Robot Kinematics Research Articles

Task priority based design optimization of a kinematic redundant robot

This paper presents and defines a new design optimization method for kinematic redundant robot manipulators based on their applications. Kinematic redundant manipulators can reach a pose with an infinite number of postures. So, identifying the best robot design and configuration for a set of desired tasks is a highly complex non-linear problem. This approach employs a task priority control algorithm to perform a task oriented robot design optimization. The design parameters are replaced by controllable prismatic or revolute virtual joints and controlled by the algorithm to accomplish the desired tasks. Therefore, this new method finds an optimal robot design for a set of tasks taking advantage of the robot kinematic redundancy. This method is evaluated on a highly kinematic redundant manipulator, which tracks a set of paths with its end-effector while maintaining good kinetostatic performance.

A Travelling Salesman Problem Approach to Efficiently Navigate Crop Row Fields with a Car-Like Robot

In recent years, interest in the use of mobile robots in the agricultural industry has increased, both to address labor shortages in rural areas and to increase food production in a more sustainable way. In order to have an efficient navigation system to cover long crop row fields, a path planner algorithm must consider maneuvering restrictions of the targeted robot. Most state-of-the-art works in agricultural navigation systems are intended for robots with a high degree of maneuverability that can typically make in-place turnings. This work aims to fill the gap in terms of the development of an efficient navigation system for car-like robots with limited turning radius in crop row fields. For this, we combine the global path planner A* and the local trajectory planner Timed Elastic Band (TEB). Additionally, we state the problem of finding an optimal path that covers the entire field as a Travelling Salesman Problem (TSP) that is based on the different turning maneuvers the robot can perform at field headlands. The solution of the TSP results in a time efficient coverage strategy that aligns with the robot's kinematics. Experiments performed in the Gazebo simulation environment show a reduction in field completion times of up to 20%, compared to trivial coverage paths. On the other hand, deviation of the robot with respect to the center of the field furrows was in all cases less than 10cm, which proves that the entire system operates with sufficient accuracy to avoid damaging the crops.

Data-Driven Control for Continuum Robots Based on Discrete Zeroing Neural Networks

The effectiveness of continuous-time zeroing neural network (CZNN) in continuum robot control has been preliminarily verified. However, CZNN is not friendly to digital devices and hardware implementation. Although discrete ZNN (DZNN) has been investigated in other tasks, all the existing DZNNs are designed with fixed time steps, which cannot meet the needs of different tasks. This motivates us to develop a generic DZNN model with variable number of time steps and apply it to continuum robot control. In this paper, we present a data-driven scheme based on CZNNs to learn the unknown kinematics of continuum robots and solve the kinematic control problem. Further, a unified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -step forward discretization formula is derived to discretize the CZNN-based scheme into DZNN-based control algorithms. Finally, we take the 1-step and the 3-step algorithms as examples to show the discretization process, and verify their efficacy by simulations and experiments.

Machine learning-based framework for optimally solving the analytical inverse kinematics for redundant manipulators

Solving the analytical inverse kinematics (IK) of redundant manipulators in real time is a difficult problem in robotics since its solution for a given target pose is not unique. Moreover, choosing the optimal IK solution with respect to application-specific demands helps to improve the robustness and to increase the success rate when driving the manipulator from its current configuration towards a desired pose. This is necessary, especially in high-dynamic tasks like catching objects in mid-flights. To compute a suitable target configuration in the joint space for a given target pose in the trajectory planning context, various factors such as travel time or manipulability must be considered. However, these factors increase the complexity of the overall problem which impedes real-time implementation. In this paper, a real-time framework to compute the analytical inverse kinematics of a redundant robot is presented. To this end, the analytical IK of the redundant manipulator is parameterized by so-called redundancy parameters, which are combined with a target pose to yield a unique IK solution. Most existing works in the literature either try to approximate the direct mapping from the desired pose of the manipulator to the solution of the IK or cluster the entire workspace to find IK solutions. In contrast, the proposed framework directly learns these redundancy parameters by using a neural network (NN) that provides the optimal IK solution with respect to the manipulability and the closeness to the current robot configuration. Monte Carlo simulations show the effectiveness of the proposed approach which is accurate and real-time capable ($\approx$ \SI{32}{\micro\second}) on the KUKA LBR iiwa 14 R820.

Hierarchical Optimization for Control of Robotic Knee Prostheses Toward Improved Symmetry of Propulsive Impulse.

Automatically personalizing complex control of robotic prostheses to improve gait performance, such as gait symmetry, is challenging. Recently, human-in-the-loop (HIL) optimization and reinforcement learning (RL) have shown promise in achieving optimized control of wearable robots for each individual user. However, HIL optimization methods lack scalability for high-dimensional space, while RL has mostly focused on optimizing robot kinematic performance. Thus, we propose a novel hierarchical framework to personalize robotic knee prosthesis control and improve overall gait performance. Specifically, in this study the framework was implemented to simultaneously design target knee kinematics and tune 12 impedance control parameters for improved symmetry of propulsive impulse in walking. In our proposed framework, HIL optimization is used to identify an optimal target knee kinematics with respect to symmetry improvement, while RL is leveraged to yield an optimal policy for tuning impedance parameters in high-dimensional space to match the kinematics target. The proposed framework was validated on human subjects, walking with robotic knee prosthesis. The results showed that our design successfully shaped the target knee kinematics as well as configured 12 impedance control parameters to improve propulsive impulse symmetry of the human users. The knee kinematics that yielded best propulsion symmetry did not preserve the normative knee kinematics profile observed in non-disabled individuals, suggesting that restoration of normative joint biomechanics in walking does not necessarily optimize the gait performance of human-prosthesis systems. This new framework for prosthesis control personalization may be extended to other wearable devices or different gait performance optimization goals in the future.

工业机器人和机器视觉在中国白酒智能上甑工艺中的应用

In traditional Chinese Baijiu (white wine) brewing factory, most processes were usually done manually. In order to promote industrial automation in liquor brewing industry and alleviate highly repetitive manual labor, an industrial robot operating system equipped with machine vision was designed to accomplish caldron feeding process during the distillation of Chinese Baijiu. Firstly, Modified D-H method was used to establish robot kinematics model. Shovel actuators and peripheral devices were designed to spread and transport grains during caldron feeding process. After that, an online detection system based on machine vision and master computer were designed to control key parameters of caldron feeding process, including grains height, caldron pressure, robot speed and waiting time, etc. Finally, multiple brewing experiments were carried out on crushed grains and spent grains. The experimental results showed that robot system maximum working space was 3.15 m, which could feed 3 caldrons at the same time. During Chinese Baijiu distillation, air pressure average error in caldron was 8.28 %. High proportion of first-class liquor and second-class liquor was obtained, which met engineering production requirements.

Continuum robots: Developing dexterity evaluation algorithms using efficient inverse kinematics

Performance evaluation and inverse kinematics (IK) for continuum robots are always hard and time-consuming. In this paper, an efficient dexterity evaluation algorithm based on IK (DEAIK) for continuum robots is proposed. The IK model is established using an oval curve equation to improve computational efficiency. The relationship between length distributions and dexterity distribution is obtained by the simulation in this paper. Length distribution of the two-segment continuum robot is optimized under the guidance of the dexterity indices using the fruit fly algorithm. Theoretical analysis and numerical simulations demonstrate that the dexterity of the structure-optimized continuum robot is better than that of the traditional continuum robot. The simulation shows that the DEAIK algorithm is 3.68 times faster than the algorithm based on forward kinematics. The IK algorithm in this paper is 3 203 times faster than the IK algorithm based on the Levenberg-Marquardt algorithm in the same accuracy. This work is significant for designing a high-performance continuum robot.

Application of hybrid robotic systems in crop harvesting: Kinematic and dynamic analysis

Harvesting of agricultural crops, especially when the ripe products have to be picked one by one, has always been a major concern of farmers and growers. Consequently, the use of robots and the mechanization of harvesting operations have been considered as the effective means of tackling such challenges. In the present research, the use of hybrid robots (which combine open-loop and closed-loop manipulators) has been proposed for this purpose; because these robots help reduce the size of the workforce and thus slash the harvesting costs, and they also boost the speed and accuracy of harvesting operations. However, using this type of robots without paying attention to some important factors (e.g., using motors of optimal power at the actuated joints of a robot and choosing appropriate lengths for robot links) can have a substantial adverse effect on the manufacturing cost, working space and the effective functioning of such robots. So the main focus of this paper is on the kinematic and dynamic analysis of hybrid robotic systems. The main advantage of this research over similar works is the generality by which it describes the mechanical behavior of these robots; so that by using the recursive algorithms presented in this paper, all kinds of robots in the hybrid class can be analyzed. However, due to the complexity in the analysis of such systems, a hybrid robot consisting of three closed loops and one open loop was considered in this paper as a sample case study, so that we could explain, visually and graphically, all the steps involved in deriving its motion equations. The simulation results for this robotic system demonstrate the efficacy of the proposed method in the kinematic and dynamic analysis of hybrid robots, as efficient and functional crop harvesting equipment.

Nonlinear kinematics of space parallel robot with Bricard mechanism

The space parallel robot with Bricard mechanism (SPRBM) is a complex mechanical device used for capturing free-floating space debris. It consists of a 3-RRS (Revolute-Revolute-Spherical Joints) parallel mechanism and a Bricard mechanism. Space debris can be captured by the interior space of the SPRBM. Compared with the conventional manipulator system, the SPRBM can capture a target with an arbitrary shape and motion state if the target can be enwrapped by the mechanism. In addition, it does not impose any stringent requirement on the locations of the capture points on the debris. However, the SPRBM has complex configurations and therefore nonlinear kinematics. This paper presents a detailed kinematics analysis of the SPRBM. The analysis involves the degrees of freedom, singularity, forward and inverse kinematics, workspace, and the trajectory planning method. These analyses and the results provide the fundamentals for modeling and controlling the SPRBM.

Stiffness Modeling and Deformation Analysis of Parallel Manipulators Based on the Principal Axes Decomposition of Compliance Matrices

Abstract This paper presents a general equivalent approach to solve the stiffness modeling or load-deformation problem of non-redundant parallel mechanisms. Based on the principal axes decomposition of structure compliance matrices, an equivalent six-degrees-of-freedom (6-DOFs) serial mechanism is established to approximate the load-deformation behavior of each flexible link in the mechanism. Hence, each limb of the parallel mechanism can be equivalent to a serial redundant rigid body mechanism with passive elastic joints, and the load-deformation problem can be transformed to the equilibrium configuration calculation of the equivalent mechanism. The main advantage of the proposed method is that the robotic kinematics and statics, rather than the elastic mechanics, can be directly adopted to solve the equilibrium configuration of the parallel mechanism under external load. Besides, a closed form solution of the corresponding deformation can be obtained, which can be solved by the gradient-based searching algorithm. Therefore, the final deformation will no longer be linear to the external load, which makes this method more accurate and more suitable for the deformation prediction and compensation in real industrial working conditions. In order to verify the effectiveness and correctness of this method, a 3PRRU parallel manipulator will be introduced as an example, to compare the load-deformation results with the finite element analysis (FEA) simulation and matrix calculation methods, so the nonlinearity feature can be shown in an intuitive manner.

Solving redundant inverse kinematics of CMOR based on chaos-driven particle swarm optimization algorithm

The China Fusion Engineering Test Reactor (CFETR) multipurpose overload robot (CMOR) is used to maintain the vacuum chamber inner components damaged by heat loads, electromagnetic fields, and nuclear radiation. The CMOR is a redundant robot whose configuration does not meet the condition of the Pieper criterion. Redundant kinematic problems can only be solved by numerical methods, not analytical methods. Conventional numerical iterative methods include considerable computational load, accumulated errors, and the Jacobian matrix's singularity. To solve redundant inverse kinematics efficiently, we compared the behaviors of 25 versions of particle swarm optimization (PSO) algorithms with 12 one-dimensional chaotic maps under unimodal and multimodal test functions. Moreover, we selected three chaos-driven PSO algorithms with optimal convergence performance to address CMOR inverse kinematics. The experimental results indicated that chaos-driven PSO algorithms have higher computational efficiency and can effectively improve the speed and accuracy of algorithm convergence. This proposed algorithm delivers a novel and efficient method for inverse kinematics of redundant robots based on chaotic maps.

Inverse Kinematics for Serial Robot Manipulators by Particle Swarm Optimization and POSIX Threads Implementation

Inverse kinematics is a fundamental problem in manipulator robotics: a set of joint angles must be calculated so that the robot arm can be manipulated to the corresponding desired end effector position and orientation (also known as “pose”). Traditional solution techniques include analytical kinematics solvers, which provide the closed-form expressions for the joint positions as functions of the end-effector pose. When analytical inverse kinematics solvers are not possible due to the manipulator structure, numerical methods such as Newton–Raphson or Jacobian inverse can be used to achieve the task, but at a much slower speed due, to the iterative nature of the computation. Recent swarm intelligence technology has also contributed to manipulator inverse kinematics solutions. In this paper, the use of the Particle Swarm Optimization (PSO) approach in solving the inverse kinematics problem is investigated for the general serial robotic manipulators. Many of the reviewed robotic manipulator inverse kinematics solvers using swarm intelligence only deal with end effector position and not its orientation. Our PSO approach provides the convergence of a complete end-effector pose and will be demonstrated using the Baxter Research Robot, which has two seven-joint arms, although the method is applicable to any general serial robotic manipulator. For computational efficiency, the inverse kinematic calculations were implemented in parallel using Portable Operating Interface (POSIX) threads to take advantage of the independent swarm particle dynamics.

Precise Robot Calibration Method-Based 3-D Positioning and Posture Sensor

This article presents a distance accuracy technique for zero-offset industrial robot calibration based on 3-D positioning and posture (3DPP) sensor along with modified Denavit–Hartenberg (MDH) for robot kinematics identification. Industrial robots’ distance and positioning accuracy tend to diminish rapidly due to repetitive tasks and heavy loads. Many aspects can be involved in reducing the distance and positioning accuracy of the industrial robot; therefore, using the proper sensors and identification methods is essential to increase or restore the accuracy back to the original state. Normally, maintenance requires apparatus made specifically for robot kinematic calibration which are costly, time-consuming, and must be done by trained operators, while traditional calibration techniques suffer from cumulative error between the calibration device coordinates and robot coordinates. In this article, a 3DPP measurement device and MDH robot kinematic identification method is proposed for a six-degree-of-freedom (6DOF) industrial manipulator calibration, and a distance error calibration model is applied. The results indicate that the proposed 3DPP sensor and calibration model based on MDH improved the absolute position accuracy and distance accuracy greatly, the positioning maximum and average error were reduced by 94% and 96% respectively, and the distance maximum and average error were reduced by 95% and 96% respectively. The experimental results confirm the efficiency of the sensor and proposed method.

MP26-11 PROJECT SURGICAL RADAR: MASTERY REGISTRY OF OBJECTIVE PERFORMANCE INDICATORS DURING A REALISTIC NERVE-SPARING ROBOTIC RADICAL PROSTATECTOMY SIMULATION

You have accessJournal of UrologyCME1 Apr 2023MP26-11 PROJECT SURGICAL RADAR: MASTERY REGISTRY OF OBJECTIVE PERFORMANCE INDICATORS DURING A REALISTIC NERVE-SPARING ROBOTIC RADICAL PROSTATECTOMY SIMULATION Nathan Schuler, Lauren Shepard, Andrea Smith, Sue Kulason, Andrew Yee, and Ahmed Ghazi Nathan SchulerNathan Schuler More articles by this author , Lauren ShepardLauren Shepard More articles by this author , Andrea SmithAndrea Smith More articles by this author , Sue KulasonSue Kulason More articles by this author , Andrew YeeAndrew Yee More articles by this author , and Ahmed GhaziAhmed Ghazi More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000003254.11AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Erectile dysfunction after nerve-sparing robot-assisted radical prostatectomy (NS-RARP) ranges from 8% to 49% even by high-volume surgeons with no consensus on granular tasks that result in the best potency. Studies found robotic kinematic data to correlate with select RARP clinical outcomes. We analyzed Objective Performance Indicators (OPIs) from a mastery registry of high-volume urologists performing a NS-RARP simulation to identify which metrics can predict operative expertise. METHODS: Thirty urologists (avg. RARP volume=734) performed a NS-RARP on a validated, realistic, hydrogel model with force sensors embedded in the neurovascular bundle (NVB). Robotic kinematic (51 OPIs) and force (number of force peaks, variability, average and total force) data were collected. These OPIs were mapped to broader technical skill categories and used to build logistic regression classifiers that predict the probability of case volume and force data. Data was fit to a Gaussian Mixture model, then separated based on OPIs. Tukey’s HSD test was used to determine significance in inter-group comparisons of overall robotic, RARP volume and force metrics RESULTS: Urologists were clustered into 3 groups based on mean total Overall robotic and RARP volume into 7 Super-Users (SU), 14 High Volume (HV), and 9 Low Volume (LV) surgeons (Figure 1a). OPIs were categorized into 3 broad categories: robotic efficiency, tool motion, and operative fluidity. A significant difference (<0.05) across groups was seen in 31 OPIs (18 across all 3 groups, 10 when comparing SU vs HV and HV vs LV, and 6 when comparing HV vs LV groups). Force metrics showed significant difference comparing SU and HV to LV (Figure 1a). A significant negative correlation was found between force metrics and robotic volume, RARP volume and 13 OPIs (5 robotic efficiency , 4 tool motion, and 4 operative fluidity OPIs) (Figure 1b). CONCLUSIONS: Experienced urologists demonstrated a higher incidence of operative fluidity and tool efficiency OPIs, which are also predictive of less force data applied to the NVB during the simulation. These specific actions specific to super users may improve surgical efficiencies through structured feedback. Source of Funding: None © 2023 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 209Issue Supplement 4April 2023Page: e356 Advertisement Copyright & Permissions© 2023 by American Urological Association Education and Research, Inc.MetricsAuthor Information Nathan Schuler More articles by this author Lauren Shepard More articles by this author Andrea Smith More articles by this author Sue Kulason More articles by this author Andrew Yee More articles by this author Ahmed Ghazi More articles by this author Expand All Advertisement PDF downloadLoading ...

Multimodal Learning of Keypoint Predictive Models for Visual Object Manipulation

Humans have impressive generalization capabilities when it comes to manipulating objects and tools in completely novel environments. These capabilities are, at least partially, a result of humans having internal models of their bodies and any grasped object. How to learn such body representations for robots remains an open problem. In this work, we present a self-supervised learning approach that extends a robot's kinematic model for object manipulation from visual latent representations. Our framework comprises two components: First, we present our multimodal keypoint detector: A neural network autoencoder architecture that fuses proprioception and vision during learning to predict visual key points on an object; second, we show how we can learn an extension of the kinematic chain of the robot by regressing virtual joints from the visual keypoints predicted by our multimodal keypoint detector. Our evaluation shows that our approach learns to consistently predict visual keypoints on objects in the manipulator's hand and, thus, can easily facilitate learning an extended kinematic chain to include the object grasped in various configurations, from a few seconds of visual data. Finally, we show that this extended kinematic chain lends itself for object manipulation tasks such as placing a grasped object and present experiments in simulation and on hardware.

Path tracking control of differential drive mobile robot based on chaotic-billiards optimization algorithm

&lt;span lang="EN-US"&gt;Mobile robots are typically depending only on robot kinematics control. However, when high-speed motions and highly loaded transfer are considered, it is necessary to analyze dynamics of the robot to limit tracking error. The goal of this paper is to present a new algorithm, chaotic-billiards optimizer (C-BO) to optimize internal controller parameters of a differential-drive mobile robot (DDMR)-based dynamic model. The C-BO algorithm is notable for its ease of implementation, minimal number of design parameters, high convergence speed, and low computing burden. In addition, a comparison between the performance of C-BO and ant colony optimization (ACO) to determine the optimum controller coefficient that provides superior performance and convergence of the path tracking. The ISE criterion is selected as a fitness function in a simulation-based optimization strategy. For the point of accuracy, the velocity-based dynamic compensation controller was successfully integrated with the motion controller proposed in this study for the robot's kinematics. Control structure of the model was tested using MATLAB/Simulink. The results demonstrate that the suggested C-BO, with steady state error performance of 0.6 percent compared to ACO's 0.8 percent, is the optimum alternative for parameter optimizing the controller for precise path tracking. Also, it offers advantages of quick response, high tracking precision, and outstanding anti-interference capability.&lt;/span&gt;

Simultaneous Online Registration-Independent Stiffness Identification and Tip Localization of Surgical Instruments in Robot-assisted Eye Surgery.

Notable challenges during retinal surgery lend themselves to robotic assistance which has proven beneficial in providing a safe steady-hand manipulation. Efficient assistance from the robots heavily relies on accurate sensing of surgery states (e.g. instrument tip localization and tool-to-tissue interaction forces). Many of the existing tool tip localization methods require preoperative frame registrations or instrument calibrations. In this study using an iterative approach and by combining vision and force-based methods, we develop calibration- and registration-independent (RI) algorithms to provide online estimates of instrument stiffness (least squares and adaptive). The estimations are then combined with a state-space model based on the forward kinematics (FWK) of the Steady-Hand Eye Robot (SHER) and Fiber Brag Grating (FBG) sensor measurements. This is accomplished using a Kalman Filtering (KF) approach to improve the deflected instrument tip position estimations during robot-assisted eye surgery. The conducted experiments demonstrate that when the online RI stiffness estimations are used, the instrument tip localization results surpass those obtained from pre-operative offline calibrations for stiffness.

Development of a Novel 4-DOF Flexible Endoscopic Robot Using Cable-Driven Multisegment Continuum Mechanisms

Abstract This paper presents the design, analysis, and development of a novel four degrees of freedom (4-DOF) endoscopic robot with cable-driven multisegment flexible continuum mechanisms. The endoscopic robot is mainly composed of the passive positioning arm, cable-pulley system, and 3-DOF flexible continuum mechanism. The forward and inverse kinematics of the endoscopic robot are derived based on the constant curvature assumption, and its working space, flexibility, and preoperative incision determination method are analyzed as well. Based on the hardware structure of the robot system, a control strategy and a control software are developed, and the continuum mechanism is kinematically calibrated to carry out the trajectory planning experiment and simulated surgery experiment. The experimental results show that the calibrated constant curvature model can be used for the motion control of the continuum mechanism, and the 4-DOF endoscopic robot can meet the visual field requirements of minimally invasive surgery.

Study on the Multi-Level Experiment Teaching of NAO Robot Combined with Virtual and Reality

NAO is the most well-known and widely used humanoid robot in education and research in the world. With the development of computer technology and cloud computing, and the establishment of artificial intelligence specialty, robot teaching is more important in artificial intelligence specialty. The artificial intelligence specialty is a new specialty just established in the talent training program of Chinese universities. Its purpose is to cultivate application-oriented talents in China's artificial intelligence industry and promote the construction of first-class disciplines of artificial intelligence. MATLAB is the commercial mathematical software. It is a high-level technical computing language and interactive environment for algorithm development, data visualization, data analysis and numerical calculation. It is widely used in the experimental teaching of NAO robot. In the teaching of robot theory, kinematics, trajectory planning, dynamics and control are relatively abstract and complex, and there are a lot of formula derivation and differential equations. In order to improve the quality of NAO experiment teaching, this paper analyzes the current situation and existing problems in NAO experiment teaching. In the exploration and practice of robot experiment teaching, NAO robot is taken as the research object, and MATLAB is used to carry out robot simulation and control experiments. The paper presents a multi-level experimental teaching method of NAO robot based on the combination of virtual and Reality. The multi-level includes three levels: basic experiment, design experiment and innovative experiment. The paper describes the design and implementation process of the three-level experimental teaching in detail, and gives the assessment methods, characteristics and effectiveness evaluation of the experimental teaching. The teaching results show that the teaching effect is good, can stimulate students' enthusiasm for learning, and help to cultivate students' programming ability and innovation ability.

Jacobian methods for dynamic polarization control in optical applications.

Dynamic polarization control (DPC) is beneficial for many optical applications. It is often realized via tunable waveplates to perform automatic polarization tracking and manipulation. Efficient algorithms are essential to realize an endless polarization control process at high speed. However, the standard gradient-based algorithm is not well analyzed. Here, we model the DPC with a Jacobian-based control theory framework that finds a lot in common with robot kinematics. We then give a detailed analysis of the condition of the Stokes vector gradient as a Jacobian matrix. We identify the multi-stage DPC as a redundant system enabling control algorithms with null-space operations. An efficient, reset-free algorithm can be found. We anticipate more customized DPC algorithms to follow the same framework in various optical systems.