Flexible Links Research Papers

Trajectory Control of a Single Link Rigid-Flexible Manipulator

2024

The primary goal of this work is to control the end-point position of a planar single link flexible robot manipulator,from point A to point B. We first introduce the non-minimum phase problem of flexible manipulators. The simulation tools... more

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Determining load carrying capacity of a manipulator by game theory: Closed-loop nonzero-sum differential game approach

by Roozbeh Dargahi

2024, Journal of Mechanical Science and Technology

The aim of this study is to determine the load capacity of a manipulator in closed-loop form and specified path. In open-loop mode, due to the lack of controller in the system, end effector accuracy constraint is considered as the main... more

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Trajectory Control of a Single Link Rigid-Flexible Manipulator

by Engenharia Mecânica

2023

The primary goal of this work is to control the end-point position of a planar single link flexible robot manipulator,from point A to point B. We first introduce the non-minimum phase problem of flexible manipulators. The simulation tools... more

Figure 3.12: Simulation tracking error for an external input torque (Amplitude 0.1Nm)

produces an undesired response of the closed-loop torque. Figure 3.13: Simulated close-loop torque with noise

Figure 3.14: Simulation output with noise

Figure 3.16: Simulated close-loop torque with miscalculated zeros

Figure 4.20: Second Path validation - Deformation of the end-effector (External step input torque with

Figure 3.15: Simulation tracking error with noise

Figure 3.17: Simulation output with miscalculated zeros End-effector motion planning for a one link non-minimum phase robot

Figure 3.18: Simulation tracking error with miscalculated zeros

Number of articles indexed each year in a pubmed search containing the words “robot” or robotic.

Figure 1.2: The Robodoc surgical system.

Figure 2.1: IST planar flexible link robot manipulator Figure 2.2: Mechanical model of a single flexible link The kinetic energy of the system can deduced as [16]

Figure 2.3: The first two mode shapes 2.2 IST planar flexible robot manipulator The state space representation of the flexible robot manipulator can be represented as,

Throughout this work, the above linear model is used for controller design, whereas a non-linear model (quadratic displacements) is used for simulation.

End-effector motion planning for a one link non-minimum phase robot Figure 3.1: Location of the zeros and poles for transfer function (3.16) associated to the initial conditions:

Note that these conditions were chosen to force the desired torque to be symmetric. The desired a coefficients are then directly obtained, solving the linear system above. In this way it’s possible to calculate 3.2 Stable inversion of the IST planar flexible link robot manipulator

End-effector motion planning for a one link non-minimum phase robot

Figure 3.2: Desired input 1. A partial state feedback, based on the joint position and velocity variables.

The method was applied for tr = 2.78 and yr = —0.35m (fig. 3.2 and fig. 3.3). The closed loop control

Figure 3.5: Simulated displacement of the End-Effector

End-effector motion planning for a one link non-minimum phase robot Fig. 3.6 presents the output torque of the close-loop form. Indiscernible in this figure, are the smal

Figure 3.7: Simulated output with an external input torque (Amplitude 0.002Nm)

Figure 3.8: Simulated close-loop torque with an external input torque (Amplitude 0.002Nm)

Figure 3.9: Simulation tracking error for an external input torque (Amplitude 0.002Nm) End-effector motion planning for a one link non-minimum phase robot

Figure 3.10: Simulated close-loop torque with an external input torque (Amplitude 0.1Nm)

Figure 3.11: Simulation output with an external input torque (Amplitude 0.1Nm)

Figure 4.21: Second Path validation - Simulation Tracking error (White noise with the noise power of 1)

End-effector motion planning for a one link non-minimum phase robot _ The calculation of the proportional part of the controller is obtained by the following transfer function

End-effector motion planning for a one link non-minimum phase robot As in the previous validation, the effects of external factors, such as noise, external torque input, and miss calculation of the transfer function zeros will be analyzed here.

Figure 3.24: Simulated output with an external input torque (Amplitude 0.002Nm) Introducing an input torque starting at t = 1s and ending at t = 2.4s, with an amplitude of 0.002N/m,

Figure 3.25: Simulated close-loop torque with an external input torque (Amplitude 0.002Nm)

Figure 3.26: Simulation tracking error for an external input torque (Amplitude 0.002Nm)

Figure 3.27: Simulated output with an external input torque (Amplitude 0.1Nm) external input was applied (t=1s and t=1.4s).

Figure 3.28: Simulated close-loop torque with an external input torque (Amplitude 0.1Nm)

Figure 3.30: Simulated close-loop torque with noise

Figure 3.29: Simulation tracking error for an external input torque (Amplitude 0.1Nm)

Figure 3.31: Simulation output with noise End-effector motion planning for a one link non-minimum phase robot

Figure 3.32: Simulation tracking error with noise

Figure 3.33: location of zeros and poles in a closed loop form for miss calculation of transfer function The location of the zeros and the poles is represented in fig. 3.33: As before the miscalculation of the transfer function zeros for this case does not pose a significan

Figure 3.34: Simulated closed-loop torque with miscalculated zeros End-effector motion planning for a one link non-minimum phase robot

Figure 3.35: Simulation output with miscalculated zeros

Path-Following for one link non-minimum phase robot one solution to achieve a path controller for (4.1), such that the close loop state is bounded detectable, the number of inputs is as large as the number of outputs (mm > q) and the zeros of (A, B, C, D

Figure 4.2: The path in x, y coordinates the obtained path in x, y coordinates is the arc represented in fig. 4.2. while a2 smoothens the square wave, and the m, variable constrains the maximum speed. As expected For the results presented in the next section, the following data was used. The system state space where the parameter a, sets the width of the low speed regions at the beginning and the end of the path, epresentation in continuous mode is

In fig. 4.3 we present the desired yg versus the 7 variable. Figure 4.3: First Path - Evolution of the desired output versus the path variable

Figure 4.4: First Path - Simulation tracking error

Figure 4.5: First Path - Deformation of the end-effector

Figure 4.6: First Path - Evolution of the desired path variable versus the real path variable Path-Following for one link non-minimum phase robot Second path simulation results

Path-Following for one link non-minimum phase robot Figure 4.9: Second Path - Evolution of the desired path variable versus the real path variable

Figure 4.10: Second Path - Deformation of the end-effector

Figure 4.12: Second Path - Deformation of the end-effector

Figure 4.13: Second Path - Simulation tracking error

Figure 4.14: First Path validation - Simulation input torque (White noise with the noise power of 0.1) These values can be considered as a reasonable external perturbation. introduced. The external perturbation torque is 5 times greater that the normal input torque (fig. 4.14)

Figure 4.15: First Path validation - Deformation of the end-effector (White noise with the noise power of 4.1 Path-Following method with internal model control

Figure 4.16: First Path validation - Simulation output (White noise with the noise power of 0.1)

Figure 4.17: First Path validation - Simulation tracking error (White noise with the noise power of 0.1) Path-Following for one link non-minimum phase robot

Figure 4.18: Second Path validation - Simulation tracking error (External step input torque with amplitude \ step input torque with the periodicity of 2s and amplitude of 1Nm has been added to the system input

Figure 4.19: Second Path validation - Simulation input torque (External step input torque with amplitude 4.1 Path-Following method with internal model control

Figure 4.22: Second Path validation - Simulation input torque (Path properties: a; = 1, ag = 0.1 and external input torque remains the same

Figure 4.23: Second Path validation - Deformation of the end-effector (Path properties: aj = 1, ag = 0.1 values increase, at the same time as the desired theta limitation occurs.

Figure 4.24: Second Path validation - Simulation tracking error (Path properties: aj = 1, ag = 0.1 anc

Path-Following for one link non-minimum phase robot Figure 4.25: Second Path validation - Evolution of the desired path variable versus the real path variable

Figure 4.26: Second Path validation - Simulation output (Path properties: a; = 1, ag = 0.1 and m, = 1

Figure 4.27: Second Path validation - Joint velocity versus error (Path properties: a; = 1, ag = 0.1 anc Normal Form Path Following

which controls the leader’s motion in the following system The trajectories of system (4.27) - (4.29) are related to the leader's motion via the error coordinates MIMO and SISO systems to strictly proper systems (Normal form representation). lind this representation can be found in [19]. There, it is explained the structural decomposition of

functions, the M matrix is assumed with the following structure torque vector. Here we have included a damping matrix C. Furthermore to contemplate any type of shape

For n sample times during trajectory execution, data for g, g , ¢ and 7 must be acquired. Renaming Transforming the system equations into state space representation we obtain and the T vector represents the system input

Path-Following for one link non-minimum phase robot therefore using a Recursive Least Square (RLS) method (Appendix B), where all the data acquired during the time of trajectory execution was used, we may obtain the following variables.

Figure 4.28: Output values of flexible link identification

Figure 4.30: Flexible link identification error

Figure 4.29: Flexible link identification validation

To identify the system, an ARMAX model with a time sample of 0.001, and a Tustin transformatior The identification of the system was obtained using an input step with amplitude of 0,02rad and a period from discrete to continuos model was used. This model had the following properties. The system transfer function considered had a 6 order denominator, a 5 order numerator, considering a noise transfer function with a 6 order denominator and a 4 order numerator and with one time sample delay

Figure 5.3: Location of zeros and poles in a close loop form Fig. 5.3 presents zeros and poles of the system.

Figure 5.5: End-effector vibration at high average velocity

Figure 5.4: End-effector vibration at low average velocity

Figure 5.6: Input torque at low average velocity

Figure 5.7: Input torque at high average velocity

[he new state space, input and output of the decomposed system are described by the following set of

Where (Aaa) contains all the system invariant zeros and nq is the relative degree of T Figure A.1: Interpretation of structural decomposition of a SISO system

is nonsingular. Zo can be chosen where whose rows form a basis of the null space of Za. Being 2. Construction of a preliminary state transformation Let Z) be an ng X n constant matrix, where n,

being k the total number of assumed modes. The mass matrix and the stiffness matrix are given by The inertial displacement at a distance x from the frame origin in the OX direction is given by Here, 7; is the modal amplitude of the ith mode considering the assumed modes discretization procedure

Table 2.1: IST flexible robot manipulator characteristics suffice. In table 2.1 the most important properties of the IST planar manipulator are presented.

Simplifying equation (4.49) yields M;'= = Gi;, from equation (4.45) and (4.48) the following equation is obtained:

algorithm: problem of inverting the matrix at each sample time that reduces the computational efficiency of the Equation (B.14) gives a direct update from B(t — 1) to B(t), but the update of P(t — 1) emerge the

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A New Approach to Finite Element Modeling and Simulation of Flexible Robot Manipulators

by Ahmad Reza Khoogar

2023

Traditional robot manipulators that have large links need powerful actuators and their massive structures strongly limit their operating speed. Flexible manipulators having lightweight links are designed to overcome these disadvantages... more

3. Numerical Solution and Simulation Results In this section, the numerical solutions and _ the simulation resu presented. solving difference metho he numerical was executed wi and 2. The pro used in t he sys his simu condition, angular velocity of the two links are considered. Figures 4 and 5 show the first and second joint angle variations with réspect to time. Fig’s 6-7 show the displacement of two points located at the end of the The rotation of the tangent line at end points is shown in Fig’s 8-9. Fig’s 10- 15 show the rate of the mentioned quantities. These are the angular velocities of the first and second links, the inear velocity of the ends of two links, and their wo. links. ts of em o d. Tab h quad files o varia a flexible manipulator, is A computer program was developed for equations using the ‘finite le 1 indicates the parameters solution. The computer code ratic torques appliedyat joints f the quadratic torque used in ation are shown in Fig’s 2-3. Under this ion oO the joint angles and the angular velocity of the tangent line at the end, resnectivelv.

Fig. (3): Actuating torque applied to joint two.

Fig. (7): End point displacement of the second link.

Fig. (8): Rotation of tangent line at the end point of first link.

Fig. (4): Joint angle versus time at first joint.

Fig. (5): Joint angle variations versus time at second joint.

Fig. (6): End point displacement of the first link.

Fig. (9): Rotation of tangent line at the end point of second link.

Fig. (10): Angular velocity of the first link.

Fig. (11): Angular velocity of the second link.

Fig.(12): Linear velocity of the end point of first link.

Fig. (13): Linear velocity of the endpoint of second link.

Fig. (14): Angular velocity at the end of first link.

Fig. (15): Angular velocity at the end of Second link.

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Effect of rigid-flexible coupling and bending-torsional vibrations on the dynamics of an articulated two degree of freedom manipulator

by Natraj Mishra

2023, Research Square (Research Square)

In the present, dynamic analysis of a Two-Link Flexible manipulator is carried out using 11 Lagrangian dynamics and finite element method. The flexible links are susceptible to both bending 12 and torsional vibrations. Euler-Bernoulli... more

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Implementation of a neural network tracking controller for a single flexible link: comparison with PD and PID controllers

by Luis B . Gutierrez

2023, IEEE Transactions on Industrial Electronics

The objective of this paper is to show the results of the practical implementation of a neural network (NN) tracking controller on a single flexible link and compare its performance to that of proportional derivative (PD) and proportional... more

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Implementation of a neural network tracking controller for a single flexible link: comparison with PD and PID controllers

by LUIS DIEGO BARAJAS GUTIERREZ

2023, IEEE Transactions on Industrial Electronics

The objective of this paper is to show the results of the practical implementation of a neural network (NN) tracking controller on a single flexible link and compare its performance to that of proportional derivative (PD) and proportional... more

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Finite-Time Feedback Linearization (FTFL) Controller Considering Optimal Gains on Mobile Mechanical Manipulators

by Shahab Kazemi

2022, Journal of Intelligent & Robotic Systems

The aim of this paper is to implement a novel finite-time feedback linearization (FTFL) controller considering optimal gains on mobile manipulators with differential wheels theoretically and experimentally. To achieve the finite time... more

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Determining load carrying capacity of a manipulator by game theory: Closed-loop nonzero-sum differential game approach

by habib esfandiar

2022, Journal of Mechanical Science and Technology

The aim of this study is to determine the load capacity of a manipulator in closed-loop form and specified path. In open-loop mode, due to the lack of controller in the system, end effector accuracy constraint is considered as the main... more

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Determining load carrying capacity of a manipulator by game theory: Closed-loop nonzero-sum differential game approach

by habib esfandiar

2022, Journal of Mechanical Science and Technology

The aim of this study is to determine the load capacity of a manipulator in closed-loop form and specified path. In open-loop mode, due to the lack of controller in the system, end effector accuracy constraint is considered as the main... more

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Implementation of a neural network tracking controller for a single flexible link: comparison with PD and PID controllers

by Luis B. Gutierrez

2022, IEEE Transactions on Industrial Electronics

The objective of this paper is to show the results of the practical implementation of a neural network (NN) tracking controller on a single flexible link and compare its performance to that of proportional derivative (PD) and proportional... more

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Optimal Balancing of Spatial Suspended Cable Robot in Point-to-Point Motion using Indirect Approach

by amin nikoobin

2022

In this paper, a method based on the indirect solution of optimal control problem is presented to specify the optimal trajectory of spatially suspended cable robot in point to point motion with considering the counterweights. In fact, an... more

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Maximum Allowable Dynamic Payload for Flexible Mobile Robotic Manipulators

by amin nikoobin

2022

Finding the full load motion for a point-to-point task can maximize the productivity and economic usage of the mobile manipulators. The presented paper proposes a technique to determine the maximum allowable dynamic payload for flexible... more

Fig. 1 - A two links mobile manipulator with flexible links and joints.

In this case w is assumed to be constant at w=1. The ro- bot path planning problem is investigated by increasing the payload mass until the maximum allowable payload be determined. The maximum payload is obtained as m,= 8.4 kg (case 4). The obtained angular positions, an- gular velocities and torque curves graphs for a range of m, given in Table 2, are shown in Figs. 4-6. It can be found that increasing the m, results to enlarge the veloc- ity values. Also, as shown in figures, increasing the pay- oad increases the required torque until the maximum payload obtained. So that for the last case the torque curves lay on their limits. Hence, it is the most possible values of the torques and increasing the payload is im- possible, because it leads to violate the boundary condi- ions. n the first case, effects of changing in performance in- dex in the path planning problem are investigated. The payload is assumed to be | Kg and simulation is done for the different values of W=(w, w, 0, 0, w, w) and op- imal paths for a given payload are obtained. Note that in this vector, w refers to the velocities of joints and ac- uators and zero refers to the velocities of mode shapes. The simulations are performed for different values of w as 0, 1, 100 and 1000.

Fig. 2 - Angular velocities of joints with respect to time

Fig. 5- Angular velocities of joints with respect to time In this case, the maximum payload of flexible mobile manipulator is calculated and corresponding optimal tra- jectory at point-to-point motion is illustrated for differ-

ent values of w given in Table 3. The computed torqueses for these cases are plotted in Fig. 7. As it can be seen, increasing w causes to increase oscillatory be- haviors of the systems that results to reduce the maxi- mum dynamic payload as shown in Table 3. Simulation D The effect of time on the calculating of maximum pay- oad is investigated in this section. w is fixed at w=1. The range of times and corresponding calculated maxi- mum payload are given in Table 4. It can be concluded from table, raising the execution time leads to consider- able increase of the obtained maximum payload value. The reason of this issue is that by increasing the execu- ion time the acceleration and velocity of the system are reduced. Hence, the centrifugal and Coriolis forces which applied to the system are decreased and it causes o increase the maximum payload.

reduced. So, the link flexibility affects the robot vibra- tions significantly. Another simulation is carried out to investigate the effect of joint stiffness in the obtained maximum pay- load of the systems. Simulation is performed for a flexi- ble joint manipulator with different values of joint stiff- ness while EI is remained constant at EI=100 kg.m’/s’. K, and the obtained The values of spring constants, maximum payloads are given in ities of links and motors in the Table 6. hat by increasing the joint stiffness va mum payload is increased. Angular positions and veloc- first cas Figs. 10 and 11. It shows that both the lin! ions and velocities have deviations from their respec- ive motor angular positions and veloci It is observed ue, the maxi- e are given in k angular posi- ies. It can be concluded that link and joint flexibility significantly af- fect on the system oscillation and robot behavior. Hence neglecting of such important affects in modeling and control of robots, results in the considerable and irrepa- rable errors in the applicable usage.

Fig. 11- Angular velocities of links and motors with respect to time After that, maximum allowable dynamic payload has been determined along with the obtained ories in the various simulations. The ob dex, operating time and robot charac maximum payload trajectory. As a resul posed method, the designer can choose priate path among the various optimal o designing necessities. Also, the op payload trajectory obtained using this he closed-loop control of the robot mani optimal trajec- ained results il- ustrate the influences of the problem performance in- eristics on the , using the pro- he most appro- paths according imal maximum method can be used as a reference signal and feed forward command in pulators.

n this section, various numerical simulations are pre- sented based on the mathematical model describes in the section 2. In these simulations, the robot optimal trajec- ories and corresponding maximum payloads are de- fined based on various parameters such as operating ime, robot characteristics and the problem performance index. Note that in all simulations, the payload is calcu- ated with the accuracy of 0.1 Kg. Also, in all simula- ions the performance index is defined for minimizing he actuating inputs in addition bounding the joint ve- ocities. For this reason, it form as en

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The Comparative Assessment of Modelling and Control of Mechanical Manipulator

by amin nikoobin

2022

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Dynamic Modeling Of A Robot For Playing A Curved 3D Percussion Instrument Utilizing A Finite Element Method

by Prakash Persad

2022

The Finite Element Method is commonly used in the analysis of flexible manipulators to predict elastic displacements and develop joint control schemes for reducing positioning error. In order to preserve simplicity, regular geometries,... more

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Implementation of a neural network tracking controller for a single flexible link: comparison with PD and PID controllers

by Luis Alejandro Herrera Gutierrez

2022, IEEE Transactions on Industrial Electronics

The objective of this paper is to show the results of the practical implementation of a neural network (NN) tracking controller on a single flexible link and compare its performance to that of proportional derivative (PD) and proportional... more

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Trajectory Optimization of Space Manipulators with Flexible Links Using a New Approach

by Reza Jamilnia

2021

In this paper, the problem of trajectory optimization of flexible space robots is considered and an efficient approach is proposed for solving it. The space robots are considered as mechanical manipulators having rigid and flexible links.... more

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Trajectory tracking for robot manipulators using differential flatness Trajectory tracking for robot manipulators using differential flatness

by Max Dutra

2021, INGENIERÍA E …

This paper proposes applying differential flatness to robot manipulator trajectory tracking. The trajectories for each generalised coordinate are proposed as a function and the corresponding input must be found to guarantee tracking. It... more

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Trajectory Optimization of Space Manipulators with Flexible Links Using a New Approach

by Mohsen Bahrami

2021

In this paper, the problem of trajectory optimization of flexible space robots is considered and an efficient approach is proposed for solving it. The space robots are considered as mechanical manipulators having rigid and flexible links.... more

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Deployable space manipulator closed-loop control, ideas and possibilities of using GPS as a sensor

by Rade Vignjevic

2021, Advances in Space Research

In the area of study of the dynamics and control of large flexible spacecraft one of the most challenging areas is that of large space-based robotic manipulators. The Deployable Space Manipulator is a large manipulator concept for Low... more

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Trajectory Tracking for Robots Manipulators using Differential Flatness

by Omar Lengerke

2021

This paper proposes applying differential flatness to robot manipulator trajectory tracking. The trajectories for each generalised coordinate are proposed as a function and the corresponding input must be found to guarantee tracking. It... more

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Trajectory tracking for robot manipulators using differential flatness Trajectory tracking for robot manipulators using differential flatness

by Omar Lengerke

2021, INGENIERÍA E …

This paper proposes applying differential flatness to robot manipulator trajectory tracking. The trajectories for each generalised coordinate are proposed as a function and the corresponding input must be found to guarantee tracking. It... more

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Implementation of a Neural Network Tracking Controller for a Single Flexible Link: Comparison with PD and PID Controllers

by Luis B . Gutierrez

2021, IEEE Transactions on Industrial Electronics

The objective of this paper is to show the results of the practical implementation of a neural network (NN) tracking controller on a single flexible link and compare its performance to that of proportional derivative (PD) and proportional... more

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Trajectory Optimization of Space Manipulators with Flexible Links Using a New Approach

by International Journal of Robotics K.N. Toosi University of Technology

2016

In this paper, the problem of trajectory optimization of flexible space robots is considered and an efficient approach is proposed for solving it. The space robots are considered as mechanical manipulators having rigid and flexible links.... more

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Dynamic Load Carrying Capacity of Mobile- Base Flexible-link Manipulators: Feedback Linearization Control Approach

by International Journal of Robotics K.N. Toosi University of Technology

2016

This paper focuses on the effects of closed-control on the calculation of the dynamic load carrying capacity (DLCC) for mobile-base flexible-link manipulators. In previously proposed methods in the literature of DLCC calculation in... more

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