A spacecraft cluster trajectory planning method applied to high-density environment

Abstract

The application discloses a spacecraft cluster trajectory planning method applied to a high-density environment, which plans a trajectory for a spacecraft through a sequential convex optimization algorithm, and makes collision avoidance constraints based on a reachable set and the geometric shape of the spacecraft itself, can effectively solve the collision avoidance problem of the spacecraft cluster trajectory planning at an intermediate time, and reduces the conservativeness of the collision constraint, and meanwhile, the method reduces fuel consumption of the trajectory planning. The application can solve the deficiency that the traditional method uses a minimum safe allowable distance to reduce the collision avoidance probability at the intermediate time. The reachable set is used to accurately calculate the position of the spacecraft at a future time, so that the spacecraft cluster can be closer to the planned distance of the neighbor spacecraft during the trajectory planning, and no collision occurs, thereby improving the success rate of the trajectory planning, reducing fuel consumption, and without significant increase in the calculation time.

Description

Technical Field

[0001] This invention relates to the field of spacecraft swarm trajectory planning technology, specifically to a method for spacecraft swarm trajectory planning applied in high-density environments. Background Technology

[0002] In recent years, with the rapid development of technologies such as spacecraft modularization and multi-satellite launches, spacecraft swarms are becoming smaller, more numerous, and denser, performing increasingly complex space missions. To reduce dependence on ground systems, enable rapid adjustments to spacecraft swarm configurations, and ensure safe operation, spacecraft swarm trajectory planning technology has attracted increasing attention. Micro-spacecraft, with their limited payload and fuel load, possess numerous physical and dynamic constraints, making them well-suited for trajectory planning using optimal control methods.

[0003] In recent years, direct methods for solving optimal control problems have emerged because they can discretize state and control variables, transforming infinite-dimensional optimal control problems into finite-dimensional convex optimization problems. These problems can then be solved using mixed-integer programming, convex optimization, and sequential convex optimization. In trajectory planning, sequential convex optimization can provide a nominal trajectory, linearizing nonlinear dynamic equations and convexizing non-convex collision avoidance constraints. Furthermore, iterative solutions and asymptotically convergent trust region constraints enable lossless information transfer between adjacent spacecraft in distributed trajectory planning.

[0004] It is worth noting that the control sequences obtained by the discretization method can only guarantee that the constraints of the problem are valid only at the sampling time. Between sampling times, it is highly likely that the constraints will be violated, especially in high-density environments where the probability of collisions between sampling times will be greatly increased. Traditional collision avoidance constraints in existing research methods generally reduce the probability of collisions by setting a spherical collision avoidance zone consisting of a minimum safe distance. Its physical meaning is the distance traveled by two spacecraft moving towards each other at their maximum thruster-allowed speed over a sampling time plus the actual physical radii of the two spacecraft.

[0005] Although the radius of the minimum safe distance can be shortened by adaptively reducing the length of the sampling time, thereby increasing the size of the feasible solution space, when the number of spacecraft is large, each additional sampling point will add a large number of constraints, which will greatly increase the time for us to solve for the optimal trajectory and the communication burden between spacecraft.

[0006] Therefore, without increasing the communication and computing burden, reducing the conservatism of the spherical collision avoidance zone has become an important challenge in solving spacecraft trajectory planning in high-density environments. Summary of the Invention

[0007] The following provides a general overview of one or more aspects to address the basic understanding of these aspects. This overview cannot elaborate on all the conceived aspects; its sole purpose is to present some concepts of one or more aspects in a simplified form to prepare for the more detailed descriptions that follow.

[0008] The purpose of this invention is to solve the above-mentioned problems and provide a spacecraft cluster trajectory planning method for high-density environments. This method uses a sequential convex optimization algorithm to plan the trajectory of spacecraft and applies collision avoidance constraints based on the reachability set and the geometry of the spacecraft itself. This method can effectively solve the collision avoidance problem at intermediate moments in the trajectory planning of multiple spacecraft clusters and reduce the conservatism of collision constraints. At the same time, this method reduces the fuel consumption of trajectory planning.

[0009] This invention proposes a spacecraft swarm trajectory planning method applicable to high-density environments, comprising:

[0010] Step 1: Establish the optimal control model for the trajectory planning of the cluster spacecraft. This optimal control model includes relative orbital dynamics constraints, initial state constraints, target state constraints, collision avoidance constraints, actuator boundary constraints, and the optimal fuel consumption objective equation. Transform the continuous optimal control model into a discrete form to obtain the discrete optimal control model.

[0011] Step 2: The discretized optimal control model can only guarantee that the constraints are effective at discrete points. To increase safety, traditional methods propose the concept of minimum safe allowable distance, using the farthest distance the spacecraft can travel within the sampling time to determine the minimum safe allowable distance. However, the actual operation of a spacecraft is not a uniform sphere. This method leads to increased conservatism in spacecraft cluster trajectory planning, and the success rate of trajectory planning in high-density environments is greatly reduced. This method accurately predicts the possible positions of the spacecraft in the future based on the reachable set, and proposes a collision avoidance constraint based on the reachable set, that is, no collision at the current time and no intersection with the reachable set at the next time. In addition, the reachable set is obtained based on the dynamic equation of the mass. Considering the size of the spacecraft body, the Minkowski sum of the spacecraft body and the reachable set is solved as the reachable region of the spacecraft body. To reduce the computational cost of the non-intersection problem of the reachable region, the minimum volume ellipsoidal envelope of the Minkowski sum is solved, transforming the non-intersection constraint of the reachable set into the non-intersection constraint of the minimum volume ellipsoidal envelope.

[0012] Step 3: For the optimization problem of non-intersecting outer envelopes of the minimum volume ellipsoid, it is equivalently transformed into the problem of a point mass not intersecting with another large ellipsoid. This operation will facilitate the subsequent use of Taylor expansion for convexity transformation. In the trajectory planning problem, it is necessary to ensure that there is no collision between spacecraft at the current time and no collision between spacecraft at the next sampling time. The former can be solved by the distance between spacecraft being greater than twice the physical radius, while the latter can be transformed into convex constraints through the above equivalent conditions. Then, sequential convex optimization can be used to solve the trajectory and control sequence.

[0013] According to a method for spacecraft cluster trajectory planning in a high-density environment based on the present invention, step one includes the following steps:

[0014] Step 1: Establish two coordinate systems for the spacecraft: the ECI coordinate system (O1-XYZ) and the LVLH coordinate system (O2-xyz). The x-axis represents the direction from the Earth's center to the reference spacecraft, the z-axis represents the direction of the reference spacecraft's angular velocity, and the y-axis follows the right-hand screw rule with the other two axes. Determine the orbital angular velocity of the reference spacecraft. The swarm trajectory planning system considered in this invention is located in a near-Earth, near-circular orbit, and the distance between spacecraft is much smaller than the orbital semi-major axis. To form a stable swarm configuration, the orbital semi-major axes of the reference spacecraft and the slave spacecraft should be the same. Based on this assumption, establish an approximate relative motion dynamic equation for slave spacecraft j, namely the CW equation:

[0015]

[0016] Where w is the orbital angular velocity of the reference spacecraft, let Represents the state of spacecraft j, u j =[u jx u jy u jz [T represents the control acceleration of the spacecraft in the LVLH coordinate system, and T represents the transpose of the matrix.] The first derivative represents the coordinate x. Let x represent the second derivative of the coordinate x, then the above expression can be transformed into:

[0017]

[0018] in Represents the state variable ζ j The first derivative, where t represents time. The system representing the state transition matrix, I 3×3 It represents the three-dimensional identity matrix.

[0019] Step 2: Determine the spacecraft's own physical parameters, including the maximum control acceleration of the thrusters. The spacecraft's true physical radius R realBased on this, the discrete form of the trajectory planning optimal control problem is established as follows:

[0020]

[0021] The objective equation is the sum of the thrusts of each spacecraft, denoted by ||·||1, and includes dynamic constraints and initial state constraints ζ. j，o Final state constraint ζ j，t Maximum thrust constraint Collision avoidance constraints at the current moment r i [k] represents the position of spacecraft i in the k-th sampling sequence, r j [k] represents the state of spacecraft j in the kth sequence, for a total of N. s A spacecraft, Δt=t k+1 -t k Indicates the sampling time. Indicates rounding down, A d =e AΔt , Here, e represents the base of the natural logarithm, and τ represents the integrand. This indicates the distance R at which spacecraft j is detected. d Neighboring spacecraft, i.e.

[0022]

[0023] t0 represents the initial time;

[0024] According to a method for spacecraft swarm trajectory planning in a high-density environment based on the present invention, step two includes the following further steps:

[0025] Step 1: Solve for the reachable set of the spacecraft at a sampling time. For a continuous system (1.1), the reachable set represents the spacecraft's state under the influence of initial state and bounded inputs at time t. f The possible positions at time t are expressed mathematically as {r j (t)|(r j (t)-c j (t0)) T N(t)(r j (t)-c j (t0))≤1}, where the symmetric positive definite matrix is... The reachable set based on the CW equation has a three-dimensional ellipsoid shape. For the maximum control acceleration constraint of the spacecraft, Let H represent the drift position of spacecraft j after time t-t0, and H satisfy...

[0026]

[0027] in

[0028]

[0029]

[0030]

[0031]

[0032]

[0033] H 23 =H 31 =H 32 =0 and

[0034] Step 2: To ensure that the two spacecraft do not intersect at time k+1, add a non-intersecting reachability set constraint for the next time step. However, the aforementioned reachability set is obtained based on the solution of a point mass. To increase safety and consider the size of the spacecraft itself, the minimum ellipsoidal approximation of the reachability set and the actual physical radius of the spacecraft can be iteratively solved.

[0035] {r j [k+1]|(r j [k+1]-c j [k]) T M(Δt)(r j [k+1]-c j [k])≤1} (1.4)

[0036] in, It is also a symmetric positive definite matrix, where β n The value conforms to the iterative formula λ is an eigenvalue of matrix N, which allows the disjoint constraint of the reachable set to be transformed into a disjoint constraint of the minimum ellipsoidal approximation.

[0037] Based on the spacecraft swarm trajectory planning problem constructed above, sequential convex optimization is used for solution. Step three includes the following further steps:

[0038] Step 1: To avoid approximate non-intersection between the two minimum ellipsoids and thus ensure the safety of spacecraft trajectory planning, the minimum distance between the two ellipsoids should be greater than 0. This is a computationally complex optimization problem. To reduce the computational load, this invention utilizes the concept of Minkowski sums. This method can equivalently transform the non-intersection constraint of two convex sets into the non-intersection constraint between a point mass and a convex set, and equivalently transform the non-intersection constraint of two ellipsoidally reachable sets into...

[0039]

[0040] Step 2: To ensure that subsequent trajectories gradually converge to the optimal value, a trust region constraint is constructed. Specifically, as the number of iterations in the sequential convex optimization increases, the distance between two adjacent iterations gradually decreases, as shown in the following equation:

[0041]

[0042] Where L m Represents the trust domain.

[0043] Step 3: To accelerate the trajectory planning solution, the non-convex collision avoidance constraints are made convex using Taylor's formula, resulting in the following formula:

[0044]

[0045] and

[0046]

[0047] The optimal control problem is ultimately obtained as follows:

[0048]

[0049] Step 4: Solve the above formula based on sequence convex optimization. The algorithm flow is shown in the specific implementation method, where the reference state... and Using the information obtained from the previous sequence convex optimization, the detailed steps of sequence convex optimization can be found in the specific implementation plan.

[0050] Compared with existing technologies, the spacecraft swarm trajectory planning method based on reachability set collision avoidance constraints proposed in this invention can overcome the shortcomings of traditional methods that use minimum safe allowable distances to reduce the collision avoidance probability at intermediate moments. Traditional methods increase the minimum safe allowable distance to ensure collision avoidance between sampling moments. However, with the same sampling time, the algorithm proposed in this patent utilizes reachability sets to accurately calculate the spacecraft's position at future moments. Therefore, during trajectory planning, the spacecraft swarm can plan closer distances to neighboring spacecraft without collisions, thereby improving the success rate of trajectory planning, reducing fuel consumption, and without significantly increasing computation time. Specific effects are shown in the attached figures. Attached Figure Description

[0051] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments of this application and their descriptions are used to explain this application and do not constitute an undue limitation of this application.

[0052] Figure 1This represents the trajectory change in this method.

[0053] Figure 2 Trajectory changes according to traditional methods;

[0054] Figure 3 To reconstruct trajectory planning diagrams for square and straight formations of spacecraft swarms in high-density environments;

[0055] Figure 4 The superiority of the algorithm of this invention compared with traditional methods is illustrated in the figure.

[0056] Figure 5 Compared with traditional methods, the algorithm of this invention improves the restricted area;

[0057] Figure 6 This diagram illustrates the reachable set, modified reachable set, and approximate modified reachable set of a spacecraft. The outermost set represents the approximate modified reachable set, the middle set represents the modified reachable set, and the innermost set represents the reachable set. Detailed Implementation

[0058] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. It should be noted that the aspects described below with reference to the accompanying drawings and specific embodiments are merely exemplary and should not be construed as limiting the scope of protection of the present invention in any way.

[0059] Step 1: Establish the optimal control problem form for spacecraft cluster trajectory planning, which includes relative orbital dynamics constraints, initial state constraints, target state constraints, collision avoidance constraints, actuator boundary constraints, and the objective equation for optimal fuel consumption. Transform the continuous optimal control into a discrete form, and then transform the problem except for the collision avoidance constraint into a convex problem.

[0060] Step 2: Improve traditional collision avoidance constraints based on reachable sets. To ensure the safety of reachable sets, solve the Minkowski equations of the spacecraft body and the reachable set at the next time step after discretization, and the reachable region of the spacecraft body. To reduce the computational cost of the non-intersecting reachable region problem, solve the minimum volume ellipsoidal envelope of the reachable region.

[0061] Step 3: The optimization problem of two non-intersecting ellipsoids is equivalently transformed into the problem of a point mass and another ellipsoid not intersecting. This operation makes it easier to expand the constraints into affine constraints using Taylor formula.

[0062] Step 4: Consider that no collision occurs between spacecraft at the current moment, and use Taylor's formula to transform the obstacle avoidance constraint into a convex constraint. Consider that no collision occurs between spacecraft at the next moment, which is the collision avoidance constraint based on reachability set proposed in Step 2 and Step 3.

[0063] Step 5: Solve the convex problem based on sequential convex optimization. The specific implementation process is as follows;

[0064] Step 6: Set the orbital parameters, physical radius, thrust range of the thrusters, collision detection range and number of spacecraft in the cluster. Randomly generate the initial state and target state of each spacecraft in the cluster. Determine the sampling time of the spacecraft. Solve for the geometry of the minimum volume envelope of the cluster and the main body after a certain sampling time according to the formula.

[0065] Step 7: Ignoring collision avoidance constraints, use convex optimization to generate the control sequence and state sequence of each spacecraft from the initial state to the target state;

[0066] Step 8: Using the control sequence and state sequence obtained from the previous convex optimization solution as the reference sequence, considering the spacecraft whose reference position is within the detection range, the affine form of the collision avoidance constraint is obtained based on the method proposed in Step 4, and the distance between the current iteration and the previous iteration state is gradually reduced to solve the convex problem containing the affine constraint.

[0067] Step 9: Repeat step 8 until the distance between the states of two adjacent iterations satisfies the convergence constraint.

[0068] Figure 1 , Figure 2 This is a graph showing the trajectory changes between the proposed method and the traditional method.

[0069] Figure 3 To reconstruct trajectory planning diagrams for square and straight formations of spacecraft swarms in high-density environments;

[0070] Figure 4 The superiority of the algorithm of this invention compared with traditional methods is illustrated in the figure.

[0071] Figure 5 Compared with traditional methods, the algorithm of this invention improves the restricted area;

[0072] This system has been experimentally proven to quickly find the optimal fuel solution using the proposed algorithm. The average experimental results of this algorithm compared with those of the ordinary spherical collision avoidance constraint algorithm across 300 sets of data are shown below. Figure 6 As shown, under the same sampling time, the trajectory planning algorithm for the ordinary spherical collision avoidance area has a much lower success rate than this algorithm, and the average fuel consumption is greater than that of this algorithm. When the sampling time is reduced to 1 / 5 of the original sampling time, the success rate of solving the spherical collision avoidance area increases, but it is still less than that of this algorithm, and the required computation time is much greater than that of this algorithm.

[0073] This diagram illustrates the trajectory planning algorithm results for spacecraft with a number of 10 to 30 spacecraft in a limited target space. It shows that the algorithm can still find the optimal solution for fuel consumption despite the increase in spacecraft density. In the cluster trajectory planning of micro and nano spacecraft, fuel consumption has a significant impact on the success of the mission and the lifespan of the spacecraft.

[0074] To simplify the explanation of this method, the above diagrams and text have been described as a series of steps. This method also has different strategies for two different tasks, and those skilled in the art can understand its principles.

[0075] Although the illustrative specific embodiments of the present invention have been described above in order to enable those skilled in the art to understand them, the present invention is not limited to the scope of the specific embodiments. Those skilled in the art can make various modifications or variations within the scope of the claims, as long as such variations are within the spirit and scope of the present invention as defined and determined by the appended claims.

Claims

1. A spacecraft cluster trajectory planning method applied to a high-density environment, characterized in that, The implementation steps of the method are as follows: step one: establishing an optimal control model for cluster spacecraft trajectory planning, the optimal control model including relative orbit dynamics constraints, initial state constraints, target state constraints, collision avoidance constraints, actuator boundary constraints and optimal fuel consumption target equation, converting the continuous optimal control model into a discrete form to obtain a discrete form of the optimal control model; Step two: the discretized optimal control model can only guarantee that the constraints are effective at discrete points, the reachable set is accurately predicted based on the reachable set of the spacecraft at the future time, and the collision avoidance constraint based on the reachable set is proposed, that is, the current time does not collide and the next time reachable set is disjoint; in addition, the reachable set is obtained based on the dynamics equation of the particle, considering the size of the spacecraft body, the Minkowski sum of the spacecraft body and the reachable set is solved as the reachable area of the spacecraft body, in order to reduce the calculation amount of the non-intersection problem of the reachable area, the minimum volume ellipsoid envelope of the Minkowski sum is solved, and the non-intersection constraint of the reachable set is converted into the non-intersection constraint of the minimum volume ellipsoid envelope; Step three: for the optimization problem of minimum volume ellipsoid envelope non-intersection, it is equivalent to convert it into a particle and another large ellipsoid non-intersection problem, and Taylor expansion is used for convexization; in the trajectory planning problem, it is ensured that the spacecraft does not collide at the current time and the next sampling time, the former is that the distance between the spacecraft is greater than twice the physical radius, and the latter is converted into a convex constraint through the equivalent condition, and then the sequence convex optimization is adopted to solve the trajectory and control sequence; In step two, the following steps are included: Step 1: Solving the reachable set of the spacecraft at a sampling time For continuous systems, the reachable set represents the positions that the spacecraft can take at a time instant under the influence of the initial state and the bounded input, which is expressed as ​ ； where k is the sampling time; denotes the state of the spacecraft at the t-th sequence; S is a symmetric positive definite matrix The shape of the reachable set based on C-W equation is a three-dimensional ellipsoid, is the maximum control acceleration constraint of the spacecraft, is the maximum control acceleration of the thruster; denotes the state of the spacecraft after the time elapsed, A is the matrix of the state space equation; is the state of the spacecraft , and satisfies ； In step one, the following steps are included: ， ， ， ， ， ， , and ; is the reference spacecraft's orbital angular velocity; Step 2: To ensure the two spacecrafts are disjoint at the time instant, the disjoint constraint of the reachable set at the next time instant is added The reachable set is solved based on the mass point. To increase the safety, the size of the spacecraft body is considered, and the minimum ellipsoid outer approximation of the reachable set and the real physical radius of the spacecraft is iteratively solved ； where is also a symmetric positive definite matrix, where the value of , is the eigenvalue of the matrix , the disjoint constraints of the reachable set are transformed into the disjoint constraints of the outer approximation of the minimum ellipsoid.

2. The spacecraft cluster trajectory planning method for high-density environment of claim 1, wherein, In step three, the following further steps are included: Step 1: Establish two coordinate systems for the spacecraft, namely the ECI coordinate system ( ) and LVLH coordinate system ( ),in The axis is the direction from the Earth's center to the reference spacecraft. The axis is the direction of the angular velocity of the reference spacecraft. The direction of the reference axis and the other two axes satisfy the right-hand screw rule; the orbital angular velocity of the reference spacecraft is determined. The swarm trajectory planning system under consideration is located in a near-Earth, near-circular orbit, and the distance between the spacecraft is much smaller than the orbital semi-major axis. To form a stable swarm configuration, the orbital semi-major axes of the reference spacecraft and the slave spacecraft are the same, and an approximate slave spacecraft is established. The equations of relative motion dynamics, namely the CW equations: ； in To reference the orbital angular velocity of the spacecraft, let represent The status of the spacecraft. Represents the control acceleration of the spacecraft in the LVLH coordinate system. Represents the transpose of a matrix. Representing coordinates The first derivative, Representing coordinates The second derivative of the above equation is then transformed into: ； wherein a representative state quantity a first derivative, a representative time, a system representative of a state transition matrix, a three-dimensional identity matrix; Step 2: Determine the spacecraft's own physical parameters, including the maximum control acceleration of the thrusters , the true physical radius of the spacecraft , and establish the discrete form of the trajectory planning optimal control problem as follows: ； where the target equation is the thrust and of each spacecraft , which represents the initial state constraint , the final state constraint , the maximum thrust constraint , the collision avoidance constraint at the current time , represents the position of the th spacecraft at the th sampling sequence represents the state of the th spacecraft at the th sequence represents the sampling time , represents the floor function , is the discretized dynamics parameter represents the base of the natural logarithm, and τ represents the integrand represents the neighbor spacecraft within the detection distance of the th spacecraft , i.e.​​​​​​ ； Represents the initial moment.

3. The spacecraft cluster trajectory planning method for high-density environment of claim 2, wherein, Step 1: In order to avoid the non-intersection of the two minimum ellipsoid envelopes and ensure the safety of the spacecraft trajectory planning, the minimum distance between the two ellipsoids is greater than 0; by using the concept of Minkowski sum, this method can equivalent convert the non-intersection constraint of two convex sets into the non-intersection constraint of a particle and a convex set, and equivalent convert the non-intersection constraint of two ellipsoid reachable sets into Step 2: In order to ensure that the subsequent trajectory can gradually converge to the optimal value, the trust region constraint is constructed; with the increase of the iteration number of the sequence convex optimization, the distance between the adjacent two iterations is gradually reduced, as follows: ； Step 3: In order to speed up the solving speed of the trajectory planning, Taylor formula convexization is used for the non-convex collision avoidance constraint, and the following formula is obtained: ； wherein represents a trust domain; And ； Finally, the following optimal control problem is obtained: ； ​ ； Step 4: Solving based on sequence convex optimization, reference state and the information obtained from the last sequence convex optimization.

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