Multi-degree-of-freedom mechanical arm control method based on quantum particle swarm optimization of migration strategy

Abstract

The invention discloses a multi-degree-of-freedom mechanical arm control method based on quantum particle swarm optimization of a migration strategy. The method comprises the steps that system modeling is conducted; mPC to-be-adjusted parameter definition and constraint processing are carried out; adopting, adopting, and introducing penalty terms to construct a composite fitness function; updating particle positions by adopting a basic or enhanced quantum updating mode; triggering conditions are judged in a group diversity measurement mode, and when the conditions are met, a migration strategy and dynamic subgroup division are executed; the global optimal solution and the fitness value are loaded into a real-time MPC controller, and online adjustment is carried out; a prediction equation is constructed, tracking errors and energy consumption optimization are converted into a standard quadratic programming problem, and the solving precision is dynamically adjusted in combination with Cholesky pre-decomposition, a structured sparse solver and a warm-start and early stop strategy; and a closed-loop adaptive control system is constructed. According to the method, the precision, robustness and real-time performance of trajectory tracking control of the mechanical arm can be effectively improved.

Description

Technical Field

[0001] This invention belongs to the field of robot intelligent control technology, specifically a multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy. Background Technology

[0002] In recent years, with the rapid development of robotics and intelligent manufacturing, multi-degree-of-freedom (DOF) robotic arms have been widely used in various fields, such as industrial assembly, precision machining, medical rehabilitation, and service robots. Six-DOF robotic arms, due to their large workspace, motion flexibility, and strong task adaptability, have become indispensable key equipment in automated production lines. However, as a complex system with multiple inputs and outputs and high nonlinearity, six-DOF robotic arms often face numerous engineering challenges in practical applications, such as parameter uncertainty, frictional nonlinearity, joint coupling, and external disturbances. These challenges necessitate meeting the requirements of high-precision trajectory tracking, smooth motion, and real-time performance while also considering energy consumption and safety, placing higher demands on the robotic arm's control system.

[0003] In these complex environments, Multi-Process Control (MPC) is an effective control method widely used in robotic arm trajectory tracking tasks due to its ability to explicitly handle system state and input constraints and to make systematic trade-offs among multiple objectives. MPC is based on model prediction and generates control commands by solving finite-time domain optimization problems online, demonstrating significant advantages in multi-objective tasks such as tracking accuracy, smoothness, and energy consumption. However, MPC has high requirements for model accuracy and online solution efficiency, especially when dealing with highly coupled hyperparameters, where tuning these parameters becomes a complex problem. Specifically, the hyperparameters in MPC constitute a highly coupled search space, and manually tuning these parameters is both time-consuming and prone to getting trapped in local optima. Furthermore, the computational burden of online solution for MPC also limits its deployment in real-time systems.

[0004] To address these problems, intelligent optimization algorithms offer an effective solution. Intelligent optimization algorithms are a class of global search methods inspired by biological behavior or physical processes in nature. Algorithms developed based on this principle include Genetic Algorithm (GA), Differential Evolution (DE), Ant Colony Algorithm (ACO), Simulated Annealing (SA), and Quantum Behavioral Particle Swarm Optimization (QPSO). These algorithms search for optimal or near-optimal solutions in complex solution spaces through mechanisms such as simulated evolution, group cooperation, or energy reduction. Compared to gradient-based local optimization methods, intelligent optimization algorithms do not require the differentiability of the objective function and can better handle non-convex, multimodal, and high-dimensional optimization problems. Therefore, they are widely used in tasks such as controller parameter optimization, trajectory smoothing parameter tuning, and multi-objective tradeoffs. Quantum Behavioral Particle Swarm Optimization (QPSO), as an advanced intelligent optimization algorithm, features a quantum probabilistic cloud update mechanism and maintains population diversity through global mean (mbest), thus performing exceptionally well in high-dimensional control parameter tuning tasks where objective evaluation is costly. However, standard QPSO may face trade-offs between convergence speed and local refinement in complex search spaces, as well as sensitivity to parameter tuning, which may make it difficult for the algorithm to balance global exploration and local exploitation in certain situations.

[0005] Therefore, there is an urgent need to provide a comprehensive optimization control method that can simultaneously optimize MPC hyperparameters, improve control accuracy, smoothness, and energy consumption, solve high-dimensional coupling problems, avoid local optimum traps, and be effectively deployed in real-time environments, so as to improve the accuracy, robustness, and real-time performance of six-degree-of-freedom robotic arm trajectory tracking. Summary of the Invention

[0006] To address the problems existing in the prior art, this invention provides a multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy. This method is simple to implement and has low implementation costs, and it can effectively improve the accuracy, robustness, and real-time performance of trajectory tracking control for a six-degree-of-freedom robotic arm.

[0007] To achieve the above objectives, this invention provides a multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy, comprising the following steps: Step 1: System Modeling; S11: Establish a nonlinear dynamic model of a six-degree-of-freedom robotic arm, define the state vector, and map joint errors to end-effector pose errors using the Jacobian matrix; S12: Based on the working point, the nonlinear model is locally linearized and discretized into a step-by-step prediction model, generating trajectory sequences of joint angles, torques and end poses; Step 2: Definition and constraint handling of MPC parameters to be adjusted; The hyperparameters to be tuned include the control time-domain parameters, control input sequence, constraints, weight matrix, and parameter boundaries; constraint handling achieves flexible control through linear inequalities or slack variables, while performance is improved by combining dynamic objective optimization. Step 3: Construct a composite fitness function; use Measuring trajectory accuracy Assess joint errors, Controlling smoothness, Energy consumption is estimated, and constraints are handled using the Huber loss penalty term. Finally, a composite fitness function is constructed by combining various indicators with a weighted scalar quantization. ; Step 4: Optimize MPC parameters using DM-QPSO; S41: Depending on the different optimization stages, the particle position is updated using either basic quantum update or enhanced quantum update. S42: The triggering condition is determined by measuring the population diversity. When the triggering condition is met, the migration strategy and dynamic subgroup partitioning are executed. During the execution of the migration strategy, the worst particle is partially replaced or completely replaced. At the same time, the population is divided into multiple subgroups according to the diversity index. Quantum updates are performed independently in each subgroup, and the optimal solutions between subgroups are exchanged periodically. The process ends when the maximum number of iterations is reached or the convergence accuracy requirement is met. S43: Output the global optimal solution and the corresponding fitness value as the parameters after DM-QPSO optimization; Step 5: Load the optimal parameters into the real-time MPC; The global optimal solution optimized by DM-QPSO and the corresponding fitness value are loaded into the real-time MPC controller; during real-time operation, trajectory tracking is performed using the parameters optimized by DM-QPSO, and online adjustments are made to ensure a smooth transition of the system when switching to new parameters without causing transient instability. Step Six: Online Scrolling MPC Control; The prediction equation is constructed using the current linearized discrete model, transforming the optimization of tracking error and control energy consumption into a standard quadratic programming problem, and linear constraints are applied to ensure feasibility. Simultaneously, Cholesky pre-decomposition, a structured sparse solver, and warm-start and early-stop strategies are employed to dynamically adjust the solution accuracy during peak computational loads. Finally, the initial control input is achieved through rolling optimization. Real-time execution; Step 7: Construct a closed-loop adaptive control system; Real-time tracking of key performance indicators under the sliding window; if detected When the threshold is exceeded, the high-frequency energy ratio of the control input exceeds the limit, or a constraint violation occurs in several consecutive samples, the incremental DM-QPSO online parameter optimization mechanism is triggered. The current parameter neighborhood is used to start a small-range search or local sensitivity parameter optimization strategy to dynamically adjust the MPC parameters while ensuring real-time performance. The MPC parameters are updated smoothly through hot switching technology. At the same time, historical parameter configurations are retained for quick rollback, forming a closed-loop adaptive control system of monitoring-evaluation-optimization.

[0008] As a preferred option, in step one, the specific process of S11 is as follows: Establish a nonlinear dynamic model of the six-degree-of-freedom robotic arm according to formula (1); define a 12-dimensional state vector according to formula (2). According to formula (3), the joint space error is... Through the Jacobian matrix Mapped to end-effector pose error This enables the quantitative representation of task space errors. (1); (2); (3); In the formula, It is a quality matrix; It is the Coriolis force matrix; It is the gravity matrix; For friction; It is a control input; This is the joint position vector; This is the joint velocity vector; This is the joint acceleration vector; This represents the joint space error.

[0009] As a preferred option, in step one, the specific process of S12 is as follows: At the discrete sampling time, based on the current operating point... The nonlinear dynamic model is expanded by first-order Taylor expansion. A locally linearized approximation model is established according to formula (4). The continuous model is then discretized into a step-by-step prediction model according to formula (5). Finally, the trajectory sequence of joint angle, torque and end pose is generated using the discrete model to provide data support for optimization calculation. (4); (5); In the formula, This is the state transition matrix; This represents the state error at the current moment. The input matrix; This is for input error; Here is the state transition matrix of a continuous-time system. , The sampling period; For the first Error state at time, ; Let be the input matrix of the discrete-time system. ; For the first Input error at time, ; This is process noise.

[0010] As a preferred embodiment, in step two, the time-domain parameters include the prediction range. and control range Prediction range Decide to optimize window length and control range The balance between control strength and stability; control input sequence The constraints are obtained according to formula (6); the constraints include hard constraints and soft constraints. Hard constraints include joint limits, speed limits, and control input limits; soft constraints are obtained through relaxation variables. The transformation uses linear inequalities; the weight matrix satisfies symmetric positive definiteness to ensure the convexity of the cost function; parameter boundaries: all parameters are set with upper and lower bounds to ensure feasibility; (6); In the formula, This represents the control input in the control time domain. , and These are the lower and upper limits for controlling the input, respectively.

[0011] As a preferred option, in step three, a composite fitness function is constructed. The process is as follows: S31: Obtain the end position and attitude error according to formula (7) root mean square error To measure trajectory accuracy; the cumulative absolute error of each joint is obtained according to formula (8). The rate of change of control input is obtained according to formula (9). To measure smoothness; the input energy measure is obtained according to formula (10). ; (7); (8); (9); (10); In the formula, , for The actual value at time, for Reference value at any given time; For the first The attitude error of each variable; For the first Time-based control input; S32: Introduce penalty items for constraint violations Convert hard constraints to soft constraints to enhance robustness; S33: The composite fitness function is obtained by using a scalar weighted combination of various indices according to formula (11). ; (11); In the formula, To constrain the penalty for violations; weighting Used for scale normalization or preference setting.

[0012] As a preferred option, in step S41 of step four, when updating the particle position using the basic quantum update method, a dual-guidance mechanism of individual historical optimal position and global optimal position is used to jointly drive the particle motion, so as to achieve a balance between local development and global exploration; at the same time, the first quantum position is obtained according to formula (12). The particle in the first Substitute position ; (12); In the formula, For the first The particle in the first The position of the generation; This represents the optimal position in the particle's history. The globally optimal position; and These are two different acceleration constants; and Two distinct random numbers; symbol This indicates element-wise multiplication.

[0013] As a preferred option, in step S42 of step four, when updating particle positions using an enhanced quantum update method, the average optimal position is introduced as the group knowledge center. The convergence factor of the quantum tunneling effect is dynamically adjusted through logarithmic transformation, encouraging exploration in the early stage and strengthening development in the later stage. At the same time, the first position is obtained according to formula (13). The particle in the first Substitute position ; (13); In the formula, As the local center, ; The convergence factor; Indicates the first The average optimal position of the particles. , Population size.

[0014] As a preferred option, in step four, S42, the migration strategy is executed as follows: S42-1: Obtain the first according to formula (14) The average Euclidean distance between each particle and all other particles in the population. ; (14); In the formula, , For the first and the The position vectors of each particle; S42-2: Obtain the average individual diversity of the entire population according to formula (15). ; (15); S42-4: When Below the preset threshold Or, several consecutive generations of globally optimal improvement Below the stagnation convergence threshold When the triggering condition is met; S42-4: According to formula (16) for the worst particle Perform partial replacement, or directly replace with the global optimal solution according to formula (17); (16); (17); In the formula, This is a dynamically adjusted factor.

[0015] As a preferred option, the online rolling MPC control process in step six is ​​as follows: S61: At each sampling time, based on the current linearized discrete model and Construct state transition matrices respectively The cumulative effect matrix of the input matrix And construct the stacking prediction equation for the system dynamics according to formula (18); (18); In the formula, To predict the stacked vector for all states in the time domain; Current moment The initial state increment vector; To control the stacked vector of control increments within the time domain; This is the cumulative effect of measurable interference; S62: Obtain the tracking error and control energy consumption optimization target according to formula (19); (19); In the formula, To predict the first in the time domain The system state vector at time t; To predict the first in the time domain The reference state vector at time t; This is the state error weight matrix; To control the input weight matrix; To predict the length of the time domain; S63: Transform formula (19) into the standard quadratic programming form in formula (20) by matrix transformation; (20); In the formula, Represents the Hessian matrix; This is the vector of linear term coefficients; A constant term that is independent of the optimization variables; S64: Construct linear inequality constraints according to formula (21), and ensure that the control quantity meets the physical constraints by applying linear constraints; (twenty one); In the formula, This is the constraint matrix; To constrain the right-hand side; This is the state deviation influence matrix; S64: Execute the first-step control after real-time QP solution. The system then moves forward one step and repeats the process. S65: Yes Perform Cholesky pre-decomposition on the matrix to reduce solution complexity; utilize To mitigate the block diagonal sparsity of the matrix, a dedicated solver is used to reduce computational load. The QP solution is initialized based on the previous time-optimal solution. If the convergence condition is met early in the iteration, the QP solution is terminated early. If the CPU load is too high, the solution accuracy is reduced to prioritize the stability of the control cycle. As a preferred option, in step seven, the key performance indicators include RMSE error, IAE growth rate, high-frequency energy ratio of control input, number of constraint violations, and QP solution delay.

[0016] Traditional control methods face numerous challenges in trajectory tracking of six-DOF robotic arms. First, the high nonlinearity and strong coupling of the robotic arm model make it difficult for traditional control algorithms to find the global optimum in the high-dimensional parameter space. Second, while MPC (Predictive Control Programming) can effectively handle state and input constraints and balance multiple objectives, the adjustment of its hyperparameters, such as prediction window length, control window, and weight matrix, is highly coupled, and traditional manual tuning methods are time-consuming and prone to getting trapped in local optima. Meanwhile, online solution computation is heavy, and real-time performance issues often limit the application of MPC in complex control tasks. Therefore, existing control methods struggle to achieve an ideal trade-off between accuracy, smoothness, energy consumption, and real-time performance, especially in environments with dynamic uncertainties, external disturbances, and noise. To address this, this invention proposes a quantum particle swarm optimization algorithm based on a diverse migration strategy for automatic tuning of predictive control parameters and trajectory tracking of a six-DOF robotic arm model. Firstly, during system modeling, the accuracy of the end-effector pose is ensured through a nonlinear model and Jacobian matrix mapping, making it suitable for high-precision control tasks. Meanwhile, local linearization reduces the computational burden, and the discretized model is suitable for embedded systems to achieve rapid iterative control. The step-by-step prediction model can dynamically compensate for external disturbances, improving system stability. Generating trajectory sequences supports complex path planning, meeting diverse operational needs. Furthermore, through the collaborative design of parameter tuning and constraint handling, efficient control of complex systems is achieved. Simultaneously, by combining data optimization and real-time feedback, performance and robustness are balanced. Moreover, through fusion... , , and Four metrics are used, and a penalty term is introduced to handle constraints. Factors such as end-effector position / attitude RMSE, cumulative absolute error (IAE) of each joint, control input change rate, and energy consumption estimation are considered, achieving a reasonable trade-off between tracking accuracy, joint error, motion smoothness, and energy consumption. Furthermore, key hyperparameters of the MPC are used as optimization variables, and the DM-QPSO algorithm, particularly suitable for optimization tasks with multiple hyperparameters and complex cost functions, is employed for optimization. The DM-QPSO algorithm maintains population diversity through a diversity migration strategy and enhances global search capability, effectively improving the convergence and robustness of the optimization process, thus avoiding the local optimum traps that may occur in traditional QPSO in complex search spaces. Simultaneously, by dynamically dividing the subgroup and adjusting the subgroup structure during iteration, combined with information exchange between subgroups and a quantum update strategy based on the mean best position (mbest), population diversity is maintained and global escape capability is enhanced, thus exhibiting more robust convergence and optimization performance in high-dimensional and strongly constrained problems. The DM-QPSO optimizer, which integrates dynamic subgroup partitioning, diversity measurement based on average Hamming distance, and a migration and replacement mechanism, balances population diversity and convergence efficiency, solving the problems of high dimensionality and computational burden in MPC hyperparameter tuning. Thus, by introducing the DM-QPSO optimization algorithm to dynamically optimize key MPC hyperparameters, such as prediction time domain, control time domain, state / input weight matrix, and constraint relaxation coefficients, and combining them with a composite fitness function to comprehensively consider multiple objectives such as end-position / attitude error, cumulative absolute error (IAE) of each joint, and rate of change of control input, this avoids the manual tuning, high-dimensional coupling, and local optima problems of traditional methods. It significantly improves the adaptive optimization efficiency and robustness of MPC parameters under complex constraints, and achieves a balance between control accuracy, smoothness, and energy consumption. Furthermore, the globally optimal parameters optimized by DM-QPSO are loaded into the real-time MPC controller, and a smooth transition is achieved through online adjustment strategies. This not only utilizes optimized parameters to improve trajectory tracking accuracy and system stability but also avoids transient instability caused by parameter switching, thereby ensuring the robustness and adaptability of the real-time control system. Subsequently, the optimized parameters are applied to a real-time linearized finite-time domain MPC controller, and combined with constraint relaxation and an efficient quadratic programming solution strategy to address model uncertainties and external disturbances, significantly improving the computational efficiency and dynamic response capability of the real-time control system. Finally, by real-time monitoring of RMSE, control input energy ratio, and constraint violations, incremental DM-QPSO online parameter optimization is triggered. Combined with local sensitivity analysis and hot-switching technology, dynamic adjustment and smooth updates of MPC parameters are achieved, while retaining historical configurations to support rapid rollback, significantly improving the system's real-time performance, robustness, and adaptability under complex operating conditions.

[0017] This invention accelerates convergence through quantum behavior sampling, dynamic subgroup partitioning, and dimensionality reduction search. Simultaneously, it explicitly introduces a control rate-of-change term into the fitness function to suppress chattering, smooth control input, and reduce energy consumption. Ultimately, it achieves automated parameter tuning and engineering reproducibility for six-degree-of-freedom (DOF) robotic arm trajectory tracking, effectively solving the inefficiencies of traditional methods that rely on empirical parameter tuning and the problem of being trapped in local optima in high-dimensional non-convex parameter spaces. It achieves comprehensive improvements in tracking accuracy, control smoothness, and robustness. The invention was validated in a simulation environment incorporating model uncertainty, external disturbances, sensor noise, and control delay. The validation results demonstrate that this invention exhibits superior tracking accuracy, robustness, and real-time performance, significantly improving the trajectory tracking accuracy, system robustness, and real-time performance of a six-DOF robotic arm. It provides an efficient, automated, and reliable optimization scheme for the practical engineering application of six-DOF robotic arms.

[0018] This method is simple to implement and has low implementation costs. It can effectively improve the accuracy, robustness and real-time performance of trajectory tracking control for six-degree-of-freedom robotic arms. Attached Figure Description

[0019] Figure 1 This is the method flow of the present invention; Figure 2 This is a system structure diagram of the model proposed in this invention; Figure 3 This is the optimized convergence curve of the algorithm proposed in this invention; Figure 4 This is a comparison diagram of the control inputs of the model proposed in this invention and the traditional control algorithm; Figure 5 This is a comparison chart of the outputs of the model proposed in this invention and the traditional control algorithm. Detailed Implementation

[0020] The invention will now be further described with reference to the accompanying drawings.

[0021] like Figure 1 As shown, this invention provides a multi-degree-of-freedom robotic arm control method based on a migration strategy and quantum particle swarm optimization, comprising the following steps: Step 1: System Modeling; S11: Establish a nonlinear dynamic model of a six-degree-of-freedom robotic arm to describe the motion equations of the robotic arm; define the state vector and map the joint error to the end-effector pose error through the Jacobian matrix to realize the quantification of task space error; S12: Based on the working point, the nonlinear model is locally linearized and discretized into a step-by-step prediction model, generating trajectory sequences of joint angles, torques and end poses, providing data support for optimization calculations; Step 2: Definition and constraint handling of parameters to be tuned in MPC (Model Predictive Control); The hyperparameters to be tuned include the control time-domain parameters, control input sequence, constraints, weight matrix, and parameter boundaries; constraint handling achieves flexible control through linear inequalities or slack variables, while performance is improved by combining dynamic objective optimization. Step 3: To comprehensively evaluate indicators such as trajectory tracking accuracy, control smoothness, and energy consumption, a composite fitness function is constructed. ; use Measuring trajectory accuracy Assess joint errors, Controlling smoothness, Energy consumption is estimated, and constraints are handled using the Huber loss penalty term. Finally, a composite fitness function is constructed by combining various indicators with a weighted scalar quantization. ; Step 4: Optimize MPC parameters using DM-QPSO (Diversity Migration-based Quantum Behaved ParticleSwarm Optimization); S41: After population initialization, depending on the optimization stage, particle positions are updated using either basic quantum update or enhanced quantum update. The updated candidate parameters are then mapped within the boundary for evaluation. And update and As a preferred approach, the basic quantum update method is used to update particle positions during the regular iteration process, while the enhanced quantum update method is used when the population polymorphism is insufficient. S42: To avoid premature convergence and improve global escape capability, triggering conditions are determined by measuring population diversity. When the triggering conditions are met, a migration strategy and dynamic subgroup partitioning are executed. During the migration strategy, the worst-performing particle is partially replaced or completely replaced. Simultaneously, the population is divided into multiple subgroups according to the diversity index. Quantum updates are performed independently within each subgroup, and the optimal solutions between subgroups are periodically exchanged to balance development intensity and exploration breadth. The process ends when the maximum number of iterations is reached or the convergence accuracy requirement is met. S43: Output the global optimal solution and the corresponding fitness value as the parameters after DM-QPSO optimization; Step 5: Load the optimal parameters into the real-time MPC; The global optimal solution optimized by DM-QPSO and the corresponding fitness value are loaded into the real-time MPC controller; during real-time operation, trajectory tracking is performed using the parameters optimized by DM-QPSO, and online adjustments are made to ensure a smooth transition of the system when switching to new parameters without causing transient instability. Step Six: Online Scrolling MPC Control; By constructing prediction equations using the current linearized discrete model, the optimization of tracking error and control energy consumption is transformed into a standard quadratic programming (QP) problem, with linear constraints applied to ensure feasibility. Simultaneously, Cholesky pre-decomposition, a structured sparse solver, and warm-start and early-stop strategies are employed to dynamically adjust the solution accuracy during peak computational loads. Finally, the initial control input is achieved through rolling optimization. Real-time execution; Step 7: Construct a closed-loop adaptive control system; Real-time tracking of key performance indicators under the sliding window is preferred. These key performance indicators include RMSE error, IAE growth rate, high-frequency energy ratio of control input, number of constraint violations, and QP solution delay. When the threshold is exceeded, the high-frequency energy ratio of the control input exceeds the limit (characterizing chattering), or constraints are violated in several consecutive samples, the incremental DM-QPSO online parameter optimization mechanism is triggered. The current parameter neighborhood is used to perform a small-range warm-start search or a local sensitivity parameter optimization strategy (to perform local optimization on several key parameters identified by sensitivity analysis). The MPC parameters are dynamically adjusted while ensuring real-time performance, and the MPC parameters are smoothly updated through hot switching technology. At the same time, historical parameter configurations are retained for quick rollback, forming a closed-loop adaptive control system of monitoring-evaluation-optimization.

[0022] As a preferred option, in step one, the specific process of S11 is as follows: A nonlinear dynamic model of the six-degree-of-freedom robotic arm is established according to formula (1) to represent the kinematic and dynamic relationships of the robotic arm using standard nonlinear dynamic equations; a 12-dimensional state vector is defined according to formula (2). This is used for spatial state description and provides a basis for state space analysis; according to formula (3), the joint space error is... Through the Jacobian matrix Mapped to end-effector pose error This enables the quantitative expression of task space error and facilitates the calculation of subsequent evaluation indicators. (1); (2); (3); In the formula, It is a quality matrix; It is the Coriolis force matrix; It is the gravity matrix; For friction; It is a control input; This is the joint position vector; This is the joint velocity vector; This is the joint acceleration vector; This represents the joint space error.

[0023] As a preferred option, in step one, the specific process of S12 is as follows: At the discrete sampling time, based on the current operating point... A first-order Taylor expansion is performed on the nonlinear dynamic model. A locally linearized approximation model is established according to formula (4). Then, according to formula (5), the sampling period is calculated. The continuous model is discretized into a step-by-step prediction model for MPC prediction and simulation evaluation. The discretized linear model is used to generate state and input response sequences for fitness evaluation. Finally, the discrete model is used to generate trajectory sequences of joint angles, torques and end poses to provide data support for fitness evaluation and optimization calculation. (4); (5); In the formula, This is the state transition matrix; This represents the state error at the current moment. The input matrix; This is for input error; Here is the state transition matrix of a continuous-time system. , The sampling period; For the first Error state at time, ; Let be the input matrix of the discrete-time system. ; For the first Input error at time, ; This is process noise.

[0024] As a preferred embodiment, in step two, the time-domain parameters include the prediction range. and control range Prediction range Decide to optimize window length and control range The balance between control strength and stability; control input sequence The constraints are obtained according to formula (6); the constraints include hard constraints and soft constraints. Hard constraints include joint limits, speed limits, and control input limits. Soft constraints are obtained through relaxation variables. The transformation uses linear inequalities; the weight matrix satisfies symmetric positive definiteness to ensure the convexity of the cost function; parameter boundaries: all parameters are set with upper and lower bounds to ensure feasibility; (6); In the formula, This represents the control input in the control time domain. , and These are the lower and upper limits for controlling the input, respectively.

[0025] As a preferred option, in step three, a composite fitness function is constructed. The process is as follows: S31: Obtain the end position and attitude error according to formula (7) root mean square error To measure trajectory accuracy; the cumulative absolute error of each joint is obtained according to formula (8). (Integrated absolute error); The rate of change of control input is obtained according to formula (9). To measure smoothness; the input energy measure is obtained according to formula (10). ; (7); (8); (9); (10); In the formula, , for The actual value at time, for Reference value at any given time; For the first The attitude error of each variable; For the first Time-based control input; S32: Introduce penalty items for constraint violations Converting hard constraints to soft constraints enhances robustness; as a preferred option, Huber loss or other robust loss functions can be used to reduce the interference of outliers. S33: The composite fitness function is obtained by using a scalar weighted combination of various indices according to formula (11). ; (11); In the formula, To constrain the penalty for violations; weighting Used for scale normalization or preference setting; As a preferred option, in step S41 of step four, when updating the particle position using the basic quantum update method, a dual-guidance mechanism of individual historical optimal position and global optimal position is used to jointly drive the particle motion, so as to achieve a balance between local development and global exploration; at the same time, the first quantum position is obtained according to formula (12). The particle in the first Substitute position ; (12); In the formula, For the first The particle in the first The position of the generation; This represents the optimal position in the particle's history. The globally optimal position; and These are two different acceleration constants; and Two distinct random numbers; symbol This indicates element-wise multiplication.

[0026] As a preferred option, in order to preserve the quantum guidance characteristics, an enhanced quantum update method can also be used when generating randomized seeds for candidates; in step S42 of step four, when updating particle positions using the enhanced quantum update method, the average optimal position is introduced as the group knowledge center, and the convergence factor of the quantum tunneling effect is dynamically adjusted through logarithmic transformation, encouraging exploration in the early stage and strengthening development in the later stage; at the same time, the first is obtained according to formula (13). The particle in the first Substitute position ; (13); In the formula, As the local center, ; The convergence factor; Indicates the first The mean best position (mbest) of a generation of particles. , Population size.

[0027] As a preferred option, in step four, S42, the migration strategy is executed as follows: S42-1: Obtain the first according to formula (14) The average Euclidean distance between each particle and all other particles in the population. ; (14); In the formula, , For the first and the The position vectors of each particle; Population size; S42-2: Obtain the average individual diversity of the entire population according to formula (15). , used to quantify the uniformity of population distribution; (15); S42-4: When Below the preset threshold Or, several consecutive generations of globally optimal improvement Below the stagnation convergence threshold When the triggering condition is met; S42-4: According to formula (16) for the worst particle Perform partial substitution to achieve asymptotic improvement, or directly replace with the global optimal solution according to formula (17), which is suitable for premature convergence scenarios; (16); (17).

[0028] In the formula, This is a dynamically adjusted factor.

[0029] As a preferred option, the online rolling MPC control process in step six is ​​as follows: S61: At each sampling time, based on the current linearized discrete model and Construct state transition matrices respectively The cumulative effect matrix of the input matrix And construct the stacking prediction equation for the system dynamics according to formula (18); (18); In the formula, To predict the stacked vector for all states in the time domain; Current moment The initial state increment vector; To control the stacked vector of control increments within the time domain; This is the cumulative effect of measurable interference; S62: Obtain the tracking error and control energy consumption optimization target according to formula (19); (19); In the formula, To predict the first in the time domain The system state vector at time t; To predict the first in the time domain The reference state vector at time t; This is the state error weight matrix; To control the input weight matrix; To predict the length of the time domain; S63: Transform formula (19) into the standard quadratic programming (QP) form in formula (20) by matrix transformation; (19); In the formula, Represents the Hessian matrix; This is the vector of linear term coefficients; A constant term that is independent of the optimization variables; S64: Construct linear inequality constraints according to formula (21), and ensure that the control quantity meets the physical constraints by applying linear constraints; (twenty one); In the formula, This is the constraint matrix; To constrain the right-hand side; This is the state deviation influence matrix; S64: Execute the first-step control after real-time QP solution. The system then moves forward one step and repeats the process. S65: To meet real-time requirements, Matrix pre-decomposition using Cholesky methods reduces solution complexity; [the text abruptly ends here, likely due to an incomplete sentence or missing information.] To mitigate the block diagonal sparsity of the matrix, a dedicated solver is used to reduce computational load. The QP solution is initialized based on the previous time-optimal solution. If the convergence condition is met early in the iteration, the QP solution is terminated early. If the CPU load is too high, the solution accuracy is reduced in exchange for real-time response, prioritizing the stability of the control cycle. Experimental Analysis: The proposed algorithm was implemented on a simulation platform and compared with other traditional control methods. The tracking accuracy, control input smoothness, and system robustness were evaluated to verify the effectiveness and engineering feasibility of the proposed method. Specifically, the Denavit-Hartenberg (DH) method was used to establish the kinematic model of a six-DOF robotic arm. It was assumed that the mass of both joints of the robotic arm was concentrated at the end effector. First, a kinematic model was established as follows... Figure 2 The coordinate system shown is defined. ( =1, 2, 3, 4, 5, 6) about the coordinate system The position and orientation are given by the parameters in Table 1: First, establish such as Figure 2 The coordinate system shown is defined. ( =1, 2, 3, 4, 5, 6) about the coordinate system The position and orientation are given by the parameters in Table 1.

[0030] The object of study is a six-degree-of-freedom robotic arm. Its DH parameters are shown in Table 2.

[0031] The range of joint limitation is When using the DM-QPSO algorithm for controller parameter optimization tuning, the algorithm parameters are set as follows: , , , , , and The resulting MPC weight matrix is: ; ; The reference input signal is set as follows: ; To verify the effectiveness of the proposed method, this experiment compares the quantum particle swarm optimization (DM-QPSO) algorithm based on diversity migration strategy with the sliding mode control (VSC) algorithm based on the reaching law. The experimental results demonstrate the performance of the two algorithms in different control tasks.

[0032] First, the convergence curve of the DM-QPSO optimized MPC algorithm, as shown... Figure 3 As shown, the algorithm converges quickly and stably during the iteration process, without premature convergence. This indicates that DM-QPSO effectively avoids the local optima problem in traditional optimization methods while maintaining good global search capability. Furthermore, the control inputs of different algorithms, such as... Figure 4 As shown, the input signal of the VSC controller exhibits significant fluctuations, while the MPC control input optimized by DM-QPSO is relatively smooth, with a smaller rate of change in the control input, thus avoiding excessive input adjustments. The method proposed in this invention helps reduce system energy consumption and extend the service life of the robotic arm.

[0033] Tracking outputs of different algorithms, such as Figure 5As shown, it is evident that VSC-based tracking exhibits significant high-frequency jitter during the tracking process, especially in complex dynamic environments, leading to decreased robotic arm accuracy and control signal jitter. The control method designed in this invention effectively avoids jitter, achieving significantly better tracking performance than VSC. The DM-QPSO-optimized MPC algorithm can smoothly track the desired trajectory, maintaining stable and high-precision tracking performance even when facing challenges such as model uncertainty, external disturbances, and noise. This demonstrates that the control algorithm of this invention has significant advantages in control input smoothness, tracking accuracy, and system robustness.

[0034] Therefore, the experimental results fully verify the effectiveness of the quantum particle swarm optimization algorithm based on the diversity migration strategy proposed in this invention in MPC optimization. Compared with the traditional VSC method, it not only avoids the jitter phenomenon, but also significantly improves the trajectory tracking performance of the six-degree-of-freedom robotic arm in complex environments, and has better application prospects and engineering value.

[0035] Traditional control methods face numerous challenges in trajectory tracking of six-DOF robotic arms. First, the high nonlinearity and strong coupling of the robotic arm model make it difficult for traditional control algorithms to find the global optimum in the high-dimensional parameter space. Second, while MPC (Predictive Control Programming) can effectively handle state and input constraints and balance multiple objectives, the adjustment of its hyperparameters, such as prediction window length, control window, and weight matrix, is highly coupled, and traditional manual tuning methods are time-consuming and prone to getting trapped in local optima. Meanwhile, online solution computation is heavy, and real-time performance issues often limit the application of MPC in complex control tasks. Therefore, existing control methods struggle to achieve an ideal trade-off between accuracy, smoothness, energy consumption, and real-time performance, especially in environments with dynamic uncertainties, external disturbances, and noise. To address this, this invention proposes a quantum particle swarm optimization algorithm based on a diverse migration strategy for automatic tuning of predictive control parameters and trajectory tracking of a six-DOF robotic arm model. Firstly, during system modeling, the accuracy of the end-effector pose is ensured through a nonlinear model and Jacobian matrix mapping, making it suitable for high-precision control tasks. Meanwhile, local linearization reduces the computational burden, and the discretized model is suitable for embedded systems to achieve rapid iterative control. The step-by-step prediction model can dynamically compensate for external disturbances, improving system stability. Generating trajectory sequences supports complex path planning, meeting diverse operational needs. Furthermore, through the collaborative design of parameter tuning and constraint handling, efficient control of complex systems is achieved. Simultaneously, by combining data optimization and real-time feedback, performance and robustness are balanced. Moreover, through fusion... , , and Four metrics are used, and a penalty term is introduced to handle constraints. Factors such as end-effector position / attitude RMSE, cumulative absolute error (IAE) of each joint, control input change rate, and energy consumption estimation are considered, achieving a reasonable trade-off between tracking accuracy, joint error, motion smoothness, and energy consumption. Furthermore, key hyperparameters of the MPC are used as optimization variables, and the DM-QPSO algorithm, particularly suitable for optimization tasks with multiple hyperparameters and complex cost functions, is employed for optimization. The DM-QPSO algorithm maintains population diversity through a diversity migration strategy and enhances global search capability, effectively improving the convergence and robustness of the optimization process, thus avoiding the local optimum traps that may occur in traditional QPSO in complex search spaces. Simultaneously, by dynamically dividing the subgroup and adjusting the subgroup structure during iteration, combined with information exchange between subgroups and a quantum update strategy based on the mean best position (mbest), population diversity is maintained and global escape capability is enhanced, thus exhibiting more robust convergence and optimization performance in high-dimensional and strongly constrained problems. The DM-QPSO optimizer, which integrates dynamic subgroup partitioning, diversity measurement based on average Hamming distance, and a migration and replacement mechanism, balances population diversity and convergence efficiency, solving the problems of high dimensionality and computational burden in MPC hyperparameter tuning. Thus, by introducing the DM-QPSO optimization algorithm to dynamically optimize key MPC hyperparameters, such as prediction time domain, control time domain, state / input weight matrix, and constraint relaxation coefficients, and combining them with a composite fitness function to comprehensively consider multiple objectives such as end-position / attitude error, cumulative absolute error (IAE) of each joint, and rate of change of control input, this avoids the manual tuning, high-dimensional coupling, and local optima problems of traditional methods. It significantly improves the adaptive optimization efficiency and robustness of MPC parameters under complex constraints, and achieves a balance between control accuracy, smoothness, and energy consumption. Furthermore, the globally optimal parameters optimized by DM-QPSO are loaded into the real-time MPC controller, and a smooth transition is achieved through online adjustment strategies. This not only utilizes optimized parameters to improve trajectory tracking accuracy and system stability but also avoids transient instability caused by parameter switching, thereby ensuring the robustness and adaptability of the real-time control system. Subsequently, the optimized parameters are applied to a real-time linearized finite-time domain MPC controller, and combined with constraint relaxation and an efficient quadratic programming solution strategy to address model uncertainties and external disturbances, significantly improving the computational efficiency and dynamic response capability of the real-time control system. Finally, by real-time monitoring of RMSE, control input energy ratio, and constraint violations, incremental DM-QPSO online parameter optimization is triggered. Combined with local sensitivity analysis and hot-switching technology, dynamic adjustment and smooth updates of MPC parameters are achieved, while retaining historical configurations to support rapid rollback, significantly improving the system's real-time performance, robustness, and adaptability under complex operating conditions.

[0036] This invention accelerates convergence through quantum behavior sampling, dynamic subgroup partitioning, and dimensionality reduction search. Simultaneously, it explicitly introduces a control rate-of-change term into the fitness function to suppress chattering, smooth control input, and reduce energy consumption. Ultimately, it achieves automated parameter tuning and engineering reproducibility for six-degree-of-freedom (DOF) robotic arm trajectory tracking, effectively solving the inefficiencies of traditional methods that rely on empirical parameter tuning and the problem of being trapped in local optima in high-dimensional non-convex parameter spaces. It achieves comprehensive improvements in tracking accuracy, control smoothness, and robustness. The invention was validated in a simulation environment incorporating model uncertainty, external disturbances, sensor noise, and control delay. The validation results demonstrate that this invention exhibits superior tracking accuracy, robustness, and real-time performance, significantly improving the trajectory tracking accuracy, system robustness, and real-time performance of a six-DOF robotic arm. It provides an efficient, automated, and reliable optimization scheme for the practical engineering application of six-DOF robotic arms.

[0037] This method is simple to implement and has low implementation costs. It can effectively improve the accuracy, robustness and real-time performance of trajectory tracking control for six-degree-of-freedom robotic arms.

Claims

1. A multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy, characterized in that, Includes the following steps: Step 1: System Modeling; S11: Establish a nonlinear dynamic model of a six-degree-of-freedom robotic arm, define the state vector, and map joint errors to end-effector pose errors using the Jacobian matrix; S12: Based on the working point, the nonlinear model is locally linearized and discretized into a step-by-step prediction model, generating trajectory sequences of joint angles, torques and end poses; Step 2: Definition and constraint handling of MPC parameters to be adjusted; The hyperparameters to be tuned include the control time-domain parameters, control input sequence, constraints, weight matrix, and parameter boundaries; constraint handling achieves flexible control through linear inequalities or slack variables, while performance is improved by combining dynamic objective optimization. Step 3: Construct a composite fitness function; use Measuring trajectory accuracy Assess joint errors, Controlling smoothness Energy consumption is estimated, and constraints are handled using the Huber loss penalty term. Finally, a composite fitness function is constructed by combining various indicators with a weighted scalar quantization. ; Step 4: Optimize MPC parameters using DM-QPSO; S41: Depending on the different optimization stages, the particle position is updated using either basic quantum update or enhanced quantum update. S42: The triggering condition is determined by measuring the population diversity. When the triggering condition is met, the migration strategy and dynamic subgroup partitioning are executed. During the execution of the migration strategy, the worst particle is partially replaced or completely replaced. At the same time, the population is divided into multiple subgroups according to the diversity index. Quantum updates are performed independently in each subgroup, and the optimal solutions between subgroups are exchanged periodically. The process ends when the maximum number of iterations is reached or the convergence accuracy requirement is met. S43: Output the global optimal solution and the corresponding fitness value as the parameters after DM-QPSO optimization; Step 5: Load the optimal parameters into the real-time MPC; The globally optimal solution optimized by DM-QPSO and its corresponding fitness value are loaded into the real-time MPC controller; During real-time operation, the system tracks the execution trajectory using parameters optimized by DM-QPSO and makes online adjustments to ensure a smooth transition when switching to new parameters without causing transient instability. Step Six: Online Scrolling MPC Control; The prediction equation is constructed using the current linearized discrete model, transforming the optimization of tracking error and control energy consumption into a standard quadratic programming problem, and linear constraints are applied to ensure feasibility. Simultaneously, Cholesky pre-decomposition, a structured sparse solver, and warm-start and early-stop strategies are employed to dynamically adjust the solution accuracy during peak computational loads. Finally, the initial control input is achieved through rolling optimization. Real-time execution; Step 7: Construct a closed-loop adaptive control system; Real-time tracking of key performance indicators under the sliding window; if detected When the threshold is exceeded, the high-frequency energy ratio of the control input exceeds the limit, or a constraint violation occurs in several consecutive samples, the incremental DM-QPSO online parameter optimization mechanism is triggered. The current parameter neighborhood is used to start a small-range search or local sensitivity parameter optimization strategy to dynamically adjust the MPC parameters while ensuring real-time performance. The MPC parameters are updated smoothly through hot switching technology. At the same time, historical parameter configurations are retained for quick rollback, forming a closed-loop adaptive control system of monitoring-evaluation-optimization.

2. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step one, the specific process of S11 is as follows: Establish the nonlinear dynamic model of the six-degree-of-freedom robotic arm according to formula (1); define the 12-dimensional state vector according to formula (2). ; According to formula (3), the joint space error Through the Jacobian matrix Mapped to end-effector pose error This enables the quantitative representation of task space errors. （1）； （2）； （3）； In the formula, It is a quality matrix; It is the Coriolis force matrix; It is the gravity matrix; For friction; It is a control input; This is the joint position vector; This is the joint velocity vector; This is the joint acceleration vector; This represents the joint space error.

3. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step one, the specific process of S12 is as follows: At discrete sampling times, based on the current operating point The nonlinear dynamic model is expanded by first-order Taylor expansion. A locally linearized approximation model is established according to formula (4). The continuous model is then discretized into a step-by-step prediction model according to formula (5). Finally, the trajectory sequence of joint angle, torque and end pose is generated using the discrete model to provide data support for optimization calculation. （4）； （5）； In the formula, This is the state transition matrix; This represents the state error at the current moment. The input matrix; This is for input error; Here is the state transition matrix of the continuous-time system. , The sampling period; For the first Error state at time, ; Let be the input matrix of the discrete-time system. ; For the first Input error at time, ; This is process noise.

4. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step two, the time-domain parameters for control include the prediction range. and control range Prediction range Decide to optimize window length and control range The balance between control strength and stability; control input sequence The constraints are obtained according to formula (6); the constraints include hard constraints and soft constraints. Hard constraints include joint limits, speed limits, and control input limits; soft constraints are obtained through relaxation variables. The transformation uses linear inequalities; the weight matrix satisfies symmetric positive definiteness to ensure the convexity of the cost function; parameter boundaries: all parameters are set with upper and lower bounds to ensure feasibility; （6）； In the formula, This represents the control input in the control time domain. , and These are the lower and upper limits for controlling the input, respectively.

5. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step three, a composite fitness function is constructed. The process is as follows: S31: Obtain the end position and attitude error according to formula (7) root mean square error To measure trajectory accuracy; the cumulative absolute error of each joint is obtained according to formula (8). The rate of change of control input is obtained according to formula (9). To measure smoothness; the input energy measure is obtained according to formula (10). ; （7）； （8）； （9）； （10）； In the formula, , for The actual value at time, for Reference value at any given time; For the first The attitude error of each variable; For the first Time-based control input; S32: Introduce penalty items for constraint violations Convert hard constraints to soft constraints to enhance robustness; S33: The composite fitness function is obtained by using a scalar weighted combination of various indices according to formula (11). ; （11）； In the formula, To constrain the penalty for violations; weighting Used for scale normalization or preference setting.

6. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step S41 of step four, when updating the particle position using the basic quantum update method, a dual-guidance mechanism of individual historical optimal position and global optimal position is used to jointly drive the particle motion, so as to achieve a balance between local development and global exploration; at the same time, the first quantum position is obtained according to formula (12). The particle in the first Substitute position ; （12）； In the formula, For the first The particle in the first The position of the generation; This represents the optimal position in the particle's history. The globally optimal position; and These are two different acceleration constants; and Two distinct random numbers; symbol This indicates element-wise multiplication.

7. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step S42 of step four, when updating particle positions using an enhanced quantum update method, the average optimal position is introduced as the group knowledge center. The convergence factor of the quantum tunneling effect is dynamically adjusted through logarithmic transformation, encouraging exploration in the early stage and strengthening development in the later stage. At the same time, the first position is obtained according to formula (13). The particle in the first Substitute position ; （13）； In the formula, As the local center, ; The convergence factor; Indicates the first The average optimal position of the particles. , Population size.

8. The method for controlling a multi-degree-of-freedom robotic arm based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step S42 of step four, the migration strategy is executed as follows: S42-1: Obtain the first according to formula (14) The average Euclidean distance between each particle and all other particles in the population. ; （14）； In the formula, , For the first and the The position vectors of each particle; S42-2: Obtain the average individual diversity of the entire population according to formula (15). ; （15）； S42-4: When Below the preset threshold Or, a global optimal improvement over several consecutive generations. Below the stagnation convergence threshold When the triggering condition is met; S42-4: According to formula (16) for the worst particle Perform partial replacement, or directly replace with the global optimal solution according to formula (17); （16）； （17）； In the formula, This is a dynamically adjusted factor.

9. A multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step six, the online rolling MPC control process is as follows: S61: At each sampling time, based on the current linearized discrete model and Construct state transition matrices respectively The cumulative effect matrix of the input matrix And construct the stacking prediction equation for the system dynamics according to formula (18); （18）； In the formula, To predict the stacked vector for all states in the time domain; Current moment The initial state increment vector; To control the stacked vector of control increments within the time domain; This is the cumulative effect of measurable interference; S62: Obtain the tracking error and control energy consumption optimization target according to formula (19); （19）； In the formula, To predict the first in the time domain The system state vector at time t; To predict the first in the time domain The reference state vector at time t; This is the state error weight matrix; To control the input weight matrix; To predict the length of the time domain; S63: Transform formula (19) into the standard quadratic programming form in formula (20) by matrix transformation; （20）； In the formula, Represents the Hessian matrix; This is the vector of linear term coefficients; A constant term that is independent of the optimization variables; S64: Construct linear inequality constraints according to formula (21), and ensure that the control quantity meets the physical constraints by applying linear constraints; （21）； In the formula, This is the constraint matrix; To constrain the right-hand side; This is the state deviation influence matrix; S64: Execute the first-step control after real-time QP solution. The system then moves forward one step and repeats the process. S65: Yes Perform Cholesky pre-decomposition on the matrix to reduce solution complexity; utilize To mitigate the block diagonal sparsity of the matrix, a dedicated solver is used to reduce computational load. The QP solution is initialized based on the previous time-optimal solution. If the convergence condition is met early in the iteration, the QP solution is terminated early. If the CPU load is too high, the solution accuracy is reduced to prioritize the stability of the control cycle.

10. A multi-degree-of-freedom robotic arm control method based on quantum particle swarm optimization using a migration strategy, as described in claim 1, is characterized in that... In step seven, key performance indicators include RMSE error, IAE growth rate, high-frequency energy ratio of control input, number of constraint violations, and QP solution delay.

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