A control method for suppressing the side slip and hysteresis of a battery swapping robot

Abstract

The present invention discloses a control method for suppressing the sideslip and hysteresis of a battery swapping robot, comprising the steps of: 1. Establishing a kinematic model of the battery swapping robot, estimating the equivalent sideslip angle based on the pose information at the previous moment and the current moment, and compensating the desired steering angle; 2. Estimating the pose information at the future moment according to the speed and pose information of the battery swapping robot at the current moment, correcting the pose information at the current moment, calculating the new lateral error and heading error, and optimizing the desired steering angle calculated by the pure pursuit algorithm; 3. Simultaneously inputting the optimized desired steering angle and the compensated sideslip angle into the steering controller to mitigate the errors caused by the hysteresis and sideslip of the control system of the battery swapping robot. This method is applicable to the robot path tracking process with hysteresis and sideslip phenomena, can overcome the reduction in path tracking accuracy and safety caused by sideslip interference, and solve the problem of the reduction in path tracking stability caused by system hysteresis.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot path tracking control, and specifically relates to a control method for suppressing the side slip and hysteresis of a battery swapping robot. Background Art

[0002] As an important direction for the transformation and upgrading of the automotive industry, electric vehicles have received great attention from countries around the world. In recent years, they have developed rapidly and many countries have put the full promotion of electric vehicles on the agenda. Although electric vehicles are a major option to replace fuel vehicles and can reduce the negative impact on the environment and dependence on petroleum energy, there are still the following problems in their large-scale promotion: 1) The energy density of power batteries is limited, and the driving range of vehicles is short; 2) Power batteries are expensive, and the vehicle purchase cost is high; 3) The battery charging speed is slow; 4) The charging process has a great impact on the power grid load. In order to solve the bottleneck faced by the large-scale promotion of electric vehicles on the basis of existing technologies, the electric vehicle battery swapping operation mode has emerged. In the battery swapping mode, the battery replacement process depends on a series of battery swapping automation devices to complete. Among them, the research on the planning and control of battery swapping robots is a current hot topic.

[0003] Path tracking is a key technology to enable the battery swapping robot to carry the battery to the target point according to the planned path. However, the battery swapping robot has the characteristics of large volume, mass, inertia and load, and the control system and actuator have obvious hysteresis and nonlinearity, making it difficult to operate. At the same time, due to the working environment, side slip is likely to occur, affecting the accuracy and stability of its automatic operation. Existing research on the hysteresis and nonlinearity of the steering system and system model requires a relatively accurate mathematical model of the system, and needs to re-model for different steering mechanisms, with poor generality, difficult parameter adjustment, large algorithm calculation amount and high complexity, which makes it difficult and insufficient to specifically implement on intelligent battery swapping robots; while existing research on side slip compensation is mostly based on dynamic models, and dynamic models have the disadvantages of complex mathematical models, many parameters and large calculation amounts. Therefore, two key factors need to be considered during the path tracking of the battery swapping robot: overcoming the problem of decreased path tracking accuracy caused by side slip and suppressing the decrease in path tracking stability caused by the hysteresis of the system. Summary of the Invention

[0004] To solve the above technical problems, the present invention proposes a control method for suppressing the side slip and hysteresis of a battery swapping robot. This method overcomes the negative impact of path tracking caused by vehicle side slip and suppresses the problem of decreased path tracking stability caused by the hysteresis of the system, and can significantly improve the path tracking effect of the battery swapping robot.

[0005] To achieve the above object, the technical solution of the present invention is as follows:

[0006] A control method for suppressing the sideslip and hysteresis of a battery swapping robot, comprising the following steps:

[0007] (1) Ignoring the action of the lateral force, regarding the tire as a rigid wheel, and establishing a two-wheeled vehicle model;

[0008] (2) Calculating the equivalent sideslip angle:

[0009]

[0010] Among them, the front-wheel sideslip angle θ sf is defined as the included angle between the actual speed direction θ h and the sideslip direction θ s , with the unit of rad; the rear-wheel sideslip angle θ sr is defined as the included angle between the actual speed direction θ h and the theoretical sideslip direction θ h-theory , with the unit of rad; θ se is the equivalent sideslip angle, with the unit of rad;

[0011] (3) Predicting the pose information at the future moment: The battery swapping robot receives the pose information at point m, calculates the desired steering angle, and only executes the command when it travels to point n. Calculate the lateral deviation and heading deviation at point n to optimize the desired steering angle calculated by the pure pursuit algorithm;

[0012] (4) Calculating the optimized desired steering angle and the compensated sideslip angle to correct the pose information at the current moment, thereby overcoming the problem of sideslip generated during the path tracking process and correcting the hysteresis and nonlinear problems of the battery swapping robot.

[0013] As a further improvement of the present invention, in step (1), in the navigation coordinate system X - O - Y, (x, y) is the front-wheel coordinate point, with the unit of m; v is the speed of the battery swapping robot, with the unit of m·s -1 ; L wb is the wheelbase of the front and rear axles of the battery swapping robot, with the unit of m; θ h is the speed direction of the battery swapping robot, that is, the heading angle, with the unit of rad; θ ws is the steering angle of the rear wheel of the battery swapping robot, with the unit of rad. From the kinematic constraints and the relationship in the figure, the kinematic model of the battery swapping robot is obtained:

[0014]

[0015] As a further improvement of the present invention, the calculation of the equivalent sideslip angle θ se in step (2) includes the following steps:

[0016] (2 - 1) From the front-wheel sideslip angle θ sf and the rear-wheel sideslip angle θ srThe definition gives:

[0017]

[0018] In the formula, μ s is the sideslip correction coefficient, dimensionless; θ h-theory is the theoretical velocity direction, in rad; θ s is the sideslip direction, in rad;

[0019] (2-2) The theoretical velocity direction is calculated based on the pose information of the previous moment. Assuming that the velocity and the rear-wheel steering angle do not change within the control period T, the theoretical velocity direction is:

[0020]

[0021] In the formula, θ h-prev is the actual velocity direction of the previous moment, in rad;

[0022] (2-3) The sideslip direction uses the inertial component IMU to obtain the acceleration of the battery swapping robot at the current moment, and combines with the pose information to get:

[0023]

[0024] In the formula, a x , a y is the acceleration value of the battery swapping robot in the navigation coordinate system, in m·s -2 ;

[0025] (2-4) The sideslip correction coefficient measures the severity of the sideslip. By estimating the theoretical pose information at the current moment through the actual pose information of the previous moment and comparing it with the actual pose information at the current moment. Assuming that the velocity and the heading angle do not change within the control period T, the theoretical position at the current moment is:

[0026]

[0027] In the formula, x theory , y theory is the theoretical position at the current moment, x prev , y prev is the actual position of the previous moment. The severity of the sideslip is expressed as:

[0028]

[0029] (2-5) The global satellite positioning system using the real-time kinematic carrier phase differential technology is used to obtain the pose information. Considering the influence of the positioning error, the relationship between μ s and d diff is expressed as:

[0030]

[0031] wherein, d th1 , d th2 are the judgment thresholds of the severity of sideslip, with the unit of m;

[0032] (2-6) According to the kinematic model of the battery swapping robot, the influence of the actual steering angle of the battery swapping robot on the steering radius by the equivalent sideslip angle, the front wheel sideslip angle, and the rear wheel sideslip angle satisfies the relationship:

[0033] r e = r r + r f

[0034] wherein, r e is the steering radius affected by the equivalent sideslip angle, r r is the steering radius affected by the rear wheel sideslip angle, r f is the steering radius affected by the front wheel sideslip angle, and the unit of all is m. According to the kinematic model, there is:

[0035]

[0036] After Taylor expanding the above formula and ignoring the high-order terms, that is, the relationship between the equivalent sideslip angle and the front and rear wheel sideslip angles:

[0037]

[0038] As a further improvement of the present invention, in step (3), a correction method based on the prediction idea is adopted. By the length of the speed estimation delay time, the pose information at the future moment is predicted to correct the pose information at the current moment;

[0039] wherein l mn is the arc length of the battery swapping robot running from point m to point n, with the unit of m; (x m , y m ), (x n , y n ) are the coordinates of point m and point n in the navigation coordinate system X-O-Y; d mn is the distance between point m and point n, with the unit of m; Δθ c is the change amount of the heading angle during the driving time, with the unit of rad;

[0040] (3-1) Let the delay time of the battery swapping robot running from point m to point n be t d , t dIt consists of fixed time and dynamic time. The fixed time is mainly related to the algorithm complexity of the battery swapping robot controller, the mechanical structure of the battery swapping robot itself, the steering control mechanism, etc.; the dynamic time is generally related to the working environment of the battery swapping robot. In the pure pursuit algorithm based on the kinematic model, changes in the ground state, changes in the load of the battery swapping robot, etc. will ultimately be reflected in the change of speed. Adaptive adjustment of the delay time according to speed and acceleration can better improve the adaptability of the battery swapping robot. If a linear relationship is adopted, the delay time t d and its relationship with the speed v and acceleration a of the battery swapping robot is:

[0041] t d = μ v v + μ a a + t ft

[0042] In the formula, μ v is the speed correction coefficient, with the unit of s 2 ·m, μ a is the acceleration correction coefficient, with the unit of s 3 ·m, t ft is the fixed time, with the unit of s;

[0043] (3 - 2) After estimating the time, predict the position of point n. Assuming that the speed and steering angle do not change during this period, the change in the heading angle traveled during this period is:

[0044]

[0045] In the formula, l mn is the arc length of the battery swapping robot running from point m to point n, with the unit of m;

[0046] (3 - 3) The distance between point m and point n is expressed as:

[0047]

[0048] (3 - 4) Therefore, in the vehicle body coordinate system, the position relationship of point n relative to point m is expressed as:

[0049]

[0050] In the navigation coordinate system, the position relationship of point n relative to point m is expressed as:

[0051]

[0052] Among them is the coordinate of point n in the vehicle body coordinate system.

[0053] As a further improvement of the present invention, in step (4), the pose information at the current moment is corrected by the optimized desired steering angle and the compensated sideslip angle, specifically as follows:

[0054] After predicting the position of point n in step (3), the new lateral deviation and heading deviation are calculated using point n and substituted into the following formula to optimize the desired steering angle calculated by the pure pursuit algorithm:

[0055]

[0056] In the formula, d XTE is the lateral deviation of the battery swapping robot, which is defined as the distance between the current position of the battery swapping robot and the desired path. Its sign is defined as positive when leading to the right side of the desired path and negative otherwise, with the unit of m; θ e is the heading deviation of the battery swapping robot, which is defined as the angle between the current speed direction of the battery swapping robot and the desired path direction. Its sign is defined as positive when rotating counterclockwise and negative otherwise, with the unit of rad.

[0057] The optimized desired steering angle and the equivalent sideslip angle calculated in step (2) are combined and input into the steering controller of the battery swapping robot, that is, to overcome the negative impact of sideslip on path tracking and suppress the decline in path tracking stability caused by the hysteresis of the system.

[0058] Beneficial effects: The present invention discloses a control method for suppressing the sideslip and hysteresis of a battery swapping robot. When kinematic modeling, the concept of sideslip angle is introduced. The controller estimates the theoretical position at the current moment based on the actual position of the battery swapping robot at the previous moment, combines the actual position at the current moment and the IMU to estimate the equivalent sideslip angle, and compensates the desired steering angle calculated by the pure pursuit algorithm to overcome the problem of reduced tracking accuracy caused by sideslip; for the nonlinearity and hysteresis problems of the control system, the controller will estimate the pose information at the future moment according to the speed and pose information of the battery swapping robot at the current moment, correct the pose information at the current moment, calculate the new lateral error and heading error, and optimize the desired steering angle calculated by the pure pursuit algorithm; the optimized desired steering angle and the compensated sideslip angle are simultaneously input into the steering controller to alleviate the error caused by the hysteresis and sideslip of the control system of the battery swapping robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 is a flowchart of the method disclosed in the present invention;

[0060] Figure 2 is the two-wheeled vehicle model of the battery swapping robot in the method disclosed in the present invention;

[0061] Figure 3 is a schematic diagram of the hysteresis of the battery swapping robot in the method disclosed in the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0062] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments:

[0063] The present invention discloses a control method for suppressing the side slip and hysteresis of a battery swapping robot. The specific flowchart is as Figure 1 shown, including the following steps:

[0064] Step 1: Ignore the action of the lateral force, regard the tire as a rigid wheel, and establish a simplified two-wheeled vehicle model as Figure 2 shown. In the navigation coordinate system X-O-Y, (x, y) is the coordinate point of the front wheel, with the unit of m; v is the speed of the battery swapping robot, with the unit of m·s -1 ; L wb is the wheelbase of the front and rear axles of the battery swapping robot, with the unit of m; θ h is the speed direction of the battery swapping robot, that is, the heading angle, with the unit of rad; θ ws is the steering angle of the rear wheel of the battery swapping robot, with the unit of rad. From the kinematic constraints and the relationships in the figure, the kinematic model of the battery swapping robot can be obtained:

[0065]

[0066] Step 2: Calculate the equivalent side slip angle. It includes the following steps:

[0067] (2-1) From the definitions of the front wheel side slip angle θ sf and the rear wheel side slip angle θ sr , we can get:

[0068]

[0069] In the formula, μ s is the side slip correction coefficient, dimensionless; θ h-theory is the theoretical speed direction, with the unit of rad; θ s is the side slip direction, with the unit of rad.

[0070] (2-2) The theoretical speed direction can be calculated based on the pose information of the previous moment. Suppose that within the control period T, the speed and the rear wheel steering angle do not change, then the theoretical speed direction is:

[0071]

[0072] In the formula, θ h-prev is the actual speed direction of the previous moment, with the unit of rad.

[0073] (2-3) The side slip direction uses inertial components (IMU) to obtain the acceleration of the battery swapping robot at the current moment, and combines the pose information to get:

[0074]

[0075] In the formula, a x , a y is the acceleration value of the battery swapping robot in the navigation coordinate system, with the unit of m·s -2 .

[0076] (2 - 4) The sideslip correction coefficient measures the severity of sideslip. The theoretical pose information at the current moment is estimated through the actual pose information at the previous moment and compared with the actual pose information at the current moment. Assuming that the speed and heading angle do not change within the control period T, the theoretical position at the current moment is:

[0077]

[0078] In the formula, x theory , y theory is the theoretical position at the current moment, x prev , y prev is the actual position at the previous moment. The severity of sideslip can be expressed as:

[0079]

[0080] (2 - 5) The global satellite positioning system (Real-time kinematic - Global Navigation Satellite System, RTK - GNSS) using real-time kinematic carrier phase differential technology is used to obtain pose information. Considering the influence of positioning errors, the relationship between μ s and d diff can be expressed as:

[0081]

[0082] In the formula, d th1 , d th2 are the judgment thresholds for the severity of sideslip, with the unit of m.

[0083] (2 - 6) According to the kinematic model of the battery swapping robot, the influence of the actual steering angle of the battery swapping robot on the turning radius by the equivalent sideslip angle, the front wheel sideslip angle, and the rear wheel sideslip angle satisfies the relationship:

[0084] r e = r r + r f

[0085] In the formula, r e is the turning radius affected by the equivalent sideslip angle, r r is the turning radius affected by the rear wheel sideslip angle, r fThe steering radius affected by the front wheel sideslip angle, with the unit of m for all. According to the kinematic model, there is:

[0086]

[0087] After Taylor expanding the above formula and neglecting the high-order terms, that is, the relationship between the equivalent sideslip angle and the front and rear wheel sideslip angles:

[0088]

[0089] Step 3: Predict the pose information at the future moment. Figure 3 It is a schematic diagram of the lag of the electric robot. It includes the following steps:

[0090] (3-1) Assume that the delay time for the battery swapping robot to travel from point m to point n is t d , generally t d is mainly composed of a fixed time and a dynamic time. The fixed time is mainly related to the algorithm complexity of the battery swapping robot controller, the mechanical structure of the battery swapping robot itself, the steering control mechanism, etc.; the dynamic time is generally related to the operating environment of the battery swapping robot. In the pure pursuit algorithm based on the kinematic model, changes in the ground state, changes in the load of the battery swapping robot, etc. will ultimately be reflected in the change of speed. Adjusting the delay time adaptively according to speed and acceleration can better improve the adaptability of the battery swapping robot. If a linear relationship is adopted, the delay time t d and the relationship with the speed v and acceleration a of the battery swapping robot is:

[0091] t d = μ v v + μ a a + t ft

[0092] In the formula, μ v is the speed correction coefficient, with the unit of s 2 ·m, μ a is the acceleration correction coefficient, with the unit of s 3 ·m, t ft is the fixed time, with the unit of s.

[0093] (3-2) After estimating the time, predict the position of point n. Assume that the speed and steering angle do not change during this period, then the change amount of the heading angle traveled during this period is:

[0094]

[0095] In the formula, l mn is the arc length of the battery swapping robot traveling from point m to point n, with the unit of m.

[0096] (3-3) The distance between point m and point n can be expressed as:

[0097]

[0098] (3-4) Therefore, in the vehicle body coordinate system, the positional relationship of point n relative to point m can be expressed as:

[0099]

[0100] In the navigation coordinate system, the positional relationship of point n relative to point m can be expressed as:

[0101]

[0102] Where is the coordinate of point n in the vehicle body coordinate system.

[0103] Step 4: Calculate the optimized desired steering angle and the compensated sideslip angle to correct the pose information at the current moment. After predicting the position of point n in step (3), use point n to calculate the new lateral deviation and heading deviation, and substitute them into the following formula to optimize the desired steering angle calculated by the pure pursuit algorithm:

[0104]

[0105] In the formula, d XTE is the lateral deviation of the battery swapping robot, which is defined as the distance between the current position of the battery swapping robot and the desired path. Its sign is defined as positive when leading to the right side of the desired path, and negative otherwise, with the unit of m; θ e is the heading deviation of the battery swapping robot, which is defined as the angle between the current speed direction of the battery swapping robot and the desired path direction. Its sign is defined as positive when rotating counterclockwise, and negative otherwise, with the unit of rad.

[0106] Combining the optimized desired steering angle and the equivalent sideslip angle calculated in step (2) and inputting them into the steering controller of the battery swapping robot can overcome the negative impact of sideslip on path tracking and suppress the decline in path tracking stability caused by the system lag.

[0107] The above is only a preferred embodiment of the present invention, and it is not any other form of limitation to the present invention. Any modification or equivalent change made according to the technical essence of the present invention still belongs to the scope protected by the present invention.

Claims

1. A control method for suppressing the side slip and hysteresis of a battery swapping robot, characterized in that, It includes the following steps: (1) Ignoring the action of lateral force, regarding the tire as a rigid wheel, and establishing a two-wheeled vehicle model; (2) Calculating the equivalent sideslip angle: ； Among them, the front wheel sideslip angle is defined as the angle between the actual speed direction and the sideslip direction , with the unit of ; The rear wheel sideslip angle is defined as the angle between the actual speed direction and the theoretical sideslip direction , with the unit of ; is the equivalent sideslip angle, with the unit of ; (3) Predicting the pose information at a future moment: The battery swapping robot receives the pose information at point m, calculates the desired steering angle, and only finishes executing the command when it travels to point n. Calculate the lateral deviation and heading deviation at point n to optimize the desired steering angle calculated by the pure pursuit algorithm; (4) Calculating the optimized desired steering angle and the compensated sideslip angle to correct the pose information at the current moment, thereby overcoming the problem of sideslip generated during the path tracking process and correcting the hysteresis and nonlinear problems of the battery swapping robot; In step (4), the pose information at the current moment is corrected by the optimized desired steering angle and the compensated sideslip angle, specifically as follows: After predicting the position of point n in step (3), use point n to calculate the new lateral deviation and heading deviation, and substitute them into the following formula to optimize the desired steering angle calculated by the pure pursuit algorithm: ； In the formula, is the lateral deviation of the battery swapping robot, which is defined as the distance between the current position of the battery swapping robot and the desired path. Its sign is defined as positive when leading to the right side of the desired path and negative otherwise, with the unit of ; is the heading deviation of the battery swapping robot, which is defined as the angle between the current speed direction of the battery swapping robot and the desired path direction. Its sign is defined as positive when rotating counterclockwise and negative otherwise, with the unit of .

2. A control method for suppressing the sideslip and hysteresis of a battery swapping robot according to claim 1, characterized in that: The step (1) is in the navigation coordinate system where is the coordinate point of the front wheel, with the unit of ; is the speed of the battery swapping robot, with the unit of ; is the wheelbase of the front and rear axles of the battery swapping robot, with the unit of ; is the speed direction of the battery swapping robot, i.e., the heading angle, with the unit of ; is the steering angle of the rear wheel of the battery swapping robot, with the unit of . The kinematic model of the battery swapping robot is obtained from kinematic constraints: 。 3. A control method for suppressing the sideslip and hysteresis of a battery swapping robot according to claim 2, characterized in that: The calculation of the equivalent sideslip angle in step (2) includes the following steps: (2-1) From the definitions of the front wheel sideslip angle and the rear wheel sideslip angle it follows that: ； In the formula, is the sideslip correction coefficient, dimensionless; is the theoretical velocity direction, with the unit of ; is the sideslip direction, with the unit of ; The theoretical velocity direction is calculated based on the pose information of the previous moment. Suppose that within the control period neither the velocity nor the rear wheel steering angle changes, then the theoretical velocity direction is: ； In the formula, is the actual velocity direction at the previous moment, with the unit of ; (2-3) In the sideslip direction, use the inertial component IMU to obtain the acceleration of the battery swapping robot at the current moment, and combine the pose information to obtain: ； Wherein, is the acceleration value of the battery swapping robot in the navigation coordinate system, with the unit of ; The (2-4) sideslip correction coefficient measures the severity of sideslip. It is obtained by estimating the theoretical pose information at the current moment through the actual pose information at the previous moment and comparing it with the actual pose information at the current moment. Suppose that within the control cycle the speed and heading angle do not change, then the theoretical position at the current moment is: ； In the formula, is the theoretical position at the current moment, is the actual position at the previous moment, and the severity of sideslip is expressed as: ； (2-5) The global satellite positioning system using real-time kinematic carrier phase differential technology is used to obtain pose information. Considering the influence of positioning errors, and The relationship of is expressed as: ； In the formula, , are the judgment thresholds for the severity of sideslip, with the unit of ; (2-6) According to the kinematic model of the battery swapping robot, the influence of the actual steering angle of the battery swapping robot on the steering radius by the equivalent sideslip angle, the front wheel sideslip angle, and the rear wheel sideslip angle satisfies the relationship: ； In the formula, is the turning radius affected by the equivalent sideslip angle, is the turning radius affected by the rear-wheel sideslip angle, is the turning radius affected by the front-wheel sideslip angle, and the unit is all , according to the kinematic model, there is: ； After Taylor expanding the above formula and ignoring the high-order terms, that is, the relationship between the equivalent sideslip angle and the front and rear wheel sideslip angles: 。 4. A control method for suppressing the sideslip and hysteresis of a battery swapping robot according to claim 1, characterized in that: In step (3), a correction method based on the prediction idea is adopted. By estimating the length of the speed estimation delay time, predict the pose information at a future moment to correct the pose information at the current moment; Among them is the arc length of the battery swapping robot running from point m to point n, with the unit of ; and are the coordinates of point m and point n in the navigation coordinate system ; is the distance between point m and point n, with the unit of m; is the change in the heading angle during the driving time, with the unit of ; (3-1) Let the delay time for the battery swapping robot to travel from point m to point n be , which consists of a fixed time and a dynamic time. The fixed time is related to the algorithm complexity of the battery swapping robot controller, the mechanical structure of the battery swapping robot itself, and the steering control mechanism; the dynamic time is related to the operating environment of the battery swapping robot. In the pure pursuit algorithm based on the kinematic model, changes in the ground state and changes in the load of the battery swapping robot will ultimately be reflected in the change of speed. Adjusting the delay time adaptively according to speed and acceleration can better improve the adaptive ability of the battery swapping robot. If a linear relationship is adopted, the delay time and the speed of the battery swapping robot and acceleration are related as follows: ； wherein, is the speed correction coefficient, with the unit of , is the acceleration correction coefficient, with the unit of , is the solid time, with the unit of ; (3-2) After estimating the time, predict the position of point n. Assuming that the speed and steering angle do not change during this period, the change amount of the heading angle traveled during this period is: ； In the formula, is the arc length of the battery swapping robot running from point m to point n, with the unit of ; (3-3) The distance between point m and point n is expressed as: ； (3-4) Therefore, in the vehicle body coordinate system, the position relationship of point n relative to point m is expressed as: ； In the navigation coordinate system, the position relationship of point n relative to point m is expressed as: ； wherein is the coordinate of point n in the vehicle body coordinate system.

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