CN116561951A - Real-time interactive high-precision rope simulation method and system - Google Patents
Real-time interactive high-precision rope simulation method and system Download PDFInfo
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Abstract
A real-time interactive high-precision cable simulation method and system comprises an initialization stage and a simulation stage, wherein the initialization stage is used for discretely processing and storing continuous cables according to the initial state of the cables to be simulated and then setting simulation parameters; the simulation stage adopts an improved projection mechanics algorithm, uses matrix decomposition and buffer acceleration according to motion constraint deduced based on a Cosseat rope rod model, and outputs the motion state of each logic frame of the rope on the basis of real-time performance, so that the physical precision enhancement of the rope simulation is realized. The invention adds the direction and the angular velocity into the original projection mechanical frame based on the improved projection mechanical algorithm, utilizes matrix decomposition and buffer acceleration, can simulate the rope stress of various materials in real time and the running state after collision, can ensure to achieve the accuracy similar to the off-line simulation of commercial finite element software, and improves the reality of simulating the phenomena of stretching, shearing, bending and twisting of the rope in real-time interactive application.
Description
Technical Field
The invention relates to a technology in the field of material science, in particular to a real-time interactive rope simulation method and system with average error lower than three percent.
Background
Cables are one of the most common devices in life. The use of computers to simulate the motion state of a cable has been of great interest. In the related art, a series of developments have been made in simulation algorithms and techniques with high physical precision for ropes, such as the commercial software Abaqus, using the Finite Element Method (FEM), etc. A common feature of such classical algorithms is that based on force-pushing the equations of motion and solving these nonlinear partial differential equations using Finite Element Methods (FEM) or Finite Difference Methods (FDM), generally physically accurate results can be obtained. However, the main disadvantage of this type of force-based method is that the calculation efficiency for solving the rigid differential equation is very low and cannot be achieved in real time.
Some studies focusing on real-time interactive applications in the graphics field, in order to reduce the computational overhead, quick simulation methods such as position-based dynamics (PBD) have been proposed that sacrifice physical accuracy in place of higher performance. The method is commonly used for simulating ropes in real-time graphic applications such as games and the like, has higher calculation efficiency, but has the defect that the physical meaning of parameters is not clear, and the physical precision is required to be improved. Therefore, on PC equipment with less calculation force resources, real-time physical simulation of the rope with high physical precision is carried out, and certain challenges still exist.
Disclosure of Invention
Aiming at the problem that the physical precision of the cable simulation is not high enough in the existing real-time graphic application, the invention provides a real-time interactive high-precision cable simulation method and system, which are based on an improved projection mechanics algorithm, add the direction and angular velocity into the original projection mechanics frame, utilize matrix decomposition and buffer acceleration, simulate the stress and the running state of the cable of various materials in real time after collision, ensure the precision similar to the commercial finite element software offline simulation, and improve the reality of the phenomena of cable stretching, shearing, bending and torsion in the real-time interactive application.
The invention is realized by the following technical scheme:
the invention relates to a real-time interactive high-precision rope simulation method, which comprises an initialization stage and a simulation stage, wherein: the initialization stage is to discretely process and store the continuous ropes according to the initial state of the ropes to be simulated and then set simulation parameters; the simulation stage adopts an improved projection mechanics algorithm, uses matrix decomposition and buffer acceleration according to motion constraint deduced based on a Cosseat rope rod model, outputs the motion state of each logic frame of the rope on the basis of real-time performance, and realizes physical precision enhancement of rope simulation.
The invention relates to a real-time interactive rope simulation system for realizing the method, which comprises the following steps: interaction module, emulation module, rendering module, input device and display device, wherein: the interaction module acquires the input of a user in real time, and changes the motion state of the rope or other objects in the whole virtual scene according to the input; the simulation module simulates the motion state of the rope in real time according to the rope simulation method; the rendering module restores the geometric model of the cable in real time in the virtual scene according to the motion state and parameters of the cable and outputs a rendering picture; the input device is used for receiving user input; the display device outputs the rendered rope motion animation.
Technical effects
According to the method, through an improved projection mechanics algorithm, constraints affecting the motion performance of the rope are split into potential energy constraints derived based on a Cosseat model and constraints dynamically generated such as collision and animation, and the performance bottleneck in the potential energy constraints is solved by the projection mechanics, namely a high-dimensional linear equation set is solved, so that the problem of solving a sparse matrix and a vector product is generalized, and the problem of solving the sparse matrix is benefited by the fact that the sparse matrix is unchanged after the split, the solving process can be greatly accelerated by decomposing the sparse matrix and buffering the decomposition result, and the instantaneity is achieved; the remaining constraints are solved using a location-based dynamics algorithm. The method has the technical effects that the method obtains the precision similar to the non-real-time simulation of the near-commercial finite element software, the average error is lower than three percent, and the performance is improved by one order of magnitude, so that the real-time performance is satisfied; compared with the simulation method commonly used in real-time graphic application such as games, the accuracy is greatly improved, and parameters, weights and the like have real physical significance.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a flow chart of an initialization process of the present invention;
FIG. 3 is a schematic diagram of a reconstructed cable model of the present invention;
FIG. 4 is a flow chart of a simulation process of the present invention;
FIG. 5 is a system diagram of the present invention;
fig. 6 to 9 are schematic views of effects of the embodiment.
Detailed Description
As shown in fig. 1, this embodiment relates to a real-time interactive high-precision cable simulation method, which includes an initialization stage and a simulation stage, wherein: the initialization stage is to discretely process and store the continuous ropes according to the initial state of the ropes to be simulated and then set simulation parameters; the simulation stage adopts an improved projection mechanics algorithm, the orientation of a discrete rope segment is introduced, matrix decomposition and buffer acceleration are used according to potential energy constraint deduced based on a Cosseat rope rod model, the motion state of each logic frame of the rope is output on the basis of real-time performance, and physical precision enhancement of rope simulation is achieved.
As shown in fig. 2, the initialization stage specifically includes:
the first step, a control point which is input by a user and used for defining the center line of the rope is obtained: the user inputs the radius of the rope and the control point according to the initial shape of the rope in the eyes of the user; after the control points in the three-dimensional space are obtained, a Catmull-Rom spline curve with C1 continuity and passing through all the control points is selected as the center line of the cable in the embodiment.
Secondly, acquiring a sampling frequency N input by a user: according to the definition of Cosseat theory model, the central line of the rope with length L is recorded asAccording to arc coordinates on a central lines uniformly samples N points: r (0), r (L/N-1), r (2L/N-1.) r (L), the cable is divided into N-1 segments, each segment being divided by two vertices { x } i ，x i+1 And a four element u representing a rotation i And (5) defining. By u i And rotating the world coordinate system to obtain a local coordinate system in the Cosseat rope rod model, wherein N vertexes are particles, and N-1 segments divided by the rope are discrete rope segments, and the discrete rope segments are used as a data structure stored in a computer.
The particles have quality, position and speed attributes, the initial position is set at a sampling point, and the initial speed is 0; the discrete rope portion has the properties of orientation, angular velocity, geometric moment of inertia and initial length, wherein the initial orientation is set as the tangential direction of the central line of the rope at the initial particle of the discrete rope portion, the initial angular velocity is 0, and the initial length is the distance between two corresponding vertexes of the discrete rope portion.
Thirdly, the user inputs the geometric parameters and the material parameters of the rope: as shown in fig. 3, according to the cross-section radius input by the user, a circular surface with a corresponding radius is drawn at each particle, and the normal direction of the circular surface is the tangential direction of the central line at the point; the points on the circular surfaces are sequentially connected to form a triangular grid, so that the geometric model of the rope is restored.
The geometric parameters include initial length and cross-sectional radius.
The material parameters include density, young's modulus and poisson's ratio.
The geometric moment of inertia of the discrete rope portion is obtained by multiplying the average value of the lengths of the discrete rope portions with the particles as a starting point and an ending point according to the density of the rope input by a user, and then the geometric moment of inertia of the discrete rope portion can be obtained according to the radius of the cross section.
Fourth, accepting simulation parameters input by a user, including: the overall duration of the simulated motion required by the simulation, the time step of each change of the motion state of the simulated rope, and the iteration times of each state change calculation.
And fifthly, storing the geometric parameters, the material parameters and the information of the particles and the discrete rope parts of the rope.
As shown in fig. 4, the simulation phase specifically includes the steps of:
step 1) dividing the whole process of simulating the rope movement into a plurality of logic frames to be calculated according to the setting.
Adjacent logical frames are separated by a set time step.
Step 2) inputting the motion state of a frame on the cable in each logic frame, and calculating the motion state of the frame by using an improved projection mechanics algorithm, wherein the method specifically comprises the following steps:
2.1 Predicting cable position and orientation: predicting the position of a frame of the particle according to the speed and the position of the frame and the combined force applied to the frame, wherein the method specifically comprises the following steps:wherein: />The position of all particles predicted for this frame, x (t) For the positions of all particles in the previous frame, h is the set time step, v (t) For the velocity of all particles of the previous frame, M is a diagonal matrix of all particle masses, f ext Is the resultant force to which the particles are subjected.
2.2 According to the angular speed and the orientation of a frame on the discrete rope portion and the resultant moment to which the frame is subjected, predicting the orientation of the frame of the discrete rope portion, specifically:wherein: />For the orientation of the frame of the discrete rope portion u (t) For the orientation of the frame of discrete rope portions, ω (t) The angular speed of the discrete rope portion is obtained from the motion state of the previous frame, tau is the total torque, and tau is obtained by user input, collision processing or animation design. Diagonal matrix-> Representing the geometrical moment of inertia of the rope portion on the central line, the definition of J can be known by combining the quaternion definition because the quaternion is regarded as a four-dimensional vector in the process of linear solving n ＝l n ρdiag(0，J 1 ，J 2 ，J 3 ) Wherein l n For the initial length of the rope portion. When the whole rope is uniform, the same density ρ is used everywhere. J (J) 1 ，J 2 ，J 3 Is the geometric moment of inertia of the cross section, and can be determined byAnd (5) calculating.
2.3 Initializing constraints: constraints include potential energy constraints derived from the costerat model, and collision constraints and animation driven constraints additionally added to enhance physical manifestation realism and interactivity, wherein the potential energy constraints are no longer changed after the determination of the cable parameters, only need to be initialized when the cable parameters are modified; collision constraints and animation driven constraints change with cable motion and user interaction, requiring re-initialization at each frame, including in particular: i) Initializing constraint deduced from Cosseat model tensile and shear potential energy asii) initializing the constraint derived from the Cosseat model bending and torsional potential energy to +.>iii) Initializing a crash constraint to +.> Iv) initializing animation constraints toWherein: w (W) sEi (q，p i ) For the current particle p i A tensile and shear potential difference between the state and the target state, min representing the constraint to minimize the potential difference; Γ -shaped structure n To measure the strain of the degree of stretching, shearing, is equal to the direction vector of the particle on the centerline tangent, minus the direction vector of the discrete rope segments starting with the particle.
The direction vector on the centerline tangent is approximately the difference between the positions of adjacent particles in the discrete unit divided by the initial length, i.e Thus->Wherein: im (q) represents the imaginary part of the four elements q; w (W) BTi (q，p i ) For the current discrete rope portion p i A bending and torsional potential difference between the state and the target state, min representing the constraint to minimize the potential difference; omega shape n To measure the strain of the degree of stretching, shearing, the rate of change of curvature of the discrete rope portion and the adjacent discrete rope portion, i.e +.>c j Is the intensity of the collision constraint; delta j Is an indicator function, 1 when the collision volume of the particle still coincides with the other collision volumes, and 0 when it no longer coincides; r is the radius of the cross section of the rope; />For the position of particle j in three dimensions, +.>The position of the particle projected onto the surface of other collider; c j Intensity of animation constraint is adopted, and the user is added with->Is particles at theThe position set in the animation frame.
2.4 Using a position-based dynamics method (PBD) to solve for animation constraints: and determining the time of the animation key frame, the state information of the key frame and the interpolation mode, and calculating the state information of the time point of each logic frame.
The state information includes: the position of the particle controlled by the animation, the orientation of the discrete rope segments, and the solving method are similar to Macklin solving the fixed point constraint in the extended position dynamics method (XPBD), except that the target fixed point position is set to the position of the particle controlled by the animation in the state information.
2.5 For all constraints deduced from potential energy, solving the projection position of the constraint, namely, the position which meets the constraint i and has the minimum generalized distance with the current motion state q, specifically: solving for the current motion state q and a position p under constraint i i Is a broad distance of (2)Take the minimum pi, the projected location of the constraint, where: A. b and S are constant matrices constructed according to constraints, w i To constrain the corresponding non-negative weights, δ C As an indication function if and only if the projection position p i The constraint state is 0 when satisfied, and the constraint state is infinite when not satisfied.
The solution mode of the projection position in the stretching and shearing constraint state is specifically as follows: at the projection pi meeting the constraints of tensile and shear potential energy, the strain measuring the degree of tension and shear of the discrete unit should be 0, i.e Wherein: particle position variation-> The two particle positions of the discrete unit in the iteration, the normal of the cross section of the discrete unit +.>Since the position of the particles and the orientation of the rope portion are independent of each other, a further split into two optimization problems, respectively the optimization of the particle position +.>And optimization of the local orientation of the rope portion: />The solution for particle position optimization is at Δx n ＝d 3 The difference between the positions of the particles is parallel to the normal line of the cross section, and the distance between the particles is the same as that in the initial state, that is, the tensile strain of the discrete unit is 0. The solution of the local orientation optimization of the rope segment is +. >Obtained when the shear deformation of the discrete units is 0, wherein +.>Is represented by Deltax n To d 3 Rotated quaternions. From this, the solution of the constraint of tensile and shear potential energy is obtained
The solution mode of the projection position in the bending and torsion constraint state is specifically as follows: projection p satisfying bending and torsional potential energy constraints i Where the rate of change of curvature Ω of the degree of bending and torsion of the discrete unit is measured n Should be 0, in measuring the system state q to projection p i Distance of (u), i.e. u n ，u n+1 And (3) withWhen the distance between the two is calculated, the quaternion is simply regarded as a four-dimensional vector, and the Frobenius norm (Frobenius norm) which is the same as the calculated position distance is adopted. From the following componentsThe solution of the optimization can be found as +.>and/>The solution of the constraint of the bending potential energy and the torsional potential energy is obtained as A i ＝I 8 ， B i ＝I 8 ，/>
2.6 Based on the constraint derived from potential energy, the motion state of the current frame, that is, the vector q representing the motion state is obtained, so that the projection position in the implementation step 2.5 is minimized, specifically:wherein: q is the current frame motion state sought, containing the positions of all particles and the orientation of all discrete rope portions, i.e. q= [ x1, ], x N ，u 1 ，...，u N-1 ]The diagonal matrix M, the diagonal elements are the moments of inertia for all particle masses and all discrete rope segments. The remaining symbols are the same as in step 2.5.
Preferably, solving the current frame motion state q is equivalent to solving a linear equation, the left matrix described above when the particle mass is fixed, the time step is fixed, the number of constraints and parameters are fixedIs unchanged. Before the first calculation, LU decomposition is adopted for the matrix of the left formula, and the decomposition result is stored. Use in computationThe upper triangular matrix and the lower triangular matrix after LU decomposition are calculated, so that the speed can be obviously improved.
The motion state of the current frame is equivalent to solving a linear equation set A * x=b, wherein: sparse matrix When the linear equation set A is directly solved by adopting the primordial elimination method * The time complexity of x=b is O (n 3 ) N is matrix A * Is a dimension of (c). Since the four elements of the position and representative orientation are represented using three-dimensional and four-dimensional vectors, respectively, A * The dimension of (2) is the sum of 3 times of particle number and 4 times of discrete rope segment number, is a high-dimension sparse matrix, and has no small calculation cost when directly solving the linear equation set.
The embodiment adopts an improved solving method, and specifically comprises the following steps:
i) For the storage of the sparse matrix, when a traditional array storage scheme is adopted, elements in the matrix can be randomly acquired in constant time, but a lot of unnecessary memory is occupied, and the algorithm is not beneficial to accelerating operation by utilizing sparsity. The sparse matrix is stored in a compressed sparse column (Compressed Sparse Column, CSC) format capable of efficiently acquiring a certain row of non-zero elements of the matrix and supporting sparse matrix arithmetic operation, and one sparse matrix is stored by storing non-zero values, a range of columns and a row index.
ii) although the vector b changes in each iteration due to the change in projection result of step 2.5, as long as the selection matrix S of any constraint i i Constant matrix A i Weight w i And all particle mass, cable moment of inertia and time step are unchanged, sparse matrix A * Nor does it change. Therefore, constraints generated in real time according to the motion state and input are split, potential energy constraints derived by a Cosseat model are reserved, and weights and constant matrixes of the constraints remain unchanged; so as long as the geometric parameters and material parameters of the ropeAnd the simulation parameters remain unchanged, the sparse matrix A * Will remain unchanged.
iii) Decomposing a sparse matrix A which is kept unchanged by using an LU decomposition method * . The matrix can be proved to be a positive definite symmetric matrix by the construction process, so that the matrix A can be obtained * The product of a unique lower triangular matrix L and an upper triangular matrix U, i.e. PA * Q=LU, wherein P and Q are permutation matrices obtained by adopting minimum degree ordering and other methods, which is beneficial to subsequent decomposition of the sparse matrix A * Filling as little as possible of non-zero elements in the process of (2), then solving the system of linear equations A * x=b translates to solving ly=b and LUx =b, respectively, because of the triangular nature of L and U, the process of solving for the elimination can be performed at O (n 2 ) And (5) finishing the process.
iv) buffering the lower triangular matrix L and the upper triangular matrix U of the decomposition result, and multiplexing in the subsequent simulation. Although the decomposition itself requires O (n 3 ) But benefits from the method to sparse matrix a * Remain unchanged and therefore only need to be disassembled once at cable initialization. The method has the technical effects that the time complexity of solving the vector q representing the motion state in each logic frame is reduced by one order of magnitude compared with the method of directly solving the linear equation set by a primordial elimination method and the like.
v) for other constraints after resolution than the potential energy constraints derived by the coserat model, a position-based dynamic method (PBD) is used for pre-or post-treatment, as shown in steps 2.4 and 2.7. After splitting constraint, the sparse matrix A is obtained * The condition that the particle mass, the moment of inertia of the cable and the time step are unchanged is kept unchanged, so that when a user changes the geometric parameter, the material parameter or the simulation parameter of a simulation algorithm of the cable, the sparse matrix A * Will be reconfigured and broken down. Most of the constraints are related to scene interaction (such as collision constraint) and user interaction (such as animation constraint), are irrelevant to phenomena caused by internal stress such as stretching, shearing, bending and torsion of the rope, and have little influence on the physical precision of the rope simulation.
2.7 Solving collision constraints using a position-based dynamics method (PBD) to solve the self-collision of the ropes; collision and processing of the cable with other objects within the scene is typically performed by the graphics engine after the triangular mesh has been reconstructed using, but not limited to, macklin's method of extended position dynamics (XPBD).
2.8 Using the motion state q solved above, updating the velocity of the particles and the angular velocity of the discrete rope portions, comprising: the velocity of the particles is set to the difference between the new and old positions obtained, divided by the time step. The angular velocity of the discrete rope portion is set to the difference between the solved new orientation and the old orientation divided by the time step.
2.9 Outputting simulation results including the position and the speed of each particle and the orientation and the angular speed of each discrete rope portion.
As shown in fig. 5, this embodiment relates to a real-time interactive cable simulation system for implementing the method, which includes: interaction module, emulation module, rendering module, input device and display device, wherein: the interaction module acquires the input of a user in real time, and changes the motion state of the rope or other objects in the whole virtual scene according to the input; the simulation module simulates the motion state of the rope in real time according to the rope simulation method; the rendering module restores the geometric model of the cable in real time in the virtual scene according to the motion state and parameters of the cable and outputs a rendering picture; the input device is used for receiving user input; the display device outputs the rendered rope motion animation.
The interaction module acquires the input of a user, and modifies the geometric parameters and the material parameters of the rope according to the input of the user, or changes the motion state of the rope, or creates animation for the motion of the rope, and the like. Because the physical simulation method of the rope in the embodiment has real-time performance, the interaction module comprises: a geometric parameter modification unit, a material parameter modification unit, a simulation parameter modification unit, a scene control unit and an animation creation unit, wherein: the geometric parameter modification unit performs cable reinitialization according to cable geometric parameter information, such as parameters of a cable center line control point, a cable cross section radius, a cable sampling frequency and the like, which are adjusted by a user in the visual editor, and comprises the following steps: and generating a catmull-Rom spline curve according to the control points, uniformly sampling the arc coordinates according to the sampling frequency to obtain sampling points and discrete rope portions therebetween, and restoring the cross section and the connected triangular meshes according to the radius of the cross section. The result is to update the geometric model of the rope, and update the information stored in the computer after the rope is discretized; and the material parameter modification unit updates parameters in the simulation algorithm according to the cable material parameter information, such as cable density, young modulus, poisson rate and the like, which is adjusted by the user in the visual editor. Comprising the following steps: updating a particle mass matrix, updating the inertia moment of a discrete rope portion, and updating the weight of the constraint of stretching, shearing, bending and torsion potential energy; the simulation parameter modification unit updates the operation of the cable simulation algorithm according to parameters, such as time step, iteration times and the like, of the simulation algorithm adjusted by the user in the visual editor. Comprising the following steps: modifying the time from the motion state updated by each rope to the last motion state, and solving the iteration times from the predicted motion state to the minimum weighted sum of the constraint projection distances; the scene control unit updates the simulated scene in the computer according to the input information of the user, such as key operation, mouse operation and the like, and comprises the following steps: moving characters in the scene, directly modifying the motion state of the cable in the scene, and the like, and waiting for the updated virtual scene; the animation creation unit creates animation constraints in corresponding animation and simulation algorithm according to the key frame time and key frame state of the user, and obtains the states of particles and discrete rope segments controlled by the animation in the current logic frame by linear interpolation between key frames, and obtains the rope motion result controlled by part or all of the animation.
The simulation module comprises: motion prediction unit, projection position calculation unit, projection distance optimization unit and motion state updating unit, wherein: the motion prediction unit includes, based on previous frame motion information: the positions, the speeds and the combined external forces of all particles, the orientations, the angular speeds and the combined external moments of all discrete rope portions are processed according to a motion equation of classical mechanics, and a motion result of the rope under the condition that the internal stress is ignored is obtained; the projection position calculation unit calculates projection of the predicted rope motion state to each constraint in parallel according to the predicted motion state; the projection distance optimizing unit is used for solving a linear equation set according to the predicted motion state and projection information reaching all constraints so as to optimize the distance weighted sum reaching all projections and obtain a new predicted motion state result; and the motion state updating unit updates the motion state of the rope according to the predicted motion state meeting the iteration times or the error tolerance, obtains the motion result of the frame and transmits the motion result to the rendering module.
The rendering module calculates a round surface with the radius being the radius of the preset rope at each particle of the rope and represents the transverse section of the rope; the normal direction of the cross section is the same as the tangential direction of the discrete rope portion where the particles are located. A certain number of points are sampled on the cross section where each particle is located, and the sampling points on each cross section are connected to form a triangular grid which represents the surface of the rope. And rendering the triangular mesh by using a preset rope material, so that the moving rope can be restored in the virtual scene. The rendering module is integrated in the graphics engine Unity.
Through specific practical experiments, the cable simulation algorithm is started by the parameters in the table 3 under the hardware environment of the table 1 and the software environment of the table 2, and before specific experiments are carried out, accurate solutions serving as references need to be prepared before the physical accuracy of the cable simulation is measured. Because most of the scenes of rope movement are difficult to calculate the accurate value of integration without analytic solution in mathematical sense, the same scenes are restored in the commercial finite element simulation software ABAQUS which is tested for a long time, simulation is carried out by adopting a very small time step, and a numerical solution with very high accuracy is obtained as a reference.
TABLE 1
| Hardware device | Configuration of |
| CPU | Amd5800 |
| Memory | 16GB |
| Hard disk | Western number SN550 |
| Display card | NVIDIA3070 |
TABLE 2
| Software name | Version of |
| Unity3D | 2020.2.3 |
| Abaqus | 2021 |
TABLE 3 Table 3
| Picture numbering | FIG. 7 | Fig. 8 and 9 |
| R(mm) | 10 | 6 |
| ρ(g/m 3 ) | 1.3 | 1.3 |
| E(MPa) | * | 10 |
| v | 0.25 | 0.25 |
| L(m) | 2 | 2 |
| h(ms) | 5 | 5 |
| g y (m/s 2 ) | 9.861 | 9.861 |
In Table 3, R is the radius of the cross-sectional area (mm) of the cable, ρ is the density (g/m) of the cable 3 ) E is Young's modulus (MPa), v is poisson ratio, L is total length of the cable(m), h is the time step (ms), g y Is the component (m/s) of the gravitational acceleration in the Y-axis direction 2 ) Representative of the parameters varied during the experiment. In Abaqus as reference solution, only as high accuracy as possible is considered without performance, so part of the parameters are not consistent, e.g. very small time steps: h is a Abaqus ＝1ms。
As shown in fig. 6, to simulate the two end points of different cables, the cables are naturally drooping under the action of gravity, the cables are initially placed horizontally, and the length is a scene of natural length (without any elastic deformation). The comparative approach is the PBD approach widely used in real-time graphics applications, employing implementation of the most popular cable analog plug-in obipe over Unity. The Y-axis of the image corresponds to the component of the position of each discrete element of the cable in the Y-direction. Since the deformation in the Y-axis direction is smaller than the original length of the rope under the condition of only gravity except the rope with extremely high elasticity, the X-axis of the image is selected to be the particle serial number X e 1, N, instead of the component of the particle position in the X-axis, wherein N is the number of discrete units in Abaqus, which is a fixed value 600 in the experiment. Since the output of Abaqus is a relative deformation, it is added to the height of the initial state of the cable. In fig. 6, cables of different materials are simulated, one for each row. (a) is a rubber rope with e=1 MPa, (b) is a rubber rope with e=10 MPa, and (c) is a hemp rope with e=100 MPa. It is known that the weights in the PBD are derived from empirical tuning and do not have physical significance. In the experiment, the weight parameters of the PBD are carefully adjusted so that the PBD can better simulate a rubber rope with the Young modulus of 10MPa, but the modification of physical parameters can not be fed back into the parameters of the PBD, so that the PBD can not simulate ropes of other materials. The algorithm better simulates the performance of ropes made of different materials, and the average error is lower than three percent.
As shown in fig. 7, to verify that the method can be driven and process collisions correctly by animation, and can obtain relatively accurate results with a small number of iterations, the experiment is designed to fix the starting end point of the rope, and the following animation is designed for the end point: two key frames, the first key frame has a natural length from the starting end point of the cable, and the second key frame has a coincident position with the starting end point of the cable. The inter-key frame interval is 3s, which is set to 30s for higher accuracy in Abaqus as a reference solution. The horizontal and vertical coordinates in the figure are components of the position of the rope in the world space in the X axis and the Y axis, respectively, and the unit is m. As in the previous experiment, since the output of Abaqus is relatively variable, it is added to the initial position of the cable by post-processing. The effect of different iteration times is represented in fig. 7: (a) 4 times, (b) 8 times, and (c) 16 times, it can be seen that the algorithm converges in a smaller number of iterations, and the constraint strength is determined by the physical parameters, irrespective of the number of iterations. In the PBD method, as the iteration number increases, the rigidity of each constraint increases, and the physical accuracy is not necessarily improved, and the cable shows too high elasticity when the iteration number is smaller in the experiment, and also shows too high rigidity after the iteration number is higher, so that the disadvantage that the relatively correct visual performance can be obtained only by adjusting parameters according to experience is caused. In addition, for a scenario where multiple physical parameters determine physical performance at the same time, tuning of the PBD becomes more difficult. For example, in this experimental scenario, the minimum position of the sag and the degree of flexibility exhibited by the cable are determined by the young's modulus and the poisson's ratio together, and it is difficult for the PBD method to simulate correctly at the same time.
As shown in fig. 8, in the case of self-collision after the movement of the end point of the rope in the scene of fig. 7, collision constraint and animation driving constraint can be seen, and the movement of the rope can be accurately controlled and the collision caused by the movement can be handled. In collision detection, the generated collision constraint is better than the PBD by detecting an implicit curved surface determined by the center line of a particle and Cosseat theoretical model. In fig. 9, some scenes provided by Obi Rope are supplemented, and the adopted algorithm simulates the phenomena of breakage, common force application and torsion of the Rope. The physical performance of the rope can be sufficiently restored.
The performance data in the first two experiments are summarized as shown in table 4. The PD algorithm framework (Standard-PD) only added with the Rope orientation and the angular momentum, the PD (Cache-PD) algorithm after matrix decomposition and buffering and the PBD algorithm (realized by the Obi Rope team) are adopted to complete the test under the same scene, the performance of the Rope is updated only by the statistical algorithm, the degree of freedom of the system state and the iteration times are unified, and the other parameters are the same as those in the table 2. With 30 frames per second as real-time requirement, the time consumption of each operation is less than 33ms. The algorithm proved by the method achieves real-time performance, and the performance is obviously improved compared with that before optimization.
TABLE 4 Table 4
| Scene(s) | N | Standard-PD | The invention is that | PBD |
| FIG. 6 | 280 | 45.8ms | 17.46ms | 0.92ms |
| FIG. 7 | 280 | 61.2ms | 18.70ms | 1.30ms |
As shown in table 5, performance data in a cable simulation method which is close to the physical accuracy of the method in recent years is shown, wherein: the SOLER method is also based on projection mechanics, and although calculation is faster, animation and collision constraint are not added, and interactivity and realism are discounted; the method shown by WEN can automatically downsample the cable in real time to sacrifice a certain accuracy for the guarantee of real-time performance, so that the actual accuracy of the method shown by WEN is discounted; after the GAZZOLA is discrete, a classical mechanical derivation motion equation is adopted and is calculated through an explicit Euler method, so that the stability of the algorithm in the process of high deformation of the rope cannot be ensured.
TABLE 5
Compared with the prior art, the method obtains excellent balance of physical precision and calculation efficiency of the cable simulation; the physical precision is similar to that of a non-real-time finite element method, the physical precision is greatly improved compared with the PBD method in the current real-time graphic application, and the calculation efficiency is improved compared with other cable simulation algorithms in recent years. According to the invention, the angular momentum is added into the original projection mechanics algorithm, the original projection mechanics constraint is split into the fixed rope constraint and the dynamically generated collision and animation constraint, and the linear equation set is solved by matrix decomposition and cache acceleration, so that the real-time performance can be achieved while the original precision is maintained, the problem of insufficient physical precision of the simulated rope in the current common real-time graphic application is solved, and the rope simulation performance is enhanced. The projection force algorithm can be expanded to simulate various phenomena of the cable by using the potential energy of stretching, shearing, bending and torsion of the cable as constraint, and the physical precision similar to that of commercial finite element software is obtained.
The foregoing embodiments may be partially modified in numerous ways by those skilled in the art without departing from the principles and spirit of the invention, the scope of which is defined in the claims and not by the foregoing embodiments, and all such implementations are within the scope of the invention.
Claims (10)
1. The real-time interactive high-precision rope simulation method is characterized by comprising an initialization stage and a simulation stage, wherein: the initialization stage is to discretely process and store the continuous ropes according to the initial state of the ropes to be simulated and then set simulation parameters; the simulation stage adopts an improved projection mechanics algorithm, uses matrix decomposition and buffer acceleration according to motion constraint deduced based on a Cosseat rope rod model, and outputs the motion state of each logic frame of the rope on the basis of real-time performance, so that the physical precision enhancement of the rope simulation is realized.
2. The real-time interactive high-precision cable simulation method according to claim 1, wherein the initializing stage comprises:
the first step, a control point which is input by a user and used for defining the center line of the rope is obtained: inputting a control point by a user; after the control points in the three-dimensional space are obtained, a Catmull-Rom spline curve with C1 continuity and passing through all the control points is selected as the center line of the cable;
Secondly, acquiring a sampling frequency N input by a user: according to the definition of Cosseat theory model, the central line of the rope with length L is recorded asEvenly sampling N points on the central line according to the arc coordinates s: r (0), r (L/N-1), r (2L/N-1) … r (L), the cable is divided into N-1 segments, each segment being divided by two vertices { x } i ，x i+1 And a four element u representing a rotation i Definition; by u i Rotating the world coordinate system to obtain a local coordinate system in the Cosseat rope rod model, wherein N vertexes are particles, and N-1 segments into which the rope is divided are discrete rope segments, and the discrete rope segments are used as a data structure stored in a computer;
thirdly, the user inputs the geometric parameters and the material parameters of the rope: drawing a circular surface with a corresponding radius at each particle according to the radius of the cross section input by a user, wherein the normal direction of the circular surface is the tangential direction of the center line at the point; the points on the circular surfaces are sequentially connected to form a triangular grid, so that the geometric model of the rope is restored;
fourth, accepting simulation parameters input by a user, including: the overall duration of the simulated motion required by the simulation, the time step of each change of the motion state of the simulation rope, and the iteration times of each state change calculation;
and fifthly, storing the geometric parameters, the material parameters and the information of the particles and the discrete rope parts of the rope.
3. The real-time interactive high-precision rope simulation method according to claim 2, wherein the particles have mass, position and speed attributes, the initial position is set at a sampling point, the initial speed is 0, and the mass is the rope density multiplied by the average value of the lengths of adjacent discrete rope segments; the discrete rope portion has the properties of orientation, angular velocity, geometric moment of inertia and initial length, wherein the initial orientation is set as the tangential direction of the central line of the rope at the initial particle of the discrete rope portion, the initial angular velocity is 0, the initial length is the distance between two corresponding vertexes of the discrete rope portion, and the geometric moment of inertia is calculated according to the radius of the cross section;
the geometric parameters comprise initial length and cross-section radius;
the material parameters include density, young's modulus and poisson's ratio.
4. The real-time interactive high-precision rope simulation method according to claim 1, wherein the simulation phase comprises the following specific steps:
step 1), dividing the whole process of simulating the rope movement into a plurality of logic frames to be calculated according to the setting;
step 2) inputting the motion state of a frame on the cable in each logic frame, and calculating the motion state of the frame by using an improved projection mechanics algorithm, wherein the method specifically comprises the following steps:
2.1 Predicting cable position and orientation: predicting the position of a frame of the particle according to the speed and the position of the frame and the combined force applied to the frame, wherein the method specifically comprises the following steps:wherein: />The position of all particles predicted for this frame, x (t) For the positions of all particles in the previous frame, h is the set time step, v (t) For the velocity of all particles of the previous frame, M is a diagonal matrix of all particle masses, f ext The particle is subjected to an external force;
2.2) According to the angular speed and the orientation of a frame on the discrete rope portion and the resultant moment born by the frame, the orientation of the frame of the discrete rope portion is predicted, specifically:wherein: />For the orientation of the frame of the discrete rope portion u (t) For the orientation of the frame of discrete rope portions, ω (t) The angular speed of the discrete rope portion is obtained from the motion state of the previous frame, tau is the total torque, and tau is obtained by user input, collision processing or animation design; diagonal matrix-> Representing the geometrical moment of inertia of the rope portion on the central line, the definition of J can be known by combining the quaternion as a four-dimensional vector in the process of linear solving n ＝l n ρdiag(0，J 1 ，J 2 ，J 3 ) Wherein l n For the initial length of the rope portion; when the whole rope is uniform, the same density rho is adopted everywhere; j (J) 1 ，J 2 ，J 3 For the geometric moment of inertia of the cross section, this can be the case when the cross section is circular>Calculating;
2.3 Initializing constraints: constraints include potential energy constraints derived from the costerat model, and collision constraints and animation driven constraints additionally added to enhance physical manifestation realism and interactivity, wherein the potential energy constraints are no longer changed after the determination of the cable parameters, only need to be initialized when the cable parameters are modified; crash constraints and animation driven constraints change with cable motion and user interaction, requiring re-initialization at each frame, in particularComprising the following steps: i) Initializing constraint derived from Cosseat model tensile and shear potential energy asii) initializing the constraint derived from the Cosseat model bending and torsional potential energy to +.>iii) Initializing crash constraints as And iv) initializing animation constraints to +.>Wherein: w (W) SEi (q，p i ) For the current particle p i A tensile and shear potential difference between the state and the target state, min representing the constraint to minimize the potential difference; Γ -shaped structure n For measuring the strain of the stretching and shearing degree, the direction vector of the particle on the tangent line of the central line is subtracted by the direction vector of the discrete rope part taking the particle as the initial particle;
2.4 Using a position-based dynamics method (PBD) to solve for animation constraints: determining animation key frame time, state information of key frames and interpolation mode, and calculating state information of time points of each logic frame;
The state information includes: the position of the particle controlled by the animation, the orientation of the discrete rope portion, and the solving method are similar to Macklin solving the fixed point constraint in the extended position dynamics method (XPBD), except that the target fixed point position is set as the position of the particle controlled by the animation in the state information;
2.5 For all the constraints deduced by potential energy, solving the projection position of the constraint, namely, the position which meets the constraint i and has the minimum generalized distance with the current motion state q is specifically: solving for the causeObtaining the current motion state q and a position p under constraint i i Is a generalized distance of (2)Take the minimum value p i I.e. the projection position of the constraint, wherein: A. b and S are constant matrices constructed according to constraints, w i To constrain the corresponding non-negative weights, δ C As an indication function if and only if the projection position p i 0 when the constraint state is satisfied, and infinity when the constraint state is not satisfied;
2.6 Based on the constraint derived from potential energy, the motion state of the current frame, that is, the vector q representing the motion state is obtained, so that the projection position in the implementation step 2.5 is minimized, specifically:wherein: q is the current frame motion state calculated, including the positions of all particles and the orientation of all discrete rope portions, i.e. q= [ x 1 ，…，x N ，u 1 ，…，u N-1 ]A diagonal matrix M, the diagonal elements being the mass of all particles and the moment of inertia of all discrete rope segments; the other symbols are the same as those in the step 2.5;
2.7 Solving collision constraints using a position-based dynamics method (PBD) to solve the self-collision of the ropes; collision and processing of the cable and other objects in the scene is generally performed by a graphic engine after reconstructing the triangular mesh, and the collision detection and processing is performed by a method of Macklin in an expanded position dynamics method (XPBD);
2.8 Using the motion state q solved above, updating the velocity of the particles and the angular velocity of the discrete rope portions, comprising: the velocity of the particles is set to the difference between the solved new and old positions divided by the time step; the angular velocity of the discrete rope portion is set to the difference between the solved new orientation and the old orientation divided by the time step;
2.9 Outputting simulation results including the position and the speed of each particle and the orientation and the angular speed of each discrete rope portion.
5. The method of claim 4, wherein the direction vector on the centerline tangent is approximately the difference between the positions of adjacent particles in the discrete unit divided by the initial length, i.e., the Thus->Wherein: im (q) represents the imaginary part of the four elements q; w (W) BTi (q，p i ) For the current discrete rope portion p i A bending and torsional potential difference between the state and the target state, min representing the constraint to minimize the potential difference; omega shape n To measure the strain of the degree of stretching, shearing, the rate of change of curvature of the discrete rope portion and the adjacent discrete rope portion, i.e +.>c j Is the intensity of the collision constraint; delta j Is an indicator function, 1 when the collision volume of the particle still coincides with the other collision volumes, and 0 when it no longer coincides; r is the radius of the cross section of the rope;for the position of particle j in three dimensions, +.>The position of the particle projected onto the surface of other collider; c k Intensity of animation constraint is adopted, and the user is added with->The position of the particle in the animation frame is set.
6.The real-time interactive high-precision cable simulation method of claim 4, wherein the solving mode of the projection position in the stretching and shearing constraint state is as follows: projection p satisfying constraints of tensile and shear potential energy i Where the strain, which measures the degree of stretching, shearing of the discrete units, should be 0, i.e Wherein: particle position variation-> The two particle positions of the discrete unit in the iteration, the normal of the cross section of the discrete unit +. >Since the position of the particles and the orientation of the rope portion are independent of each other, a further split into two optimization problems, respectively the optimization of the particle position +.>And optimization of the local orientation of the rope portion:the solution for particle position optimization is at Δx n ＝d 3 The position difference of the particles is parallel to the normal line of the cross section, the distance between the particles is the same as the initial state, namely the stretching deformation of the discrete units is 0; the solution of the partial orientation optimization of the rope portion is +.>Obtained at the time, i.e. the shears of the discrete unitsWhen the cutting deformation is 0, wherein +.>Is represented by Deltax n To d 3 A rotated quaternion; thereby obtaining the solution of the constraint of stretching and shearing potential energy as +.>
The solution mode of the projection position in the bending and torsion constraint state is specifically as follows: projection p satisfying bending and torsional potential energy constraints i Where the rate of change of curvature Ω of the degree of bending and torsion of the discrete unit is measured n Should be 0, in measuring the system state q to projection p i Distance of (u), i.e. u n ，u n+1 And (3) withWhen the distance between the four-dimensional vector is calculated, the quaternion is simply regarded as a four-dimensional vector, and the Fu Luo Beini Usne norm with the same calculated position distance is adopted; by->Can solve for the optimization and/>The solution of the constraint of the bending potential energy and the torsional potential energy is obtained as A i ＝I 8 ，B i ＝I 8 ，/>
7. The real-time interactive high-precision rope simulation method according to claim 4, wherein solving the current frame motion state q is equivalent to solving a linear equation, and the left matrix is obtained when the particle quality is fixed, the time step is fixed, the constraint quantity and the parameter are fixed Unchanged; before the first calculation, LU decomposition is adopted for the matrix of the left formula, and the decomposition result is stored; when the LU decomposition is used for calculation, the upper triangular matrix and the lower triangular matrix after LU decomposition are used for calculation, so that the speed can be obviously improved;
the motion state of the current frame is equivalent to solving a linear equation set A * x=b, wherein: sparse matrix When the linear equation set A is directly solved by adopting the primordial elimination method * The time complexity of x=b is O (n 3 ) N is matrix A * Is a dimension of (2); since the four elements of the position and representative orientation are represented using three-dimensional and four-dimensional vectors, respectively, A * The dimension of (2) is the sum of 3 times of particle number and 4 times of discrete rope segment number, which is a high-dimension and sparse matrix, and the direct solving of the linear equation system has no small calculation cost.
8. The real-time interactive high-precision cable simulation method according to claim 7, wherein said solving the motion state of the current frame comprises:
i) Storing a sparse matrix by adopting a compressed sparse column format, and storing a sparse matrix by storing non-zero values, a range of columns and a row index;
ii) splitting constraints generated in real time according to the motion state and input, and reserving potential energy constraints derived from a Cosseat model;
iii) Decomposing a sparse matrix A which is kept unchanged by using an LU decomposition method * : matrix A * The product of a unique lower triangular matrix L and an upper triangular matrix U, i.e. PA * Q=lu, where P and Q are permutation matrices obtained by minimum degree ordering and the like, to solve the linear equation set a * x=b translates to solving ly=b and LUx =b, respectively;
iv) buffering the lower triangular matrix L and the upper triangular matrix U of the decomposition result and multiplexing in the subsequent simulation;
v) pre-or post-processing the other constraints after resolution in step ii) except potential energy constraints derived from the costerat model by using a position-based kinetic method.
9. A real-time interactive cable simulation system implementing the method of any of claims 1-8, comprising: interaction module, emulation module, rendering module, input device and display device, wherein: the interaction module acquires the input of a user in real time, and changes the motion state of the rope or other objects in the whole virtual scene according to the input; the simulation module simulates the motion state of the rope in real time according to the rope simulation method; the rendering module restores the geometric model of the cable in real time in the virtual scene according to the motion state and parameters of the cable and outputs a rendering picture; the input device is used for receiving user input; the display device outputs the rendered rope motion animation.
10. The real-time interactable cable simulation system according to claim 9, wherein the interaction module comprises: a geometric parameter modification unit, a material parameter modification unit, a simulation parameter modification unit, a scene control unit and an animation creation unit, wherein: the geometric parameter modification unit reinitializes the cable and updates the geometric model of the cable according to the cable geometric parameter information adjusted by the user in the visual editor, and updates the information stored in the computer after the cable is discretized; the material parameter modification unit updates parameters in the simulation algorithm according to the cable material parameter information adjusted by the user in the visual editor; the simulation parameter modification unit updates the operation of the cable simulation algorithm according to the simulation algorithm parameters adjusted by the user in the visual editor; the scene control unit updates the simulation scene in the computer according to the input information of the user; the animation creation unit creates animation constraints in corresponding animation and simulation algorithms according to the key frame time and key frame state of a user, and obtains the states of particles and discrete rope segments controlled by the animation in the current logic frame by linear interpolation among key frames, thereby obtaining the rope motion result controlled by part or all of the animation.
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