Real-time skill trajectory generation method based on dynamic motion primitive and artificial potential field method

Abstract

The invention discloses a real-time skill trajectory generation method based on a dynamic motion primitive and an artificial potential field method, and the method comprises the steps: 1, collecting a group of object trajectories as demonstration trajectories, and obtaining the trajectory information; 2, based on the track information, a dynamic motion primitive algorithm is adopted to learn a demonstration track; 3, according to the position of the target object and the original point position of the task object coordinate system at the current moment, generating the original point position of the task object coordinate system corresponding to the next moment in real time; and 4, carrying out collision detection on the original point position of the generated task object coordinate system at the next moment, if collision occurs, generating an obstacle avoidance gradient according to an artificial potential field method, and adjusting the next track point position until the collision does not occur, otherwise, returning to the step 3 until the task is completed. According to the method, the skill demonstration track can be simulated and learned, so that important track characteristics of the skill are coded, unnecessary mirroring capability is avoided, excessive amplification of the track is reduced, and the tail end track similarity is improved.

Description

Technical Field

[0001] The present invention belongs to the field of machine skill imitation learning, and relates to a real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method. Background Art

[0002] With the development of technology, robots have gradually been applied to the service industry. Traditional skills are mostly achieved through manual programming by humans, which is time-consuming and labor-intensive, and requires reprogramming when the scene and task objectives change. Therefore, robot imitation learning methods that learn skill strategies from human demonstrations have begun to emerge. The skills obtained based on imitation learning can adapt to different scenes and target states, and can adaptively generate the required skill trajectories to complete the execution of skill sequences.

[0003] Commonly used skill learning algorithms, such as the Dynamic Movement Primitives (DMP) algorithm, have good generalization capabilities. However, when the starting point and the target point in the demonstration trajectory are close, subsequent generalization may cause over-amplification problems, and DMP will cause trajectory mirroring problems and end trajectory similarity problems. Summary of the invention

[0004] In order to solve the above technical problems existing in the prior art, the present invention proposes a real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method, which avoids unnecessary mirroring capabilities, reduces excessive amplification of trajectories, and improves the similarity of terminal trajectories. The specific technical solution is as follows:

[0005] A real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method, comprising:

[0006] Step 1: Collect a set of object trajectories as demonstration trajectories;

[0007] Step 2: Use the dynamic motion primitive algorithm to learn the demonstration trajectory;

[0008] Step 3: Generate the origin position of the task object coordinate system corresponding to the next moment in real time based on the position of the target object and the origin position of the task object coordinate system at the current moment;

[0009] Step 4: Perform collision detection on the origin position of the generated task object coordinate system at the next moment. If a collision occurs, generate an obstacle avoidance gradient according to the artificial potential field method, and adjust the next trajectory point until there is no collision. Otherwise, return to step 3 until the task is completed.

[0010] Furthermore, in step 1, an object coordinate system is constructed based on the key points of the object, and the object coordinate system information at each moment is collected to form a trajectory to represent the skill. The trajectory is expressed in the robot base coordinate system. The expression of the demonstration trajectory is as follows:

[0011]

[0012] ξ i ={p i ,q i},

[0013]

[0014]

[0015]

[0016]

[0017] Where i represents the index of the trajectory point, T is the total number of trajectory points, and t i represents the time corresponding to the i-th trajectory point; ξ i ={p i ,q i} represents the pose trajectory point, including position information p = [x, y, z] T And posture information q = [q0, q1, q2, q3] T , x, y, z represent the three-dimensional coordinates corresponding to the origin of the object coordinate system, q0, q1, q2, q3 represent the unit quaternion corresponding to the rotation matrix of the object coordinate system; Represents the derivative of the pose trajectory point with respect to time, including the position derivative ω i represents the angular velocity, dt is the time interval between adjacent trajectory points, norm([q1,q2,q3] T ) represents the modulus of the vector [q1,q2,q3]; Represents the second-order derivative of the pose trajectory point with respect to time, including the second-order derivative of position Angular acceleration ξ goal ={p goal ,q goal} represents the pose data of the target object coordinate system, and its definition form is consistent with the pose trajectory point.

[0018] Furthermore, in step 2, weighted learning of the trajectory shape is performed by calculating the weight of the nonlinear term in the dynamic motion primitive algorithm formula.

[0019] Furthermore, the weight of the nonlinear term is calculated, and the specific expression is as follows:

[0020]

[0021]

[0022]

[0023]

[0024]

[0025] Where N represents the number of basis functions in the nonlinear term, ψ j (s) represents the jth basis function, w j is the weight corresponding to the basis function, s i is the phase corresponding to time i, τ is the total duration of the demonstration trajectory, c j represents the center of the Gaussian basis function, h j represents the width of the Gaussian basis function, and α is the speed adjustment coefficient;

[0026] Construct the following matrix:

[0027]

[0028]

[0029]

[0030] Where φ is the basis function matrix, F is the nonlinear term matrix, and W is the weight matrix, then the following formula is obtained:

[0031] φ*W=F,

[0032] In the above formula, the basis function matrix φ is directly determined by the prior parameter definition, and the nonlinear term matrix F is calculated by the demonstration trajectory data. The calculation formula is as follows:

[0033] The formula for the nonlinear term in the x-direction is as follows:

[0034]

[0035] where α x , β x is the second-order system parameter, x i is the x coordinate of the demonstration trajectory at time i, x goal is the x coordinate of the origin of the target coordinate system;

[0036] The formula for the nonlinear term in the y direction is as follows:

[0037]

[0038] where α y, β y is the second-order system parameter, y i is the y coordinate of the demonstration trajectory at time i, y goal is the y coordinate of the origin of the target coordinate system;

[0039] The formula for the nonlinear term in the z direction is as follows:

[0040]

[0041] z structure,i =z i -z0*f control,i -z goal *(1-f control,i ),

[0042]

[0043] where α z , β z is the second-order system parameter, z i is the z coordinate of the demonstration trajectory at time i, z0 is the z coordinate of the demonstration trajectory at the initial time, z goal is the z coordinate of the origin of the target coordinate system, f control,i is the control coefficient at time i;

[0044] After obtaining the F matrix, the approximate solution of the weight matrix W is further obtained by the pseudo-inverse method:

[0045] W=(φ T φ) -1 φ T F.

[0046] Furthermore, in step 3, the origin position of the task object coordinate system at the next moment is obtained by calculating the coordinate values ​​in the x, y, and z directions. The calculation expression of the value at the next moment in the x direction is:

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053] In the above formula, x newgoal is the x coordinate of the origin of the new target object coordinate system, x newstartis the x coordinate of the origin of the new starting object coordinate system, x goal , x0 are the target x coordinate and the starting x coordinate in the demonstration trajectory, t c is a manually selected moment, the subsequent trajectory will be close to the demonstration trajectory, k controlx (t) is the generated control parameter, is its derivative with respect to time, x new,i is the x coordinate of the origin of the current object coordinate system, is its derivative with respect to time, is the second-order derivative with respect to time;

[0054] To calculate the value of the y direction at the next moment, just replace the above x parameter with the parameter corresponding to the y direction;

[0055] The calculation expression of the value at the next moment in the z direction is:

[0056]

[0057]

[0058]

[0059]

[0060] z new,i+1 =z newstructure,i+1 +z newstart *f controlz,i +z newgoal *(1-f controlz,i ),

[0061]

[0062] where z newstructure,i , is the new z coordinate constructed at time i and the corresponding first-order and second-order time derivatives, z new,i+1 It is the z coordinate of the origin of the object coordinate system actually generated at time i+1.

[0063] Furthermore, the step 4 specifically includes:

[0064] Calculate the shortest distance D(o) between the object and the obstacle at the i+1th moment i+1 ), o i+1 Represents the object coordinate system information at the i+1th moment: o i+1 =[x new,i+1 ,y new,i+1 ,z new,i+1 ];

[0065] The collision detection is judged as D(oi+1 )<Q * When the obstacle avoidance gradient is calculated, the object coordinate system information o at the i+1th moment is updated. i+1 To adjust the position of the trajectory point, the object coordinate system information o at the i+1th moment is updated i+1 The expression is as follows:

[0066] o i+1 =o i+1 -F(o i+1 )*step,

[0067] Where step is compensation, Q * is the obstacle influence range, F(o i+1 ) is the obstacle avoidance gradient;

[0068] After the update is completed, the closest distance D(o i+1 );

[0069] The collision detection is judged as D(o i+1 )>Q * If , it ends and returns to step 3, otherwise, the obstacle avoidance gradient is recalculated according to the above formula and o is continued to be updated. i+1 .

[0070] Furthermore, the obstacle avoidance gradient is calculated by the artificial potential field method, and the expression is as follows:

[0071]

[0072]

[0073] F att (o i+1 )=-▽U att (o i+1 )=-ζ(o i+1 -o goal ),

[0074]

[0075]

[0076] Where ζ is the target attraction coefficient, ρ(o i+1 ,o goal ) is the modulus of the difference between two vectors, o goal is the origin coordinate of the target coordinate system, U att (o i+1 ) is the target attraction potential field, η is the obstacle potential field coefficient, U rep (o i+1 ) is the obstacle avoidance potential field, F att (oi+1 ) is U att (o i+1 ) relative to the gradient in the x, y, and z directions, F rep (o i+1 ) is U rep (o i+1 ) with respect to the gradient in the x, y, and z directions, ▽D(o i+1 ) is D(o i+1 ) with respect to the gradient in the x, y, and z directions.

[0077] The advantages and beneficial effects of the present invention are:

[0078] The method of the present invention can perform imitation learning on the skill demonstration trajectory to encode important trajectory features of the skill, avoid unnecessary mirroring capabilities, reduce excessive amplification of the trajectory, and improve the similarity of the terminal trajectory. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 It is a flow chart of the real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method of the present invention. DETAILED DESCRIPTION

[0080] In order to make the purpose, technical scheme and technical effect of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments.

[0081] like Figure 1 As shown, the real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method of the present invention comprises the following steps:

[0082] Step 1: Collect a set of object trajectories as demonstration trajectories. The information of the demonstration trajectory includes: the position, velocity, and acceleration of the origin of the object coordinate system at each moment, the unit quaternion corresponding to the rotation matrix of the object coordinate system at each moment, and the position and posture of the target object coordinate system.

[0083] Specifically, an object coordinate system is constructed based on the key points of the object, and the object coordinate system information at each moment is collected to form a trajectory to represent the skill. The trajectory is expressed in the robot base coordinate system. The expression of the skill demonstration trajectory expressed by time, position, velocity, acceleration, quaternion posture, angular velocity, and angular acceleration information is as follows:

[0084]

[0085] ξ i ={p i ,q i},

[0086]

[0087]

[0088]

[0089]

[0090] Where i represents the index of the trajectory point, T is the total number of trajectory points, and t i represents the time corresponding to the i-th trajectory point; ξ i ={p i ,q i} represents the pose trajectory point, including position information p = [x, y, z] T And posture information q = [q0, q1, q2, q3] T , x, y, z represent the three-dimensional coordinates corresponding to the origin of the object coordinate system, q0, q1, q2, q3 represent the unit quaternion corresponding to the rotation matrix of the object coordinate system; Represents the derivative of the pose trajectory point with respect to time, including the position derivative ω i represents the angular velocity, dt is the time interval between adjacent trajectory points, norm([q1,q2,q3] T ) represents the modulus of the vector [q1,q2,q3]; Represents the second-order derivative of the pose trajectory point with respect to time, including the second-order derivative of position Angular acceleration ξ goal ={p goal ,q goal} represents the pose data of the target object coordinate system, and its definition form is consistent with the pose trajectory point.

[0091] Step 2: Based on the trajectory information obtained above, the dynamic motion primitive algorithm is used to learn the demonstration trajectory.

[0092] Specifically, calculate the weight of the nonlinear term in the dynamic motion primitive formula:

[0093] The nonlinear term f at the i-th moment i The expression is as follows:

[0094]

[0095]

[0096]

[0097]

[0098]

[0099] Where N represents the number of basis functions in the nonlinear term, ψ j (s) represents the jth basis function, w j is the weight corresponding to the basis function, s i is the phase corresponding to time i, τ is the total duration of the demonstration trajectory, c j represents the center of the Gaussian basis function, h j represents the width of the Gaussian basis function, and α is the speed adjustment coefficient; in this embodiment, N is 350, τ is 661*0.01, and α is 48.

[0100] Construct the following matrix:

[0101]

[0102]

[0103]

[0104] Where φ is the basis function matrix, F is the nonlinear term matrix, and W is the weight matrix, then the following formula can be obtained:

[0105] φ*W=F,

[0106] In the above formula, the basis function matrix φ can be directly determined by the prior parameter definition, and the nonlinear term matrix F needs to be calculated through the demonstration trajectory data. The calculation formula is as follows:

[0107] The formula for the nonlinear term in the x-direction is as follows:

[0108]

[0109] where α x , β x is the second-order system parameter, x i is the x coordinate of the demonstration trajectory at time i, x goal is the x coordinate of the origin of the target coordinate system;

[0110] The formula for the nonlinear term in the y direction is as follows:

[0111]

[0112] where α y , β y is the second-order system parameter, y i is the y coordinate of the demonstration trajectory at time i, y goal is the y coordinate of the origin of the target coordinate system;

[0113] The formula for the nonlinear term in the z direction is as follows:

[0114]

[0115] z structure,i =z i -z0*f control,i -z goal *(1-f control,i ),

[0116]

[0117] where α z , β z is the second-order system parameter, z i is the z coordinate of the demonstration trajectory at time i, z0 is the z coordinate of the demonstration trajectory at the initial time, z goal is the z coordinate of the origin of the target coordinate system, f control,i is the control coefficient at time i.

[0118] In this embodiment, α is uniformly taken x =α y =α z =48,β x =β y =β z =12, after obtaining the F matrix, we can further obtain the W approximate solution through the pseudo-inverse method:

[0119] W=(φ T φ) -1 φ T F.

[0120] Step 3: Generate the origin position of the task object coordinate system corresponding to the next moment in real time based on the position of the target object and the origin position of the task object coordinate system at the current moment.

[0121] The generation methods of the x direction and the y direction are similar. Taking x as an example, the value of the x direction at the next moment is:

[0122]

[0123]

[0124]

[0125]

[0126]

[0127]

[0128] where x newgoal is the x coordinate of the origin of the new target object coordinate system, x newstart is the x coordinate of the origin of the new starting object coordinate system, x goal, x0 are the target x coordinate and the starting x coordinate in the demonstration trajectory, t c is a manually selected moment, the subsequent trajectory will be close to the demonstration trajectory, k controlx (t) is the generated control parameter, is its derivative with respect to time, x new,i is the x coordinate of the origin of the current object coordinate system, is its derivative with respect to time, is the second-order derivative with respect to time, and other parameters not mentioned are the same as before.

[0129] For the y direction, just replace the above x parameters with the parameters corresponding to the y direction.

[0130] The next moment value in the z direction:

[0131]

[0132]

[0133]

[0134]

[0135] z new,i+1 =z newstructure,i+1 +z newstart *f controlz,i +z newgoal *(1-f controlz,i ),

[0136]

[0137] where z newstructure,i , is the new z coordinate constructed at time i and the corresponding first-order and second-order time derivatives, z new,i+1 It is the z coordinate of the origin of the object coordinate system actually generated at time i+1. Other parameters not mentioned are the same as before.

[0138] Step 4: Perform collision detection on the origin position of the generated task object coordinate system at the next moment. If a collision occurs, calculate the obstacle avoidance gradient according to the artificial potential field method and adjust the next trajectory point until there is no collision. Otherwise, return to step 3 until the task is completed.

[0139] After step 3 is completed, the object coordinate system x at the i+1th moment will be obtained. new,i+1 ,y new,i+1 ,z new,i+1 Coordinates, based on which the object coordinate system at time i+1 is generated, using o i+1 Represents the object coordinate system information at the i+1th moment: o i+1=[x new,i+1 ,y new,i+1 ,z new,i+1 ].

[0140] Specifically, calculate the i+1 time o i+1 In the object coordinate system, the shortest distance D(o i+1 ), if D(o i+1 )<Q * Then calculate the obstacle avoidance gradient, otherwise return to step 3 until the task is completed. The obstacle avoidance gradient is calculated as follows:

[0141]

[0142]

[0143] F att (o i+1 )=-▽U att (oi +1 )=-ζ(o i+1 -o goal ),

[0144]

[0145]

[0146] Where ζ is the target attraction coefficient, ρ(o i+1 ,o goal ) is the modulus of the difference between two vectors, o goal is the origin coordinate of the target coordinate system, U att (o i+1 ) is the target attraction potential field, Q * is the obstacle influence range, η is the obstacle potential field coefficient, U rep (o i+1 ) is the obstacle avoidance potential field, F att (o i+1 ) is U att (o i+1 ) relative to the gradient in the x, y, and z directions, F rep (o i+1 ) is U rep (o i+1 ) with respect to the gradient in the x, y, and z directions, ▽D(o i+1 ) is D(o i+1 ) with respect to the gradient in the x, y, and z directions.

[0147] Calculate the obstacle avoidance gradient F(o i+1 ) and then update the object coordinate system information o at the i+1th moment i+1 as follows:

[0148] o i+1 =o i+1 -F(o i+1 )*step,

[0149] Where step is the compensation, and after the update is completed, the closest distance D(o) between the current object and the obstacle is recalculated. i+1 );

[0150] If D(o i+1 )>Q * Then return to step 3, otherwise recalculate F(o i+1 ) and continue to update o i+1 .

[0151] The above is only a preferred implementation case of the present invention and does not limit the present invention in any form. Although the implementation process of the present invention is described in detail above, for those familiar with the art, they can still modify the technical solutions recorded in the above examples, or replace some of the technical features therein with equivalents. All modifications, equivalent replacements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A real-time skill trajectory generation method based on dynamic motion primitives and artificial potential field method, characterized in that: include: Step 1: Collect a set of object trajectories as demonstration trajectories; Step 2: Use the dynamic motion primitive algorithm to learn the demonstration trajectory; Step 3: Generate the origin position of the task object coordinate system corresponding to the next moment in real time based on the position of the target object and the origin position of the task object coordinate system at the current moment; Step 4: Perform collision detection on the origin position of the generated task object coordinate system at the next moment. If a collision occurs, generate an obstacle avoidance gradient according to the artificial potential field method, and adjust the next trajectory point until there is no collision. Otherwise, return to step 3 until the task is completed.

2. The real-time skill trajectory generation method according to claim 1, characterized in that: In step 1, an object coordinate system is constructed based on the key points of the object, and the object coordinate system information at each moment is collected to form a trajectory to represent the skill. The trajectory is expressed in the robot base coordinate system. The expression of the demonstration trajectory is as follows: ξ i ={p i ,q i }, Where i represents the index of the trajectory point, T is the total number of trajectory points, and t i represents the time corresponding to the i-th trajectory point; ξ i ={p i ,q i } represents the pose trajectory point, including position information p = [x, y, z] T And posture information q = [q0, q1, q2, q3] T , x, y, z represent the three-dimensional coordinates corresponding to the origin of the object coordinate system, q0, q1, q2, q3 represent the unit quaternion corresponding to the rotation matrix of the object coordinate system; Represents the derivative of the pose trajectory point with respect to time, including the position derivative ω i represents the angular velocity, dt is the time interval between adjacent trajectory points, norm([q1,q2,q3] T ) represents the modulus of the vector [q1,q2,q3]; Represents the second-order derivative of the pose trajectory point with respect to time, including the second-order derivative of position Angular acceleration ξ goal ={p goal ,q goal } represents the pose data of the target object coordinate system, and its definition form is consistent with the pose trajectory point.

3. The real-time skill trajectory generation method according to claim 2, characterized in that: In step 2, weighted learning of the trajectory shape is performed by calculating the weight of the nonlinear term in the dynamic motion primitive algorithm formula.

4. The real-time skill trajectory generation method according to claim 3, characterized in that: Calculate the weight of the nonlinear term. The specific expression is as follows: Where N represents the number of basis functions in the nonlinear term, ψ j (s) represents the jth basis function, w j is the weight corresponding to the basis function, s i is the phase corresponding to time i, τ is the total duration of the demonstration trajectory, c j represents the center of the Gaussian basis function, h j represents the width of the Gaussian basis function, and α is the speed adjustment coefficient; Construct the following matrix: Where φ is the basis function matrix, F is the nonlinear term matrix, and W is the weight matrix, then the following formula is obtained: φ*W=F, In the above formula, the basis function matrix φ is directly determined by the prior parameter definition, and the nonlinear term matrix F is calculated by the demonstration trajectory data. The calculation formula is as follows: The formula for the nonlinear term in the x-direction is as follows: where α x , β x is the second-order system parameter, x i is the x coordinate of the demonstration trajectory at time i, x goal is the x coordinate of the origin of the target coordinate system; The formula for the nonlinear term in the y direction is as follows: where α y , β y is the second-order system parameter, y i is the y coordinate of the demonstration trajectory at time i, y goal is the y coordinate of the origin of the target coordinate system; The formula for the nonlinear term in the z direction is as follows: With structure,i =from i -z0*f control,i -With goal *(1-f control,i ), where α z , β z is the second-order system parameter, z i is the z coordinate of the demonstration trajectory at time i, z0 is the z coordinate of the demonstration trajectory at the initial time, z goal is the z coordinate of the origin of the target coordinate system, f control,i is the control coefficient at time i; After obtaining the F matrix, the approximate solution of the weight matrix W is further obtained by the pseudo-inverse method: W=(φ T f) -1 f T F.

5. The real-time skill trajectory generation method according to claim 3, characterized in that: In step 3, the origin position of the task object coordinate system at the next moment is obtained by calculating the coordinate values ​​in the x, y, and z directions. The calculation expression of the value at the next moment in the x direction is: In the above formula, x newgoal is the x coordinate of the origin of the new target object coordinate system, x newstart is the x coordinate of the origin of the new starting object coordinate system, x goal , x0 are the target x coordinate and the starting x coordinate in the demonstration trajectory, t c is a manually selected moment, the subsequent trajectory will be close to the demonstration trajectory, k controlx (t) is the generated control parameter, is its derivative with respect to time, x new,i is the x coordinate of the origin of the current object coordinate system, is its derivative with respect to time, is the second-order derivative with respect to time; To calculate the value of the y direction at the next moment, just replace the above x parameter with the parameter corresponding to the y direction; The calculation expression of the value at the next moment in the z direction is: With new,i+1 =from newstructure,i+1 +from newstart *f controlz,i +from newgoal *(1-f controlz,i ), where znewstructure,i, is the new z coordinate constructed at time i and the corresponding first-order and second-order time derivatives, and znew,i+1 is the z coordinate of the origin of the object coordinate system actually generated at time i+1.

6. The real-time skill trajectory generation method according to claim 5, characterized in that: The step 4 specifically includes: Calculate the shortest distance D(o) between the object and the obstacle at the i+1th moment i+1 ), o i+1 Represents the object coordinate system information at the i+1th moment: o i+1 =[x new,i+1 ,y new,i+1 ,z new,i+1 ]; The collision detection is judged as D(o i+1 )<Q * When the obstacle avoidance gradient is calculated, the object coordinate system information o at the i+1th moment is updated. i+1 To adjust the position of the trajectory point, the object coordinate system information o at the i+1th moment is updated i+1 The expression is as follows: o i+1 =o i+1 -F(o i+1 )*step, Where step is the compensation, Q* is the obstacle influence range, F(oi +1 ) is the obstacle avoidance gradient; After the update is completed, the closest distance D(o i+1 ); The collision detection is judged as D(o i+1 )>Q * If , it ends and returns to step 3, otherwise, the obstacle avoidance gradient is recalculated according to the above formula and o is continued to be updated. i+1 .

7. The real-time skill trajectory generation method according to claim 6, characterized in that: The obstacle avoidance gradient is calculated by the artificial potential field method, and the expression is as follows: Where ζ is the target attraction coefficient, ρ(o i+1 ,o goal ) is the modulus of the difference between two vectors, o goal is the origin coordinate of the target coordinate system, U att (o i+1 ) is the target attraction potential field, η is the obstacle potential field coefficient, U rep (o i+1 ) is the obstacle avoidance potential field, F att (o i+1 ) is U att (o i+1 ) relative to the gradient in the x, y, and z directions, F rep (o i+1 ) is U rep (o i+1 ) with respect to the gradient in the x, y, and z directions, D(o i+1 ) with respect to the gradient in the x, y, and z directions.

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