WO2021051952A1 - 基于并联杆系多维力传感器的多维力获取方法 - Google Patents

基于并联杆系多维力传感器的多维力获取方法 Download PDF

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WO2021051952A1
WO2021051952A1 PCT/CN2020/099609 CN2020099609W WO2021051952A1 WO 2021051952 A1 WO2021051952 A1 WO 2021051952A1 CN 2020099609 W CN2020099609 W CN 2020099609W WO 2021051952 A1 WO2021051952 A1 WO 2021051952A1
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coordinate system
displacement
local
force
local coordinate
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PCT/CN2020/099609
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English (en)
French (fr)
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马洪文
邢宇卓
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马洪文
邢宇卓
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L5/00—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
    • G01L5/16—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes for measuring several components of force
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L1/00—Measuring force or stress, in general
    • G01L1/16—Measuring force or stress, in general using properties of piezoelectric devices
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L1/00—Measuring force or stress, in general
    • G01L1/20—Measuring force or stress, in general by measuring variations in ohmic resistance of solid materials or of electrically-conductive fluids; by making use of electrokinetic cells, i.e. liquid-containing cells wherein an electrical potential is produced or varied upon the application of stress
    • G01L1/22—Measuring force or stress, in general by measuring variations in ohmic resistance of solid materials or of electrically-conductive fluids; by making use of electrokinetic cells, i.e. liquid-containing cells wherein an electrical potential is produced or varied upon the application of stress using resistance strain gauges
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L5/00—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
    • G01L5/16—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes for measuring several components of force
    • G01L5/165—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes for measuring several components of force using variations in capacitance
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L5/00—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
    • G01L5/16—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes for measuring several components of force
    • G01L5/167—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes for measuring several components of force using piezoelectric means

Definitions

  • the invention belongs to the technical field of sensor measurement, and relates to a method for acquiring the force of a multidimensional force sensor.
  • the multi-dimensional force sensor can detect the information of the force acting in space.
  • a typical six-dimensional force sensor can obtain the 3 component forces and 3 moments formed by the force in the space coordinate system.
  • aerospace manipulators aerospace docking, collaborative robots, bionic machinery, medical auxiliary equipment, walking robots, wind tunnel force measurement, aerospace engine thrust testing, propeller thrust testing, automobile collision testing, minimally invasive surgical robots, polishing and polishing equipment, stirring
  • the six-dimensional force sensor plays an important role, and the accuracy of the six-dimensional force obtained directly affects the working performance and control accuracy of the system.
  • the six-dimensional force sensor can be mainly divided into the overall elastic structure type, the Stewart parallel structure type, the piezoelectric crystal type, the frictionless guide type (air suspension, magnetic suspension), the flexible structure type, etc., among which the commercial small six-dimensional type
  • the field of force sensors and MEMS mainly adopts overall elastic structure, including three vertical ribs, I-beam building blocks, double ring, barrel type, Maltese cross beam, eight vertical ribs, non-radial three beams, E structure, T structure, double cantilever Beam and other structures.
  • Large-scale six-dimensional force sensors mainly adopt Stewart parallel structure and ball-supported parallel structure.
  • Piezoelectric crystal type mainly has 4, 6, and 8 plane layout structures, which are used in the field of high-frequency dynamic measurement.
  • the frictionless guide rail type is used because of the large structural volume.
  • the flexible structure is mainly used for the gripping of the end of the mechanical finger, and the accuracy is very low.
  • the overall elastic structure generally uses a flexible hinge or a flexible flat structure instead of a physical hinge. Its accuracy is slightly higher, but the structural rigidity is small, and due to the coupling effect of the flexible body part, the accuracy is generally 2% to 5%, and because of its low structural rigidity , Generally need to increase the safety structure (to avoid excessive deformation and damage) or use rubber filling to avoid excessive vibration.
  • the Stewart parallel structure has higher rigidity, but due to the use of physical hinges, there is greater friction and its accuracy is very low.
  • There are also ways to use flexible hinges instead of physical hinges in the structure but it will cause low structural rigidity and flexibility The coupling moment in the hinge cannot be eliminated, and the accuracy is also not high.
  • Piezoelectric crystal type generally adopts a plane multi-group arrangement, each group contains three wafers to measure three axial forces respectively, and the torque is calculated by the multi-group axial forces.
  • the force measurement frequency response is high, but the force measurement accuracy is low. , And it is not suitable for static measurement due to charge drift.
  • the main sensitive components used are strain gauge type (resistance strain, semiconductor strain, grating strain, etc.), piezoelectric crystal type, optical sensor type (triangular light, confocal light, astigmatism, etc.) ) And electrical sensor type (capacitance, inductance, eddy current, etc.).
  • strain gauge type resistance strain, semiconductor strain, grating strain, etc.
  • piezoelectric crystal type optical sensor type (triangular light, confocal light, astigmatism, etc.)
  • electrical sensor type capacitortance, inductance, eddy current, etc.
  • the overall elastic structure sensor mostly uses strain gauges, optical sensors, and electrical sensors.
  • Piezoelectric crystal sensors mostly use multiple stacked piezoelectric crystals (generally 3 pieces, which measure 3 axial forces perpendicular to each other).
  • the sensitive components of Stewart sensors mostly use strain gauges and piezoelectric crystals.
  • Decoupling matrices can be obtained by experimental methods or finite element methods. This method considers the strain and deformation of certain parts of the elastic structure based on the principle of minimum energy. It must be proportional to the six-dimensional force, but in fact this assumption often has large errors.
  • the six-dimensional force sensor of Stewart parallel structure is usually calculated by spiral theory. This calculation assumes that the hinge part of all parallel mechanisms is an ideal hinge without any friction (torque), and the friction (torque) that is ignored in the calculation is usually It has a very large influence on the result, so its accuracy is very poor.
  • the piezoelectric crystal structure six-dimensional force sensor usually arranges all the piezoelectric crystals in a plane. It is considered that the piezoelectric crystal only bears the force in 3 directions, and the torque on it is ignored, which will also cause large calculation errors. And the static measurement effect is not good due to the charge attenuation.
  • the existing six-dimensional force sensor is calculated by neglecting certain deformation stress of the corresponding variable beam (or piezoelectric crystal), or neglecting the friction force of the mechanical hinge, and then simplifying the model to find a solution
  • the specific performance is that the inter-dimensional coupling (Crosstalk, Crosstalk, Dimentional coupling, Class II error) is very large.
  • a very flexible structure is adopted in some dimensions, and then As a result, the rigidity of the sensor in some dimensions is very small, and the stability during operation is very poor. It is even necessary to adopt a protective structure during large deformation. Even if the rigidity is small, the problem of excessive coupling between dimensions is difficult to fundamentally solve, and it is still as high as 2 % ⁇ 5%.
  • the existing six-dimensional force sensor has low accuracy and low rigidity, it is almost difficult in the commercial field, except for the low accuracy of the sensor required for grinding, polishing, clamping, automobile crash test, wheel force measurement, etc.
  • the above-mentioned applications such as grinding, polishing, and clamping can be easily replaced with pneumatic and elastic components, so there are not many applications.
  • the current real commercial force control robots, such as iiwa almost all use single-axis force sensors instead, but because each axis of the robot needs to use a single-axis force Sensors cause the robot structure to be extremely complex and costly, and it is extremely difficult to calculate the inertial force during high-speed motion.
  • the invention aims to solve the problems of low accuracy of the multidimensional force obtained by the existing multidimensional force sensor (mainly due to excessive coupling between dimensions), low sensor structure rigidity, and too complex calculation process and poor real-time performance.
  • the multi-dimensional force acquisition method based on the parallel rod system multi-dimensional force sensor includes the following steps:
  • the load platform produces displacement under the global coordinate system oxyz under the action of external force among them, Is the linear displacement in the global coordinate system oxyz; Is the angular displacement in the global coordinate system oxyz;
  • the origin of the local coordinate system is the center of the contact surface between the strain beam and the supporting platform Establish the local coordinate system of the supporting platform respectively
  • the strain beam produces deformation and displacement under the force of among them Is the linear displacement in the local coordinate system o i x i y i z i; Is the angular displacement in the local coordinate system o i x i y i z i ; it is determined according to the force-deformation relationship of the strain beam with
  • the coincidence point of the load platform and the corresponding local coordinate system origin is obtained in the local coordinate system
  • the local deformation displacement along the measurement axis is called the observable measurement; according to the space vector transformation, the local deformation displacement in the partial local coordinate system can be used to observe the measurement and calculate the displacement of the load platform under the global coordinate system oxyz Then calculate the local displacement of all strain beams in the local coordinate system
  • the load platform and support platform regard the load platform and support platform as a semi-elastic space; define the flexibility matrix of the load platform at the local coordinate origin o i as Define the local coordinate origin of the supporting platform on the supporting platform
  • the flexibility matrix is Use finite element analysis method or test method to obtain flexibility matrix average with Or, use Boussinesq, Mindlin or modern high-order semi-elastic space theory to derive and determine the flexibility matrix with Approximate value of
  • T i g represents the spatial transformation matrix from the coordinate system o i x i y i z i to the coordinate system oxyz (which can be abbreviated as g);
  • the external force borne by the load platform in the global coordinate system is The displacement of the load platform in the global coordinate system when subjected to external forces is
  • the generalized force consisting of three forces and three moments is They are the forces on the x-axis, y-axis, and z-axis when acting on the origin o of the global coordinate system under the global coordinate system g; They are the moments on the x-axis, y-axis, and z-axis when acting on the origin o of the global coordinate system under the global coordinate system g.
  • the generalized displacement deformation including three displacement deformations and three corner deformations is They are the displacement and deformation along the x-axis, y-axis, and z-axis when acting on the origin o of the global coordinate system under the global coordinate system g; They are the rotation angle deformations around the x-axis, y-axis, and z-axis when acting on the origin o of the global coordinate system under the global coordinate system g.
  • the micro-displacement of the load platform under the action of the external force is measured, and then the external load force, that is, the multi-dimensional force obtained by the multi-dimensional force sensor is obtained, and the multi-dimensional force includes three-dimensional force and six-dimensional force.
  • the multi-dimensional force acquisition method based on the parallel rod system multi-dimensional force sensor includes the following steps:
  • the generalized force includes force and moment, and the generalized deformation displacement includes linear displacement and angular displacement;
  • the generalized force When the multidimensional force is a six-dimensional force, the generalized force includes 3 forces and 3 moments, and the generalized deformation displacement includes 3 linear displacements and 3 angular displacements; when the multidimensional force is a plane three-dimensional force, the generalized force includes 2 forces and 1 Moment, generalized deformation displacement includes 2 linear displacements and 1 angular displacement.
  • the coincidence point of the load platform and the corresponding local coordinate system origin is obtained in the local coordinate system
  • the local deformation displacement along/around the measurement axis is called the observable measurement
  • the deformation displacement of the load platform in the global coordinate system calculate all the local deformation displacements of the corresponding local coordinate origin of each strain beam in the local coordinate system, including three linear displacements and three angular displacements;
  • the local generalized forces of all strain beams in the local coordinate system are translated to the origin of the global coordinate system according to the vector transformation relationship between the local coordinate system and the global coordinate system and summed to obtain the multi-dimensional force of the multi-dimensional force sensor.
  • the above process can also be used for adjustment, and can also be used as a multi-dimensional force acquisition method, that is, a multi-dimensional force acquisition method based on a parallel rod-based multi-dimensional force sensor, including the following steps:
  • the generalized force includes force and moment, and the generalized deformation displacement includes linear displacement and angular displacement;
  • the generalized force When the multidimensional force is a six-dimensional force, the generalized force includes 3 forces and 3 moments, and the generalized deformation displacement includes 3 linear displacements and 3 angular displacements; when the multidimensional force is a plane three-dimensional force, the generalized force includes 2 forces and 1 Moment, generalized deformation displacement includes 2 linear displacements and 1 angular displacement;
  • the local stiffness matrix and flexibility matrix of each beam are translated to the origin of the global coordinate system and summed to obtain the global stiffness matrix and flexibility matrix of the multi-dimensional force sensor;
  • the coincidence point of the load platform and the corresponding local coordinate system origin is obtained in the local coordinate system
  • the local deformation displacement along/around the measurement axis is called the observable measurement
  • the generalized force and the generalized deformation displacement are directly linear, that is, the generalized force is equal to the global stiffness matrix multiplied by the generalized deformation displacement. According to the global stiffness matrix and the generalized deformation displacement obtained above, the generalized force of the multidimensional force sensor can be obtained.
  • the theoretical mechanics calculation in the calculation method of the present invention uses the Newton-Euler principle.
  • the same calculation method as the present invention can also be calculated by using, for example, screw theory. Due to the actual nature of the screw theory, The above is also based on the Newton-Euler principle, and the derivation method is essentially the same.
  • the invention can greatly improve the measurement accuracy of the multi-dimensional force sensor, and at the same time can effectively improve the structural rigidity of the multi/six-dimensional force sensor through a parallel rod system (strain beam).
  • the calculation method is to solve the linear equations, its calculation It is extremely fast and can be used for real-time measurement and feedback control.
  • the calculation method of the present invention takes into account all the six-dimensional deformation of the strain beam and its stress and the six-dimensional elastic deformation of the support platform and the load platform, and further eliminates the inevitable dimensionality in the traditional calculation method based on the principle. Coupling.
  • the elastic sensitive element adopts the structure of multiple strain beams in parallel between two pseudo-rigid bodies, which avoids the coupling of the elastic structure in the traditional structure, so that the six-dimensional force sensor not only has high accuracy, but also has great structural rigidity.
  • the inter-dimensional coupling is generally greater than 2%.
  • the inter-dimensional coupling can be less than 1 ⁇ , due to the redundant parallel link structure .
  • the calculation method uses linear equations to solve, the calculation speed is extremely fast, and it is suitable for obtaining the force measurement results in real time.
  • the calculation method of the present invention is finally manifested as the solution of linear equations. Under the condition that the calculation accuracy is high, the calculation speed is extremely fast, and it has no influence on the measurement bandwidth of the multidimensional force sensor.
  • Figure 1 is a schematic diagram of a parallel rod system multi-dimensional force sensor, the loading platform is the loading platform, and the supporting platform is the supporting platform;
  • Figure 2 is the local coordinate system of the i-th beam in the sensor, and the strain gauge in the figure is a strain gauge;
  • Figure 3 shows the formation of the beam local coordinate system and the establishment of Coordinate Ma
  • Figure 4 is a schematic diagram of deformation of a strain beam under force in a local coordinate system
  • Figure 5 is a schematic diagram of the structure of a strain beam of arbitrary shape
  • Figure 6 is a schematic diagram of the force on the rigid plane of the elastic half-space
  • Fig. 7 is a schematic diagram of stiffness matrix coordinates, Fig. 7(a) is a curved beam, and Fig. 7(b) is a straight beam;
  • Figure 8 is a schematic diagram of parallel plane strain beams
  • Figure 9 is a schematic diagram of the sensor measuring along the measurement axis
  • Figure 9 (a) is a schematic diagram of the capacitive sensor measuring the displacement of the load platform
  • Figure 9 (b) is a schematic diagram of the triangular light sensor measuring the displacement of the load platform
  • Figure 10 is a schematic diagram of displacement sensor layout and coordinate establishment
  • Figure 10 (a) is a schematic diagram of displacement sensor layout
  • Figure 10 (b) is a schematic diagram of displacement sensor coordinate system establishment
  • Figure 11 is a schematic diagram of a strain gauge set on a strain beam
  • Figure 12 is a schematic diagram of strain gauge measurement
  • Figure 12(a) is a schematic diagram of strain gauges 1, 3
  • Figure 12(b) is a schematic diagram of strain gauges 2, 4;
  • Figure 13 is a schematic diagram of the deformation of the strain beam under force in the local coordinate system of the plane strain beam
  • FIG. 14 is a schematic diagram of force transformation between the local coordinate system and the global coordinate system corresponding to the multidimensional force sensor;
  • Figure 15 is a designed three-dimensional force sensor with a plane 8-beam structure
  • Figure 16 is a stress cloud diagram under a specific load
  • Figure 17 shows the force along the x, y axis and the moment around the z axis
  • Figure 18 shows the force along the x, y axis and the moment error around the z axis
  • Figure 19 shows the random error of the force along the x, y axis and the moment around the z axis
  • Figure 20 shows the force along the x, y axis and the moment around the z axis
  • Figure 21 shows the force along the x, y axis and the moment error around the z axis
  • Figure 22 shows the random error of the force along the x, y axis and the moment around the z axis
  • Fig. 23 is a plane three-dimensional force sensor using piezoelectric crystal as a strain beam;
  • Fig. 23(a) is a schematic diagram of the force of the piezoelectric crystal along its own axis x i ;
  • Fig. 23(b) is a schematic diagram of the pretension of the piezoelectric crystal;
  • Figure 24 is a six-dimensional force sensor using piezoelectric crystals as strain beams
  • Figure 25 is a plane three-dimensional force sensor using a capacitive sensor
  • Fig. 26 is a six-dimensional force sensor using a capacitive sensor; Fig. 26(a) is an overall schematic diagram; Fig. 26(b) is a schematic diagram of the internal layout of the capacitive sensor.
  • Loading platform loading platform; Supporting platform: supporting platform; Strain gauge: strain gauge; Fixed on supporting platform: fixed on supporting platform; Initial state: initial state; Rotation about x/y/z: rotation about x/y/ axis; Transformation along x,y,z: move along the x,y,z axis; Connection with loading platform: connect with the loading platform; Displacement of loading platform: displacement of the loading platform; Bending deformation by*: bending deformation caused by *; Shear deformation by*: shear deformation caused by *; View A: view A; High stiffness along x i : high stiffness along the x i axis; Low stiffness along x i : low stiffness along the x i axis; Elastic half-space: elastic half-space Space; Rigid plane: rigid plane; Beam_i: beam_i; Local force rotation from local coordinate system to global coordinate system: local force rotation from local coordinate system to global coordinate system; Forces transformation from o i to o: force from
  • the support platform and the load platform are rigid bodies (in practical applications, the support platform and the load platform are pseudo-rigid bodies, that is, the approximate rigid body with minimal deformation), the load platform and the support platform pass Parallel bars (multiple strain beams) are connected, and the strain beams in the parallel bars are used as elastic sensitive elements;
  • a micro-displacement measuring sensor is arranged between the supporting platform and the load platform to measure the micro-displacement between the two (due to the deformation of the strain beam).
  • the micro-displacement measuring sensor includes electrical, optical displacement sensors and other non-contact or micro-force contact sensors; Displacement sensors generally use electrical sensors such as capacitance, inductance, and eddy current, and optical displacement sensors generally use optical sensors such as triangular light, confocal light, and astigmatism;
  • strain gauge is pasted on the elastic sensitive element (strain beam).
  • the strain gauges include resistance strain gauges, semiconductor strain gauges, and optical strain gauges (Bragg grating FBG), etc.;
  • piezoelectric crystals as elastic sensitive components (strain beams);
  • a micro-displacement measurement sensor can be arranged between the support platform and the load platform, or a strain gauge can be pasted on the elastic sensitive element, or a piezoelectric crystal can be used as the elastic sensitive element, or a combination of the three can be used.
  • the main body of the symbol represents the space vector, Q represents the generalized force including force and moment, F represents force, and M represents moment; ⁇ represents generalized deformation including displacement and corner deformation, ⁇ D represents displacement deformation, ⁇ represents corner deformation; r represents strain beam The distance between the origin of the local coordinate system and the origin of the global coordinate system in the global coordinate system, ⁇ represents the rotation angle of the strain beam local coordinate system around the three axes of the global coordinate system;
  • the upper corner in the upper left corner represents the coordinate system
  • the upper corner in the upper left corner is marked as g to indicate that the corresponding parameter is a parameter in the global coordinate system (oxyz)
  • the upper corner in the upper left corner is marked as i to indicate that the corresponding parameter is the local coordinate system ( o i x i y i z i ) under the parameters;
  • the subscript in the lower left corner represents the point where the vector acts.
  • the subscript in the lower left corner is marked as o, which means that the corresponding vector acts on the origin o of the global coordinate system (oxyz).
  • the subscript in the lower left corner is marked as o i means that the corresponding vector acts on the local coordinates. Is the origin o i of (o i x i y i z i );
  • the upper corner of the upper right corner is marked with i, which means the applicator is the i-th strain beam, g or blank means the global quantity, that is, the applicator is the external force on the load platform;
  • the subscript in the lower right corner represents the direction of the vector, and the subscript in the lower right corner is marked as x, which means along the x axis in the upper left corner coordinate system, such as In the upper left corner of the middle, g refers to the x axis along the global coordinate system oxyz, and if the upper left corner is i, it refers to the x i axis along the local coordinate system o i x i y i z i .
  • the lower corner of the lower right corner is marked as y means along the y axis in the upper left corner coordinate system
  • the lower right corner is marked as z means along the z axis in the upper left corner coordinate system
  • the lower right corner marked with F and M means that the variable is caused by force or moment, not The band finger is caused by the combined action of force and moment
  • the subscript blank in the lower right corner indicates the vector formed by the xyz axis.
  • E.g Represents the i-th beam, in the global coordinate system g (oxyz), the force F acting on the o i point along the x direction of the global coordinate system; Represents the i-th beam, acting on point o i under the local coordinate system i (o i x i y i z i ), the linear displacement ⁇ D along the z i direction of the local coordinate system under the action of the torque M ;
  • the present invention is applicable to any form of multi-dimensional force sensor structure in the first embodiment.
  • the multi-dimensional force acquisition method based on the parallel rod system multi-dimensional force sensor includes the following steps:
  • the coordinate system is fixedly connected to the supporting platform and does not move, but for display convenience, the origin of the coordinate system is generally placed at the center o of the force-bearing part of the load platform.
  • the global coordinate system in the figure is oxyz, abbreviated as xyz; the y-axis is perpendicular to the x-axis, and the z-axis is perpendicular to the plane y-x;
  • the local coordinate system in the figure is o i x i y i z i , abbreviated as x i y i z i , where i represents the i-th beam; the strain beam and load
  • the center of the contact surface of the platform is the origin o i of the local coordinate system; as shown in Figure 2, the center line of the strain beam is the local coordinate system x i axis, the y i axis is perpendicular to the x i axis, and the y i axis is in the end face of the strain beam,
  • the z i axis is perpendicular to the plane y i -x i .
  • each local coordinate system and the global coordinate system can be represented by three rotation angles and three translation distances, denoted as with As shown in Figure 3, Figure 3 shows the process of determining the relationship between each local coordinate system and the global coordinate system, that is, the method of establishing the beam local coordinate system; that is: the initial state is that the local coordinate system coincides with the global coordinate system.
  • the strain beam rotates around the x-axis relative to the initial position And then rotate around the y axis And then rotate around the z axis Then translate along the xyz axis respectively Then connect the two ends of the strain beam to the load platform and the support platform respectively; when the load platform is displaced by the force, the coincidence point of the load platform with the origin of the global coordinate system moves from o to o′; the strain beam and the local coordinate system The coincidence point of the origin o i is moved to o i ′.
  • the present invention refers to the establishment of this local coordinate system as Coordinate Ma;
  • the deformation diagram of the strain beam under force in the local coordinate system is shown in Figure 4; when Euler beams are used (Timoshenko beams or other high-order beams can also be used), according to the force relationship of the strain beams:
  • E is the elastic modulus
  • G is the shear modulus
  • l i is the length of the strain beam
  • a i is the cross-sectional area of the strain beam
  • the representation of the space vector symbol above is the same. The subscript in the lower right corner represents the direction of the vector.
  • the subscript in the lower right corner is marked as x to indicate along the x axis
  • the subscript in the lower right corner is marked as y to indicate along the y axis.
  • the lower corner in the lower right corner is marked with z to indicate that it is along the z axis; the presence of other parameters in the lower corner in the lower right corner indicates the amount of the corresponding parameter on the corresponding axis.
  • the lower corner in the lower right corner is marked as Mz, which means it is on z due to M ⁇ The amount.
  • the flexibility matrix of the strain beam at the local coordinate origin o i is defined as:
  • the strain beam can be a strain beam of any shape (including the bending beam in FIG. 7), as shown in FIG. 5.
  • the flexibility matrix of the strain beam at the origin o i of the local coordinates can be obtained by using finite element or test methods; for straight-rod strain beams of constant cross-section, the strain beam can also be subjected to the aforementioned force-deformation relationship, and further based on Euler-Bernoulli beam theory writes the flexibility matrix (which can also be obtained from Timoshenko beam and other modern beam theories) as:
  • the schematic diagram of the rigid plane force of the elastic half space is shown in Figure 6.
  • the load platform and the support platform can be regarded as the elastic half space, and the flexibility matrix of the connection with the strain beam can be passed through the elastic half space.
  • the force-displacement-deformation relationship of the rigid plane is obtained;
  • the flexibility matrix of the load platform at the local coordinate origin o i is defined as:
  • the origin of the local coordinate system is the center of the contact surface between the strain beam and the supporting platform Establish the local coordinate system of the supporting platform (similar to the local coordinate system established at the center of the contact surface between the strain beam and the load platform); the supporting platform is at the origin of the local coordinate of the supporting platform
  • the flexibility matrix is defined as:
  • the finite element or test method can be used to obtain the average flexibility matrix with
  • Flexibility matrix corresponding to strain beam Flexibility matrix corresponding to the load platform Flexibility matrix corresponding to the supporting platform Both need to be processed on point o i and summed; therefore, the flexibility matrix corresponding to the support platform Move to point o i ;
  • Rot() refers to spatial rotation transformation; its inverse transformation is:
  • Fig. 7 The coordinate schematic diagram of the stiffness matrix transformation is shown in Fig. 7, the beam in Fig. 7(a) is a curved beam, and the beam in Fig. 7(b) is a straight beam;
  • T i g represents the space transformation matrix from the coordinate system i to the coordinate system g, the angle between the coordinate system i and the coordinate system g is ⁇ i , and the distance between the origins is r i ;
  • the stiffness matrix sum of all strain beams, load platforms, and support platforms at the origin in the global coordinate system is The spatial six-dimensional force sensor shown in Figure 1 is completely consistent with it;
  • the external generalized force borne by the load platform in the global coordinate system is The displacement of the load platform in the global coordinate system when subjected to external forces is
  • the stiffness matrix is only related to the actual structure, all the structural parameters have been obtained in advance.
  • the six-direction micro-displacement of the load platform under the action of the external force is measured, the six components of the external load force can be obtained.
  • the size of the load is measured by the micro-displacement measuring sensor arranged between the support platform and the load platform, and/or the strain gauge pasted on the elastic sensitive element, and/or the piezoelectric crystal is used as the sensitive element to measure the load
  • the six-direction micro-displacement of the platform under the action of external force can obtain the multi-dimensional force obtained by the multi-dimensional force sensor, including three-dimensional force, six-dimensional force and other dimensional forces.
  • the micro-displacement measurement sensor arranged between the support platform and the load platform measures the micro-displacement of the load platform in six directions under the action of external force, and the specific process is as follows:
  • the displacement measurement of the load platform can use a non-contact electrical displacement sensor, an optical displacement sensor, or a micro-force contact displacement sensor:
  • Figure 9(a) is a capacitive sensor, only the displacement on the x j axis has an effect on the capacitive displacement sensor;
  • Figure 9(b) is a triangular light sensor, and only the displacement on the x j axis has an effect on the triangular light Displacement sensor is useful;
  • each displacement sensor can Measure the displacement of the load platform along the x j axis of the local coordinate system of the displacement sensor;
  • three displacement sensors can completely measure the three displacements of the load platform.
  • the least square method can be used for calculation;
  • 6 displacement sensors can completely measure the 6 displacements of the load platform.
  • the least square method can be used for calculation.
  • the displacement relationship of the displacement sensor in the local coordinate system of the displacement sensor and in the global coordinate system is:
  • one of them Can be transformed by determine, It can be measured by the displacement sensor; thus, the displacement of the load platform with six degrees of freedom can be obtained.
  • the pose of each sensor in the global coordinate system can be determined by The only certainty, because this type of sensor rotates around its own x-axis is meaningless, so it can be regarded as According to It can be obtained:
  • c ⁇ means cos( ⁇ )
  • s ⁇ means sin( ⁇ );
  • Formula (21) is completely consistent with formula (22). Since ⁇ j and r j in formula (22) are known parameters, it can be seen that in formula (21) All are known parameters.
  • the six-dimensional force sensor if you want to obtain the six-dimensional force, you can set at least six displacement sensors at the same time to obtain the displacement of the load platform with six degrees of freedom;
  • a ⁇ is a matrix determined according to the known pose parameters of the sensor in the global coordinate system, B ⁇ is the displacement measured by each sensor along its own x j axis, and v ⁇ is the six-degree-of-freedom displacement of the load platform to be solved;
  • Fig. 26(a) and Fig. 26(b) are schematic diagrams of a six-dimensional force sensor with a plurality of capacitive sensors arranged between the support platform and the load platform.
  • the measurement axis of the capacitive sensor is along its local coordinate system o j x j
  • the x j axis of y j z j the displacement of the load platform can be measured by multiple capacitance sensors along each of its own measurement axes It is calculated; that is, in the specific process of obtaining the three-dimensional force by the planar three-dimensional force sensor, according to To calculate, use formula (21) to calculate.
  • the micro-displacement of the load platform in six directions under the action of external force is measured by the strain gauge pasted on the elastic sensitive element, and the specific process is as follows:
  • the strain gauges are symmetrically attached to the four (or two) sides of the strain beam, as shown in Figure 11;
  • the tensile force of the strain beam along the x i axis of its own local coordinate system can be obtained by the following differential circuit, as shown in Figure 12 (a) and Figure 12 (b), where E is the reference voltage and e is the measured voltage .
  • E is the reference voltage
  • e is the measured voltage
  • the tensile flexibility coefficient along the x i axis is E is the modulus of elasticity, A i is the cross-sectional area of the strain beam, and l i is the length of the strain beam; alternatively, it is determined by the finite element analysis method or the test method.
  • the displacement between the load platform and the support platform along the x i axis of the local coordinate system can be regarded as equal to the tensile (or compression) deformation of the strain beam along the x i axis.
  • the flexibility coefficients of the aforementioned strain beam, load platform and support platform can also be measured by finite element or test methods.
  • the above method is equivalent to treating the strain beam as a sensor that can measure the deformation along the x i axis of its own local coordinate system, which is consistent with the measurement effect of the corresponding capacitive sensor or optical sensor, and the subsequent calculations are exactly the same.
  • this embodiment provides a specific process of obtaining a plane three-dimensional force.
  • the above-mentioned space transformation symbol Write it out with specific formulas.
  • the sensor uses strain gauges as the deformation measurement element, and the load platform and support platform are both considered as pure rigid bodies.
  • E is the elastic modulus
  • a i is the cross-sectional area of the strain beam
  • I i is the moment of inertia
  • G is the shear modulus
  • the force on the strain beam is transformed from o i to the origin o of the global coordinate system, and the additional torque is
  • the torque applied to point o by the strain beam i is:
  • the above force transformation formula is also a specific expression using T i g transformation, which is completely consistent with the aforementioned displacement transformation expression.
  • Variables include global forces Local force And global displacement
  • the last 3 lines can be directly used to calculate the displacement of the load platform (rigid body) in the global coordinate system This is to convert the data measured by the strain gauge into the displacement along the x i axis of the corresponding strain beam, so the strain beam can be regarded as a sensor that can measure the displacement along the x i axis of the local coordinate system.
  • strain gauges are used as the deformation measuring element, and the load platform and support platform are both considered as pure rigid bodies.
  • the following calculation method sets the rotation transformation angle of the strain beam along the x-axis in the space Coordinate Ma transformation to 0. Because the following calculation only uses the strain beam along the x i axis of the local coordinate system as the initial calculation condition, this This setting is reasonable. When the torque around the y i axis and z i axis of the local coordinate system of the strain beam is also used as the initial calculation conditions in the calculation, the rotation transformation angle along the x axis cannot be ignored, then All rotation angles in the Coordinate Ma transformation need to be calculated in detail.
  • the corresponding force transformation can be specifically derived:
  • the above transformation is the specific expression formula of the space transformation T i g that includes the antisymmetric operator on the force transformation.
  • the corresponding displacement transformation can be specifically derived:
  • each displacement of the local origin o i Can be expressed as the global displacement of the global origin Linear function.
  • the overall force can be calculated in three steps.
  • Step (1) calculate the overall displacement of the load platform:
  • Formula (67) is used to construct A ⁇ and B ⁇ .
  • the global displacement of the load platform can be calculated using formula (74).
  • Step (2) calculate the local force in the local coordinate system:
  • formulas (63) and (66) can be used to calculate all local displacements in the local coordinate system
  • Equation (48) can be used to find the local force of each strain beam i, and all the equations for all beams can also be written as:
  • formula (85) can be used to directly find the local forces of all strained beams.
  • Step (3) calculate the global force on the loading platform:
  • each global force and moment applied by each beam i to the global origin o can be calculated. Therefore,
  • the six-dimensional force of the sensor can be calculated by the above method, and it can also be characterized by a unified linear equation group.
  • Formula (95) is also a way of writing Principle Ma that considers global displacement, global force, and local force.
  • This embodiment is a method for obtaining multi-dimensional force, and the multi-dimensional force is a three-dimensional force or a six-dimensional force.
  • the piezoelectric crystal is used as the strain beam to obtain the multi-dimensional force.
  • the calculation method of the plane three-dimensional force sensor to obtain the three-dimensional force is basically the same as that of the fifth embodiment;
  • piezoelectric crystal is used as the strain beam, and the load platform and support platform are both considered as pure rigid bodies.
  • the piezoelectric crystal is installed in a pre-compressed manner to ensure that it will not be out of contact with the support platform and the load platform during the stress process.
  • the force applied by the piezoelectric crystal along its axis x i can be converted into the change in the charge of the piezoelectric crystal.
  • the positive of each piezoelectric crystal along the x i direction can be obtained.
  • pressure Then obtain the deformation along the x i axis in the local coordinate system o i x i y i z i at the coincidence point of the load platform and o i at this time
  • the other calculation methods are completely the same as those in the foregoing embodiment.
  • the pre-tightening method of the piezoelectric crystal is shown in Figure 23(b).
  • the supporting platform is divided into two parts: the upper supporting platform and the lower supporting platform.
  • the two parts are arranged in multiple groups.
  • the bolts are fixed and the preload is applied at the same time
  • Fig. 24 The principle diagram shown in Fig. 24 can be used for the way of obtaining the six-dimensional force in space.
  • the material of the structure is aluminum alloy 7075, and its performance is shown in Table 1.
  • the first loading curve is along the x axis
  • the second loading curve is around the axis z.
  • the input force of the first loading curve is expressed as
  • the input force is applied to the load platform to calculate the stress of the structure. Thereafter, the stress of the beam along the axis x i is measured in the finite element software. Next, input the measured stress into the Principle Ma-based program to calculate the force exerted on the load platform. Finally, the calculated force can be used for comparison with the initial input force.
  • the calculated results, calculated errors and calculated random errors are shown in Figure 17, Figure 18 and Figure 19, respectively.
  • the calculated force and torque error is the difference between the input force and the calculated force, as shown in Figure 18. It can be observed that the error is linear. Therefore, the error can be corrected by linear correction.
  • the random error after linear correction is shown in Figure 19.
  • the effective bit of the stress data from the FE software is 6, as shown in Table 3. Therefore, the truncation error is 0.0005%.
  • the random error shown in FIG. 22 is similar to that shown in FIG. 19. This means that in the calculation results of the above-mentioned solution method in the FE software, it is almost impossible to observe the error (including type I error, that is, the error along the force loading axis, and type II error, that is the coupling error between dimensions).
  • type I error that is, the error along the force loading axis
  • type II error that is the coupling error between dimensions

Abstract

基于并联杆系多维力传感器的多维力获取方法，属于传感器测量技术领域，能够解决现有多维力传感器获得的多维力维间耦合大、精度低、传感器结构刚度小等问题。多维力获取方法包括以下步骤：建立局部坐标系统和全局坐标系统间的矢量变换关系矩阵，根据负载平台坐标系原点在局部坐标系下沿/绕测量轴线的局部变形位移获得可观测量，并计算负载平台在全局坐标系统下的变形位移，根据负载平台在全局坐标系统下的变形位移计算每个应变梁相应局部坐标原点在局部坐标系统下的所有局部变形位移，并根据每个应变梁在局部坐标系统下的局部变形位移计算每个应变梁在局部坐标系统下的局部广义力；通过求解线性方程组得到多维力传感器的多维力。

Description

基于并联杆系多维力传感器的多维力获取方法 技术领域
本发明属于传感器测量技术领域，涉及多维力传感器力的获取方法。
背景技术
多维力传感器能检测力在空间作用的信息，其中典型的六维力传感器可以获取作用力在空间坐标系所形成的3个分力和3个力矩。在航天机械臂、航天对接、协作机器人、仿生机械、医疗辅助器械、步行机器人、风洞测力、航空航天发动机推力测试、螺旋桨推力测试、汽车碰撞测试、微创手术机器人、打磨抛光器械、搅拌摩擦焊、机床加工力测量等众多领域，六维力传感器发挥着重要作用，其获得的六维力的准确性直接影响着系统的工作性能和控制精度。
从六维力传感器结构上分析，六维力传感器主要可以分为整体弹性结构式，Stewart并联结构式，压电晶体式、无摩擦导轨式(气浮、磁悬浮)，柔性结构式等，其中商用小型六维力传感器和MEMS领域主要采用整体弹性结构式，包括三垂直筋、I型梁积木、双环、桶型、马尔蒂斯十字梁、八垂直筋、非径向三梁、E结构、T结构，双悬臂梁等结构。大型六维力传感器主要采用Stewart并联结构式及球支撑并联结构，压电晶体式主要有4、6、8组平面布置结构，用于高频动态测量领域，无摩擦导轨式由于结构体积太大应用极少，柔性结构式主要用于机械手指端抓取，精度很低。
整体弹性结构式一般采用柔性铰链或柔性平板结构代替物理铰链，其精度稍高，但结构刚度小，且由于柔性体部分的耦合影响，精度一般在2％～5％，且由于其结构刚度较低，一般还需增加安全结构(避免变形过大损坏)或采用橡胶填充以避免振动过大。Stewart并联结构式结构刚度较大，但由于采用了物理铰链，有较大的摩擦力影响，其精度很低，也有采用柔性铰链代替结构中的物理铰链的方式，但会造成结构刚度低，且柔性铰链中的耦合力矩也无法消除，精度同样不高。压电晶体式一般采用平面多组布置，每一组包含三个晶片分别测量三个轴向力，转矩由多组轴向力进行推算，测力频响较高，但测力精度较低，且由于电荷漂移不适合静态测量。
从六维力传感器所用敏感元件分析，主要采用的敏感元件有应变片式(电阻应变、半导体应变、光栅应变等)，压电晶体式、光学传感器式(三角光、共焦光、像散光等)和电学传感器式(电容、电感、电涡流等)。一般情况下，整体弹性结构传感器多采用应变片、光学传感器、电学传感器。压电晶体式传感器多采用多片叠加(一般为3片，分别测量相互垂直的3个轴向力)的压电晶体。Stewart式传感器敏感元件多采用应变片和压电晶体。
绝大多数整体弹性结构多维力传感器采用解耦矩阵进行六维力计算，解耦矩阵可以采用实验法或者有限元法获得，这种方法认为根据能量最小原理弹性结构体的某些部位的应变变形一定与六维力呈正比，但实际上这种假设经常存在较大误差。Stewart并联结构六维力传感器通常采用螺旋理论进行计算，这种计算假定所有并联机构的铰链部分为理想铰链，不存在任何摩擦力(力矩)，而在计算中被忽略的摩擦力(力矩)通常对结果产生非常大的影响，因而其精度很差。压电晶体结构六维力传感器通常将所有压电晶体布置在一个平面内，认为压电晶体内只承受3个方向的力，而忽略其上的力矩，这同样会导致很大的计算误差，且由于电荷衰减静态测量效果不好。
综上所述，现有六维力传感器在计算方法上都是对应变梁(或压电晶体)的某一些变形应力进行忽略，或者对机械铰链摩擦力进行忽略，进而将模型简化找到求解方法，这就导致其计算精度很差，具体表现为维间耦合(Crosstalk，Crosscoupling，Dimentional coupling，Class II error)非常大，为了解决该类问题而在某些维度采用了柔性非常大的结构，进而导致了传感器在某些维度的刚度又非常小，工作时稳定性很差，甚至必须采用大变形时保护结构，且即使刚度很小，维间耦合过大问题也难以根本解决，依然要高达2％～5％。
现有的商用六维力传感器制造商主要有ATI，JR3，AMTI，Kistler，HBM，Schunk，Sunrise Instruments(宇立)，ME，Optoforce(Onrobot)，Hypersen(海伯森)等，这些公司的产品基本上采用的也是上述的一些基本结构，其中以整体弹性式和压电晶体式为主，其传感器精度基本上都在2％～5％，由于计算方法限制，进而导致结构限制，很难进一步提高。
由于现有的六维力传感器精度很低且其刚度很小，在商用领域，除打磨、抛光、夹持、汽车碰撞试验、车轮力测量等所需传感器精度很低的情况外，几乎很难进行大规模商业应用，而上述的打磨、抛光、夹持等应用场合可以很容易地用气动、弹性等元器件进行替代，因此应用也不多。以需要高精度测力的协作力控机器人为例，当前真正的商品化力控机器人，例如iiwa，几乎都采用单轴力传感器进行替代，但由于机器人的每一个轴都需要采用一个单轴力传感器，造成机器人结构极其复杂，成本极高，且造成高速运动时惯性力解算极其困难。以需要高精度测力的医疗微创手术机器人为例，几乎所有操作医生均认为手术过程中的力反馈对操作者影响很大，但由于现有六维力传感器精度太低，因而其只在研究中有所应用，所有当前真正商用化的微创手术机器人，例如da Vinci，都放弃了采用六维力传感器而只采用图像传感器。
现有的三维力传感器，包括三个轴向力和平面内两个轴向力和一个转矩类型，以及其它多维力传感器(两维、四维、五维)同样存在精度不高的问题，解决最通用的空间六维力传感器及平面三维力传感器的高精度求解问题则可以通过忽略一些维度力的方法解决所 有多维力传感器高精度求解问题。
发明内容
本发明为了解决现有多维力传感器获得的多维力精度低(主要是维间耦合过大)、传感器结构刚度小、计算过程太复杂实时性差等问题。
基于并联杆系多维力传感器的多维力获取方法，包括以下步骤：
建立附着于支撑平台上的全局坐标系统oxyz；
外力作用下负载平台在全局坐标系统oxyz下产生位移
Figure PCTCN2020099609-appb-000001
其中，
Figure PCTCN2020099609-appb-000002
为全局坐标系oxyz下的直线位移；
Figure PCTCN2020099609-appb-000003
为全局坐标系oxyz下的转角位移；
分别建立以应变梁与负载平台接触面中心为局部坐标系原点o i，附着于应变梁上的局部坐标系统o ix iy iz i；i表示应变梁的序号；
以应变梁与支撑平台接触面中心为局部坐标系原点
Figure PCTCN2020099609-appb-000004
分别建立支撑平台局部坐标系统
Figure PCTCN2020099609-appb-000005
根据应变梁和支撑平台的结构关系确定每个局部坐标系统与全局坐标系统的关系；
局部坐标系统中应变梁在力作用下产生变形位移，
Figure PCTCN2020099609-appb-000006
其中
Figure PCTCN2020099609-appb-000007
为局部坐标系o ix iy iz i下的直线位移；
Figure PCTCN2020099609-appb-000008
为局部坐标系o ix iy iz i下的转角位移；根据应变梁的受力变形关系确定
Figure PCTCN2020099609-appb-000009
和
Figure PCTCN2020099609-appb-000010
根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿测量轴线的局部变形位移，称为可观测量；根据空间矢量变换利用所述的部分局部坐标系下的局部变形位移，即可观测量，计算出负载平台在全局坐标系统oxyz下产生位移
Figure PCTCN2020099609-appb-000011
进而计算出所有应变梁局部坐标系下的局部位移
Figure PCTCN2020099609-appb-000012
根据所述的得到的所有应变梁的
Figure PCTCN2020099609-appb-000013
得到所有应变梁的局部广义力
Figure PCTCN2020099609-appb-000014
其中
Figure PCTCN2020099609-appb-000015
为局部坐标系o ix iy iz i 下的力，
Figure PCTCN2020099609-appb-000016
为局部坐标系o ix iy iz i下的力矩；
定义应变梁在局部坐标原点o i的柔度矩阵为
Figure PCTCN2020099609-appb-000017
采用有限元分析的方法或者试验方法获得应变梁的在局部坐标原点o i的柔度矩阵
Figure PCTCN2020099609-appb-000018
或者，采用Euler、Timoshenko或高阶现代梁理论，根据该应变梁的受力变形关系确定柔度矩阵
Figure PCTCN2020099609-appb-000019
对于负载平台和支撑平台，将负载平台和支撑平台看做半弹性空间；定义负载平台在局部坐标原点o i的柔度矩阵为
Figure PCTCN2020099609-appb-000020
定义支撑平台在支撑平台局部坐标原点
Figure PCTCN2020099609-appb-000021
的柔度矩阵为
Figure PCTCN2020099609-appb-000022
采用有限元分析的方法或试验方法获得柔度矩阵均
Figure PCTCN2020099609-appb-000023
和
Figure PCTCN2020099609-appb-000024
或者，采用Boussinesq、Mindlin或现代高阶半弹性空间理论推导确定柔性矩阵
Figure PCTCN2020099609-appb-000025
和
Figure PCTCN2020099609-appb-000026
的近似值；
将支撑平台对应的柔度矩阵
Figure PCTCN2020099609-appb-000027
移动到点o i，根据在局部坐标o i处的柔度和矩阵
Figure PCTCN2020099609-appb-000028
表示从局部坐标系
Figure PCTCN2020099609-appb-000029
到局部坐标系o i的空间变换矩阵；进而获得柔度和矩阵的逆矩阵，即其刚度矩阵
Figure PCTCN2020099609-appb-000030
将局部坐标系转换到全局坐标下得到全局坐标下刚度矩阵
Figure PCTCN2020099609-appb-000031
T i g表示从坐标系o ix iy iz i到坐标系oxyz(可简写为g)的空间变换矩阵；
则在全局坐标系下的原点处的所有应变梁、负载平台、支撑平台的刚度矩阵和为
Figure PCTCN2020099609-appb-000032
全局坐标系下负载平台承受的外部合力为
Figure PCTCN2020099609-appb-000033
全局坐标系下负载平台在承受外部力时的位移为
Figure PCTCN2020099609-appb-000034
多维力是六维力时，包括三个力和三个力矩组成的广义力为
Figure PCTCN2020099609-appb-000035
分别为在全局坐标系统g下作用在全局坐标系的原点o时x轴、y轴、z轴上的力；
Figure PCTCN2020099609-appb-000036
分别为在全局坐标系统g下作用在全局坐标系的原点o时x轴、y轴、z轴上的力矩。包括三个位移变形和三个转角变形的广义位移变形为
Figure PCTCN2020099609-appb-000037
分别为在全局坐标系统g下作用在全局坐标系的原点o时沿x轴、y轴、z轴的位移变形；
Figure PCTCN2020099609-appb-000038
分别为在全局坐标系统g下作用在全局坐标系的原点o时绕x轴、y轴、z轴的转角变形。
广义力、广义位移变形和刚度矩阵的关系为
Figure PCTCN2020099609-appb-000039
在多维力传感器实际测量时，由于刚度矩阵只与实际结构相关，事先已经得到；
测量出负载平台在外力作用下的的微位移，进而得到外部负载力，即多维力传感器获得的多维力，所述多维力包括三维力、六维力。
上述过程可以直接表达为：基于并联杆系多维力传感器的多维力获取方法，包括以下步骤：
建立附着于支撑平台上的全局坐标系统；
分别建立基于应变梁和位移传感器的局部坐标系统，建立之后应变梁和位移传感器分别对应的局部坐标系统不随应变梁和位移传感器运动；
根据空间矢量变换法则建立局部坐标系统和全局坐标系统间的矢量变换关系矩阵，包括广义力变换关系、广义变形位移变换关系和位移传感器变换关系；
所述的广义力包括力和力矩，所述广义变形位移包括直线位移和转角位移；
多维力为六维力时，广义力包括3个力和3个力矩，广义变形位移包括3个直线位移和3个转角位移；多维力为平面三维力时，广义力包括2个力和1个力矩，广义变形位移包括2个直线位移和1个转角位移。
根据理论力学、材料力学和弹性力学建立局部坐标系统下的应变梁、支撑平台及负载平台的变形和受力的关系矩阵，即局部刚度矩阵和局部柔度矩阵；
根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿/绕测量轴线的局部变形位移，称为可观测量；
根据所述负载平台与相应局部坐标系原点重合点在局部坐标系沿/绕测量轴线的局部变形位移计算负载平台在全局坐标系统下的变形位移，包括三个直线位移和三个转角位移；
根据负载平台在全局坐标系统下的变形位移计算每个应变梁相应局部坐标原点在局部坐标系统下的所有局部变形位移，包括三个直线位移和三个转角位移；
根据每个应变梁在局部坐标系统下的局部变形位移计算每个应变梁在局部坐标系统下的局部广义力，包括三个力和三个转矩；
将所有应变梁在局部坐标系统下的局部广义力根据局部坐标系统和全局坐标系统间的矢量变换关系平移到全局坐标系统原点并求和，得到多维力传感器的多维力。
实际上上述过程也可以用进行调整，也能够作为多维力获取方法，即基于并联杆系多维力传感器的多维力获取方法，包括以下步骤：
建立附着于支撑平台上的全局坐标系统；
分别建立基于应变梁和位移传感器的局部坐标系统，建立之后应变梁和位移传感器分别对应的局部坐标系统不随应变梁和位移传感器运动；
根据空间矢量变换法则建立局部坐标系统和全局坐标系统间的矢量变换关系矩阵，包括广义力变换关系、广义变形位移变换关系和位移传感器变换关系；
所述的广义力包括力和力矩，所述广义变形位移包括直线位移和转角位移；
多维力为六维力时，广义力包括3个力和3个力矩，广义变形位移包括3个直线位移和3个转角位移；多维力为平面三维力时，广义力包括2个力和1个力矩，广义变形位移包括2个直线位移和1个转角位移；
根据理论力学、材料力学和弹性力学建立局部坐标系统下的应变梁、支撑平台及负载平台的变形和受力的关系矩阵，即局部刚度矩阵和局部柔度矩阵；
根据空间矢量变换关系，将每个梁的局部刚度矩阵和柔度矩阵平移到全局坐标系统原点并求和，得到多维力传感器的全局刚度矩阵和柔度矩阵；
根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿/绕测量轴线的局部变形位移，称为可观测量；
根据所述负载平台与相应局部坐标系原点重合点在局部坐标系沿/绕测量轴线的局部变形位移计算负载平台在全局坐标系统下的广义变形位移，包括三个直线位移和三个转角位移；
由于广义力与广义变形位移直接呈现线性关系，即广义力等于全局刚度矩阵乘以广义变形位移，根据上述得到的全局刚度矩阵和广义变形位移即可得到多维力传感器的广义力。
本发明中的计算方法中的理论力学计算采用的是牛顿-欧拉原理，实际上采用例如螺旋理论(Screw theory)等也可以计算得到与本发明同样的计算方法，由于螺旋理论等实际上本质上也是基于牛顿-欧拉原理得到，推导方法本质上并无不同。
本发明的有益效果：
本发明可以极大地提高多维力传感器的测量精度，同时可以通过并联杆系(应变梁)方式有效地提高多/六维力传感器的结构刚度，另外由于本计算方法为求解线性方程组，其计算速度极快，可用于实时测量和反馈控制。
本发明的计算方法考虑到了应变梁的所有六维方向的变形及其应力及支撑平台和负载平台的六维弹性变形，进而从原理出发在计算方法上消除了传统计算方法中必然存在的维间耦合，基于该种方法，弹性敏感元件采用了两伪刚体间并联多根应变梁结构，避免了传 统结构中弹性结构的耦合，令六维力传感器不但精度高，且结构刚度极大。传统结构中无论是整体弹性结构还是Stewart结构、压电晶体结构，其维间耦合一般都大于2％，而采用本发明，可以令维间耦合小于1‰，由于采用了冗余并联杆系结构，通过适当设计，可以非常容易令结构刚度很大，计算方法采用的是线性方程组求解，计算速度极快，适于实时得到测力结果。
本发明的计算方法最终表现为线性方程组求解，在保证计算精度高的情况下，计算速度极快，对多维力传感器的测量带宽没有任何影响。
附图说明
图1为并联杆系多维力传感器示意图，loading platform为负载平台，supporting platform为支撑平台；
图2为传感器中的第i根梁局部坐标系统，图中strain gauge为应变片；
图3为梁局部坐标系统的形成及Coordinate Ma的建立方式；
图4为局部坐标系统中应变梁在力作用下的变形示意图；
图5为任意形状应变梁结构示意图；
图6为弹性半空间刚性平面受力示意图；
图7为刚度矩阵坐标示意图，图7(a)为弯梁，图7(b)为直梁；
图8为平面应变梁并联示意图；
图9为传感器沿测量轴线测量示意图，图9(a)为电容传感器测量负载平台位移示意图，图9(b)为三角光传感器测量负载平台位移示意图；
图10为位移传感器布置及坐标建立示意图，图10(a)为位移传感器的布置示意图；图10(b)为位移传感器坐标系的建立示意图；
图11为应变梁上设置应变片的示意图；
图12为应变片测量示意图，图12(a)为1、3应变片测量示意图，图12(b)为应变片2、4测量示意图；
图13为平面应变梁局部坐标系统中应变梁在力作用下的变形示意图；
图14为多维力传感器对应的局部坐标系和全局坐标系的力变换示意图；
图15为设计的平面8梁结构的三维力传感器；
图16为特定载荷下的应力云图；
图17为沿x，y轴的力和绕z轴的力矩；
图18为沿x，y轴的力和绕z轴的力矩误差；
图19为沿x，y轴的力和绕z轴的力矩随机误差；
图20为沿x，y轴的力和绕，z轴的力矩；
图21为沿x，y轴的力和绕z轴的力矩误差；
图22为沿x，y轴的力和绕z轴的力矩随机误差；
图23为采用压电晶体作为应变梁的平面三维力传感器；图23(a)为压电晶体沿自身轴线x i的受力示意图；图23(b)为压电晶体的预紧示意图；
图24为采用压电晶体作为应变梁的六维力传感器；
图25为采用电容传感器的平面三维力传感器；
图26为采用电容传感器的六维力传感器；图26(a)为整体示意图；图26(b)为电容传感器内部布置示意图。
图中中英文对照说明：
Loading platform:负载平台；Supporting platform:支撑平台；Strain gauge:应变片；Fixed on supporting platform:固定于支撑平台；Initial state:初始状态；Rotation about x/y/z:绕x/y/轴旋转；Transformation along x,y,z:沿x,y,z轴移动；Connection with loading platform:与负载平台连接；Displacement of loading platform:负载平台位移；Bending deformation by*:由*引起的弯曲变形；Shear deformation by*:由*引起的剪切变形；View A:视图A；High stiffness along x i:沿x i轴高刚度；Low stiffness along x i:沿x i轴低刚度；Elastic half-space:弹性半空间；Rigid plane:刚性平面；Beam_i:梁_i；Local force rotation from local coordinate system to global coordinate system:局部力从局部坐标系统到全局坐标系统的旋转；Forces transformation from o i to o:力从o i到o的变换；Zone forces imposed on:力施加区域；The thickness of this structure:结构体厚度；Piezoelectrical crystal:压电晶体；Fixed and prestressed with bolts:采用螺栓固联并预紧；Differential capacitive sensor:差动电容传感器；
具体实施方式
具体实施方式一：
并联杆系多维力传感器，如图1所示，支撑平台和负载平台为刚体(实际应用中支撑平台和负载平台都是伪刚体，即变形量极小的近似刚体)，负载平台和支撑平台通过并联杆系(多根应变梁)连接，并联杆系中的应变梁作为弹性敏感元件；
在支撑平台和负载平台之间布置有测量两者之间微位移(由于应变梁变形导致)的微位移测量传感器，微位移测量传感器包括电学、光学位移传感器及其它非接触或微力接触传感器；电学位移传感器一般采用电容、电感、电涡流等类型电学传感器，光学位移传感器一般采用三角光、共焦光、像散光等类型光学传感器；
和/或，
在弹性敏感元件(应变梁)上粘贴有应变片(strain gauge)，应变片包括电阻应变片、半导体应变片和光学应变片(布拉格光栅FBG)等；
和/或，
采用压电晶体作为弹性敏感元件(应变梁)；
实际上就是，可以在支撑平台和负载平台之间布置微位移测量传感器，或者在弹性敏感元件上粘贴有应变片，或者采用压电晶体作为弹性敏感元件，或者三者结合。
具体实施方式二：
在说明本实施方式之前首先对本发明的空间矢量符号的表示形式进行说明，例如
Figure PCTCN2020099609-appb-000040
Figure PCTCN2020099609-appb-000041
每个参数的整体作为一个形式进行说明；
符号的主体表示空间矢量，Q表示包括力和力矩的广义力，F表示力，M表示力矩；Δ表示包括位移和转角变形的广义变形，ΔD表示位移变形，Δθ表示转角变形；r表示应变梁局部坐标系原点在全局坐标系下的与全局坐标系原点的距离，β表示应变梁局部坐标系绕全局坐标系三个轴的转角；
左上角的上角标代表坐标系统，左上角的上角标为g表示对应的参数为全局坐标系(oxyz)下的参数，左上角的上角标为i表示对应的参数为局部坐标系(o ix iy iz i)下的参数；
左下角的下角标代表矢量作用的点，左下角的下角标为o表示对应的矢量作用在全局坐标系(oxyz)的原点o，左下角的下角标为o i表示对应的矢量作用在局部坐标系(o ix iy iz i)的原点o i；
右上角的上角标为i表示施加者为第i根应变梁，g或者空白表示为全局量，即施加者为负载平台上的外力；
右下角的下角标代表矢量的方向，右下角的下角标为x表示沿着左上角坐标系中的x轴，如
Figure PCTCN2020099609-appb-000042
中左上角为g则指沿着全局坐标系oxyz的x轴，如果左上角为i则指沿着局部坐标系o ix iy iz i的x i轴，同样，右下角的下角标为y表示沿着左上角坐标系中的y轴，右下角的下角标为z表示沿着左上角坐标系中的z轴，右下角标带有F和M是指变量由力或力矩引起，不带指由力和力矩共同作用引起，右下角的下角标空白表示xyz轴共同构成的矢量。
例如，
Figure PCTCN2020099609-appb-000043
表示第i根梁，在全局坐标系统g(oxyz)下，作用于o i点，沿全局坐标系统x方向的力F；
Figure PCTCN2020099609-appb-000044
表示第i根梁，在局部坐标系统i(o ix iy iz i)下，作用于o i点，在转矩M的作用下导致的沿该局部坐标系统z i方向的线性位移ΔD；
基于并联杆系多维力传感器获取多维力，本发明适用于实施方式一中的任何形式的多维力传感器结构。
基于并联杆系多维力传感器的多维力获取方法，包括以下步骤：
首先建立各坐标系：
建立附着于支撑平台上的全局坐标系统，即该坐标系统固联于支撑平台不运动，但为了显示方便，一般将坐标系原点放置于负载平台受力部分中心o。如图2所示，图中全局坐标系为oxyz，简记为xyz；y轴与x轴垂直，z轴与平面y-x垂直；
建立表达应变梁局部変形的的局部坐标系统，图中局部坐标系为o ix iy iz i，简记为x iy iz i，其中i表示第i根梁；以应变梁与负载平台接触面中心为局部坐标系原点o i；如图2所示，以应变梁中心线为局部坐标系x i轴，y i轴与x i轴垂直，且y i轴处于应变梁端面内，z i轴与平面y i-x i垂直，该局部坐标系统建立后即视为在全局坐标系统中固定，并不随应变梁变形而改变，其具体建立方式如下所述：
每个局部坐标系统与全局坐标系统的关系都可以用三个旋转角度和三个平移距离来表示，记为
Figure PCTCN2020099609-appb-000045
和
Figure PCTCN2020099609-appb-000046
如图3所示，图3表示每个局部坐标系统与全局坐标系统的关系确定的过程，即梁局部坐标系统的的建立方式；即：初始状态为局部坐标系与全局坐标系重合，先将应变梁相对初始位置绕x轴旋转
Figure PCTCN2020099609-appb-000047
再绕y轴旋转
Figure PCTCN2020099609-appb-000048
再绕z轴旋转
Figure PCTCN2020099609-appb-000049
然后沿xyz轴分别平移
Figure PCTCN2020099609-appb-000050
再将应变梁两端分别连接到负载平台和支撑平台上；当负载平台受力产生位移后，负载平台上与全局坐标系原点的重合点从o移动到o′；应变梁上与局部坐标系原点o i的重合点移动到o i′，本发明将这种局部坐标系统的建立方式称为Coordinate Ma；
局部坐标系统中应变梁在力作用下的变形示意图如图4所示；采用Euler梁时(也可采用Timoshenko梁或其它高阶梁)，根据应变梁的受力关系可知：
Figure PCTCN2020099609-appb-000051
Figure PCTCN2020099609-appb-000052
Figure PCTCN2020099609-appb-000053
Figure PCTCN2020099609-appb-000054
Figure PCTCN2020099609-appb-000055
Figure PCTCN2020099609-appb-000056
E为弹性模量，G为剪切模量；l i是应变梁长度；A i为应变梁横截面面积；
Figure PCTCN2020099609-appb-000057
为绕y轴的惯性矩；
Figure PCTCN2020099609-appb-000058
为绕z轴的惯性矩；
Figure PCTCN2020099609-appb-000059
(其实就是
Figure PCTCN2020099609-appb-000060
一般写为
Figure PCTCN2020099609-appb-000061
)为绕x轴的惯性矩，即极惯性矩；
Figure PCTCN2020099609-appb-000062
与上述空间矢量符号的表示形式是相同的，右下角的下角标代表矢量的方向，仍然是右下角的下角标为x表示沿着x轴，右下角的下角标为y表示沿着y轴，右下角的下角标为z表示沿着z轴；右下角的下角标中还出现其它参数就表示对应参数在相应轴上的量，例如右下角的下角标为Mz就表示由于M导致在z上的量。
应变梁在局部坐标原点o i的柔度矩阵定义为：
Figure PCTCN2020099609-appb-000063
应变梁可以为任意形状应变梁(包括图7中弯梁)，如图5所示。对于任意形状的应变梁，可以采用有限元或试验方法获得应变梁的在局部坐标原点o i的柔度矩阵；对于等截面直杆应变梁，还可以根据前述的该应变梁受力变形关系，进一步根据Euler-Bernoulli梁理论将柔度矩阵(该矩阵也可以根据Timoshenko梁及其它现代梁理论获得)写为：
Figure PCTCN2020099609-appb-000064
弹性半空间刚性平面受力示意图如图6所示，对于负载平台和支撑平台，可以将负载平台和支撑平台看做弹性半空间，其与应变梁连接处的柔度矩阵可以通过弹性半空间上的刚性平面受力位移变形关系得到；
负载平台在局部坐标原点o i的柔度矩阵定义为：
Figure PCTCN2020099609-appb-000065
以应变梁与支撑平台接触面中心为局部坐标系原点
Figure PCTCN2020099609-appb-000066
建立支撑平台局部坐标系统(与建立在应变梁与负载平台接触面中心的局部坐标系统相似)；支撑平台在支撑平台局部坐标原点
Figure PCTCN2020099609-appb-000067
的柔度矩阵定义为：
Figure PCTCN2020099609-appb-000068
可以采用有限元或试验方法获得柔度矩阵均
Figure PCTCN2020099609-appb-000069
和
Figure PCTCN2020099609-appb-000070
也可以采用Boussinesq和Mindlin等的弹性半空间理论推导该柔性矩阵近似值：
Figure PCTCN2020099609-appb-000071
式中：E-弹性模量；μ-泊松比；A-刚性平面面积；I p-刚性平面绕x轴极惯性矩；r p-刚性平面绕x轴极惯性半径；s-刚性平面沿z轴边长；w-刚性平面沿y轴边长；
应变梁对应的柔度矩阵
Figure PCTCN2020099609-appb-000072
负载平台对应的柔度矩阵
Figure PCTCN2020099609-appb-000073
支撑平台对应的柔度矩阵
Figure PCTCN2020099609-appb-000074
均需要在点o i上进行处理，并对其求和；所以将支撑平台对应的柔度矩阵
Figure PCTCN2020099609-appb-000075
移动到点o i；
为了后续统一理解，现将一种通用的空间矢量变换矩阵进行说明，说明中的两个坐标系统o px py pz p和o qx qy qz q为泛指，实际应用中可以用任意坐标系统替代：
Figure PCTCN2020099609-appb-000076
为坐标系o px py pz p到坐标系o qx qy qz q的空间变换矩阵，其中o p，x p，y p，z p分别表示坐标系o px py pz p的坐标原点，x轴，y轴和z轴，o q，x q，y q，z q分别表示坐标系o qx qy qz q的坐标原点，x轴，y轴和z轴，γ＝[γ x,γ y,γ z] T为坐标系o px py pz p和坐标系o qx qy qz q在坐标系o qx qy qz q内绕x，y，z的空间夹角，d＝[d x,d y,d z] T为坐标系o px py pz p和坐标系o qx qy qz q坐标原点在坐标系o qx qy qz q内沿x，y，z的距离，其具体含义如下：
Rot(γ)＝Rot(z,γz)Rot(y,γ y)Rot(x,γ x)       (13)
Rot()指空间旋转变换；其逆变换为：
Rot T(γ)＝Rot T(x,γ)Rot T(y,γ)Rot T(z,γ)        (14)
Figure PCTCN2020099609-appb-000077
代表矢量d＝[d x,d y,d z] T对应的反对称算子；该算子也可以看做叉乘算子，即力与力臂叉乘转换为力矩，及转速(微转角或转角差分)与转动半径叉乘转换为直线速度(微位移或位移差分)；
在具体应用
Figure PCTCN2020099609-appb-000078
时，可以将o px py pz p和o qx qy qz q替换为其它具体的坐标系统，将γ替换为具体的两个坐标系统的夹角，将d替换为具体的两个坐标系统的原点距离即可，例如后面所述的
Figure PCTCN2020099609-appb-000079
即为从梁与支撑平台相交处的坐标系
Figure PCTCN2020099609-appb-000080
到梁与负载平台相交处的坐标系o ix iy iz i的空间变换，T i g即为从梁与负载平台相交处的坐标系o ix iy iz i到全局坐标系统g的空间变换。
刚度矩阵变换坐标示意图如图7所示，图7(a)中梁为弯梁，图7(b)中梁为直梁；
在局部坐标o i处的柔度和矩阵
Figure PCTCN2020099609-appb-000081
Figure PCTCN2020099609-appb-000082
表示从局部坐标系
Figure PCTCN2020099609-appb-000083
到局部坐标系o i的空间变换矩阵；
Figure PCTCN2020099609-appb-000084
为两个局部坐标系o ix iy iz i和
Figure PCTCN2020099609-appb-000085
的坐标轴夹角，
Figure PCTCN2020099609-appb-000086
为两个局部坐标系o ix iy iz i和
Figure PCTCN2020099609-appb-000087
的原点间距离；
当应变梁为直梁时，
Figure PCTCN2020099609-appb-000088
Figure PCTCN2020099609-appb-000089
代表矢量l＝[l x，l y，l z] T对应的反对称算子；
其中
Figure PCTCN2020099609-appb-000090
代表两个局部坐标系o ix iy iz i和
Figure PCTCN2020099609-appb-000091
原点在局部坐标系o ix iy iz i中的距离；后面所述的
Figure PCTCN2020099609-appb-000092
变换都采用类似定义，将不再详述。
对于每一根应变梁i，在其局部坐标系原点的柔度矩阵都可以采用上述方法获得；
单根应变梁以及分别与负载平台、支撑平台连接处的柔度和矩阵的逆矩阵，即其刚度矩阵
Figure PCTCN2020099609-appb-000093
局部坐标系转换到全局坐标下刚度矩阵的转换公式为：
Figure PCTCN2020099609-appb-000094
Figure PCTCN2020099609-appb-000095
T i g表示从坐标系i到坐标系g的空间变换矩阵，坐标系i与坐标系g之间的夹角为β i，原点间距离为r i；
以图8所示的平面三维力传感器为例，在全局坐标系下的原点处的所有应变梁、负载平台、支撑平台的刚度矩阵和为
Figure PCTCN2020099609-appb-000096
图1所示的空间六维力传感器与其完全一致；
全局坐标系下负载平台承受的外部广义力为
Figure PCTCN2020099609-appb-000097
全局坐标系下负载平台在承受外部力时的位移为
Figure PCTCN2020099609-appb-000098
力和位移、刚度的关系可以写为：
Figure PCTCN2020099609-appb-000099
在多维力传感器实际测量时，由于刚度矩阵只与实际结构相关，所有结构参数事先已经得到，只要测量出负载平台在外力作用下的六个方向的微位移，即可以得到外部负载力六个分量的大小，即：只要通过支撑平台和负载平台之间布置的微位移测量传感器，和/或，弹性敏感元件上粘贴有的应变片，和/或，采用压电晶体作为敏感元件，测量得到负载平台在外力作用下的六个方向的微位移，即可以得到多维力传感器获得多维力，包括三维力，六维力及其它维度力。
本发明将该计算方法称为Principle Ma。
具体实施方式三：
本实施方式所述通过支撑平台和负载平台之间布置的微位移测量传感器测量负载平台在外力作用下的六个方向的微位移，具体过程如下：
负载平台位移的测量可以采用非接触电学位移传感器或光学位移传感器或者微力接触位移传感器：
如图9所示，图9(a)为电容传感器，只有x j轴上的位移对电容位移传感器有作用；图9(b)为三角光传感器，也只有x j轴上的位移对三角光位移传感器有作用；
如图10(a)和图10(b)所示，建立附着于位移传感器上的位移传感器局部坐标系统，位移传感器测量的发生位移的方向为局部坐标系统x j轴；每一个位移传感器都可以测量出沿位移传感器局部坐标系统x j轴的负载平台位移；
对于平面的三维力传感器而言，有3个位移传感器即可完全测量出负载平台的3个位移，当采用更多位移传感器时可以采用最小二乘法进行计算；
对于立体的六维力传感器而言，有6个位移传感器即可完全测量出负载平台的6个位移，当采用更多位移传感器时可以采用最小二乘法进行计算。
针对于六维力传感器而言，位移传感器在位移传感器局部坐标系下与在全局坐标系下的位移关系为：
Figure PCTCN2020099609-appb-000100
其中，
Figure PCTCN2020099609-appb-000101
Figure PCTCN2020099609-appb-000102
表示从传感器局部坐标系j到全局坐标系g的空间变换矩阵，坐标系j与坐标系g之间的夹角为β j，原点间距离为r j；
上式是一种通用的表达，对于平面三维力和其它维度传感器(二维、四维、五维)而言，实际上只要做适度简化，其公式表达完全一致。
定义如下全局坐标系原点与位移传感器局部坐标系原点的位移协调关系矩阵为
Figure PCTCN2020099609-appb-000103
其中的
Figure PCTCN2020099609-appb-000104
可以由变换
Figure PCTCN2020099609-appb-000105
确定，
Figure PCTCN2020099609-appb-000106
可以由位移传感器测量得出；从而获得负载平台六个自由度的位移。
每一个传感器的全局坐标系下的位姿都可由
Figure PCTCN2020099609-appb-000107
唯一确定，由于该类传感器绕自身x轴旋转并无意义，因此可视为
Figure PCTCN2020099609-appb-000108
此时根据
Figure PCTCN2020099609-appb-000109
即可以得出：
Figure PCTCN2020099609-appb-000110
式中cβ表示cos(β)，sβ表示sin(β)；
公式(21)与公式(22)完全一致，由于公式(22)中的β j和r j都为已知参数，可见公式(21)中
Figure PCTCN2020099609-appb-000111
都为已知参数。
针对于六维力传感器而言，想要获得六维力，可以同时设置至少六个位移传感器获得负载平台六个自由度的位移；
当位移传感器m＝6，即采用6个传感器时，
Figure PCTCN2020099609-appb-000112
则可定义
Figure PCTCN2020099609-appb-000113
Figure PCTCN2020099609-appb-000114
Figure PCTCN2020099609-appb-000115
则上述方程组可写为矩阵形式：
A Δv Δ＝B Δ           (27)
A Δ为根据传感器全局坐标系下已知位姿参数确定的矩阵，B Δ为每一个传感器沿自身x j轴测得的位移量，v Δ为待求解的负载平台六自由度位移；
因此根据A Δv Δ＝B Δ，通过六个传感器测得的位移量即可获得负载平台六个自由度的位移。
如图26(a)和图26(b)为在支撑平台和负载平台之间布置有多个电容传感器的六维力传感器示意图。
针对于三维力传感器而言，如果在支撑平台和负载平台之间布置有多个电容传感器，如图25所示的平面三维力传感器，电容传感器的测量轴线为沿其局部坐标系统o jx jy jz j的x j轴，负载平台的位移可以通过多个电容传感器沿着每个自身测量轴线得到的测量量
Figure PCTCN2020099609-appb-000116
经过计算得到；即，平面三维力传感器获得三维力的具体过程中，根据
Figure PCTCN2020099609-appb-000117
进行计算， 即采用公式(21)进行计算，现将公式(21)重新写在下面：
Figure PCTCN2020099609-appb-000118
具体对于平面三维传感器可以写为：
Figure PCTCN2020099609-appb-000119
当有3个以上的传感器时，则可以列出方程组，每一个
Figure PCTCN2020099609-appb-000120
都可以由电容传感器直接测量得到，进而可以得到负载平台的总体位移
Figure PCTCN2020099609-appb-000121
接下来的计算方法与前面所述的计算方法完全一致。
其他步骤与具体实施方式二相同。
具体实施方式四：
本实施方式所述通过弹性敏感元件上粘贴有的应变片测量负载平台在外力作用下的六个方向的微位移，具体过程如下：
采用应变片时，应变片对称粘贴于应变梁的四个(或者两个)侧面，如图11所示；
则应变梁沿其自身局部坐标系统的x i轴的拉伸力可以由以下差动电路获得，如图12(a)和图12(b)所示，其中E为参考电压，e为测量电压，可以只用应变片1、3或者应变片2、4求得应变梁沿x i的拉伸力
Figure PCTCN2020099609-appb-000122
也可以同时采用应变片1、2、3、4求得，此时将1、3和2、4分别求得的力做平均值。
此时由于应变梁的柔度系数已知，沿x i轴的拉伸柔度系数为
Figure PCTCN2020099609-appb-000123
E为弹性模量，A i为应变梁横截面面积，l i是应变梁长度；或者，采用有限元分析方法或者试验方式测量确定
Figure PCTCN2020099609-appb-000124
则应变梁沿x i轴的拉伸(或压缩)变形为
Figure PCTCN2020099609-appb-000125
如果将负载平台和支撑平台都视为刚体，则负载平台与支撑平台之间沿局部坐标系统x i轴的位移就可以视为等于上述应变梁沿x i轴的拉伸(或压缩)变形。
如果考虑负载平台和支撑平台的柔度，由于负载平台和支撑平台在x i轴柔度均为已知，即：
Figure PCTCN2020099609-appb-000126
A为应变梁横截面面积，μ是泊松比；
则负载平台与支撑平台之间沿局部坐标系统x i轴的位移为：
Figure PCTCN2020099609-appb-000127
上述的应变梁、负载平台和支撑平台的柔度系数也可以采用有限元或者试验方式测量 的出。
上述方式就相当于将应变梁看做一个可以测量沿其自身局部坐标系统x i轴变形的传感器，与相应的电容传感器或者光学传感器测量的效果一致，接下来的计算则完全相同。
其他步骤与具体实施方式二相同。
具体实施方式五：
上面的分析中采用了空间变换矩阵简化符号
Figure PCTCN2020099609-appb-000128
等，；
为了更清楚地说明计算方法，本实施方式给出一种平面的三维力获取的具体过程，在该过程中，将上述空间变换符号
Figure PCTCN2020099609-appb-000129
等用具体公式写出。
平面的三维力传感器获得平面三维力的具体过程：
该传感器采用应变片作为变形测量元件，且负载平台和支撑平台都考虑为纯刚体。
针对于平面的的三维力传感器而言，多维力传感器中局部坐标系统中应变梁在力作用下的图4所示变形示意图简化为图13所示；
在局部坐标系x iy i下
Figure PCTCN2020099609-appb-000130
E为弹性模量，A i为应变梁横截面面积；
Figure PCTCN2020099609-appb-000131
I i为惯性矩；G为剪切模量；
Figure PCTCN2020099609-appb-000132
Figure PCTCN2020099609-appb-000133
那么，在局部坐标系x iy i和全局坐标系xy下的o i位移的关系为：
Figure PCTCN2020099609-appb-000134
上述关系中的
Figure PCTCN2020099609-appb-000135
以及公式中与
Figure PCTCN2020099609-appb-000136
的叉乘关系即为
Figure PCTCN2020099609-appb-000137
具体形式；
局部坐标系和全局坐标系的力示意图如图14所示；
在局部坐标系x iy i下和全局坐标系xy下的应变梁受到力的关系为
Figure PCTCN2020099609-appb-000138
应变梁上的力从o i变换到全局坐标系的原点o，附加扭矩是
Figure PCTCN2020099609-appb-000139
由应变梁i施加在点o上的扭矩为：
Figure PCTCN2020099609-appb-000140
上面力的变换公式同样为采用T i g变换的具体表达，与前述位移变换的表达完全一致。
因此，负载平台上的总力为
Figure PCTCN2020099609-appb-000141
基于以上的公式，基本方程组可以写成
Figure PCTCN2020099609-appb-000142
方程组中的变量是：
Figure PCTCN2020099609-appb-000143
变量包括全局力
Figure PCTCN2020099609-appb-000144
局部力
Figure PCTCN2020099609-appb-000145
和全局位移
Figure PCTCN2020099609-appb-000146
以应变梁n＝3为例，
Figure PCTCN2020099609-appb-000147
视为观测变量，即可由粘贴于应变梁上的应变片测量得出，方程组(37)中有12个未知变量，即
Figure PCTCN2020099609-appb-000148
线性方程系统可以写成
Av＝B           (38)
其中，
Figure PCTCN2020099609-appb-000149
Figure PCTCN2020099609-appb-000150
其中，sβ＝sin(β)，cβ＝cos(β)；
观察线性方程系统Av＝B，最后3行可以直接用来计算全局坐标系中负载平台(刚体)的位移
Figure PCTCN2020099609-appb-000151
这里即为将应变片测量得到的数据转换为沿相应应变梁的x i轴的位移，因此可以将应变梁视为一个可以测量沿局部坐标系x i轴位移的传感器。
然后用A的中间6行计算局部力；最后用前3行计算全局力；该方法可进一步提高计算速度。
这可以整理为3个公式：
A Δv Δ＝B Δ            (39)
A iv i＝B i            (40)
A Fv F＝B F            (41)
其中
Figure PCTCN2020099609-appb-000152
通过A Δv Δ＝B Δ即可以求解出负载平台的位移量。
当其中观测变量大于3个，即应变梁数量大于3或
Figure PCTCN2020099609-appb-000153
中的某一些量能够被观测，该线性方程组转化为超定线性方程组；许多方法可以用来求解这样一个超定方程组。这里给出了一个简单的最小二乘法：A TAv＝A TB；
这意味着无论在刚性平台上附加多少根应变梁，计算整体力都是非常容易的，通常由于均化效应，随着梁数的增加，计算精度也会提高。
具体实施方式六：
本实施方式中，空间六维力传感器获得六维力的具体过程，上述空间变换矩阵
Figure PCTCN2020099609-appb-000154
(具体有
Figure PCTCN2020099609-appb-000155
等)将用具体公式写出。
下面的方法中采用应变片作为变形测量元件，且负载平台和支撑平台都考虑为纯刚体。
下面计算方法将应变梁在空间的Coordinate Ma变换中的沿x轴的旋转变换角度设为0，由于下面的计算中只采用应变梁沿局部坐标系x i轴的拉力作为初始计算条件，因此这种设定是合理的，当计算中也要将绕应变梁的局座坐标系的y i轴和z i轴的转矩作为初始计算条件时，则沿x轴的旋转变换角度不能忽略，则Coordinate Ma变换中的所有旋转角度均需要具体计算。
当沿x轴的旋转变换角度设为0时，如图3所示的Coordinate Ma的变换的旋转变换及逆变换为：
Figure PCTCN2020099609-appb-000156
Figure PCTCN2020099609-appb-000157
上述变换与T i g变换是一致的。
相应的力变换可以具体推导：
Figure PCTCN2020099609-appb-000158
Figure PCTCN2020099609-appb-000159
施加在负载平台原点o点上的力矩应分为两部分：
Figure PCTCN2020099609-appb-000160
Figure PCTCN2020099609-appb-000161
表示由应变梁o i点上的力矩
Figure PCTCN2020099609-appb-000162
产生的力矩，
Figure PCTCN2020099609-appb-000163
表示由应变梁点o i上的力
Figure PCTCN2020099609-appb-000164
产生的力矩；因此，
Figure PCTCN2020099609-appb-000165
Figure PCTCN2020099609-appb-000166
根据coordinate Ma，
Figure PCTCN2020099609-appb-000167
Figure PCTCN2020099609-appb-000168
当应变梁上的局部力在全局坐标系中从o i转换为o时，由转换引起的o上的扭矩为：
Figure PCTCN2020099609-appb-000169
根据方程(50)和(51)，
Figure PCTCN2020099609-appb-000170
上述变换即为包括了反对称算子的空间变换T i g对力变换的具体表达公式。
相应的位移变换可以具体推导：
在全局坐标系xyz中，负载平台的原点o的位移
Figure PCTCN2020099609-appb-000171
Figure PCTCN2020099609-appb-000172
可写为
Figure PCTCN2020099609-appb-000173
Figure PCTCN2020099609-appb-000174
在全局坐标系xyz中，应变梁上的原点o i的位移
Figure PCTCN2020099609-appb-000175
Figure PCTCN2020099609-appb-000176
可写为
Figure PCTCN2020099609-appb-000177
Figure PCTCN2020099609-appb-000178
在局部坐标系x iy iz i中，应变梁的原点o i的位移
Figure PCTCN2020099609-appb-000179
Figure PCTCN2020099609-appb-000180
可写为
Figure PCTCN2020099609-appb-000181
Figure PCTCN2020099609-appb-000182
加载平台的整体变形
Figure PCTCN2020099609-appb-000183
与应变梁i的局部变形
Figure PCTCN2020099609-appb-000184
之间的关系需要在下面计算。可以使用多种方法来计算关系。为了简化分析，下面应用矢量变换法，也就是前面所述的包括反对称算子(或叉乘算子)的T i g变换的具体表示。
Figure PCTCN2020099609-appb-000185
Figure PCTCN2020099609-appb-000186
Figure PCTCN2020099609-appb-000187
Figure PCTCN2020099609-appb-000188
Figure PCTCN2020099609-appb-000189
Figure PCTCN2020099609-appb-000190
Figure PCTCN2020099609-appb-000191
将方程(65)引入到方程(64)中，可得：
Figure PCTCN2020099609-appb-000192
因此，在应变梁i的局部坐标系x iy iz i中，局部原点o i的每个位移
Figure PCTCN2020099609-appb-000193
Figure PCTCN2020099609-appb-000194
都可以表示为全局原点的全局位移
Figure PCTCN2020099609-appb-000195
Figure PCTCN2020099609-appb-000196
的线性函数。
根据上述推导可以进行求解，其方法如下：
全局力可分为三步计算。
步骤(1)、计算负载平台的整体位移：
根据公式(66)，局部坐标系x iy iz i中应变梁沿x i的变形为
Figure PCTCN2020099609-appb-000197
采用公式(67)来构造A Δ和B Δ。
Figure PCTCN2020099609-appb-000198
Figure PCTCN2020099609-appb-000199
则公式(67)可以写为：
Figure PCTCN2020099609-appb-000200
应变梁为6个，即n＝6，因此可以令i＝1～6，共构造6个方程，组成方程组，写成矩阵形式：
Figure PCTCN2020099609-appb-000201
Figure PCTCN2020099609-appb-000202
Figure PCTCN2020099609-appb-000203
A Δv Δ＝B Δ            (74)
相关参数可以参考公式(38)中的矩阵A和向量B获得；
可以使用公式(74)计算负载平台的全局位移。
当应变梁大于6，即n>6时，
Figure PCTCN2020099609-appb-000204
Figure PCTCN2020099609-appb-000205
用最小二乘法来计算v Δ：
Figure PCTCN2020099609-appb-000206
步骤(2)、计算局部坐标系中局部力：
在计算全局位移Δ的情况下，可以使用公式(63)和(66)计算局部坐标系中的所有局部位移；
然后，根据公式(1)至(6)用于计算局部坐标系中的所有局部力。
也可以采用矩阵方式进行求解：
Figure PCTCN2020099609-appb-000207
Figure PCTCN2020099609-appb-000208
Figure PCTCN2020099609-appb-000209
A Niv Ni＝B Ni            (81)
可以用公式(48)分别求每个应变梁i的局部力，所有梁的全部方程式也可以写作：
Figure PCTCN2020099609-appb-000210
Figure PCTCN2020099609-appb-000211
Figure PCTCN2020099609-appb-000212
A Nv N＝B N            (85)
因此，可以使用公式(85)直接求出所有应变梁的局部力。
步骤(3)、计算加载平台上的全局力：
根据方程(49)和(52)，可以计算出每个梁i对全局原点o施加的每个全局力和力矩。因此，
Figure PCTCN2020099609-appb-000213
Figure PCTCN2020099609-appb-000214
公式(86)和(87)也可以写作矩阵形式
Figure PCTCN2020099609-appb-000215
Figure PCTCN2020099609-appb-000216
v F＝[F x，F y，F z，M x，M y，M z] T          (90)
A Fv F＝B F                 (91)
采用上述方式即可以计算出传感器六维力，其也可以用一个统一线性方程组来表征。
以上公式(74)、(85)和(91)可以合并成一个方程系统；
Figure PCTCN2020099609-appb-000217
B＝[B 1,0,B 3] T             (93)
v＝[v F T,v N T,v Δ T] T           (94)
Av＝B           (95)
公式(95)也是Principle Ma考虑了全局位移、全局力和局部力的一种书写方式，当n＝6时，即有6个可观测量时，可以直接求解该齐次线性方程组；如果它是超定线性方程组，即n>6时，则可以使用最小二乘法A TAv＝A TB来计算结果。
具体实施方式七：
本实施方式为一种多维力获取方法，多维力为三维力或六维力。本实施方式中，平面三维力传感器获得三维力的具体过程中，或空间六维力传感器获得六维力的具体过程中，采用压电晶体作为应变梁获取多维力。
平面三维力传感器获得三维力的计算方法与具体实施方式五基本一致；
下面的方法中采用压电晶体作为应变梁，且负载平台和支撑平台都考虑为纯刚体。压电晶体采用预压紧方式安装，保证受力过程中不会与支撑平台和负载平台脱离接触。
如图23(a)所示压电晶体沿自身轴线x i的受力可以转换为压电晶体电荷的变化量，测量其点和变化量既可以得到每个压电晶体沿x i方向的正压力
Figure PCTCN2020099609-appb-000218
进而得到此时负载平台与o i重合点在局部坐标系统o ix iy iz i中沿x i轴的变形量
Figure PCTCN2020099609-appb-000219
其它的计算方法与前述实施例 中的计算方法完全相同。
压电晶体的预紧方法如图23(b)所示，将支撑平台分为上装支撑平台(Upper supporting platform)和下装支撑平台(Lower supporting platform)两部分，两部分用多组交叉布置的螺栓进行固定并同时施加预加载力；
对于空间六维力获取方式可以采用图24所示的原理图。
实施例
如图15所示，为设计的平面8梁结构的三维力传感器。
该结构的材料是铝合金7075，其性能如表1所示，
表1
Figure PCTCN2020099609-appb-000220
平面8梁结构的三维力传感器结构参数如表2所示
表2
Figure PCTCN2020099609-appb-000221
仿真中，在负载平台上施加两条单轴加载曲线。第一个加载曲线沿x轴，第二个加载曲线绕轴z。
第一加载曲线的输入力表示为
F y＝50(N)；M z＝5(Nm)；F x＝[-100，-90，…，-10，-5，-1，0，1，5，10，…，90，100](N).
加载曲线上特定点的计算数据如表3所示。特定载荷下的应力云图如图16。
表3
Figure PCTCN2020099609-appb-000222
Figure PCTCN2020099609-appb-000223
在有限元软件中将输入力施加在负载平台上以计算结构的应力。此后，在有限元软件中测量梁沿轴x i的应力。接下来，将测量的应力输入到基于Principle Ma的程序中，以计算施加在负载平台上的力。最后，计算出的力可用于与初始输入力进行比较。
计算结果，计算误差和计算出的随机误差分别如图17，图18和图19所示。计算出的力和扭矩误差是使用输入力和计算力之间的差值，如图18所示。可以观察到误差是线性的。因此，可以通过线性校正来校正误差。
线性校正后的随机误差如图19所示。沿轴x和沿轴y的力的随机误差约为0.0006(N)，测量范围的标度为±100(N)，因此F.S的相对误差是0.0006/200＝0.0003％。绕轴z的扭矩随机误差约为0.000012(Nm)，测量范围的标度为±10(Nm)，因此F.S的相对误差是0.000012/20＝0.00006％。
来自FE软件的应力数据的有效位为6，如表3所示。因此截断误差为0.0005％。
可以看出，力和扭矩的计算随机误差甚至小于来自FE软件的初始计算数据的截断误差。原因在于均化效应，在结构中使用了8个应变梁。沿轴x的加载对沿轴y和绕轴z的计算结果具有小的线性影响。线性影响可以很容易地纠正。校正后几乎没有观察到维间耦合。
沿轴y的输入力的结果与沿轴x的输入力的结果完全相同，因此这里将不再示出结果。第二曲线的输入扭矩表示为F x＝30(N)；F y＝50(N)；M z＝[-10，-9，…，-1，-0.5，-0.1，0，0.1，0.5，1，…，9，10](Nm)。
加载曲线上特定点的计算数据如表4所示，
表4
Figure PCTCN2020099609-appb-000224
Figure PCTCN2020099609-appb-000225
计算结果，计算误差和计算出的随机误差分别如图20，图21，图22所示。
图22中所示的随机误差与图19中所示的类似。这意味着在FE软件中使用上述求解方法的计算结果几乎无法观测到误差(包括I类误差，即沿力加载轴线误差，和II类误差，即维间耦合误差)。使用不同的FE软件(Solidworks simulation，Ansys，Abaqus)或使用不同的网格时(梁部位网格密度需要足够大)，结果基本类似。

Claims (10)

  1. 基于并联杆系多维力传感器的多维力获取方法，其特征在于，包括以下步骤：
    建立附着于支撑平台上的全局坐标系统；
    分别建立基于应变梁和位移传感器的局部坐标系统，建立之后应变梁和位移传感器分别对应的局部坐标系统不随应变梁和位移传感器运动；
    根据空间矢量变换法则建立局部坐标系统和全局坐标系统间的矢量变换关系矩阵，包括广义力变换关系、广义变形位移变换关系和位移传感器变换关系；
    所述的广义力包括力和力矩，所述广义变形位移包括直线位移和转角位移；
    根据理论力学、材料力学和弹性力学建立局部坐标系统下的应变梁、支撑平台及负载平台的变形和受力的关系矩阵，即局部刚度矩阵和局部柔度矩阵；
    根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿/绕测量轴线的局部变形位移，称为可观测量；
    根据所述负载平台与相应局部坐标系原点重合点在局部坐标系沿/绕测量轴线的局部变形位移计算负载平台在全局坐标系统下的变形位移，包括三个直线位移和三个转角位移；
    根据负载平台在全局坐标系统下的变形位移计算每个应变梁相应局部坐标原点在局部坐标系统下的所有局部变形位移；
    根据每个应变梁在局部坐标系统下的局部变形位移计算每个应变梁在局部坐标系统下的局部广义力；
    将所有应变梁在局部坐标系统下的局部广义力根据局部坐标系统和全局坐标系统间的矢量变换关系平移到全局坐标系统原点并求和，得到多维力传感器的多维力。
  2. 根据权利要求1所述的基于并联杆系多维力传感器的多维力获取方法，其特征在于，所述方法的具体过程如下：
    建立附着于支撑平台上的全局坐标系统oxyz；
    外力作用下负载平台在全局坐标系统oxyz下产生位移
    Figure PCTCN2020099609-appb-100001
    其中，
    Figure PCTCN2020099609-appb-100002
    为全局坐标系oxyz下的直线位移；
    Figure PCTCN2020099609-appb-100003
    为全局坐标系oxyz下的转角位移；
    分别建立以应变梁与负载平台接触面中心为局部坐标系原点o i，附着于应变梁上的局部坐标系统o ix iy iz i；i表示应变梁的序号；
    以应变梁与支撑平台接触面中心为局部坐标系原点
    Figure PCTCN2020099609-appb-100004
    分别建立支撑平台局部坐标系统
    Figure PCTCN2020099609-appb-100005
    局部坐标系统中应变梁在力作用下产生变形位移，
    Figure PCTCN2020099609-appb-100006
    其中
    Figure PCTCN2020099609-appb-100007
    为局部坐标系o ix iy iz i下的直线位移；
    Figure PCTCN2020099609-appb-100008
    为局部坐标系o ix iy iz i下的转角位移；
    根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿测量轴线的局部变形位移，称为可观测量；根据空间矢量变换利用所述的部分局部坐标系下的局部变形位移，即可观测量，计算出负载平台在全局坐标系统oxyz下产生位移
    Figure PCTCN2020099609-appb-100009
    进而计算出所有应变梁局部坐标系下的局部位移
    Figure PCTCN2020099609-appb-100010
    根据所述的得到的所有应变梁的
    Figure PCTCN2020099609-appb-100011
    得到所有应变梁的局部广义力
    Figure PCTCN2020099609-appb-100012
    其中
    Figure PCTCN2020099609-appb-100013
    为局部坐标系o ix iy iz i下的力，
    Figure PCTCN2020099609-appb-100014
    为局部坐标系o ix iy iz i下的力矩；
    定义应变梁在局部坐标原点o i的柔度矩阵为
    Figure PCTCN2020099609-appb-100015
    获得应变梁的在局部坐标原点o i的柔度矩阵
    Figure PCTCN2020099609-appb-100016
    对于负载平台和支撑平台，将负载平台和支撑平台看做半弹性空间；定义负载平台在局部坐标原点o i的柔度矩阵为
    Figure PCTCN2020099609-appb-100017
    定义支撑平台在支撑平台局部坐标原点
    Figure PCTCN2020099609-appb-100018
    的柔度矩阵为
    Figure PCTCN2020099609-appb-100019
    确定柔性矩阵
    Figure PCTCN2020099609-appb-100020
    和
    Figure PCTCN2020099609-appb-100021
    的近似值；
    将支撑平台对应的柔度矩阵
    Figure PCTCN2020099609-appb-100022
    移动到点o i，在局部坐标o i处负载平台、支撑平台的弹性变形部分与应变梁的柔度矩阵之和为：
    Figure PCTCN2020099609-appb-100023
    表示从局部坐标系
    Figure PCTCN2020099609-appb-100024
    到局部坐标系o ix iy iz i的空间矢量变换矩阵；进而获得柔度和矩阵的逆矩阵，即其刚度矩阵
    Figure PCTCN2020099609-appb-100025
    将局部坐标系转换到全局坐标下得到全局坐标下刚度矩阵
    Figure PCTCN2020099609-appb-100026
    表示从坐标系o i到坐标系o的空间矢量变换矩阵；
    则在全局坐标系下的原点处的所有应变梁、负载平台、支撑平台的刚度矩阵和为
    Figure PCTCN2020099609-appb-100027
    全局坐标系下负载平台承受的外部广义力为
    Figure PCTCN2020099609-appb-100028
    全局坐标系下负载平台在承受外部力时的广义位移为
    Figure PCTCN2020099609-appb-100029
    全局坐标系统下广义力和广义位移、刚度的关系为：
    Figure PCTCN2020099609-appb-100030
    全局坐标系统下的广义力和应变梁局部坐标系统下的局部广义力的关系为：
    Figure PCTCN2020099609-appb-100031
    根据上述获得的
    Figure PCTCN2020099609-appb-100032
    和
    Figure PCTCN2020099609-appb-100033
    采用公式
    Figure PCTCN2020099609-appb-100034
    计算六维力
    Figure PCTCN2020099609-appb-100035
    或者根据上述获得的
    Figure PCTCN2020099609-appb-100036
    采用公式
    Figure PCTCN2020099609-appb-100037
    计算六维力
    Figure PCTCN2020099609-appb-100038
  3. 根据权利要求2所述的基于并联杆系多维力传感器的多维力获取方法，其特征在于，所述每个局部坐标系统o ix iy iz i与所述全局坐标系统oxyz的关系以及
    Figure PCTCN2020099609-appb-100039
    和o ix iy iz i的关系确定方法如下：
    假设初始状态为局部坐标系与全局坐标系重合，将应变梁相对初始位置沿着x旋转
    Figure PCTCN2020099609-appb-100040
    再沿着y旋转
    Figure PCTCN2020099609-appb-100041
    再沿着z旋转
    Figure PCTCN2020099609-appb-100042
    然后沿着xyz坐标轴分别平移
    Figure PCTCN2020099609-appb-100043
    每个局部坐标系统o ix iy iz i与全局坐标系统oxyz的关系都用三个旋转角度和三个平移距离来表示，记为
    Figure PCTCN2020099609-appb-100044
    和
    Figure PCTCN2020099609-appb-100045
    同理，
    Figure PCTCN2020099609-appb-100046
    和o ix iy iz i也同样用三个旋转角度和三个平移距离来表示，记为
    Figure PCTCN2020099609-appb-100047
    和
    Figure PCTCN2020099609-appb-100048
  4. 根据权利要求2所述的基于并联杆系多维力传感器的多维力获取方法和权利要求3所述的坐标系之间的关系，其特征在于，所述
    Figure PCTCN2020099609-appb-100049
    Figure PCTCN2020099609-appb-100050
    Rot(·)指坐标旋转，S(r i)和S(l i)分别代表矢量r i和l i对应的反对称算子。
  5. 根据权利要求2、3或4所述的基于并联杆系多维力传感器的多维力获取方法，其特征在于，所述负载平台与相应局部坐标系原点重合点在局部坐标系下沿测量轴线的局部 变形位移通过支撑平台和负载平台之间布置的微位移测量传感器获得，具体过程如下：
    a、针对六维力传感器的微位移测量传感器测量负载平台在外力作用下的各个方向的微位移：
    负载平台位移的测量采用电学微位移传感器或光学微位移传感器：
    建立附着于微位移传感器上的位移传感器局部坐标系统，位移传感器测量的发生位移的方向为局部坐标系统x j轴；测量出沿位移传感器局部坐标系统x j轴的负载平台位移；
    位移传感器在位移传感器局部坐标系下与在全局坐标系下的位移关系为
    Figure PCTCN2020099609-appb-100051
    具体针对沿局部坐标系统x j轴微位移，上述变换关系具体写为：
    Figure PCTCN2020099609-appb-100052
    Figure PCTCN2020099609-appb-100053
    方程中sβ＝sin(β)，cβ＝cos(β)，上式即为全局坐标系原点与位移传感器局部坐标系原点的位移协调关系方程，该方程进一步简写为：
    Figure PCTCN2020099609-appb-100054
    式中，
    Figure PCTCN2020099609-appb-100055
    分别为方程系数，可见这些系数完全由
    Figure PCTCN2020099609-appb-100056
    和
    Figure PCTCN2020099609-appb-100057
    确定，
    Figure PCTCN2020099609-appb-100058
    由位移传感器测量得出；该方程称为沿局部坐标系统x j轴方向的协调关系方程；
    当有6个以上的传感器，即j≥6时，则由上述协调关系方程组成方程组，方程组称为负载平台微位移求解方程组，每一个
    Figure PCTCN2020099609-appb-100059
    由传感器直接测量得到，进而获得负载平台六个自由度的微位移；
    b、针对三维力传感器的微位移测量传感器测量负载平台在外力作用下的各个方向的微位移，协调关系方程写为：
    Figure PCTCN2020099609-appb-100060
    当有3个以上的传感器时，则列出负载平台微位移求解方程组，每一个
    Figure PCTCN2020099609-appb-100061
    都由传感器直接测量得到，进而得到负载平台的位移。
  6. 根据权利要求2、3或4所述的基于并联杆系多维力传感器的多维力获取方法，其特征在于所述负载平台与相应局部坐标系原点重合点在局部坐标系下沿测量轴线的局部变形位移通过弹性敏感元件上粘贴有的应变片实现获得，具体过程如下：
    采用应变片时，应变片对称粘贴于四个应变梁侧面或者两个相对的应变梁侧面；
    则应变梁沿其自身局部坐标系统的x i轴的拉伸力由以下差动电路获得，只用两个相对的侧面的应变片求得应变梁沿x i的拉伸力
    Figure PCTCN2020099609-appb-100062
    或者采用四个侧面的应变片求得，即对两组相对的侧面确定的
    Figure PCTCN2020099609-appb-100063
    求平均值；
    应变梁的柔度系数已知，沿x i轴的拉伸柔度系数为
    Figure PCTCN2020099609-appb-100064
    E为弹性模量，A i为应变梁横截面面积，l i是应变梁长度；或者，采用有限元分析方法或者试验方式测量确定
    Figure PCTCN2020099609-appb-100065
    则应变梁沿x i轴的拉伸变形为
    Figure PCTCN2020099609-appb-100066
    如果将负载平台和支撑平台都视为刚体，则负载平台与支撑平台之间沿局部坐标系统x i轴的位移视为等于上述应变梁沿x i轴的拉伸变形；从而获得负载平台六个自由度的位移；
    如果考虑负载平台和支撑平台的柔度，由于负载平台和支撑平台在x i轴柔度均为已知有
    Figure PCTCN2020099609-appb-100067
    A i是应变梁的截面面积，μ是泊松比；或者，采用有限元分析方法或者试验方式测量确定
    Figure PCTCN2020099609-appb-100068
    则负载平台与支撑平台之间沿局部坐标系统x i轴的位移为
    Figure PCTCN2020099609-appb-100069
    从而获得负载平台六个自由度的位移。
  7. 基于并联杆系多维力传感器的多维力获取方法，其特征在于，包括以下步骤：
    建立附着于支撑平台上的全局坐标系统；
    分别建立基于应变梁和位移传感器的局部坐标系统，建立之后应变梁和位移传感器分别对应的局部坐标系统不随应变梁和位移传感器运动；
    根据空间矢量变换法则建立局部坐标系统和全局坐标系统间的矢量变换关系矩阵，包括广义力变换关系、广义变形位移变换关系和位移传感器变换关系；
    所述的广义力包括力和力矩，所述广义变形位移包括直线位移和转角位移；
    根据理论力学、材料力学和弹性力学建立局部坐标系统下的应变梁、支撑平台及负载平台的变形和受力的关系矩阵，即局部刚度矩阵和局部柔度矩阵；
    根据空间矢量变换关系，将每个梁的局部刚度矩阵和柔度矩阵平移到全局坐标系统原点并求和，得到多维力传感器的全局刚度矩阵和柔度矩阵；
    根据应变梁上粘贴的应变片，或/和压电晶体作为应变梁，或/和负载平台和支撑平台之间设置的位移传感器，得到负载平台与相应局部坐标系原点重合点在局部坐标系下沿/绕测量轴线的局部变形位移，称为可观测量；
    根据所述负载平台与相应局部坐标系原点重合点在局部坐标系沿/绕测量轴线的局部 变形位移计算负载平台在全局坐标系统下的广义变形位移；
    由于广义力与广义变形位移直接呈现线性关系，即广义力等于全局刚度矩阵乘以广义变形位移，根据上述得到的全局刚度矩阵和广义变形位移即可得到多维力传感器的广义力。
  8. 采用并联杆系的三维力传感器的三维力获取方法，所述的三维力传感器的负载平台和支撑平台位于同一平面内，其特征在于，包括以下步骤：
    采用应变片测量负载平台在外力作用下的各个方向的微位移；
    针对于平面的三维力传感器而言，在局部坐标系x iy i下
    Figure PCTCN2020099609-appb-100070
    左上角的上角标代表坐标系统，左上角的上角标为g表示对应的参数为全局坐标系下的参数，左上角的上角标为i表示对应的参数为局部坐标系下的参数；
    左下角的下角标代表矢量作用的点，左下角的下角标为o表示对应的矢量作用在全局坐标系的原点o，左下角的下角标为o i表示对应的矢量作用在局部坐标系的原点o i；
    右上角的上角标为i表示施加者为第i根应变梁，g或者空白表示为全局量，即施加者为负载平台上的外力；
    右下角的下角标代表矢量的方向，右下角的下角标为x表示沿着左上角坐标系中的x轴，左上角为g则指沿着全局坐标系oxyz的x轴，如果左上角为i则指沿着局部坐标系o ix iy iz i的x i轴；同样，右下角的下角标为y表示沿着左上角坐标系中的y轴，右下角的下角标为z表示沿着左上角坐标系中的z轴，右下角标带有F和M是指变量由力或力矩引起，不带指由力和力矩共同作用引起，右下角的下角标空白表示xyz轴共同构成的矢量；
    参数的含义如下：
    E为弹性模量；A为应变梁横截面面积，A i为第i根应变梁的截面面积；l是应变梁长度，l i是第i根应变梁的长度；ΔD为位移变形，
    Figure PCTCN2020099609-appb-100071
    为第i根应变梁局部坐标系下沿着x轴的位移变形；F为力，
    Figure PCTCN2020099609-appb-100072
    为第i根应变梁局部坐标系下沿着x轴的力；
    Figure PCTCN2020099609-appb-100073
    I i为惯性矩；G为剪切模量；右下角角标中的Fy1表示剪切变形对应的参数，Fy2表示弯曲变形对应的参数；
    Figure PCTCN2020099609-appb-100074
    Figure PCTCN2020099609-appb-100075
    那么，在局部坐标系x iy i和全局坐标系xy下的o i位移的关系为：
    Figure PCTCN2020099609-appb-100076
    上述关系中的
    Figure PCTCN2020099609-appb-100077
    以及公式中与
    Figure PCTCN2020099609-appb-100078
    的叉乘关系即为
    Figure PCTCN2020099609-appb-100079
    具体形式；
    在局部坐标系x iy i下和全局坐标系xy下的应变梁受到力的关系为
    Figure PCTCN2020099609-appb-100080
    应变梁上的力从o i变换到全局坐标系的原点o，附加扭矩是
    Figure PCTCN2020099609-appb-100081
    由应变梁i施加在点o上的扭矩为：
    Figure PCTCN2020099609-appb-100082
    因此，负载平台上的总力为
    Figure PCTCN2020099609-appb-100083
    基于以上的公式，基本方程组为：
    Figure PCTCN2020099609-appb-100084
    方程组中的变量是：
    Figure PCTCN2020099609-appb-100085
    变量包括全局力
    Figure PCTCN2020099609-appb-100086
    局部力
    Figure PCTCN2020099609-appb-100087
    和全局位移
    Figure PCTCN2020099609-appb-100088
    Figure PCTCN2020099609-appb-100089
    视为观测变量，即可由粘贴于应变梁上的应变片测量得出，方程组(37)中有12个未知变量，即
    Figure PCTCN2020099609-appb-100090
    线性方程系统写成
    Av＝B       (38)
    其中，
    Figure PCTCN2020099609-appb-100091
    Figure PCTCN2020099609-appb-100092
    其中，sβ＝sin(β)，cβ＝cos(β)；
    观察线性方程系统Av＝B，最后3行直接用来计算全局坐标系中负载平台的位移
    Figure PCTCN2020099609-appb-100093
    Figure PCTCN2020099609-appb-100094
    即为将应变片测量得到的数据转换为沿相应应变梁的x i轴的位移；
    整理为3个公式：
    A Δv Δ＝B Δ     (39)
    A iv i＝B i      (40)
    A Fv F＝B F        (41)
    其中
    Figure PCTCN2020099609-appb-100095
    通过A Δv Δ＝B Δ求解出负载平台的位移量；通过公式(38)中的A和B即可进而得到A i、B i、A F和B F，进一步可以得到三维力。
  9. 采用并联杆系的六维力传感器的六维力获取方法，其特征在于，包括以下步骤：
    采用应变片测量负载平台在外力作用下的各个方向的微位移；
    coordinate Ma：假设初始状态为局部坐标系与全局坐标系重合，将应变梁相对初始位置沿着x旋转
    Figure PCTCN2020099609-appb-100096
    再沿着y旋转
    Figure PCTCN2020099609-appb-100097
    再沿着z旋转
    Figure PCTCN2020099609-appb-100098
    然后沿着xyz坐标轴分别平移
    Figure PCTCN2020099609-appb-100099
    当沿x轴的旋转变换角度设为0时，旋转变换及逆变换为：
    Figure PCTCN2020099609-appb-100100
    Figure PCTCN2020099609-appb-100101
    上述变换与
    Figure PCTCN2020099609-appb-100102
    变换是一致的；
    有：
    Figure PCTCN2020099609-appb-100103
    Figure PCTCN2020099609-appb-100104
    施加在负载平台原点o点上的力矩应分为两部分：
    Figure PCTCN2020099609-appb-100105
    Figure PCTCN2020099609-appb-100106
    表示由应变梁o i点上的力矩
    Figure PCTCN2020099609-appb-100107
    产生的力矩，
    Figure PCTCN2020099609-appb-100108
    表示由应变梁点o i上的力
    Figure PCTCN2020099609-appb-100109
    产生的力矩；因此，
    Figure PCTCN2020099609-appb-100110
    Figure PCTCN2020099609-appb-100111
    根据coordinate Ma，
    Figure PCTCN2020099609-appb-100112
    Figure PCTCN2020099609-appb-100113
    当应变梁上的局部力在全局坐标系中从o i转换为o时，由转换引起的o上的扭矩为：
    Figure PCTCN2020099609-appb-100114
    进而有：
    Figure PCTCN2020099609-appb-100115
    上述变换即为包括了反对称算子的空间变换
    Figure PCTCN2020099609-appb-100116
    对力变换的具体表达公式；
    在全局坐标系xyz中，负载平台的原点o的位移
    Figure PCTCN2020099609-appb-100117
    写为
    Figure PCTCN2020099609-appb-100118
    Figure PCTCN2020099609-appb-100119
    在全局坐标系xyz中，应变梁上的原点o i的位移
    Figure PCTCN2020099609-appb-100120
    写为
    Figure PCTCN2020099609-appb-100121
    Figure PCTCN2020099609-appb-100122
    在局部坐标系x iy iz i中，应变梁的原点o i的位移
    Figure PCTCN2020099609-appb-100123
    写为
    Figure PCTCN2020099609-appb-100124
    Figure PCTCN2020099609-appb-100125
    应用矢量变换法有
    Figure PCTCN2020099609-appb-100126
    Figure PCTCN2020099609-appb-100127
    Figure PCTCN2020099609-appb-100128
    将方程(65)引入到方程(64)中，可得：
    Figure PCTCN2020099609-appb-100129
    因此，在应变梁i的局部坐标系x iy iz i中，局部原点o i的每个位移
    Figure PCTCN2020099609-appb-100130
    都能够表示为全局原点的全局位移
    Figure PCTCN2020099609-appb-100131
    的线性函数；
    然后确定全局力，具体过程如下：
    计算负载平台的整体位移：
    根据公式(66)，局部坐标系x iy iz i中应变梁沿x i的变形为
    Figure PCTCN2020099609-appb-100132
    采用公式(67)来构造A Δ和B Δ，
    Figure PCTCN2020099609-appb-100133
    Figure PCTCN2020099609-appb-100134
    则公式(67)写为：
    Figure PCTCN2020099609-appb-100135
    当应变梁为6个，即n＝6，令i＝1～6，共构造6个方程，组成方程组，写成矩阵形式：
    Figure PCTCN2020099609-appb-100136
    Figure PCTCN2020099609-appb-100137
    Figure PCTCN2020099609-appb-100138
    A Δv Δ＝B Δ     (74)
    使用公式(74)计算负载平台的全局位移；
    当当应变梁大于6，即n>6时，
    Figure PCTCN2020099609-appb-100139
    Figure PCTCN2020099609-appb-100140
    用最小二乘法来计算v Δ：
    Figure PCTCN2020099609-appb-100141
    使用上述方法计算负载平台的全局位移；
    最后根据负载平台的位移量得到六维力。
  10. 根据权利要求9所述的采用并联杆系的六维力传感器的六维力获取方法，其特征在于，根据负载平台的位移量得到六维力的过程如下：
    计算局部坐标系中局部力：
    在计算全局位移Δ的情况下，使用公式(63)和(66)计算局部坐标系中的所有局部位移；
    采用矩阵方式进行求解所有应变梁的局部力：
    Figure PCTCN2020099609-appb-100142
    Figure PCTCN2020099609-appb-100143
    Figure PCTCN2020099609-appb-100144
    A Niv Ni＝B Ni     (81)
    用公式(48)分别求每个应变梁i的局部力，所有梁的全部方程式写作：
    Figure PCTCN2020099609-appb-100145
    B N＝[B N1 T,…,B Ni T,…,B Nn T] T  (83)
    v N＝[v N1 T,…,v Ni T,…,v Nn T] T   (84)
    A Nv N＝B N    (85)
    使用公式(85)直接求出所有应变梁的局部力；
    计算加载平台上的全局力：
    根据方程(49)和(52)，计算出每个梁i对全局原点o施加的每个全局力和力矩；因此，
    Figure PCTCN2020099609-appb-100146
    Figure PCTCN2020099609-appb-100147
    公式(86)和(87)写作矩阵形式
    Figure PCTCN2020099609-appb-100148
    Figure PCTCN2020099609-appb-100149
    v F＝[F x，F y，F z，M x，M y，M z] T     (90)
    A Fv F＝B F     (91)
    采用上述方式计算出传感器六维力，采用一个统一线性方程组来表征。
    以上公式(74)、(85)和(91)合并成一个方程系统；
    Figure PCTCN2020099609-appb-100150
    B＝[B 1,0,B 3] T     (93)
    v＝[v F T,v N T,v Δ T] T   (94)
    Av＝B(95)
    根据公式(95)最终确定六维力。
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