Advances in Flexible Robotic Manipulator Systems — Part II: Planning, Control, Applications, and Perspectives

This work offers a review of techniques related to flexible robotic manipulator (FRM) systems. Our review is divided into two parts. The first part presents the background of FRMs, a survey of previous surveys, and FRM dynamic modeling methods. This article is the second part, which reviews the advances in FRM motion planners and controllers, along with the emerging applications and perspectives. Compared with previous FRM surveys, our work presents a more detailed categorization and extensive review of emerging research trends since 2020. The ongoing development of FRMs is marking advancements in the field of robotics.

the accuracy of FRM dynamic modeling, efficiency of motion planning, and stability of control.Our survey reviews the recent advances in FRM modeling, planning, and control methods, underscoring their applications and future potentials.The survey is divided into two parts, and this article is the second part, which reviews the FRM motion planning and control methods, together with applications and perspectives.
The rest of this article is organized as follows.Section II reviews FRM motion planning methods.Section III reviews FRM tracking controllers.Section IV explores the emerging applications of FRMs.Section V presents our future outlook.Finally, Section VI concludes this article.

II. FRM MOTION PLANNING METHODS
FRM motion planning is about generating time-dependent trajectories for FRM joints, which move an FRM from one configuration to another within the manipulator's capability.Since the number of feasible solutions to a motion planning problem is infinite, a criterion is needed to select the optimal solution among all candidates.The primary criterion used in FRM motion planning is the vibration suppression performance.Other minimization criteria include the FRM motion execution time and the kinetic energy consumption.Notably, FRM motion planning process is also denoted as trajectory planning, which emphasizes the operation to generate a spatio-temporal curve centered on the FRM's end-effector, especially when users only care about the point-to-point motion of the end-effector.This article prefers the term "motion planning", indicating that we not only plan the motion simply for the end-effector spot but the whole multibody system.The motion planning result, consisting of the end-effector trajectory and each joint's torque trajectory, is sent to the low-level controller for tracking control.Some previous studies regarded motion planning as feedforward control [2], thus, did not particularly review the FRM motion planning methods.However, motion planning methodologies are obviously different from control methodologies, although their function targets are sometimes similar.Thus this work separates them into different topics in Sections II and III, respectively.
A generic FRM motion planning scheme is described as an optimal control problem (OCP), which contains a cost function (also known as the objective function, fitness function, or performance index) and several types of constraints such as FRM dynamic equations, physical constraints applied to joint torques, tow-point boundary conditions, and environmental restrictions.The solution to this OCP includes the optimized joint angle profiles, which yield the deformable motion of the whole FRM system via forward simulation.
FRM motion planners are basically classified by the solution space used for planning, namely the Cartesian space and joint space.A joint-space planner determines the joint trajectories of the FRM before deriving the spatial motion of the FRM in the Cartesian space.Conversely, a Cartesian-space motion planner determines the spatial motion of the FRM (usually the trajectory of the FRM's end effector) in the Cartesian space before inferring the joint actions via inverse dynamic analysis.We categorize the existing planners by their inherent features, wherein a motion planning process is described by how it describes the shape of an FRM trajectory, what criterion it adopts to sort a qualified solution among all candidates, and how to sort.

A. Template-Based Planners
Template-based FRM motion planners refer to those who use predefined template patterns to define the shapes of a joint angle function or its derivatives.Such planners run fast because the planning process does not have any degree of freedom (DOF).
Bayo and Paden [3] noted that using bang-bang control for the manipulator tip acceleration lead to sharp spikes in joint torque, thus is not suitable for flexible manipulators.They proposed a Gaussian function for the tip velocity profile to reduce the execution time and high-frequency vibrations.Once the tip velocity profile is determined in the Cartesian space, the joint torque profiles are derived by inverse dynamics.In solving a quasi-static planning problem along a reference trajectory, Cho and Park [4] proposed a time-varying impulse shaping method to convolve a sequence of impulses to the predefined reference trajectory to plan deflection-suppressed motions for the FRM's joint torques.Mimmi and Pennacchi [5] provided a comparison among different impulse shaping methods used for FRM motion planning.Bilal et al. [6] described FRM joint angle profiles as predefined quintic polynomials and determined the unknown coefficients in the polynomials analytically via boundary conditions.This method guarantees the real-time computation performance and motion smoothness.

B. Direct Optimization-Based Planners
FRM motion planning problems, when nominally described as OCPs, are hard to solve analytically [7].Alternatively, the variables in an OCP can be parameterized as functions of time so that the OCP is transformed into a mathematical program (MP) problem.Direct methods solve the MP via gradient-based optimizers or analytical methods.As opposed to direct methods, indirect methods utilize the MP's optimality conditions to formulate a new MP problem before solving it via a gradientbased optimizer.This and the following sections review the direct/indirect optimization-based methods, respectively.Sarkar et al. [8] described joint angle profiles as B-spline curves in the OCP and planned FRM trajectories via a hillclimbing method.Mohri et al. [9] proposed a motion planner to determine the B-spline joint position profiles along a reference path in the Cartesian space to minimize the residual energy at the goal point via the aforementioned hill-climbing method.At the point of cost function design, K. J. Park and Y. S. Park [10] pointed out that simply minimizing the magnitude of the final location error does not really reduce residual vibration, thus, they shaped each joint profile as a finite cosine Fourier series and a quintic polynomial for fast convergence and accurate satisfaction of boundary conditions.Pond and Sharf [11] shaped the joint acceleration profiles as cubic splines in formulating an OCP to minimize the strain energy (i.e., the elastic deformation level) throughout the entire motion process.Quasi-Newton optimization is adopted to solve this OCP numerically.Xi and Fenton [12] used the Newton-Raphson method to solve point-to-point quasi-static FRM motion planning problems, wherein quasistatic constraints limit the manipulator to move slowly without link deflections.Yue et al. [13] used the cost function of the OCP to maximize the dynamic payload-carrying capacity while the vibrations are restricted by constraints within the OCP.Eisler et al. [14] adopted the Powell's recursive quadratic programming method to solve the formulated OCP numerically.
Most gradient-based optimizers only achieve local optimality instead of global optimality.Thus, an initial guess close to global optimality is needed, otherwise gradient optimization-based FRM motion planning easily converges to low-quality local optima.Serna and Bayo [15] generated by graphic enumeration, i.e., to discretize a promising subset of the whole solution space into grids and greedily select the best one.Liao et al. [16] adopted a random searcher to coarsely generate an initial guess to warm-start the used numerical optimal control solver named sequential quadratic programming algorithm.Eisler et al. [17] proposed a homotopic initialization strategy that incrementally migrates from a rigid manipulator motion planning solution to an FRM motion planning solution.Lismonde et al. [18] proposed a similar incremental warm-starting strategy.
Besides exploring a good initial guess, some studies focused on how to simplify the constraints in the OCP or deploy computational strategies to reduce the FRM motion planning runtime.Palleschi et al. [19] used McCormick envelopes to replace the nonconvex constraints in the nominal OCP with bilinear/quadratic approximations and solved the simplified problems sequentially until a feasible solution to the original OCP is found.Yue [20] introduced an iterative computation framework wherein the OPB-1 optimizer determines a joint angle candidate and the candidate is then simulated to assess end-effector deformation.Minor deformations conclude the iteration, while significant deformations prompt feedback to the optimizer for the next iteration.

C. Indirect Optimization-based Planners
Korayem et al. [21] used Pontryagin's minimum principle (PMP) to derive the optimality conditions of the concerned OCP, thereby formulating a two-point boundary-value problem solved by a BVP4C solver in MATLAB toolbox.A similar idea was proposed by Boscariol and Gasparetto [22], which minimized the joint angular jerk profiles.Compared with [21], [23] additionally modeled joint flexibility-related constraints in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I OVERVIEW OF TYPICAL FRM MOTION PLANNING METHODS
the motion planning problem before formulating a two-point boundary-value problem via PMP.

D. Bioinspired Planners
Bioinspired planners refer to those who utilize the intelligence of search from nature to find qualified FRM motion planning solutions in the large solution space.Similar to Section II-B, a bioinspired planner typically describes the joint profiles or end-effector trajectory as parameterized functions before determining the parameters/coefficients in those functions via bioinspired search.Different from gradient-based optimization, bioinspired search is insensitive to an initial guess and has underlying chances to explore the whole solution space for global optimality.More importantly, bioinspired planners can easily handle FRM motion planning problems with nondifferentiable dynamic constraints while gradient-based optimizers become inefficient to deal with nondifferentiable cost functions or constraints.
Kojima and Kibe [24] described the angular velocities of FRM joints as cubic polynomials and used genetic algorithm (GA) to determine the unknown coefficients in those polynomials with the purpose of minimizing accumulative residual vibrations.Similarly, Wu et al. [25] used particle swarm optimization (PSO) method to plan optimal motions for a flexible dual-arm space robot.Faris et al. [26] described FRM joint angles as fourth-order polynomials and adopted GA to determine the coefficients in those polynomials.Abe [27] used cubic spline functions to parametrically describe FRM joint angle variables before optimizing them via PSO for residual vibration suppression vibrations.Similar studies include [28], [29], [30], [31].Particularly, Xiao et al. [30] introduced collision-avoidance constraints in the formulated motion planning problem because the end of the concerned FRM is equipped with a fragile monitoring camera in narrow environments.

E. Learning-Based Planners
This section reviews the FRM motion planners that involve machine learning.Abe [32] deployed an artificial neural network to learn critical coefficients that shape a trajectory of the end-effector with minimum displacement.Li et al. [33] used quintic polynomials to describe joint angle profiles in an FRM motion planning problem and adopted PSO to solve it offline; the offline optimization performance is used to train a back-propagation neural network for online motion planning.Wang et al. [34] proposed a learning-based FRM motion planner, wherein visually observed demonstration motion data is encoded as time-series configurations via Gaussian mixture model and a planner is built via Gaussian mixture regression.
Table I summarizes the primary motion planners reviewed in this section.
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III. FRM CONTROL METHODS
This section reviews the FRM tracking control methods.The primary objective of FRM control is to suppress vibrations during the FRM motion.Typical control goals that appeared in the literature are summarized in Table II [35].
FRM control strategies are broadly classified into passive and active categories [36].Passive control, initially adopted for vibration suppression during tracking and execution in FRMs, entails the integration of mechanical dampers to mitigate elastic oscillations.Despite being straightforward, this method leads to augmented mass, heightened energy use, and elevated costs.Crucially, passive damping offers limited adaptability in adjusting control responses, thus prompting a shift in research toward active FRM control strategies.Active controllers are further categorized into model-based and model-free methods, details of which are presented in the rest part of this section.

A. Model-Based Controllers
Model-based FRM controllers utilize the analytical FRM system dynamics to define control laws.In fact, most of the FRM modeling methods in the literature ultimately serve as foundations of FRM controller designs.
1) Linear Quadratic Regulator (LQR) Control: An LQR controller, grounded in optimal control theories, aims to minimize a cost function encompassing both state errors and control efforts.This dual focus is particularly suitable for an FRM, where precise position tracking needs to be balanced against vibration suppression.
Konno and Uchiyama [37] proposed a memory-free LQR for controlling a two-link seven-joint FRM in the 3-dim space.Kayastha et al. [38] proposed a composite controller that integrates predictive control and LQR for postimpact motion control of a space FRM.Wang et al. [39] proposed an LQR-based motion and vibration synthesized controller for a multilink FRM under uncertain disturbances with arbitrary frequencies and unknown magnitudes.
2) Boundary Control: Distinct from other FRM control strategies, boundary control directly interacts with an FRM's edges, offering an advantage in mitigating wave propagation and enhancing control stability.This technique is particularly suitable for reducing vibrations at the extremities of the manipulator by actively controlling the boundary inputs.
He and Ge [40] proposed a barrier controller to suppress vibrations in a flexible Euler-Bernoulli manipulator under boundary output constraints.Liu et al. [41] applied the Hamiltonian principle to model the FRM dynamics as a set of ordinary differential equations (ODEs) and partial differential equations (PDEs), and then proposed a boundary controller to regulate orientation and suppress elastic vibrations in an FRM.For suppressing the vibrations in an FRM with input backlash, He et al. [42] proposed a boundary controller embedded with PDE-based manipulator dynamics.Zhao and Liu [43] developed a finite-time convergence boundary controller for a Timoshenko-type FRM to suppress vibrations and disturbances.Liu et al. [44] developed boundary control laws for vibration suppression and angular position tracking of an FRM with a length-varying manipulator.Ren et al. [45] integrated neural networks with boundary controllers to deal with input constraints, model uncertainties, and external disturbances in an FRM.Zhao et al. [46] proposed a boundary disturbance observer-based control for a single-link FRM under external disturbances.This controller is derived based on rigorous analysis with no model reduction and is uniformly bounded stable.Cao and Liu [47] proposed a boundary controller based on coupled ODE-PDEs without neglecting the high-frequency modes.
3) Sliding Mode Control (SMC): SMC stands out for its robustness to deal with internal uncertainties and external disturbances in an FRM.SMC is featured by its capability to maintain system stability and performance under varying conditions.
Zhu et al. [48] developed an adaptive global SMC without gain overestimation for FRM joint tracking control.To improve tip positioning accuracy and trajectory tracking accuracy for an FRM, Long et al. [49] developed model-based SMCs to output driving torques and deployed a learning method to output small compensation torques.Xu [50] adopted singular perturbation to separate an FRM into fast and slow subsystems and used a sliding mode controller to deal with fast dynamics.
4) Adaptive Control: Zhao et al. [51] developed an adaptive fault-tolerant control for an FRM considering the effects of actuator failures, backlash-like hysteresis, and external disturbances.Yang et al. [52] proposed a model-based nonlinear adaptive controller for an FRM.Asymptotical stability and effectiveness are validated via Lyapunov-based analysis and real experiments.

5) Singular Perturbation Control:
As pointed out by Siciliano and Book [53], the full-order FRM dynamics cannot lead to a nonlinear feedback control law because the control input dimension is lower than that of the output dimension.This limitation is conquered by model order reduction based on singular perturbation.Subudhi and Morris [54] described FRM dynamics via Euler-Lagrange formulation and assumed mode method (AMM) before singularly perturbing the derived model to achieve a reduced-order controller.

B. Model-Free Controllers
As FRM systems become complex both in their internal dynamics and in response to external disturbances, limitations of model-based controllers become apparent, often resulting in delayed responses and reduced control accuracy.This has led to the rise of model-free controllers, wherein the FRM dynamics is not explicitly considered.Model-free FRM controllers are adept at dealing with complex dynamics and disturbances.However, a model-free FRM control system relies heavily on the data collected by sensors and observers, thus, the generated control commands are delayed and even unstable if the data collection process is slow.The rest of this section reviews some typical model-free FRM controllers.
1) Proportional-Integral-Derivative (PID) Control: Sarkhel et al. [55] used a PID controller for vibration suppression in a fishing rod-type FRM and explored the best location to place the controller along the flexible rod.Shang et al. [56] developed a tunable PI controller, wherein the key parameters in the controller are dynamically tuned online by AMM-based pole placement.
2) Strain Feedback Control: Ge et al. [57] proposed a nonlinear strain feedback controller to enhance the tip regulation performance of a simple joint PD controller for an FRM.Mohamed et al. [58] proposed a direct strain feedback controller for vibration suppression of an FRM.
3) Prescribed Performance Control: Kostarigka et al. [59] proposed a prescribed performance controller for an FRM with time-varying and unknown joint stiffness to track a reference trajectory without reliance on knowledge of FRM dynamics.
4) Fuzzy Control: Ling et al. [60] proposed an adaptive fuzzy dynamic surface controller for a single-link FRM with input saturation.Xie et al. [61] proposed an adaptive fuzzy eventtriggered fixed-time controller for an FRM with flexible joints, where fuzzy logics are used to model the highly nonlinear FRM dynamics.Sun et al. [62] proposed an adaptive fuzzy tracking controller to deal with the unknown dynamics in an FRM.
5) Neural Network Control: He et al. [63] proposed a neural network-based controller to suppress the vibrations of an FRM with input dead zone.In that work, the input dead zone dynamics and other unknown dynamics are approximated by radial basis function neural networks.Liu et al. [64] proposed an adaptive neural network controller for vibration-reducing and angle-tracking problems of a flexible spacecraft system.Ouyang et al. [65] proposed an adaptive controller with actorcritic networks for tracking the motion of an FRM, where the critic network measures control metrics and the actor network utilizes the critic results as reinforcement signal to generate control commands with FRM uncertainties considered.
Gao et al. [66] proposed an adaptive neural network controller for a two-link FRM to track desired trajectories and suppress vibrations.
6) Reinforcement Learning Control: Qiu et al. [31] proposed a multiagent reinforcement learning vibration controller for vibration suppression of a movable double piezoelectric flexible beam coupling system, where the reinforcement learner was trained by offline trajectory planning samples.For tip trajectory tracking of a two-link FRM, Pradhan and Subudhi [67] used a reinforcement learning-based controller with dynamically updated actor and critic weights in response to payload variations.In that method, a discrete-time PD controller is deployed as a supplementary to guarantee control robustness.
Table III summarizes the advantages and disadvantages of typical controllers reviewed in this section.In the domain of FRM, selecting the right controller is guided by the specific features of the tracking control objective, FRM type, and operation environment.For FRM systems with multivariable dynamic models prone to errors or uncertainties, techniques like fuzzy control or neural network control are appropriate, leveraging their ability to handle uncertainties and nonlinearities.When dealing with infinite-dimensional systems or situations where economical solutions with fewer actuators and sensors are needed, boundary control is a good choice.In environments with high uncertainties and low-frequency disturbances, robust methods like sliding mode control are advantageous, offering rapid response and resilience, though they may not perform well where mechanical wear or noise is a concern.For systems with time-varying parameters or those requiring adaptability and self-improvement, adaptive control is beneficial, despite that it involves a time-consuming FRM system identification process [91].Scenarios involving time-constant dynamics and disturbances, where controller simplicity and stability are needed, may benefit from PID control, but they might struggle in complex disturbances and nonlinear FRM dynamics.In cases where vibration suppression of tip position control is a primary concern, strain feedback control can be effective.For applications that demand both transient and steady-state performance, prescribed performance control is a promising approach.In real-world industrial applications where safety and operational sensitivity are critical, extensive simulations are necessary before employing intelligent controllers such as reinforcement learning control.Overall, the choice of control method should be aligned with the specific FRM dynamics, constraints, and performance metrics of the FRM system, ensuring that the selected controller effectively addresses the challenges in each application.

IV. FRM APPLICATIONS
Emerging FRM applications, especially those absent from previous surveys, are reviewed in this section.

A. Minimally Invasive Surgery
The evolution of minimally invasive surgery has been influenced by the advent of FRMs.Characterized by their adaptability and dexterity, FRMs are suitable for navigating through Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.narrow and complex pathways, making them ideal for intricate surgical operations [68].Wu et al. [69] developed a novel FRM for minimally invasive surgery, integrating two-DOF vertebrae with a ball-and-socket joint for enhanced torsional stiffness and smooth bending [see Fig. 1(a)].This design renders reduced positional errors and high torsional stiffness, showing promise in improving dexterity for torsional movements.Wang et al. [70] introduced a miniature spring-based cable-driven FRM, optimizing microsurgical procedures like fetal and maxillary sinus surgeries.

B. Industrial Manufacturing
In industrial manufacturing, FRMs bring enhanced precision and adaptability to tasks including deburring, drilling, milling, grinding, turning, impact cutting, and polishing.The flexibility Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
enables precise and delicate handling, offering an advantage over traditional rigid manipulators [71].Lv et al. [72] developed a flexible robotic system for precise belt grinding of complex blade edges without overcutting effects [see Fig. 1(b)].This system accounts for the elastic deformation at the contact wheelworkpiece interface and adaptively plans grinding points in response to the curvature changes of the free-form surface.

C. Agriculture
In agriculture, FRMs revolutionize fruit harvesting with their soft touches and precise movements, which ensure the integrity of sensitive fruits.Qin et al. [73] developed an FRM system equipped with tactile sensing for kiwifruit harvesting, adept at identifying fruit firmness before setting an appropriate picking strength [see Fig. 1(c)].Yeshmukhametov et al. [74] developed an FRM for harvesting cherry tomatoes, addressing the need for safe and efficient operations in confined agricultural spaces.

D. Nuclear Maintenance
FRMs are useful in nuclear maintenance tasks such as decontamination and steam generator maintenance.Compared with traditional rigid manipulators, FRMs can touch the inner wall of a tiny and irregularly shaped reaction vessel with a required pressure.Qin et al. [75] developed an FRM system for the China Fusion Engineering Test Reactor (CFETR), focusing on inspection and flaw detection tasks.This FRM system is adept at maneuvering through the confined and complex spaces of CFETR, thus ensuring nuclear maintenance flexibility.Zhang et al. [76] developed a composite multijoint flexible robotic platform [see Fig. 1(d)] for nuclear fusion reactor vessel maintenance, improving load capacity and maneuverability in complex nuclear fusion reactors.

E. Aerospace
Research in space robotics has increasingly focused on the development of lightweight FRM systems, essential for on-orbit servicing missions such as assembly, berthing, docking, orbital debris removal, refueling, repairing, rescuing, transporting, and upgrading.Stolfi et al. [77] developed a dual-arm flexible space manipulator system for postgrasping manipulation of passive objects in space.The system's flexible arms, each with seven DOFs, enable precise motions in on-orbit missions.

F. In-Fluid Turbine Design
The use of flexible responsive materials in turbine design enhances adaptability and efficiency for a robotic manipulator in frequently varying wind or water conditions.For wind turbines, this flexibility allows blades to adjust to changing winds, optimizing energy capture and reducing structural stress for helicopter rotor blades, aircraft wings, and wind-turbine blades [78], [79].When the fluid is water, flexible components in propellers and underwater robotics adapt to fluid dynamics, improving maneuverability, and efficiency.Shang et al. [80] developed an underwater flexible telescopic manipulator, the flexibility of which enables it to handle dynamic underwater currents and obstacles in challenging tasks such as deep-sea exploration and resource extraction [see Fig. 1(e)].

G. Bioinspired Robot Design
Motivated by the natural flexibility of biological organisms, bioinspired robot design enables machines to perform with adaptability, especially when rigid robots are inefficient.Gao et al. [81] developed a bionic flapping-wing robotic aircraft system, wherein the wings are notable for their lightweight, high flexibility, and low energy consumption.Wang et al. [82] developed a flexible underwater vehicle-manipulator system for shallow sea aquafarm tasks.Propelled by flexible flippers, this system imitates the adaptive movements of marine life w.r.t.efficient thrust and precise grasping [see Fig. 1(f)].Inspired by the movement of fishtails, Sun et al. [83] addressed the uneven stress distribution issue in the compliant joints of tendon-driven continuum robots.

V. PERSPECTIVES OF FRMS
This section discusses the emerging trends and perspectives of FRM systems, providing an outlook on the advancements and innovations that are shaping the future route of FRM research and applications.

A. Designing FRM Systems in a Holistic Way
The existing FRM systems are typically developed in a pipeline way, i.e., to establish an FRM hardware system first, followed by FRM dynamic modeling and planner/controller design based on the formulated dynamic model.The upstream modules in this pipeline, if inappropriately designed, would constrain the full potential of downstream modules.A remedy for this is to conduct a holistic design strategy, i.e., considering and deciding all modules of the FRM system simultaneously [84], [85].Sufficient computing hardware resources and powerful simulation software tools are needed to achieve such holism.

B. Inheriting Human Intangible Civilization Through FRMs
FRMs, designed to mirror human musculoskeletal flexibility, are naturally suitable to replicate complex human behaviors.In the past, experts in fields such as cooking, art, music, nursing, or martial arts typically shared their skills through videos or verbal descriptions.These skills, often regarded as part of human intangible civilization, face challenges in transmission because only watching videos or reading descriptions cannot provide a comprehensive learning experience.The advent of FRMs, in conjunction with fundamental models that learn from multimodal human action data, offers a new solution to this problem.By digitally encoding these expert skills into an FRM system, skill copying and sharing are largely facilitated [86].This idea enables people to directly experience these skills through the flexible nature of an FRM, effectively preserving and propagating human intangible cultural heritage.
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C. Emphasizing the Importance of FRM Motion Planning
According to Table I, the goal of a motion planning process is similar to that of a tracking controller.That may be the reason why FRM motion planning was regarded as less important and ignored in some previous studies.Notably, most of the motion planners listed in Table I do not have a collision-avoidance function.However, as the duties of FRMs are becoming more complicated, motion planning and control purposes would be different.Specifically, motion planning is responsible for finding task-aware, dynamics-aware, collision-free, fault-tolerant, vibration-suppressed, and smooth motions while control is only responsible for tracking such open-loop motions.In this sense, FRM motion planning methods have potential to be improved.

D. Developing Micro/Nano FRMs for Material Design
Micro/nano FRMs may bring new research chances to the realm of material design.Such small FRMs, when added to materials, would respond to external stimuli or activate intrinsic energy with flexibility [87], enabling the materials to exhibit desirable functionalities such as energy harvesting, sensing, and memorable shaping, thus opening new avenues for material design and applications.

E. Advancing Human-Robot Interaction Techniques With Assistance of Large Language Models
In human-robot collaboration, a challenge for FRMs is the comprehensive interpretation of human intentions expressed by natural languages.Given the inherent ambiguity and subtlety in human languages, FRMs should be prepared to process and standardize these linguistic nuances effectively.This involves not just executing direct commands, but also understanding and integrating suggestions, evaluations, and critiques into their operational system.Such a language processing procedure enables FRMs to function like loyal apprentices, adapting and responding in a nuanced and context-aware manner.This evolution in language processing is crucial to make the human-robot interaction seamless and intuitive, especially when communication complexity and precision are paramount [88], [89].Another interesting topic is how to make an FRM improve human trust during human-robot interactions via voices and/or flexible motions [90].

VI. CONCLUSION
This article has examined the development and applications of FRMs, charting their progression from theoretical concepts to practical tools from various aspects.From the initial theoretical foundations to their diverse applications, FRMs have demonstrated remarkable growth in both capability and adaptability.FRMs are at the cutting edge of a robotics revolution, reshaping our understanding of what is possible for more effective, safe, and sustainable practices.