Dissertations / Theses on the topic 'Planar Parallel Manipulator'

Cable Suspended Parallel Manipulators represent an emerging field of study due to the complexities predicting their pose. Despite this issue, suspended cable manipulators possess several advantages over fully-constrained cable manipulators. These include, among others, ease of setup and fewer cables. The reduction in cables relieves excess force computation and likelihood of cable interference.

Two planar Cable Suspended Parallel Manipulator models were created. One has one end-effector connection point, a pendulum type

CSPM

, and the other has two connection points, a suspended plate type

CSPM

. The model's dynamic properties were explored to create system input commands that limited residual vibration. Simulations were run demonstrating the effectiveness of the control methods.

The simulations were verified using experimental data. The pendulum type

CSPM

experiments were performed on a small-scale

CSPM

setup, while the platform type

CSPM

experiments were performed on a full-scale bridge-inspecting robot. The control method created for both experiments proved to reduce the vibration opposed to no control method. The

CSPM

model was also used to create a cooperative input control method, which reduced the risetime of the control command, while still providing vibration reduction.

Robotic automation has been the major driving force in modern industrial developments. High speed pick-and-place operations find their place in many manufacturing applications. The goal of this project is to develop a class of high speed robots that has a planar workspace. The presented robots are intended for pick-and-place applications that have a relatively large workspace. In order to achieve this goal, the robots must be both stiff and light. The design strategies adapted in this study were expanded from the research work by Prof Khajepour and Dr. Behzadipour. The fundamental principles are to utilize a parallel mechanism to enhance robot stiffness and cable construction to reduce moving inertia. A required condition for using cable construction is the ability to hold all cables under tension. This can only be achieved under certain conditions. The design phase of the study includes a static analysis on the robot manipulator that ensures certain mechanical components are always held under tension. This idea is extended to address dynamic situations where the manipulator velocity and acceleration are bounded. Two concept robot configurations, 2D-Deltabot, and 2D-Betabot are presented. Through a series of analyses from the robot inverse kinematic model, the dynamic properties of a robot can be computed in an effective manner. It was determined that the presented robots can achieve 4g acceleration and 4m/s maximum speed within their 700mm by 100mm workspace with a pair of 890W rotary actuators controlling two degrees of freedom. The 2D-Deltabot was chosen for prototype development. A kinematics calibration algorithm was developed to enhance the robot accuracy. Experimental test results had shown that the 2D-Deltabot was capable of running at 81 cycles per minute on a 730mm long pick-and-place path. Further experiments showed that the robot had a position accuracy of 0. 62mm and a position repeatability of 0. 15mm, despite a few manufacturing errors from the prototype fabrication.

This thesis presents the mechanical design of the prototype of a planar three degree-of-freedom parallel manipulator with revolute joints and the design and implementation of a closed-loop control circuit which controls the motors of the constructed manipulator.
Mechanical design here is understood as the decision-making process involved in tasks such as: material selection, structural design and drive selection. In this study the aforementioned tasks for the three degree-of-freedom parallel manipulator are described in detail and their conclusions are used for the construction of the manipulator.
Finally a motor speed/control system for the control of the selected motors, to form a complete microprocessor-controlled DC motor servopositioning system, is designed. The final application circuit which includes additional hardware as well, is implemented connected and tested with the motors of the constructed prototype.
In addition an introduction to parallel processing and an interface proposal are made as a first step towards the next phase of this project.

The application of a parallel processing system (hardware and software) in the solution of the kinematics and dynamics to control a three degree of freedom parallel manipulator is the subject of this thesis.
Parallel processing systems are introduced and analysed for this application. A parallel microcomputer system is used and a new method of direct kinematics for displacement analysis is implemented. The selected microcomputer system is integrated in an IBM 286 compatible personal computer.
Finally, a parallel software program is implemented, which allows for efficient control of three actuators driving the three degree of freedom planar manipulator. Hence, introducing a new processing method for the control of this manipulator.

This thesis presents a detailed kinematic analysis of a 3-degree-of-freedom planar parallel manipulator with holonomic higher pairs. The manipulator consists of a circular disk which rolls without slip on the non-grounded rigid links of each of three 2R serial legs.
The first portion of the thesis is devoted to the review of the geometric and mathematical tools used in the kinematic analysis. Planar isometries and group theory are used in the developmental of the inverse kinematics (IK) algorithm. Kinematic mapping and Grobner bases are important for the forward kinematics (FK) algorithm.
After six important geometric properties of the manipulator are identified, the IK algorithm is developed. It is based on the decomposability and commutativity of planar displacements. The four step algorithm provides closed form analytic solutions. The algorithm may be used on similar parallel manipulators with any number of 2R legs, and hence, applies to a whole class of manipulators. It will be shown that there can be no more than 4$ sp{n}$ real solutions, where n is the number of 2R legs. Three numerical examples are given.
The FK problem is solved using kinematic mapping. To employ a technique from the literature, pseudo inputs must be used to specify joint parameter inputs. The resulting set of three non-linear equations in three unknowns is solved using Grobner bases theory. A numerical example is given.
Finally, velocity and acceleration analysis are performed after the determination of the Jacobian matrix.

Manipuladores paralelos apresentam inércia reduzida, o que lhes permite alcançar altas acelerações e melhor eficiência energética. Porém, frequentemente seu espaço de trabalho possui desempenho pouco uniforme. De fato, a presença de singularidades paralelas é um forte limitante para essa arquitetura robótica. A redundância de atuação já é conhecida como uma alternativa para essa questão. Enquanto que a redundância cinemática ainda não possui consequências claras. Um manipulador com esse tipo de redundância apresenta um número de graus de liberdade do end-effector menor que o número de graus de liberdade do mecanismo como um todo. Considerando essa lacuna, o objetivo desse mestrado é analisar a influência da redundância cinemática no desempenho de manipuladores paralelos através de simulações e testes experimentais. Tal estudo não é trivial, uma vez que com o maior número de atuadores e graus de liberdade, é adicionada também inércia ao sistema. Foram definidas métricas para avaliar o quão favorável é uma dada posição e, com elas, estratégias de resolução de redundância foram analisadas. A estratégia principal proposta se compõe por duas etapas. No primeiro passo, é definida uma movimentação na qual a posição a cada instante é ótima segundo uma dada métrica de desempenho multiobjetivo. Isso resulta em um movimento de referência que em geral possui altas acelerações. Na segunda etapa, aplica-se uma otimização global, procurando manter um compromisso de proximidade com o movimento de referência e com os níveis de aceleração. Além deste, foram aplicados diversos métodos de otimização local (onde a cinemática inversa é resolvida para cada instante isoladamente) e uma estratégia global truncada. Essas opções foram comparadas numericamente e experimentalmente, trazendo uma resposta objetiva da influência da redundância cinemática no manipulador paralelo. A campanha experimental foi realizada em uma protótipo construído no Laboratório de Dinâmica da Escola de Engenharia de São Carlos. Esse protótipo consiste em um manipulador paralelo planar com 6 graus de liberdade, tendo assim, até 3 graus de redundância para a movimentação no plano. Têm-se 6 motores rotativos para atuá-los, sendo 3 deles acoplados a guias lineares com fusos para obtenção de atuação linear. O acionamento ou não destas guias define o grau de redundância do sistema, garantindo a versatilidade do protótipo.
As a consequence of their reduced inertia, parallel manipulators present superior energetic efficiency and they are able to reach high accelerations. Nevertheless, their workspace has a poorly uniform performance. Indeed, the presence of parallel singularities is a strong limitant for this kind of robots. On the one hand, actuation redundancy is well-known as a good choice in an effort to solve this issue. On the other hand, kinematic redundancy still have unclear consequences on this matter. A kinematically redundant manipulator presents an end-effector with fewer degrees of freedom than the mechanism as a whole. Considering this gap, the objective of this research is to analyze the influence of kinematic redundancy on the performance of parallel manipulators through simulations and experimental tests. This issue turns out to be complex, once traveling actuators sum additional inertia to the system. Metrics are defined in order to evaluate how favourable is a given position, and redundancy resolution strategies are analyzed using them. The main proposed strategy is composed of two steps. In the first one, a movement is defined so that the position for each instant is optimum for a given multiobjective performance metric. This procedure delivers a refererence movement which generally presents high accelerations. On the second step, a global optimization is applied, seeking for a trade-off between the proximity to the reference and the acceleration levels. In addition, several local methods (which resolve the inverse kinematics for each instant independently) and one truncated global strategy were addressed. These configurations were compared numerically and experimentally, delivering a objective analysis of the influence of kinematic redundancy on the performance of the parallel manipulator. The experimental campaign was executed with a physical prototype built at the São Carlos School of Engineering. This is a planar manipulator with 6 degrees of freedom, consequently presenting up to 3 degrees of redundancy. The mechanism is actuated by 6 rotating motors, of which 3 are coupled to leadscrews, resulting in linear actuators. These leadscrews can be locked, defining different degrees of redundancy and granting the versatility of the prototype.

碩士
國立成功大學
機械工程學系
86
Although parallel manipulators have many advantages in applications, such as good precision and high rigidity, the applications of parallelmanipulators are limited because of their branching and singularity problem. To overcome these problems, the motions of parallel manipulatorsmust be well- planned, and there must be indices to indicate the occurrenceof the singular configurations. In this thesis, the motion planning of the three-degree-of-freedom planar parallel manipulator is studied. Thegoal is to develop a scheme to guide the end- effector from one positionto another without passing through any singular configuration. First, we discuss the forward kinematics of the planar parallel manipulator,including the branching problem and the conditions of the occurrence of singularity. The singularity is identified when the determinant of the manipulator''s Jacobian matrix is zero. Second, a three- dimensional illustrationof the singularity loci, named the singularity surface, is developed. Third, the finite motion of the end-effector is planned in the same space as that of the singularity surface to guide the end-effector without penetratingthe singularity surface. To overcome the branching problem, the end-effector is guided by many via points. Finally, we adjust the velocities of the joints to obtain smooth motions at the via points. In this thesis, a C++ program is developed to perform the motion planningof the manipulator. The motion is then simulated in the software package Working Model to validate the planned path. We have successfully planned the paths of the end-effector of the manipulator to avoid branching and singularity expect for the cases in which it is theoretically impossible to achieve the goal.

Industrial robots and automation technology have advanced rapidly in the last several decades. New types of manipulators that uses parallel mechanisms are becoming more popular due to their high speed and high stiffness. This thesis focuses on a sub-class of parallel manipulators that uses cables to replace rigid links for further increase in speed. The design strategies in this study were expanded from research works by Khajepour. Behzadipour, and Edmon Chan. This thesis presents analysis and development of a new cable-based planar parallel manipulator that is based on a previous prototype built by Edmon Chan. The new manipulator design added a new rotational DOF to the end-effector, and the number of cables are doubled in order to increase the stiffness. New methods for kinematics and dynamics analysis are formulated to make the procedure more systematic. A new mathematical formulation for stiffness matrix of the end-effector is presented. The resultant stiffness matrix is equivalent to the stiffness matrix formulated by Behzadipour. Additional stiffness analysis is conducted on valid range of stiffness calculation and comparison of different cable configurations. A multi-objective optimization problem is formulated in order to search for the best set of design parameters for the manipulator, and it is solved with an exhaustive complete search method. A physical prototype of the manipulator is modelled and manufactured with the help of partners from Conestoga college. Experiments with the manipulator show that more powerful motors are needed to run the robot at full speed. Motor torque measurements show that the dynamics analysis of the manipulator is valid. Stiffness of the manipulator is measured by applying external force to the end-effector, and it is shown to be strong. The manipulator is able to demonstrate a sort and pick-and-place operation at 60 cycles per minute while running at 70% of the maximum speed, with an acceleration of 2.8 g and velocity of 4 m/s.

碩士
國立臺灣科技大學
自動化及控制研究所
107
The main purposes of this thesis are to use five different control methods to position the three-degree-of-freedom planar parallel manipulator (3-RRR PPM),to find out the better control scheme based on the comparison results, and to explore the feasibility of the future and improve the goal. This thesis first introduces the model of three-degree-of-freedom planar parallel manipulator, derives its kinematics and dynamics equations, and analyzes the working space and singular points of the robot arm to prevent the manipulator from exceeding the working space and causing the damage of hardware. In the control theory, this thesis uses five different controllers for position control, proportional-integral-derivative (PID) control, computed torque control (CTC), resolved acceleration control (RAC), visual servoing resolved acceleration control (VSRAC), and convolutional neural network resolved acceleration control (CNN-RAC), where the latter two controllers are proposed. The VSRAC is a variant of the RAC and the VSRAC reduces the computational complexity of the RAC and improves the accuracy of the tracking trajectory, where the image information is added for correction. However the image processing still requires much computing time. In order to solve this problem, this thesis proposes the control method of CNN-RAC, which learns the relationship between an image and the end-effector by convolutional neural network. This method can effectively reduce the time required for image processing and reduce the noise in image processing. The image could not be crawled. In the hardware and experimental part, this thesis uses a brushless DC motor (BLDC) as the actuator of the robot arm, which is controlled by Texas Instruments' digital signal processor (DSP) for motor control and programmed with the development software CCS. RS485 is used as the communication architecture between the computer and the DSP, and the computer is the output command of the C# development tool computing controller. The simulation part for manipulator uses the MATLAB mathematics software, and the derived dynamics is used as the system model, which brings the data of the actual hardware. Finding the convergence and parameters based on the simulation results as the basis of the experimental architecture. Finally, the simulation and experimental results of five control theories of proportional-integral-derivative control, computed torque control, resolved acceleration control and visual servoing resolved acceleration control, convolutional neural network resolved acceleration control are compared and analyzed. According to the results, a suitable three-degree-of-freedom planar parallel manipulator control method is proposed and the future can be further studied.

Given the advantages of parallel manipulators and lightweight manipulators, a 3-PRR planar parallel manipulator with three lightweight intermediate links has been developed to provide an alternative high-speed pick-and-place positioning mechanism to serial architecture manipulators in electronic manufacturing, such as X-Y tables or gantry robots. Lightweight members are more likely to exhibit structural defection and vibrate due to the inertial forces from high speed motion, and external forces from actuators. Structural flexibility effects are much more pronounced at high operational speeds and accelerations. Therefore, this thesis presents the dynamics and vibration control of a 3-PRR parallel manipulator with three flexible links. Firstly, a procedure for the generation of dynamic equations for a 3-PRR parallel manipulator with three flexible intermediate links is presented based on the assumed mode method. The dynamic equations of the parallel manipulator with three flexible intermediate links are developed using pinned-pinned boundary conditions. Experimental modal tests are performed using an impact hammer and an accelerometer to identify the mode shapes, frequencies, and damping ratios of flexible intermediate links. The mode shapes and frequencies, obtained from experimental modal tests, match very well the assumed mode shapes and frequencies obtained based on pinned-pinned boundary conditions, and therefore the dynamic model developed is validated. Secondly, this thesis presents the investigation on dynamic stiffening and buckling of the flexible links of a 3-PRR parallel manipulator by including the effect of longitudinal forces on the modal characteristics. Natural frequencies of bending vibration of the intermediate links are derived as the functions of axial force and rigid-body motion of the manipulator. Dynamic stiffening and buckling of intermediate links is investigated and configuration-dependant frequencies are analyzed. Furthermore, using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight into the design and control of the parallel manipulator with flexible intermediate links. Thirdly, an active vibration control strategy is developed for a moving 3-PRR parallel manipulator with flexible links, each of which is equipped with multiple PZT control pairs. The active vibration controllers are designed using the modal strain rate feedback (MSRF). The amplification behavior of high modes is addressed, and the control gain selection strategy for high modes is developed through modifying the IMSC method. The filters are developed for the on-line estimation of modal coordinates and modal velocity. The second compensator is used to cut off the amplified noises and unmodeled dynamics due to the differentiation operation in the developed controller. The modal coupling behavior of intermediate links is examined with the modal analysis of vibrations measured by the PZT sensors. The error estimation of the moving platform is examined using the measurement of PZT sensors. Finally, an active vibration control experimental system is built to implement the active vibration control of a moving 3-PRR parallel manipulator with three flexible links. The smart structures are built through mounting three PZT control pairs to each intermediate flexible link. The active vibration control system is set up using National Instruments LabVIEW Real-Time Module. Active vibration control experiments are conducted for the manipulator moving with high-speed, and experimental results demonstrate that the vibration of each link is significantly reduced.

Robots are now moving from their conventional confined habitats such as factory floors to human environments where they assist and physically interact with people. The requirement for inherent mechanical safety is overarching in such human-robot interaction systems. We propose a dual actuator called Parallel Force/Velocity Actuator (PFVA) that combines a Force Actuator (FA) (low velocity input) and a Velocity Actuator (VA) (high velocity input) using a differential gear train. In this arrangement mechanical safety can be achieved by limiting the torque on the FA and thus making it a backdriveable input. In addition, the kinematic redundancy in the drive can be used to control output velocity while satisfying secondary operational objectives. Our research focus was on three areas: (i) scalable parametric design of the PFVA, (ii) analytical modeling of the PFVA and experimental testing on a single-joint prototype, and (iii) generalized model formulation for PFVA-driven serial robot manipulators. In our analysis, the ratio of velocity ratios between the FA and the VA, called the relative scale factor, emerged as a purely geometric and dominant design parameter. Based on a dimensionless parametric design of PFVAs using power-flow and load distributions between the inputs, a prototype was designed and built using commercial-off-the-shelf components. Using controlled experiments, two performance-limiting phenomena in our prototype, friction and dynamic coupling between the two inputs, were identified. Two other experiments were conducted to characterize the operational performance of the actuator in velocity-mode and in what we call ‘torque-limited’ mode (i.e. when the FA input can be backdriven). Our theoretical and experimental results showed that the PFVA can be mechanical safe to both slow collisions and impacts due to the backdriveability of the FA. Also, we show that its kinematic redundancy can be effectively utilized to mitigate low-velocity friction and backlash in geared mechanisms. The implication at the system level of our actuator level analytical and experimental work was studied using a generalized dynamic modeling framework based on kinematic influence coefficients. Based on this dynamic model, three design case studies for a PFVA-driven serial planar 3R manipulator were presented. The major contributions of this research include (i) mathematical models and physical understanding for over six fundamental design and operational parameters of the PFVA, based on which approximately ten design and five operational guidelines were laid out, (ii) analytical and experimental proof-of-concept for the mechanical safety feature of the PFVA and the effective utilization of its kinematic redundancy, (iii) an experimental methodology to characterize the dynamic coupling between the inputs in a differential-summing mechanism, and (iv) a generalized dynamic model formulation for PFVA-driven serial robot manipulators with emphasis on distribution of output loads between the FA and VA input-sets.
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The wrench capability of a manipulator is defined as the maximum forces and moments that can be applied (or sustained) by the manipulator. Understanding these kinetostatic performances of a manipulator is crucial in design and control of manipulators. A novel analytical methodology for the determination of the force and moment capabilities of non-redundantly and redundantly actuated planar parallel manipulators is presented. This new methodology is compared to a previous methodology that used optimization and it is shown that the new methodology is computationally more efficient and yields more accurate results. Based on this methodology. five force and moment capability indices are proposed. The proposed indices are used to generate the wrench capabilities of nine different non-redundantly and redundantly actuated planar parallel manipulator architectures.

Parallel manipulators are prone to have force-unconstrained configurations. If the resultant. forces together do not span the system of forces to be applied or sustained, the manipulator is degenerate and is force unconstrained. Physically. the mobile platform can have motion even if all the actuated ,joints are locked, i.e., the manipu¬lator can instantaneously gain one or more degrees of freedom that are unconstrained by the actuators. Since force-unconstrained configurations are uncontrollable, the identification and elimination of such configurations become critical. Two methodologies for identifying the force-unconstrained poses are analyzed. The first method involves the differentiation of the nonlinear kinematic constraints of the input and output variables with respect to time. The second method makes use of the reciprocal screws associated with the actuated joints. Force-unconstrained poses of planar manipulators are analyzed depending on their actuation: Non Redundant Planar Parallel Manipulators.- Force-unconstrained poses of non-redundantly actuated planar parallel manipulators can be mathematically expressed as a function of the three variables that define the dimensional space of the ma¬nipulator. i.e., the location and orientation of the mobile platform. As a conse- Quence, these poses can be plotted as surfaces in the mentioned three dimensional space. i.e., there are two orders of infinity of force-unconstrained poses. Examples of force-unconstrained poses of parallel manipulators are presented: 3-RPR, 3-PRR, and 3-RRR, where the underscore indicates the actuated joint. For the 3-RPR ma¬nipulator, a comparison and discussion between both methodologies is carried out. For the 3-PRR and 3-RRR manipulators, an efficient technique for identifying their force-unconstrained poses, based upon having joint displacements as known values, is presented. Planar Parallel Manipulators with In-Branch Redundancy.- Force-unconstrained poses of planar parallel manipulators with actuated joints replacing passive joints lead to conditions of the joint displacements that have to be satisfied. In particular, the RRR - 2RRR. PRR - 2PRR. and RRR - 2RRR layouts are analyzed. In addi¬tion, equivalent mechanisms, whose motions describe the path of continuous force-unconstrained poses. are presented. The force-unconstrained poses of the analyzed layouts with in-branch redundancy represent curves in the three dimensional space, i.e., there is one order of infinity of force-unconstrained poses. Planar Parallel Manipulators with, Additional Actuated Branches.- Force-uncon-strained poses of planar parallel manipulators with the inclusion of actuated branches, beyond three. lead to a system of multivariable polynomials. Elimination methods are used to reduce the multivariable polynomials to a single polynomial in terms of one variable. In particular, Gröbner Bases and dialytic elimination methods are employed. The actuation layouts 4-RPR. 4-PRR. and 4-RRR are analyzed. The force-unconstrained poses of the analyzed planar parallel manipulators with additional actuated branches also represent curves in the mentioned three dimensional space, i.e.. there is one order of infinity of force-unconstrained poses.

碩士
國立成功大學
機械工程學系
87
The Workspace and Singularity Analysis of the Planar Parallel R-Joint Manipulators Parallel Manipulators can provide higher stiffness and accuracy than serial manipulators, but their geometric constraints are usually complicated in analysis. All of these constraint equations are highly nonlinear. As a result, velocity analysis equations are even more complicated if we differentiate the geometric constraint equations with respect to time. For this reason, the workspace and the singularity analysis of parallel manipulators are relatively difficult, and thus the applications of parallel manipulators are limited. To overcome these problems, we need to derive concise velocity equations and regard these equations as a basis of the workspace and singularity analysis of parallel manipulators. In this paper, we are concerned with the analysis of planar parallel R-joint manipulators. We first define branches and configurations of the manipulators. And we analyze the characteristics of the planer 5R manipulator and the planar 3RRR manipulator, including their forward and inverse kinematics, workspaces, and singularities. In the singularity analysis, the relative velocity approach is used to derive concise equations. In order to avoid singularity configurations of the manipulators, we study the singularities for various combinations of actuated joints. This enables us to find the best actuation scheme during motion planning. Two examples of actuation planning are given in this thesis. According to the analyses of planar parallel R-Joint manipulators, this thesis also find the relationships among their workspaces, singularity curves(surfaces), branches, and configurations. We also utilize the concept of force analysis and Grassman geometry to verify the derived equations.

The goal of the research of this Dissertation is using actuation redundancy to reduce backlash in parallel manipulators (PMs.) Initially, 3-RRR and 3-RPR PM layouts where 3 is the number of branches, R is a revolute joint and P is a prismatic joint, are introduced. Actuated joints will later be underlined in the PM desciptions. A method for determining PM working area for rotated payload platforms, based on a mechanism inversion, is presented. Force solutions for non-redundantly actuated 3-RRR, 3-RRR, 3-RPR and 3-RPR PMs are formulated in terms of screw coordinates. The reciprocal product of screw coordinates is demonstrated to be invarient under changes in reference location and orientation. As examples, the PMs execute basic circle, logarithmic spiral and arc displacement and force trajectories. All non-redundantly-actuated PMs, encounter two backlash-prone zero-actuator-output configurations when executing any of the trajectories. Therefore, non-redundantly actuated PMs are found inadequate for precision applications. Force-uncertainties, where PMs cannot sustain or apply forces in uncertain directions, are examined. For typically actuated 3-RRR and 3-RPR PMs, force uncertainties are identified using screw system arguments based on the existance of 3 actuated forces forming degenerate (rank = 2) planar pencils of forces. These degenerate force pose make arbitrary force and moment application impossible and cause singularities in the force solutions. The working area of the 3-RRR PM is found compatible with all trajectories. This compatibility is due to zero minimum branch length being possible with the limitless angular displacements possible with stacked R joints. In comparison, the 3-RPR PM with minimum joint lengthes imposed on the P joints, has a smaller working area, and is not compatible with any of the trajectories. A P joint modification allowing relative length minimums of zero and a compatible working area identical to the 3-RRR PM, is considered. To address inadequacies, symmetric actuation-redundant 3-RRR and 3-RPR PMs are considered. Pseudo (right Moore-Penrose) inverse of the 3×6 ARS (associated reciprocal screw) matrix is considered to solve for the required actuation. This solution, while providing a minimum 2-norm of the vector of required actuator outputs, does not reduce backlash-prone configurations with all actuators still having two backlash-prone zero-output configurations. An algorithm for reducing backlash, using MATLAB’s constrained optimization routine FMINCON is applied. Minimizing the 2-norm of the vector of actuator outputs, subject to the backlash-free constraint of having outputs ≥ 0 or ≤ 0 depending on the initial values, is considered. Actuators providing the best conditioned ARS matices are utilized for the particular solutions.
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