reflective optical sensor for use with aspects of the disclosure. 
         FIG. 80  illustrates an ERM with a line of sight sensor for use with aspects of the disclosure. 
         FIG. 81  illustrates an ERM with a Hall effect sensor for use with aspects of the disclosure. 
         FIG. 82  illustrates a vibration device with four ERMs arranged in a row for use with aspects of the disclosure. 
         FIG. 83  illustrates time steps of a waveform with cancellation of forces according to aspects of the disclosure. 
         FIG. 84  illustrates a vibration device with two pairs of ERMs that share the same center. 
         FIGS. 85A-B  illustrate an ERM pair with interleaved masses having varying thickness according to aspects of the disclosure. 
         FIGS. 86A-C  illustrate an ERM pair with interleaved masses having support bearing according to aspects of the disclosure. 
         FIG. 87  illustrates haptic feedback within a system having a visual display according to aspects of the disclosure. 
         FIG. 88  illustrates another example of haptic feedback within a system having a visual display according to aspects of the disclosure. 
         FIG. 89  illustrates a vibration device with sensor feedback according to aspects of the disclosure. 
         FIG. 90  illustrates a locomotion device for use with aspects of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The foregoing aspects, features and advantages of the present disclosure will be further appreciated when considered with reference to the following description of preferred embodiments and accompanying drawings, wherein like reference numerals represent like elements. 
     As used herein, an actuator is a device that can generate mechanical motion or force. Actuators can convert a source of energy into mechanical motion or force. The source of energy can be electrical, pneumatic, hydraulic, or another source. Examples of actuators include rotary and linear motors. Examples of electric actuators include DC, AC, and stepper motors. 
     The term “direction” includes the orientation of an axis, also referred to as vector direction. A vector aligned with a specific direction can be either in the positive direction along the axis or the negative direction along the axis. As used herein, the term direction may distinguish between all angles in a circle, such as 0 to 360 degrees. And vibration control may distinguish between positive and negative directions along a single axis. Furthermore, the term “controller” is used herein in some situations to reference to game controller, and in other situations to a real-time controller of actuators, such as a microprocessor or an ASIC. 
     In this disclosure, the term “General Synchronized Vibration” refers to control of the timing, and in some cases also control of amplitude, of multiple vibration forces, torques, or forces and torques. The sources of these vibration forces and torques can be electromagnetic, electrostatic, magnetic, spring forces, inertial forces such as centripetal forces, piezoelectric, pneumatic, hydraulic, or other force and torque sources. The sources of these vibration forces and torques can include those described in the text “Engineering Haptic Devices: A Beginner&#39;s Guide for Engineers” by Thorsten A. Kern, © 2009 (the entire disclosure of which is hereby expressly incorporated by reference herein). These vibration forces and torques can be generated from separate Vibration Actuators or from actuators that generate multiple force, torques, or forces and torques. In General Synchronized Vibration the forces, torques, or forces and torques are vectorially combined so that they generate a combined force, torque, or force and torque onto an object. The vector combination of force and torque vectors is also referred to as superposition. General Synchronized Vibration results in a combined vibration force, a combined vibration torque, or a combined vibration force and vibration torque onto an object. A force applied onto an object can also apply a torque onto that object. Accordingly, the references in this document to force also apply to force and torque unless explicitly described otherwise. 
     In the event that there is a difference in the usage of terminology between the instant application and any wholly included reference identified herein, the usage of the differing term definitions will be governed by the use in the present disclosure. 
     A vibration (or vibratory) actuator can impart repeated forces onto an object. These repeated forces can repeat a similar force profile over time during each repetition. Examples include rotary motors with eccentric masses, and linear actuators which move masses back and forth. These actuators can be DC, AC, stepper, or other types of actuators. A vibration actuator can repeat a similar force profile (waveform) in each cycle, or there can be variations in force profiles between cycles. Variations between cycles can be in amplitude, frequency, phase, and profile shape. 
     When a force is generated in a repeated cycle it can generate a vibratory force. The profile (also referred to as a waveform) of a repeated force cycle can be in a sinusoidal shape, triangular wave, a square wave, or other repeated profile as shown in  FIG. 1 . The frequency of vibration describes how frequently a vibration cycle is repeated. A frequency of vibration, f, is defined as the number of vibrations per unit time, and often is given in Hertz whose units are cycles per second. The period of vibration, T, is the duration of each cycle in units of time. The mathematical relationship between frequency and period of vibration is given by the following equation:
 
 f= 1 /T   (1)
 
     A vibration force, F, is in a repeated cycle when
 
 F ( t+T )= F ( t )  (2)
 
where T is the period of vibration and t is time.
 
     For purposes of vibration devices it is sufficient for the period of vibration to be approximate, and therefore a vibration is considered to be in a repeated cycle when:
 
 F ( t+T )≈ F ( t )  (3)
 
     One vibration waveform is a sinusoidal waveform, where the vibration force can be given by:
 
 F ( t )= A  sin(ω t +φ)  (4)
 
     Here, F(t) is force as a function of time. A is the maximum amplitude of force. ω is the frequency of vibration in radians per second (the frequency in Hertz is f=ω/(2π)). And φ is the phase of vibration in radians. When ωt=2π the force profile repeats itself. 
     A vibration actuator may impart repeated forces onto an object. Due to the dynamics of an actuator, a single actuator can impart forces at multiple frequencies at the same time. However, for the purposes of analyzing vibrations and describing vibration devices herein, the primary frequency of an actuator&#39;s motion means the frequency having the largest component of kinetic energy in it. 
     The period of vibration can be defined by the time elapsed between the beginning of one vibration cycle and beginning of the next cycle. Thus to identify the period of vibration it is useful to identify the beginning of a cycle. One method for defining the beginning of cycle is to define the beginning of the cycle as the point with maximum amplitude in the profile.  FIG. 1  is an amplitude versus time chart  10  showing the vibration profiles of a sine wave  12 , a triangle wave  14 , an arbitrarily shaped profile  16 , and a square wave  18 . The period for each of these profiles is designated by T. 
     The sine wave  12 , triangle wave  14 , and arbitrary profile wave  16  all have a unique point of maximum amplitude during each repeated cycle, and this point of maximum amplitude is used to define the beginning of the cycle. The square wave  18  does not have a unique point of maximum amplitude within a cycle; in such cases a repeated point on the profile can be selected to designate the beginning of the cycle. In  FIG. 1 , the point at which the square wave  18  transitions from a low value to a high value is designated at the beginning point of the cycle, and used use to define the period of the repeated profile. Thus, any profile that can be represented as repeated cycles can represent a vibration. 
     A frequency of vibration can also be identified when the shape of signal does not consist of exactly repeated profiles. Variations in amplitude of the cycle and small changes in the shape of a cycles profile still allow one to identify a unique point that designates the beginning of the cycle. As long as a repeated point in the profile can be identified, then the beginning of each cycle, a vibration period, and vibration frequency can be determined. 
     The phase of vibration defines the timing of the beginning of a cycle of vibration. A phase difference between two vibration waveforms is defined as the difference between the beginning of a vibration cycle in one waveform and the beginning of a vibration cycle in the other waveform. If there is a nonzero difference in the phase of vibration between two profiles, then the beginning of the cycles do not coincide in time.  FIG. 2  is an amplitude versus time chart  20  showing two vibration profiles,  22  and  24 , with a phase difference Δ between them. The phase difference Δ can be given in units of time, such as shown in  FIG. 2 . Alternatively, the phase of vibration can also be given in radians for sinusoidal vibrations. When the phase difference Δ between two waveforms is zero, then the two waveforms are considered to be in-phase, as shown in the amplitude versus time chart  30  of  FIG. 3 . 
     As long as it is possible to identify the beginning of a cycle it is possible to identify a phase of vibration, even when the amplitude and frequency of vibration change between cycles of vibration. 
     One implementation of synchronized vibration is a vibration force formed by the superposition of two or more vibration waveforms where each of the waveforms include peaks that coincide in time with the peaks of the other waveforms on a regularly repeating basis. In a preferred embodiment, each of the waveforms would have the same frequency and a specified phase difference between them. Superposition can preferably be the vector sum of forces, torque, or forces and torque. Typically, the sources of these vibration waveforms are different vibration actuators. Often in synchronous vibration the waveforms have a zero phase difference between them, and thus the vibration waveforms are in-phase and in synchronous vibration. As used herein, specified phase difference may range between and including 0° and 360°. In some embodiments, the specified phase difference is 0° or 180°. In synchronized vibration, the various vibration waveforms can have different amplitudes.  FIG. 3  illustrates two vibration waveforms of triangular profile that are synchronized. Both of these waveforms have the same frequency, they have different amplitudes, and the waveforms are in-phase. The maximum amplitude of both waveforms in  FIG. 3  occurs at the same time. 
     Typically, synchronized vibration profiles will have similar shaped profiles. However, vibration actuators with different shaped vibration profiles can also be vibrated synchronously by matching frequency of vibration and specifying the phase difference between the waveforms. The matching of phase and frequency of vibration can be done approximately and still result in synchronized vibration. 
     Synchronized vibration can be generated by adding two vibration profiles together, where the amplitude of the second vibration profile is a multiple of the amplitude of the first vibration profile. This multiplying factor can be either positive or negative. 
     If there are two or more vibrating actuators where the peak amplitude of force of each vibrating actuator occurs repeatedly at approximately the same time, then these actuators are in-phase and in synchronous vibration. The peak amplitude of force can be either in the positive or negative direction of the vibration actuators&#39; or vibration device&#39;s coordinate system. Thus if a positive peak amplitude from one actuator occurs at approximately the same time as the negative peak amplitude of another actuator, then these actuators are in-phase and are in synchronous vibration. 
     An exemplary linear motion vibration actuator  100  is shown in  FIG. 4 . As shown, the linear motion vibration actuator  100  contains a moving mass  102  and a base  104 . The moving mass  102  moves relative to the base  104  in a back and forth linear motion. Force can be applied from the base  104  to the moving mass  102  and in a similar fashion from the moving mass  102  onto the base  104 . The force transfer can occur, for instance, via magnetic forces, spring forces, and/or lead screw forces. Examples of linear actuators suitable for use in accordance with the present disclosure are described in U.S. Pat. Nos. 5,136,194 and 6,236,125, and in U.S. patent application Ser. No. 11/325,036, entitled “Vibration Device,” the entire disclosures of which are hereby incorporated by reference herein. 
     As the moving mass  102  in the linear motion vibration actuator  100  moves back and forth, forces are generated between the moving mass  102  and the base  104 . These forces can be transmitted through the base  104  of the actuator  100  to an object that the actuator is mounted to (not shown). The moving mass  102  may also be attached to an object, such as a handle (not shown), that is external to the actuator  100 , and may transmit forces directly to an object external to the actuator  100 . 
     The forces in the linear motion vibration actuator  100  may be magnetic forces, such as with a voice coil. The moving mass  102  may contain, for instance, a permanent magnet, electromagnet, ferromagnetic material, or any combination of these. The base  104  may contain, for instance, a permanent magnet, an electromagnet, ferromagnetic material, or any combination of these. Magnetic forces may be generated between base  104  and the moving magnet that generate acceleration and motion of the moving mass  104 . A force in the linear motion vibration actuator  100  generated with an electromagnet can be modulated by controlling the current flowing through the electromagnet. 
     One embodiment of linear motion vibration actuator  100  in accordance with the present disclosure is shown in  FIGS. 5A-B  as linear motion vibration actuator  110 . Actuator  110  preferably contains a moving mass  112  that comprises an electromagnet, as well as a permanent magnet  116  attached to the base  114 . The motion of the moving mass  112  is along the x axis as shown in the side view in  FIG. 5A . The magnetization polarity of the permanent magnet  116  is along the x axis as shown by the North and South poles on the permanent magnet  116 . The electromagnet is preferably configured as a coil wound about the x axis. As shown in the end view of  FIG. 5B , in the present embodiment the shape of the electromagnet is desirably cylindrical and the shape of the permanent magnet  116  is desirably tubular, although the electromagnet and the permanent magnet  116  may have any other configuration. In this embodiment both the electromagnet and the permanent magnet  116  may have ferromagnetic material placed adjacent to them to increase the force output of the actuator  110 . 
     In this embodiment, the force in the actuator  110  can be modulated by controlling the current in the electromagnet. When the current in the electromagnet flows in one direction, then the magnetic force will push the moving mass  112  towards one side of the actuator. Conversely when the current in the electromagnet flows in the other direction, then the moving mass  112  will be pushed to the other side of the actuator  110 . Increasing the amount of current in the electromagnet will increase the amount of force applied onto the moving mass  112 . 
     Another embodiment of the linear motion vibration actuator  100  in accordance with the present disclosure is shown in  FIGS. 6A-B . Here, linear motion vibration actuator  120  preferably contains a moving mass  122  that comprises a permanent magnet, as well as an electromagnet magnet  126  attached to base  124 . The motion of the moving mass  122  is along the x axis as shown in the side view in  FIG. 6A . The magnetization polarity of the permanent magnet is along the x axis as shown by the North and South poles on the permanent magnet. The electromagnet  126  is preferably a coil wound about the x axis. As shown in the end view of  FIG. 6B , in this embodiment the shape of the electromagnet  124  is tubular and the shape of the permanent magnet is cylindrical. 
     In this embodiment both the electromagnet  124  and the permanent magnet of the moving mass  122  may have ferromagnetic material placed adjacent to them to increase the force output of the actuator  120 . The force in the actuator  120  can be modulated by controlling the current in the electromagnet  124 . When the current in the electromagnet  124  flows in one direction, then the magnetic force will push the moving mass  122  towards one side of the actuator  120 . Conversely when the current in the electromagnet flows in the other direction, then the moving mass  122  will be pushed to the other side of the actuator  120 . Increasing the amount of current in the electromagnet will increase the amount of force applied onto the moving mass  122 . 
     Another embodiment of the linear motion vibration actuator  100  in accordance with aspects of the present disclosure is shown in  FIGS. 7A-B , which is similar to the embodiment shown in  FIGS. 6A-B . Here, actuator  130  includes a moving mass  132  and a base  134 . The moving mass  132  preferably comprises a permanent magnet. An electromagnet  136  at least partly surrounds the moving mass  132 . The electromagnet  136  is desirably connected to the base  134 . Unlike the actuator  120 , the actuator  130  in this embodiment preferably includes one or more springs  138  that are attached to the base  134  and to the moving magnet  132  at either end, as shown in the side view of  FIG. 7A . The springs  138  are operable to generate forces in a direction that returns the moving mass  132  to a center position, for instance midway between either end of the electromagnet  136 . 
     The springs  138  function to keep the moving mass  132  close to the center position when the actuator power is off, and to provide a restoring force when the moving mass  132  is at one end of travel of the actuator  130 . The stiffness of the springs  138  can be selected so that the natural frequency of the actuator  130  increases the amplitude of vibration at desired natural frequencies. This spring effect can be generated from a single spring, from a nonlinear spring, from extension springs, as well as compression springs. A number of such spring configurations which may be employed with the present disclosure are described in the aforementioned U.S. patent application Ser. No. 11/325,036. 
     Another embodiment of the linear motion vibration actuator  100  according to aspects of the present disclosure is shown in  FIGS. 8A-B . This embodiment is similar to the embodiments shown in  FIGS. 6A-B  and  7 -B in that actuator  140  includes a moving mass  142  including a permanent magnet, a base  144 , and an electromagnet  146  coupled to the base  144  and at least partly surrounding the moving mass  142 . The electromagnet  146  may be, e.g., rigidly or semi-rigidly coupled such that a vibration force is transmitted from the actuator  140  to the base  144 , for instance to enable a user to perceive the vibration force. In this embodiment, a pair of permanent magnets  148  is attached to the base and are in operative relation to the moving magnet  142  at either end as shown in the side view of  FIG. 8A . The permanent magnets  148  have poles, as shown by the N and S in  FIG. 8A , which are configured to repel the moving mass  142  and to generate forces in a direction that returns the moving mass  142  to a center position. The permanent magnets  148  function to keep the moving mass  142  close to a center position when the actuator power is off, and to provide a restoring force when the moving mass  142  is at one end of travel of the actuator  140 . 
     The size of the permanent magnets  148  attached to the base  144  can be selected so that the natural frequency of the actuator  140  increases the amplitude of vibration at desired natural frequencies. The actuator  140  may be controlled so that one or more natural frequencies are selected during different modes or times of operation. Use of repulsive magnetic forces as shown in  FIG. 8A  to generate centering forces on the moving permanent magnet of the moving mass  142  can provide lower friction than use of springs  138  as shown in  FIG. 7A , and thus can generate increased actuator efficiency and smoothness. A number of configurations showing use of permanent magnets to center a moving mass, which are suitable for use in the present disclosure, are described in the aforementioned “Vibration Device” patent application. 
     Alternative embodiments of linear motion vibration actuators that may also be utilized with the present disclosure include both springs and magnets, either alone or in combination, that return a moving mass towards the center of range of motion of the actuator. 
     A further alternative embodiment of the linear motion vibration actuator  100  in accordance with the present disclosure is shown in  FIG. 9 . This embodiment comprises actuator  150 , which is similar to a solenoid in that it has a ferromagnetic moving plunger  152  for moving relative to a base  154 . The plunger  152  is pulled into an electromagnetic coil  156  when current flows through the coil  156 . The coil  156  is coupled to the base  154 . A ferromagnetic end piece  158  can be located within or at the end of the coil  156  to increase the force output of the actuator  150 . A spring device  160  may be positioned opposite the end piece  158 . The spring device  160  is preferably employed to retract the plunger  152  out of the coil  156 . As shown in  FIG. 9 , both an end of the coil  156  and an end of the spring  160  are desirably fixed to the base  154  of the actuator  150 . The coil  156  and the spring  160  may be fixed to a single base at different sections thereon, or may be fixed to separate base elements that are coupled together. The current in the coil  156  can be turned on and off to generate a vibration force. 
     A preferred embodiment of a vibration device  200  according to the present disclosure is shown in  FIG. 10 . In this embodiment, the vibration device  200  preferably includes two linear motion vibration actuators mounted on to it, namely actuator  202  and actuator  204 . The actuator  202  includes a moving mass  206  and the actuator  204  includes a moving mass  208 . The vibration actuators  202 ,  204  are attached to the vibration device  200  in a manner that transmits the force from the vibration actuators  202 ,  204  to the vibration device  200 . Preferably the vibration device  200  has an enclosure or base (not shown) to which the vibration actuators  202 ,  204  are connected. 
     The vibration actuators  202 ,  204  are desirably attached in a relatively rigid fashion to the vibration device enclosure or base. Rigid attachment provides a common base to the vibration device  200 , upon which forces from both vibration actuators  202 ,  204  are applied. In this embodiment, the two actuators  202 ,  204  are mounted at approximately right angles to each other. The force generated by actuator  202  is shown as force vector F 1 , and the force vector from actuator  204  is shown as F 2 . As expressed herein, vectors and matrices are designated by bold font and scalars are designated without bolding. The combined force generated by the vibration device  200  is the vector sum of the vibration forces from both of the actuators  202 ,  204 , and is shown in  FIG. 10  as vector F combined . 
     The combined force, F combined , applied by the vibration actuators  202  and  204  onto the vibration device  200  is a superposition of the vibration forces from each actuator, and is a function of time, t. The force vector can F combined (t) is given by the vector equation:
 
 F   combined ( t )= F   1 ( t )+ F   2 ( t )  (5)
 
where F 1 (t) is the force vector from actuator  202  as a function of time, and F 2 (t) is the force vector from actuator  204  as a function of time.
 
     Both actuators  202 ,  204  can be operated in a vibratory fashion. For the case of a sine wave vibration, the actuator forces can be given by:
 
 F   1 ( t )= a   1   A   1  sin(ω 1   t+φ   1 )  (6)
 
and
 
 F   2 ( t )= a   2   A   2  sin(ω 2   t+φ   2 )  (7)
 
respectively, where A 1  and A 2  are the respective amplitudes of vibration, a 1  and a 2  are the unit vectors corresponding to the respective directions of vibration, ω 1  and ω 2  are the respective frequencies of vibration, φ 1  and φ 2  are the respective phase angles, and t is time. Other profile vibrations including square waves, triangle waves, and other profiles can also be implemented with each actuator.
 
     In the example shown in  FIG. 10 , actuator  202  is aligned with the y axis, and thus the unit vector a 1  is represented by: 
                     a   1     =     [         0           1         ]             (   8   )               
and the unit vector a 2  aligned with the x axis and is represented by:
 
                     a   2     =     [         1           0         ]             (   9   )               
The combined force vector, F combined , is given by the superposition of forces form the actuators  202  and  204 , and thus is given by:
 
 F   combined ( t )= a   1   A   1  sin(ω 1   t+φ   1 )+ a   2   A   2  sin(ω 2   t+φ   2 )  (10)
 
     It is possible to vibrate actuators  202  and  204  shown in  FIG. 10  in a manner that is in-phase and in synchronous vibration. Under such vibration, there will be a single vibration frequency, ω and a single phase φ Accordingly, F combined  can be given by:
 
 F   combined ( t )=[ a   1   A   1   +a   2   A   2 ] sin(ω t +φ)  (11)
 
     With such in-phase and synchronous vibration the vibration is synchronized, then the peak forces from both linear motion vibration actuators will occur at the same instances during each cycle of vibration. The net direction of vibration force is the vector combination of [a 1 A 1 +a 2 A 2 ]. Thus, in synchronized vibration and in-phase vibration, the vibration device generates a vibration force at a specified frequency in a specified direction that results from the vector combination of forces from the direction and magnitude of each of the actuators in the device. It is possible to control the magnitude of vibration in each linear motion vibration actuator, and thereby control the net direction of vibration of F combined . 
     In a preferred example, the vibration frequency, w, phase φ, and waveform of each actuator are substantially identical. For instance, ω 2  may be set to be substantially equal to ω 1  and φ 2  may be set to be substantially equal to φ 1 . By way of example only, ω 2  may be set to within 10% of the value of ω 1 , more preferably to within 5% of the value of ω 1 . Similarly, by way of example only, φ 2  may be set to within 10% of the value of ω 1 , more preferably to within 5% of the value of φ 1 . In another example, the frequencies and/or phases may be set exactly equal to one another. Alternatively, the frequencies, phases, and/or waveforms of each actuator may be set so that a user would not be able to notice the difference in frequency, phase or waveform. In a further alternative, if the vibration device is used in a haptic application to generate force sensations on the user, small variations may occur which may not be detected by the user or which cannot be significantly felt by the user. In other instances, force sensations in a haptic application or in a vibratory feeder application may vary minutely so that user performance in the haptic application or performance of the vibratory feeder is not significantly changed. 
     It is also possible to apply equation 11 to a vibration profile/waveform of arbitrary shape. Here, waveform p(t) may be used to represent the waveform shape over time t. A period of vibration may be represented by p(t)=p(t+nT), where n=1, 2, 3, etc. and T is the period of vibration. In this case, an arbitrarily shaped synchronized vibration profile may be represented as:
 
 F   combined ( t )=[ a   1 ( t ) A   1 ( t )+ a   2 ( t ) A   2 ( t )] p ( t )  (11.1)
 
When the direction of vibration force for each actuator is substantially constant relative to a base member, the arbitrarily shaped synchronized vibration profile may be represented as:
 
 F   combined ( t )=[ a   1   A   1 ( t )+ a   2   A   2 ( t )] p ( t )  (11.2)
 
     To illustrate how the direction of F combined  can be controlled, the peak magnitudes, A 1  and A 2 , are represented in  FIGS. 10 and 11  by the location of the moving masses  206  and  208  within each of the actuators  202  and  204 , respectively. In  FIG. 10 , both actuator  202  and actuator  204  are desirably vibrated at the same amplitude, and the corresponding F combined  is at approximately a 45 degree angle between the actuators  202 ,  204 . 
     By varying the magnitude of the vibration force in the actuators  202 ,  204 , it becomes possible to control the direction of vibration of the combined force effect. In  FIG. 11 , the actuator  202  is vibrating at peak amplitude as illustrated by the peak position of moving mass  206  at the end of travel limits of actuator  202 . However, actuator  204  is vibrating at a lower peak amplitude, as illustrated by the peak position of moving mass  208  closer to the middle of travel limits of actuator  204 . The lower peak force is also illustrated in  FIG. 11  by the shorter length vector for F 2 . The direction of the combined force, F combined , is the result of vector addition of F 1  and F 2 , and for vibrations illustrated in  FIG. 11  is rotated counterclockwise relative to the direction shown in  FIG. 10 . 
     In a similar fashion, the direction of combined force can be rotated in the clockwise direction as shown in  FIG. 12 . The vibration case illustrated in  FIG. 12  shows the peak amplitude of vibration of actuator  202  reduced relative to that shown in  FIG. 10 , while the peak amplitude of actuator  204  remains high. In this case, the vector addition of F 1  and F 2  results in a clockwise rotation of F combined  in  FIG. 12  relative to the direction shown in  FIG. 10 . 
     It is also possible to change the direction of F combined  to an adjacent quadrant. As shown in  FIG. 13 , the sign of the F 2  has changed be in the direction of the negative x axis, relative to the positive x direction that shown in  FIG. 10 . The change in sign of F 2  can be achieved by changing the sign of A 2  in equation 11 above. It should be noted that one could achieve a similar representation of the combined force equation by defining actuator  204  vibration as at 180 degrees out of phase of actuator  202 . However, changing the sign on the actuators vibration amplitude maintains the form of equation of synchronous vibration shown in equation 11. Thus, vibration that can be represented as 180 degrees out of phase can also be represented as in-phase vibration but with a negative amplitude of vibration. 
     An alternative embodiment of a vibration device in accordance with the present disclosure is shown in  FIG. 14 . Here, vibration device  210  includes a first actuator  212  and a second actuator  214 , having respective moving masses  216  and  218 .  FIG. 14  represents a two dimensional embodiment where two linear motion vibration actuators  212 ,  214  are aligned with an xy plane. In this embodiment, it is not necessary for the actuators  212 ,  214  to be orthogonal to each other. A 1  and A 2  are respectively the amplitudes of vibration of actuators  212  and  214 , while a 1  and a 2  are respectively the unit vectors specifying the direction of vibration of actuators  212  and  214 . 
     The unit vector a 1  is given by: 
                     a   1     =     [           cos   ⁡     (   α   )                 sin   ⁡     (   α   )             ]             (   12   )               
where the angle α describes the orientation of actuator  1  relative to the x axis as shown in  FIG. 14 . The unit vector a 2  is given by:
 
                     a   2     =     [           cos   ⁡     (   β   )                 sin   ⁡     (   β   )             ]             (   13   )               
where the angle β describes the orientation of actuator  2  relative to the x axis as shown in  FIG. 14 .
 
     For a given vibration waveform the maximum magnitude of force vectors, F 1, max  and F 2, max , from actuators  212  and  214  in  FIG. 14  can be given by equations:
 
 F   1,max   =A   1   a   1   (14)
 
 F   2,max   =A   2   a   2   (15)
 
     When actuators  212  and  214  are vibrated synchronously and in-phase (e.g. with the same frequency and with zero phase difference), then the maximum force amplitude occurs at the same time. Thus the maximum combined force vector, F combined, max , is given though superposition of the force vectors, and is given by:
 
 F   combined,max   =F   1,max   +F   2,max   (16)
 
     A matrix of actuator directions, D L , can be created where each of its columns is a unit vector that corresponds to the direction of vibration of a linear motion vibration actuator in a vibration device. For a vibration device with two linear motion vibration actuators, such as the one shown in  FIG. 14 , the matrix D L  is given by:
 
 D   L   =[a   1   |a   2 ]  (17)
 
where a 1  and a 2  are column vectors.
 
     A matrix representation of the combined force is given by: 
                     F     combined   ,   max       =       D   L     ⁡     [           A   1               A   2           ]               (   18   )               
where A 1  and A 2  are scalars. For the case of vibration in a plane, the vectors a 1  and a 2  will be 2×1 vectors and the matrix D L  will be 2×2.
 
     When the direction matrix, D L , is invertible then the amplitude of vibration in the individual actuators that corresponds to a desired combined force vector, F combined , is given by: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                       
                       
                         
                           
                             A 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       D 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     Fcombined 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     When the actuators are aligned orthogonally, then the direction matrix, D L , is orthonormal and its inverse is given by its transpose as shown below:
 
 D   −1   =D   T   (20)
 
     When the direction matrix, D L , in not invertible because there are more vibration actuators than directions of force being controlled, then a pseudo inverse of matrix D L  can be used. For example, if there are 3 vibration actuators in the xy plane, and the control objective is only to control a two dimensional force, the D L  matrix is given by:
 
 D   L   =[a   1   |a   2   |a   3 ]  (21).
 
where a 1 , a 2 , and a 3  are 2×1 column vectors.
 
     The pseudo inverse is described in “ Introduction to Linear Algebra”,  3rd Edition by Gilbert Strang, published in 2003 by Wellesley-Cambridge Press, the entire disclosure of which is incorporated by reference herein. 
     One method for calculating a pseudo inverse, D L   + , is given by:
 
 D   L   +   =D   L   T ( D   L   D   L   T ) −1   (22)
 
     In such a case the amplitude of vibration for each actuator can be given by: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                       
                       
                         
                           
                             A 
                             2 
                           
                         
                       
                       
                         
                           
                             A 
                             3 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       D 
                       L 
                       + 
                     
                     ⁢ 
                     Fcombined 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     It is possible to specify the combined force vector, F combined , in terms of a direction of vibration and amplitude. For a two dimensional embodiment the combined amplitude of vibration can be specified by the scalar A combined , and the direction of vibration can be specified by an angle, theta, as shown in  FIG. 14 . In this two dimensional embodiment F combined  can be given by: 
     
       
         
           
             
               
                 
                   
                     F 
                     combined 
                   
                   = 
                   
                     
                       A 
                       combined 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 theta 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 theta 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Thus, it can be seen that the amplitudes of vibration, A 1  and A 2 , can be represented in terms of the direction of vibration, theta, combined amplitude of vibration, A combined , and direction matrix, D L , as given by: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             1 
                           
                         
                       
                       
                         
                           
                             A 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       D 
                       L 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       Acombined 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   theta 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   theta 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Equation 25 provides the scalar magnitude of A 1  and A 2 . When the sign of A 1  is different than the sign of A 2  then vibration waveform can be generated directly using the results of Eq. Avec. Alternatively, the waveform can be generated using absolute values of A 1  and A 2  but with one waveform completely out of phase with the other waveform. A sine wave is defined to be completely out of phase when it is 180 degrees out of phase. General waveforms are defined to be completely out of phase when the maximum positive amplitude of vibration of one waveform concedes with the maximum negative amplitude of the other waveform. A depiction of two actuators vibrating completely out of phase is shown in  FIG. 13 . Two actuators vibrating completely out of phase are also considered to be in synchronized vibration. 
     It is also possible to specify the combined direction of vibration in terms of a unit vector, a combined , as shown in  FIG. 14 . The vector F combined  can be given by:
 
 F   combined   =A   combined   ×a   combined   (26)
 
     Another configuration according to aspects of the present disclosure is a three dimensional configuration, where there are at least 3 linear motion vibration actuators as shown in  FIG. 15 . 
     In the vibration device  220  of  FIG. 15 , actuators  222 ,  224  and  226  each include a moving mass  228 ,  230  and  232 , respectively. The actuators  222 ,  224  and  226  are preferably orthogonal to each other and aligned with an xyz coordinate system. In an alternative three dimensional embodiment the actuators are not necessarily orthogonal to each other; yet the force vectors of the actuators span the three dimensional vector space. With such an alternative, an arbitrary direction of three dimensional force can be generated. In the three dimensional cases, the combined direction of vibration can be specified by the 3×1 unit vector, a combined . The three dimensional combined force can be given by the same equations for the 2 dimensional case, as shown below
 
 F   combined   =A   combined   ×a   combined   (27)
 
where a combined  and F combined  are 3 dimensional vectors.
 
     Vibration devices according to the present disclosure may include an arbitrary number of actuators in arbitrary locations and orientations. 
       FIG. 16  illustrates a vibration device  240  having a pair of actuators  242  and  244 . The actuators  242  and  244  include moving masses  246  and  248 , respectively. In this embodiment, vibration device housing  250  is configured as a hand held game controller for computer or video games. Linear motion vibration actuator  242  is shown as being located in the left handle and linear motion vibration actuator  244  is shown as being located in the right handle. The actuators  242  and  244  need not be orthogonal, and need not be in the same plane. 
     Another alternative embodiment of a vibration device according to the present disclosure is shown in  FIG. 17 , where vibration device  260  includes a first linear motion vibration actuator  262  and a second linear motion vibration actuator  264 . As shown, the actuators  262 ,  264  are located on top of each other. An advantage of such a configuration is that the actuators  262 ,  264  create little torque about the center of the vibration device  260 , which may be desirable in some vibration applications. 
     In a variation of  FIG. 17 ,  FIG. 18  illustrates a game controller  270  having two linear actuators,  272  and  274  disposed perpendicular to each other. The actuators  272  and  274  are preferably rigidly mounted to case  276  of a game controller. The actuators  272  and  274  could be mounted in a plane of any angle; however, they are preferably mounted in a horizontal plane of the case  276 . The actuators  272  and  274  do not have to be located one on top of the other; rather they can be attached to the same rigid body, such as the case  276  of a game controller. Of course, one could attach three or more linear actuators to the case  276 , preferably at right angles to each other to create force vectors than span the three dimensional space of the case  276 . Moreover, the actuators do not have to be at right angles to each other. Desirably, the actuators are positioned relative to one another with different orientations. 
     A further embodiment of a vibration device according to the present disclosure is shown in  FIG. 19 . Here, vibration device  280  includes two linear motion vibration actuators,  282  and  284 , which are aligned in their orientation but separated by a distance D. Actuator  282  includes moving mass  286  and actuator  284  includes moving mass  288 . The actuators  282 ,  284  may be vibrated such that the moving mass  286  in actuator  282  is at a negative extreme along the y axis when the moving mass  288  in actuator  284  has a positive extreme along the y axis. In this fashion the two actuators  282 ,  284  generate a combined torque when vibrated in a synchronous fashion. The embodiment shown in  FIG. 19  could be operated, in one example, such that the moving masses  286  and  288  move in the same direction when synchronized, and thereby generate a combined force along the y axis. In this fashion the configuration shown in  FIG. 19  could be used to generate a combined torque, a combined force, or a combination of force and torque. 
     An alternative embodiment of a vibration device  290  in accordance with aspects of the present disclosure is shown in  FIG. 20 . Here, three linear motion vibration actuators  292 ,  294  and  296 , each having a moving mass, are orientated on an xy plane. In this embodiment it is possible to generate a combined force and a combined torque. It is also possible to independently control the combine force and torque by modulating the amplitude of vibration in each of the actuators  292 ,  294  and  296 . The combined torque and force are superpositions of the forces and torques generated by each actuator. Since there are three actuators that can be controlled independently, the components of the force along the x axis, the force along the y axis, and the torque about a selected point on the xy plane can all be modulated independently. 
     In the vibration device embodiments described herein the vibration actuators may be attached to the vibration device in a rigid, a semi-rigid or a non-rigid fashion. Even when vibration actuators are attached in a non-rigid fashion to a vibration device, the vibration device is operable to transmit the superposition of forces from all vibration actuators. When vibration actuators are attached in a rigid fashion to a vibration device, the combined force applied by the vibration device becomes less dependent on the location where the vibration device transmits force and torques to other bodies. In addition, the more rigid the attachment between the vibration actuators and the vibration device, the more uniform the timing of the force superposition becomes at all points of the vibration device. 
     In an example, it is possible to attach the actuators directly onto a person&#39;s hand and body, for instance as shown in U.S. Pat. Nos. 6,275,213 and 6,424,333. In uses of the present disclosure where actuators are directly attached or indirectly coupled to the hand or body, the vibration force from each actuator may be felt directly at different locations on the body, yet a synchronized combined force vector can still be applied onto the body by synchronizing the operation of the actuators. 
     Vibration devices in accordance with the present disclosure can be built with rotary vibration actuators as well as with linear motion vibration actuators. In some cases the cost to manufacture rotary vibration actuators is less than the cost to manufacture linear motion vibration actuators. Thus, if cost is a factor, it may be desirable to utilize rotary vibration actuators in place of or in combination with linear motion vibration actuators. However, in order to generate synchronized vibration with rotary vibration actuators, it is necessary to control the rotary position of the actuators along with the rotary velocity. 
     A rotary vibration actuator may comprise, for example, a DC motor, a rotary solenoid, a rotary stepper motor, a servo motor, or other type of rotary actuator. One advantage of rotary actuators is their relatively low cost. The servo motor uses a position sensor and/or a velocity sensor for feedback. In some situations the rotary stepper motor may be more desirable because it allows for control of position and velocity without the use of a sensor. 
       FIG. 21  shows a rotary vibration actuator  300  suitable for use with the present disclosure. The actuator  300  includes an eccentric mass  302  coupled to a rotary actuator  304  along a shaft  306 . As the rotary actuator  304  is rotated, a centrifugal force is generated in the radial direction aligned with the eccentric mass  302  as shown by the vector CF in  FIG. 21 . 
     Many existing vibrators utilize rotary vibration actuators with eccentric masses, but not with synchronized vibration. In accordance with the present disclosure, a pair of rotary vibration actuators can be configured to achieve a vibration force that is aligned with a single direction of motion. Accordingly, a pair of such rotary actuators can be used when a vibration force in a specified direction is required. 
     For instance, a vibration device according to the present disclosure can be built, by way of example only, with two rotary vibration actuators that rotate in opposite directions, as shown in  FIG. 22 . As shown, the vibration device  310  includes a pair of rotary vibration actuators  312  and  314 , each having an eccentric mass  316  and  318 , respectively. Actuator  312  preferably rotates clockwise, and actuator  314  preferably rotates counterclockwise. In the orientation shown the centrifugal force vectors from both actuators are aligned with the y axis and superimpose to create a combined force vector, CVF, in the y direction. 
     With rotary vibration actuators it is possible to create synchronized vibration in an analogous fashion to the synchronized vibration described with linear motion vibration actuators. With rotary vibrating actuators, synchronized vibration is defined to occur where two rotary actuators rotate in approximately the same plane at the same angular velocity in opposite directions, and where the relative angle between the actuators is controlled, such that the actuator centrifugal force vectors align repeatedly in the direction of desired vibration force. 
     The direction of vibration force can be controlled with a pair of rotary (or rocking) vibration actuators by controlling the angle at which the centrifugal force vectors become aligned. Therefore, it is possible to control the direction of combined force with rotary actuators in a fashion analogous to how the direction of combined force can be controlled with multiple linear vibration actuators. 
       FIG. 23  shows the embodiment of two rotary vibration actuators as described with respect to  FIG. 22 , wherein the actuators are controlled in synchronized vibration for a number of positions. As shown in  FIG. 23 , the combined force vector, CFV, remains in the y axis, and its magnitude changes according to the rotary position of the actuators. The maximum combined force vector occurs when the centrifugal force from both rotary actuators are aligned. 
     An alternative type of rotary actuator suitable for use with the present disclosure is a rotary actuator with a pivoting mass.  FIGS. 24A-C  illustrate respective front, side and bottom views of an exemplary pivoting actuator  400 , which includes a mass  402  operable to pivot relative to a rotary actuator  404 . The mass  402  is connected to the rotary actuator  404  via a shaft  406 . The center of mass of the mass  402  can be located anywhere on the body of the mass  402 . Thus, the center of mass may be concentric with the axis of rotation, or eccentric to the axis of rotation. The pivoting actuator  400  may be configured to function in a manner similar to the rotary vibration actuators discussed above. 
     As seen in  FIGS. 25A-C , the rotary actuator  404  may be affixed to a support  408 , which, in turn, may connect to another object (not shown). Preferably a spring device  410  couples the pivoting mass  402  to a support  412 , which may be the same or a different support than the support  408 .  FIG. 25A  illustrates the pivoting actuator  400  when the spring device  410  is in a rest state when the pivoting mass  402  is in a central position. 
     The mass  402  may pivot in either a clockwise or counterclockwise manner.  FIG. 25B  illustrates counterclockwise operation. Here, the spring device  410  is in a compressed state. In the present embodiment as shown, the spring device  410  is under a compression force that is primarily linear and is applied toward the right hand side of the figure.  FIG. 25C  illustrates clockwise operation of the mass  402 . Here, the spring device  410  is in an uncompressed state in response to a force that is primarily linear and is applied toward the left hand side of the figure. 
     Vibration forces and/or torques can be generated with the pivoting actuator  400  as shown in  FIGS. 25A-C . The pivoting actuator  400  can be activated to pivot the pivoting mass  402  first clockwise and then counterclockwise, or vice versa. As the pivoting mass  402  rocks back and forth, the spring device  410  generates a vibration force, a torque, or both a vibration force and torque onto the object to which it is affixed via the support  408 . In this fashion, if the pivoting mass  402  has a center of mass concentric with the axis of rotation, the pivoting mass  402  can be used to generate a vibration torque. Also in this fashion, if the pivoting mass  402  has a center of mass eccentric with the axis of rotation, the pivoting mass  402  can be used to generate a vibration force. 
     Vibration forces and/or torques can be generated by moving a mass back and forth. It is possible to define the beginning of a vibration waveform as an instance at which a mass reverses its direction of motion. For linear actuators, the reversal of direction is a reversal of translation. For rotary actuators, the reversal of direction is a reversal of rotation. In general, the reversal of motion of a mass in an actuator may include both translation and rotation. 
     In actuators having a spring device attached to a moving mass, energy can be built up in the spring device, especially when the mass is moved back and forth close to a natural frequency of the mass and spring system. In such cases, the maximum vibration force can occur at the maximum deformation of the spring device, which can occur when the mass reaches its maximum excursion and reverses its direction. Accordingly, moving masses in two (or more) actuators, that are operating in synchronized vibration, can reverse direction at approximately the same time. 
     An alternative method for generating vibration would be to operate the pivoting actuator  400  in a clockwise (or counterclockwise) direction and then deactivate the pivoting actuator  400  while allowing the spring device  410  to rotate the pivoting mass  402  in the counterclockwise (or clockwise) direction. This approach would allow one to use pivoting actuators and control circuitry that only operates in a single direction. 
       FIG. 26  illustrates a variation of the pivoting actuator  400 , namely pivoting actuator  400 ′, which desirably includes the pivoting mass  402  operable to pivot relative to the rotary actuator  404 , and which is connected thereto via the shaft  406 . As above, the rotary actuator  404  may be affixed to the support  408 , which, in turn, may connect to another object (not shown). Preferably a first spring device  410   a  couples the pivoting mass  402  to a first support  412   a , and a second spring device  410   b  also couples the pivoting mass  402  to a second support  412   b . The supports  412   a  and  412   b  may be a single support, separate supports that are physically connected, or physically disconnected supports. One or both of the supports  412   a,b  may be the same or a different support than the support  408 . 
     One type of pivoting actuator  400  that could be employed is a DC motor. However, not all the components of the DC motor are necessary for this application, because the output shaft does not rotate continuously. Accordingly it is not necessary to have motor brushes, which can reduce cost as well as electrical power losses and frictional losses. In a preferred example, the pivoting actuator  400  may essentially include a stator and a rotor. The stator may be stationary and desirably contains permanent magnets and/or electromagnets. The rotor is operable to pivot and can contain permanent magnets and/or electromagnets. The polarity of the magnets in the stator and rotor can be configured so that activation of the electromagnets causes an electromagnetic torque to be exerted onto the rotating mass  402 . 
     In the embodiment of  FIGS. 25A-C , the spring device  410  is configured to operate in a generally linear fashion. However, In order to generate large magnitude of vibration forces with small actuators, it can be advantageous to utilize the resonance of a system. The embodiments shown in  FIGS. 25A-C  have both a mass and a spring, and thus have a resonant frequency. If the actuator is excited at or close to this resonant frequency large amplitude vibrations can build up. However, it can be desirable to operate the vibration device at a range of frequencies. It is possible for a device to have a variable resonant frequency with use of nonlinear spring forces, as discussed in the aforementioned “Vibration Device” patent application. Accordingly, one could use a nonlinear spring in the vibration device to achieve larger amplitude of vibration over a range of frequencies. 
     It is possible to generate nonlinear spring force, even with use of a linear spring element. Consider the embodiment shown in  FIG. 27A . Here, pivoting actuator  420  has a mass  422  operable to pivot relative to a rotary actuator  424 . The mass  422  is connected to the rotary actuator  424  via a shaft  426 . The rotary actuator  424  may be affixed to a support  427 , which, in turn, may connect to another object (not shown). Preferably a spring device  428  couples the pivoting mass  422  to a support  427 ′, which may be the same or a different support than the support  427 . 
     As shown in  FIG. 27A , the spring device  428  is desirably placed in-line with the pivoting mass axis. When the pivoting mass  422  is rotated a small amount about the center position very little lengthening occurs in the spring device  428 . Accordingly, the effective spring constant is low and the resonant frequency is low. 
     Low frequency operation is desirable in some situations, for instance in games that have low frequency effects. For instance, games may generate actions or events in the sub-200 Hertz range, such as between 15 and 150 Hertz. In certain cases the actions or events may be as low as 20-50 Hertz or lower, such as about 10-20 Hertz. Examples of such actions/events include gunshots, automobile related sounds such as a car spinning out of control, and helicopter related sounds such as the whirring of the rotor blades. Eccentric mass actuators may not be suitable to generate a haptic sensation in this frequency range, but pivoting actuators or linear actuators may generate such frequencies. 
     As the magnitude of rotation of the pivoting mass  422  increases, the lengthening of the spring device  428  increases as shown in  FIGS. 27B and 27C . Accordingly, for larger amplitudes of rotation, the effective spring constant is higher and the natural frequency of the system is higher. In order to quickly ramp up the vibration amplitude when a nonlinear spring force is used, the excitation frequency can be varied so that it always matches the natural frequency of the vibration device. 
       FIG. 27D  illustrates a rotating actuator  430  having a rotating mass  432  coupled to rotary actuator  434  via shaft  436 . The rotary actuator  434  is desirably coupled to a support  437 , which, in turn, may connect to another object (not shown). In this alternative, a spring device such as a torsion spring  438  is attached between the rotating mass  432  and the rotary actuator  434 . As shown, one end or tang  439   a  of the torsion spring  438  is attached to the rotating mass  432 , and the other end or tang  439   b  is attached to the support  437  (or, alternatively, to the rotary actuator  434  itself). Torsion spring  438  may be employed because such spring devices permit a large degree of rotation of the rotating mass  432  relative to the rotary actuator  434  and the support  437 . 
       FIGS. 27E and 27F  illustrate a further rotating actuator, namely rotating actuator  440 . The rotating actuator  440  includes a rotating mass  442  having a slot  443  therein, a rotary actuator  444 , and a shaft  446  coupling the rotating mass  442  to the rotary actuator  444 . The rotary actuator  444  is desirably coupled to a support  447 , which, in turn, may connect to another object (not shown). In this embodiment a pin  445  is held within the slot  443 . A spring device  448  is coupled at one end or tang  449   a  to the pin  445 . The spring device  448  is coupled at the other end or tang  449   b  to a support  447 ′. The support  447 ′ is preferably different from the support  447 , or, alternatively, is preferably a different section of the support  447  from where the rotary actuator is coupled. 
       FIG. 27E  shows the spring device  448  in a “rest” position.  FIG. 27F  shows the spring device  448  in a “compressed” position. Here, by way of example only, the rotating mass  442  may be rotating in a clockwise direction. As the rotating mass  442  rotates, the pin  445  moves relative to the slot  443 , but the spring device  448  remains in substantially the same orientation relative to the support  447 ′. In this fashion, the force applied onto the fixed  447 ′ remains in relatively the same direction as the moving mass  442  rotates. It is possible to incorporate a gap between the slot  443  and the pin  445  that would allow for some rotation of the shaft  446  before the spring device  448  is extended or compressed from its rest position. The gap would create a non-linear force effect on the rotating mass  442 , which could aid in increasing the magnitude of vibration. The gap would allow the shaft  446  to more quickly reach higher speeds and for the rotating actuator  440  to more quickly build up rotating inertia. 
     While several types of actuators have been described above that may be used with the present disclosure, other types of actuators may also be employed so long as they can be controlled as described herein. For instance, piezoelectric devices without separate or distinct “moving” and “stationary” masses may be employed either alone or in combination with other actuator types to impart vibratory forces in the manners described herein. 
       FIG. 28  illustrates a synchronized vibration system  450 , which may comprise two vibration devices  452  and  454 , such as any of those of  FIGS. 24A-C ,  25 A-C,  26  and/or  27 A-F. Of course, more that two vibration devices may be provided. The vibration devices  452  and  454  are preferably mounted onto a base plate  456  in a generally orthogonal manner as shown, although orthogonality is not required. The vibration device  452  is preferably a horizontal vibrator that desirably has a spring device  458  which applies primarily horizontal forces onto the base plate  456 . The vibration device  454  is preferably a vertical vibrator that desirably has a spring device  460  that applies primarily vertical forces onto the base plate  456 . As long as the directions of the vibration forces of the different vibration devices are not aligned, it is possible to control the combined direction of vibration using the synchronized vibration methods as described herein as well as in the aforementioned “Vibration Device” patent application. 
     An alternative embodiment of the present disclosure includes two rotary vibration actuators whose planes of vibration are not the same; however, in this case the two planes are not orthogonal to each other. In this embodiment, the component of centrifugal force from one actuator that can be projected onto the plane of the other actuator can be used to achieve a component of synchronous vibration. 
     In one example, two or more vibration devices may be mounted devices into a game controller, as shown in  FIG. 29A . Here, a game controller  470  includes a pair of vibration devices  472  and  474  mounted in both the right and left handles, respectively, of housing  476 . The directions of vibration of the vibration devices  472  and  474  are preferably not aligned, and thus it is possible to control the direction of vibration using the synchronized vibration approach discussed herein. 
     There are many orientations of both the rotary actuators and springs that can be used to achieve an embodiment where synchronized vibration is possible. For instance, the axis of rotation of both actuators can be aligned while the spring direction can vary, allowing an alternative configuration for synchronized vibration.  FIG. 29B  illustrates a game controller  480  having a pair of vibration devices  482  and  484  within a housing  486  where the axes of the rotating shafts in both rotary actuators are aligned, yet the spring forces are not aligned. 
       FIG. 30  illustrates yet another variation similar to the rotary and pivoting vibration devices. Here, a rocking actuator  490  preferably includes a rocking weight  492  rotatable about a shaft  494 . Desirably, one end of the rocking weight  492  is operatively coupled via a first spring device  496   a  to a first support  498   a . The same end of the rocking weight  492  is also desirably operatively coupled via a second spring device  496   b  to a second support  498   b . The supports  498   a  and  498   b  may be a single support, separate supports that are physically connected, or physically disconnected supports. The rocking actuator  490  may be implemented in a device such as a game controller in any of the configuration described above. 
     A controller for synchronized vibration of a pair of rotary vibration actuators specifies the angular position of each rotating shaft, such that the angle where the centrifugal force vectors are aligned is the desired direction of force vibration and the angular position is incremented such that the rotational velocity matches the desired vibration frequency. 
     A system  500  having a controller for one or more vibration devices that use linear motion vibration actuators is shown in  FIG. 31 . Vibration device controller  502  specifies the desired vibration effect and one or more driver circuit(s)  504   a ,  504   b , . . . ,  504 N provide the necessary power to actuators  506   a ,  506   b , . . . ,  506 N. While each actuator  506  is shown as being powered by a separate driver circuit  504 , it is possible for multiple actuators  506  to be driven by one driver circuit  504 . 
     The controller  502  may be, by way of example only, a microprocessor and the driver circuit(s)  504  may be, for instance, one or more electrical amplifiers. The controller  502  and drive circuit  504  may be integrated into a single microprocessor or single electrical circuit. The control method in this figure is for a configuration with N actuators, where N is an arbitrary number of actuators. Some of the figures showing various control methods in the instant application illustrate only two actuators. However, it should be understood that control methods according to the present disclosure can be extended to include an arbitrary number of actuators, as shown in  FIG. 31 . 
       FIG. 32  shows a control method for two actuators. Here the controller  502  specifies the desired vibration amplitude, A, frequency, f, and phase, p, for each actuator  506 . The amplitude, frequency, and phase of actuator  506   a  (A 1 , f 1 , p 1 ) may differ from the amplitude, frequency, and phase of actuator  506   b  (A 2 , f 2 , p 2 ). The profile/waveform of the desired vibration force may be a sine wave, square wave, triangle wave, or other profile, such as is discussed above with regard to  FIG. 1 . The actual vibration profiles/waveforms of the actuators  506   a,b  may differ from the desired vibration profiles due the dynamics of the drive circuits  504   a,b  and actuators  506   a,b.    
       FIG. 33  shows a control method where the frequency of vibration, f, is the same for both actuators  506   a,b .  FIG. 34  shows a control method where the frequency of vibration, f, and the phase of vibration, p, are the same for both actuators  506   a,b . In this embodiment, the actuators  506   a,b  are desirably driven synchronously such that the peak amplitude of vibration will occur approximately at the same time for both actuators  506   a,b . The amplitude of vibration may differ between the actuators  506   a,b.    
       FIG. 35  shows a control embodiment in accordance with the present disclosure where the vibration device controller  502  includes an internal direction and amplitude controller  508 , an internal frequency controller  510 , and an internal vibration controller  512 . The direction and amplitude controller  508  desirably specifies the combined vibration amplitude, Acombined, and the direction of vibration theta. The frequency controller  510  desirably specifies the vibration frequency, f. The vibration controller  512  uses the inputs of theta, Acombined, and f to output vibration commands to the individual actuators  506   a,b . The vibration controller  512  is operable to output various waveforms including sine waves, square waves, triangle waves, or other profiles as discussed herein. 
     The output from the vibration device controller  502  shown in  FIG. 35  provides the magnitude of vibration as a function of time to each drive circuit  504   a,b . In the case where the profile of vibration is a sine wave, the amplitude of vibration for each actuator as a function of time is given by the equation shown below: 
                     [             A   1     ⁡     (   t   )                   A   2     ⁡     (   t   )             ]     =       D     -   1       ⁢     Acombined   ⁡     [           cos   ⁡     (   theta   )                 sin   ⁡     (   theta   )             ]       ⁢       sin   ⁡     (       ω   ⁢           ⁢   t     +   p     )       .               (   28   )               
Here, t is time and ω is the vibration frequency in radians per second. The parameter p is the phase of vibration and may be set to zero. The value of ω in terms of frequency f in vibrations per second is given by ω=2πf.
 
     When the vibration actuators have a linear relationship between the command magnitude and the magnitude of vibration, the output A 1 (t) and A 2 (t) from equation 28 can be applied directly to the vibration actuators to generate a combined vibration direction corresponding to the angle theta. However some vibration actuators may have a nonlinear relationship between the command magnitude and the magnitude of vibration. For such nonlinear actuators it is possible to generate vibration in the direction theta by using a linearization function that adjusts the magnitude of A 1  and A 2  to compensate for the nonlinearity of the actuator, as shown in the following equation. 
                     [             A   1     ⁡     (   t   )                   A   2     ⁡     (   t   )             ]     =     linearziation_function   ⁢     {       D     -   1       ⁢       A   combined     ⁡     [           cos   ⁡     (   theta   )                 sin   ⁡     (   theta   )             ]       ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     +   p     )         }               (   29   )               
The linearization equation described above can be a lookup table or a scaling algorithm or other type of function.
 
     The ability to control the direction of vibration over time, such as though use of equations 28 and 29, is an important advantage of the present disclosure. The ability to control vibration direction can be used in vibratory feeders to direct parts in a desired direction. In addition, there are numerous advantages of using the disclosure for haptic devices as described herein. 
       FIG. 36A  illustrates a system  550  showing the input of various input parameters of amplitude, phase and position (or time) for a pair of linear actuators. A computer  552  receives input of the parameters, which are preferably entered using a computer keyboard (not shown); however, the parameters also could be input using a graphical user interface, analog potentiometers, or many other means generally known to those skilled in the art. The appropriate output waveforms for linear actuators  554   a  and  554   b  are then computed using the computer  552 . Each waveform is preferably independent. While computation may be performed using an analog computer, a digital computer is preferred. 
     If a digital computer is used, the digital output for each actuator  554   a,b  is then preferably fed into respective digital-to-analog (“DAC”) converters  556   a  and  556   b , which convert the output to the appropriate analog waveform. The analog waveforms are then fed into the appropriate driver circuits  558   a  and  558   b . Those skilled in the art could use other means to modulate the linear vibrations of each actuator  554   a  and  554   b , for example via pulse width modulated (“PWM”). Varying the parameters produces an extremely broad range and rich set of haptic sensations for the end user. 
     In addition to creating varying force effects, one could control the direction of vibration—that is to say the direction of vibration could remain stationary. The resultant force effects can be of lower frequency than the frequency of vibration. 
     There are also useful applications for generating precise patterns of vibrations from simple parameters. Such patterns include circles, ellipses and straight lines. Furthermore, the amplitude and duration of the patterns may be precisely controlled over time. Moreover, a sequence of patterns may be generated as desired. 
       FIG. 36B  illustrates the system  550  where the input of various input parameters includes input of pattern number, amplitude, duration and start-time for the vibration device using compound vibrations. The parameters are preferably entered using a computer keyboard. The appropriate output waveforms for each linear actuator are then computed at computer  552 . As described above, the digital output for each actuator  554   a  and  554   b  is then fed into DACs  556   a  and  556   b  for conversion to the appropriate analog waveforms. The waveforms are then fed into the driver circuits  558   a  and  558   b . Again, the various parameters produce an extremely broad and rich set of haptic sensations for the end user. 
     Each of the vibration devices described herein according to the present disclosure can be used as a haptic interface. Haptic interfaces provide force sensation to a user. Haptic interfaces include computer gaming controllers, robot controllers, surgical tool controllers, as well as other devices where a force sensation is provided to a user. 
     An embodiment  600  of the present disclosure with a haptic interface application is shown in  FIG. 37 . In this embodiment a systems controller  602  provides force commands to a haptic interface  604  which generates forces which result in force sensations to user  606 . The systems controller  602  may be microprocessor, a central processing unit, an ASIC, a DSP, a game controller, an analog controller, or other type of controller or any combination thereof. The user  606  can input commands to the haptic interface  604  that are transmitted as user commands back to the system controller  602 . The user commands can be input through pressing buttons, moving joysticks, squeezing the haptic interface at various level forces, moving the haptic interface, applying force and torque onto the haptic interface and through other means. 
     In the embodiment shown in  FIG. 37 , there is preferably a graphical display  608  which receives an image command from the system controller  602  and displays a visual image to the user  606 . The graphical display  608  may be, for instance, a computer monitor, a television monitor, an LCD display, a plasma display, a combination of light sources, or other type of means for generating a graphical image. A haptic interface application can also be implemented without a graphical display  608 . 
     A haptic interface application can include a simulation of a virtual environment or representation of a real environment to the user  606 . A systems controller method of control can be based upon this real or virtual environment. Typical simulated environments include games, driving and flight simulations, surgical simulations, and other types of simulations. Typical real world environments include control of robots and remote machines, long distance interactions, and other types of environments. It is often desirable that a haptic interface provide force sensations that correlate with the real or simulated environment in which the haptic interface is being used. 
     Another embodiment  620  having a haptic interface application is shown in  FIG. 38 . This embodiment is similar to the one of  FIG. 37 , and includes a systems controller  622 , which provides force commands to a haptic interface  624  that generates forces which result in force sensations being received by user  626 . A graphical display  628  is also provided for receiving image commands from the system controller  622  and for displaying a visual image to the user  626 . 
     In the embodiment of  FIG. 38 , the haptic interface  624  desirably includes a vibration device  630  having vibration actuators (not shown), a vibration controller  632 , driver circuits  634  which drive the vibration device actuators, and an input device  636 , which can detect user input and which can include buttons, joysticks, and pressure sensors. The components of the haptic interface  624  may be of any of the configurations described herein. In this embodiment the graphical display  628  preferably presents a two dimensional image. The graphical display  628  shows an object of interest at a direction specified by the angle theta. It is may be desirable that the force sensation felt by the user  626  correspond to the image on the graphical display in terms of direction, such as theta, and other attributes. 
     The embodiment shown in  FIG. 38  can be utilized so that the force sensations felt by the user  626  are generated by the vibration device controller  632  specifically to correspond to the image on the graphical display  628 . The vibration device controller  632  may specify one or more of the amplitude of vibration, Acombined, direction of force, theta, and frequency of vibration, f, as described above. The values of Acombined, theta, and/or f can be selected to correspond to the image on the graphical display  628  and the environment being used by the system controller  622 . The complete force effect (including frequency, amplitude, combined direction of force and torque, and duration of force effect) generated by the vibration device may correlate events within a graphical computer simulation. Several examples of such operation follow. 
     A first example involves the simulation of a user firing a gun. In this simulation, the vibration device controller  632  could specify the angle theta to represent the direction of a gun firing, the amplitude of vibration, Acombined, to represent the amplitude of the gun recoil, and the frequency of vibration, f, to represent the frequency of bullets leaving the gun. 
     A second example involves an impact between objects. In this simulation the vibration device controller  632  may specify the angle theta to represent the direction of impact, and the amplitude of vibration, Acombined, to represent the amplitude of impact. 
     A third example involves driving a vehicle. In this simulation the vibration device controller  632  could specify the angle theta to represent the direction of vehicle motion, the frequency of vibration, f, to represent the frequency of vehicle vibration as it drives over bumps in the road or the speed of the vehicle, and the amplitude of vibration, Acombined, to represent the amplitude of bumps in the road. 
     A fourth example involves a car or spacecraft spinning out of control. In this simulation the vibration device controller  632  could specify an angle theta that represents the vehicle&#39;s orientation. To represent the vehicle spinning, the angle theta can vary over time. The rate at which the angle theta can be different than the vibration frequency. Typically the frequency at which a vehicle spins would be significantly lower than typical vibration frequencies. 
     An algorithm that can be used to create the vehicle spinning described above varies the direction of vibration continually. The direction of vibration may be rotated at a rate of β radians per second, using the equation below: 
     
       
         
           
             
               
                 
                   
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                                 sin 
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                   30 
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     Equation 30 illustrates that the frequency of direction change, β, can be modified independently from the frequency of vibration ω. A user such as user  606  or  626  can sense both the frequency of vibration and the direction of vibration. In this fashion, sensations at both the β and ω frequencies can felt by the user. It is possible to set the frequency β much lower than the frequency ω, thereby overcoming a limitation of known devices. By way of example only, ω may vary between 10 Hz and 100 Hz while β may be on the order of 1 Hz. In another instance, β may vary from between about 5% to 20% of ω. Of course, in other instances ω and β may be similar or the same, or, alternatively, β may be larger than ω. All of these examples will depend on the specific effect that is desired. 
     Low frequency operation is desirable in some situations, for instance in games that have low frequency effects. For instance, games may generate actions or events in the sub-200 Hertz range, such as between 1 and 150 Hertz. In certain cases the actions or events may be as low as 2 Hertz or lower, such as about 0.5-1 Hertz. Examples of such actions/events include gunshots, automobile related sounds such as corresponding to a car spinning out of control, and helicopter related sounds such as the whirring of the rotor blades. A traditional eccentric mass actuator may not be suitable to generate a haptic sensation in this frequency range; however, two or more vibration actuators operated in synchronized vibration may generate such frequencies. 
     β is not limited to any particular rate or range of rates. For instance, β may be a relatively low rate to represent a slow spinning action, e.g., of a car spin out at less than 10 miles per hour, or β may be a relatively high rate to represent a fast spinning action, e.g., of a car spin out at a speed in excess of 40 miles per hour. Similarly, ω is not limited to any particular frequency of vibration. Preferably, ω is set within a range of frequencies that can be felt or otherwise detected by a user. 
     Equation 30 may be modified by changing the vibration profile from a sine wave to a square wave, triangle wave, or other profile. In addition, the amplitude of vibration, Acombined, can be varied over time. The frequencies β and ω can also be varied over time. In this fashion a wide range of force effects can be created. 
     Vibration actuators can be used to provide haptic sensations either through synchronized vibration or otherwise. Actuators can be vibrated without synchronization when there is no need to convey directional information, and then the actuators can be switched to synchronous vibration when there is a need to convey directional information though the haptic interface. 
     Many linear motion vibration actuators take advantage of resonance to achieve relatively high level of forces with low power requirements. However, to achieve these high levels of forces a number of vibration cycles have to occur before the peak magnitude of vibration occurs. In addition when the actuator is shut off, the moving mass in the actuator may continue to oscillate for a number of cycles. Thus the dynamics of the actuator prevents instantaneous response of the actuator to increase or decrease the magnitude of vibration. 
     When synchronous vibration is used to control the direction of combined force, the actuator dynamics may limit the speed at which the direction of combined force can be changed. One of the examples presented above describes implementation of a haptic force sensation that corresponds to the spinning of a car. However, the actuator dynamics may limit the rate at which such spinning effect can be generated. As will be described in detail below, it is possible to provide a method that can increase the rate at which the direction of force can be changed for a system of vibration actuators that are synchronously vibrated. 
     Equation 25 above defines the required amplitude of vibration of actuators to achieve a combined force direction corresponding to an angle theta. For a given actuator in a vibration device, the required amplitude of vibration is defined as Ades, which indicates the desired amplitude of vibration of that actuator. If the actuator is at rest or at a lower level of vibration than Ades, then it may be desirable to initially drive the actuator at a higher level of vibration to more quickly raise the amplitude of vibration to Ades. Conversely if the actuator is already vibrating at an amplitude higher than Ades it may be desirable to initially drive the actuator at a lower level or even brake the actuator to more quickly lower the amplitude of vibration to Ades. These variations in the amplitude at which the actuator is driven are defined as corrections to the commanded vibration magnitude. 
     One method of determining the proper corrections to the vibration magnitude is to model the dynamics of the actuator. This approach allows one to predict the dynamic states of the actuator and optimal commands to most quickly generate the desired amplitude of vibration. 
     An alternate method of determining the corrections to the vibration magnitude does not require a dynamic model of the actuator or explicitly predicting the dynamic states of the actuator. In this method a counter is maintained to track the recent number of vibrations of the actuator and the corresponding commands sent to the actuator during these recent vibrations. The command to the actuator at the k th  vibration is given by the following equation:
 
 A   com     —     k   =A   des     —     k   +A   cor     —     k  
 
     A des     —     k  represents the desired actuator amplitude for the k th  vibration of the actuator. A cor     —     k  represents the correction to the command for the k th  vibration. And A com     —     k  represents the actual amplitude of the command sent to the actuator for the k th  vibration. 
     If the desired amplitude at the k th  vibration is greater than the amplitude during the previous vibration, then most likely the vibration level needs to be increased. Accordingly, the correction to the command at vibration k, A cor     —     k , can be chosen to be proportional to the difference between the current desired amplitude, A des     —     k , and the previous commanded amplitude A com     —     k-1 . An equation which described this approach for calculation A cor     —     k  is:
 
 A   cor     —     k   =K *( A   des     —     k   −A   com     —     k-1 )  (31)
 
     Here, K is a gain chosen based upon actuator performance. This same equation works for reducing the magnitude of vibration quickly. When A des     —     k  is less than the value of A com     —     k-1 , it indicates that most likely the level of vibration needs to be reduced and the correction A cor     —     k  is negative. If the large reduction in vibration amplitude is commanded, then the negative magnitude of A cor     —     k  may be greater than A des     —     k  and the actual command sent to the actuator, A com     —     k , will be negative resulting in braking of the moving mass in the actuator. 
     Another approach to correcting the magnitude of vibration takes into consideration the two previous commanded amplitudes, and is given by the following equation:
 
 A   cor     —     k   =K   1 *( A   des     —     k   −A   com     —     k-1 )+ K   2 *( A   des     —     k   −A com k-2 )  (32)
 
     Here K 1  is a gain that corresponds to the k−1 vibration command, and K 2  is a gain that corresponds to the k−2 vibration command. In a similar fashion even more prior commands can be incorporated into the correction algorithm. The following equation shows how “m” prior commands can be incorporated into an actuator command.
 
 A   cor     —     k   =K   1 *( A   des     —     k   −A   com     —k-1   )+ K   2 *( A   des     —     k   −A   com     —     k-2 )+ . . . + K   m *( A   des     —     k   −A   com     —     k-m )  (33)
 
     Alternative methods of control for multiple vibrating actuators may include modified synchronization. One method of modified synchronization is for one actuator to vibrate at a frequency that is an integer multiple of the vibration frequency of another actuator.  FIG. 39  is a plot  650  presenting two vibration profiles,  652  and  654 , showing such a control method. The vibration frequency of profile  654  is twice the vibration frequency of profile  652 . The beginning of cycles of vibration can be controlled to occur at the same time only ever other cycle for profile  2 . Thus the superposition of, peak amplitudes only occurs ever other cycle for profile  654 . This modified synchronization method can be applied for arbitrary integer multiples of vibration frequency, arbitrary vibration profiles, and an arbitrary number of actuators. 
     One advantage of such a modified synchronization method is that multiple vibration frequencies can occur at the same time while still providing for some superposition or peak amplitudes. The superposition of peak amplitudes allows for control of direction of vibration, in a similar fashion to how the direction for vibration is controlled for synchronized, vibration. With this modified method of synchronized vibration, it is possible to specify the direction of combined force only during a portion of the vibration cycle. Nevertheless, a direction component to the vibration can be controlled in the duration close to the time where the superposition of peaks occurs. Close to the time at which there is superposition of peaks in the vibrations, the combined force vector, F combined , can be approximated by:
 
 F   combined   =a   1   A   1   +a   2   A   2   (34)
 
     Here, a 1  and a 2  are the unit vectors aligned with the directions of actuator  1  and actuator  2 , respectively. A 1  and A 2  are the amplitudes of force of actuator  1  and actuator  2 , respectively, near the duration of the superposition of peaks. By modifying the amplitudes A 1  and A 2  it is possible to modify the amplitude and direction of the combined force vector, F combined . A similar approach can be used when there are more than two vibration actuators. 
     If there are two or more vibrating actuators where repeatedly the peak amplitude of force of these vibrating actuators occurs at approximately the same time, then the combined direction of force of these actuators can be controlled near the time when these repeated peak amplitudes occur. In this case, the combined direction of force can be controlled by modifying the amplitude of vibration of the actuators. 
     An alternative modified synchronization is to drive two vibration actuators at the same frequency but one vibration actuator at a phase where its peak magnitude of force occurs when a second vibration actuator is at zero force, which is at 90 degrees out of phase for a sinusoidal vibration. In such a modified synchronization the combined force direction rotates in a circle or ellipsoid during each vibration period. 
     Additional methods for modified synchronization of vibration may include the superposition of profiles as described in the “Jules Lissajous and His Figures” (“Lissajous”), appearing in chapter 12 of “Trigonometric Delights” by Eli Maor, published in 1998 by Princeton University Press. The entire disclosure of Lissajous is hereby incorporated by reference. Lissajous describes how profiles can be combined through various combinations of frequencies, phases, amplitudes, and profiles to generate a wide range of output figures. These are also known as Bowditch curves. Lissajous also describes how geometric shapes can be created from multiple vibration sources. These combinations of vibrations can be applied to haptic devices and vibration devices in accordance with aspects of the present disclosure. Thus, the concepts of superposition described in Lissajous can be applied by vibration actuators to yield a wide range of force sensations. 
     Electric actuators often require a driver circuit separate from a controller. The driver circuit provides sufficient current and voltage to drive the Actuators with the necessary electrical power. A wide range of driver circuits have been developed for electrical actuators and specifically for vibration actuators, and are known to those skilled in the field. Such driver circuits include linear drivers, PWM drivers, unipolar drivers, and bipolar drivers. A circuit block diagram for a vibration actuator  700  according to the present disclosure includes a vibration controller  702 , a driver circuit  704 , and an actuator  706 , as shown in  FIG. 40 . 
     The vibration controller  702  shown in  FIG. 40  can be located on the vibration device itself or could be located remotely, where the vibration signals are transmitted to the driver circuit  704  through wired or wireless communication. 
     It is often desirable to control a vibration device or actuators from a digital controller such as a microprocessor or other digital circuit. Digital control circuits often have low level power output, and therefore require a higher power driver circuit to drive an actuator. In addition, low cost digital controllers often have digital outputs, but do not have analog outputs. To simplify the vibration controller circuitry and lower cost, the vibration signal can be a binary logic directional signal which signals the moving mass to move either forward or backwards. In this configuration, the vibration signal can be in the form of a square wave to generate the desired vibration effect. Even with such a square wave control signal, the actual motion and vibration force of the vibration actuator will most likely not follow a square wave exactly due to the dynamics of the actuator. 
     To further simplify the vibration controller circuitry and lower cost, the amplitude of the vibration signal can be modulated with a PWM signal, where the duty cycle of the signal is proportional to the amplitude of vibration. An embodiment  710  with such a digital vibration controller  712  for one actuator  716  is shown in  FIG. 41 . In this embodiment, the output of the digital vibration controller  712  includes an amplitude signal in PWM form and a direction signal, for instance in the form of a logic bit, both of which preferably are sent to a driver circuit  714 . The driver circuit  714 , in turn, sends electrical power to the actuator  716 . 
     Digital control circuitry can be used to control a complete vibration device in synchronized vibration. In synchronized vibration the frequency and phase of two or more actuators are the same. Accordingly, a single square wave can be used to control the direction of the vibration actuators that are in synchronized vibration. The amplitude of vibration can be controlled independently for each actuator, with separate PWM signals. 
       FIG. 42  shows an embodiment  720  where a vibration device controller  722  generates one directional signal (“dir”), which may be in the form of a square wave. The dir signal is preferably provided to a pair of drive circuits  724   a  and  724   b . The vibration device controller  722  desirably generates separate amplitude signals, A 1  and A 2 , in PWM form to the drive circuits  724   a,b  for a pair of actuators  726   a  and  726   b . The vibration device controller  722  preferably includes a direction and amplitude controller  728 , a frequency controller  730  and a vibration controller  732  as in the embodiment described above with regard to  FIG. 35 . The direction and amplitude controller  728 , the frequency controller  730  and the vibration controller  732  may be configured in hardware, software, firmware or a combination thereof, and may be implemented either as separate components or processes, or may be implemented as a single component or process. 
     The embodiment  720  of  FIG. 42  may be used to control in synchronous vibration the vibration devices with two actuators, for instance as described above with regard to  FIGS. 10-20 . Embodiment  720  can also be used to vibrate two or more actuators completely out of phase, which occurs during synchronized vibration when equation 25 provides results with the sign of A 1  being different than the sign of A 2 . To vibrate two actuators completely out of phase, the binary direction signal dir can be inverted for one of the actuators. The inversion of the directional signal dir can occur at a driver circuit  724   a  or  724   b , or the vibration controller  732  can output two directional signals, with one being the inverse of the other. The case where two actuators are being driven completely out of phase is shown in  FIG. 13 . 
     Electric actuators in accordance with the present disclosure can be driven with unipolar or bipolar drivers. A unipolar driver will generate current in an actuator in a single direction. A unipolar driver is well suited for actuators where the moving mass is ferromagnetic and an electromagnetic coil only generates attractive magnetic forces, such as the actuator  150  shown in  FIG. 9 . One example of a unipolar driver circuit is a Darlington array, such as the ULN2803A DARLINGTON TRANSISTOR ARRAY manufactured by Texas Instruments. 
     A bipolar driver can generate current in two directions. Bipolar drivers are well suited for actuators where the moving mass is magnetic and where reversing the direction of current in an electromagnetic coil can reverse the direction of force on the moving mass. Examples of such actuators are presented in  FIGS. 5A-B  through  8 A-B. One example for a bipolar driver circuit is an H bridge, such as the L298 manufactured by ST Microelectronics. Alternative H bridges are the 3958 and 3959 drivers manufactured by Allegro Microsystems. 
     In vibrating circuits it can be advantageous to increase power output of the driver circuits through use of a charge pump capacitor as used in 3958 and 3959 drivers manufactured by Allegro Microsystems. It can also be advantageous to incorporate a capacitor in series with a linear motion vibrating actuator to benefit from a resonance effect and temporary storage of energy in the capacitor, as described in the aforementioned U.S. patent application entitled “Vibration Device.” 
     As detailed herein, vibration actuators can be used in a variety of methods to create haptic effects. Vibration actuators can be operated continuously throughout the duration of a specified haptic effect, or can be pulsed on and, off during the haptic effect. By pulsing vibration actuators on and off the user feels only a small number of vibrations, then feels a pause, and then the vibration resumes. In this fashion it is possible to generate secondary sensations associated with the frequency of pulsing the actuators on and off. Examples of how such pulse effects can be used are described in U.S. Pat. Nos. 6,275,213 and 6,424,333. 
     Any of the actuators described herein may be used in accordance with the present disclosure to produce a wide variety of haptic effects. While some actuators such as linear actuators and rocking mass actuators may be particularly suited for low frequency operation, all actuators herein may provide synchronized feedback. Such feedback may be employed in games, virtual reality equipment, real-world equipment such as surgical tools and construction equipment, as well as portable electronic devices such as cellular phones and pagers. By way of example only, cellular phones and pagers may implement different vibration effects to identify different callers or different actions. Synchronized vibration may provide directional feedback, for instance, with the impact or recoil of a gun in a game, or to distinguish between frontal and side impacts in driving games. Synchronized vibration may also provide a continual rotation of a vibration force vector in a game to simulate a car spinning out of control. Synchronized vibration may also be used in endless other applications and situations to provide a rich haptic experience to a user. 
     As mentioned above, other aspects of the disclosure include General Synchronized Vibration. General Synchronized Vibration differs from non-synchronized vibration in that the frequency and phase of multiple vibration forces are controlled. Embodiments with multiple Vibration Actuators that are not controlled with the General Synchronized Vibration approach will often have inconsistent frequency, amplitude, or relative phase between the actuators. With General Synchronized Vibration the frequency and phase of the Vibration Actuators may vary during the start-up and transitions between various waveforms. However, once the actuators are synchronized, each actuator is controlled to a specific frequency and phase. 
     Often each actuator is controlled to a fixed frequency and phase for a given duration of time. This duration of time depends on the application, but is typically longer than the period of the highest frequency vibration force that is being synchronized. In haptic applications this duration of time is typically along enough for a person to sense the effect. However, there are some implementations of General Synchronized Vibration where the desired waveform of vibration varies quickly, such as a quickly changing direction used to provide a sensation of spinning. In such quickly varying waveforms, the desired frequency and phase of a vibration actuator may be changing in a duration that is shorter than the period of the vibration of that actuator. A common characteristic of General Synchronized Vibration is that the frequency and relative phase of multiple vibration actuators are explicitly controlled to desired values rather than randomly selected values. 
     In General Synchronized Vibration there is typically a consistent correlation between frequency and phase of the actuators and desired vibration effects. For example, a haptics effect library for software developers may have a routine labeled “spin,” which generates a sequence of desired frequency and phase for a plurality of Vibration Actuators. Each time the spin effect is executed, a similar sequence of frequency and phase and generated by the plurality of Vibration Actuators. 
     Embodiments of this disclosure include a Vibration Device comprised of multiple Vibration Actuators mounted onto a mounting platform such as a base plate, sub-frame, housing, or enclosure. For example the mounting platform could be the housing of a game controller, or the housing of a Vibration Actuator. The mounting platform transfers force and torque between the Vibration Actuators and thereby allows the vibration forces and torques to be superimposed upon each other. The mounting platform is preferably rigid, but can also be relatively rigid component, or a semi-rigid component. The mounting platform could be made of separate pieces. The mounting platform could include components of an object upon which vibration forces are being applied. For example if multiple Vibration Actuators are mounted onto a person&#39;s arm or other body parts and forces are transmitted from these actuators through the arm or body parts, then the arm or body parts can serve as the mounting platform. This disclosure pertains to any configuration where the forces and torques from multiple Vibration Actuators can be vectorially combined to generate a net vibration force, vibration torque, or vibration force and torque. 
     The mounting platform is typically attached to a number of items such as battery, control circuit board, and the stationary parts of the Vibration Actuators including housing and stator. The combined mass of the mounting platform and items that are attached to it is defined as a “Reference Mass”. The vibration force and torques are transferred from Vibration Actuators to the Reference Mass. If the mounting platform is able to move, the vibration forces may shake the Reference Mass. Typically the Reference Mass is in contact with an “External Object”, and forces and torques are transmitted between the Reference Mass and the External Object. For example, a game controller held in a user&#39;s hand would transfer forces and torques from the game controller&#39;s Reference Mass onto a user&#39;s hands, which in this case is an External Object. The mounting platform may be attached to the Earth, which would also be an External Object. A Vibration Device attached to the Earth is sometimes termed a “Shaker” or a “Shaker Device”. 
     A preferred embodiment uses two aligned LRAs, as shown in  FIG. 43 . LRA  1102   a  and LRA  1102   b  are attached to mounting platform  1100  and are aligned in the axis of vibration that they generate. Each LRA has a moving mass,  1108 , and a housing  1106  which is attached to the Mounting platform  1100 . This configuration of vibration actuators is referred to as an LRA Pair. The vibration forces from each LRA are combined together through the mounting platform  1100 . The vibration force generated by LRA  102   a  is designated as F 1  and the vibration force generated by LRA  1102   b  is designated by F 2 . 
     For the embodiment shown in  FIG. 43 , one method of generating an asymmetric vibration force is to operate LRA  1102   b  at twice the frequency of LRA  1102   a , with a specified phase difference of either 90 or −90 degrees. The vibration forces in such an embodiment with sinusoidal vibrations can be given by:
 
 F   1   =B   1  sin(ω 1   t+φ   1 )
 
 F   2   =B   2  sin(ω 2   t+φ   2 )
         Where ω 2 =2ω 1      φ 1 =0, and φ 2 =−90       

     The combined force for the LRA Pair is given by:
 
 F   LRA     —     Pair   =B   1  sin(ω 1   t+φ   1 )+ B   2  sin(ω 2   t+φ   2 )  (35)
 
     Typically it is not critical to control vibration effects relative to absolute time. Accordingly, when implementing the vibration effect described in Eq. 35 above, it is not critical to control both the phase φ 1  and φ 2 , but rather the relative phase between the two actuators. Therefore in some implementations one could set phase φ 1  to zero and control only φ 2 . Alternatively one could directly control the phase difference between the actuators. In this application typically the phases of all the actuators are shown in the equations. However, without loss of utility only the relative phase of the actuators can be controlled. Thus the phase of Vibration Actuators  2 ,  3 ,  4 , etc. would be controlled relative to the phase of actuator  1 ; thereby eliminating the need to control the phase of actuator  1  relative to absolute time. 
     A feature of this disclosure includes the use of superposition of synchronized vibration waveforms. When multiple vibration forces are generated on a single vibration device, the Combined Vibration Force for the device is the superposition of the multiple waveforms. An example with two synchronized sine waves described by Eq. 35 is shown in  FIG. 44 . As shown, waveform  2  has twice the frequency of waveform  1 . The phase of both waveforms is set such that at a time of zero the peaks of both waveforms have their maximum value in a positive direction, and the forces magnitudes are added together (also referred to as constructive interference or positive interference). 
     Furthermore, at the time when waveform  1  is at its negative peak then waveform  2  is at a positive peak, and the forces magnitudes are subtracted from each other (also referred to as destructive interference or negative interference). Due to this synchronization the combined vibration waveform is asymmetric, meaning that the force profile for positive force values is different than the force profile for negative force values. In the asymmetric waveform shown in  FIG. 44  there is a higher peak positive force and a lower peak negative force. In haptic applications the larger force in the positive direction can generate more of a force sensation than the lower magnitude force in the negative direction, even though the duration of force in the negative direction is longer. In this fashion asymmetric vibrations can be used to generate a haptic cue in a specific direction with a vibration device. 
     In an LRA, a moving mass moves relative to the actuator housing, and a restoring spring transfers force between the moving mass and the actuator housing. The force imparted by an LRA onto a mounting platform is a combination of the force from the restoring spring, and the electromagnetic force between the stator and moving mass. The restoring spring can be, for example: a mechanical spring or a magnetic spring. As resonance builds up in an LRA, the magnitude of the spring restoring force increases and becomes the dominant portion of the actuator force. Accordingly, the peak force imparted by a LRA onto the mounting platform typically occurs at or near the peak excursion point of the moving mass. 
     In  FIG. 43  the moving masses are graphically depicted as towards the right side of the LRAs to indicate actuator forces being applied to the right. Accordingly, when the embodiment shown in  FIG. 43  is controlled to follow the waveform described by Eq. 35, then the moving mass of LRA  1102   a  is at its peak excursion to the right at the same time when the moving mass of LRA  1102   b  is at its peak excursion to the right resulting in a large combined force to the right, yet when the moving mass of LRA  1102   a  is at or near its peak excursion to the left then the moving mass of LRA  1102   b  is at or near its peak excursion to the right (since it is vibrating at twice the frequency) resulting in force cancellation and a low combined force to the left. 
     Thus, in this embodiment the timing of the moving masses is an indication of an asymmetric vibration waveform. In  FIG. 45 , the embodiment shown in  FIG. 43  is shown at various time steps as it implements the vibration waveform shown in  FIG. 44 . In  45 , the top LRA vibrates at twice the frequency and generates lower forces that the bottom LRA, the position of the moving masses indicates the forces generated by each LRA, and the combined force vector is shown between the LRAs. Each time step in  FIG. 45  is labeled according to the period, T, of the slower LRA. 
     In the embodiment shown in  FIG. 43 , the alignment of the actuators does not have to be precise. Indeed, in haptic applications having the two actuators are not precisely aligned may not deter from the primary haptic effect that is being generated. 
     A variation of this embodiment is shown in  FIG. 46 . The actuators  1102   a  and  1102   b  are attached directly to each other to provide an even more compact configuration. Also the LRAs can share housings, shafts, power supplies, and other components to make the device even more compact. 
     Another variation of this embodiment is shown in  FIG. 47 . The actuators  1102   a  and  1102   b  are attached in line with each other. In this embodiment, the forces of each LRA are collinear, and create no net torque along the axis of the LRAs. This embodiment is useful where pure force output is desired without any torque output. 
     The timing of vibration force within a Vibration Actuator can be correlated with a number of physical properties. For example, in many LRAs a spring applies a restoring force onto a moving mass and the vibration force is largely correlated with the position of the moving mass. In ERMs the direction of the vibration force largely correlates to the angular position of a rotating eccentric mass. Linkage mechanisms can be used to generate vibrations, such as a slider-crank vibration actuator  1110  shown in  FIG. 48 , where a rotating motor  1114  moves a mass  1120  back and forth. With such linkages the vibration force can be correlated with the acceleration of a moving mass. Since the vibration force can be correlated with a number of physical properties, General Synchronized Vibration can also be characterized by control of the frequency and phase of the position or acceleration of moving masses within Vibration Actuators. 
     A feature of this disclosure includes combining vibration waveforms from multiple Vibration Actuators to generate a more complex vibration waveform. The asymmetric vibration described by Eq. 35 and shown in  FIG. 44  is only one such type of combined vibration waveform. A more general embodiment shown  FIG. 49  has a set of N LRAs all aligned with the same axis. According to one aspect, in General Synchronized Vibration a plurality of Vibration Actuators are synchronized in phase and frequency, and in some cases amplitude. A wide range of vibration effects can be generated by controlling the frequencies and phases of all N actuators. 
     A vibration force, F, is in a repeated cycle over a period T when F(t+T)=F(t). The vibration force of an ith actuator in a repeated cycle can be given by:
 
 F   i ( t+Δ   i   +T   i )= F   i (Δ i   +t ),
 
where Δ i  is the phase and T i  is the period of the ith actuator. For the embodiment shown in  FIG. 49 , there is a set of N LRAs all aligned with the same axis. If all actuators are operated at set frequencies and phases, then the combined vibration force can be given by:
 
 F   AlignedSet   =F   1 (Δ 1   +t )+ F   2 (Δ 2   +t )+ . . . + F   N (Δ N   +t )  (36)
 
     In the general case, the waveform shapes of F i  can be a wide range of waveforms including sine waves, triangle waves, square waves, or other waveforms. In some embodiments, the frequency of the actuator with the lowest frequency is defined as the fundamental frequency, ω 1 , and the remaining actuators vibrate at integer multiples of the fundamental frequency. In these embodiments the period of the fundamental frequency is given by T 1  and the remaining vibration periods are given by such that:
 
 T   1 =2 T   2   ,T   1 =3 T   3   , . . . T   1   =NT   N  
 
     When all the vibration actuators vibrate at integer multiples of the fundamental frequency, then the combined waveform has a repeated waveform with a period of the fundamental frequency. The fundamental frequency is also referred to as the first harmonic. 
     One method of implementing General Synchronized Vibration is to use sinusoidal vibrations in each actuator of an aligned set, and use Fourier Waveform Synthesis to select the phase, frequency, and amplitude of each actuator to approximate a desired vibration waveform. For a set of N aligned actuators with sinusoidal waveforms, the combined force of an Aligned Set, F AlignedSetFourier , is given by:
 
 F   AlignedSetFourier   =B   1  sin(ω 1   t+φ   1 )+ B   2  sin(ω 2   t+φ   2 )+ . . . + B   N  sin(ω N   t+φ   N )  (37)
 
     A wide range of additional waveforms can be synthesized from a set (a plurality) of vibration waveforms. Fourier synthesis is a method whereby an arbitrary waveform can be approximated from a combination of sine waves, including both symmetric and asymmetric waveforms. It is advantageous to use actuators vibrating at frequencies that are integer multiples of the frequency of vibration of other actuators. The lowest frequency in the set is referred to as the fundamental frequency or the first harmonic, the second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental frequency, and so on. 
     An advantage of using harmonics is that all the waveforms in the set repeat at the period of the fundamental frequency, thereby providing a repeating waveform profile of the combined waveform. In many vibration applications each vibration actuator generates a force with a repeated waveform that has a zero DC component and the combined force is described by Eq. 37. Accordingly, the combined vibration force does not have a DC component. Fourier synthesis is widely used in create a wide range of waveforms. One example waveform is a Sawtooth waveform, which creates a sudden change of force in one direction. In this manner, the Sawtooth waveform can be used to generate directional haptic cues. When the set of waveforms consists of three sine waves, the Sawtooth waveform can be generated with the first harmonic at relative amplitude 1, the second harmonic is at relative amplitude of ½, and the third linear sine wave with a relative amplitude of ⅓. With Fourier waveform synthesis, arbitrary waveforms can be approximated including both symmetric and asymmetric waveforms. When using Fourier waveform synthesis, both constructive and destructive interference can occur for both the positive and negative forces amplitudes. 
     An operating advantage of an LRA is to use resonance to generate high magnitude vibration forces from a relatively low power input, and an LRA can be designed and manufactured to have a specific resonant frequency by optimizing its spring stiffness and moving mass. In embodiments of General Synchronized Vibration, it can be advantageous to select a set of LRAs with resonant frequencies that correspond to at least some of the harmonics of a desired waveform. For example for a vibration device such as that in  FIG. 49  with a set of n LRAs, the first LRA  1102   a  could have a specified resonant frequency of ω 1 , the second LRA  1102   b  could have a specified resonant frequency of 2ω 1 , the third LRA could have a specified resonant frequency of 3ω 1 , and so on through the nth LRA  1102   n.    
     Although LRAs are generally designed to operate at their resonant frequency, one can operate LRAs at other frequencies with lower amplitude force output per input command signal. Since lower amplitude force output is typically required at higher harmonics, once could build a Vibration Device with LRAs that all have the same resonant frequency, but operate them at different frequencies. For example for a vibration device with a set of 2 LRAs, both LRAs could have a specified resonant frequency of (3/2)ω 1 , where the first LRA is driven at ω 1 , and the second LRA is driven at 2ω 1 . In this configuration both LRAs are amplifying the input signal, but less than if they were driven at the resonant frequency of the LRAs, which is (3/2)ω 1  for this example. 
     Asymmetric Vibration waveforms are useful for generating directional haptic cues, and can be synthesized using Fourier synthesis. For instance, an example of a method for selecting frequency, phase, and amplitude of sinusoidal vibrations to generate a high level of asymmetry is discussed below. Vibration parameters are specified for a set of 2, 3, and 4 actuators. In addition a process is presented for identifying parameters for waveforms with a high level of vibration asymmetry for any number of actuators. It should be noted that high levels of asymmetry may be achieved even if the values specified by this example are only approximately implemented. For instance, in the case of superposition of two sine waves, if there is a 30% error in the amplitude of vibration then 90% of desired asymmetry effect will still be realized. 
     Fourier synthesis allows one to approximate an arbitrary waveform with a superposition of sinusoidal waves. However, it is advantageous in some applications to generate asymmetric waveforms that have higher peak magnitudes in the positive direction than in the negative direction (or vice versa). The question then becomes what is the best function to approximate that will maximize the amount of asymmetry for a given number of superimposed sine waves? It is of special interest to consider asymmetric waveforms that have a zero DC component and thus can be composed solely of sine waves. Waveforms with a zero DC component can be used to generate vibrations from a set of vibrators since each vibrator will typically have a zero DC component. An asymmetric pulse train is illustrated in  FIG. 50 . The pulse-train is just one example of an asymmetric waveform, but it is a useful example. For the pulse-train to have a zero DC component, the area above the axis. Thus: 
                 W   ·   P     =       (     T   -   W     )     ⁢   V       ,   and                 V   =       W   ·   P       (     T   -   W     )         ,         
where W is the pulse width, V is valley amplitude, T is period of repeated pulse, and P is peak amplitude.
 
     The amount of asymmetry in a pulse-train can be defined by the percentage increase of P over V. One could increase the amount of asymmetry by reducing W, which would generate a thin and high pulse. However, if W is too small, the waveform would not be well-approximated with a small number of sine waves. Accordingly, an analytical question is, “What is the optimal value of W for a waveform composed of N sine waves?” 
       FIG. 51  illustrates a pulse-train with zero DC component. Given this waveform, one may find its Fourier coefficients according to the Fourier series: 
                 f   ⁡     (   t   )       =       a   0     +       ∑     n   =   1     N     ⁢     (         a   n     ⁢     sin   ⁡     (     2   ⁢   π   ⁢           ⁢   n   ⁢           ⁢   t     )         +       b   n     ⁢           ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢   n   ⁢           ⁢   t     )           )           ,         
where f(t) is an arbitrary waveform and when a 0 =0 it have zero DC component. The Fourier coefficients can be calculated by multiplying both sides of the above equation by sin(2 π n t) or cos(2 π n t) and then canceling out terms. The coefficients are:
 
     
       
         
           
             
               
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                             ⁢ 
                             nT 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           P 
                           + 
                           V 
                         
                         ) 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             nW 
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     In a similar fashion: 
     
       
         
           
             
               
                 b 
                 n 
               
               2 
             
             = 
             
               
                 
                   ∫ 
                   0 
                   W 
                 
                 ⁢ 
                 
                   P 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
               + 
               
                 
                   ∫ 
                   W 
                   T 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       - 
                       V 
                     
                     ) 
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 b 
                 n 
               
               2 
             
             = 
             
               
                 ( 
                 
                   1 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     n 
                   
                 
                 ) 
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       ( 
                       
                         P 
                         + 
                         V 
                       
                       ) 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           W 
                         
                         ) 
                       
                     
                   
                   - 
                   
                     V 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           nT 
                         
                         ) 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     By substituting in v from the equation above, the result is: 
     
       
         
           
             
               
                 a 
                 n 
               
               2 
             
             = 
             
               
                 
                   
                     - 
                     PT 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         W 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   PW 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         T 
                       
                       ) 
                     
                   
                 
                 + 
                 PT 
                 - 
                 PW 
               
               
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   nT 
                 
                 - 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   nW 
                 
               
             
           
         
       
       
         
           
             
               
                 b 
                 n 
               
               2 
             
             = 
             
               - 
               
                 
                   
                     PW 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         T 
                       
                       ) 
                     
                   
                   - 
                   
                     PT 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           W 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     nT 
                   
                   - 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     nW 
                   
                 
               
             
           
         
       
     
       FIG. 52  is a flow diagram illustrating a process for maximizing asymmetry. As shown in the flow diagram, the process includes selecting a number of sine waves, and then guessing (estimating) values for W. Fourier coefficients are then calculated, and the time domain of the wave, f(t), is generated according to the equation set forth above. The amount of asymmetry in f(t) is then calculated. The process may be repeated with different values for W, and the value for W is selected that gives the most asymmetry. 
     Fourier coefficients can be represented by a n  and b n  as: 
     
       
         
           
             
               f 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 a 
                 0 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   N 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         a 
                         n 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         b 
                         n 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     An alternative representation using sine waves and phase is: 
     
       
         
           
             
               f 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 A 
                 0 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   N 
                 
                 ⁢ 
                 
                   
                     A 
                     n 
                   
                   ⁢ 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         + 
                         
                           ϕ 
                           n 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     To relate the two representations, the addition of sines formula:
 
sin(α+β)=sin(α)cos(β)+cos(α)sin(β)
 
may be used with:
 
α=2 πnt  
 
β=φ n  
 
 A   n  sin(2 πnt+φ   n )= A   n  cos(φ n )·sin(2 πnt )+ A   n  sin(φ n )cos(2 πnt )
 
     Let 
     
       
         
           
             
               a 
               n 
             
             = 
             
               
                 
                   A 
                   n 
                 
                 ⁢ 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       ϕ 
                       n 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   b 
                   n 
                 
               
               = 
               
                 
                   
                     
                       A 
                       n 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           ϕ 
                           n 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ∴ 
                   
                     A 
                     n 
                   
                 
                 = 
                 
                   
                     
                       
                         a 
                         n 
                         2 
                       
                       + 
                       
                         b 
                         n 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             A 
                             n 
                             2 
                           
                           ⁢ 
                           
                             
                               cos 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 n 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             A 
                             n 
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               sin 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                     
                     = 
                     
                       A 
                       n 
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               ϕ 
               n 
             
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     b 
                     n 
                   
                   
                     a 
                     n 
                   
                 
                 ) 
               
             
           
         
       
     
     In one scenario, the process shown in  FIG. 52  was implemented for a range of sine waves according to the table below. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE I 
               
               
                   
               
               
                 NACT 
                 W 
                 Asym 
                 A 1   
                 φ 1   
                 A 2   
                 φ 2   
                 A 3   
                 φ 3   
                 A 4   
                 φ 4   
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 2 
                 0.33 
                 100% 
                 1 
                 30° 
                 0.5 
                 −30° 
                   
                   
                   
                   
               
               
                 3 
                 0.25 
                 189% 
                 1 
                 45° 
                 0.71 
                  0° 
                 0.33 
                 −45° 
               
               
                 4 
                 0.2 
                 269% 
                 1 
                 54° 
                 0.81 
                  18° 
                 0.54 
                 −18° 
                 0.25 
                 −54° 
               
               
                   
               
            
           
         
       
     
     The variable “NACT” in Table I is used to define the number of sine waves since it can also represent the number of actuators. For two sine waves, an asymmetry of 100% can be achieved, which indicates there is twice the magnitude in the positive direction (or vice versa). Higher numbers of sine waves can provide even higher amounts of asymmetry as shown in Table I. One example is shown in  FIG. 44 . Other examples are shown in  FIGS. 53-55 . 
     General Synchronized Vibration can be performed with a set of non-sinusoidal waveforms. Even without use of Fourier synthesis, asymmetric waveforms can be generated by synchronizing the waveforms to create positive interference of two or more waveforms in one direction, and negative interference of two or more waveforms in the opposite direction. Embodiments with non-sinusoidal waveforms can still have the peaks of two or more waveforms occur simultaneously with positive interference in one direction and also occur simultaneously with negative interference in the opposite direction. 
       FIG. 56  shows two triangular waveforms that are synchronized together. Profile  1112   a  has twice the amplitude of profile  1112   b , while profile  1112   b  vibrates at twice the frequency of profile  1112   a . The peaks of profile  1112   a  and  1112   b  occur simultaneously, at times with positive interference and at times with negative interference. The combined waveform of profile  1112   a  and  1112   b  will generate an asymmetric waveform in a similar fashion that the combined waveform in  FIG. 44 . 
     To create an especially distinct vibration effect, some LRA vibration actuators can be operated at an amplitude high enough to push the moving mass into the travel stops, thereby creating an impact force during each oscillation. The impact with the travel stops will generate a vibration waveform that is not sinusoidal. Multiple such actuators can be synchronized together to generate positive and negative interference as instances of impacts of masses with travel stops. This configuration can generate sharp peaks of vibration force, where direction of vibration is controllable. These sharp peaks of vibrations could be used to generate haptic sensations corresponding to impacts such as simulating the recoil of a gun. A wide range of vibration effects can be generated with non-sinusoidal vibrations. Examples are presented herein that use sine wave vibration waveforms, with the understanding that similar approaches could be generated with other waveforms. 
     One waveform that can be simulated is referred to as a “missing fundamental” waveform, which takes advantage of a phenomenon of human perception. As explained in “Music and Connectionism” by Peter M. Todd, D. Gareth Loy, MIT Press 2003, humans may perceive that a sound contains pitch of a certain frequency even though that frequency is not present in the sound if the sound contains higher frequencies that are integer multiples of the low frequency. In haptic applications, low frequency vibrations may be difficult to generate due to size and power constraints, while it may be easier to generate higher frequency vibrations. A vibration waveform can be generated that does not contain a desired low frequency, but does include higher frequencies at integer multiples of the desired low frequency. A person may perceive the desired low frequency vibration, just as they perceive the missing fundamental in a sound. The perception of a missing fundamental in vibration can be enhanced by including audio or visual effects at the desired low frequency. 
     The embodiment shown in  FIG. 57  can generate asymmetric torques about the mounting platform. A pair of LRAs  1116   a  and  1116   b  are mounted towards the top of the mounting platform  1100 . A second pair of LRAs  1118   a  and  1118   b  are mounted towards the bottom of mounting platform  1100 . When the top pair of LRAs is operated with the same magnitude but opposite direction force than the bottom pair, a pure torque is generated on the mounting platform. When both the top and bottom pair vibrate with an asymmetric waveform, such as that shown in  FIG. 44 , then the torque vibration is also asymmetric and can apply a higher peak torque in the clockwise direction than the counterclockwise direction (or vice versa). Furthermore, the amplitude of the asymmetric torque vibration may be controlled by proportionally controlling the peak force in each LRA. 
     LRAs generate vibration forces along an axis and thus are described as “Linear Force Actuators.” Other Linear Force Actuators include slider-crank vibrators, rack and pinion vibrators, linear actuators that do not use resonance, pistons, and solenoids. Rocking actuators and pivoting actuators (such as described in U.S. patent application Ser. No. 11/476,436) generate forces that are approximately along an axis and for many applications can be considered Linear Force Actuators. Indeed, any embodiment described herein as employing LRAs can also be implemented with Linear Force Actuators or other actuators that generate forces that are approximately along an axis. 
     A controller for General Synchronized Vibration of a pair of Linear Force Actuators is shown in  FIG. 58 , which could control embodiments such as that shown in  FIG. 43 . A vibration device controller generates commands of frequency, f, commanded amplitudes, Ac, and commanded phase pc. A driver circuit generates the voltage and current that drives the actuators. The driver circuit may output a waveform of a sine wave, square wave, triangle wave, or other waveform. The actuator may generate a force waveform that is similar to the waveform output of the driver circuit. Alternatively, the actuator may generate a force waveform that differs from the waveform output of the driver circuit. For example, the driver circuit may output a square wave but the actuator may generate a force that is mostly a sine wave due to the physics of the actuator. 
     Both LRA and ERM Vibration Actuators take some time to ramp up to speed to generate their maximum force output. Embodiments described herein include controllers that may or may not synchronize the actuators during the ramp up period. In addition, a Vibration Device may be commanded to transition from one vibration effect to another vibration effect. During this transition time interval, the controller may or May not synchronize the actuators. 
     A vibration device controller can be a microprocessor or other programmable device. For each actuator in the vibration device, the vibration device controller can modify the frequency of vibration, the phase of vibration, the amplitude of vibration, or any combination of these parameters. The ability to change these parameters allows for a single vibration device to generate a wide range of waveforms. 
     The phase and amplitude of the force output of a Vibration Actuator depends on both the control signal and the physical characteristics of the actuator. For example there is often a phase lag between the control signal and the force output of the actuator. To distinguish between the waveform of the actuator outputs and the waveform of the control signal, the subscript “c” notation is used to designate the control waveform. Thus the commanded amplitudes, Ac, and the commanded phase pc are not necessarily a direct correlation to the actual amplitude and phase of the actuator force. For example, the command voltage, V, of a vibration device controller of an LRA actuator driven with a sinusoidal voltage signal at a frequency ω, with a command phase of φ c , and a voltage peak magnitude A c , given by:
 
 V=A   c  sin(ω t+φ   c )  (38)
 
     However, due to the phase lag inherent in the actuator and frequency response of the actuator, the steady state force output of the actuator, F a , may be given by:
 
 F   a   =A  sin(ω t +φ)  (39)
 
     The phase lag is the difference between φ and φ c . The frequency response is reflected in the ratio between A c  and A. Both the phase lag and the frequency response are functions of the actuator physics that can vary with vibration frequency, and which is often represented by an actuator specific Bode plot. For effective implementation of synchronized vibration it can be advantageous to take into consideration the phase lag inherent in each vibration actuator. This can be done by adding an equal but opposite phase offset to the controller waveform so that the actuator phase lag does not impact synchronization. 
     One method to implement this offset is to use a look up table, Bode plot, or algorithm for each actuator that determines the appropriate phase offset for a given vibration frequency. In addition, it can be advantageous to use a lookup table, Bode plot, or algorithm to determine the required voltage magnitude needed to generated the desired vibration force magnitude. The Fourier synthesis approach and the approach of matching positive and negative peaks of vibration described herein are implemented in reference to the actual phase of the actuator force output rather than the phase of the waveform from the actuator drive circuits. In order to simplify notation herein, the phase lag due to the actuator physics is generally not included in the equations relating to synchronization. Rather a more compact notation is used which represents the vibration force output, F, with the understanding that the appropriate command signal is generated to provided that output. The command signal includes the necessary phase lag and magnitude adjustment as needed based upon the actuator physics. The magnitude control can be implemented with a voltage, current, PWM signal of voltage or current, or other type of command used to drive said actuator. The Fourier synthesis approach and the approach of matching positive and negative peaks of vibration describe specific target frequency and phase of vibration for actuators within the vibration device; however, even if these target frequency and phase are not exactly met, the overall vibration effect often is close enough to the desired waveform to achieve a desired effect. 
     Due to manufacturing variations, two actuators that are built on the same assembly line may have different physical characteristics that affect their Bode plot, including phase lag, amplitude characteristics or resonant frequency. In some embodiments a sensor or sensors can be used to detect the phase of an actuator, the amplitude of vibration of an actuator, or the amplitude and phase. Such a sensor could be an optical sensor, Hall-effect sensor or other type of sensor that detects when a moving mass passes the midpoint or other point of vibration. One such embodiment is shown in  FIG. 59 , where a sensor  1128  is integrated into to a Linear Force Actuator  1124  and detects when the moving mass  1126  reaches passes a midpoint position. A sensor integrated into an actuator can provide continuous, continual or periodic measurement of actuator performance and be used to update calibration parameters while the device is in use and does not require a specified calibration period. 
     Another method of sensing is to attach actuators  1124   a  and  1124   b  to the Mounting Platform  1100  of the vibration device  1134  as shown in  FIG. 60 . This sensor  1136  could be an accelerometer or other sensor that measures the combined motion or combined force of the mounting platform. 
     The sensor measurements can be used to self-calibrate the vibration devices. A test pattern can operate each actuator separately to identify the actuator phase lag, force amplitude characteristics, and resonant frequency. These characteristics can be used to update a lookup table, Bode plot, or algorithm used to generate the voltage commands to the actuators. The combined force of multiple actuators can also be measured to confirm that the desired force effects are being achieved. Accordingly, the vibration device controller can use the sensor measurements to update the commanded amplitude, phase, and frequency as shown in  FIG. 61 . 
     Embodiments of the disclosure also include configurations with multiple sets of aligned vibration actuators. One such configuration is shown in  FIG. 62  that includes two sets of actuators. Set  1  consists of two LRAs  1138   a  and  1138   b  that are both aligned with the x axis of the vibration device  1134 . Set  2  consists of two LRAs  1140   a  and  1140   b  that are both aligned with the y axis of the vibration device. Set  1  generates force F S1B1  from LRA  1138   a , and generates force F S1B2  from LRA  1138   b . Set  2  generates force F S2B1  from LRA  1140   a , and generates force F S2B2  from LRA  1140   b.    
     In the embodiment shown in  FIG. 62 , the combined vibration force is the vector sum of all the vibration actuators. Using the notation of U.S. patent application Ser. No. 11/476,436, a 1  and a 2  are unit vectors aligned with the forces from set  1  and set  2  respectively. In one control approach for the embodiment shown in  FIG. 62 , the waveforms of both sets are controlled to have similar shapes but with different magnitudes. Magnitude coefficients are designated by the variable A, where the scalar A 1  multiplies the waveform of set  1  and the scalar A 2  multiplies the waveform of set  2 . The combined force vector, F combined , with this control approach with sinusoidal waveforms is given by:
 
 F   combined   =a   1   A   1 ( B   1  sin(ω 1   t+φ   1 )+ B   2  sin(ω 2 φ 2 ))+ a   2   A   2 ( B   1  sin(ω 1   t+φ   1 )+ B   2  sin(ω 2   t+φ   2 ))  (40)
 
     As described in U.S. patent application Ser. No. 11/476,436, there are methods for selecting the magnitude of A 1  and A 2  that will generate a desired direction for the vector F combined , yet these methods may only specify the axis of vibration and not whether the magnitude of force is positive or negative and thus limit the range of unique direction of vibrations to a range of 180 degrees. According to one aspect of the disclosure, an embodiment allowing control of the direction of vibration in all 360 degrees of the plane of the Mounting Platform, may have the following parameter relationships: 
     ω 2 =2ω 1 ; 
     φ 1 =0 and φ 2 =−90 for a direction between −90 and +90 degrees; 
     φ 1 =0 and φ 2 =90 for a direction between 90 and 270 degrees; 
     A 1  and A 2  specified by equation 19 above 
     Numerous other embodiments are possible with multiple sets of aligned vibration actuators. Each set of aligned actuators can generate an arbitrary waveform, p AlignedSet . Embodiments of synchronized vibrations created from arbitrary shaped profiles are described above. Many such embodiments show a single actuator generating each waveform. However, it is also possible to have a set of aligned actuators create these waveforms. Therefore, such embodiments can be expanded to include configurations where a set of aligned actuators take the place of a single actuator. In these configurations, the arbitrary waveform profiles would take the form of the arbitrary waveform, p AlignedSet  as discussed herein. 
     Accordingly, embodiments of asymmetric vibration include 3D configurations and non-orthogonal configurations. An example of two non-orthogonal LRA Pairs is shown in  FIG. 63 . These LRAs can generate waveforms in desired directions throughout the xy plane. The actuators in each aligned set can be LRAs, rocker actuators, and other sets of actuators that generate approximately linear forces. An equation describing the combined force vector for M aligned sets with all sets having similar shaped waveforms but potentially different magnitudes is given by:
 
 F   combined   =a   1   A   1 ( p   AlignedSet )+ a   2   A   2 ( p   AlignedSet )+ . . . + a   M   A   M ( p   AlignedSet )  (41)
 
     The approaches used to determine the values of A described above can be applied to these configurations as well. A variety of Lissajous vibration patterns are also described above, including lines, circles, ellipses, parabolas, etc. Asymmetric vibration waveforms can be used to produce larger peak forces during one part of the Lissajous vibration pattern than another part. 
     Turning to another aspect of the disclosure, an ERM is depicted in  FIG. 64 . A basic ERM includes a motor  1204 , a shaft  1208 , and an eccentric mass  1206 . The motor  1204  could be a DC brushed motor, a DC brushless motor, an AC induction motor, stepper motor, or any other device that turns electrical energy into rotary motion. The shaft  1208  is a power transmission element that transmits the rotary motion of the motor into rotary motion of the eccentric mass. However, alternate power transmission methods could be any means of transmitting the rotary motion of the motor  1204  into rotary motion of the eccentric mass  1206 , such as a belt, gear train, chain, or rotary joint. The eccentric mass  1206  could be any body that spins on an axis that is not coincident with its center of mass. Furthermore, the power transmission element may include geometry such that the axis of rotation of the eccentric mass  1206  is not necessarily coincident or parallel to the rotation axis of the motor  1204 , and the eccentric mass  1206  does not necessarily rotate at the same angular velocity as the motor  1204 . 
     One method of generating vibration forces is with an ERM where an eccentric mass is attached to motor shaft. As the motor rotates, centrifugal forces are generated onto the motor. General Synchronized Vibration can be applied to multiple ERMs by controlling the frequency and phase of rotation of the eccentric masses.  FIG. 65  shows one embodiment for a vibration device  1200  that uses an arbitrary number M ERMS; the first two being ERM  1210   a ,  1210   b  and the last being  1210   m . All ERMs are attached to a mounting platform  1202  and the combined vibration force of the device is the vector sum from all ERMs. 
     For the ith ERM, A i  is the amplitude of the vibration force, ω i  is the frequency of vibration, and φ i  is the phase of vibration. The combined vibration force of the ERMs in  FIG. 65  is given in the x and y coordinates by:
 
 F   Ex   =A   1  cos(ω 1   t+φ   1 )+ A   2  cos(ω 2   t+φ   2 )+ . . . + A   M  cos(ω M   t+φ   M )
 
 F   Ey   =A   1  sin(ω 1   t+φ   1 )+ A   2  sin(ω 2   t+φ   2 )+ . . . + A   M  sin(ω M   t+φ   M )
 
       FIG. 66  shows one embodiment for a vibration device  1200  that uses four ERMS  1212   a ,  1212   b ,  1214   a , and  1214   b . All four ERMs are attached to a mounting platform  1202  and the combined vibration force of the device is the vector sum from all four ERMs. 
     The force and torque imparted by an ERM onto a mounting platform are due to a combination of the centrifugal force from the rotating eccentric mass, the torque between the stator and rotor of the motor and other inertial forces such as gyroscopic effects. As the speed of the ERM increases the centrifugal force increases and typically becomes the dominant portion of the vibration force. Accordingly, once an ERM has sped up, the vibration force imparted by an ERM onto the mounting platform is close to the centrifugal force imparted by the rotating eccentric mass. 
     In one embodiment, the ERMs are configured in counter-rotating pairs, where each ERM in a pair has the same eccentric mass and operates at the same angular speed but the ERMs rotate in opposite directions from each other.  FIG. 4  ERM shows such an embodiment with a first counter-rotating pair consisting of ERM  1212   a  and ERM  1212   b . The combined vibration force of just this first pair is given by:
 
 F   E1x   =A   1  cos(ω 1   t+φ   1 +σ 1 )+ A   1  cos(−ω 1   t−φ   1 σ 1 )
 
 F   E1y   =A   1  sin(ω 1   t+φ   1 +σ 1 )+ A   1  sin(−ω 1   t−φ   1 +σ 1 )
 
     The phase difference between ERM  1212   a  and ERM  1212   b  is represented by two variables, φ 1  and σ 1 , where φ 1  represents a temporal phase and is half of the difference in overall phase and σ 1  represents a geometric angle and is half of the average of the overall phase difference. For an ERM the magnitude of the vibration force, A, is equal to mrω 2 , where m is the mass, r is the radius of eccentricity, and ω is the speed of angular rotation in radians per second. Through trigonometric identities, this combined vibration force vector of the first ERM pair can be represented by the equation below. In this configuration, the force from a single counter-rotating pair generates a sinusoidal vibration force aligned with an axis of force direction defined by the angle σ 1 . 
     
       
         
           
             
               
                 
                   
                     F 
                     
                       E 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       A 
                       1 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ω 
                               1 
                             
                             ⁢ 
                             t 
                           
                           + 
                           
                             ϕ 
                             1 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ⌊ 
                       
                         
                           
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   σ 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   σ 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ⌋ 
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     The embodiment in  FIG. 66  has a second counter-rotating pair formed by ERM  1214   a  and ERM  1214   b , with both ERMS having the same eccentric mass as each other and operating at the same angular speed as each other but in opposite directions. This second counter-rotating pair generates a combined vibration force of:
 
 F   E2x   =A   2  cos(ω 2   t+φ   2 +σ 2 )+ A   2  cos(−ω 2   t−φ   2 +σ 2 )
 
 F   E2y   =A   2  sin(ω 2   t+φ   2 +σ 2 )+ A   2  sin(−ω 2   t−φ   2 +σ 2 )
 
     In one control method, σ 1  and σ 2  are set equal to the same value, σ, and therefore both ERM pairs generate a vibration along the same axis and the combined vibration force vibration force vector of all four ERMs is given by: 
     
       
         
           
             
               
                 
                   
                     F 
                     E 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         A 
                         1 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ω 
                                 1 
                               
                               ⁢ 
                               t 
                             
                             + 
                             
                               ϕ 
                               1 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                         
                         ⌋ 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         A 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ω 
                                 2 
                               
                               ⁢ 
                               t 
                             
                             + 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                         
                         ⌋ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     In another control method, σ 2  is set equal to n+σ 1  and therefore both ERM pairs generate a vibration along the same axis but the contribution from the second ERM pair has a negative sign. With this method the combined vibration force vibration force vector of all four ERMs is given by: 
     
       
         
           
             
               
                 
                   
                     F 
                     E 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         A 
                         1 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ω 
                                 1 
                               
                               ⁢ 
                               t 
                             
                             + 
                             
                               ϕ 
                               1 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                         
                         ⌋ 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         A 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ω 
                                 2 
                               
                               ⁢ 
                               t 
                             
                             + 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                             
                           
                         
                         ⌋ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     There are similarities between the application of General Synchronized Vibration to Linear Force Actuators and ERMs. In both cases, the combined vibration force can be composed of a superposition of sine waves, and in both cases it is possible to implement asymmetric vibrations. One embodiment asymmetric vibration uses the relative magnitudes and phases for superposition of two sinusoidal waves. In this embodiment, the amplitude of the fundamental frequency is twice that of the second harmonic. For the embodiment shown in  FIG. 66 , a configuration for high asymmetry is shown in Table II below. The Geometric Angle, σ, can be selected arbitrarily based upon the desired direction of vibration. The eccentricity of the second ERM pair is represented relative to the eccentricity of the first ERM pair. The speed of rotation of the second ERM pair is twice the speed of rotation of the first ERM pair. It should be noted that high levels of asymmetry may be achieved even if the values specified in Table II are only approximately implemented. For example, in the case of superposition of two sine waves, if there is a 30% error in the amplitude of vibration, then 90% of desired asymmetry effect may still be realized. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE II 
               
               
                   
               
               
                   
                 Centrifugal 
                   
                 Geometric 
                 Frequency of 
                 Temporal 
               
               
                   
                 Force 
                 Eccentric- 
                 Angle, σ 
                 Rotation 
                 Phase, φ 
               
               
                 ERM 
                 Magnitude 
                 ity 
                 (radians) 
                 (radians/sec) 
                 (radians) 
               
               
                   
               
             
            
               
                 1212a 
                 A 1   
                 m 1 r 1   
                 σ 
                  ω 1   
                 0 
               
               
                 1212b 
                 A 1   
                 m 1 r 1   
                 σ 
                 −ω 1   
                 0 
               
               
                 1214a 
                 (½)A 1   
                 (⅛)m 1 r 1   
                 σ 
                 2ω 1   
                 0 (or n) 
               
               
                 1214b 
                 (½)A 1   
                 (⅛)m 1 r 1   
                 σ 
                 −2ω 1   
                 0 (or n) 
               
               
                   
               
            
           
         
       
     
     Steps of General Synchronized Vibration are shown in  FIG. 67  for the case of a configuration shown in Table II. The time, t, is represented in terms of the period of the fundamental frequency, where T 1 =2π/ω 1 . As seen in the uppermost illustration of  FIG. 67 , at time t=0, the forces of all ERMs are aligned with the axis of vibration in the positive direction, and the position of the eccentric masses are all aligned in the same orientation. Accordingly, at t=0 the combined vibration force has a large magnitude. At t=T 1 /4 as shown in the center illustration, the combined force vector is in the negative direction along the axis of vibration, yet the negative magnitude is not at a peak value since contribution only occurs from ERM  1214   a  and ERM  1214   b , while the forces from ERM  1212   a  and ERM  1212   b  cancel each other out. At t=T 1 /2, as shown in the bottom illustration, the combined force vector is also in the negative direction along the axis of vibration, yet the negative magnitude is not at a peak value since there is negative interference between the first ERM pair (ERM  1212   a  and  1212   b ) and the second ERM pair (ERM  1214   a  and  1214   b ). At t=T 1 /2 the forces of the first ERM pair are in the opposite direction of the forces from the second ERM pair, and the orientation of the eccentric masses of the first ERM pair is 180 degrees opposite the orientation of the eccentric masses of the second ERM pair. Accordingly, asymmetric vibration is generated with a larger peak force occurring along the positive direction aligned with the axis of vibration. As shown in Table II the temporal phase of ERMs  1214   a  and  1214   b  can also be set to n, in which case asymmetric vibration will occur with a larger peak force along the negative direction aligned with the axis of vibration. 
     Embodiments are possible with a plurality of ERM pairs, as shown in  FIG. 68  which has N ERM pairs; the first two pairs being  1216   a  and  1216   b , and the last pair  1216   n . In one control method the first ERM pair  1216   a  is rotated at a fundamental frequency, the second ERM pair  1216   b  is rotated at twice the fundamental frequency, and so on through all N pairs with the Nth pair  1216   n  rotating at N times the fundamental frequency. Using Fourier synthesis it is possible to approximate a wide range of waveforms. 
     In the embodiment shown in  FIG. 68 , each ERM within a pair can have the same eccentricity, and each pair can be controlled so that one ERM in the pair rotates in the opposite direction of the other ERM with the same rotational speed. Asymmetric vibrations can be generated that have a higher peak force in a direction relative to the peak force in the opposite direction. High amounts of asymmetry can be generated using the process discussed above with regard to  FIG. 52  (and Table I), which specifies magnitudes and phases for each harmonic sine wave. The magnitude of vibration of an ERM is the product of the eccentricity, mr, and the angular velocity, ω, squared. Accordingly, the eccentricity of the ith ERM as a function of the relative sine wave amplitude is given by:
 
 m   n   r   n =( A   n   /A   1 ) m   1   r   1   /n   2   (45)
 
     The phases may be represented relative to the starting time of a specific waveform of pulse-trains being approximated. In some implementations it is more convenient to set the phase of the first harmonic to zero and represent the phases of the other harmonics relative to the first harmonic. An equation that converts the phase of the nth harmonic, φ n , to a phase of the nth harmonic relative to the first harmonic, is given by:
 
φ rn =φ n −(ω n /ω 1 )φ 1   (46)
 
     In addition, the phases may be defined relative a series of sine waves, while the ERM vibration equation Eq. 42 is specified in terms of a cosine wave. A cosine wave is a sinusoidal wave, but the phase is shifted by 90 degrees from a sine wave. Table I shows parameters for embodiments that superimpose 2, 3, and 4 sine waves. These parameters can be converted to relevant parameters for embodiments with 2, 3, and 4 ERM pairs, using Eq. 45 and Eq. 46 along with the 90-degree shift for the cosine representation. Table III, provided below, shows these parameters for ERM pairs which generate high levels of asymmetry. The method described in  FIG. 52  can be used to specify parameters for any number of ERM pairs. 
     
       
         
           
               
               
               
               
             
               
                 TABLE III 
               
               
                   
               
               
                 Number of ERM Pairs 
                 2 
                 3 
                 4 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 Pair 1 Amplitude: A1 
                 1 
                 1 
                 1 
               
               
                 Pair 1 Eccentricity 
                 m 1 r 1   
                 m 1 r 1   
                 m 1 r 1   
               
               
                 Pair 1 Relative Phase φ r1   
                 0 
                 0 
                 0 
               
               
                 (degrees) 
               
               
                 Pair 2 Amplitude: A2 
                 0.5 
                 0.71 
                 0.81 
               
               
                 Pair 2 Eccentricity 
                 0.125 m 1 r 1   
                 0.1775 m 1 r 1   
                 0.2025 m 1 r 1   
               
               
                 Pair 2 Relative Phase φ r2   
                 180 
                 180 
                 180 
               
               
                 (degrees) 
               
               
                 Pair 3 Amplitude: A3 
                   
                 0.33 
                 0.54 
               
               
                 Pair 3 Eccentricity 
                   
                 0.0367 m 1 r 1   
                  0.060 m 1 r 1   
               
               
                 Pair 3 Relative Phase φ r3   
                   
                 270 
                 270 
               
               
                 (degrees) 
               
               
                 Pair 4 Amplitude: A4 
                   
                   
                 0.25 
               
               
                 Pair 4 Eccentricity 
                   
                   
                 0.0156 m 1 r 1   
               
               
                 Pair 4 Relative Phase φ r4   
                   
                   
                 0 
               
               
                 (degrees) 
               
               
                   
               
            
           
         
       
     
     Implementing General Synchronized Vibration with ERMs has an advantage that a wide range of vibration frequencies can be generated without being restricted to a specific resonance range. As the ERM frequency increases the centrifugal forces increase, the ratio of waveform amplitudes of A 1  and A n  remains constant. Accordingly, high levels of asymmetric vibrations can be generated with a single ratio of eccentricity, as shown in Table II and Table III, over an arbitrary frequency. 
     An embodiment with four ERMS is shown in  FIG. 69 . ERMS  1222   a ,  1222   b ,  1224   a  and  1224   b  are stacked vertically inside a tube  1220 , which serves as the mounting platform  1202 . This embodiment could be used as a user input device which is grasped by the hand, similar to how the PlayStation® Move motion controller is grasped. Configurations with stacked ERMs are convenient for a wide range of hand held devices and to apply vibration forces to a wide range of body parts. 
     Steps of General Synchronized Vibration are shown in  FIG. 70  for the case of a configuration shown in  FIG. 69 . The parts shown in  FIG. 69  are the same parts as shown in  FIG. 70 , but part numbers are not called out in  FIG. 70 . Each frame of  FIG. 70  shows the eccentric masses of the ERMs and a line extending from each mass indicates the centrifugal force vector that the mass generates. The combined force vector of all ERMs is shown by the thicker line under the ERMs. In the embodiment shown in  FIG. 70  the top two ERMs  1222   a  and  1224   b  are rotating clockwise from the top view perspective, and the bottom two ERMs  1222   a  and  1224   b  are rotating counter-clockwise. Furthermore the top  1224   b  and bottom  1224   a  ERMs have lower eccentric masses and are rotating at twice the frequency of the middle two ERMs  1222   a  and  1222   b.    
     Other embodiments are possible with different frequency and mass relationships. The time, t, is represented in terms of the period of the fundamental frequency, where T 1 =2π/ω 1 . As seen in  FIG. 70 , at time t=0, the forces of all ERMs are aligned with the axis of vibration in the positive direction, and the position of the eccentric masses are all aligned in the same orientation. Accordingly, at t=0 the combined vibration force has a large magnitude. At t=2T 1 /8 the combined force vector is in the negative direction along the axis of vibration, yet the negative magnitude is not at a peak value since contribution only occurs from ERM  1224   a  and ERM  1224   b , while the forces from ERM  1222   a  and ERM  1222   b  cancel each other out. At t=4T 1 /8 the combined force vector is also in the negative direction along the axis of vibration, yet the negative magnitude is not at a peak value since there is negative interference between the first ERM pair (ERM  1222   a  and  1222   b ) and the second ERM pair (ERM  1224   a  and  1224   b ). At t=4T 1 /8 the forces of the first ERM pair are in the opposite direction of the forces from the second ERM pair, and the orientation of the eccentric masses of the first ERM pair is 180 degrees opposite the orientation of the eccentric masses of the second ERM pair. The magnitude of the combined vibration force is shown by the line beneath the eccentric masses at each point in time. 
     Another vibration device is shown in  FIG. 71 , in which the device contains two ERMs  1230   a  and  1230   b  attached to a mounting platform  1202  that are rotating in the same direction. When the rotational speed and eccentricity of both ERMS are the same, this configuration is referred to as a Co-Rotating Pair, or “CORERM Pair”. The center between the ERM eccentric masses is referred to as the center of the COREMR Pair. When the angle between the two ERMs is kept at a fixed value of angle, c, the CORERM Pair generates a combined centrifugal force that is equivalent to a single ERM. However, the magnitude of centrifugal force of the CORERM Pair is a function of the angle c. When c is equal to zero the combined force magnitude is twice that of a single ERM and when c is equal to 180 degrees then the centrifugal force magnitude is equal to zero since there is no overall eccentricity. Accordingly, when c is close to 180 degrees, the centrifugal force may not be the dominant force output of the CORERM Pair. Instead, gyroscopic or torque effects may take on a larger proportion of the force and torques applied onto the Vibration Device. Where A is the magnitude of force from just one of the ERMs in the pair, ω is the rotational speed, and φ is the phase of rotation, then the combined vibration force generated by a CORERM Pair is given by: 
     
       
         
           
             
               
                 
                   
                     F 
                     CORERM 
                   
                   = 
                   
                     2 
                     ⁢ 
                     A 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             c 
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     A single vibration device could operate similar to ERMs as either counter-rotating pairs or co-rating pairs. There are a number of advantages of operating a vibration device in a mode where some of the ERMs function as CORERMs. One advantage is that the magnitude of vibration can be increased by using a CORERM pair. Another advantage is that legacy vibration effects can be generated that simulate a single ERM rotating. For example, a haptic interface could be operated at one time to generate asymmetric vibration forces and at another time to simulate a single ERM. If users are accustomed to haptic signals from a single ERM, the CORERM pair allows for such familiar effects to be generated. 
     A large number of co-rotating ERMs could by synchronized together in with no phase offset such that their force magnitudes combine to create a vibration effect similar to a single large ERM. If all the co-rotating ERMs are CORERM pairs with co-located centers, then the center for the combined force would be the same as for a single large ERM. 
     Another advantage of using CORERM pairs is that they allow for Fourier syntheses of a wider range waveforms. One such embodiment is to replace each ERM in  FIG. 66  with a CORERM pair, which is shown in  FIG. 72 . ERM  1212   a ,  1212   b ,  1214   a , and  1214   b  in  FIG. 66  correspond to CORERM  1232   a ,  1232   b ,  1234   a , and  1234   b  in  FIG. 72 . Such an embodiment would be similar to the original configuration of  FIG. 66 , but where the magnitude of centrifugal force from each ERM could be adjusted independently of the speed of rotation (by adjusting the angle c within CORERM pairs). Fourier synthesis allows arbitrary waveforms to be approximated with a superposition of sine waves where the amplitude, phase, and frequency of the sine waves can be adjusted. With a sufficiently large number of CORERM pairs, any waveform with a zero-DC offset could be approximated. The embodiment in  FIG. 72  also allows the direction of vibration to be controlled. 
     Control of amplitude of vibration force can be especially useful in asymmetric vibrations used for haptic applications. A vibration device can be grasped by one hand, two hands, held with other body parts, attached to any body part, or placed in contact with any body part. Generally at least two sides of a haptic vibration device are in contact with a user, and each side contacts the user at somewhat different locations on their body. These different locations could be the different sides of a grip of a tube vibration device, such as shown in  FIG. 69 . 
     Human perception often requires that a threshold be exceeded before a sensory event is perceived. In one embodiment, the magnitude of an asymmetric waveform is adjusted so that on one side of a Vibration Device low vibration forces are generated that are below a threshold of perception and on the opposite side higher peak forces are generated that are above a threshold of perception. In this manner, a vibration force may be perceived on mostly one location that is in contact with the vibration device, even though the vibration device is in contact with a number of locations on the body. As the direction of vibration is varies, the location on the body at which vibration is perceived may also vary. This approach uses vibration to generate effects that are vary significantly according to the direction of vibration, and thus are useful for indicating directional cues. 
     An embodiment with 2 ERMs in a tube is shown in  FIG. 73 . In  FIG. 73A  the ERMs  1222   a  and  1222   b  are mounted close to the center of the tube  1220  and thereby reduce the torque vibration that is due to the distance between the ERMs. One way of controlling this configuration is to operate the ERMs in a counter-rotating mode and generate force in a specified direction, with only a small torque vibration so as to minimize distraction from the force effect. In  FIG. 73B  the ERMs  1222   a  and  1222   b  are mounted close to the ends of the tube  1220  and thereby increase the torque vibration that is due to the distance between the ERMs. One way of controlling this configuration is to operate the ERMs in a counter-rotating mode and generate force in a specified direction, while simultaneously generating a large torque vibration effect. 
     The embodiment in  FIG. 69  can also be operated with CORERM pairs. ERMs  1224   a  and  1224   b  can form one pair, and ERMs  1222   a  and  1222   b  can form another pair. When both of these CORERM pairs are rotating in the same direction and have a 180 phase difference, there will be no net force or net torque on the vibration device. However, this embodiment will create a gyroscopic effect with minimal force or torque vibrations. This implementation could be used to generate the sensation of moving a sword or a heavy mass in a video game or other type of simulation. 
     The forces between an ERM and a mounting platform include both centrifugal forces and the motor torque generated between the motor stator and rotor. When an ERM is rotating at operating speed, the centrifugal forces are typically large and dominate the effect from the motor torque. However, some embodiments can bring effects from the motor torque to the forefront. When two ERMs with parallel axes are operated as a co-rotating pair with a phase offset of 180 degrees, the two eccentric masses balance each other out and the centrifugal forces cancel each other out. In this embodiment, the torque about the axes of rotation can be felt more prominently. The torque about the axis of rotation is felt during the acceleration and deceleration of the rotating masses. Higher torques can often be generated by periodically reversing the applied voltage to the motor, since the electromagnetic force (back EMF) in the motor can add to the reverse voltage being applied. 
     Even higher torques about the axis of rotation can be generated by using a brake to cause a sudden deceleration to a rotating mass. This approach is known as a reaction-wheel method for generating torques, and is useful when there is no grounded actuator to apply a torque effect.  FIG. 74  shows an eccentric mass  1206  configured for use as a reaction wheel. A rim  1242  is attached to the eccentric mass  1206 , and creates a surface for a brake  1244  to contact. When the brake  1244  is actuated a relatively high torque can be generated. The reaction-wheel configuration is another example of the wide range of effects that General Synchronized Vibration can generate. A single vibration device can have ERMs that are operated in counter-rotating modes, co-rotating modes, and as reaction-wheels. 
     One embodiment of an ERM Pair uses interleaved masses, an example of which is shown in  FIG. 75 . In this embodiment, the shapes of the eccentric masses are implemented so that the masses can be interleaved within one another yet still rotate independently. With interleaved Masses, both ERMs can share the same axis of rotation. In addition, a mass distribution can be implemented such that the eccentric forces share the same plane (which can be indicated by the height in the side view in  FIG. 75 ). Each ERM in the pair has a rotating mass that includes both an eccentric component, and a symmetric component such as the motor&#39;s rotor. The center of mass of the eccentric mass refers to the center of mass of only the eccentric component of the rotating mass. The center of mass of the eccentric mass rotates about the axis of rotation of the ERM, yet its position can be projected (in a linear algebra sense) onto a single point on the axis of rotation. With interleaved masses the geometry and density of the eccentric masses can be selected such that the center of mass of the eccentric masses from both ERMs are projected onto the same position on the axis of rotation. In this configuration the eccentric forces from both ERMs share the same plane. In  FIG. 75 , ERM 1   1250   a , contains a motor  1252   a  and an eccentric mass  1254   a  which is shaped with a semi-circle cross section, and ERM 2   1250   b , contains a motor  1252   b  and an eccentric mass  1254   b  which is shaped with an arc cross-section. Other shapes of eccentric masses are possible that allow for independent rotation of two masses. 
     In an embodiment with interleaved masses, the ERM pair can generate centrifugal forces without generating a torque due to the distance between the ERMs. Interleaved ERM pairs are useful for generating pure force vibrations without torque vibrations. Interleaved ERM pairs can be operated as a co-rotating pair, and thereby vary the amplitude of vibration independently from the frequency of vibration. A co-rotating interleaved pair can switch between a 180 degree angle between the ERMs and a 0 degree angle to rapidly turn the vibration effect on or off. Since there are no torque effects, the complete vibration sensation will be turned off when the ERMs have a relative phase angle of 180 degrees. In addition, such a configuration can generate a gyroscopic effect without generating torque vibrations. 
     An interleaved ERM pair can also be operated as a counter-rotating pair, and thereby generate a vibration force along an axis. By controlling the phase of the interleaved ERMS, the direction of the vibration force can be controlled. 
     Embodiments with 3 ERMs are shown in  FIGS. 76A-B . In  FIG. 76A , a mounting platform  1202  shaped as a tube, holds a center ERM,  1312 , an ERM  1314   a  is located above the Center ERM, and an ERM  1314   b  is located below the center ERM. All 3 ERMs are aligned such that their axis of rotation is collinear. In this figure, the dimension A is the distance along the axis of rotation between the projection of the center of the rotating eccentric mass of ERM  1312  onto the axis of rotation and the projection of the center of the rotating eccentric mass of  1314   a  onto the axis of rotation. In a similar fashion, ERM  1314   b  is located such that it is at a distance B along the axis of rotation between the projection of the center of its eccentric mass onto the axis of rotation and that of the projection of the center of the rotating eccentric mass of ERM  1312  onto the axis of rotation. Furthermore, the ERMs  1314   a  and  1314   b  can be synchronized to operate at the same frequency and same phase, which will generate a combined force centered along the axis of rotation. 
     When the distance A times the eccentricity of ERM  1314   a  is equal to the distance B times the eccentricity of ERM  1314   b , then the combined vibration force from synchronized ERMs  1314   a  and  1314   b  is projected onto the axis of rotation at the same point along this axis that the center of the eccentric mass of ERM  1312  is projected onto. In this configuration the combined vibration force from all 3 ERMs share the same plane. With this configuration, a vibration force can be generated by all 3 ERMs without generating a torque. Accordingly, the embodiment with 3 ERMs in  FIG. 76A  can be operated in a mode where it is functionally similar to the embodiment with 2 ERMs shown in  FIG. 75 , but the embodiment in  FIG. 76A  uses standard shaped eccentric masses. The embodiment in  FIG. 76A  can be operated in a co-rotation mode, where all 3 ERMs rotate in the same direction and with the same frequency. ERMs  1314   a  and  1314   b  can be operated with the same phase, and this phase can be adjusted relative to the phase of the center ERM,  1312 , which will modulated the amplitude of the vibration force. 
     If the eccentricity of ERM  1314   a  plus the eccentricity of ERM  1314   b  is equal to the eccentricity of ERM  1312 , then complete cancellation of the vibration forces can occur when all 3 ERMs are rotating. This complete cancellation allows for rapid on and off control of vibration forces. The embodiment in  FIG. 76A  can also be operated in a counter-rotation mode, where the direction of rotation and phase of ERMs  1314   a  and  1314   b  are the same, yet the center ERM,  1312 , is operated in the opposite direction. In the counter-rotating mode, vibration forces along an axis can be generated, and the direction of the vibration can be controlled by modulation the relative phase of the ERMs. The embodiment in  FIG. 76A , also can be operated in a mode that is not similar to the interleaved embodiment in  FIG. 75 ; here, the center ERM can be turned off and ERM  1314   a  can be operated out of phase with ERM  1314   a  to create a rocking torque in the device. In addition, each ERM in  FIG. 76A  can be operated at a different frequency. ERMs with smaller eccentric masses often can be operated at higher top frequencies, and thereby the embodiment in  FIG. 76A  can create even a wider range of vibration effects. 
     Another embodiment with 3 ERMs is shown in  FIG. 76B . A mounting platform  1202  shaped as a tube holds a center ERM,  1312 , an ERM  1314   a  is located above the center ERM, and an ERM  1314   b  is located below the center ERM. All 3 ERMs are aligned such that their axis of rotation is collinear. In  FIG. 76B , the dimension A is the distance along the axis of rotation between the center of the rotating eccentric mass of ERM  1312  and the center of the rotating eccentric mass of  1314   a . ERM  1314   b  is located at the same distance A along the axis of rotation between its center of the rotating eccentric mass and that of the center of the rotating eccentric mass of ERM  1312 . 
     When the eccentricity of ERMs  1314   a  and  1314   b  are half the eccentricity of the center ERM  1312 , and the ERMs  1314   a  and  1314   b  are synchronized to operate at the same frequency and same phase, then complete cancellation of vibration forces and torques can occur at a phase offset of 180 degrees. Thus, the embodiment in  FIG. 76B  can have the same functional advantages as the embodiment in  76 A. A further advantage of the embodiment of  FIG. 76B  is that two ERMs have identical specifications and thus can be more easily manufactured. 
     An additional embodiment with 3 ERMs is shown in  FIG. 77 . A mounting platform  1202 , holds a center ERM,  1312 , an ERM  1314   a  is located to one side of the center ERM, and an ERM  1314   b  is located to the other side the Center ERM. All 3 ERMs are aligned such that their axes of rotation are parallel. When all 3 ERMs are rotating in the same direction, the embodiment in  FIG. 77  can create similar vibration effects as the embodiments in  FIGS. 76A-B ; the frequency off all 3 ERMs can be the same, the phase of ERMs  1314   a  and  1314   b  can be the same, and the relative phase with the center ERM  1312  will determine the magnitude of the vibration force. 
     To provide complete cancellation of the vibration force, the eccentricity of the rotating mass of ERMs  1314   a  and  1314   b  can be selected to be half that of the center ERM  1312 . Complete cancellation of vibration torques can occur in the co-rotating mode when the center ERM  1312 , is located in the center between ERMs  1314   a  and  1314   b . The embodiment in  FIG. 77  can also be operated in a counter-rotating mode, where the ERMs  1314   a  and  1314   b  rotate in the same direction with the same phase, and the center ERM  1312  rotates in the opposite direction. This counter-rotating mode provides a vibration force along an axis, and the direction of the vibration force can be controlled by the phases of the ERMs. However, in the embodiment in  FIG. 77 , there will be a vibration torque during the counter-rotating mode since the axes of the ERMs are not collinear. 
     The embodiment in  FIG. 77  can also be operated in a counter-rotating mode, where the ERMs  1314   a  and  1314   b  rotate in the same direction with the same phase, and the Center ERM  1312  rotates in the opposite direction. This counter-rotating mode provides a vibration force along an axis, and the direction of the vibration force can be controlled by the phases of the ERMs. However, in the embodiment in  FIG. 77 , there will be a vibration torque during the counter-rotating mode since the axes of the ERMs are not collinear. 
     General Synchronized Vibration of ERMS requires control of both the frequency and phase of rotating eccentric masses. One method is to use a motor, such as a stepper motor, where the position and speed can be defined open-loop by specifying a desired series of steps. Another method is to use closed loop control with a sensor or sensors that measure frequency and phase. An ERM with a sensor  1260  is shown in  FIG. 78 . The sensor  1262  can be a continuous position sensor that measures the position of the eccentric mass at frequent intervals. Continuous sensors could be encoders, potentiometers, a Hall Effect sensor that detects a series of gear teeth or other feature of a rotating object, or other types of position sensors. The velocity of the eccentric mass could be calculated from the time interval between subsequent rotations, through taking the derivative of position measurements, or directly through use of a tachometer. 
     Another method to sense frequency and phase is to use a discrete sensor that detects when the motor shaft spins by a set position relative to the motor housing, or a number of set positions relative to the motor housing. Such discrete sensors can use reflective optical sensors that reflect off a rotating object coupled to the motor shaft, line-of-sight optical sensors that detect when a rotating object coupled to the motor shaft interrupts the line of site, hall effect sensors that detect a discrete component that is coupled to the rotating shaft, or other method of discrete detection of the shaft position. 
       FIG. 79  shows an ERM with a reflective optical sensor  1264  which detects light reflecting off an eccentric mass  1206 . A light source  1268 , such as an LED, is shining onto the pathway of the eccentric mass  1206 . When the eccentric mass  1206  rotates by the sensor  1266 , light reflects off the eccentric mass  1206  into the light sensor  1266 . For each rotation of the eccentric mass  1206  the light sensor  1266  will detect when the eccentric mass  1206  comes into the range of the sensor  1266  and begins to reflect light, and when the eccentric mass  1206  leaves the range of the sensor  1266  and stops to reflect light. The velocity of the ERM  1264  can be determined between the intervals of each rotation, such as the time when the eccentric mass  1206  begins to reflect light. Alternatively the velocity of the ERM  1264  can be calculated by the duration of time that the eccentric mass  206  reflects light. The phase of the eccentric mass  1206  can be determined by the timing of a specific event such as the rising or, falling edge of the light sensor  1266  which corresponds to the time when the eccentric mass  1206  begins and stops reflecting light. 
       FIG. 80  shows an ERM with a line-of-sight optical sensor  1270 . The light sensor  1266  detects when the eccentric mass  1206  interrupts the light path. A light source  1268 , such as an LED, is shining onto the pathway of the eccentric mass  1206 . When the eccentric mass  1206  rotates through the light path, the sensor  1266  detects the interruption.  FIG. 81  shows an ERM with a Hall Effect sensor  1272 . The Hall Effect sensor  1274  is triggered when the eccentric mass  1206  rotates by. 
     Implementing General Synchronized Vibration with ERMs requires that the frequency and phase be controlled for each ERM that is used to synthesize the desired waveform. Both the frequency, ω, and phase, φ, can be controlled by controlling the position, θ, of the rotating shaft of the ERM to be at a desired position as a function of time. Accordingly, control of frequency and phase can also be equivalent to control of the position of an eccentric mass to a desired position trajectory over time. Measurement of the shaft position can be performed continuously or at discrete instances such as when the shaft passes a certain position. Continuous measurements could be made with an encoder or other type of sensor that measures positions at frequency intervals. Discrete measurements could be made with an optical sensor that detects when the eccentric mass passes by. Discrete measurements could be made at a single position of motor rotation or at multiple positions. Discrete measurements can be augmented with a second sensor that also measures the direction of rotation. A direction sensor could be a second optical sensor mounted close to the first optical sensor. The direction of rotation can be determined by which optical sensor is triggered first. 
     A wide range of methods can be used for real-time control the position and speed of an ERM. One method is Proportional-Integral-Control. Another method includes time optimal control as described by “Optimal Control Theory: An Introduction”, by Donald E. Kirk, Dover Publications 2004. One real-time control approach is presented below for controlling a set of synchronized ERMs. The approach is written for use with a discrete sensor, but can also be applied with a continuous sensor. When a continuous sensor is used, the dynamic performance of the system can be improved by more accurately updating the commands to the motor continuously. 
     An exemplary control approach for a system with M ERMs is now discussed. For each ERM for i=1 to M, define the desired frequency, ω des, i , and desired phase, φ des, i . The desired direction of rotation is defined as dir des, i =sign(ω des, i ). Initialize the following variables:
         a. Time, t=0   b. Number of revolutions of each ERM, nrev i =0 (for all i)       

     Next, start motors turning by providing an open-loop command, V open     —     loop, i  to each ERM corresponding to the desired frequency, ω des, i . The open-loop command can be determined by the motor&#39;s torque-speed curve and correspond to the voltage that will generate a terminal velocity as the desired value. An optional startup operation is to turn on the motors at a high or maximum voltage to reduce the startup time. Since sensors exist to detect speed of rotation, the voltage can be reduced to a desired level when the ERMs reach an appropriate speed. In this fashion the sensors used for synchronization can also be used to reduce the startup time of the overall vibration device. As each ERM passes its discrete sensor:
         c. Measure the time and record: t meas, i =t   d. Calculate the desired position at the measured time:
 
θ des,i =ω des,i   t   meas,i +φ des,i  
   e. Calculate the measured position, θ meas, i  at the measured time:
           Increment the number of revolutions: nrev i =nrev i +1
 
θ meas,i =2πdir i   n rev i +θ sensor     —     offset,i  
   θ sensor     —     offset, i  is based upon the mounting location of the discrete sensor, and is often equal to zero.   dir i  is the actual direction of the ERM rotation. Typically the ERM will be rotating in the direction of the initial open-loop command. However, it is also possible to use a second sensor input to measure the direction of rotation, or use the time history of the motor command to calculate the direction.   
           f. Calculate the error in position, θ error, i , for each ERM:
 
θ error,i =θ des,i −θ meas,i  
       

     A control law may be implemented to reduce the position error of each ERM. There are a wide range of control of control approaches in the field of control, including:
         g. Proportional, Integral, Derivative (“PID”) based upon the calculated error in position. The command to the motor would be:
 
 V   com,i   =K   p,i θ error,i   +K   I,i ∫θ error,i   dt+K   D,i   dθ   error,i   /dt  
   h. Use the open-loop command as a baseline command to the ERM, since it is based upon the motor&#39;s characteristics, and apply PID to correct for remaining errors. The command to the motor would be:
 
 V   com,i   =V   open     —     loop,i   +K   p,i θ error,i   +K   I,i ∫θ error,i   dt+K   D,i   dθ   error,i   /dt  
           The use of the open-loop command can reduce the need for a large integral control gain, and improve dynamic performance.   
           i. State-space control approach. The physical state of each ERM is a function of both its position and velocity. Each time an ERM passes its discrete sensor, the speed of revolution can be calculated from the time interval since the last sensor measurement. The state-space approach uses both the position and velocity to determine an appropriate control signal. For the durations where no sensor measurements are made, a state observer can be used to estimate the motor&#39;s position and speed, where the model of the state observer is based upon the physical properties of the motor and rotating mass.   j. Use bang-bang control, which operates the motor at maximum forward command and maximum reverse commands for specified durations of time. For example, if an ERM is operating at the correct speed but position has a phase lag, then the motor should be accelerated for a duration of time and then decelerated back to the original speed for a second duration of time. A physical model of the motor dynamics can be used to determine the appropriate durations of acceleration and deceleration.       

     With all control approaches a bidirectional or unidirectional motor driver could be used. An advantage of using bidirectional motor drivers is that high levels of deceleration can be applied to an ERM by applying a reverse voltage, even if the motor never changes direction of rotation. This approach can reduce the time it takes to synchronize the ERMs. Another advantage of using bidirectional motor drivers is that ERMs could be operated in both counter-rotating and co-rotating modes. 
     An alternative method of calculating the position error is discussed below. Where the desired force is represented by Ai sin(ωit+φi) and the desired position is represented by θ i (t)=ω i t+φ 1 , start all ERMs at open loop voltages corresponding to ω i . Let the motors spin up to speed when ERM  1  passes the sensor so that it starts in phase, then reset the timer so t=0. See Table IV below 
     
       
         
           
               
             
               
                 TABLE IV 
               
             
            
               
                   
               
               
                 Control of ERM 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Time at 
                 θ 
                 θ 
                   
                 θ 
                 Change 
                   
                   
               
               
                 which sensor 
                 measured 
                 desired 
                   
                 desired 
                 in θdes 
               
               
                 is triggered 
                 (θ meas ) 
                 (θ des) 
                 Δ t 
                 (θ des ) 
                 (Δθdes) 
                 Δθ meas   
                 θ error   
               
               
                   
               
               
                 t 1   
                 0 
                 ωt 1  + φ 
                   
                   
                   
                   
                 θ des  − θ meas   
               
               
                 t 2   
                 2π 
                 ωt 2  + φ 
                 t 2  − t 1   
                 ωΔt + 
                   
                 2π 
                 θ error  = 
               
               
                   
                   
                   
                   
                 θ des     —     prev   
                   
                   
                 θ error     —     prev  + 
               
               
                   
                   
                   
                   
                   
                   
                   
                 Δθ des  − 
               
               
                   
                   
                   
                   
                   
                   
                   
                 Δθ meas   
               
               
                 t 3   
                 4 π 
                 ωt 2  + φ 
                 t 3  − t 2   
                 ωΔt + 
                 θ des  − 
                 2π 
               
               
                   
                   
                   
                   
                 θ des     —     prev   
                 θ des     —     prev   
               
               
                   
               
            
           
         
       
     
     In a digital system, ERM control may include the following. First, set rotation counts per revolution (e.g., 256 or 512). Correct for timer overflow so Δt=t i −t i-1  is always correct. Define ω in terms of rotation counts per timer counts. And use interrupts (or other operations) to avoid missing when an ERM passes by a sensor. 
     Some embodiments of Synchronized Vibration Devices can be controlled such that the combined force and torque sum to zero. In such an embodiment the force and torques from individual Vibration Actuators balance each other out to generate a net zero force and torque. An advantage of such an embodiment is that Vibration Actuators can be brought up to speed and put into a mode when no vibration effects are generated. When vibration effects are desired, they can be quickly implemented by modifying the phase of the vibration, without the lag for bringing the actuators up to speed. This embodiment is referred to as “Spinning Reserve”, and is analogous to the same term used for kinetic energy in an electric utility power plant that is held in reserve to quickly provide power when needed. The spinning reserve approach allows vibration to be quickly turned on and off. Spinning Reserve embodiments can include ERM actuators that are spinning in such a manner that the combined forces and torque sum to zero. Spinning Reserve embodiments can also include with LRA actuators and other resonant actuators that are vibrating in such a manner that the combined forces and torque sum to zero. 
     The spinning reserve approach has the advantage of fast on and off response times, but also can require increased power consumption since the vibration actuators are operated even when no overall vibration effects are generated. To reduce the added power consumption, the vibration actuators can be spun up to speed at the first indication that a need for vibration force is imminent. Such indications could be a keystroke, computer mouse motion, user touching a touch-screen, movement detection via a sensor of a game controller, beginning of a game portion where vibration effects are used, or any other event that would indicate that a desired vibration effect would be imminent. In a similar fashion power can be conserved by spinning down and stopping the actuators once the need for vibration is no longer imminent. Indications to spin down the actuators could include passage of a set amount of time where no user input is registered, transition to a new phase of a computer program where vibration effects are no longer needed, or other indication. During the spin up and spin down of the actuators, the actuators can by synchronized so that they operate in a spinning reserve mode and do not generate a combined vibration force. In this fashion, the user will not feel the spin up and spin down of the vibration actuators. 
     A spinning reserve embodiment with 4 ERMs is shown in  FIG. 82 . Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where the combined forces and torque cancel each other out. In one such embodiment the eccentricity and rotational inertia of ERMs  1190   a ,  1190   b ,  1192   a  and  1192   b  are equal to each other. In one such control method all 4 ERMs rotate in the same direction. The frequency and phase could be as shown in Table V below. The synchronized phases within a set of ERMs, can be controlled relative to each other and not just relative to absolute time. Accordingly, the phases shown in Table V and other tables in this document only represent one set of phases in absolute time that achieve the described effect. Other phase combinations can achieve similar effects. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE V 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 ω 
                 ω 
                 ω 
               
               
                 Phase 
                 −90° 
                 90° 
                 90° 
                 −90° 
               
               
                   
               
            
           
         
       
     
       FIG. 83  shows the forces of the ERMs from Table V as the ERMs progress through time, wherein each row of images illustrates one time slice (8 slices in all). The parameters for frequency and phase shown in Table V correspond to the force vectors shown in  FIG. 83 . In a similar fashion other configurations and control methods of vibrations devices can also be simulated. 
     Another method of Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where the combined forces and torque cancel each other out. In such a control method ERM  1190   a  rotates in the opposite direction of ERM  1190   b , and ERM  1192   a  rotates in the opposite direction of ERM  1192   b . The frequency and phase could be as shown in Table VI. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE VI 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 ω 
                 −ω 
                 −ω 
               
               
                 Phase 
                 −90° 
                 90° 
                 90° 
                 −90° 
               
               
                   
               
            
           
         
       
     
     When ERMs are rotating they generate a gyroscopic effect due to the angular inertia of the motor rotor and rotating mass. When the angular velocity of the ERMs is large this gyroscopic effect can be used to generate a haptic sensation in response to changes in orientation of the vibration device. The implementation of spinning reserve as shown in Table V has a gyroscopic effect since all ERMs are rotating in the same direction and their angular inertia combined. The implementation of spinning reserve as shown in Table VI does not have a gyroscopic effect since half the ERMs are rotating in the opposite direction of the other half, and therefore angular inertias cancel each other out when rotational inertias are equal. The mode of implementation of spinning reserve can be selected according to the desired gyroscopic effect. 
     Another method of Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where the combined forces generate a force along the x axis and the torques cancel each other out. The frequency and phase could be as shown in Table VII. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE VII 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 −ω 
                 −ω 
                 ω 
               
               
                 Phase 
                 0° 
                 0° 
                 0° 
                 0° 
               
               
                   
               
            
           
         
       
     
     Another method of Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where the combined forces generate a force along the y axis and the torques cancel each other out. The frequency and phase could be as shown in Table VIII. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE VIII 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 −ω 
                 −ω 
                 ω 
               
               
                 Phase 
                 90° 
                 90° 
                 90° 
                 90° 
               
               
                   
               
            
           
         
       
     
     Indeed Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where the combined forces generate a force along any axis in the XY plane. The control that implements an axis at 30 degree and the torques cancel each other out is shown in Table IX. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE IX 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 −ω 
                 −ω 
                 ω 
               
               
                 Phase 
                 30° 
                 30° 
                 30° 
                 30° 
               
               
                   
               
            
           
         
       
     
     Another method of Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where a combined torque is generated and the forces cancel each other out. One such pure torque embodiment generates equal amplitudes torque in the clockwise and counterclockwise directions, and is referred to as a symmetric torque implementation. The frequency and phase that generates a symmetric torque could be as shown in Table X. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE X 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 −ω 
                 ω 
                 −ω 
                 ω 
               
               
                 Phase 
                 −90° 
                 −90° 
                 90° 
                 90° 
               
               
                   
               
            
           
         
       
     
     Another implementation of pure torque can produce an asymmetric torque, where the peak torque in the clockwise direction is larger than the peak torque in the counterclockwise direction, or vice versa. One such asymmetric torque implementation for a 4 ERM configuration could be as shown in Table XI. This is achieved by operating ERMs  1192   a  and  1192   b  at twice the frequency of ERMs  1190   a  and  1190   b , and controlling the phase appropriately. For the configuration shown in  FIG. 82 , when all ERMs have the same eccentricity, the amount of asymmetry in the torque can be increased by placing ERMs  1192   a  and  1192   b  at a distance of ⅛th from the center relative to the distances of ERMs  1190   a  and  1190   b . 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE XI 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 2ω 
                 2ω 
                 ω 
               
               
                 Phase 
                 −90° 
                 −90° 
                 90° 
                 90° 
               
               
                   
               
            
           
         
       
     
     Yet another method of Synchronized Vibration can be applied to the embodiment shown in  FIG. 82 , where all ERMs rotate together and forces do not cancel each other out. This implementation generates an effect of one large ERM that would have the eccentricity of all ERMs combined. The frequency and phase that generates a symmetric torque could be as shown in Table XII. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE XII 
               
               
                   
                   
               
               
                   
                 ERM 1190a 
                 ERM 1192a 
                 ERM 1192b 
                 ERM 1190b 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Frequency 
                 ω 
                 ω 
                 ω 
                 ω 
               
               
                 Phase 
                 0° 
                 0° 
                 0° 
                 0° 
               
               
                   
               
            
           
         
       
     
     A wide range of haptic effects can be generated by switching between the various effects described herein. When the ERMs are rotating at the same speed in two different effects, the change between effects (including the no-vibration spinning reserve) can be achieved quickly. In many cases the change in effect only requires a positive or negative phase change of 90 degrees in specific ERMs. 
     Embodiments with 4 ERMs that are not aligned along the same axis also can generate many useful effects. Fig. DIAMOND_OF — 4ERMS shows an embodiment of 4 ERMs. When this embodiment is implemented with 4 ERMs with the same eccentricity, a spinning reserve effect can be generated with same frequency and phases shown in Table V. In  FIG. 84  the center of ERM pair  1194   a  and  1194   b , have the same center as ERM pair  1196   a  and  1196   b . Indeed, any embodiment with 2 pairs of ERMs that share the same center can be controlled in a spinning reserve mode. 
     The embodiment shown in  FIG. 84  can also be controlled to generate a pure force vibration along a specified direction, where the torques cancel each other out. The same frequency and phase as shown in Table VII, Table VIII, and Table IX can be used. A symmetric torque can be generated with this embodiment as well, but with a frequency and phase as defined in Table V, and replacing ERMs  1190   a ,  1190   b ,  1192   a , and  1192   b  with ERMs  1194   a ,  1194   b ,  1196   a , and  1196   b , respectively. 
     As discussed above with regard to  FIG. 75 , interleaved ERM pairs may be employed according to aspects of the disclosure. Another embodiment of an interleaved ERM pair is shown in  FIGS. 85A-B . As shown in  FIG. 85A , an inner eccentric mass  1320   a  is driven by motor  1322   a  and an outer eccentric mass  1320   b  is driven by motor  1322   b . The outer eccentric mass  1320   b  is shaped so that the walls get thicker going away from the motor  1322   b . This extra thickness compensates for the material required for structural support of the eccentric mass near the motor. As shown in the side view of  FIG. 85B , the inner eccentric mass  1320   a  fills the void inside eccentric mass  1320   b . The result is that both eccentric masses  1320   a  and  1320   b  share the identical center of mass, which eliminates unwanted torque effects. 
     Another embodiment of an interleaved ERM pair is shown in  FIGS. 86A-C . Here, an inner eccentric mass  1330   a  is driven by motor  1332   a  and an outer eccentric mass  1330   b  is driven by motor  1332   b . The end of eccentric mass  1330   a  that is furthest from the motor  1332   a  is supported by a bearing  1334   b , which is installed into eccentric mass  1330   b . The end of eccentric mass  1330   b  that is furthest from the motor  1332   b  is supported by a bearing  1334   a , which is installed into eccentric mass  1330   a . The bearings  1334   a  and  1334   b  allow for the spinning eccentric masses  1330   a  and  1330   b  to be supported on both ends. This allows the eccentric masses  1330   a  and  1330   b  to spin faster without deflection due to cantilever loads, and helps reduce friction in the motors  1330   a  and  1330   b.    
     The performance of almost any vibration device can be improved by applying the methods and embodiments of General Synchronized Vibration discussed herein. This approach toward synchronization allows for a wide range of waveforms to be generated including asymmetric waveforms that generate larger peak forces in one direction than the opposing direction. Applications range from seismic shakers and fruit tree harvesters, to vibratory feeders and miniature vibration applications. The embodiments described herein can replace more expensive actuation devices that are used to generate complex waveforms of vibrations. Such applications include seismic shakers that are simulating specific earthquake profiles, and voice coils that are used to generate complex haptic effects. 
     Haptic applications described herein can be used to augment any device that has a visual display including computer gaming, television including 3D television, a handheld entertainment system, a smartphone, a desktop computer, a tablet computer, a medical device, a surgical instrument, an endoscope, a heads-up display, and a wristwatch. Implementation of haptic feedback within a system that has a visual display is shown in  FIG. 87 . 
     As described herein, Vibration Force cues can be generated in specific directions, and these directions can be chosen to correspond to direction that is relevant to an object or event that is being shown on a graphic display.  FIG. 88  shows a graphic display with an image that has a direction of interest specific by an angle σ. The Vibration Device shown in  FIG. 88  can generate haptic cues in the same direction to provide multi-sensory input and enhance the overall user experience. 
     Moreover, it is be useful to generate haptic cues of directionality for applications where a person does not have visual cues, such as to guide a blind person or applications where vision is obscured or preoccupied with another task. For example, if a person had a handheld device such as a mobile phone that could generate directional haptic cues through vibration, and the mobile phone knew its absolute orientation as it was being held and the orientation the person should be in to move forward to a goal, then the mobile phone could communicate directional haptic cues through vibration (a force, a torque, or a combined force and torque) that corresponded to the direction and magnitude of the change in orientation the person holding the mobile phone needed to make. 
     The Vibration Devices describe herein can be used to improve the performance of existing devices that use vibration. For example vibration is used in fruit tree harvesting. By allowing the operator to generate complex waveforms and control the direction of vibration a higher yield of ripe fruit could be harvested, while leaving unripe fruit on the tree. Vibratory feeders are used in factory automation, and typically involve a significant amount of trial an error to achieve the desired motion of the parts. By allowing the operator to generate complex waveforms and control the direction of vibration it can be easier to generate the desired part motion and a wider range of parts could be processed with vibratory feeders. 
     The Vibration Devices described herein allow for a wide and continuous adjustment in areas such as vibration magnitude, frequency, and direction. To improve performance of a Vibration Device, sensor feedback can be used, as shown in  FIG. 89 . With this approach a Vibration Device applies forces onto an object, and a sensor measures a feature or features of the object. The sensor information is provided to the Vibration Device Controller, which can then modify the vibration waveform to improve overall system performance. One area of application could be a vibratory parts feeder, where a sensor measures the rate at which parts move along a pathway, and the waveform is modified to improve the part motion. Another area of application could be preparation and mixing of biological and chemical solutions. A sensor could measure the effectiveness of the mixing and the vibration waveforms could be adjusted accordingly. 
     One application is to use General Synchronized Vibration for locomotion.  FIG. 90  shows an embodiment where a Vibration Device  1200  rests on a surface  1282 . There exists friction between the surface  1282  and the Vibration Device  1200 . Accordingly, motion of the Vibration Device  1200  will only occur if a force parallel to the surface  1282  exceeds a friction threshold. In this embodiment, an asymmetric waveform is being generated so that the peak positive force exceeds the friction threshold and the peak negative force is less than the friction threshold. Accordingly in each vibration cycle the Vibration Device  1200  can be pushed in the positive x direction when the peak force in the positive x direction exceeds the friction threshold. 
     However, there will generally be no motion in the negative x direction, since the friction threshold is not exceeded. In this fashion, the Vibration Device  1200  will take steps in the positive x direction. The direction of motion along the x axis can be reversed by changing the synchronization of the Vibration Actuators and generating an asymmetric waveform that has a larger peak force in the negative direction. A location device can be made to move in arbitrary directions on a surface  1282  by using a Vibration Device  1200  where the direction of vibration can be controlled on a plane, such as those shown in  FIG. 62  and  FIG. 66 . In a similar fashion a locomotion device can be made to rotate by generating asymmetric torque vibrations, such as the one shown in  FIG. 57 . 
     Vibration is also used for personal pleasure products such as Jimmyjane&#39;s Form 2 Waterproof Rechargeable Vibrator. Vibration is also used for personal massager products such as the HoMedics® Octo-Node™ Mini Massager. Vibration is also used for beauty products such as Estée Lauder&#39;s TurboLash and Lancôme&#39;s Ôscillation mascara applicators. INOVA produces the AHV-IV Series Vibrators for Vibroseis seismic exploration. General Synchronized Vibration can be used to improve the performance of such products by allowing the user to customize the vibration waveforms and direction of peak vibration forces. 
     General Synchronized Vibration may also be used in therapeutic medical applications. For example a Vibration Device could vibrate a patient&#39;s stomach to aid in digestion, and the patient or a sensor could determine how to adjust the vibration over time. 
     Although aspects of the disclosure have been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present disclosure. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present disclosure as defined by the appended claims. By way of example only, it is possible to vary aspects of the embodiments herein to some degree while achieving General Synchronized Vibration and other benefits of the disclosure. For instance, the frequency of vibration, amplitude of vibration, profile or waveform of vibration, phase of vibration, timing of vibration, alignment of actuators, rigidity of the vibration device, rigidity of the attachment between the actuators and the vibration device, and design and control parameters may all be adjusted, either independently or in any combination thereof.