Abstract:
A gravity gradiometer having at least three differential accelerometers with a low response to linear accelerations and at least six angular accelerometers that give it the capability of measuring angular rates by integrating the angular accelerations. Both types of accelerometers are based on a compliant mechanism with very low and adjustable stiffness that is achieved by using flexures under compressive load that contribute a negative stiffness to the total elastic response of the mechanism. Both types of accelerometers are operated in a servo-compensation feedback mode so that at no time is the mechanism far from its equilibrium position.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to gravity gradiometry and, more particularly, to quadrupole gravity gradiometers. 
     2. Description of the Related Art 
     Measuring the gravitational field involves quantifying the gravitational forces acting on test masses. Gravimeter instruments measure the acceleration of gravity and are sensitive to linear accelerations. Doing measurements with an instrument carried by a non-inertial moving platform requires evaluating the inertial forces acting on test masses and subtracting their effect from the measured signal. Also, platform induced forces are many orders of magnitude higher than the variations of acceleration of gravity measured and an appropriate suspension is required on order to reduce such forces to a level that can be handled by the dynamic range of the instrument. For that reason gravity gradiometer instruments are more useful, since they have low sensitivity to linear acceleration by their operating principle. 
     The gravity potential generated by mass m at distance r is 
     
       
         
           
             Φ 
             = 
             
               G 
               ⁢ 
               
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                 r 
               
             
           
         
       
     
     where G=6.674×10 −11  m 3  kg −1  s −2  is the Newtonian constant of gravitation. The acceleration of gravity in that potential field is 
     
       
         
           
             g 
             = 
             
               
                 
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     The tensor of the gravity acceleration gradient is 
     
       
         
           
             Γ 
             = 
             
               
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                         Γ 
                         xx 
                       
                     
                     
                       
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     Obviously, the order of differentiation is irrelevant, therefore the tensor is symmetric. It also has zero trace (the sum of diagonal elements is zero) since gravity obeys Laplace&#39;s equation in free space. Therefore only five components of the tensor are independent. 
     The gravity acceleration gradient is measured in Eötvös units (E) defined as 1 E=10 −9  s −1 . Current commercial gradiometers have a sensitivity of a few Eötvös units. 
     BRIEF SUMMARY OF THE INVENTION 
     In view of the above, it is an object of the present invention to provide a gravity gradiometer instrument for full gradient tensor measurement, suitable for operation from a moving platform such as an airplane. The instrument according to the present invention has inherent low sensitivity to linear accelerations and is capable of measuring and compensating the centrifugal forces resulting from the non-inertial movement of the platform carrying it. 
     A gravity gradiometer according to the present invention consists of at least three differential accelerometers that, by their symmetric construction, have low response to linear accelerations. As any gravity gradiometer has inherent sensitivity to centrifugal forces due to non-inertial movement of the carrier platform, the present invention further comprises mounting the gradiometer in a suspension system able to isolate rotations. However, no isolation system is perfect and some rotations are sensed by the instrument causing noise and reducing its sensitivity. The gravity gradiometer according to the present invention also includes at least six angular accelerometers that give it the capability of measuring angular rates by integrating the angular accelerations. Both types of accelerometers are based on a compliant mechanism with very low and adjustable stiffness. This is achieved by using flexures under compressive load that contribute a negative stiffness to the total elastic response of the mechanism. The compliant mechanism is machined out of a single plate of metal by milling and electro-discharge machining. Compared to other construction methods this eliminates uncontrollable friction forces and vibrations, reduces the weight, and eliminates assembly tolerances. Both types of accelerometers are operated in a servo-compensation feedback mode so that at no time is the mechanism far from its equilibrium position. This arrangement ensures linear response over a very large range of forces. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S) 
       The present invention will be more fully understood and appreciated by reading the following Detailed Description in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a kinematic diagram of a differential accelerometer mechanism according to the present invention. 
         FIG. 2  is a kinematic diagram of a differential accelerometer according to the present invention. 
         FIG. 3  is a kinematic diagram of an angular accelerometer. 
         FIGS. 4A and 4B  are a perspective view and a close-up view, respectively, of an exemplary embodiment of a compliant mechanism achieving the kinematic behavior of a differential accelerometer according to the present invention. 
         FIG. 5  is a perspective view of a balancing assembly according to the present invention. 
         FIGS. 6A and 6B  show a side view and a perspective view, respectively, of a differential accelerometer according to the present invention. 
         FIG. 7  is a perspective view of a test mass for an angular accelerometer according to the present invention. 
         FIG. 8  is a perspective view of an angular accelerometer according to the present invention. 
         FIG. 9  is a diagram of exemplary configuration of an accelerometer subsystem according to the present invention. 
         FIG. 10  is a perspective view of three differential accelerometers forming part of a gradiometer according to the present invention. 
         FIG. 11  is a perspective view of two angular accelerometers forming part of a gradiometer according to the present invention. 
         FIG. 12  is a diagram of an exemplary configuration of a full tensor gradiometer according to the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to the drawings, wherein like numbers refer to like parts throughout, there is seen in  FIG. 1  a differential accelerometer mechanism  10  according to the present that is constrained to move in the plane of the figure. Mechanism  10  includes a linkage comprising four rigid beams  11  of equal length, articulated at a series of nodes  12 . Nodes  12  are constrained to move perpendicularly to the symmetry axes of mechanism  10 . The constraint is achieved by suspensions, symbolically represented by a series of guides  13 , attached to a fixed frame. While nodes  12  execute rectilinear movements, the beam midpoints  14  move along a circle centered at the geometric center of mechanism  10 . If an elastic element, such as a spring  15 , and a test mass (not shown) are added to mechanism  10 , it becomes an accelerometer. 
     It is obvious that mechanism  10  has one degree of freedom. Therefore any combination of springs, in any positions, can be represented symbolically by spring  15 . Spring  15  works both in extension and compression and, if there are no loads present, maintains the mechanism in its equilibrium position, which is a position where the adjacent beams are perpendicular to each other forming a square. By adding two or more test masses, a differential accelerometer may be created. Several configurations that should be obvious to those skilled in the art are possible. 
     An exemplary embodiment of a differential accelerometer according to the present invention may be seen  FIG. 2 . Two equal test masses  16  are attached to two of the opposing nodes of mechanism  10  of  FIG. 1  where, for clarity, the spring and constraints are not shown. This arrangement forms a differential accelerometer with its sensitive axis passing through the nodes having the test masses  16 . A difference in the acceleration of gravity along the sensitivity axis will cause a displacement of the test masses from their equilibrium position. If we take into account centrifugal forces caused by rotations, the accelerometer static response, namely the displacement of the nodes, δl is: 
               δ   ⁢           ⁢   l     =         m   ⁢           ⁢   l     k     ⁢     (       Γ   zz     +     ω   x   2     +     ω   y   2       )             
where m is the test mass, l is the distance between opposing nodes, also called the accelerometer base, k is the elastic constant of the mechanism, and ω x  and ω y  are the angular rates of rotations around axes x and y respectively, of a coordinate system attached to the accelerometer frame, relative to an inertial frame, as shown in  FIG. 2 .
 
     An exemplary embodiment of an angular accelerometer may be seen in  FIG. 3 , which depicts mechanism  10  of  FIG. 1  having two equal masses  16  attached to the centers  14  of opposite beams  11 . This arrangement forms an angular accelerometer. For the orientation in  FIG. 3 , the accelerometer response is: 
               δ   ⁢           ⁢   l     =         m   ⁢           ⁢   l     k     ⁢     (       Γ   xz     +     ω   y     -       ω   x     ⁢     ω   z         )             
where m is the test mass, l is the distance between mass centers of test masses, k is the elastic constant of this mechanism, and angular rates have the same meaning as in  FIG. 2 .
 
     An exemplary embodiment of a mechanism achieving the function of the model in  FIG. 1  is presented in  FIGS. 4A and 4B . This structure is built as a compliant mechanism  17  and comprises a rigid frame  18  supporting the mechanism itself consisting of the blocks  20  connected to the beams  22  by means of the flexible sections  23  that accomplish the function of joints  12  in  FIG. 1 . The function of guides  13  in  FIG. 1  is accomplished by the long flexures  19  having one end connected to the blocks  20  and another supported by the suspensions  21 , which are double parallelogram mechanisms, allowing longitudinal movement of the flexure ends  19 , while restraining their transversal displacement. 
     The details of suspension  21  are shown in  FIG. 4B . Four flexible elements  25  on each side of the suspension are connected to a shuttle beam  26 . Two of the flexible elements  25  have their other end attached to the rigid frame and the other two support the mobile base of the suspension. A number of holes  24  are provided in the node blocks and the frame for attaching various components. The mechanism can be built out of a single block using a combination of milling and electro-discharge machining. It is advantageous to make the mechanism out of a metal with a high yield strength and low elastic modulus, such as high strength aluminum. The high thermal conductivity of aluminum has the additional benefit of minimizing temperature gradients, which is important for dimensional stability. A compliant mechanism of about 0.3 m in diameter may be sufficient for a gradiometer with a sensitivity of a few Eötvös. A larger mechanism can achieve higher sensitivity, but the tradeoff is dictated by the design constraints of the application. 
     Even though the compliant mechanism in  FIGS. 4A and 4B  can be built with high dimensional accuracy, there will always be small geometrical deviations and mass differences from a perfect symmetry. These have to be compensated by a means to balance the mechanism after machining.  FIG. 5  shows an exemplary embodiment of a balancing assembly  30  and comprises two screws  33  that clamp a back plate  32  and a block  31  on beams  22 . Block  31  carries a number of screws  34  of different sizes and orientations. Screws  34  and their locking nuts  35  are used to move the position of the center of mass of assembly  30  along the beam  22 . The weight, pitch, and angle of screws  34  are selected such that they have overlapping ranges of moving the center of mass and effects ranging from coarse to very fine. By moving the center of mass of one of assemblies  30  towards one block  20 , a user can compensate for that block being lighter than its opposite block. 
     An exemplary embodiment of a differential accelerometer  40  may be seen in  FIGS. 6A and 6B  that shows the compliant mechanism  17  of  FIG. 4A  to which sensors and actuators have been added. Voice-coil actuators are mounted with the permanent magnet  25  fixed to block  20  and to coil  26  attached to frame  18 . The voice-coil actuators are used to apply a force that restores the equilibrium position when an external disturbance moves the mechanism away from it, causing sensors  27  to detect a displacement. Capacitive or other kind of actuators can be used instead, provided they can apply small enough forces and have sufficient dynamic range. The displacement sensors can be capacitive, inductive (eddy current) or any other type with a sensitivity of the order of nanometers or better. Fine adjusting screws  29  are used to apply a compressive force on the flexures  19 . Piezoelectric actuators  28  positioned between screws  29  and suspension  21  are used for extremely fine adjustments in the compression of flexures  19 . Compressive loads on flexures  19  add a negative stiffness component to the spring constant of the mechanism by bringing the flexures close to their buckling point. This way, the sensitivity and dynamic response of the accelerometer can be adjusted and controlled. Balancing assemblies  30  clamped on the beams  22  are used to compensate the differences in the masses and geometric imperfections of the accelerometer.  FIG. 6B  shows top and bottom blocks  20  loaded with plates  41 , preferably out of a denser metal such as brass, bolted by means of screws  42 . Blocks  20 , plates  41 , and screws  42  together form the test masses of the differential accelerometer in  FIG. 2 . 
     For an angular accelerometer, test masses must be placed on beams  22 .  FIG. 7  shows an exemplary embodiment of a test mass assembly  50  for the angular accelerometer. Two screws  53  are screwed into threaded holes in back block  51  and are used to clamp block  52  on beams  22 . Block  52  carries a number of screws  54  of different sizes and orientations with their locking nuts  55 . Their weight, pitch, and angle are selected such that they have overlapping ranges of moving the center of mass of assembly  50 . The center of mass has to be adjusted such that equal accelerations on masses  50  would produce equal torque moments. 
     An exemplary embodiment of an angular accelerometer  60  may be seen in  FIG. 8 . Angular accelerometer  60  comprises compliant mechanism  17  of  FIG. 4A  to which sensors and actuators were added just as for differential accelerometer  40 . Voice-coil actuators and sensors are mounted the same way as in  FIG. 6A . Test mass assemblies  50  are clamped on beams  22  and screws  54  are used to compensate the differences in the masses and geometric imperfections of the accelerometer. 
     Both types of accelerometers are operated in a feedback loop as exemplified in  FIG. 9 . A comparator  85  calculates the difference between the reference position, which is the equilibrium position, and the actual position as indicated by sensors  89  mounted on the accelerometer. A controller  86  applies to actuators  87  the excitation required to restore the equilibrium position. Actuators  87  are initially calibrated, so that from their excitation one can calculate the restoring force required to balance the force sensed by the accelerometer mechanism  88  due to the gravity gradient and rotations. 
     By combining three differential accelerometers, as seen in  FIG. 10 , one can measure the diagonal, or in-line, components of the gravity gradient tensor. If the accelerometers  40  are mounted with their sensitive axes, represented by arrows  70 , along gradiometer axes x, y and z, one obtains the signals:
 
 S   xx =Γ xx +ω y   2 +ω z   2  
 
 S   yy =Γ yy +ω x   2 +ω z   2  
 
 S   zz =Γ zz +ω x   2 +ω y   2  
 
where S xx , S yy , S zz  are the signals coming from the accelerometers with sensitive axes oriented along axes x, y and z, respectively. We assume the signals are all scaled to be proportional to the physical quantities they represent with the same proportionality constant.
 
     If angular rates are known the three diagonal gradient components can be calculated. To measure angular rates one can use angular accelerometers. When mounted as in  FIG. 11 , a pair of such accelerometers will produce the signals:
 
 S   xyz =Γ xy −ω z −ω x ω y  
 
 S   yxz =Γ xy +ω z −ω x ω y  
 
where S xyz , S yxz  are signals from the accelerometers rotating around the z axis with test masses aligned with axes x and y, respectively, and {dot over (ω)} z  is the angular acceleration about the z axis.
 
     The difference of these signals yields the angular acceleration and their sum gives the signal
 
 S   xyz   +S   yxz =2(Γ xy −ω x ω y )
 
     Three such pairs would provide all angular accelerations and off-diagonal elements of the gradient tensor. The angular acceleration can be integrated to obtain the angular rates that can be used then to solve the equations for all tensor elements. 
       FIG. 12  illustrates an exemplary configuration of a gradiometer capable of measuring all gradient tensor components and all components of the rotation vector. Angular accelerometers  91  are grouped in pairs, each pair consisting of accelerometers in the same plane. The signals from each pair are subtracted and summed by the differential and summing amplifiers  92  and  94 . The difference signal is integrated by an integrator  93  and sent to a microprocessor  96  along with the signals from the summing amplifiers  94 . 
     The system, as illustrated in  FIG. 12 , is overdetermined, since there are nine signals and only eight unknown quantities. Thus, a least squares algorithm may be used to solve the equation system and allows calculating an estimate of the error. It should be recognized by those of skill in the art that one can build a gradiometer with improved sensitivity by adding more accelerometers, thereby adding more redundancy to the system.