## Advances in Robot Kinematics: Analysis and Control

On the other hand and caused by the unilateral constraints of the wires, the workspace of wire robots is primarily limited by the forces which may be exerted by the wires. The unilateral constraints necessitate positive forces. Practically, long wires will sag at low tensions which makes kinematical computations more complicated and may lead to vibration problems.

Hence, the minimum allowed forces in the wires should never fall below a predefined positive value. Against, high forces lead to increased wear and elastic deformations. Thus, a description of the force distribution in the wires for given end effector poses and wrenches is needed. Here a convenient description of the force distribution will be presented, while in Bruckmann et al.

The force and torque equilibrium at the end effector gives according to figure 5. Hence, the force and torque equilibrium can be written in matrix form. In the following the matrix A T is called structure matrix. It is noteworthy that the structure matrix can also be derived as the transpose of the Jacobian of the inverse kinematics, but generally, it is easier to construct it based on the force approach Verhoeven, In practical applications knowledge of the workspace of the robot under consideration is essential.

In contrast to conventional parallel manipulators using rigid links, the workspace of a wire robot is not mainly limited by the actuator strokes, since the length of the wires is not the main limiting factor, just restricted by the drum capacity. In fact, the workspace of a wire robot is limited anyway by the wire force limits f min and f max.

Additionally further criteria, like stiffness or wire collisions, can be taken into account. Different methods to calculate the workspace of a wire robot are available. Here discrete methods as well as a continuous method using interval analysis are discussed. Further methods exist as for example presented in Bosscher and Ebert-Uphoff, , where the workspace boundaries are computed.

In order to perform a discrete workspace analysis at first an assumed superset of the workspace is discretized. Mostly an equidistant discretization is desired. This leads to a set of points, which is then tested with respect to the chosen workspace requirements. This is a widely used approach, but nevertheless, some considerations should be taken into account:.

The calculation of the workspace conditions for the grid points generally requires the verification of a valid wire force distribution. Since it is sufficient to identify any valid distribution, fast calculation methods as presented in section Bruckmann et al. For some parallel kinematic mechanisms, typically symmetrical configurations are singular, leading to uncontrollable d. Thus, it is recommended to explicitly test at symmetrical poses of the end effector. Generally, it is desired to rule out gaps in the workspace. Using a discrete approach, this is intrinsically impossible, but for practical usage, one may try to increase the grid resolution. Clearly this leads to a dramatical increase of the number of points to be checked and thus to extremely long computation times.

To come up against this, parallelisation of the calculation by partitioning the workspace and allocation to different processing units is helpful and especially for this problem very efficient due to the independency of the workspace parts. Nevertheless, up from a specific resolution, continuous methods as presented in the next section should be considered. In this section a method to compute the workspace of a wire robot, formulating this task as a constraint satisfaction problem CSP , is shown.

The CSP can be solved using interval analysis. However, other solving algorithms are also conceivable. The presented formulation can also be used for design just by interchanging the roles of the variables Bruckmann et al. This fact simplifies the generally complicated and complex task of robot design. For details see section 5.

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In Gouttefarde et al. A criteria for the solvability of the interval formulation of eqn. The solvability of those 2 n systems of linear inequalities guarantees the existance of at least one valid wire force distribution. Based on this criteria a bisection algorithm is presented. This approach is beneficial in terms of the number of variables on which bisections are performed since no verification or existance variables are required. Here, however the CSP approach is presented due to its straight forward transferability to robot design. It will be shown later that for a description of the workspace, this problem can to be extended to.

Examining eqn. Since A T has a non-squared shape, this is usually done using the Moore-Penrose pseudo inverse. Thus, the calculated forces will be a least squares solution. In fact, not a least squares result but a force distribution within predefined tensions is demanded. Now, the resulting force distribution can be calculated as. Force equilibrium workspace of plain manipulator, 2 translational d. Hence eqns. To calculate a workspace for a specific robot, the following variable set for the CSP is used:. Optionally, the exerted external wrench w and desired platform orientations can be set as verification variables.

The workspace for a fix orientation of the platform is called constant orientation workspace according to Merlet, On the other hand, sometimes free orientation of the platform within given ranges must be possible within the whole workspace. The resulting workspace is called the total orientation workspace. In fig. Interval Analysis is a powerful tool to solve CSPs. Therefore a short introduction is given in the following section. Then b is called the supremum and a the infimum of I.

Parallel Continuum Robots: Modeling, Analysis, and Actuation-Based Force Sensing

A n-tupel of intervals is called box or interval vector. This phenomenon is called overestimation and causes additional numerical effort to get sharp boundaries. For sure the same holds for min and Inf. Thus for input intervals I 0 , This evaluation is guaranteed to include all possible solutions, e. As shown in detail in Pott, , a CSP can be solved using interval analysis which guarantees reliable solutions Hansen, , Merlet, b , Merlet, According to eqn. Additionally, available implementations for interval analysis computations are robust against rounding effects.

To use it for the special problem of analyzing wire robots, they have been extended. Details are described in the next sections. Here the domain Xv is represented by the list of boxes L T v. Thus, the result can be valid, invalid, undefined or finite. If at least one box is invalid, the whole search domain does not fulfill the required properties and is therefore invalid.

Algorithm Verify. Define a search domain in the list L T v. In the simplest case, L T v contains one search box. If the result is valid, goto 2. If the box is invalid, return with invalid. If the box is finite, goto Thus, the box is valid. Goto 2. Thus, the box is invalid. Return with invalid. Divide the box on a verification variable and add the parts to L T v. Existence is a modification of Verify. Here the domain X e is represented by the list of boxes L T e The result can be valid, invalid or finite. If at least one box is valid, the whole search domain fulfills the required properties and is therefore valid.

Algorithm Existence. Define a search domain in the list L T e. In the simplest case, L T e contains one search box. If L T e is empty, the algorithm is finished with invalid. Return with valid. Divide the box on an existence variable and add the parts to L T e. It uses Existence or Verify to identify valid boxes within the search domain. Thus, the result is a list L s of valid boxes and optionally the lists L I for invalid boxes and L F for finite boxes, respectively.

Algorithm Calculate. Define a search domain in the list L T c. In the simplest case, L T c contains one search box. L s for solution boxes,. L I for invalid boxes,. L F for finite boxes. Goto 3. If the result of Verify is valid, move the box to the solution list L s. If the result of Verify is invalid, move the box to the invalid list L I. If the result of Verify is finite, move the box to the finite list L F. In order to determine L s , Calculate is called with the search domain L T c. Within Calculate, Verify is called. Since existence variables are present, Existence is called in order to validate the current calculation box Otherwise in Verify the CSP would be directly evaluated.

In the Existence algorithm the CSP is evaluated and the result is rated. In case that the result is undefined, the current box is divided on an existence variable. In case that the Existence algorithm returns with finite, the calling algorithm divides on its own variables and calls Existence again. If the result is valid or invalid, the result is directly returned to the calling algorithm. The same calling sequence and return behaviour is used in Calculate calling Verify. For an effective CSP solver the return scheme should be more advanced in the way that not one variable is bisected until the box under consideration is finite, but a more sophisticated bisection distribution is used.

Since solving the force equilibrium is a computationally expensive task, favorable prechecks are demanded to reduce computation time.

## Advances in Robot Kinematics: Analysis and Control

The resulting preliminary workspace is an outer estimate and excludes poses which are not treated furthermore. Another possibility to reduce the computation time is to take symmetries into account. If symmetry axes as well as a symmectrical load range are present it is sufficient to compute only one part of the workspace and to complete the workspace by proper mirroring. Besides the force equilibrium, additional workspace conditions can be applied. Due to the high elasticity of the wires using plastic material, e. Thus, for practical applications, especially if a predefined precision is required, it may be necessary to guarantee a given stiffness for the whole workspace.

Otherwise, the compensation of elasticity effects by control may be required. Generally, this should be avoided as far as possible by an appropriate design.

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As shown in Verhoeven, , the so-called passive stiffness can be described as the reaction of a mechanical system onto a small pertubation, described by a linear equation:. Here, L is the diagonal matrix of the wire lengths and k ' is the proportionality factor force per relative elongation , treating the wires as linear springs. For the calculation, the inverse problem. This equation can again be treated as a CSP. However, stiffness can also be checked performing a discrete workspace analysis. The stiffness workspace for a simple plain manipulator with 2 translational d.

Therefore all wire robots with pure translational d. For a wire robot with rotational and translational d. Since within the workspace analysis discrete or continuous typically a system of linear equations is solved, the singularity criteria eqn. Mechanically, at singular poses certain d. Often this happens in symmetrical configurations. In analogy to the problem of link collisions for conventional parallel manipulators, wire collisions have to be avoided. Due to their normally small diameter one possibility is to consider the wires as lines.

In Merlet, a an algorithm is proposed to determine the regions in which collisions between wires as well as the collisions between wires and the end-effector occur. Practically, wires have certain diameter and thus, a predefined minimum distance at least the wire diameter should be always ensured. Therefore, the well-known problem of determining the smallest distance between two lines arises. Since the lines are known after solving the inverse kinematics this is a very basic task but may be computational expensive. Clearly, the distance condition has to be formulated as a inequality.

Hence, this criteria can be easily included in the CSP formulation. While workspace analysis examines the properties of already parametrized manipulators which allows to determine the applicable use cases, robot design describes the opposite task of finding the optimal robot for a given task. Generally, the task is abstracted e. To identify the optimal robot, usually different designs have to be compared with respect to the desired properties which makes the design process generally a computationally expensive task. Finally, one or more designs turn out as most favourable.

In parallel to the analysis methods, again both discrete as well as continuous methods are available and show differences in the analysis quality and the calculation effort. For the continuous approach the CSP formulation can be used again which is amongst others advantageous in terms of implementation effort. The interchanging of the roles of the variables turns the workspace analysis just into a design task.

According to Merlet, , the design or synthesis task can be divided into two separated subtasks:. In particular, the number and type of d. For the special case of a wire robot, the structure synthesis covers different aspects: While the link topology itself is fixed, one has to choose the number of wires wisely. Additionally, the concurrence of at least two in the planar case or three in the spatial case platform connection points may be prudential:. The workspace is comparably large Fang, After completion of the structure synthesis a dimensional synthesis can be performed.

For a wire robot this is nothing but the identification of feasible base points. This section is addressed to dimensional synthesis mainly. Discrete methods are widely used for wire robot design. In Fattah and Agrawal, and Pusey et al. Then for every point on the resulting parameter grid the discretized workspace is computed and its volume is determined by counting the points on the grid fulfilling all workspace conditions.

The approaches share the same concept:. For every parameter set, specify a superset of the workspace and discretize it by an equidistant grid. Loop through all grid points of step 2. For every point, determine if a valid wire force distribution according to eqn. Obtain the maximum volume workspace, i. Instead of the volume of the workspace a different optimization criterion can be employed. To increase the practical usability and the robustness of the design, a dexterity criterion is proposed, which uses the condition number of the structure matrix A T. These approaches have two drawbacks.

Since the design variables are discretized, every combination of parameters is checked. Hence, this method is computationally intensive. Furthermore, no desired workspace can be guaranteed by the obtained design. Hay and Snyman use a special optimizer instead of a grid of the design variables Hay and Snyman, , Hay and Snyman, Again, in this approach a desired workspace is not guaranteed by the obtained optimal design. An imaginable choice is. The winch poses and platform fixation points are the calculation variables.

Thus, the calculation delivers robot designs solving the CSP. The platform coordinates are verification variables. Hence, the workspaces of all resulting robot designs will cover the set given in X v for the platform coordinates for sure. Optionally, the exerted external wrench w and desired platform orientations can be set as verification variables to extend the applicability of the emerged designs for certain process wrenches and tasks. The suggested choice of variables leads to a CSP, whose solutions are robot designs. Furthermore, each obtained robot can reach every point given in X v for the platform coordinates with every orientation and wrench given in X v. Generally, the design task is deemed to be more complicated than the analysis. Here, the methods and formulations are inherited and just adapted to the design problem.

### 1. Introduction

Nevertheless, robot design is a computationally intensive task. The use of parallel computations is strongly advised. Solving the CSP is advantageous due to the following reasons:. Optimization is always performed with respect to a cost function. The book presents the state of the art and recent advances in the area of kinematics of robots and mechanisms. It consists of about fifty outstanding contributions dedicated to various aspects of kinematic modelling and control, emphasising in particular the kinematic performances of robots and mechanisms, workspace and trajectory analysis, numerical and symbolic computational methods and algorithms, analysis, simulation and optimisation.

The book is of interest to researchers, graduate students, and engineers specialising in the kinematics of robots and mechanisms. It should also be of interest to those engaged in work relating to kinematic chains, mechatronics, mechanism design, biomechanics and intelligent systems. Help Centre. My Wishlist Sign In Join. Husty Editor. Be the first to write a review.

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