JP2016043092A - Movement measuring device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a movement measuring device that can calculate the motion of thighs as the movement on the three-dimensional coordinate in the case of walking motion.SOLUTION: A quaternion q is calculated from angular velocity components on 3-axes obtained from an angular velocity sensor 12 attached to the thighs, and a quaternion z is calculated from acceleration components on 3-axes obtained from an acceleration sensor 13. The quaternion q of the angular velocity components is corrected by the quaternion z of the acceleration components using a Kalman filter. Further, the quaternion corrected by the Kalman filter is converted into rotational matrices. The rotational matrices are provided to the reference vector of vectors set in the thighs, and thereby the rotational motion of the vector at the present can be obtained as information on three dimensions.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a motion measuring device that measures the motion of a walking motion unit using an angular velocity sensor and an acceleration sensor.

  Patent Document 1 describes an invention related to a walking characteristic evaluation system. This walking characteristic evaluation system measures human walking characteristics, particularly three-dimensional walking characteristics in patients who are difficult to walk for a long time without restriction, and obtains information to be used for diagnosis of medical conditions and therapeutic effects.

  This walking characteristic evaluation system includes a plurality of body-mounted sensors, a portable data recorder, and an analysis device. The body acceleration sensor measures the acceleration and angular velocity of the foot.

  Measurement data of acceleration and angular velocity is calculated by an analysis device. In this calculation, the free leg state for one step is searched from the angular velocity waveform, and when the free leg state is searched, the posture / frame matrix of the foot is calculated. This calculation is performed using the gravitational acceleration detected as the acceleration. Further, the rotation amount of the frame matrix during the sampling period is obtained from the measured angular velocity after the start of the free leg, and the coordinate conversion and integration of the measured acceleration are performed.

  After the coordinate conversion and integration of acceleration are performed, the frame matrix at the time of the free leg is corrected so that the frame matrix at the end of the free leg and the frame at the time of the subsequent standing match.

JP 2010-110399 A

  In the walking measurement evaluation system described in Patent Document 1, acceleration data and angular velocity data are obtained as Euler angle measurement information, respectively, and the posture of the foot is calculated by a frame matrix method using the Euler angle measurement information. ing. In the frame matrix method, a three-dimensional acceleration matrix is obtained from three-axis acceleration components, and the rotation amount of the three-dimensional acceleration matrix is calculated from three-axis angular velocity components.

  In this calculation method, since there are a plurality of eigenvectors (eigenfunctions), degeneration occurs in the calculation of the physical quantity, so that it takes time for the calculation, and when calculating a three-dimensional walking event, etc. There is a drawback that the error becomes large.

  Moreover, the calculation result obtained by the walking measurement evaluation system described in Patent Document 1 is a change in the movement distance of the toe, etc. From this calculation result, how the walking motion unit moves in the three-dimensional space It is difficult to grasp three-dimensionally.

  The present invention solves the above-described conventional problems, and can calculate walking motion by a quick calculation. Further, from the measurement result, it is easy as three-dimensional information on how the walking motion section moves in a three-dimensional space. It is an object of the present invention to provide a motion measurement device that can be easily grasped.

The present invention relates to a motion measuring device having an angular velocity sensor and an acceleration sensor installed in a walking motion unit.
The angular velocity sensor detects angular velocity components in three axial directions orthogonal to each other, and the acceleration sensor detects acceleration components in three axial directions orthogonal to each other,
A calculation unit is provided to which the angular velocity component detection signal and the acceleration component detection signal are provided,
In the calculation unit, the motion of the walking motion unit is calculated as a vector motion from the detection signal of the angular velocity component and the detection signal of the acceleration component.

In the motion measuring apparatus according to the present invention, in the arithmetic unit, the step of converting the angular velocity component in the three-axis direction into a quaternion, the step of converting the acceleration component in the three-axis direction into a quaternion, and the quaternion converted from the acceleration component And correcting the quaternion converted from the angular velocity component by applying a Kalman filter,
A rotation matrix is calculated from the quaternion corrected by the Kalman filter, and the vector is obtained from the rotation matrix and the gravity direction vector.

  In the motion measuring device of the present invention, a reference vector is set, and in walking motion, the current vector is obtained using the rotation matrix with the reference vector as a reference.

  In the motion measuring apparatus of the present invention, a reference posture is set, and a relationship between the reference posture and the reference vector is obtained by the acceleration sensor. From this relationship, a relationship between the vector and the current posture, and the reference posture A relationship with the current posture is required.

  In the motion measuring apparatus of the present invention, the angular velocity sensor and the acceleration sensor are provided in the thigh, and the vector to be calculated is set parallel to the femur and directed to the knee joint.

  In the motion measuring apparatus according to the present invention, the angular velocity sensor and the acceleration sensor are also provided on the lower leg, the angular velocity component detection signal and the acceleration component detection signal in the lower leg, and the angular velocity component detection signal in the thigh. Further, from the detection signal of the acceleration component, it is possible to calculate the motion of the lower leg as a vector motion.

  Further, the angular velocity sensor and the acceleration sensor are also provided on the foot, and from the detection signal of the angular velocity component and the acceleration component in the foot, and the detection signal of the angular velocity component and the acceleration component in the crus The foot motion can be calculated as a vector motion.

  Furthermore, in the motion measuring device of the present invention, the locus of the tip of the vector can be obtained three-dimensionally on the three-dimensional coordinates. Further, an operation locus obtained by projecting the locus of the tip of the vector onto a plane is obtained, and further, the operation locus is projected onto three planes orthogonal to each other.

  Since the motion measuring apparatus of the present invention calculates the motion of the walking motion unit as a vector motion from the angular velocity component in the 3 axis direction and the acceleration component in the 3 axis direction, the motion of the walking motion unit is three-dimensionally represented in 3D coordinates. Can be grasped.

  In the motion measuring apparatus of the present invention, the angular velocity component in the triaxial direction is converted into quaternion, the acceleration component in the triaxial direction is converted into quaternion, and the rotation matrix is obtained from the quaternion. For example, the vector of the walking motion unit obtained based on the gravity direction vector or the detection output of the acceleration sensor is set as the reference vector, and then the rotation matrix is applied so that the current walking motion unit vector is compared with the reference vector. It is possible to calculate the rotation state relatively easily.

  By grasping the motion of the walking motion unit as a three-dimensional motion of the vector, when applied to the walking motion of a person, it becomes possible to perform a comparative measurement between the walking motion of a healthy person and the walking motion of a person with motor impairment, It can be effectively used for treatment and rehabilitation of people with motor impairment.

The block diagram which shows the processing step of the movement measuring apparatus in embodiment of this invention, The block diagram which shows the state with which the movement measuring apparatus of this invention was mounted | worn to the thigh of the human body, A three-dimensional diagram representing the walking motion of one subject's thigh as a three-dimensional trajectory of the tip of the vector, A three-dimensional diagram representing the walking motion of the thighs of multiple subjects as a three-dimensional locus of the tip of the vector, A two-dimensional diagram obtained by projecting the three-dimensional trajectory of FIG. 4 onto xy coordinates; A two-dimensional diagram obtained by projecting the three-dimensional trajectory of FIG. A two-dimensional diagram obtained by projecting the three-dimensional trajectory of FIG. Two-dimensional diagram of xz coordinates comparing the difference of vector trajectories between healthy and gait disabled people The block diagram which shows the development example with which the movement measuring apparatus of this invention was mounted | worn to the thigh of the human body, the lower leg, and the foot | leg part,

  The motion measuring device 10 according to the embodiment of the present invention measures the motion of the walking motion unit. In the following embodiments, a case where a walking motion unit of a human body is measured by the motion measurement device 10 will be described as an example.

  FIG. 2 schematically shows the human body 1 during walking. The walking motion part of the human body is roughly divided into a waist part (trunk part) 2, a thigh part 3, a crus part 4, and a foot part 5. In this embodiment, an inertial sensor St is fixed to the thigh part 3. Has been. In FIG. 2, the inertial sensor St is attached to the front part of the thigh 3, but the inertial sensor St is located at any part of the thigh 3 as long as the movement of the thigh 3 can be accurately detected. It may be installed in any orientation. In order to measure the movement of the thigh 3 and grasp the walking state of the subject, it is preferable that an inertial sensor St is attached to the thigh 3 of both legs. However, in the following, for the sake of brevity, as shown in FIG. 2, attention is paid only to the inertial sensor St attached to the thigh 3 of the left leg, and the operation of the thigh 3 of the left leg is described. The analysis will be described.

  The motion measuring apparatus 10 according to the embodiment of the present invention includes the inertial sensor St and the calculation unit 11. The detection output from the inertial sensor St is given to the calculation unit 11. The calculation unit 11 includes a CPU and a memory. The calculation unit 11 may be attached to the human body 1 or may be located away from the human body 1 and the detection output from the inertial sensor St is transmitted to the calculation unit 11 by wire or wirelessly. There may be.

FIG. 1 is a block diagram showing processing steps of the motion measuring apparatus 10.
The inertial sensor St is composed of an angular velocity sensor 12 and an acceleration sensor 13, and a detection output ω of the angular velocity sensor 12 and a detection output a of the acceleration sensor 13 are given to the calculation unit 11, and the calculation unit 11 in FIG. The calculation processing shown in step 1 (ST1) to step 6 (ST6) is performed.

The angular velocity sensor 12 is, for example, a vibration gyroscope, and the angular velocity is detected using Coriolis force. The angular velocity sensor 12 detects angular velocity components (ω _x , ω _y , ω _z ) in three axial directions (around three axes) orthogonal to each other. For example, the acceleration sensor 13 is supported by a support portion having a flexible mass, and a displacement of the mass due to the acceleration is detected by a piezoelectric element or the like as a deflection amount of the support portion. In the acceleration sensor 13, orthogonal three-axis directions of the acceleration component _{(a x, a y, a} z) are detected together.

In the arithmetic processing shown in FIG. 1, the motion of the thigh 3 is calculated as a vector motion. That is, the operation of the thigh section 3 are represented as three-dimensional rotation of the thigh 3 to the set vector e _T. In order to accurately grasp the walking motion of the thigh 3 as three-dimensional information, the vector e _T is parallel to the femur of the thigh 3 and its base is directed to the hip joint that is the root of the thigh 3 is, front portion of the vector e _T is preferably directed to the knee joint of the boundary between the thigh part 3 and the lower leg portion 4.

  As shown in FIG. 2, in the three-dimensional coordinates of xyz, the y-axis direction is set as the gravity direction, the z-axis direction is set as the walking direction of the human body 1 as the subject, and the x-axis direction is set as the y-axis. And a direction perpendicular to the z-axis.

To measure the vector e _T on the three-dimensional coordinates, first, set the reference vector e _T0 based on the coordinate axes of the three-dimensional coordinates. Next, the relationship between the reference posture of the thigh 3 and the reference vector is obtained from the detection output of the acceleration sensor 13. Subsequent walking motion, with reference to the reference vector e _T0, quaternion at each time, and obtains a vector e _T by obtaining the rotation matrix. Then, based on the relationship between the reference position and the reference vector, it is possible to measure the attitude of the thigh section 3 from the vector e _T. In the example illustrated in FIG. 2, the stationary upright posture of the subject is set as the reference posture, the y-axis direction in FIG. 2 is set as the reference vector e _T0 , and the reference posture and the reference vector are determined from the tilt angle of the femur detected by the acceleration sensor 13. e We are looking for a relationship with _T0 . Note that the relationship between the reference posture and the reference vector _eT0 may be obtained by measuring the tilt angle of the femur with respect to the direction of gravity and obtaining the difference from the tilt angle of the femur detected by the acceleration sensor 13.

In step 1 (ST1) of the calculation process of the calculation unit 11 shown in FIG. 1, the quaternion is calculated based on all the angular velocity components in the three axial directions detected by the angular velocity sensor 12. The quaternion is a four-dimensional vector represented by q (q ₀ , q ₁ , q ₂ , q ₃ ), “q ₀ ” means the angle component θ, and “q ₁ ” “q ₂ ” “q ₃ ” Means the component of the rotating shaft.

The quaternion q is calculated from the angular velocity components in the three-axis directions by integrating the quaternion time change, and is obtained by a time evolution formula. The processing steps shown in the block diagram shown in FIG. 1 are repeated, for example, at a cycle of 10 milliseconds. Assuming that a certain time is k, the quaternion q _{k + 1} obtained from the angular velocity component at the next sampling time _{k + 1} is expressed by the following equation (1).

In the above, ω _xk , ω _yk , and ω _zk are three-axis (around three axes) angular velocity components obtained from the angular velocity sensor 12 at time k. Δt is a sampling time.

In ST2, the acceleration components of 3 axes measured by the acceleration sensor _{13 (a x, a y,} a z) for calculating a quaternion _{z k} at time k determined from. a _k is the gravitational acceleration obtained from the acceleration components in three axial at time k, a ₀ is the gravitational acceleration detected when the vector e _T coincides with the reference vector e _T0. θ _k is the rotation amount obtained by measuring the accelerations a _k and a ₀ of gravity, and A _k is the rotation axis at time k.

At time k, the quaternion z _k obtained from the acceleration component is expressed by the following formula 2.

  Note that either of the conversion processing of ST1 and ST2 into a quaternion may be performed first or may be performed as a simultaneous step.

  When calculating the quaternion q from the angular velocity component, the equation of time evolution is integral, so that the offset drift of the angular velocity sensor 12 is accumulated, and the quaternion q obtained from the angular velocity component includes an accumulated error. It is. Therefore, in ST3 shown in FIG. 1, the Kalman filter is applied to the quaternion q calculated from the angular velocity component, and at this time, the quaternion z calculated from the acceleration component is used as an observation value, thereby correcting the error.

  That is, in ST3 shown in FIG. 1, from the quaternion state equation obtained from the angular velocity component shown in Equation 3 and the quaternion observation equation obtained from the acceleration component shown in Equation 4, the following Kalman filter shows the following Equation 5. As described above, the prediction value is corrected.

  In Equation 3, w represents process noise, and v in Equation 4 represents observation noise (measurement noise). K in Equation 5 represents the Kalman gain for the state equation of Equation 3, that is, the Kalman gain in the correction of the quaternion q obtained from the angular velocity component. In the Kalman filter, the Kalman gain K is determined by a noise ratio based on a value related to process noise and a value related to observation noise. For example, the Kalman gain K increases as the noise ratio decreases, and the Kalman gain K decreases as the noise ratio increases. 1 and 5, the quaternion estimated by the Kalman filter is expressed as q with a hat.

  The noise ratio can be set so that the Kalman gain K decreases as the noise ratio decreases, and the Kalman gain K increases as the noise ratio increases.

In ST5 of the calculation process of the calculation unit 11 shown in FIG. 1, the quaternion q (with a hat) corrected by the Kalman filter is converted into a rotation matrix R. The rotation matrix R is expressed by Equation 6 below. In Equation 6, q ₀ , q ₁ , q ₂ , and q ₃ are quaternion components corrected by the Kalman filter, and “q ₀ ” means the angle θ component as described above, and “q ₁ ” “q” “ ₂ ” and “q ₃ ” mean components of the rotation axis.

The vector e _T that appears in the xyz coordinates shown in FIG. 2 corresponds to the reference vector e _T0 rotated by the rotation matrix R shown in _Equation 6. Therefore, in ST5 shown in FIG. 1, if the conversion to the rotation matrix R shown in Equation 6 is performed at intervals of, for example, 10 milliseconds, the vector e _T every 10 milliseconds can be obtained.

When the vector e _T is obtained in ST5, the locus of the vector e _T can be obtained in ST6. The trajectory of the vector e _T here is a trajectory of the tip of the vector e _T with the hip joint located at the boundary between the waist 2 and the thigh 3 as the origin, and the trajectory of the tip includes the hip joint position. This is similar to the movement trajectory of the knee joint located at the boundary between the thigh 3 and the crus 4 when the origin is used. By knowing this movement trajectory, the movement trajectory of the knee joint position when the subject's hip joint position is the origin can be easily obtained as three-dimensional data on three-dimensional coordinates.

FIGS. 3 and 4 show the movement of the tip of the vector e _T calculated by the calculation unit 11 shown in FIG. 1 when the subject who fixed the inertial sensor _ST to the thigh 3 performs a walking motion. The trajectory (similar to the movement trajectory of the knee joint position when the hip joint position is the origin) is shown on the three-dimensional coordinates on xyz.

3 and 4 of the HC (heel contact) is heel of the foot 5 in the middle walking indicates the position of the tip portion of the vector e _T when landed. For example, when the acceleration detected by the acceleration sensor 13 exceeds a predetermined threshold (when an impact greater than or equal to a predetermined value is received), the calculation unit 11 recognizes that the kite has landed. In Figure 3 and Figure 4, the position of the front portion of the vector e _T when it is recognized HC are indicated by white circles.

The diagram of Figure 3, the healthy subjects and subjects, in which superposed movement locus of the tip of the measured vector e _T in walking three paces of the same person in the subject. In FIG. 3, the moving direction of the distal end portion of the vector e _T during walking operation is indicated by arrows. In the three-dimensional coordinates in FIG. 3, the minus z indicates the rear in the traveling direction. According to FIG. 3, the tip portion or the knee of the HC in the vector e _T is positioned slightly forward of the waist 2, then the knee in the stance phase is moved rearward, the knee forward swing phase The movement can be grasped in three dimensions.

FIG. 4 shows the measurement of walking motion in the same manner with three healthy persons 1, healthy persons 2, and healthy persons 3 as subjects. In FIG. 4, only the HC of the healthy person 1 is indicated by white circles. Figure 5 is a two-dimensional graph obtained by projecting the locus of the tip of the vector e _T shown in FIG. 4 in the x-y plane. 6 is a two-dimensional graph projected onto the yz plane, and FIG. 7 is a two-dimensional graph projected onto the xz plane.

The motion measurement apparatus 10 according to the embodiment of the present invention converts a quaternion corrected by a Kalman filter into a rotation matrix R. It can be obtained vector e _T by applying the rotation matrix R to the reference vector e _T0. As shown in FIG. 2, the vector e _T is therefore stereoscopic thighs from the vector e _T those parallel tip and femur in the femoral part 3 is set to face the knee joint Can understand the operation. Moreover, since it is possible to know the trajectory of the tip of the vector e _T, it is possible to grasp the movement of the knee joint position on the three-dimensional coordinates.

  Therefore, by using the movement measuring device 10, by comparing the movement of the knee joint position when a healthy person walks with the movement of the knee joint position when the movement functional person walks, the movement impaired person It can be used for diagnosis. In addition, it is possible to compare the movement trajectory of the knee joint position when a person with motor impairments walks while receiving electrical stimulation and the walking movement when no electrical stimulation is given or the walking movement of a healthy person.

FIG. 8 shows the projection of the locus of the tip of the vector e _{T on} the xz plane, similar to FIG. 7, but the test subjects were three healthy persons and a motor impairment person (one piece) shown in FIG. One person is paralyzed. 8, the trajectory of the tip of the vector e _T when motor disabilities to walk bulges in the x-axis direction, can be grasped from a so-called sentence turning gait.

FIG. 9 shows another developmental embodiment of the present invention.
In the embodiment shown in FIG. 9, the movements of the waist (trunk) 2, thigh 3, crus 4, and foot 5 of the human body 1 can be measured. An inertial sensor Sb is fixed to the waist 2, an inertial sensor St is fixed to the thigh 3, an inertial sensor Ss is fixed to the crus 4, and an inertial sensor Sf is fixed to the foot 5. Each of the inertial sensors Sb, Ss, and Sf has an angular velocity sensor 12 and an acceleration sensor 13 as in the inertial sensor St fixed to the thigh 3.

The inertial sensor Sb, the vector e _B along the spine of the human body 1 is set. A vector e _S is set for the lower leg 4 and a vector e _F is set for the foot. The vector e _S is set parallel to the lower leg bone and the tip of the vector e _S is directed to the ankle. The vector e _F is set parallel to the back body part of the foot 5 and the tip of the vector e _F is directed to the toes.

Here, paying attention to the detection output of the inertial sensor Ss fixed to the crus 4, the angular velocity component and the acceleration component obtained from the inertial sensor Ss are not due to the operation of the crus 4 alone, but the thigh 3 And the movement of the lower leg 4 are combined. Therefore, even if only the angular velocity component and acceleration component obtained from the inertial sensor Ss are used to obtain the rotation matrix from the quaternion in the same flow as shown in FIG. Instead of representing the rotation operation of only the vector e _S, the rotation operation of the vector e _{S and} the rotation operation of the vector e _T are combined.

Therefore, seeking a rotation matrix from quaternion using an angular velocity component and the acceleration component obtained from the inertial sensor Ss, when asked the rotation of a vector by applying the rotation matrix from the calculation result, the inertial sensor S _T detection output the components of the operation of the measured vector e _T by removing can calculate the rotational movement of only the set vector e _S to crus 4 based on.

As an example of the calculation, as shown in Equation 7, the Kutanion q _S indicating the posture of the crus 4 is converted into a rotation matrix R _S and applied to the reference vector e _S0 of the crus 4. The vector e _S of the crus 4 is calculated. Here, in order to eliminate the influence of the operation of the thigh section 3, and calculates an inverse quaternion q _T ^-1 of the quaternion q _T indicating the posture of the thigh 3 and converts it into the rotation matrix R _T ^-1 . By applying (accumulating) the rotation matrix R _T ⁻¹ to the vector e _S of the crus 4, it is possible to obtain a vector e _S ′ indicating only the operation of the crus 4.

Furthermore, when calculating the vector e _F set in the foot 5, than to convert a quaternion of the foot to the rotation matrix, we calculate a vector e _F with respect to the reference vector e _F0. Then, to calculate the inverse quaternion _q ^{S -1} of the quaternion _{q S} indicating the posture of the lower leg portion 4, converts it into the rotation matrix _R ^{S -1.} By applying (accumulating) the rotation matrix R _S ⁻¹ to the vector e _F of the foot 5, a vector e _F ′ indicating only the motion of the foot 5 can be obtained.

  In the above embodiment, the invention has been described on the basis of measuring the walking motion of the human body by the motion measuring device 10, but the present invention is a motion measurement of a walking motion portion of a so-called robot or a walking motion portion of a prosthetic leg, Furthermore, it can be used for angle measurement of three-dimensional motion of various machines.

1 body 2 waist 3 thigh 5 feet 10 movement measuring device 11 calculating unit 12 the angular velocity sensor 13 acceleration sensor _{e B, e T, e S} , e F vector q, z quaternion

Claims (10)

In a motion measuring device having an angular velocity sensor and an acceleration sensor installed in a walking motion unit,
The angular velocity sensor detects angular velocity components in three axial directions orthogonal to each other, and the acceleration sensor detects acceleration components in three axial directions orthogonal to each other,
A calculation unit is provided to which the angular velocity component detection signal and the acceleration component detection signal are provided,
The motion measuring device, wherein the computing unit calculates the motion of the walking motion unit as a vector motion from the angular velocity component detection signal and the acceleration component detection signal. In the calculation unit, the step of converting the angular velocity component in the three-axis direction into a quaternion, the step of converting the acceleration component in the three-axis direction into a quaternion, the quaternion converted from the acceleration component as an observation value, Correcting the converted quaternion by applying a Kalman filter,
The motion measurement apparatus according to claim 1, wherein a rotation matrix is calculated from a quaternion corrected by the Kalman filter, and the vector is obtained from the rotation matrix.   The motion measuring apparatus according to claim 2, wherein a reference vector is set, and the current vector is obtained by using the rotation matrix with the reference vector as a reference in the walking motion.   A reference posture is set, and a relationship between the reference vector and the reference posture is obtained from a detection signal of the acceleration sensor. From this relationship, a relationship between the vector and the current posture, and a relationship between the reference posture and the current posture. The motion measuring device according to claim 3, wherein the relationship is required.   The exercise according to any one of claims 1 to 4, wherein the angular velocity sensor and the acceleration sensor are provided in a thigh, and the vector to be calculated is set so as to be parallel to the femur and toward the knee joint. measuring device.   The angular velocity sensor and the acceleration sensor are also provided on the crus, from the detection signal of the angular velocity component and the acceleration component in the crus, and the detection signal of the angular velocity component and the acceleration component in the thigh The motion measurement device according to claim 5, wherein the motion of the lower leg is calculated as a motion of a vector.   The angular velocity sensor and the acceleration sensor are also provided on the foot. From the detection signal of the angular velocity component and the acceleration component in the foot, and the detection signal of the angular velocity component and the acceleration component in the crus, The motion measuring device according to claim 6, wherein the motion of the unit is calculated as a motion of a vector.   The motion measuring apparatus according to claim 1, wherein the locus of the tip of the vector is obtained three-dimensionally on three-dimensional coordinates.   The motion measuring apparatus according to claim 1, wherein an operation locus obtained by projecting a locus of a tip portion of a vector onto a plane is obtained.   The motion measuring apparatus according to claim 9, wherein the motion trajectory is projected on three planes orthogonal to each other.

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