1. Introduction

Sensing and digitizing the physical world has always been a critical topic, and the six degrees of freedom (6-DOF) rigid body motion information is one of the most important aspects. Recently, with the emergence of more and more intelligent systems, motion information of rigid bodies has shown its value as the fundamental element of a wide range of tasks. With regard to specific application scenarios, motion sensing methods can vary greatly.

For applications in common environments such as security monitoring, robot navigation, and human-machine interaction, the motions are mainly estimated by changes in the appearance of 2-D images or 3-D point clouds in different frames. For methods using 2-D images, correspondence between some invariant features [1,2], patches [3,4], or pixels [5,6] in different frames are found that enables motion to be estimated by solving a perspective-n-point problem [7] or optimizing the motion for minimized reprojection error [8,9]. Point-cloud-based methods, on the other hand, tend to optimize the motion for minimum error when merging overlapping 3-D point clouds in two different frames [10–12]. This typically leads to the matching of several salient structures in the point clouds by separately weighting the point pairs or trimming the outliers [13, 14]. These methods can be effective in a wide range of scenarios, but the robustness of such methods depends on whether the motion causes changes in the observations of the object. When the target has barely any texture or structure, or the ambient illumination is so harsh that changes cannot be seen, these methods are likely to fail.

To improve the universality, some methods try to directly measure the motion of the object by introducing the Doppler effect. Heide et al. [15,16] modulated the adaptive infrared illumination for time-of-flight (TOF) cameras so that the velocity can be measured via the frequency shift of reflected light. Thus, the velocities are densely mapped to specific positions in the scene and a 3-D flow can be calculated by the combination of the measured velocities and 2-D optical flow. Although the 3-D flow is promising for the description of 6-DOF motion, the robustness of 2-D optical flow computation still depends on the ambient light and appearance of the object. On the other hand, quantitative evaluations on the accuracy of these methods are not discussed in detail, while the poor noise performance and low resolution of commercially available TOF chips are mentioned [15].

In the context of high-accuracy machining, several methods measure multi-DOF geometric motion or motion errors via laser interferometry [17–19], laser collimation [20] or time-of-flight measurements [21]. Mirrors or reflectors are attached to the target, e.g. a linear or rotary stage, and ultra-high accuracy with translational errors in the order of micrometers and angular errors in the order of arc seconds can be achieved. However, such systems make their focus on the specific application, where reflectors are intrusively attached on the target and the motion of the targets are always to some extent limited, so that the principles also cannot be extended to non-cooperative objects with unknown motion.

The laser Doppler velocimeter (LDV), as a interferometric device, is capable for non-intrusive, high-accuracy but low-dimensional velocity measurement for non-cooperative targets. Such properties of the LDV can open up the possibility to measure high-precision motion for any diffuse object, thus showing a promising solution for the common multi-DOF motion sensing problems. The principles and configurations of LDV are well-explored in the literature [22–24], and various arrangements of the devices are introduced to deal with specific tasks such as measuring the fluid [22,25], transversal motion of a surface [26] or the rotation of a motor [27]. However, principle on using LDV for common multi-DOF motion sensing scenarios have been barely explored.

In our previous work [28], we first explored the principles on using LDV for common motion sensing scenarios and demonstrated several applications of such a system, including 3-D shape integration and user interface. The proposed system calculates motion via the distance and velocity acquired simultaneously by a laser range finder (LRF) and a LDV. By time-division measurement of several points on the target, the method can reconstruct the motion with high throughput at approximately 410 Hz and low error for certain kinds of motions bounded below 2%. However, in this method, the accuracy is sensitive to different motion components, and the accuracy may decline when there are motion components in some specific DOFs.

In this paper, we propose a method for robust, 6-DOF, contactless motion sensing of an arbitrary rigid body using multi-view LDVs. The proposed system is partially based on our previous work [28], but in this paper we fully explore the essential conditions for LDV-based 6-DOF motion sensing, and overcome the shortcomings of the previous system with a largely optimized multi-view system configuration without a LRF. With the basic assumption that the available LDVs provide one-dimensional velocity measurement with adequate precision in the direction of the laser beam, the paper begins with a discussion of the basic motion sensing principle, and then proves that at least three viewpoints are essential for LDV-based 6-DOF motion reconstruction. We acknowledge that the proposed method is a typical linear system; hence, errors are strongly related to the condition number of its measurement matrix. Thus, to find the best system configuration, numerical simulation of the condition number was performed. Based on the discussion, a simplest form of the proposed system was built and some implementation techniques are introduced. Finally, the system was validated by comparison with the marker tracking scheme of a high-speed camera. The accuracy and robustness of the proposed system were proven by demonstrating the sensed velocity as well as calculated change of position in six different motion patterns. We also evaluated the accuracy for sensing constant velocity when the distance and velocities were different. It is shown in the experiment that the mean value error is less than 1.3 deg/s in rotation and less than 3.2 mm/s in translation in the proposed system. Finally, the throughput of the proposed method is approximately 250 Hz and the latency is approximately 20 ms.

2. Principle

2.1. Formulation of LDV-based 6-DOF motion sensing

As a prior information, the LDV used in the proposed system takes the heterodyne principle [29] which is commonly adopted in today’s commercial LDV devices for measurement on solid surfaces. When the laser hits the object, only the velocity component along the direction of the laser beam is measured.

We discuss the motion of a rigid body in an orthogonal, right-handed coordinate system with axis x, y, and z, as illustrated in Fig. 1. It can be represented by six DOFs consisting of angular velocity ω = [ω_x, ω_y, ω_z] around the origin, and linear velocity v = [v_x, v_y, v_z]. ω_x, ω_y, and ω_z are the components of the angular velocity around the x, y, z axis; and v_x, v_y, v_z are the components of translation velocity along x, y, z axis. Assume that we use N non-overlapping laser beams from the LDV for measurement, namely L_i, i = 1, 2 . . . N, and the unit direction vector of L_i is l_i. The laser beam L_i hits the object at point p_i. The velocity of p_i, namely v_i consists of two parts, caused by the rotation and translation velocity of the object, respectively, as formulated in Eq. (1).

 

Fig. 1 The 6-DOF motion of a rigid body measured by a laser from the LDV. The rigid body is moving with rotational velocity ω and translational velocity v, and is hit by a laser beam L_i from the LDV at point p_i. Points on L_i can be denoted by o_i + sl_i, where o_i is any point on L_i, l_i is the direction vector, and s is a scalar. For ease of description, we refer to o_i as “viewpoint” and l_i as “direction” for each laser beam L_i. The velocity of p_i, namely v_i, consists of two parts introduced by the rotation velocity and translation velocity of the object. The measured velocity by LDV is the projection of v_i in the laser beam’s direction l_i, namely v_i = v_i · l_i.

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Let v_i be the measured velocity from L_i. Given that the LDV measures only the velocity component in the direction of the laser beam, v_i equals the length of the projection of v_i in l_i:

Eqation (2) formulates what the LDV measures when the laser beam hits an object. At first sight, this means that the measured velocity is related to the position where the laser beam hits the target. This was taken as the basic formulation in previous work [28], and an LRF was used for position measurement. However, we can rewrite Eq. (2) as

Equation (4) shows that the measured 6-DOF velocity from the laser Doppler effect is not really related to where the laser beam hits the object. In other words, the shape and position of the target object does not matter. Hence, once the laser beam is calibrated, i.e., o_i and l_i are obtained, we can get an equation where only ω and v are unknown, when v_i is measured via LDV.

Thus, combining the measurement from N non-overlapping laser beams, an equation set can be formulated with matrix representation as

Let the leftmost matrix, namely the measurement matrix, be denoted by A = [r₁^T r₂^T . . . r_N^T]^T, where r_i = [(o_i × l_i)^T l_i^T], i = 1, 2, . . ., N, is the i_th row of A. A represents the arrangement of the laser beams from the LDV in the system configuration. The rightmost vector is denoted by b, consisting of the measured velocities from the LDV. Assuming that X = (ω^T v^T)^T, the 6-DOF motion reconstruction can be formulated by

2.2. Modeling for system configuration

Equation (6) provides a universal least-squares solution for LDV-based velocity sensing. A system configuration that solves such a problem is described by the corresponding measurement matrix A, which can be further abstracted into an arrangement of several non-overlapping laser beams in the space, i.e., the corresponding N groups of o_i and l_i. In this subsection, we discuss the system configurations and their influence on the solution of Eq. (6).

2.2.1. Conditions for solution existence

The problem of whether 6-DOF motion can be robustly reconstructed is equivalent to whether a stable solution for Eq. (6) exists, which apparently relies on whether rank(A) = 6. The most direct conclusion is that N ≥ 6, but this is not sufficient. With regard to the specific system configurations, some rows in A may be linearly correlated, causing the reduction of rank(A).

For common linear systems, we need to determine all the entries of matrix A in order to analyze its rank; fortunately, the rows of A are well defined in this problem. Hence, we can analyze the dimension of the row space of A, given that rank(A) = dim(span{r₁, r₂, . . ., r_N}). Intuitively, the reduction of dim(span{r₁, r₂, . . ., r_N}) occurs when o_i and l_i are shared among different rows. While l_i can be adjusted easily with the use of controllable reflectors such as galvanometer scanners, o_i is highly related to the placement of system component. Here, we discuss the sharing of viewpoints in an enumerative way. It is worth noting again that o_i can be any point on L_i without changing r_i. Thus, the o_i of two laser beams can be regarded shared as long as the two laser beams intersect with each other, in order to simplify the following discussion.

First, let us consider the simplest condition where there are a total of N⁽¹⁾ rows in matrix A⁽¹⁾ sharing the same viewpoint o₁. The j_th row of A⁽¹⁾ can be written as

Noting that l_j is a unit vector in the 3-D space, each l_j can be represented by the linear combination of any other three linearly independent 3-D vectors. Obviously, the same property exists for any ${\mathit{r}}_{\mathit{j}}^{(\mathbf{1})}$, which is expressed as Eq. (7). Hence, we have

Following the above discussion, in order to increase dim(span{r_j}), we add N⁽²⁾ rows with a different viewpoint, namely o₁ + q₁. Then, we can write the k_th row of matrix A⁽²⁾ as

${\mathit{r}}_{\mathit{k}}^{(\mathbf{2})}$ is the summation of two terms. The first term belongs to $\mathit{span}\left\{{\mathit{r}}_{\mathit{j}}^{(\mathbf{1})}\right\}$, and the second term is a vector constrained on a two-dimensional plane. Thus, the additional viewpoint o₁ + q₁ can only add two more dimensions to the row space, i.e.,

Naturally, the last dimension can be provided by an additional viewpoint. For validation, we add N⁽³⁾ rows with viewpoint o₁ + q₂ to the matrix. In this situation, the m_th additional row can be represented by

In summary, for 6-DOF motion reconstruction represented by Eq. (6), rank(A) needs to be six.

This means that the system configuration should satisfy the following conditions:

2.2.2. Accuracy and system configuration

Following the discussion about rank(A), here we further discuss the relationship between system configuration and measurement error. As Eq. (6) denotes a typical linear system, we introduce the condition number of A, defined by κ_A = ‖A‖₂ · ‖A⁺‖₂, as the measure of stability of the corresponding system configuration. The overall error of the system can then be bounded by Eq. (12) [30,31]:

We conducted a numerical simulation of different system configurations for minimizing κ_A. The setup for this simulation is shown in Fig. 2(a). For minimum realization of the proposed method, we adopted three viewpoints and six non-overlapping laser beams (two laser beams for each viewpoint) for measurement. As the three viewpoints are always coplanar, they are fixed on the plane α_Z=0. The target is assumed to be away from the system at a distance of D = 1500 mm, which is in the middle of the expected working range of the proposed system. The details of the working range will be discussed in Sec. 3.1. During the simulation, the first viewpoint is fixed at the origin [0 0 0]. The other two viewpoints are placed on the x and y axis, respectively, at a distance of d from the first one. The color of the laser beams denotes the viewpoint they pass through, specifically, red for o₁, green for o₂, and blue for o₃. We assume that the laser beams always hit the object at some fixed points illustrated in Fig. 2(a). Thus, d will influence both o_i and l_i for each laser beam.

 

Fig. 2 Numerical simulation of different system configurations. (a) Illustration of constraints of system configuration during simulation. The three viewpoints o₁, o₂, and o₃ are on the gray plane α_z=0. In the simulation, where the laser beams hit the target is assumed to be fixed at certain points on the surface of the target. The color of a laser beam represents the viewpoint that it passes through. (b) Illustration of κ_A when d changes.

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The simulation results are illustrated in Fig. 2(b). It is shown that the condition number can be largely reduced as the distances between viewpoints increase. However, it is less effective to increase the distance when the distance is larger than 600 mm. As we also want the system to be smaller, there is a tradeoff between the size of the system and accuracy. In this work, we let d ≈ 650 mm, corresponding to κ_A at approximately 550 in the simulation.

The distribution of the points on the scanning pattern also influences the condition number. However, there is no much space to use a large scale pattern, because the lasers have to hit the object which might have a relatively small surface. In our system design and simulation, the largest pattern size at 1500 mm is limited within a circle with a diameter of 200mm. Under such constraint, changing scanning pattern is not important to the condition number, compared with the separation of viewpoints. Another numerical simulation is performed to prove this. In this simulation, the distance d = 600 mm is fixed, and we randomly generates the scanning pattern 10000 times, and evaluate the corresponding condition number. As the result, over 90% of the random generated pattern corresponds to condition numbers smaller than 1000, and around 80% of the condition number in less than 500. Even a random generated pattern can have a fairly large chance to have a fairly good performance, so that it is not too important to carefully design the points on the scanning pattern.

On the other hand, to increase the number of points on the scanning pattern can also reduce the condition number, since more fragmentary velocities on the surface of the object are used for calculation and the problem is better conditioned. Nevertheless, increasing the points on the scanning pattern also increases the delay of the system, and the 6 points on the scanning pattern which is the minimum required number, have resulted in a tolerable condition number. Thus, in the following discussion and experiments, we will only discuss about the situation of the 6-point scanning pattern.

2.2.3. Influence of coordinate system selection

It is worthwhile to point out that the motion of a rigid body has different representations in different coordinate systems, but they are equivalent, i.e., they correspond to the same physical motion. Nevertheless, if the coordinate system is changed, the measurement matrix is also changed, and it will influence the accuracy of the proposed method. This means that we can change the coordinate system in order to increase accuracy without losing information of the physical motion.

Assuming that the origin of the new coordinate system is replaced at point c, the coordinates of the viewpoints are changed but the direction vectors are not affected. Thus, the adjusted matrix A^(a)(c) will be

Here, we still use κ_A as the measure of stability. κ_A is supposed to be minimized to stabilize the calculation. Hence, the optimized center c^* can be defined by

Further discussions on the solution of Eq. (14) will be presented in Sec 3.4.

3. Implementation

3.1. System configuration

The schematic of the proposed 6-DOF motion sensing system is illustrated in Fig. 3. The laser beam emitted by the LDV will be reflected by the mirror embedded on the galvanometer scanner. The galvanometer scanner changes the direction of laser beams temporally, and will stop at different positions on the scanning pattern, namely the six points shown in Fig. 3. The LDV measures the velocity when the galvanometer scanner stops at the i_th point on the scanning pattern. The active laser beam at this time is L_i so that the measured velocity is v_i. Although only one set of LDV and galvanometer scanner is used, we enable different laser beams be reflected by different groups of mirrors (MR_1,2,3,4) to change the equivalent viewpoints of the laser beams. Specifically, laser beams No. 2 and No. 3 (denoted by green lines) are reflected by MR₂ and MR₄, and the corresponding equivalent viewpoint is o₃. laser beams No. 4 and No. 5 (denoted by blue lines) are reflected by MR₁ and MR₃, and the corresponding equivalent viewpoint is o₂. Laser beams No. 1 and No. 6 (denoted by red lines) are not reflected by mirrors, so the equivalent viewpoint o₁ is on the galvanometer scanner. When all the laser beams hit the target object, the velocities at several points on the surface of the object can be measured by the LDV using a time-division strategy. Then, the 6-DOF motion can be reconstructed using the principles discussed in Sec. 2.

 

Fig. 3 Schematic of the proposed 6-DOF motion sensing system with a single LDV.

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The proposed method was implemented by a long-range LDV (Polytec OFV-505,OFV-5000 with decoder VD-09), 2-D galvanometer scanner (GSI 6220H, silver-coated), and four silver-coated mirrors (Edmund Optics), as illustrated in Fig. 4. To verify the feasibility of the proposed 6-DOF motion sensing method, an additional CMOS camera (XIMEA, 648 × 480, 500 fps) was set up together with the proposed system for comparison.

 

Fig. 4 System implementation. The corresponding distances between viewpoints, i.e. ‖o₂ − o₁‖ and ‖o₃ − o₁‖ are 685 mm and 651 mm, respectively. The angle between the two laser beams from one viewpoint is 4.31 deg and κ_A = 520.

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3.2. Configuration of LDV

While to use a LDV device is fairly straightforward, it is important to appropriately configure the LDV in order to reduce the noise level in its output. The noises of LDVs have been investigated and results presented in the open literature [23]. The dominant noise in the LDV is the speckle noise, caused by undesired interference when the measurement is conducted on a rough surface. Works on the optical configuration have been done to reduce the speckle noise [24], and it has been proven that the speckle noise can be largely relieved by low-pass filter [23] and trimming out the undesired peaks [33].

Besides the speckle noise, the imperfect focus of the lenses and large incident angle may also cause noises sourcing from weak Doppler signal, but the effects are less important compared with speckle noise [33]. The noises caused by imperfect focus only become obvious when the mismatch in the focus is too large (e.g., larger than 500 mm), and the large incident angle only matters when the target is near-specular, as in our experimental observation.

In the context of 6-DOF motion sensing, the change of velocity is mostly in the low-frequency region. Thus in this work, a low-pass filter at 5kHz together with a tracking filter is applied for dealing with noises. While the low-pass filter limits the bandwidth of the measured velocity, the tracking filter bridges brief dropouts with large acceleration caused by the speckle noise. Note that the noises of LDV may in some cases have a profound effect on the performance of velocity sensing system, especially when oscillation at high frequency are measured. However, under the conditions applied both for simulation and experimental verification, we found the effects of speckle noise and photon noise as well as laser noise to be of minor importance.

3.3. Geometrical calibration

The system must be parameterized into the measurement matrix A for calculation. We used a pre-calibrated monochrome camera and a chessboard to independently calibrate the six laser beams on the scanning pattern in Fig. 3.

The scheme is simple. The camera is initially fixed with the system. The chessboard is placed where the camera can see it and the laser beams can hit it. The center of the laser spot (u_p, v_p) in the camera image is assumed to be the intersection between the laser beam and chessboard. We used the world coordinate system with the origin at the optical center of the camera. The rotation matrix R and translation vector T of the chessboard can be acquired by the camera with intrinsic matrix K_c [34]. The coordinate of the intersection point p_w in the world coordinate system can be denoted by

Then, the chessboard is moved and other points on the laser beam can be calculated. Assume that we have K points, namely p_w,j, j = 1, 2, . . ., K and the line equation of laser beam L_i is denoted by p = o_i + sl_i. With all the K points we can build an equation set consisting of equations with form o_i + k_j l_i = p_w,j, which is an equation set with 3K equations with 6 + K unknown parameters. Note that for this situation, the position of o_i and length of l_i remains unconstrained. Hence, we add two constraints, namely o_i = [x_o,i y_o,i 0] and l_i = [x_l,i y_l,i 1] to the equation set and solve it via a linear least-squares method. Finally, l_i will be normalized to guarantee that it is a unit vector.

3.4. Adjustment of the coordinate system

We discussed that the coordinate system influences the calculation accuracy of the proposed system in Sec 2.2.3. However, after calibration, the parameters of the system are in the camera’s coordinate system, which is not optimized for calculation. In this subsection, we describe the method to find the optimized coordinate system for calculation by solving Eq. (14).

First, to analyze the distribution of κ_A^a(c) in different coordinate systems, we performed a numerical simulation of κ_A^a(c). One-thousand different coordinate systems were evaluated in this simulation. Their axes were parallel but their origins were randomly generated in a cube with size 1000 × 1000 × 3000 mm³. The measurement matrices were built by the practical calibration parameters of the system acquired by the method described in Sec. 3.3 and were adjusted by Eq. (13). Then, κ_A^a(c) for each coordinate system was calculated.

The results of the simulation are illustrated in Fig. 5. The color of the dots denotes the condition number of the measurement matrices in the corresponding coordinate systems. It can be seen that κ_A^a(c) ranges from 500 to 50000 when we change the origin of the coordinate system, and the origins actually converge to a certain region when we attempt to minimize κ_A^a(c). Based on this observation, equation (14) is likely to be a convex problem. Hence, c^* is solved by a steepest descent method, in which the initial c is the one corresponding to the smallest κ_A^a(c) in the simulation, and the iteration terminates when the change in κ_A^a(c) is less than 10. Finally, c^* = [70 − 70 840] and κ_A^a(c^*) = 520. This new coordinate system was used in the experiments.

 

Fig. 5 Simulation of κ_A^a(c) when the origin of the coordinate system is changed. The origins are randomly generated in a cube with size 1000 × 1000 × 3000. The corresponding condition number is illustrated by the color.

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3.5. Calculation pipeline

The proposed system uses a time-division strategy for measurement. While the galvanometer scanner scans continuously, the velocities are sampled at different times, by L_i, i = 1, 2, . . ., 6, respectively. To boost the throughput and accuracy of the proposed system, a pipeline strategy was adopted in the system.

At each sampling time, the motion is calculated by one measured velocity and five interpolated velocities. Figure 6 illustrates an example of this strategy. Assume that we are trying to calculate the motion of the object at time t_k. At this time, we only know v_6,k measured by L₆, but the velocities along other laser beams are unknown. Thus, we wait until t_k+5 and calculate v′_i,k, i = 1, 2, . . ., 5 by the linear interpolation of two samples on the same laser beam. For instance, v′_5,k = v_5,k−1 + (v_5,k+5 − v_5,k−1) * (t_k − t_k−1)/(t_k+5 − t_k−1). Then, the 6-DOF motion can be reconstructed by Eq. (6). This strategy gives well approximated velocities for each time, which contributes to the accuracy of motion reconstruction without reducing the throughput. However, a latency of five time intervals is needed to perform the interpolation. With the throughput at 250 Hz in the proposed system, this latency can be as large as 20 ms. We believe that this will not be a problem in most common applications. However, both throughput and latency can be further improved by using alternative system configurations if a higher speed is desired, which will be discussed in Sec. 4.5.

 

Fig. 6 An example of the calculation pipeline.

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4. Experiments

4.1. Experimental setup

In the experiment, the target object is a white plastic board attached with a chessboard pattern. While the proposed system recorded the motion of the target, the camera captured the images simultaneously, and the motion of the object was calculated by offline analysis of these sequential chessboard images. It should be noted that the chessboard is only needed by the camera in order to provide a reference motion for evaluation, but is not necessary in the proposed method. The comparisons between the camera and proposed system are presented in the following subsections.

4.2. Experimental results for six motion patterns

As depicted in Fig. 7, the target was placed 1500 mm away from the measurement system. Six motion patterns were used to evaluate the accuracy of the proposed method in different DOFs. To be specific, in each motion pattern there is a dominant motion component, namely, ω_x, v_x, ω_y, v_y, ω_z, v_z in motion pattern 1,2,3,...,6, respectively.

 

Fig. 7 Experiment with six different motion patterns. See Visualization 1 for details. Note that the target is a chessboard in the practical experiment for the camera, but the proposed method does not need the chessboard; hence, a white board is used in the video to demonstrate this property.

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For each motion pattern, the target was moved back and forward mainly in one DOF. To mimic the motion in practical situations that are often a mixture of multiple DOFs, the target was moved manually in the experiments.

The results are shown in Fig. 8. The 10-s consistent motion sensing results, including the velocity sensing results (top) and pose/position sensing results (bottom), are illustrated. As the proposed system only measured the velocity and the camera only measured the position and pose, the position of the proposed system and velocity from the camera were calculated by temporal integration and differential, respectively. Figure 8 shows that the integrated positions and poses from the proposed system and camera are in good agreement. In the 10-s sensing process, the drift in rotation was at most 6.76 degrees and the drift in translation was at most 37.36 mm. For velocity sensing, the proposed system clearly outperformed the camera. The velocities acquired by camera vibrated largely because the slight errors in the position and pose were divided by a small time interval. This problem does not exist in the proposed system. Finally, the velocities from the proposed system were much smoother and more accurate as illustrated. As a reference, a static object was also measured using the proposed method. The results are presented in Table 1.

 

Fig. 8 Results for six motion patterns.

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Table 1. Results of 10-s motion measurements

View Table

These drifts can be attributed to two factors. The first is the error in velocity measurement and calibration, which causes static drift even when the object is not moving. The second is the change in acceleration between two velocity samples on one laser beam. Since a linear interpolation was performed to calculate the velocities, the acceleration was assumed to be constant for the two samples. However, this assumption may not be accurate. There are ways to further improve the accuracy of the proposed method. For example, if six LDVs are used to simultaneously measure the velocity from different directions, then the change in acceleration will not influence the accuracy of velocity measurement.

4.3. Velocity range performance

In this experiment, we evaluated the repeatability of motion measurement for different velocities. The target was initially placed 1500 mm away from the proposed system. DC motors were used to fix the translation and rotation velocities of the target as illustrated in Fig. 9, and the system measured the motion 500 times at each velocity. To be specific, in this experiment the rotational velocity is around z-axis, and the translational velocity is along z-axis. The true value was calculated by averaging the change in rotation and translation of the chessboard from the camera.

 

Fig. 9 Experimental setup for providing constant rotational and translational velocities.

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The mean value and standard deviation of the measured velocities are illustrated in Fig. 10. The rotational and translational speeds are represented by the norm of the corresponding speed vectors, i.e. ‖ω‖ and ‖v‖. The differences between the mean values from the proposed system and the true values were less than 1.3 deg/s in rotation and 3.2 mm/s in translation. The maximum standard deviations in the results were approximately 4.7 deg/s in rotation and 6.7 mm/s in translation. When the velocity increased, the standard deviation of the results in rotation also increased slightly. However, there was no obvious relationship between the standard deviation and translational velocity.

 

Fig. 10 Experiment results over different velocities.

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4.4. Sensitivity to different measurement distances

The distance between the system and target is not related to the accuracy of the proposed system, according to the formulation in Eq. (5). Nevertheless, when the target is far from the system, the laser beams may not be able to hit the object; hence, the configuration of the system needs to be adjusted, which can influence the accuracy of the system. In this experiment, we tried to evaluate the performance of the proposed method for measuring a fixed velocity at different distances. The target was rotating at a fixed velocity around z-axis, driven by a DC motor using the same experiment setup as in Fig. 9(a), and was placed at different distances. Because the translational motion changed with distance, we only evaluated rotational velocity measurements. The distance was changed from 1000 to 1500 mm for one system configuration, and from 1750 to 2250 mm for the other configuration, where the angle of MR₂ and MR₄ are changed. Five hundred samples of motion measurement results were taken for analysis at each distance.

The results illustrated in Fig. 11 validate the assumption that the distance has almost no influence on the accuracy of the proposed system when the system configuration was not changed. The standard deviation were nearly the same for all distances and there are only small differences in the mean value (less than 1.7 deg/s).

 

Fig. 11 Experiment over different distances when the target was rotating at constant velocity. The rotational velocity is represented by the norm of rotational speed ‖ω‖. The experiment was performed under two different system configurations, namely config1 (κ_A = 520) and config2 (κ_A = 1100). The direction of MR₂ and MR₄ were changed between these two configurations in order to enable all laser beams to hit the same target when it was farther from the system.

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As the change in the shape and position of the target only influences the distances between the viewpoints and the target, this result also proves that the proposed method is robust against targets with different shapes.

4.5. Throughput and latency

During the above experiment, the throughput of the proposed system was approximately 250 Hz and the latency was approximately 20 ms. The proposed method was not computationally expensive, and the throughput was limited mainly by the scanning speed of the galvanometer scanner. Thus, the throughput can be increased by using a smaller scanning pattern, which will affect the arrangement of laser beams, and may cause reduction of accuracy. However, there are other ways to boost throughput without affecting the accuracy. For instance, multiple LDVs can be used to simultaneously measure the velocities in different directions to reduce the time for switching between different laser beams. If six or more LDVs are used, the 6-DOF motion of the object at each time can be acquired by one shot with the proposed method. In such case, the throughput and latency would be limited only by the throughput of the LDVs and the computational time.

5. Conclusion

A robust, contactless, and 6-DOF motion sensing method based on multi-view laser Doppler measurement was presented. Benefiting from the use of the laser Doppler effect for velocity sensing, the proposed method is robust to non-cooperative objects and environments, which makes it a universal solution for 6-DOF motion sensing. The principles of the LDV-based motion sensing and the system configuration were analyzed in detail. The 6-DOF motion sensing was modeled as a linear system, and the necessity of three viewpoints for 6-DOF motion reconstruction was proven by rank analysis. The specific system configuration was optimized by minimizing the condition number of the measurement matrix, which strongly contributed to the accuracy of the proposed system. The proposed method was implemented by a simple system consisting of an LDV, a galvanometer scanner, and four mirrors, and the techniques in system configuration were introduced. The experimental results show that the proposed system can achieve accurate motion sensing for any 6-DOF motion, robust to non-cooperative target and adaptive to different distances and velocities with almost no loss on accuracy.