Robust invariant set constraint sparsification method for complex robotic systems
By sparsifying the constraints of the robust invariant set, the problem of excessive computational complexity in complex robot systems is solved, achieving a balance between robustness and efficiency in real-time control, which is applicable to mobile robots and continuum surgical robots.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NANKAI UNIV
- Filing Date
- 2025-07-01
- Publication Date
- 2026-07-21
AI Technical Summary
Robust control algorithms for complex robot systems suffer from a surge in computational complexity due to the cumulative nature of invariant set construction, making real-time solutions difficult and hindering their application in practical robot systems.
By using geometric analysis and sparsification, the number of constraints in the robust invariant set is clustered and simplified. The K-means algorithm and differential evolution algorithm are used to optimize the constraint grouping, thereby reducing computational complexity and maintaining accuracy.
It significantly reduces the computational complexity of robust control algorithms, enabling them to meet the real-time control requirements of complex robotic systems. It achieves a balance between robustness and computational efficiency and is applicable to systems such as mobile robots and continuum surgical robots.
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Figure CN120686627B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to motion control methods for complex robot systems, specifically to a robust invariant set constraint sparsity method for complex robot systems, belonging to the field of robot control technology. Background Technology
[0002] Complex robotic systems, as typical multiple-input multiple-output (MIMO) systems, often possess high-dimensional state spaces and complex dynamic characteristics. Within robust control algorithm frameworks, the number of constraints grows exponentially due to the cumulative nature of invariant set construction, easily leading to a surge in computational complexity. This makes real-time solutions to online optimization problems difficult, limiting the application of algorithms in practical robotic systems; in other words, excessive computational load hinders the deployment of algorithms in real-world robotic systems. Summary of the Invention
[0003] To overcome the shortcomings of existing technologies, this invention proposes a robust invariant set constraint sparsification method for complex robot systems. The proposed method uses geometric analysis and sparsification to accurately describe the spatial geometric characteristics of the robust invariant set with a small number of key constraints, significantly reducing computational complexity and enabling robust control algorithms to meet the real-time control requirements of complex robot systems.
[0004] A robust invariant set constraint sparsity method for complex robotic systems includes:
[0005] S1: Cluster the constraints according to the constraint distribution of the convex set and divide them into multiple groups;
[0006] S2: Gather the convex set vertices in the subgroups of each group in step S1 into a region. Since the description of the constraints by the convex set vertices is redundant, simplify the redundant constraints and obtain a simplified approximate convex set.
[0007] S3. Limit the error between the simplified approximate convex set and the original convex set to a pre-defined range to obtain a constraint set that can be directly calculated.
[0008] Furthermore, the process of clustering and dividing the constraints into multiple groups in step S1 is as follows: construct the minimum axial bounding box of the target convex set, uniformly mesh the internal space of the bounding box as the outer boundary, estimate the density of the grid cells occupied by the constrained points, preset the K-neighborhood range of the grid center, and determine the grid density estimate by the following equation:
[0009] (1)
[0010] In the formula: This represents the estimated grid density. The center position of the grid cell. To constrain the position and sum of the Kth nearest neighbors A spherical space with a radius equal to the distance. for The position of the j-th constraint in space;
[0011] According to equation (1), the density values of the regions where each grid cell is located are calculated sequentially. The sliding window algorithm is used to filter out all local peak values in the constructed grid matrix S. The area surrounding the grid where the peak value is located is the clustering area of the relevant constraints. The number of extracted local peak values N is used as the number of groups of the constraint set. Then, the K-means algorithm is used to cluster them based on the constraint distribution density to form the final constraint grouping. .
[0012] Furthermore, the process of simplifying redundant convex sets in step S2 is as follows: assuming convex sets For the original unsimplified convex set, for any subset The size of the space formed by its set Regarding convex sets The approximate accuracy is positively correlated;
[0013] Therefore, the problem of approximating the constrained convex set is transformed into the problem of convex set space. The problem of maximizing the sum of all elements is represented by the following optimization model:
[0014] (2)
[0015] In the formula The sub-constraint sets after grouping are calculated based on step S1. The index of the selected point within the current group is a positive integer that does not exceed the total number of constraint points within the group. ) represents a set The convex hull formed by all the constraint points within the convex hull.
[0016] Furthermore, the process of constructing the constraint set that can be directly computed in step S3 is as follows: First, construct the original convex set. Compared with simplified approximate convex sets difference set ,extract For all constraint points, calculate the minimum Euclidean distance between the current constraint point and the outer surface and edge of the geometric convex hull corresponding to the simplified approximate convex set, and summarize all points exceeding the preset range. External constraint points to set Merge the set Compared with simplified approximate convex sets Finally, a constraint set that satisfies the target requirements is obtained. .
[0017] The advantages of this invention compared to the prior art are:
[0018] The constraint sparsification method of this application accurately describes the spatial geometric properties of robust invariant sets with a small number of key constraints through geometric analysis and sparsification processing, significantly reducing computational complexity and enabling robust control algorithms such as Tube-MPC to meet the real-time control requirements of complex robot systems; this method can achieve accurate approximation of invariant sets with minimal loss of accuracy.
[0019] The method proposed in this application achieves a balance between computational efficiency and robustness, making it possible to apply robust control algorithms that rely on invariant set construction to typical complex systems such as mobile robots and continuum surgical robots.
[0020] The proposed solution will be further described below with reference to the accompanying drawings and embodiments: Attached Figure Description
[0021] Figure 1 A rasterized partitioning diagram of the space constrained by invariant sets;
[0022] Figure 2 This is a schematic diagram of the original robust invariant set in the embodiment;
[0023] Figure 3 This is a simplified schematic diagram of the robust invariant set in the embodiment.
[0024] Figure 4 The tracking error curve based on the unsimplified robust invariant set is shown in the example.
[0025] Figure 5 The tracking error curve based on the simplified robust invariant set is shown as an example. Detailed Implementation
[0026] The embodiments of the technical solution of the present invention will now be described in detail with reference to the accompanying drawings. Unless otherwise stated, the technical or scientific terms used in this application have the ordinary meaning as understood by those skilled in the art.
[0027] Robust Positive Invariant Set (RPI):
[0028] Given the target controlled system of the robot:
[0029] (I)
[0030] In the formula: For the controlled system in The state at any given moment, for Control input at any given time, For the disturbance term, and These are the state transition matrix and the input matrix, respectively.
[0031] The robust positive invariant set reflects the range of system uncertainty. A smaller RPI makes the target controlled system more robust and less conservative, on the one hand because it does not need to ensure robustness by excessively restricting the system state, and on the other hand, a sufficiently small RPI can resist greater external disturbances and maintain the stability of the system state.
[0032] For a robot target control system of the form (I) above, its robust invariant set can be obtained through the limit set of the following equation:
[0033] (II)
[0034] In the formula: For disturbance Zhang Cheng's invariant set, where s is the number of iterations.
[0035] In formula (I), directly calculating the RPI requires multiple Minkowski summations. The constraint points generated in the previous iteration will continue to participate in the calculation as the boundaries of the convex set in the next iteration. Therefore, the number of constraint points in the set increases exponentially with the number of iterations, which increases the computational complexity. Moreover, when s is large, a large number of RPI constraint points puts a heavy burden on the computer's memory. In addition, a large number of constraints are not suitable for the input robust constraint set. The construction is difficult because the target system (I) is a time-varying system, and the constraint set... Each control cycle requires separate calculation, and the consumption of a large amount of computing resources makes it difficult for the system to achieve a fast response to the target trajectory tracking task.
[0036] In equation (II), as the number of iterations increases... As the number of constraints increases, a large number of constraints gradually converge to multiple different regions. At this point, some constraints have extremely high similarity, and most of these constraints are repetitive and redundant in describing RPI.
[0037] To address this, this implementation scheme establishes a robust positive invariant set constraint sparsification method to simplify the redundant constraint description of convex sets. This method transforms the algebraic solution problem of RPI constraints into an optimization problem based on the vertices of the convex hull (treating the convex set as a geometric convex hull, with the RPI set constraints being the vertices of the convex hull). The proposed method significantly reduces the number of constraints in the RPI set without affecting the tracking control accuracy.
[0038] The proposed method addresses motion control of complex robot systems. It comprises constraint grouping, constraint optimization based on grouping information, and constraint set post-processing. Constraint grouping aims to decompose high-dimensional objective constraints into multiple subsets to facilitate efficient solution of the subsequent optimization problem. Due to the large number of constraint points, direct solution without grouping preprocessing would result in excessive computational complexity, potentially leading to local optima or unsolvable problems. Further constraint optimization and post-processing based on this partitioning information ensure the accuracy and reliability of the simplified RPI.
[0039] The following section, in conjunction with the implementation plan, details the implementation process of robust invariant set constraint sparsity in this application:
[0040] Part 1: Constrained Convex Set Grouping
[0041] The number of iterations s of the invariant set is controlled by the gain of the auxiliary feedback. The impact is that when s is large, the number of constraints in the invariant set is enormous, and most of the constraints converge to a few fixed regions. To reduce the burden of solving subsequent optimization problems, we cluster the constraints of the convex set into multiple groups based on their distribution. In each group, we select several representative constraint points to combine to achieve an approximate description of the convex set.
[0042] To achieve reasonable grouping of all constraints in the RPI, it is necessary to determine the appropriate number of groupings for this set. For example... Figure 1 As shown, a minimum axial bounding box for the target convex set is constructed, and its internal space is uniformly meshed using this bounding box as the outer boundary. Density estimation is performed on the grid cells occupied by constrained points. This index describes the density of the constraints around the grid cells. A K-neighborhood range for the grid center is preset, and the estimated grid cell density is determined by the following equation:
[0043] (1)
[0044] In the formula: This represents the estimated grid density. The center position of the grid cell. To constrain the position and sum of the Kth nearest neighbors A spherical space with a radius equal to the distance. for The position of the j-th constraint in space.
[0045] According to equation (1), the density values of the regions where each grid cell is located are calculated sequentially. A sliding window algorithm is used to filter out all local peak values in the constructed grid matrix S. The area surrounding the grid cell where this peak value is located is the clustering region of the relevant constraints. The number of extracted local peak values N is used as the number of constraint groupings. Then, the K-means algorithm is used to cluster the constraints based on their distribution density, forming the final constraint groupings. .
[0046] Part Two: Approximation of Constrained Convex Sets
[0047] For the constraint group G obtained from the first part of the calculation, each subgroup g i The vertices of the convex set in the model are clustered within a small region, and these vertices provide highly similar and redundant descriptions of the constraints. The purpose of this method is to simplify these redundant constraints to the greatest extent possible, thereby reducing the overall computational burden of the algorithm, without significantly negatively impacting the online solution performance of subsequent robust control algorithms.
[0048] Based on the convexity property of RPI, the following theorem is given: Assume a convex set For the original unsimplified convex set, for any subset The size of the space formed by its set It is positively correlated with its approximate accuracy with respect to B.
[0049] Therefore, the problem of approximating the constrained convex set is transformed into the problem of convex set space. The problem is to maximize the sum of all possible values. The optimization model for this problem is expressed as:
[0050] (2)
[0051] In the formula: The sub-constraint sets are calculated based on the groupings obtained in Part 1. The index of the selected point within the current group is a positive integer that does not exceed the total number of constraint points within the group. ) represents a set The convex hull formed by all the constraint points within the convex hull.
[0052] Selected from each subgroup element index As decision variables in optimization problems, the number of decision variables is significantly reduced compared to directly selecting constraint points within the group. Therefore, directly selecting the index can effectively avoid the problems of low computational efficiency and difficulty in convergence of optimization solutions caused by high-dimensional decision variables.
[0053] To further reduce the difficulty of solving the algorithm and improve the approximate accuracy of simplified convex sets under the same conditions, the grouping... The disordered constraint points within the constraint are ordered. The constraint index can be determined according to the following equation:
[0054]
[0055] In the formula for The current constraint point position within the group, M is the total number of constraint points in the original convex set, and this equation describes the distance from the current constraint point to the center of the convex set. The sorted data helps reduce the jitter of the optimization algorithm during the iteration process and avoids the solution process of equation (2) from falling into an unstable state.
[0056] Considering the characteristics of the problem to be solved, this implementation scheme uses the differential evolution algorithm (DE) to solve the optimization model (2). The DE algorithm is a metaheuristic optimization algorithm based on population iterative evolution. This algorithm has fewer parameters, and its selection and mutation mechanisms can effectively balance exploration and exploitation capabilities, thus giving it better stability and robustness.
[0057] Part Three: Refined Processing of Simplified Constraint Convex Sets
[0058] To ensure the reliability of the simplified algorithm and to extend the robust control algorithm relying on invariant sets to control tasks in other scenarios, the simplified approximate convex set generated in the second part is further refined. This method limits the error between the simplified convex set and the original convex set to a pre-defined range. Within.
[0059] First, construct the primitive convex set. Compared with simplified approximate convex sets difference set ,extract For all constraint points, calculate the minimum Euclidean distance between the current constraint point and the outer surface and edges of the corresponding geometric convex hull of the simplified approximate convex set. This distance reflects the deviation of the simplified convex set relative to the current constraint. Summarize all points exceeding the preset range. External constraint points to set Merge the set Compared with simplified approximate convex sets Finally, a constraint set that satisfies the target requirements is obtained. .
[0060] income This is a simplified set of constraints that can be directly calculated. Compared to the original set, this set removes most of the redundant constraints, thus greatly improving efficiency. The computational efficiency of the online computing process makes the online deployment of control algorithms possible.
[0061] Part Four: Testing and Verification
[0062] Given a target robust invariant set constraint, it is simplified based on the sparsification method of this constraint. Figure 2 This illustrates a given robust invariant set (the black lines represent the convex hull, indicating constraint boundaries, and the green lines connect the constraint points), which contains 768 constraints. Figure 3 The figure shows the simplified invariant set result obtained based on the proposed method (the black lines in the figure are convex hull surfaces, representing constraint boundaries, and the green lines are the lines connecting constraint points). The constraint space formed by the invariant sets is described by a three-dimensional convex hull.
[0063] Table 1. Comparison of the number of constraints before and after simplification
[0064] Standard RPI 768 -- -- Simplify RPI 58 0.007 0.052
[0065] The table shows relevant information about the convex constraint space formed by the invariant sets before and after simplification: the simplified invariant set has 58 constraints, achieving a simplification rate of 92.4%. In terms of accuracy, the average approximation error of the simplified invariant set compared to the original invariant set is 0.007 mm, and the maximum approximation error is 0.052 mm. This is negligible compared to errors introduced by measuring equipment, and therefore will not affect its implementation in actual robot control systems. Furthermore, based on these two robust invariant sets, the standard TubeMPC algorithm was used to track the same target trajectory for the controlled system according to equation (I) (control simulation, the target trajectory is a circular trajectory with a diameter of 60 mm), yielding the following results: Figure 4 and Figure 5 The error curves shown indicate that there is no significant difference in tracking error results between the two methods. However, the constraint complexity of the simplified invariant set is greatly reduced, which provides a prerequisite for the efficient online implementation of some control algorithms (such as TubeMPC).
[0066] This application discloses preferred embodiments as described above, but it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed structure and technical content to create equivalent embodiments without departing from the scope of the present invention, and all such modifications or alterations shall still fall within the scope of the present invention.
Claims
1. A robust invariant set constraint sparsity method for complex robot systems, characterized in that: The method includes: S1: Cluster the constraints according to the constraint distribution of the convex set and divide them into multiple groups; The process of clustering the constraints and dividing them into multiple groups in step S1 is as follows: Construct the minimum axial bounding box of the target convex set, and uniformly mesh its internal space using this bounding box as the outer boundary. Estimate the density of the grid cells occupied by the constrained points, and preset the K-neighborhood range of the grid center. The estimated grid density is determined by the following equation: （1） In the formula: This represents the estimated grid density. The center position of the grid cell. To constrain the position and sum of the Kth nearest neighbors A spherical space with a radius equal to the distance. for The position of the j-th constraint in space; According to equation (1), the density values of the regions where each grid cell is located are calculated sequentially. The sliding window algorithm is used to filter out all local peak values in the constructed grid matrix S. The area surrounding the grid where the peak value is located is the clustering area of the relevant constraints. The number of extracted local peak values N is used as the number of groups of the constraint set. Then, the K-means algorithm is used to cluster them based on the constraint distribution density to form the final constraint grouping. ; S2: The convex set vertices of the subgroups in the grouping of step S1 are clustered within a region. Since the description of the constraints by the convex set vertices is redundant, the redundant constraints are simplified to obtain a simplified approximate convex set. The process of simplifying redundant convex sets in step S2 is as follows: Assuming convex sets For the original unsimplified convex set, for any subset The size of the space formed by its set Regarding convex sets The approximate accuracy is positively correlated; Therefore, the problem of approximating constrained convex sets is transformed into the problem of convex set spaces. The problem of maximizing the sum of all elements is represented by the following optimization model: （2） In the formula The sub-constraint sets after grouping are calculated based on step S1. The index of the selected point within the current group is a positive integer that does not exceed the total number of constraint points within the group. ) represents a set The convex hull formed by all the constraint points within the convex hull; S3. Limit the error between the simplified approximate convex set and the original convex set to a pre-defined range to obtain a constraint set that can be directly calculated.
2. The robust invariant set constraint sparsity method for complex robot systems according to claim 1, characterized in that: Step S2 also includes grouping The disordered constraint points within are ordered, and the specific process is as follows: The index of the constraint is determined according to the following equation: In the formula: for The current constraint point position within the group, M is the total number of constraint points in the original convex set, and the DE algorithm is used to solve the optimization model (2).
3. The robust invariant set constraint sparsity method for complex robot systems according to claim 1, characterized in that: The process of constructing the constraint set that can be directly calculated in step S3 is as follows: First, construct the primitive convex set. Compared with simplified approximate convex sets difference set ,extract For all constraint points, calculate the minimum Euclidean distance between the current constraint point and the outer surface and edge of the geometric convex hull corresponding to the simplified approximate convex set, and summarize all points exceeding the preset range. External constraint points to set Merge the set Compared with simplified approximate convex sets Finally, a constraint set that satisfies the target requirements is obtained. .
4. The robust invariant set constraint sparsity method for complex robot systems according to claim 1, characterized in that: It also includes a simplified testing and verification process for robust invariant set constraints, specifically: Given a robust invariant set with 768 constraints, the constraint space formed by the invariant set is described by a three-dimensional convex hull. The simplified invariant set has 58 constraints, with a simplification rate of 92.4%. The average approximation error of the simplified invariant set compared to the original invariant set is 0.007 mm, and the maximum approximation error is 0.052 mm.