Detection method for acquiring lane and road edge by using millimeter wave radar

Abstract

The invention relates to a detection method for acquiring lane and road edges by using millimeter wave radar, which comprises the following steps of: acquiring detection points of each moving vehicle, and gathering all the acquired detection points into a point cloud set; carrying out preprocessing on the obtained point cloud set, wherein the preprocessing comprises partitioning according to distance and angle correction; performing statistical filtering processing on the preprocessed point cloud set to obtain a smooth probability-transverse distance sequence; calculating a crest value and a trough value of the probability-transverse distance sequence to obtain a lane edge coordinate and a road edge coordinate; and performing angle compensation on the lane edge coordinates and the road edge coordinates to obtain lane edge position information and road edge position information relative to the radar. The method is small in calculation amount and can be directly deployed to a road test radar end for calculation; nowadays, road test multi-target tracking is realized. According to the detection method, automatic calibration can be performed after the radar is installed, the debugging time during installation is saved, and the cost is reduced.

Description

Technical Field

[0001] The invention relates to the field of traffic radar, and in particular to a detection method for obtaining lanes and road edges by using millimeter wave radar. Background Art

[0002] In traffic scenarios, millimeter-wave radar with MIMO system has the characteristics of long measurement distance, large coverage, large number of target detection and high positioning accuracy. With the maturity of radar technology and the development of smart transportation, it is widely used in urban roads and highways for traffic flow detection and vehicle trajectory monitoring.

[0003] Due to the characteristics of the target space, the millimeter-wave radar of the MIMO system will detect a large number of detection points when detecting the target. The collection of these points is called a point cloud. The point cloud information includes the distance, speed, azimuth, pitch angle and signal-to-noise ratio of the target detection point. A large amount of time accumulation on the same road can generate sufficient detection point information. By processing these detection points, the lane edge and lane line of the road section can be calculated. The formed lane line and road edge information combined with the vehicle's trajectory information can monitor in real time which lane the vehicle is driving in and judge whether it is off the road edge, thereby providing more traffic information. Summary of the invention

[0004] The present invention provides a detection method for obtaining lane and road edges using millimeter wave radar, which can accurately determine the lane and road edges and solve the problem of difficulty in modeling lane and road edges by traffic radar.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] S1. Obtain the detection points of each mobile vehicle, and gather all the obtained detection points into a point cloud set;

[0007] S2, preprocessing the acquired point cloud set, including segmentation according to distance and angle correction;

[0008] S3, performing statistical filtering on the preprocessed point cloud set to obtain a smooth probability-lateral distance sequence;

[0009] S4. Calculate the peak value and the trough value of the probability-lateral distance sequence to obtain the lane edge coordinates and the road edge coordinates;

[0010] S5. Angle compensation is performed on the lane edge coordinates and the road edge coordinates to obtain lane edge position information and road edge position information relative to the radar.

[0011] Furthermore, the information contained in the detection point includes: distance, azimuth, speed, signal-to-noise ratio and frame number;

[0012] The content of distance segmentation in step S2 includes:

[0013] Divide the road of length L into blocks according to the interval Δl, and treat the last section less than the interval Δl as one block;

[0014] The number of blocks M is expressed as: M = cell (L / Δl), cell (*) is an upward rounding function, and the point cloud set P after block division is expressed as: P = {P1, P2, ..., P M}.

[0015] Furthermore, the content of angle correction of the point cloud data after segmentation in step S2 includes:

[0016] Define the input point cloud set [, output point cloud set q, p is any subset of the point cloud set P, that is, p = P m , m∈{1,2,3,…,M};

[0017] Where p = {p1, p2, …, p n}, q={q1,q2,…,q n}, n is any subset P of the point cloud set P after segmentation m The number of detection points contained in ;

[0018] Calculate a translation vector e (1×5) , so that q i =p i +e, where p i is the i-th detection point in the input point cloud set p, p i is the i-th detection point in the output point cloud set p;

[0019] Let e ​​= [0, δ·Δθ, 0, 0, 0], Δθ is the rotation angle, δ is the rotation angle correction coefficient, specifically:

[0020]

[0021] δ=∑snr i / ∑snr j , i and j satisfy the condition θ i <Δθ,θ j >Δθ;

[0022] Among them, θ i is the detection point p i The corresponding azimuth, snr i Detection point p i The corresponding signal-to-noise ratio, θ j is the detection point p j The corresponding azimuth, snrj Detection point p j The corresponding signal-to-noise ratio, i, j∈{1,2,3,…,n}.

[0023] Furthermore, the statistical filtering process in step S3 includes:

[0024] The detection points in the output point cloud set q are converted from the polar coordinate system to the rectangular coordinate system. The corresponding matrix expression is: q = [xyv snr t], with dx as the gradient and snr as the weight to obtain the centroid sequence g1 of the number of detection points in each gradient;

[0025] Among them, g 1(kdx) =∑snr l , l satisfies (k-0.5)·dx <x l <(k+0.5)·dx, the minimum increment range of k is in Indicates rounding down. Indicates rounding up, x min is the minimum x value among all the detection points, x max is the maximum x value among all the detection points;

[0026] The centroid probability sequence g1 is low-pass filtered using a Gaussian low-pass filter to obtain a sequence g2, g2=g1·h, where h is the Gaussian low-pass filter sequence.

[0027] Furthermore, the step S4 comprises:

[0028] Lane center calculation:

[0029] Perform multiple spline interpolations on sequence g2 to obtain sequence g. The number of peaks in sequence g is the number of lanes, and the peak position is the center position of the lane.

[0030] Let the set of all lane center coordinates corresponding to A = {A1, A2, A3, ..., A u}, the coordinates of the position of the f+1th peak from left to right are (x a(f+1) ,y a(f+1) ), f∈{1,2,3,…,u-1};

[0031] Lane Edge Computing:

[0032] Let the trough coordinates correspond to the set B = {B1, B2, B3, ..., B u-1}, the coordinates of the position of the fth trough from left to right are (x bf ,y bf ), lane edge set C = {C1, C2, C3, ..., C u-1}, the coordinates of the lane edge position between the fth lanes from left to right are (xcf ,y cf ),but:

[0033]

[0034] y cf =mΔl;

[0035] Road Edge Computing:

[0036] Let the first side road edge coordinate D1 = (x d1 ,y d1 ), the second side road edge D2 = (x d2 ,y d2 ),but:

[0037] x d1 =(2x a1 -x c1 )+ε(x a1 -x c1 ) a1 / g c1 ;

[0038] x d2 =(2x au -x c(u-1) )+ε(x au -x c(u-1) ) au / g c(u-1) ;

[0039] Among them, ε is the correction coefficient, x a1 , g a1 are the horizontal and vertical coordinates of the position of the first wave peak from left to right, respectively, au , g au are the horizontal and vertical coordinates of the position of the u-th peak from left to right, respectively, c1 , g c1 are the horizontal and vertical coordinates of the first trough position from left to right, respectively. c(u-1) , g c(u-1) are the horizontal and vertical coordinates of the position of the u-1th trough from left to right;

[0040] The y coordinate of the road edge d1 and d2 is the center distance of each block mΔl;

[0041] y d1 =y d2 =mΔl.

[0042] Furthermore, the step S5 comprises:

[0043] Lane edge coordinates and road edge coordinates for angle compensation,

[0044] Let the lane line matrix and the road edge matrix in,

[0045] x c =[x c1 ,x c2 ,…,x c(u-1) ],y c =[y c1 ,y c2, …,y c(u-1) ], x d =[x d1 ,x d2 ],y d =[y d1 ,y d2 ];

[0046] The translation vector e = [0, δ·Δθ, 0, 0, 0] is used to perform angle compensation on the lane edge and road edge coordinates. The lane edge matrix H after compensation is c and the road edge matrix H d , there is the following relationship: H c =RE c ,H d =RE d , where the rotation matrix R is:

[0047]

[0048] The present invention has the following beneficial effects:

[0049] 1. The present invention only needs the radar point cloud data for statistical fitting, and does not need to use camera data, and can be realized with only a single sensor data;

[0050] 2. Compared with using some clustering algorithms to obtain lane edges, the present invention has a small amount of calculation and can be directly deployed on the road test radar end for calculation;

[0051] 3. Currently, the calibration of the lanes of the road where the multi-target tracking radar is installed is mostly done manually, and the following information needs to be calibrated: the number of left lanes, the number of right lanes, the lane width, the lateral position of the left vehicle, and the lateral position of the right lane. It will be calibrated during installation and testing, and it is difficult to adapt to the problem of curves. The method of the present invention can be automatically calibrated after installation, saving debugging time during installation and reducing costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1A reference schematic diagram of the installation position, road position and coverage range of the millimeter wave radar based on the millimeter wave radar lane and road edge detection method in the embodiment;

[0053] Figure 2 A flowchart of the point cloud data processing process based on the acquired point cloud data in the embodiment;

[0054] Figure 3 Reference schematic diagram of the point cloud distance block algorithm in the embodiment;

[0055] Figure 4 , Figure 5 A reference schematic diagram of a point cloud angle correction algorithm in an embodiment;

[0056] Figure 6 It is a point cloud probability-lateral distance statistical diagram in the embodiment;

[0057] Figure 7 The point cloud probability-lateral distance statistical graph in the embodiment is after passing through a Gaussian low-pass filter;

[0058] Figure 8 Schematic diagram of the positions of the lane center, edge, and road edge coordinates on the statistical graph in the embodiment. DETAILED DESCRIPTION

[0059] The following is a further detailed description of a detection method for obtaining lanes and road edges using millimeter wave radar proposed by the present invention in conjunction with the accompanying drawings and specific embodiments.

[0060] The detection method provided by the present invention is mainly divided into two aspects: radar information collection and radar information processing.

[0061] Radar Information Collection Reference Figure 1 Schematic diagram of radar installation location. The radar is installed above the road, covering the entire road outside the blind spot. The normal direction of the radar is not necessarily parallel to the road direction.

[0062] It should be understood that the collection of target detection points collected by the radar is called a point cloud collection, which includes multiple detection points, and each detection point includes the following information: distance, azimuth, speed, signal-to-noise ratio and frame number.

[0063] Radar information processing is lane line and road edge detection. The detection process is as follows: Figure 2 , including: point cloud set preprocessing, lane center detection, lane edge detection, and road edge detection.

[0064] Preprocessing includes distance segmentation and angle correction after segmentation. Distance segmentation can cut the road to better adapt to the impact of curves on the road. Angle correction after segmentation can correct the angle between the radar normal direction and the road direction, which is convenient for more accurate calculation of lanes and road edges in the future. Figure 4 and Figure 5 .

[0065] Lane and road edge detection includes lane center detection, lane edge detection, and road edge detection. The principle is to use the statistical characteristics of point cloud sets for detection. Most vehicles on the road will drive in the center of the lane, which makes the intensity of radar point cloud characteristics appearing in the center of the road much greater than the intensity appearing at the edge of the lane, which gives a possibility to calculate the lane edge and road edge.

[0066] Lane and road edge detection, firstly, the point cloud set after segmentation is divided according to the horizontal coordinates and the signal-to-noise ratio is used as the weight to perform strength statistics, and the distribution characteristics of the horizontal coordinates of each detection point in the point cloud set can be obtained. Due to noise and the angle measurement error of the radar, the statistical point cloud distribution characteristics often fluctuate. Figure 6 . Applying Gaussian filtering to the point cloud distribution characteristics can filter out some noise and make the distribution characteristic curve smooth. Figure 7 The distribution characteristics of the processed point cloud have obvious peaks and troughs, and the lane edge and road edge can be calculated and fitted through the coordinate information of the peaks and troughs.

[0067] Next, the specific steps of radar information processing are explained in detail.

[0068] The block division according to distance includes: let the point cloud set be P, where each detection point in the point cloud set P is a (1×5) matrix, which is [rθv snr t], where r is the distance of the detection point, θ is the horizontal angle of the detection point, v is the speed of the detection point, snr is the signal-to-noise ratio of the detection point, and t is the frame number of the detection point.

[0069] Assume that the block distance is Δl and the road detection length is L, the road can be divided into M blocks. Figure 3 ,in:

[0070] M=cell(L / Δl)

[0071] cell(*) is a rounding function, and the distance range corresponding to the mth block is [(m-1)Δl, (m+1)Δl].

[0072] The point cloud set after block division is expressed as P = {P1, P2, ..., P M}.

[0073] For the angle correction of the detection points in each block after block division, suppose the input point cloud set p of the mth block after block division, where m∈{1,2,3,…,M}, p=P m , output point cloud set q:

[0074] p={p1,p2,…,p n}, q={q1,q2,…,q n}, where n is the number of detection points in the block after segmentation, that is, the number of elements in the input point cloud set p.

[0075] Requires a translation vector e (1×5) , so that: q i =p i +e, where i is the i-th detection point in set p, q i The i-th element in the corresponding output point cloud set q.

[0076] The input point cloud set p is rotated and translated to obtain the output point cloud set q. Since the radar normal direction is not parallel to the road direction, it is necessary to correct the angle difference between the radar normal direction and the road direction by rotation and translation so that the corrected output point cloud set p is parallel to the road direction.

[0077] By defining the rotation angle required for the input point cloud set p as Δθ and the rotation angle correction coefficient δ, the rotation angle Δθ is obtained by taking the center of gravity of the azimuth of each detection point in the input point cloud set p, and the correction coefficient δ is calculated based on the center of gravity of the number of detection points on both sides of the rotation angle. Specifically:

[0078] Where i is the i-th element in set p, i∈{1,2,3,…,n}, snr i is the signal-to-noise ratio of the i-th element in set p, θ i The horizontal angle of the i-th element in the set p.

[0079] δ=∑snr i / ∑snr j i,j satisfies the condition θ i <Δθ,θ j >Δθ.

[0080] Among them, θ i is the detection point p i The corresponding azimuth, snr i Detection point p i The corresponding signal-to-noise ratio, θ j is the detection point p j The corresponding azimuth, snr j Detection point p j The corresponding signal-to-noise ratio, i, j∈{1,2,3,…,n}, p i With p j are the i-th and j-th elements in the point cloud set p respectively.

[0081] The translation vector e is expressed as: e = [0 δ·Δθ0 0 0].

[0082] Angle correction processing effect reference Figure 4 After point cloud preprocessing calculation, the block point cloud set P = {P1, P2, ..., P M} to obtain the processed point cloud set Q = {Q1, Q2, ..., Q M}, where Q m =q, m is the mth block of point cloud after segmentation, m∈{1,2,3,…,M}, then for each distance block, the point cloud set q={q1,q2,…,q n} are processed separately to calculate the lane center position.

[0083] The statistical filtering process of the point cloud set includes: firstly, the detection points in the point cloud set q are converted from the polar coordinate system to the rectangular coordinate system, and the corresponding matrix q = [xyv snr t] is used, and the centroid sequence g1 of the number of detection points in each gradient is obtained with dx as the gradient and snr as the weight;

[0084] in, , l satisfies (k-0.5)·dx <x l <(k+0.5)·dx, the minimum increment range of k is in Indicates rounding down. Indicates rounding up, x min is the minimum x value among all the detection points, x max is the maximum x value among all detection points; l∈{1,2,3,…,n};

[0085] Due to the existence of noise, sequence g1 has noise, and a Gaussian low-pass filter is used to low-pass filter sequence g1 to obtain sequence g2. There are:

[0086] g2=g1·h, where h is the Gaussian low-pass filter sequence. Figure 6 and Figure 7 .

[0087] refer to Figure 6 , multiple spline interpolation is performed on sequence g2 to obtain sequence g. It can be considered that the number of peaks in sequence g is the number of lanes, and the peak position is the center position of the lane. The number of interpolation here is preferably 3 times.

[0088] Lane Center Calculation: Reference Figure 8 , the peak position of sequence g is the center position of the lane, and sequence g corresponds to u peaks. Let the set A corresponding to the coordinates of all lane centers = {A1, A2, A3, ..., A u}, where A f+1 is the position of the f+1th peak from left to right, and its coordinates are (x a(f+1) ,ya(f+1) ), f∈{1,2,3,…,u-1}.

[0089] Lane edge calculation: Lane edges are calculated using the coordinates of the trough and the peaks on both sides of the trough. Figure 8 , the trough corresponding set B = {B1, B2, B3, ..., B u-1}, where B f is the position of the fth trough from left to right, and its coordinates are (x bf ,y bf ), f∈[1,2,3,…,u-1]. The calculated lane edge set C={C1,C2,C3,…,C u-1}, C f is the edge position of the lane between the fth lanes from left to right, and its coordinates are (x cf ,y cf ).have:

[0090]

[0091] Get the horizontal coordinate x of the edge of the fth lane cf , the y coordinate of the lane edge cf is the distance mΔl from the center of each block in the block, that is, y cf =mΔl.

[0092] Road edge calculation: The road edge is calculated based on the coordinates of the center of two adjacent lanes and the coordinates of the lane edge between the lanes. That is, the first (left) side edge of the road D1 = (x d1 ,g d1 ) is calculated by the positions of A1 and C1, and the second (right) edge of the road D2 = (x d2 ,g d2 )Through A u ,C u-1 Position calculation. There are:

[0093] x d1 =(2x a1 -x c1 )+ε(x a1 -x c1 ) a1 / g c1 ;

[0094] x d2 =(2x au -x c(u-1) )+ε(x au -x c(u-1) ) au / g c(u-1) ;

[0095] Where ε is the correction coefficient. The horizontal coordinate x of the road edge is obtained d1 、x d2 , the ordinate y of the road edge d1 and d2 is the distance mΔl from the center of each block in the block, that is, y d1 =y d2 =mΔl.

[0096] Lane edge and road edge coordinates for angle compensation: After the above calculation, the lane line matrix can be obtained and the road edge matrix

[0097] Among them, x c =[x c1 ,x c2 ,…,x c(u-1) ],y c =[y c1 ,y c2, …,y c(u-1) ], x d =[x d1 ,x d2 ],y d =[y d1 ,y d2 ].

[0098] Using the translation vector e (1×5) The lane edge and road edge coordinates are position compensated, and the lane edge matrix H after compensation is c and the road edge matrix H d , there are: H c =RE c ,H d =RE d .

[0099] Among them, the rotation matrix R is:

[0100] Lane and road boundary data merging: After calculating the point cloud set of all blocks, the coordinate sequence of the lane edge and road edge relative to the radar position can be obtained.

[0101] In summary, the road-tested millimeter-wave radar can collect enough road point cloud information to accurately detect the contour sequence of lanes and road edges, and can provide more road information for subsequent processing.

[0102] 1. The present invention only needs the radar point cloud data for statistical fitting, and does not need to use camera data, and can be realized with only a single sensor data;

[0103] 2. Compared with using some clustering algorithms to obtain lane edges, the present invention has a small amount of calculation and can be directly deployed on the road test radar end for calculation;

[0104] 3. Currently, the calibration of the lanes of the road where the multi-target tracking radar is installed is mostly done manually, and the following information needs to be calibrated: the number of left lanes, the number of right lanes, the lane width, the lateral position of the left vehicle, and the lateral position of the right lane. It will be calibrated during installation and testing, and it is difficult to adapt to the problem of curves. The method of the present invention can be automatically calibrated after installation, saving debugging time during installation and reducing costs.

[0105] Although the content of the present invention has been described in detail through the above preferred embodiments, it should be appreciated that the above description should not be considered as a limitation of the present invention. After reading the above content, it will be apparent to those skilled in the art that various modifications and substitutions of the present invention will occur. Therefore, the protection scope of the present invention should be limited by the appended claims.

Claims

1. A method for detecting lanes and road edges using millimeter wave radar, characterized in that: The following steps are involved: S1. Obtain the detection points of each mobile vehicle, and gather all the obtained detection points into a point cloud set; S2, preprocessing the acquired point cloud set, including segmentation according to distance and angle correction; S3, performing statistical filtering on the preprocessed point cloud set to obtain a smooth probability-lateral distance sequence; S4. Calculate the peak value and the trough value of the probability-lateral distance sequence to obtain the lane edge coordinates and the road edge coordinates; S5. Angle compensation is performed on the lane edge coordinates and the road edge coordinates to obtain lane edge position information and road edge position information relative to the radar.

2. The detection method according to claim 1, characterized in that The information contained in the detection point includes: distance, azimuth, speed, signal-to-noise ratio and frame number; The content of distance segmentation in step S2 includes: Divide the road of length L into blocks according to the interval Δl, and treat the last section less than the interval Δl as one block; The number of blocks M is expressed as: M = cell (L / Δl), cell (*) is an upward rounding function, and the point cloud set P after block division is expressed as: P = {P1, P2, ..., P M }.

3. The detection method according to claim 2, characterized in that The content of angle correction of the point cloud data after segmentation in step S2 includes: Define an input point cloud set p and an output point cloud set q, where p is any subset of the point cloud set P, i.e., p = P m , m∈{1,2,3,…,M}; Where p = {p1, p2, ..., p n }, q={q1, q2, …, q n }, n is any subset P of the point cloud set P after segmentation m The number of detection points contained in ; Calculate a translation vector e (1×5) , so that q i =p i +e, where p i is the i-th detection point in the input point cloud set p, p i is the i-th detection point in the output point cloud set p; Let e ​​= [0, δ·Δθ, 0, 0, 0], Δθ is the rotation angle, δ is the rotation angle correction coefficient, specifically: δ=∑snr i / Σsnr j ,i、j meet the conditions for the i <Δθ,θ j >Dth; Among them, θ i is the detection point p i The corresponding azimuth, snr i Detection point p i The corresponding signal-to-noise ratio, θ j is the detection point p j The corresponding azimuth, snr j Detection point p j The corresponding signal-to-noise ratio, i, j∈{1,2,3,…,n}.

4. The detection method according to claim 3, characterized in that The statistical filtering process of step S3 includes: The detection points in the output point cloud set q are converted from the polar coordinate system to the rectangular coordinate system. The corresponding matrix expression is: q = [xyv snr t], with dx as the gradient and snr as the weight to obtain the centroid sequence g1 of the number of detection points in each gradient; Among them, g 1(kdx) =∑snr l , l satisfies (k-0.5)·dx <x l <(k+0.5)·dx, the minimum increment range of k is in Indicates rounding down. Indicates rounding up, x min is the minimum x value among all the detection points, x max is the maximum x value among all the detection points; The centroid probability sequence g1 is low-pass filtered using a Gaussian low-pass filter to obtain a sequence g2, g2=g1·h, where h is the Gaussian low-pass filter sequence.

5. The detection method according to claim 4, characterized in that: The step S4 comprises: Lane center calculation: Perform multiple spline interpolations on sequence g2 to obtain sequence g. The number of peaks in sequence g is the number of lanes, and the peak position is the center position of the lane. Let the set of all lane center coordinates corresponding to A = {A1, A2, A3, ..., A u }, the coordinates of the position of the f+1th peak from left to right are (x a(f+1) ,y a(f+1) ), f∈{1,2,3,…,u-1}; Lane Edge Computing: Let the trough coordinates correspond to the set B = {B1, B2, B3, ..., B u-1 }, the coordinates of the position of the fth trough from left to right are (x bf ,y bf ), lane edge set C = {C1, C2, C3, ..., C u-1 }, the coordinates of the lane edge position between the fth lanes from left to right are (x cf ,y cf ),but: and cf =mΔl; Road Edge Computing: Let the first side road edge coordinate D1 = (x d1 ,y d1 ), the second side road edge D2 = (x d2 ,y d2 ),but: x d1 =(2x a1 -x c1 )+ε(x a1 -x c1 )g a1 / g c1 ; x d2 =(2x au -x c(u-1) )+ε(x au -x c(u-1) )g au / g c(u-1) ; Among them, ε is the correction coefficient, x a1 , g a1 are the horizontal and vertical coordinates of the position of the first wave peak from left to right, respectively, au , g au are the horizontal and vertical coordinates of the position of the u-th peak from left to right, respectively, c1 , g c1 are the horizontal and vertical coordinates of the first trough position from left to right, respectively. c(u-1) , g c(u-1) are the horizontal and vertical coordinates of the position of the u-1th trough from left to right; The y coordinate of the road edge d1 and d2 is the center distance of each block mΔl; and d1 =and d2 =mΔl。 6. The detection method according to claim 5, characterized in that The step S5 comprises: Lane edge coordinates and road edge coordinates for angle compensation, Let the lane line matrix and the road edge matrix in, x c =[x c1 ,x c2 ,…,x c(u-1) ],and c =[and c1 ,and c2 ,…,and c(u-1) ],x d =[x d1 ,x d2 ],and d =[and d1 ,and d2 ]; The translation vector e = [0, δ·Δθ, 0, 0, 0] is used to perform angle compensation on the lane edge and road edge coordinates. The lane edge matrix H after compensation is c and the road edge matrix H d , there is the following relationship: H c =RE c , H d =RE d , where the rotation matrix R is:

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