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Asset's specific needs and criticality, RCM ensures that each piece of equipment receives the care it truly needs.‍Data-Driven Decision MakingRCM often integrates with advanced monitoring tools and analytics. This data-driven approach provides actionable insights, allowing businesses to make informed decisions, predict trends, and continuously refine maintenance strategies.‍Adopting Reliability Centered Maintenance is not just a strategic choice for asset management; it's a transformative decision that impacts overall operational efficiency, cost savings, and business success.‍Challenges and Overcoming Them1. Resistance to ChangeChallenge: Employees and management might resist transitioning from traditional maintenance methods to RCM, fearing the unknown or perceiving it as added complexity.Solution: Conduct training sessions and workshops to educate teams about the benefits of RCM. Highlight success stories and demonstrate the long-term advantages to gain buy-in.‍2. Initial Investment ConcernsChallenge: The upfront costs of RCM implementation, including training, tools, and software, can be daunting.Solution: Present a detailed ROI analysis showcasing the long-term savings and benefits of RCM. Highlight the cost of unplanned downtimes and reactive maintenance to contrast with proactive RCM strategies.‍3. Data OverloadChallenge: RCM often involves collecting vast amounts of data, which can be overwhelming and lead to analysis paralysis.Solution: Invest in robust data analytics tools to sift through the data and provide actionable insights. Review and refine data collection processes to focus on the most relevant metrics.‍4. Inadequate TrainingChallenge: Without proper training, teams might not fully understand or correctly implement RCM principles.Solution: Collaborate with RCM experts or consultants to provide comprehensive training. Consider periodic refresher courses to keep the team updated.‍5. Undefined Asset CriticalityChallenge: Failing to identify and prioritize the criticality of assets correctly can lead to misallocated resources.Solution: Conduct a thorough asset criticality analysis at the outset. Regularly review and update this analysis as business needs evolve.‍6. Inconsistent ImplementationChallenge: Different departments or teams might implement RCM inconsistently, leading to fragmented results.Solution: Establish clear RCM guidelines and standard operating procedures. Ensure regular audits to maintain consistency across the organization.‍7. Over-Reliance on TechnologyChallenge: While technology is a vital component of RCM, over-relying on it without human judgment can lead to missed insights.Solution: Encourage a balanced approach where technology aids decision-making, but human

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Table of ContentsThe Evolution of Maintenance StrategiesUnderstanding Reliability Centered Maintenance (RCM)Benefits of Adopting RCMChallenges and Overcoming ThemWhy Partner with an RCM Expert?Introduction to Cryotos CMMS (Computerized Maintenance Management System)To Wrap Things Up!‍Reliability Centered Maintenance (RCM) is a strategic approach to maintenance planning that focuses on maximizing asset reliability and operational performance. At its core, RCM integrates the principles of preventive and predictive maintenance, ensuring that assets receive the right kind of care at the right time. Instead of a one-size-fits-all approach, RCM emphasizes understanding each asset's unique operational context, potential failure modes, and the consequences of those failures. This results in a tailored maintenance strategy that addresses the specific needs of each asset, ensuring longevity and optimal performance.‍Adopting RCM has brought about a paradigm shift in modern asset management. Gone are the days when maintenance was seen merely as a necessary expense. With RCM, maintenance is now viewed as a strategic investment that drives operational excellence. By focusing on reliability and performance, RCM reduces unplanned downtimes and associated costs and enhances overall asset lifespan and safety. This transformative approach ensures that businesses can achieve higher operational efficiencies, reduce risks, and realize a better return on their asset investments.‍This blog will uncover the Reliability Centered Maintenance (RCM), weigh the pros and cons of it, and provide insights to help organizations make an informed decision.‍The Evolution of Maintenance StrategiesIn the past, maintenance practices primarily revolved around fixing things when they broke, often resulting in unforeseen issues, increased risks, and higher long-term costs. As time progressed and competition intensified, the realization dawned that this approach had limitations. Consequently, a shift towards a more proactive maintenance strategy was initiated. Instead of waiting for equipment to fail, regular inspections and data analysis became the norm, aimed at identifying potential problems before they escalated. The overarching objective was crystal clear: detect issues in advance, address them promptly, and circumvent major disruptions. This transition to proactive maintenance, driven by advanced technology and data analysis, curtails unexpected breakdowns and enhances operational efficiency, equipping businesses to thrive in an ever-evolving industry landscape.‍Understanding Reliability Centered Maintenance (RCM)Reliability Centered Maintenance (RCM)

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Generate and send reports with test results to physicians for patient care. LIMS are designed for use in other analytical testing environments and often associated with research laboratories and the pharmaceutical and manufacturing industries. These systems focus on the sample rather than the patient and are built to process and analyze large batches or groups of sample data. What is Revenue Cycle Management?Revenue Cycle Management (RCM) is the financial process that healthcare facilities use to identify, collect, and manage revenue from payers based on services provided. RCM includes the administrative and clinical functions associated with revenue generation through claims processing and payment, and it begins with the initial appointment or encounter. RCM ends when all claims and patient payments have been collected.How did LigoLab get its name?Ligo means ‘to unite or connect” in Latin, and that’s what the LigoLab LIS & RCM Operating Platform is designed to do. It connects, merges, and integrates every lab process and department onto one united software infrastructure that transforms laboratories and becomes the focal point for all operations and finances.How long has LigoLab been in business?LigoLab is a private and profitable company that’s based in Los Angeles. The company has been helping pathology groups and clinical laboratories grow and thrive with innovative medical software solutions since 2006. LigoLab added a Revenue Cycle Management (RCM) module to its operating platform in 2019, and a Direct-to-Consumer (D2C) Portal in 2020.Where is the LigoLab software development team based and how big is the company?LigoLab is headquartered in Los Angeles, and that’s where it’s development team resides, too. The company is growing fast, having doubled in size in just over a year. Currently, the company has 50-plus employees supporting 100-plus laboratories nationwide.What are the driving forces behind the LigoLab LIS & RCM Operating Platform? LigoLab’s driving forces are real-time workflow queues with features such as automatic coding and TC/PC splitting, regulatory compliance protocols for every stage, and a comprehensive audit trail. With this platform, laboratory executives can increase revenue with complete accounts receivable control. They gain accounting insight and can create detailed statistical reports and dashboards to monitor trends and KPI’s. All of this on a single platform that unites LIS and RCM workflows and supports automation and rule-based solutions that deliver measurable results and improve a laboratory’s profitability.How can LigoLab’s all-in-one platform help my lab become more profitable?The LigoLab platform unites LIS & RCM workflows and utilizes. RCM business RCM - Facebook Rcm. Rcm business plan - rcm marketing plan - rcm plan - rcm business plan 2025 - rcm business JayRcm Store - Mlm Guidline - http

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Focuses on making assets work reliably and efficiently throughout their life. RCM determines the best way to maintain each asset, considering how it works and what might happen if it breaks. It looks at possible problems, how they affect operations, and the best ways to stop or reduce them. RCM has two main goals: ensuring assets do their job safely and cost-effectively.‍Traditional maintenance usually waits for things to break or follows a set schedule. RCM is different. Every asset is unique and important in its own way. So, it plans maintenance based on what each asset needs, whether regular checks, predictions, or quick fixes when needed. This way, RCM makes assets work better and saves resources by focusing on what matters, setting it apart from older methods.‍Benefits of Adopting RCMEmbracing Reliability Centered Maintenance (RCM) offers businesses many advantages that transcend the traditional boundaries of asset management. Here are some of the standout benefits:‍Enhanced Asset ReliabilityAt the core of RCM is the goal to maximize the reliability of assets. By understanding each asset's operational context and potential failure modes, RCM ensures that assets consistently perform their intended functions, leading to fewer breakdowns and disruptions.‍Cost-EfficiencyWhile the initial investment in RCM might seem substantial, the long-term savings are significant. By reducing unplanned downtimes, minimizing reactive maintenance, and extending asset lifespans, RCM offers a compelling return on investment.‍Improved Safety ProtocolsSafety is paramount in any operation. RCM's systematic approach to understanding and mitigating potential failures ensures a safer work environment. By anticipating and addressing risks, RCM contributes to the overall safety and well-being of both the assets and the personnel.‍Optimized Resource AllocationRCM's targeted approach ensures that resources, be it manpower, time, or materials, are utilized where they are most needed. This results in efficient operations, where critical assets receive timely attention, and non-critical assets are maintained optimally without over-expenditure.‍Proactive ApproachRCM shifts the maintenance paradigm from being reactive to proactive. By forecasting potential issues and addressing them in advance, businesses can plan better, reduce emergency interventions, and ensure smoother operations.‍Tailored Maintenance StrategiesEvery asset is unique, and RCM recognizes this. By tailoring maintenance strategies to fit each

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NDS’s AI Technologyaccelerates your revenue cycle Our Transactional Pricing Modelensures a mutually profitable partnership Our AI Innovation reduces your RCM costs NDS's AI Automation improves the accuracy of your RCM processes NDS AI: The Vaccine for RCM Challenges Slash Write Offs by 10% Reduce RCM Cost by 20% In Just 3 Months What We Do--> NDS AI: The Ultimate RCM Champion Step into a world of unprecedented RCM success with NDS AI as your trusted ally. Witness a revolutionary partnership that reduces denials with precision, drives efficiency, and slashes costs, propelling your RCM ecosystem to unparalleled heights of excellence. Say goodbye to your RCM challenges as NDS AI emerges as the ultimate catalyst for your transformation. Our Solutions Supercharge Your RCM with the Dominance of NDS AI Our arsenal of services is meticulously crafted to deliver unmatched efficiency and profitability, delivering a knockout blow by slashing write-offs by a staggering 10% and hacking RCM costs down by a jaw-dropping 20% in a mere 3 months. AI - Charged Payment Posting Automation Efficiently convert paper EOBs and payer ERAs into electronic formats for your Practice Management System and streamline posting. AI - Powered Remittance Reconciliation Automate deposit matching with ERAs/EOBs. Ensure only funded transactions are posted, identify outstanding payments, and save time reconciling. AI - Enhanced Denial Management Optimize denial management to save time, prevent revenue loss, and boost your bottom line effectively. AI - Driven A/R Follow up & Appeals Revolutionize AR follow-up and appeals with potent AI, slashing A/R days and boosting recovery rates reliably. Autonomous Coding Solutions Automate the medical coding function, reducing impact of staffing shortages on this key aspect of your revenue cycle. About NDS At NDS we offer you AI Technovation in action! With over 20 years of expertise in continuous process improvement using cutting-edge AI technologies such as Supervised Deep Learning and NLP, NDS InfoServ is a leading AI-based technology and BPO service provider. Our two decade-long experience and deep domain expertise are catalysts for developing effective solutions. At NDS we critically examine your processes through the lens of our own experiences to identify ways to make them more cost-effective, efficient, and reliable. This might mean redesigning a process, automating parts of a process, or using smarter IT to improve overall speed and efficiency. Our innovative solutions focus on accelerating your revenue cycle and significantly reducing RCM costs. Our quality systems are defined by Six Sigma principles. The resultant learning, review, and retraining activities become a self-correcting mechanism that helps you minimize revenue leakage. To build effective solutions for our customers, we have time-tested four stages that deliver powerful solutions, customized to their specific requirements. NDS’s four stages for customer success Discovery phase: We

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The healthcare revenue cycle management (RCM) landscape is set for a major shift in the coming years. With continuous industry evolution, RCM providers and their clients must navigate a complex mix of obstacles and opportunities. Several significant trends are expected to shape RCM in 2025: Increasing Costs Due to Cybersecurity, Regulatory Compliance, and Rising Patient DebtThe financial burden on RCM providers continues to grow as they strengthen cybersecurity measures and manage intricate regulatory requirements. Compliance with FDA regulations, federal and state mandates, and even nuanced payor policies requires substantial investment. Additionally, escalating patient bad debt rates are further squeezing profit margins across the healthcare sector. Workforce ChallengesThe demand for specialized RCM skills is rising in an increasingly competitive labor market. Organizations are finding it more difficult to attract, retain, and train professionals with expertise in navigating complex payor policies and utilizing advanced technologies. These challenges make workforce development a key priority. Increasing Complexity of Payor PoliciesPayor rules are becoming more detailed and fragmented, requiring laboratories to comply with a layered regulatory framework that spans federal, state, and plan-specific guidelines. This increasing complexity underscores the need for data-driven decision-making and enhanced operational adaptability. EMR Integration and InteroperabilityAs laboratories place greater emphasis on seamless electronic medical record (EMR) integration, the need for highly interoperable systems is growing. Inefficiencies in data exchange can negatively impact claims accuracy, processing speed, and overall operational effectiveness.How RCM Customers Will Be AffectedRCM customers must address both challenges and opportunities in the evolving landscape:Cost Efficiency Becomes a PriorityWith rising compliance and cybersecurity expenses, laboratories must seek cost-saving strategies. Automation is no longer optional—it is essential for financial sustainability and operational agility. Solutions that enhance patient engagement and payment propensity will become increasingly valuable.Outsourcing to Access ExpertisePartnering with specialized RCM providers allows organizations to leverage economies of scale, cutting-edge technologies, and experienced professionals who can navigate regulatory and payor complexities efficiently. No Surprise Bill is a leader in providing expert revenue cycle management IDR services, helping healthcare organizations optimize revenue, reduce financial losses, and streamline transitions. Their experienced team ensures that practices maximize their revenue potential by leveraging advanced RCM systems and IDR processes.Greater Utilization of RCM DataRCM data will become a crucial asset, extending beyond financial operations to inform strategic decision-making across organizations.Strategies and Innovations to Overcome Emerging ChallengesTo help laboratories navigate these shifts, we are focused on delivering cutting-edge solutions and forward-thinking technologies: Leveraging Artificial Intelligence (AI) and Machine Learning (ML)While some organizations remain cautious about generative AI, many are already seeing the benefits of targeted applications of machine learning (ML) and natural language processing (NLP). We have successfully used these technologies to enhance claims processing through:Prioritizing claims based on reimbursement probability and valueRouting claims to the

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\(k=0\) is configured, then \(r_P=r_B\); when \(k=1\) is configured, then \(r_P=r_A\); when \(0, \(r_P\) is strictly between points \(r_A\) and \(r_B\). The redundancy resolution for kinematic control of the manipulator needs to finish the two tasks, 1) let the end-effector track the desired path accurately; and 2) satisfy the RCM constraint to make \(r_P\) vary in a very small range or almost static during motion planning and control. In an application scenario, the last link (e.g., \(r_A-r_B\)) of the manipulator can penetrate a small hole (e.g., \(r_P\)) and simultaneously make the end-effector (e.g., \(r_A\)) perform the path tracking task.3 Problem FormulationAs the RCM constraint makes point \(r_P\) is between \(r_A\) and \(r_B\), \(r_P\) can be depicted by the equation \(r_p-r_B=k(r_A-r_B)\). Since \(r_A\) and \(r_B\) are obtained through the forward kinematics with resolved joint angles \(\theta \), thus the state variable pair \((\theta , k)\) can describe the redundancy resolution of the manipulator with RCM constraints. By differentiating the both sides of equation \(r_p-r_B=k(r_A-r_B)\) which depicts the RCM constraint, one can obtain$$\begin{aligned} \dot{r}_P-\dot{r}_B=\dot{k}(r_A-r_B)+k(\dot{r}_A-\dot{r}_B) \end{aligned}$$ (3) When combining it with the aforementioned Eq. (1), one can further have$$\begin{aligned} \dot{r}_P-J_2\dot{\theta }=\dot{k}(r_A-r_B)+k(J_1-J_2)\dot{\theta } \end{aligned}$$ (4) As the position of RCM point \(r_P\) can be described by the following linear combination between \(r_A\) and \(r_B\)$$\begin{aligned} r_P=k(r_A-r_B)+r_B \end{aligned}$$ (5) Combining the aforementioned equations, then we have$$\begin{aligned} \dot{k}(r_A-r_B)+kJ_1\dot{\theta }+(1-k)J_2\dot{\theta }+\dot{r}_P=0 \end{aligned}$$ (6) i.e.,$$\begin{aligned} \begin{aligned} \dot{k}(r_A-r_B)+[kJ_1+(1-k)J_2]\dot{\theta }+\dot{r}_P=0 \end{aligned} \end{aligned}$$ (7) where we can call the equality above as the RCM constraint for redundancy resolution.Based on the derivations and discussions above, we propose the quadratic programming formulation for redundancy resolution of manipulators with RCM constraints as follows$$\begin{aligned} \begin{aligned}&\arg \min _{\theta ,k}~~\dot{\theta }^T\dot{\theta }/2+c_2\dot{k}^2/2\\&\text {s.t.}\left\{ \begin{array}{ll} \dot{k}(r_A-r_B)+[kJ_1+(1-k)J_2]\dot{\theta }+\dot{r}_P=0\\ J_1\dot{\theta }+c_3(r_A-r_d)-\dot{r}_d=0 \end{array} \right. \end{aligned} \end{aligned}$$ (8) where \(c_2>0\) and \(c_3>0\) denote the scaling parameters for the objective function and the last equality constraint respectively. In order to satisfy RCM constraints during redundancy resolution for kinematic control, the time-derivative \(\dot{r}_P\) of \(r_P\) should follow some rules to make \(r_P\) converge to a constant position value.As the point \(r_P\) should not hold static as much as possible to sanctify the RCM constraint, then the time derivative of \(r_P\) can be concisely depicted by$$\begin{aligned} \dot{r}_P=-c_1(r_P-r_P(0)) \end{aligned}$$ (9) to make the RCM point \(r_P\) converge to a fixed position \(r_P(0)\), where \(c_1\ge 0\) is used to scale the convergence of \(r_P\) which shows how the corresponding RCM constraint’s dynamic response changes.In the proposed optimization formulation for redundancy resolution with RCM constraints, the position of \(r_P\) is dynamically adjusted by parameter k with simultaneous positions \(r_A\) and \(r_B\). The initial value k(0) is chosen according to the specific scenarios for safe manipulation, e.g., when \(t=0\), \(r_P\) can be chosen in the middle of the line \(A-B\), and thus \(k(0)=0.5\). To maintain the RCM constraint, \({\dot{\theta }}\) and \(\dot{k}\) are used as decision variables, and the variable which really controls the manipulator is \(\dot{\theta }\) as the control input action. In practice, we can therefore solve \(\dot{k}\) and substitute it into the aforementioned proposed optimization formulation and. RCM business RCM - Facebook Rcm. Rcm business plan - rcm marketing plan - rcm plan - rcm business plan 2025 - rcm business JayRcm Store - Mlm Guidline - http

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RCM usually leaves a sole point for the manipulator performs positioning or/and insertion [21,22,23]. Such applications need to impose additional constraints in joint space to guarantee safe motion generation of the end-effector of the manipulator. Conventional pseudo-inverse based methods may neglect the necessary constraints in the RCM case, which makes it difficult to get the solutions. To remedy this, Su et al. proposed an improved and efficient recurrent neural network for RCM control of manipulators with constrained optimization [24]. Motivated by the aforementioned points, in this paper, we are making breakthroughs to propose a new model-based method based on RNN for redundancy resolution of manipulators with RCM constraints. The contributions of this work are summarized as follows: (1) The proposed RNN for manipulator control with RCM constraints is with a concise and novel neural-network architecture and can get rid of unnecessary modeling and computation, as compared with the conventional zeroing dynamics (ZD) based solvers. (2) Simulation results a 7-DoF redundant manipulator synthesized by the proposed RNN demonstrate the efficiency of the proposed method in kinematic control of manipulators with RCM constraints for different end-effector path tracking tasks. 2 PreliminariesRCM means a remote static/fixed point near the workspace with no physical revolute joint around the location [21, 22]. In order to achieve manipulation such as minimally invasive surgery into the subject’s body through small incisions or industrial implanting with sensors for detecting cracks, RCM usually leaves a sole point for the manipulator performs positioning or/and insertion. In addition to the joint physical constraints, RCM imposes an additional constraint on the motion planning and control of redundant manipulator.Let us consider a n-link manipulator with its end-effector’s position is remote from the objective RCM position \(r_P\) which is shown in Fig. 1, the redundant manipulator executes its task control operations between the two end points \(r_A\) and \(r_B\) in the workspace. Point \(r_A\) is the distal point of the end-effector, point \(r_B\) is the distal point of the \(n-1\) link. The end-effector of the redundant manipulator has to travel through the objective point \(r_P\) along the line/curve between point \(r_A\) and point \(r_B\). To describe the motion relation between joints and end-effector, we have$$\begin{aligned} \left\{ \begin{array}{ll} \dot{r}_A=J_1{\dot{\theta }}\\ \dot{r}_B=J_2{\dot{\theta }} \end{array} \right. \end{aligned}$$ (1) where \(r_A\in R^{m}\) and \(r_B\in R^{m}\) denotes the position vector of the end-effector and the position vector of the end-point of the \(n-1\)th link of the manipulator respectively, \(J_1\in R^{m\times n}\) and \(J_2\in R^{m\times n}\) respectively denotes the Jacobian matrix of the whole manipulator and the Jacobian matrix of the associated \(n-1\) link, and \(\theta \in R^{n}\) denotes the joint angle of the manipulator.Fig. 1RCM constraint for a redundant manipulator during motion planning and controlFull size imageIn Fig. 1, \(r_P\) denotes the position vector of the RCM point P, and it can be within the line \(A-B\) and the corresponding relation among \(r_P\), \(r_A\) and \(r_B\) is depicted by$$\begin{aligned} r_P-r_B=k(r_A-r_B) \end{aligned}$$ (2) where \(k\in R\) is a scaling parameter to locate the position \(r_P\) with the RCM constraint. When

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Expertise evaluates and interprets the data.‍8. Not Reviewing and AdaptingChallenge: RCM is not a one-time implementation. Failing to review and adapt strategies can lead to diminishing returns.Solution: Schedule regular RCM reviews, considering new data, business changes, and technological advancements. Adapt strategies accordingly to ensure continued relevance and effectiveness.‍Why Partner with an RCM Expert?In Reliability Centered Maintenance (RCM), working with experienced professionals can make a big difference. These experts have lots of knowledge from years of hands-on work. They help businesses avoid common problems and use the best methods. When organizations partner with RCM specialists, they get access to valuable information, from understanding how critical assets are to creating maintenance plans that fit their needs. These experts also know a lot about different industries, so they can offer custom solutions that match what a business does. They're good at using advanced tools and data to help businesses decide what to do, making assets work well and last longer. With their help, RCM becomes easier to put into action, and businesses can see the benefits faster. RCM experts aren't just advisors; they're like partners that guide businesses toward better maintenance, smoother operations, and long-term success.‍Introduction to Cryotos CMMS (Computerized Maintenance Management System)Cryotos CMMS is a cutting-edge software solution designed to revolutionize business maintenance operations. Leveraging AI and IoT technologies empowers organizations to transition from reactive to proactive maintenance, ensuring assets are consistently optimized and costly breakdowns are minimized. With a user-friendly interface and customizable workflows, Cryotos CMMS caters to various industries, making it a versatile preventive and predictive maintenance tool. It's the go-to partner for businesses seeking efficient, data-driven maintenance management that leads to enhanced asset performance and reduced operational costs.‍AI and IoT-Driven CMMS Software: Cryotos integrates Artificial Intelligence and the Internet of Things into their CMMS, which aids in predictive maintenance and real-time monitoring of assets.‍Proactive Maintenance Approach: The platform emphasizes proactive maintenance, addressing potential issues before they escalate into expensive breakdowns.‍Comprehensive Maintenance Management: Cryotos offers preventive, predictive, and proactive maintenance solutions, ensuring assets are always in optimal condition.‍Work Order Management: Cryotos revolutionizes work order management, allowing businesses to assign specific. RCM business RCM - Facebook Rcm.

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Being 0.15 m, 2) a square with its length being 0.10 m, 3 ) a tetracuspid curve and 4) an ‘8” shape (eight-character) curve. Three types of activation functions are used: 1) pure linear activation function \(\psi (u)=u\); 2) powersum activation functions, i.e., unit-coefficient powersum activation function \(\psi (u)=u+u^3+u^5\), powersum-I activation function \(y=5u+15u^3+25u^5+35u^7\) and powersum-II activation function \(y=10u+30u^3+50u^5+70u^7\); 3) hyperbolic sine (sinh) activation function \(\psi (u)=\sinh (\zeta u)\) with \(\zeta =5\) and \(\zeta =10\).Fig. 2Comprehensive performance of the proposed method for circle path tracking with the RCM constraint. The linear activation function is usedFull size imageFig. 3Comprehensive performance of the proposed method for square path tracking with the RCM constraint. The linear activation function is usedFull size imageFig. 4Comprehensive performance of the proposed method for tetracuspid path tracking with the RCM constraint. The linear activation function is usedFull size imageFig. 5Comprehensive performance of the proposed method for “8” path tracking with the RCM constraint. The linear activation function is usedFull size image5.1 Linear CaseFirstly, we utilize the linear activation function \(\psi (u)=u\) for the proposed simplified RNN method and evaluate its performance. Fig. 2 presents the comprehensive performance of the proposed simplified RNN method for the redundant manipulator for circle path tracking. Figure 2a shows the circle path tracking performance, and we can roughly observe that, the path tracking is well done with RCM constraint satisfied (a point \(r_P\) is obviously seen). Figure 2b shows the position error \([E_x,E_y,E_z]\) for the end-effector of the redundant manipulator, and one can evidently see that the position error \([E_x,E_y,E_z]\) can be lower than \(6\times 10^{-3}\) m, which is very small as compared with the radius of the circle path target. Figure 2c shows resolved joint angle for by the proposed simplified RNN method. Figure 2d shows the convergent process of the 3D coordinates of the \(r_P\) point, and we see that its 3D coordinate \(r_{p}\) can almost hold static from the beginning time instant, which indicates that when finishing the path tracking tasks the point \(r_P\) is almost made fixed. All these results verify the proposed method is efficient for the redundant manipulator on circle path tracking control with the RCM constraint satisfied.Fig. 6Comprehensive performance of the proposed method for circle path tracking with the RCM constraint. The powersum activation function is usedFull size imageFig. 7Comprehensive performance of the proposed method for square path tracking with the RCM constraint. The powersum activation function is usedFull size imageFigure 3 presents the comprehensive performance of the proposed simplified RNN method for the redundant manipulator for square path tracking. Figure 3a shows the circle path tracking performance, and we can roughly observe that, the path tracking is well done with RCM constraint satisfied (a point \(r_P\) is obviously seen). Figure 3b shows the position error \([E_x,E_y,E_z]\) for the end-effector of the redundant manipulator, and one can evidently see that the position error \([E_x,E_y,E_z]\) can be lower than \(1\times 10^{-3}\) m, which is very small as compared with the length of the square path target. Figure 3c