Orientation Workspace and Stiffness Optimization of

Orientation Workspace and Stiffness Optimization of Cable-Driven Parallel Manipulators with Base Mobility Aliakbar Alamdari ∗ SUNY at Buffalo Buffalo, NY, 14260 [email protected]

Michael Anson SUNY at Buffalo Buffalo, NY, 14260 [email protected]

Venkat Krovi Professor, ASME Fellow Mechanical and Aerospace Engineering SUNY at Buffalo Buffalo, NY, 14260 [email protected]

teed. This fully-symmetric configuration is shown to offer a variety of additional advantages: it eliminates the need to perform computationally expensive nonlinear optimization by providing a closed-form solution to the inverse kinematics problem, and it results in a convergence between kinematic singularities and wrench-closure singularities of the system. Finally, we discuss a particular limitation of this fully-symmetric configuration: the inability of the cables to obtain an even tension distribution in a loaded configuration. For this reason, it may be useful to relax the fully-symmetric cable requirement in order to yield reasonable tensions of equal magnitude.

Cable-driven parallel manipulators (CDPM) potentially offer many advantages over serial manipulators, including greater structural rigidity, greater accuracy, and higher payload-to-weight ratios. However, CDPMs possess limited moment resisting/exerting capabilities and relatively small orientation workspaces. Various methods have been contemplated for overcoming these limitations, each with its own advantages and disadvantages. The focus of this paper is on one such method: the addition of base mobility to the system. Such base mobility gives rise to kinematic redundancy, which needs to be resolved carefully in order to control the system. However, this redundancy can also be exploited in order to optimize some secondary criteria, e.g. maximizing the size and quality of the wrench-closure workspace with the addition of base-mobility. In this work, the quality of the wrench-closure workspace is examined using a TensionFactor index. Two planar mobile base configurations are investigated, and their results compared with a traditional fixed-base system. In the rectangular configuration, each base is constrained to move along its own linear rail, with each rail forming right angles with the two adjacent rails. In the circular configuration, the bases are constrained to move along one circular rail. While a rectangular configuration enhances the size and quality of the orientation workspace in a particular rotational direction, the circular configuration allows for the platform to obtain any position and orientation within the boundary of the base circle. Furthermore, if the bases are configured in such a way that the cables are fully-symmetric with respect to the platform, a maximum possible Tension-Factor of one is guaran-

∗ Corresponding

Anson

1

Introduction A cable-driven parallel manipulator, or cable robot, is a type of parallel manipulator in which the platform is driven by a set of cables in place of traditional rigid links. These cables are sometimes referred to as tendons, or wires. A winch, consisting of a tensioning motor and spool, is used to adjust the length or tension in each cable. Coordinated retraction and extension of the cables allows for the position and orientation of the platform to be controlled. Cable-driven systems offer many advantages over traditional parallel manipulators. Replacement of the heavy prismatic actuators with relatively light cables facilitates performance at higher accelerations and allows for potentially vast workspaces. Mounting of the winches to the fixed base further reduces the effective system inertia. Compared to the moving platform, the cables have relatively low mass and inertial properties and thus, their effects are oftentimes neglected; an assumption which greatly simplifies the modeling and analysis. The simple design of cable-driven robots also gives rise to a system that is recon-

author.

1

Journal of Mechanisms and Robotics

2

figurable, less expensive to construct and maintain, and easily transported, disassembled, and re-assembled. Hence, cable-driven parallel manipulators are wellsuited for a variety of applications, and have been proposed for use in: large-scale material handling and manufacturing processes [1], building construction [2], haptic devices [3] and locomotion interfaces [4], vehicle simulation [5], rescue operations [6], aerial cameras and actuated sensing applications [7], high-speed assembly and pick-and-place operations [8], large-scale radio telescopes [9], lower limb [10,11] and upper limb rehabilitation [12], among others. Despite their numerous advantages, there are several challenges associated with cable-driven robots. The fact that cables can pull but not push gives rise to a unilateral constraint wherein the cables must always be maintained in tension. Furthermore, this unilateral property gives rise to the need for actuation redundancy - i.e., more cables are needed than degrees of freedom (DoF) - in order to completely restrain and control the platform. In fact, if the number of DoF in the system is denoted by m, and the number of cables is denoted by n, it has been shown that a fully-restrained system requires at least n = m + 1 cables. This actuation redundancy adds complexity to the analysis and control scheme by creating indeterminacy in the cable tension distribution. In essence, there are an infinite number of sets of cable tensions that can achieve the desired platform motion and/or provide the required platform wrench. Past approaches at resolving redundancy have examined modulating and redistributing these internal actuation forces to satisfy secondary criteria. In addition to the challenges arising from unilateral driving constraints, cable-driven robots also suffer from limited moment resisting/exerting capabilities, and relatively small orientation workspaces. Researchers have proposed various methods to combat these limitations, including adding redundant cables, and utilizing a crossed-cable configuration [13]. A promising alternative to these options, which has been recently proposed by Zhou et al. [14], involves the incorporation of kinematic redundancy in the form of base mobility. The authors show that if the kinematic redundancy is properly resolved, base reconfiguration can be used to satisfy a variety of objectives. Their work with planar 3-DoF systems, however, was restricted to a single, rectangular base configuration. As such, the objective of the current work is to extend these concepts to a novel circular base configuration and provide a comprehensive discussion of these two general mobile cable robot configurations vs. the traditional fixed base systems. The remainder of this paper is structured as follows. In Section 2 and 3 the kinematic and static models of the two systems under consideration are developed, and the corresponding singularities identified in Section 4. With this foundation in place, a kinematic controller for trajectory tracking is developed and tested in Section 5. From the simulation results, we will see that a particular configuration in which the cables are fully-symmetric with respect to the platform elicits further investigation. This system is thus the focus Anson

Fig. 1.

KINEMATIC MODELING

Virtually Prototyping Cable-Driven Parallel Manipulators with

Base Mobility in Multi-Domain Modeling and Simulation Tools.

of Section 6. Finally, in Section 7, concluding remarks will be given and potential directions for future work will be discussed.

2

Kinematic Modeling In order to identify the kinematic relationships between the joint space and task space of the system, the loop-closure equations are first developed for each chain of the cabledriven manipulator. The notation used in this paper is as follows: the global (inertial) coordinate frame, denoted {O}, has orthonormal axes {xO , yO , zO }; a local coordinate frame, denoted {E}, is attached to a reference point at the center of the platform and has orthonormal axes {xE , yE , zE }; the vector describing the position of the platform in the global frame is given by O d, and the rotation matrix relating frame {E} and frame {O} is given by O RE ; O Li is the vector defining the ith cable, represented in the inertial frame, and is chosen to point away from the platform towards the base; O bi is the vector defining the location of the ith base in the global frame; and E ei is the vector defining the location of the ith cable attachment point on the platform, represented in the platform frame. With these definitions in mind, the loop closure equation corresponding to the ith kinematic chain can be written as:

O

d + O RE E ei +O Li −O bi = 0

(1)

Note that in the following analysis, it is assumed that the platform is rectangular in shape, and that the cables are attached at the four corners. Given that this design is commonly used in research and applications, this assignment is deemed reasonable and not overly limiting. For fixedbase systems, it is evident that O bi is constant. However, with the addition of base mobility, it becomes necessary to parametrize these vectors. The rectangular base configuration is similar to that examined in [14], while the circular base configuration offers a new and potentially more flexible design. In the rectangular configuration (Fig. 2(a)), the bases are constrained to travel along a straight line. It is thus useful to create a local reference frame in order to specify a fixed origin for each base and to simplify the description of the base locations. These local reference frames will be denoted as 2


2

{Oi }, and their x-axes chosen so as to align with the direction of feasible base motion.

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joint-space vector for the circular configuration can then be written as: T q = L1 β1 L2 β2 L3 β3 L4 β4

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Forward and Inverse Kinematics The forward kinematics problem seeks to determine the platform pose, given the cable lengths and base locations, and is necessary for the simulation of the system. While these equations are easily developed, it is in general difficult to find a closed-form analytical solution for the coupled nonlinear constraints seen in parallel manipulators. Analytical solutions have been derived for certain systems, however, these techniques involve complicated symbolic terms and require root finding of a high-order polynomial, which for order greater than four, must be performed numerically. More typically, this problem is often solved using numerical methods or through the use of additional sensors (i.e. sensor redundancy) with their own drawbacks [15]. In our case, a numerical method based on the iterative Newton-Raphson scheme is deployed . Solving Eq. (1) for Li and splitting into x and y components, and then squaring and adding these two scalar equations yields:

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Li2 = (bix −x−eix cos φ +eiy sin φ )2 +(biy −y−eix sin φ +eiy cos φ )2 (4) Collecting terms on the the left-hand side, and denoting the resulting term as fi for each of the 4 cables yields:

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T f = f1 f2 f3 f4

(5)

Taking the partial derivative of this vector with respect to the task space variables provides the Newton-Raphson Jacobian:

(b) Fig. 2. (a) Rectangular Base Configuration, (b) Circular Base Configuration

J=

h

∂f ∂f ∂f ∂x ∂y ∂φ

i

(6)

The platform pose update at the current iteration yields, xi+1 = x + ∆x = xi − J # f. This procedure is continued until ∆x is within some specified tolerance, ε, indicating that the solution has converged, i.e. k∆xk < ε. It is important to note that in the computation of the forward kinematic solutions, the cables are treated as rigid links and the tension condition of the cables is not considered. As a result, the solution may not necessarily be in the manipulator’s workspace and thus it is necessary to check the forward kinematic results with the tension conditions. The inverse kinematics problem, determining the joint positions given the position and orientation of the end effector, is necessary for realizing system control. For nonredundant parallel manipulators, this procedure is relatively straight-forward and often can be solved in closed-form. For the kinematically redundant systems, however, the inverse kinematics problem can no longer be solved directly as the

We define the angle between the ith local reference frame and the global frame as αi and the corresponding rotation matrix as O ROi . The vector defining the location of {Oi } in {O} is {O Oi }. Furthermore, the location of the ith base with respect to its local frame is Oi bi , where xi denotes the displacement along the x-direction. The joint-space vector for the rectangular configuration can then be written as: T q = L1 x1 L2 x2 L3 x3 L4 x4

(3)

2.1

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(a)

(2)

In the case of the circular base configuration (Fig. 2(b)), wherein the bases are constrained to travel along a circular path of radius R, we can parametrize the x and y position of each base using the angle βi , which represents the angle of the base with respect to the x-axis of the inertial frame. The Anson

KINEMATIC MODELING

3


2

locations, the form of A remains consistent between the two configurations. For non-redundant parallel manipulators, the forward ˙ velocity problem is typically represented as x˙ = −A−1 Bq. The matrix product A−1 B is comparable to the traditional Jacobian matrix defined for serial manipulators. Hence, this quantity is often denoted as, J = −A−1 B. For the cable-driven robots currently under investigation, the above matrix inversions cannot be performed due to the rectangular shape of A and B. Indeed, the kinematic redundancy present in these systems implies that an infinite number of solutions to Eq. (10) exist. In such cases, the Moore-Penrose pseudo-inverse can be used to provide a general way of finding the solution as:

base locations are also unknown. While the redundancy resolution scheme complicates the inverse kinematics procedure, it also affords us the opportunity to optimize some desired performance criteria, as we discuss next. 2.2

First-Order Differential Kinematics The first-order differential kinematics seeks to relate the twist of the moving platform with the actuated joint velocities and is necessary for singularity analysis and kinematic control. The twist, t, of the moving platform is defined as T follows, t = v φ˙ , where v is the Cartesian velocity vector of some reference/operating point on the end-effector and φ˙ is the angular velocity vector of the end-effector. Taking the inner product of both sides of the loopclosure equation, and differentiating the result with respect to time, and using the linear algebra identity aT b + bT a = 2aT b = 2bT a, Eq. (1) we obtain: L˙ i = Lˆ Ti (O b˙ i − O d˙ − ΩO RE E ei )

q˙ = J# x˙ + I − J# J z = q˙ P + q˙ H

x˙ − (O RE E ei × Li )φ˙ y˙

(11)

where the following definitions can be used, assuming A and B are of full rank, J3×8 = −A# B = (AT A)−1 AT B, and J#8×3 = JT (JJT )−1 . The particular solution q˙ P corresponds to the minimum norm joint-velocity solution to the inverse velocity problem.The orthogonal homogenous term q˙ H , is the manifestation of the kinematic redundancy in the manipulator which allows for the generation of internal motions within the system (i.e. joint motions) without changing the end-effector velocities. It may be computed by projecting the arbitrary vector z (whose choice will be discussed in Section 5) onto the null space of A.

(7)

where Lˆ i denotes the unit vector along the ith cable, and Ω is the skew symmetric matrix of the angular velocity vector, O ω. Re-arranging Eq. (7) such that the joint rates are on the left-hand side and the elements of the twist vector are on the right-hand side, for the rectangular base configuration, we have

˙ i − LTi O ROi Oi b˙ i = −LTi Li L

KINEMATIC MODELING

(8) 2.3

For the circular base T −Rβ˙ sin β Rβ˙i cos βi , thus

configuration,

Ob ˙i

Kinematic Singularity Analysis We begin with a discussion on the singularities present in traditional fixed-base systems to draw a contrast with those arising in mobile-base systems. Diao et al. [17] demonstrated that an architecture singularity occurs if the base and platform polygons are similar (e.g. if they are proportional rectangles) and have the same orientation (Fig. 3(a)) – this type of singularity is easily avoided at the design stage. Another singular configuration occurs if one of the cable lengths vanishes, in which case the direction defining the cable becomes undefined (Fig. 3(b)). This is a Type I singularity corresponding to a loss in rank of the B matrix. In practical fixed-based systems, while it may not be possible to obtain a true cable length of zero, B may still become ill-conditioned as the platform approaches the corners of the workspace. For kinematically redundant cable-driven systems, this analysis becomes slightly more involved. Similar to the fixed-base system, it is readily seen from Eqs. 8 and 9 that if one of the cable lengths vanishes, the two corresponding elements in the B matrix become zero, and as a result, the rank of B drops to three and is thus singular. This can be confirmed by finding the determinant of BBT and setting it equal to zero. For example, for the rectangular base configuration the determinant of this quantity is:

=

˙ T −Rβi sin βi T x˙ ˙ Li Li − Li = −Li − (O RE E ei × Li )φ˙ (9) y˙ Rβ˙i cos βi It is evident that Eqs. 8 and 9 are represented in the form needed for singularity analysis. Derivation and solving of the forward and inverse kinematic problems are presented in [16] in detail. In the case of the planar fully-restrained cable-driven manipulator with base mobility, the dimensions of each component in the first-order differential kinematic equation are:

B4×8 q˙ 8×1 = −A4×3 x˙ 3×1

(10)

The addition of base mobility into the system has the effect of increasing the number of columns of B, while the actuation redundancy has the effect of increasing the number of rows of both A and B. Furthermore, while the elements of B depend upon the parametrization used in defining the base Anson

2 2 2 2 |BBT | = (L12 + L1y )(L22 + L2x )(L32 + L3y )(L42 + L4x )

4

(12)


2

It is evident that the determinant of BBT can become zero if and only if one of the cable lengths becomes zero. A similar proof can be shown for the circular base configuration. Unlike the fixed-base system, where this Type I singularity can only occur at the corners of the base polygon, for the mobile base systems under study, this singularity can occur anywhere on the boundary of the base rectangle or circle. Examples of this are shown in Fig. 3(c),3(d).

(a)

(b)

KINEMATIC MODELING

of A drops to two. When the system is in this configuration, depicted in Figs. 4(c), 4(f), it is unable to resist any moments applied to the platform. Tangentially, we note that a simple design modification (in the form of a split cable attachment at the corners of the platform) could help mitigate the effects of such a singularity (but is not further explored here). Further, for the rectangular base configuration, the above mentioned singularities only become possible after a certain amount of rotation of the platform. In contrast, within the circular base configuration, the same types of configuration singularities can now occur at any position and orientation of the platform. For the circular base configuration, a Type III singularity is also possible, occurring when two cable lengths simultaneously vanish (Fig. 3(e)). Similar to the Type I singularity, this can occur anywhere along the boundary of the base circle. Thus, it is seen that while the addition of kinematic re࡭࡭࡭ dundancy adds dexterity to the system, it also gives rise to additional potential singularities that are not present in the fixed-base configuration. However, with the development of a suitable control scheme, base mobility provides an infinite number of ways of avoiding these kinematic singularities.

(c)

࡭࡭ ࡭

(d)

(e)

Fig. 3. (a) Kinematic singularities in the fixed-base system, architecture singularity (b) Kinematic singularities in the fixed-base system, Type I singularity (c) Type I singularities in systems with rectangular base mobility (d) Type I singularities in systems with circular base mobility, (e) Type III kinematic singularity in the circular base config-

(a)

(b)

(c)

(d)

(e)

(f)

uration

recall the A4×3 matrix, For Type IIO singularities, L cos θi L sin θi RE E ei × L . Investigating the above matrix, two singularities quickly become evident. The first occurs when all cables are parallel to each other, such that sin θi m = cos θi . In such a scenario, the first and second columns of A become linearly dependent; the second column is nothing but m times the first column (where m denotes the slope). Two typical configurations at which this occurs are illustrated in Fig. 4. When the system is in the configuration of Fig. 4(a), 4(d), it is unable to resist forces in the x-direction of the platform frame. Similarly, when the system is in the configuration of Fig. 4(b), 4(e), it is unable to resist forces in the y-direction of the platform frame. Note that in the figure, black arrows are used to denote the lines of action of the cables, and red arrows are used to denote the directions in which force or torque cannot be resisted/exerted. A second kinematic singularity is possible when the lines of action of all four cables pass through the geometric center of the platform. In such a situation, the vector describing the location of the cable attachment point in the platform frame, E ei , becomes collinear with the cable vector, Li , and thus their cross product is zero. As a result, the entries in the third column of A become zero and the rank Anson

Fig. 4. (a), (b) and (c) Common Type II Kinematic Singularities in the Rectangular Base Configuration Ͷͷ Ͷͷ Ͷͷ (d), (e) and (f) Common Type II Kinematic Singularities in the Circular Base Configuration

The full-fledged CDPM features eight actuators (four winches for the cables and four actuators for the mobile bases) and is redundant since the end-effector/platform has ͶͷͶͷ Ͷͷ system (in Sections 1 and only 3. In our formulation of the 2) we consider the full-fledged system for the configuration analyses. From Section 3 onwards, we treat the mobile baseactuation as a ”slow subsystem” (used only for reconfiguration of the overall system when active tasks are not being performed by the end-effector). The reconfiguration still has the potential to affect the quality of the workspace, which is the focus of our subsequent investigations. In all following sections, the 4 winches are assumed to form the actively controlled ”fast subsystem”, instanta5


3 3.1

Actuation Redundancy Resolution A variety of tension distribution algorithms have been proposed, with different characteristics and varying computational cost, for resolving the actuation redundancy present in cable-driven robots [18]. Perhaps the most commonly implemented method in the literature relies upon the minimization (or maximization) of some norm of the cable tensions [19]. Borgstrom et al. [20], discuss the potential limitations of various norms (such as L1 , L2 L∞ ) in the context of tension distribution of cable robots. The optimal solution can, however, be approximated using a p-norm (1 < p < ∞), and the resulting minimum-norm solution is proven to be unique and continuous except at singular configurations. The optimization problem can be formulated as:

neously equivalent to a conventional fixed-base CDPM i.e. mobile-base configuration variables are intrinsic to the formulation but the mobile-base actuation-rates do not enter the static analyses, redundancy resolution, and various trajectory tracking discussions.

3

Static Analysis For cable-driven parallel manipulators, static analysis is essential for ensuring positive cable tensions and hence a critical component for proper workspace analysis and control. Representing ti as the cable tension vector corresponding to the ith cable, and denoting the external force and moment acting on the platform by Fext and Mext , respectively, the force and moment equilibrium equations are derived as: 4

STATIC ANALYSIS

minktk p

(15)

St = −W

(16)

4

∑ ti Lˆ i + Fext = 0, ∑ (O RE E ei × ti Lˆ i ) + Mext = 0

i=1

subject to

(13)

i=1

tmin ≤ ti ≤ tmax , i = {1, 2, 3, 4}

ˆ i represents the unit The reader is reminded here that L vector along the ith cable (pointing away from the platform towards the base), and can be parameterized in terms of θi (defined as the angle between the x-axis of the global frame and the ith cable). Or, in matrix form: St = −W

Thus, the optimized cable tensions must satisfy the static equilibrium equations and remain within some specified upper and lower bounds. In general, the lower limit corresponds to the amount of tension required to keep the cables taut, while the upper bound depends on the torque capacity of the motors and/or the failure point of the cables. Under the L2 -norm the pseudo-inverse method offers an alternative equivalent solution to Eq. (14):

(14)

where

t = −S# W + (I − S# S)z

 cos θ1 cos θ2 cos θ3 cos θ4 sin θ2 sin θ3 sin θ4  S =  sin θ1 OR e × L OR e × L OR e × L OR e × L ˆ ˆ ˆ ˆ4 E 1 E 2 i E 3 E 4 1 3 

For systems with one degree of actuation redundancy, Eq. (17) can be equivalently expressed as, t = −S# W + λ N, where N represents the n × 1 dimensional null space vector of S, and λ is an arbitrary scalar that can be chosen so as to adjust the contribution of the homogeneous component to the overall solution. In an unloaded configuration, and assuming the system is within the wrench-closure workspace, the cable tensions can be computed using the null space vector and the lower tension bound, as follows:

T T t = t1 t2 t3 t4 , W = Fx Fy Mz Equation 14 allows for the computation of the set of cable tensions, t, which is needed to resist an external wrench (force and moment combination), W, exerted on the platform. These two quantities are related through the linear mapping provided by the matrix S, which is commonly referred to as the structure matrix, statics Jacobian matrix, or pulling map. The presence of actuation redundancy in the manipulator gives rise to an underconstrained system of equations. The resulting indeterminacy in the cable tensions can be exploited in order to manipulate the tension distribution so as to satisfy some secondary criteria. Careful coordination is required in order to prevent the build-up of substantial internal forces which can potentially damage the system, as discussed next. Anson

(17)

t=

tmin abs(N) min(abs(N))

(18)

The smallest element of the null space vector is thus replaced by the lower tension bound and all other cable tensions will be greater than or equal to this value. Note that the absolute value must be introduced, as these null space elements can be either positive or negative. 3.2

Kineto-Statics Duality It is interesting to note that in robotic systems there exists a link between the statics and differential kinematics of 6


4

the manipulator, a property referred to as kineto-static duality. Indeed, for a general robotic system we can write, ˙ and τ = JT W x˙ = Jq, As a result, for traditional fixed-base cable-driven robots, the following relationship exists between the structure matrix and the inverse Jacobian S = −J−T . However, for the systems under current investigation which incorporate base mobility, it is evident by inspection of Eq. (14) and the A matrix that although the structure matrix is closely related to the inverse velocity Jacobian, this relationship is not given directly by the above equation. The reason for this discrepancy is due to the fact that, while the base positions and velocities are considered in the kinematic analysis, they are not factored into the static analysis. The following relationship, however, is present, S = (diag( L11 , L12 , L13 , L14 )A)T

WRENCH-CLOSURE SINGULARITY ANALYSIS

resisted/exerted by the platform while maintaining positive cable tensions. This workspace depends only on the geometric parameters of the system - i.e. the locations of the cable attachment points on the base and platform and the pose of the platform. Several algorithms have been proposed for efficiently computing the boundary of the wrench-closure workspace. This analysis has been studied in the context of fully-restrained [21], under-constrained [22], and overrestrained [13] systems. Satisfying the necessary conditions to guarantee wrench-closure for arbitrary wrenches greatly diminishes the available workspace. If the set of wrenches that the platform will have to resist/exert are known, a more useful workspace is the wrench-feasible workspace [23]. This is the set of end-effector poses for which a specific set of wrenches can be resisted/exerted by the moving platform while maintaining control of the end-effector with (positive) cable tensions that are greater than some prescribed minimum and less than some prescribed maximum. Hence wrench-closure workspace analysis depends not only on geometric parameters, but also on the allowable tension ranges, gravitational effects, and the required wrench set. Specifically, we shall investigate the poses at which the cable robot cannot resist, or exert, an arbitrary load while maintaining equilibrium with positive cable tensions. Such configurations, although perhaps not kinematically singular, are said to be singular in terms of wrench-closure. Attention is now turned to the identification of these wrench-closure singularities.

3.3

Kinematic Reconfiguration As discussed before, the kinematic redundancy present in systems with base mobility allows for the joint positions to be optimized so as to satisfy some desired performance criteria. Here, we will investigate the use of base reconfiguration in order to maximize the tension factor. Before proceeding, it is important to keep in mind that the tension factor is intended to be used as a measure of the quality of the wrench-closure workspace. In such an analysis, only the homogeneous component of the static equilibrium equation is considered. The objective function thus seeks to maximize the ratio of the smallest null space component to the largest null space component.

4.1 max TF =

min(|N|) −min(|N|) ≡ min max(|N|) max(|N|)

Null Space Vector and Wrench Closure Singularity For cable-driven robots with one degree of actuation redundancy, the condition for wrench-closure is easily determined through evaluation of the 1-D null space vector, N, that can be constructed [24] as:

(19)

Again, the optimized cable tensions must satisfy the static equilibrium equations and remain within some specified upper and lower bounds. Additional constraints must be incorporated into these optimization routines, however, in order to take into account joint limits and base interference. In the rectangular configuration, the displacement of any given base cannot exceed the length of the side on which it travels. Denoting the length of the rectangle as bl and the width as bw , these constraints can be represented as: x1 − bw ≤ 0, x2 − bl ≤ 0, x3 − bw ≤ 0, x4 − bl ≤ 0

  (−1)r+2 |S1 | (−1)r+3 |S2 |  N = null(S) =  (−1)r+4 |S3 | (−1)r+5 |S4 |

where Si is the submatrix of S formed after removing the ith column. If the elements in this vector are either all positive or all negative, it is readily seen from Eq. (17) that, regardless of the particular solution, enough homogeneous component can be added (or subtracted) in order to ensure that all cable tensions are greater than zero. Thus, a wrench-closure singularity occurs if the signs of these elements are not the same, or, if one or more of the elements equal zero. In addition to determining the size of the wrench-closure workspace, it is useful to be able to characterize the quality of this workspace at a particular pose. The quality of the workspace can be quantified through the use of some performance metric. In min(|N|) proposed this study, the tension factor (TF) TF = max(|N|) in [21] is used. Before proceeding, we note that due to the level of kinematic redundancy present in the mobile base systems, it be-

(20)

In the circular configuration, restrictions must be placed on the base locations in order to prevent collisions. These additional constraints are represented as: β1 − β2 ≤ 0, β2 − β3 ≤ 0, β3 − β4 ≤ 0

(21)

4

Wrench-Closure Singularity Analysis The wrench-closure workspace [13] is defined as the set of end-effector poses for which any arbitrary wrench can be Anson

(22)

7


4

WRENCH-CLOSURE SINGULARITY ANALYSIS

comes difficult to define a meaningful and accurate wrenchclosure workspace. Indeed, this redundancy results in a nonunique tension factor map that is dependent upon the current base locations. Thus, in order to be able to compare the three systems at hand, the following steps are taken: first, the wrench-closure workspace and corresponding tension factor map is calculated for an example fixed-base system. Next, the discussion is separated into two components: one concerning the translation workspace subset (i.e. the constant orientation workspace subset) and another concerning the orientation workspace subset. In the translation workspace section, the wrench-closure singularities of the fixed-base system are identified for the case of φ = 0. This discussion is then qualitatively extended to the rectangular and circular base configurations. In the orientation workspace section, the three systems are compared using a quantitative example in which x = 0, y = 0.

and quality of this workspace begins to deteriorate as φ increases. The tension factor, represented through the color map, also decreases as the platform moves away from the center of the base rectangle. Now, consider the translation workspace of this system when φ = 0. Given that the cables can pull but not push, it is logical to expect that the fixed-base system will approach a wrench-closure singularity when the platform nears the boundary of the base rectangle and two cables become collinear. Under these circumstances, two elements of the null space vector become zero. These results agree with those found in [25], where the wrench-closure workspace of a fixed-base system subject to pure translation was investigated. For the specific case wherein the third and fourth cables become collinear at the right edge of the base rectangle, the wrench-closure singularity implies that infinite force would be required in these two cables in order to move the platform to the right. Analogously, in such a configuration, the system has lost its ability to resist external forces directed 4.2 Planar Fixed Base System to the left. Note that for the fixed-base system, this wrenchFor the planar fixed-base system, the presence of only closure singularity is only possible at the boundary of the ݁ ݁ ݁ ݁ three task space variables allows for the complete depiction ሺߠ െ ߠ ൅ ߠ ሻ ൅ ሺߠ െ ߠ ൅ ߠ ሻ ൅ ሺߠ െ ߠ െ ߠ ሻ‫ې‬ ‫ ʹ ۍ‬ሺߠ ൅ ߠ െ ߠ ሻ െ ʹbase rectangle (see Fig. 6(a)). ‫ۑ‬ ʹ ʹ ‫݁ێ‬ ݁ ݁ ݁ of the wrench-closure workspace in one three-dimensional ‫ ێ‬ሺߠ െ ߠ ൅ ߠ ሻ െ ሺߠ ൅ ߠ െ ߠ ሻ െ ሺߠ ൅ ߠ െ ߠ ሻ െ ሺߠ െ ߠ െ ߠ ሻ ‫ۑ‬ ʹ ʹ ʹ ܰ ൌ ‫ʹ݁ ێ‬ The results above can be‫ ۑ‬ሺ͵Ǥ͹͹ሻ extended to systems with ݁ ݁ ݁ ሺߠ െx, ߠ െyߠ ሻ െ ሺߠ െ ߠ ൅ ߠ ሻ ൅ ሺߠ ൅ ߠ െ ߠ ሻ ൅ ሺߠ െ ߠ െ ߠ ሻ ‫ۑ‬ plot. To do this, the task space is first discretized. At‫ ێێ‬each ‫ۑ‬ ʹ ʹ ʹ ʹ ݁ ݁ the fixed-base ݁base mobility. ‫݁ێ‬ ‫ۑ‬ Unlike system, however, this ሺߠ െ ߠ ൅ ߠ ሻ െ ሺߠ െ ߠ െ ߠ ሻ ൅ ሺߠ െ ߠ ൅ ߠ ሻ െ ሺߠ ൅ ߠ െ ߠ ሻ‫ے‬ and φ coordinate, the wrench-closure condition is ‫ۏ‬checked, ʹ ʹ ʹ ʹ wrench-closure singularity can now potentially occur within and, if satisfied, the corresponding tension factor is comthe workspace, as is illustrated in Fig. 6. puted. The wrench-closure workspace of the example fixedbase system is shown below in Fig. 5. ௟

ଶ

௟

ଵ

௟

ଵ

௟

ଵ

ଷ

ଷ ଶ

ଶ

௟

ସ

ଶ

௟

ସ

ଵ

௟

ସ

௟

ଷ

4.3

TF 1

3

ଵ

ଵ

ଷ

ସ

ଶ

ସ

ଷ

ଶ

ସ

ଷ

௪

௪

௪

௪

ଶ

ଷ

ଷ

ସ

ଵ

ଶ

ଷ

ଵ

ଶ

௪

ସ

ଵ

௪

௪

ସ

௪

ଶ

ଷ

ସ

ଵ

ଷ

ସ

ଵ

ଶ

ଷ

ଵ

ଶ

ସ

Planar Mobile Base Systems

‫ܤ‬ଶ

‫ܤ‬ଷ

0.9 2

0.8 0.7

φ(deg)

1

0.6 0

0.5 0.4

−1

‫ܤ‬ଵ

0.3 0.2 2 1

0 X (m)

0.1

0 −1

−2

(b)

(c)

Wrench-Closure Singularities in the Circular Base Configuration

−2 Y(m)

ͷ͹

Fig. 5. Example Wrench-Closure Workspace and Tension Factor Map for Fixed-Base System.

Although it is important to recognize the conditions under which wrench-closure singularities can occur, it is perhaps equally important to understand how the quality of the ͷͺ ͷͺ workspace (i.e. the tension factor) changes upon approaching these singularities. Implementing the optimization routines, we can evaluate the effect of base reconfiguration on the tension factor as the platform approaches the boundary of the workspace. To illustrate this, let us consider the situation in which the platform undergoes pure translation in the positive x-direction, starting from the center of the workspace. The results for the three systems under consideration are shown in Fig. 7.

The x and y-axes of this plot correspond to the x and y location of the platform reference point in the task space. The z-axis is used to represent the orientation, φ , of the platform with respect to the global frame. Each horizontal cross-section of this plot can thus be seen as the translation workspace (constant orientation workspace) at the corresponding value of φ . It is evident that, while the the translation workspace is large for very small values of φ , the size Anson

‫ܤ‬ସ

Fig. 6. (a) Example Internal Wrench-Closure Singularities in the Fixed Configuration (b) Example Internal Wrench-Closure Singularities in the Rectangular Base Configuration (c) Example Internal

−2 −3 2

(a)

8


5

KINEMATIC CONTROL FOR TRAJECTORY TRACKING

In the fixed-base system, the tension factor steadily decreases as the platform translates towards the right - reaching zero, and thus a wrench-closure singularity, at the boundary of the base rectangle. In the rectangular configuration, the addition of base mobility allows for improvement in the tension factor near the boundary of the workspace. It should be noted that, for the given scenario, this improvement is only possible when the length of the platform is greater than its width. In the example above, the platform’s length and width are 1 m and 0.5 m, respectively. It is only once the platform is within length − width = 0.5m of the boundary (in the x-direction) that the top right base is able to partially compensate for the platform translation. For translations in the y-direction, the bases in this particular system are unable to be reconfigured so as to improve the tension factor. These limitations do not exist in the circular configuration, however. Regardless of the position of the platform within the base circle, a tension factor of one is able to be achieved.

sis are shown in Figs. 8(a),8(b),8(c) and Figs. 9(a), 9(b), 9(c). It is seen that for the fixed-base system, the maximum reachable orientation is ±19◦ . A symmetric shape is observed in the plot of the tension factor, which has a maximum value of one when φ = 0, and decreases to a value of zero for both clockwise (CW) and counterclockwise (CCW) rotations of the platform (signaling the limit of the wrenchclosure workspace). At these limits, two of the null space components are positive and two are negative. The addition of base mobility into the rectangular configuration has the effect of extending the orientation workspace in the CW direction by 90◦ . Throughout this additional space, a tension factor of one is achievable given optimal base positioning. In the CCW direction, however, the limit of the workspace is the same as in the fixed-base configuration. This is attributed to the choice of base origin and platform cable attachment point locations. Indeed, in the chosen configuration, the bases starting from their specified origins are capable of moving in a CW direction.

Fixed−Base System Tension Factor Approaching X limit (Y=0, φ=0)

1

TF

0.8 0.6

ᇣᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ

0.4

஼஼ௐ ஼஼ௐ

0.2 0 0

0.5

1

(a)

ᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ ᇣ ஼ௐ ஼ௐ

1.5

X(m)

(a)

(b) Rectangular Base Configuration Tension Factor Approaching X limit (Y=0, φ=0)

ᇣᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ ஼஼ௐ ஼஼ௐ

1

TF

0.8

ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ

(b)

஼ௐ ஼ௐ

0.6 0.4 0.2 0 0

0.5

1

ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ

1.5

X(m)

(c)

஼஼ௐ ஼஼ௐ

(d) Circular Base Configuration Tension Factor Approaching X limit (Y=0, φ=0)

ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ

(c)

஼ௐ ஼ௐ

Fig. 8. Orientation Workspace: (a) Fixed-base, (b) Rectangular configuration, (c) Circular configuration

1

TF

0.8 0.6

The circular configuration removes this limitation by allowing for infinite rotation in both the CW and CCW directions. Optimal base positioning yields a constant maximum tension factor of one. In order to further investigate the capabilities of these systems in performing some desired task, such as tracking trajectory, a kinematic control scheme is developed in the next section. ͸ʹ

0.4 0.2 0 0

0.5

1

1.5

X(m)

(e)

(f) ͸Ͳ

Fig. 7. Approaching the Boundary - (a),(b) Fixed-base, (c),(d) Rect͸Ͳ angular configuration, (e),(f) Circular ͸Ͳ configuration

5

Kinematic Control for Trajectory Tracking With the kinetostatic model of the system in place, a controller can now be developed which will enable tracking of some desired trajectory. This trajectory is a time parameterized reference signal, which is defined in terms of the

We now turn to the orientation subset of the wrenchclosure workspace. In order to illustrate the effects of the addition of base mobility on the rotation capabilities of the system, a simple example for which x = 0 and y = 0 is investigated. The results of the orientation workspace analyAnson

9


5

KINEMATIC CONTROL FOR TRAJECTORY TRACKING

Rectangular Configuration with Mobile Base Orientation Workspace at X=0, Y=0

Rectangular Configuration with Fixed Bases Orientation Workspace at x=0, y=0

the tension factor The objective function to be minimized, f (q), is then:

1 1

0.8 Tension Factor

Tension Factor

0.8

0.6

0.4

0.2

0.6

8 2 1 f (q) = ∑ ki qi,d − qi i=1 2

0.4

0.2

(24)

0 −15

−10

−5

0

φ (deg)

5

10

0

15

−100

−80

−60

(a)

−40 φ(deg)

−20

0

20

The vector z is chosen as the negative gradient of this potential function i.e. z = − ∂ ∂f (q) q = Kq e. Note that the negative gradient travels in the direction of maximum function decrease, and thus seeks to minimize the joint error. The elements of the constant gain matrix, Kq , should be chosen so as to achieve the desired convergence rate while not violating joint velocity limits. It is important to note that the additional constraint specified through z has secondary priority with respect to the primary kinematic constraint [26]. A block diagram of the overall control scheme is illustrated in Fig. 10.

(b) Circular Configuration with Mobile Bases Orientation Workspace at x=0, y=0 1

Tension Factor

0.8

0.6

0.4

0.2

0 −300

−200

−100

0

φ (deg)

100

200

300

(c) Fig. 9. Wrench-Closure Orientation Workspace at x=0, y=0 - (a) Fixed-base, (b) Rectangular configuration, (c) Circular configuration

task space variables. Task space position control can readily be implemented using the inverse differential kinematics expression found in Eq. (11). Defining the error between the desired and actual trajectory as, e = xd − x, and choosing the following control law: q˙ = J# (˙xd + Kx e) + I − J# J z

(23)

our kinematic equation of motion becomes, e˙ + Kx e = 0. Thus, the new decoupled system which represents the error dynamics is linear. Typically, the proportional gain matrix is selcted to be diagonal as Kx = diag{kx , ky , kφ }. By requiring this gain matrix to be positive definite, the system will be asymptotically stable, and convergence of the tracking error to zero is guaranteed. The rate of convergence depends on the eigenvalues of the matrix Kx - the larger the eigenvalues, the faster the convergence.

Fig. 10.

A MATLAB-based graphical user interface (GUI) Fig. 11 was also developed to permit flexible specification of system reconfiguration and trajectory visualization. The user is allowed to specify various system parameters including the base and platform dimensions, current base positions, and the pose of the platform. The visual display is automatically updated when any changes are made. Several trajectory tracking options are available, each having its own adjustable parameters. While a simulation is running, a wait bar informs the user as to the overall progress. Once the simulation has completed, a variety of useful results are viewable, including plots of the tensions and tension factor, mobile base displacements and cable lengths, and task space position and orientation error. An animation of the simulation is also provided, which can be used to gain more insight into the system behavior as well as troubleshoot potential issues.

5.1

Redundancy Resolution The final task remains to determine a suitable value for the arbitrary vector, z, in Eq. (23). Recall that this vector adjusts the homogenous solution of the joint rates, and can thus be manipulated in order to reconfigure the system via active coordination of the mobile bases, without violating the primary objective of tracking the desired trajectory. A suitable secondary objective function is then to minimize the sum of the squared errors between the desired (i.e. optimal) joint positions and the actual joint positions. The optimal joint positions, denoted here as qd , are determined through the optimization scheme presented in the kinematic redundancy resolution formulation. The optimal joint positions, denoted here as qd, are determined are determined for a given (desired) platform pose through the kinematic redundancy resolution formulation (Section 3.3), which seeks to maximize Anson

Position Control Scheme for Trajectory Tracking.

5.2

Trajectory Tracking Results To demonstrate the effectiveness of the controller and the dexterity of the mobile base systems, a test trajectory is 10


6

FULLY-SYMMETRIC CABLE CONFIGURATION Table 1.

Trajectory tracking simulation parameters.

Simulation parameters Parametric equation for translation

x = xc + r cos(ωt) y = yc + r sin(ωt) (xc , yc ) = (0, 0) r = 0.75m, ω = 36◦ /sec

Parametric equation for rotation

φ = B1 + B2 sin(λt) B1 = 0; B2 = 5, λ = 108◦ /sec

Fig. 11.

developed and simulated using the above GUI, and the results for each system are recorded. The base and platform dimensions, desired trajectory, and a summary of the simulation parameters are as defined in Table 1. The desired translation of the platform reference point is chosen to be a circle, and the desired orientation of the platform with respect to the inertial frame is chosen to be a sinusoidal function. In all three cases, the systems were able to quickly converge to the desired trajectory, as illustrated in the error plots. As expected, the fixed-base system performed most poorly in terms of the tension factor and required higher tensions in general. While the rectangular base system reduced the tension requirements and improved the tension factor in certain regions of the motion, the wrench-closure workspace still suffered in quality when the platform was asked to rotate in the CCW direction. Indeed, the troughs that are present in the tension factor plot correspond to when this CCW motion was occurring. In these regions, the required tensions and resulting tension factor results merge with those of the fixed-base system. For the circular base configuration, the cable tensions are seen to quickly converge to their minimum allowable value of 0.1 N, and, as a result, the tension factor quickly approaches its optimal value of one (see Fig. 12(b)). The small oscillations present in the tension factor plot throughout the remainder of the motion can be further reduced by increasing the gain of the joint-space error. Consideration should be given to the joint velocity limits before doing so, however, as a high gain may result in unreasonable motion requirements. While effective, there are drawbacks to the optimization routine and control scheme presented above. Due to the nonlinear nature of the problem, the solution to the inverse kinematics is susceptible to local minima. Furthermore, there are no conditions currently in place to allow the system to actively avoid kinematically singular configurations. Controlling this would require modification of the objective function or necessitate the creation of additional constraints, and would thus further complicate the optimization scheme. An interesting observation from these simulations is that the quality of the wrench-closure workspace seems to be related to the ability of the system to maintain a fully symmetric cable configuration with respect to the platform, as investigated in the next section. Anson

Initial task space error

(ex , ey , eφ ) = [0.25m, 0, 1◦ ]

Task space error gain

(kx , ky , kφ ) = [1, 1, 10]

Joint space error gain

ki = 25, i = {1, 2, ..., 8}

Solver for inverse kinematics

fmincon

graphical user interface (GUI).

(a)

(b) Fig. 12. Trajectory Tracking Results: (a) Rectangular configuration, (b) Circular configuration

6

Fully-Symmetric Cable Configuration In order to investigate the existence of a relationship between full-cable symmetry and the tension factor, we approach the statics modeling in a slightly different manner. Recall that for the static equilibrium equations, the choice of the set of axes about which the forces are resolved does not matter, as long as these axes are orthogonal. Hence we now consider formulating the equilibrium equations in the mobile platform frame. Furthermore, in place of using the angle between the ith cable and the inertial frame to define the cable direction (denoted previously by θi ), the relative angle between the ith cable and the x-axis of platform frame, θi,rel , will be used. 11


7

The requirement of full-cable symmetry implies that all cable directions can be fully defined after specifying only one relative angle. For example, we can easily define θ2,rel , θ3,rel and θ4,rel in terms of θ1,rel , i.e. θ2,rel = π − θ1,rel , θ3,rel = π + θ1,rel and θ24,rel = 2π − θ1,rel . With these adjustments, the structure matrix of the system can be re-derived as:

  cos θ1,rel − cos θ1,rel − cos θ1,rel cos θ1,rel S =  sin θ1,rel sin θ1,rel − sin θ1,rel − sin θ1,rel  A B A B

STIFFNESS ANALYSIS AND OPTIMIZATION

Although it is possible for the rectangular base configuration to achieve this requirement of full-cable symmetry in a certain region of its workspace, the remaining discussion will focus on the circular base configuration for which no such limitation exists. By specifying the desired relative angle of each cable with respect the platform, the inverse kinematics problem is greatly simplified, as optimization is no longer required. Instead, an analytical solution is available using the loop closure equations. Noting that θi = θi,rel + φ , the x and y components of the ith loop closure equation can be written as:

(25)

R cos βi = Li cos(θi,rel + φ ) + x + eix cos φ − eiy sin φ (27) where A=−

R sin βi = Li sin(θi,rel + φ ) + y + eix sin φ + eiy cos φ el ew el ew cos θ1,rel + sin θ1,rel , B = cos θ1,rel − sin θ1,rel 2 2 2 2

Note that by specifying the relative angles of the cables with respect the platform, the structure matrix derived in Eq. (25) is no longer a function of x, y, and φ . This is due to the fact that, in the platform frame, the relative angles and the locations of the cable attachment points are both constant. In a manner similar to that used in the previous section, the null space of this new structure matrix is derived as:

aLi2 + bLi + c = 0

(28)

where coefficients a, b and c can be determined in terms of θi,rel . Thus, Li is solvable in terms of the task space variables and the specified value of θi,rel . Note that there is only one feasible (positive) solution to this equation. With Li now known, βi can be solved from Eq. (27). Recasting our problem in this fashion has many advantages. This closed-form solution greatly improves the computational efficiency of the inverse kinematics problem by eliminating the need for nonlinear optimization. Furthermore, the solution is no longer susceptible to local minima. Finally, by specifying the relative angles, and utilizing the controller to maintain these relative angles throughout the trajectory, the system inherently avoids kinematic singularities within the workspace. To verify the newly-developed inverse kinematics and provide a comparison with the previous optimization method, the fully-symmetric cable configuration is tested under the same conditions and with the trajectory tracking simulation parameters listed in Table 1. In this simulation, the desired value of θi,rel is set to 45◦ . As a result, the desired values of θ2,rel , θ3,rel and θ4,rel are 135◦ , 225◦ and 315◦ , respectively. Preliminary testing of more complicated trajectories suggests that this system is more robust and capable of more smoothly tracking the trajectory.

T 1111 (26) It is evident that each element in the above null space vector is equivalent. As a result, it can be concluded that any fully-symmetric cable configuration, aside from those which induce a kinematic singularity, will guarantee a tension factor of one and thereby maximize the quality of the wrenchclosure workspace. Recall that for wrench closure, the elements of the null space vector must be either all positive or all negative and the vector cannot contain any zero elements. It is readily seen from Eq. (26) that there are three conditions for which the components of the null space vector become zero. The first occurs when θ1,rel = 90◦ . In this case, all four cables become vertical with respect to the platform, and thus the system is unable to resist forces in the horizontal direction. Similarly, when θ1,rel = 0◦ all four cables become horizontal with respect to the platform, and the system is unable to resist forces in the vertical direction. Finally, when θ1,rel = tan−1 eew , the l lines of action of all four cables pass through the center of the platform, and the system is unable to resist moments. Note that these scenarios correspond to the kinematically singular configurations presented in Section 2.3. Thus, in some sense, it can be said that by limiting the potential system configurations to those which preserve full-cable symmetry, there is a convergence between the kinematic singularities and wrench-closure singularities of the system. In other words, there are no configurations that can be at a wrenchclosure singularity without also being at a kinematic singularity. This is tremendously advantageous as it eliminates the need to compute the wrench-closure workspace. N = − sin 2θ1,rel ew cos θ1,rel − el sin θ1,rel

Anson

Similar to the general case, Eq. (27) represents two equations in the two unknown joint variables, βi and Li . Squaring and summing these equations will allow for βi to be eliminated and yields the following quadratic equation in Li :

7

Stiffness Analysis and Optimization As discussed in the previous section, a fully-symmetric cable configuration always yields a tension factor of one (excluding kinematically singular configurations), and is fully defined when one relative angle is specified. Adjusting this relative angle can allow for some secondary criteria to be satisfied. Due to the importance of stiffness in maintaining 12


7

d, the force and moment equations can be separated into the following:

the stability and disturbance rejection capabilities of cabledriven parallel manipulators, it is chosen as a suitable performance criteria. The complete form the stiffness matrix, presented in [27], was termed the conservative congruence transformation, as it was shown to preserve the symmetric, positivedefinite, and conservative properties of stiffness matrices when mapping between the Cartesian and joint spaces. For the planar 3 DoF cable-driven robot system the Cartesian stiffness matrix, K, can be calculated as [28]: K=−

h

∂S ∂S ∂S ∂x t ∂y t ∂φ t

i

+ SKL ST

(29)

7.1

Stiffness Matrix Homogenization In order to assess the quality of the stiffness of the manipulator for a particular pose or configuration, it becomes necessary to define stiffness indices. However, the mixeddimensionality of the stiffness matrix for planar 3-DoF systems prevents direct evaluation of its condition number and other performance indices. Thus, we must first homogenize the stiffness matrix. To do this, the method introduced in [30], based on partitioning a unit-inconsistent matrix into homogeneous translational and rotational components, will be adopted. This method has been implemented in the evaluation of the stiffness matrix for general mechanical systems in [31], in the context of a spherical parallel manipulator in [32], and for cable-driven robots in [33]. Nguyen and Gouttefarde [34] present a recent discussion of issues of the stiffness matrix of general 6-DOF Cable-Driven Parallel Robot (CDPR), its homogenization, intuitive meaning and applicability to design problems. We begin by representing the stiffness equation corresponding to our system in symbolic form, as:

(31)

Mz = K21 d + K22 φ = Md + Mφ

(32)

F = K11 Sd ψd + K12 Sφ ψφ

(33)

Mz = K21 Hd νd + K22 Hφ νφ

(34)

Or, in a more F = GF ψ and Mz = GM ν, compact form, where GF = K11 Sd K12 Sφ and GM = K21 Hd K22 Hφ , T T and ψ = ψd ψφ , ν = νd νφ . GF is a dimensionally homogeneous coefficient matrix, of units N, which maps the dimensionless parameter vector, ψ, to the force vector, F. Similarly, GM is a dimensionally homogeneous coefficient matrix, of units N.m, which maps the dimensionless parameter vector, ν, to the moment, Mz . The eigenvectors and the square roots of the eigenvalues of matrices GF GTF and GM GTM can now be used to develop appropriate stiffness indices. Note that the square root is introduced due to the quadratic nature of the matrices. GF GTF results in a 2 × 2 matrix whose eigenvectors and eigenvalues form an ellipse corresponding to the translational portion of the stiffness. More specifically, the eigenvectors represent the directions of maximum and minimum translational stiffness, and the corresponding eigenvalues indicate the stiffness magnitudes in these directions. GM GTM reduces to a scalar value which represents the rotational part of the stiffness. We are actually converting a single 3 × 3 matrix (some of whose elements have units of N/m and some of whose elements have units of Nm) into two separate matrices: a 2 × 2 matrix with units of N, and a 1 × 1 matrix with units of Nm. Doing this allows us to determine more meaningful condition numbers. One stiffness index corresponds to the stiffness in translation, and the other to the stiffness in rotation.



(30)

Thus, the 3 × 3 stiffness matrix is split into block matrices of consistent units. Denoting the displacement vector as Anson

F = K11 d + K12 φ = Fd + Fφ

Each independent term on the right-hand side of Eq. (31), (32) can now be associated with a physically meaningful quadratic form. For example, kFβ k2 = β T KTβ Kβ β , where β is a dummy variable. The eigenvectors of the matrix KTβ Kβ define the principal axes of the ellipse and can be used to transform the space β to dimensionless intermediate spaces ψ and ν. More specifically, the following relationships are introduced, d = Sd ψd , φ = Sφ ψφ , d = Hd νd and φ = Hφ νφ , where Sd , Sφ , Hd and Hφ are in general orthogonal matrices whose columns are the eigenvectors of KT11 K11 , KT12 K12 , KT21 K21 and KT22 K22 , respectively. Note that in the planar 3-DoF case, KT12 K12 and KT22 K22 are nothing but scalars. As such, their eigenvectors are equal to 1 and their eigenvalues are equal to the scalar value itself. It can be seen that d and φ are related to the dimensionless parameters ψ and ν through a linear transformation. Substituting the expressions for d and φ into Eqs. (31), (32), we obtain:

where KL represents the cable stiffness and is oftentimes referred to as the joint stiffness matrix. As was noted in [29], cable stiffness is actually comprised of two stiffnesses in series - the cable elastic stiffness and the actuator stiffness arising from the closed-loop control system. In general, this term is highly dependent upon the pose and actuation arrangement of the manipulator. Investigating Eq. (29), it is apparent that for systems with base mobility, the Cartesian stiffness matrix can be adjusted in one of three ways: either by modifying the joint stiffness matrix, KL ; by taking advantage of the actuation redundancy in order to alter the tension distribution, t; or by adjusting the structure matrix, S, through a reconfiguration of the system. The third option will be the primary focus of this work.

   x Fx K K 11 12  Fy  = y K21 K22 Mz φ

STIFFNESS ANALYSIS AND OPTIMIZATION

13


7

7.2

Table 2.

Stiffness Modulation With the stiffness matrix partitioned into unit-consistent translational and rotational components, we can now take advantage of the kinematic redundancy in the system in order to optimize the configuration based on some chosen stiffness objective. If it was desired to provide equal resistance in all directions to an external disturbance, we would like for the system to have isotropic stiffness. If instead, we wish to align the major axis of the ellipse with a certain direction of travel, the objective turns into one which will maximize the directional stiffness. Another approach, which is the one adopted here, is to maximize stiffness in all directions. Note that for this objective, isotropicity is sacrificed for higher stiffness. If the two eigenvalues of GF GTF - corresponding to the translational stiffness - are denoted by λt1 and λt2 , and the single eigenvalue of GM GTM - corresponding to the rotational stiffness - is denoted by λr we can define the stiffness indices as follows [31, 32], p p p κt = min( λt1 , λt2 ), κr = λr

STIFFNESS ANALYSIS AND OPTIMIZATION System parameters and cable properties.

Platform length, el

1m

Platform width, ew

0.5m

Radius of base circle, R

2m

Kinematic signularities at

θ1,rel = 0, 90◦ , 26.57◦

Platform initial pose (x, y, φ )

(0, 0, 0)

Base interference at

θ1,rel = −9.46◦ , 105.95◦

Stabilizability guaranteed if

−63.41◦ ≤ θ1,rel ≤ 116.57◦

Modulus of elasticity (E)

57.3Gpa

Cable cross-sectional area

1.767 × 10−6 m2

Minimum allowable tension, tmin

5N

(35)

An objective function, f , for achieving maximum stiffness is then: f = −min(κt κr )

ߠଵ,௥௘௟

(36)

Note that weights could be given to κr and κt in order to provide preference of maximizing one over the other. In order to illustrate the effects of adjusting the relative cable angle on the above cost function, a plot of f vs. θ1,rel can be generated for the system at a particular pose. Before doing so, however, a few additional parameters must be specified. The stiffness of the ith cable is formulated as, ACi ki = LEi i+L , where Lw , E, ACi denote the length of the actuw ating winch, the modulus of elasticity of the cable and the cross-sectional area of the cable, respectively. The parameter values used in this simulation assume the cable is a 7 × 7 wire rope made of AISI 316 stainless steel. The associated properties are listed in Table 2. It would be useful to be able to limit the required search region for θ1,rel based on system characteristics. There are at least three limiting factors to consider: the kinematic singularities at θ1,rel = 0, 90◦ ,tan−1 ( eewl ), base and cable interference, and stabilizability. In [29], it was shown that for fullyconstrained planar cable-driven robots, the system is guaranteed to be stabilizable if the angle between the ith cable ˆ i , and the vector denoting the ith platform caunit vector, L ble attachment point,O RE E ei , is greater than 90◦ (for all i=4 cables). The lines or regions corresponding to these three limiting factors are indicated for θ1,rel in Fig. 13. All of these factors, with the exception of the kinematic singularities at θ1,rel = 0, 90◦ , are dependent upon the dimensions of the platform. In addition, base/cable interference depends upon the pose of the platform. The results for the system whose parameters are listed in Table 2 are shown below in Fig. 14. The range of θ1,rel was limited by the angle at which base/cable interference occurs. Anson

Kinematic Singularities Base/Cable Interference Stabilizability

Fig. 13.

Relative Angle Limits.

A few things are evident from these graphs. First, the maximum stiffness in the x-direction occurs when θ1,rel ≈ 0◦ . At roughly the same point in time, the stiffness in the ydirection is at a minimum. This agrees with intuition: when the system is in this configuration, the cables are completely horizontal and there is no ability for the cables to resist a force (e.g. a disturbance) in the vertical direction. Similarly, maximum stiffness in the y-direction and minimum stiffness in the x-direction occurs when θ1,rel ≈ 90◦ . When θ1,rel ≈ 26.57◦ , all rotational stiffness is lost, as the lines of action of all four cables pass through the geometric center of the platform. It is interesting to note that each of the above three configurations corresponds to a kinematic singularity. When θ1,rel ≈ 45◦ , the stiffness in both translational components is equivalent (i.e. isotropic). Although not shown in the figure, the maximum rotational stiffness occurs when θ1,rel ≈ 116.57◦ . In this configuration, the cables become perpendicular to the moment arm describing the locations of the platform cable attachment points. Investigating the plot of the objective function, we see that there is a local minimum which occurs at θ1,rel ≈ 13.4◦ , but the global minimum within this bounded range occurs when θ1,rel ≈ 57.3◦ . In order to examine the effects of platform dimensions 14


3

8

general effect of decreasing the optimal relative angle. For a fixed platform length, increasing the platform width has the general effect of increasing the optimal relative angle. However, at the transition point, just as one dimension becomes larger than the other, there is a discontinuity in the solution for the optimal relative angle. Clearly, the relative angle providing maximum stiffness is a function of platform dimensions. The question now is whether the optimal relative angle will change as a function of manipulator pose. To determine this, the angle is computed for a sample trajectory. The specified task is to translate along a circular path of radius 1.25m. The cost function of Eq. (36) is evaluated at each (x, y) position, and the results are given in Fig. 16(a). It is evident that the optimal angle varies throughout the trajectory. This fluctuation can be attributed to the constant change in length of the cables, which impacts the cable stiffness ki .

5 x 10 Eigenvalues of Stiffness Matrix − Translational Components

2 √ λx p λy

1 0

0 4

6

x 10

20

40

60

θ1,rel (deg)

80

Eigenvalues of Stiffness Matrix − Rotational Component

4 2

p

0

0

20

8

0

40

60

θ1,rel (deg)

λφ

80

Objective function f = − κt κr

x 10

LOADED CONFIGURATIONS

−5 −10 −15

0

20

40

60

θ1,rel (deg)

80 Relative Angle that Optimizes Stiffness Along Circular Trajectory

55

Eigenvalue and Objective Function Results for Sample Sys-

54 θ1,rel(deg)

Fig. 14. tem.

53 52 51 50 1 0

on the results, a second simulation was run in which the platform length was doubled. The minimum of the objective function in this case occurs at θ1,rel ≈ 49.4◦ . The reason for this change is due to the shift towards the left that occurs in the rotational stiffness index. Indeed, increasing the platform length has the effect of decreasing the angle at which the rotational stiffness is minimum.

−1 Y(m)

Fig. 16.

θ1,rel (deg)

0

0.5

1

(b)

Optimal Relative Angle for Sample Trajectory.

8

Loaded Configurations It is evident that a configuration in which the cables are fully-symmetric about the platform frame offers many advantages, including simplifying the inverse kinematics problem, eliminating the need to calculate the wrench-closure workspace, and allowing for additional performance criteria, such as stiffness, to be optimized. However, it is important to consider the potential drawbacks of this type of configuration. Note that the preceding analysis was based ͻ͵ on evaluating the quality of the wrench-closure workspace. However, when external loads are applied the platform, the tensions in the cables are no longer purely a function of the null space of the structure matrix. Instead, the cable tensions are heavily dependent upon the magnitude and direction of this external load, and must be distributed appropriately so as to maintain equilibrium. While it is true that the tension factor is a useful tool for characterizing the closeness of a pose to a wrench-closure singularity, its use as an indicator of even tension distribution is only meaningful in unloaded configurations. As such, while it has been shown that any non-singular fully-symmetric cable configuration guarantees a tension factor of one, this does not imply that the system can resist arbitrary loads while maintaining an even tension

65 60

60

55

50

50

40

45

30

40

20 2

35

configuration`

1.5

2 1

0.5

Platform Width (m)

Fig. 15.

30

1.5

1

25

0.5 0

0

Platform Length (m)

Effect of Platform Dimensions on Optimal Relative Angle.

To investigate this trend, the optimal relative angle is plotted as a function of platform length, el , and platform width, ew . The results, illustrated in Fig. 15, show that for a fixed platform width, increasing the platform length has the Anson

−0.5

X(m)

(a)

Effect of Platform Dimension Changes on Optimal Relative Angle

70

−1

15


8

LOADED CONFIGURATIONS

Provided that Fx > 0, the quantity cos θ1,rel + cos θ2,rel must be less than zero in order to satisfy the positivity condition on the tensions. This is true if (limiting θ1,rel and θ2,rel to be between 0 and π), π − θ2,rel < θ1,rel . Thus, the system is able to obtain an even tension distribution as long as the angle that θ1,rel makes with the positive x-axis is greater than the angle that θ2,rel makes with the negative x-axis. It is also seen from Eq. (39) that if the angle that θ1,rel makes with the positive x-axis is exactly equal to the angle that θ2,rel makes with the negative x-axis, then infinite tensions are required to obtain an even tension distribution. To further gain insight into this relationship, a contour plot is generated which illustrates the tension requirements for varying values of θ1,rel and θ2,rel (see Fig. 18(a)). The shaded region indicates the 4 4 t ∑ cos θ1,rel = −Fx , t ∑ sin θ1,rel = 0, (37) angle combinations which result in negative, and thus infeasible, tension requirements (these regions are also indicated i=1 i=1 by red lines). The boundary between the feasible and infeasible regions acts as a singularity in the system. In the lower left quadrant, this condition occurs when the constraint equat(e1x sin θ1,rel − e1y cos θ1,rel + e2x sin θ2,rel − e2y cos θ2,rel + tion relating θ1,rel and θ2,rel is violated. The optimal solution, i.e. the solution which yields the set of even cable tensions e3x sin θ3,rel −e3y cos θ3,rel +e4x sin θ4,rel −e4y cos θ4,rel ) = 0 with the smallest norm, occurs when θ1,rel and θ2,rel are both (38) 180 degrees. Indeed, at such a configuration, the applied force can be split evenly among the four cables, and no homogeneous solution is needed. It should be noted, however, that such a configuration is unstable.

distribution. In fact, as will be shown in this section, any attempt to evenly distribute the internal forces will result in infinite tension requirements. To begin, let us consider a system subjected to a unit force in the positive x-direction of the platform frame. Resolving the static equilibrium equations about the platform axes, and making use of the relative angles, we can easily obtain the three equilibrium equations. We are interested in determining under what conditions an even load distribution can be obtained. Thus, let us set t1 = t2 = t3 = t4 = t. With this constraint, equilibrium equations are obtained as (see Fig. 17):

‫ݐ‬ଶ

ߠଶǡ௥௘௟

ߠଷǡ௥௘௟

Fig. 17.

‫ܨ‬௫

‫ݐ‬ଵ ‫ݕ‬ா

‫ݔ‬ா

A similar analysis can be performed for the case in which a unit force is exerted in the positive y-direction of the platform frame. In fact, many parallels exist between this scenario and the previous one. It can be shown that, in this case, even tension distributions are achievable when the cables are symmetric about the y-axis of the platform frame and when the following quantity can be made positive,

ߠଵǡ௥௘௟

ߠସǡ௥௘௟

‫ݐ‬ଷ

‫ݐ‬ସ

t=

(40)

Unit Force Applied in x-Direction of Platform Frame.

This places the following constraint on θ1,rel and θ3,rel : θ3,rel − π > θ1,rel . The corresponding contour plot is shown in Fig. 18(b). Similar conclusions can be drawn.

We will now show that these equations are satisfied if the cables are symmetric about the x-axis of the platform. For this to be true, the following relationships must hold: θ4,rel = 2π − θ1,rel , θ3,rel = 2π − θ2,rel . Plugging these relationships into the force equilibrium equation in the y-direction, we see that the equation is satisfied. Turning to the moment equation, we must utilize the definitions of the cable attachment point locations. Provided that the platform is rectangular, we have: e4x = e1x , e3x = e2x = −e1x , e2y = e1y and e3y = e4y = −e1y . Making these substitutions, we see that the moment equilibrium equation is also satisfied. Now, solving the force equilibrium equation in the x-direction for t, yields:

t= Anson

−Fy 2(sin θ1,rel + sin θ3,rel )

−Fx 2(cos θ1,rel + cos θ2,rel )

Finally, for a unit moment exerted in the CCW direction, it is determined that even cable distributions are possible if θ3,rel is symmetric about the origin with θ1,rel and θ4,rel is symmetric about the origin with θ2,rel . It is interesting to note that this condition is dependent upon the platform geometry. Although it is slightly more difficult to obtain a relationship between θ1,rel and θ2,rel which is guaranteed to provide positive cable tensions, we may observe the behavior of the system in the contour plot of Fig. 18(c). This section has attempted to illustrate one of the weakness of the fully-symmetric system. It should be stated that, although an even tension distribution cannot be obtained when the fully-symmetric system is subjected to an external load, tension solutions are still available.

(39) 16


REFERENCES The circular configuration exhibited vast improvements both in terms of translational and rotational workspace. Despite the added dexterity of these systems, additional singular configurations become possible and must be avoided. The kinematic redundancy inherent in these systems necessitates the creation of an effective redundancy resolution scheme. Although this adds complexity to the analysis and control, it also allows for base reconfiguration in order to meet various objectives. In this work, focus was given to improving the quality of the wrench-closure workspace through evaluation of the tension factor. A kinematic controller developed for trajectory tracking was designed to take advantage of the self-motion capabilities of the mobile base systems. A GUI was developed in MATLAB which aided in setting up and animating a variety of simulations. It was observed that the cables tend to form a symmetric configuration with respect to the platform. It was later proven that a fully-symmetric cable configuration allows for a tension factor of one to always be obtained. This fullysymmetric configuration offers a variety of additional advantages including replacing the computationally expensive optimization routine with a closed-form solution to the inverse kinematics. Furthermore, these systems exhibit a convergence between kinematic singularities and wrench-closure singularities. While any non-singular fully-symmetric cable configuration can be chosen to provide a maximum tension factor, an optimal relative angle is determined by maximizing the stiffness of the manipulator. It is demonstrated that both platform pose and dimension influence the stiffness results. Finally, a limitation of the fully-symmetric system was presented. This limitation dealt with the inability of the cables to obtain an even tension distribution in loaded configurations. Because of this, it may be advantageous to relax the fully-symmetric cable requirement in order to yield reasonable tensions of equal magnitude.

(a)

(b)

Acknowledgment This work was partially supported by the National Science Foundation awards CNS-1314484 and IIS-1319084.

(c) Fig. 18.

(a) Tension requirements to evenly balance

References [1] Albus, J., Bostelman, R., and Dagalakis, N., 1993. “The nist robocrane”. Journal of Robotic Systems, 10(5), pp. 709–724. [2] Bosscher, P., Williams, R. L., Bryson, L. S., and CastroLacouture, D., 2007. “Cable-suspended robotic contour crafting system”. Automation in Construction, 17(1), pp. 45–55. [3] Kawamura, S., Ida, M., Wada, T., and Wu, J.-L., 1995. “Development of a virtual sports machine using a wire drive system-a trial of virtual tennis”. In Intelligent Robots and Systems 95.’Human Robot Interaction and Cooperative Robots’, Proceedings. 1995 IEEE/RSJ International Conference on, Vol. 1, IEEE, pp. 111–116. [4] Perreault, S., and Gosselin, C. M., 2008. “Cable-driven parallel mechanisms: application to a locomotion interface”. Journal of Mechanical Design, 130(10), p. 102301.

Fx , (b) Ten-

sion requirements to evenly balance Fy , (c) Tension requirements to evenly balance Mz

9

Conclusion In this work, planar cable-driven parallel manipulators with base mobility were investigated. More specifically, two different configurations were analyzed and compared to the traditional fixed-base system. While the rectangular configuration was shown to improve the orientation workspace in one direction, it is limited by the nature of the base setup. Anson

17


REFERENCES 912. [22] Lim, W. B., Yang, G., Yeo, S. H., and Mustafa, S. K., 2011. “A generic force-closure analysis algorithm for cable-driven parallel manipulators”. Mechanism and Machine Theory, 46(9), pp. 1265–1275. [23] Bosscher, P., Riechel, A. T., and Ebert-Uphoff, I., 2006. “Wrench-feasible workspace generation for cable-driven robots”. Robotics, IEEE Transactions on, 22(5), pp. 890–902. [24] Dai, J. S., and Jones, J. R., 2002. “Null–space construction using cofactors from a screw–algebra context”. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 458, The Royal Society, pp. 1845–1866. [25] Williams, R. L., Gallina, P., and Vadia, J., 2003. “Planar translational cable-direct-driven robots”. Journal of Robotic Systems, 20(3), pp. 107–120. [26] Siciliano, B., Sciavicco, L., Villani, L., and Oriolo, G., 2009. Robotics: modelling, planning and control. Springer Science & Business Media. [27] Chen, S.-F., and Kao, I., 2000. “Conservative congruence transformation for joint and cartesian stiffness matrices of robotic hands and fingers”. The International Journal of Robotics Research, 19(9), pp. 835–847. [28] Alamdari, A., and Krovi, V., 2015. “Modeling and control of a novel home-based cable-driven parallel platform robot: Pacer”. In Intelligent Robots and Systems (IROS), 2015 IEEE/RSJ International Conference on, IEEE, pp. 6330–6335. [29] Behzadipour, S., and Khajepour, A., 2006. “Stiffness of cablebased parallel manipulators with application to stability analysis”. Journal of mechanical design, 128(1), pp. 303–310. [30] K¨ovecses, J., and Ebrahimi, S., 2009. “Parameter analysis and normalization for the dynamics and design of multibody systems”. Journal of Computational and Nonlinear Dynamics, 4(3), p. 031008. [31] Taghvaeipour, A., Angeles, J., and Lessard, L., 2012. “On the elastostatic analysis of mechanical systems”. Mechanism and Machine Theory, 58, pp. 202–216. [32] Wu, G., 2014. “Stiffness Analysis and Optimization of a Coaxial Spherical Parallel Manipulator”. Modeling, Identification and Control, 35(1), pp. 21–30. [33] Moradi, A., 2015. “Stiffness analysis of cable-driven parallel robots”. PhD thesis, Queen’s University. [34] Nguyen, D. Q., and Gouttefarde, M., 2014. “Stiffness matrix of 6-dof cable-driven parallel robots and its homogenization”. In Advances in Robot Kinematics. Springer, pp. 181–191.

[5] Hadian, H., Amooshahi, Y., and Fattah, A., 2013. “Kinematics and dynamics modeling of a new 4-dof cable-driven parallel manipulator”. Advanced Engineering and Computational Methodologies for Intelligent Mechatronics and Robotics, p. 249. [6] Merlet, J.-P., and Daney, D., 2010. “A portable, modular parallel wire crane for rescue operations”. In Robotics and Automation (ICRA), 2010 IEEE International Conference on, IEEE, pp. 2834–2839. [7] Borgstrom, P. H., Borgstrom, N. P., Stealey, M. J., Jordan, B., Sukhatme, G. S., Batalin, M., Kaiser, W. J., et al., 2009. “Design and implementation of nims3d, a 3-d cabled robot for actuated sensing applications”. Robotics, IEEE Transactions on, 25(2), pp. 325–339. [8] Dekker, R., Khajepour, A., and Behzadipour, S., 2006. “Design and testing of an ultra-high-speed cable robot”. International Journal of Robotics and Automation, 21(1), pp. 25–34. [9] Peng, B., Nan, R., Su, Y., Qiu, Y., Zhu, L., and Zhu, W., 2001. “Five-hundred-meter aperture spherical telescope project”. Astrophysics and space science, 278(1-2), pp. 219–224. [10] Alamdari, A., 2016. “Cable-driven articulated rehabilitation system for gait training”. PhD thesis, MAE Dept., SUNY Buffalo. [11] Alamdari, A., and Krovi, V., 2016. “Design and analysis of a cable-driven articulated rehabilitation system for gait training”. Journal of Mechanisms and Robotics, 8(5), p. 051018. [12] Alamdari, A., and Krovi, V., 2015. “Parallel articulatedcable exercise robot (pacer): novel home-based cable-driven parallel platform robot for upper limb neuro-rehabilitation”. In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, pp. V05AT08A031–V05AT08A031. [13] Gouttefarde, M., and Gosselin, C. M., 2006. “Analysis of the wrench-closure workspace of planar parallel cabledriven mechanisms”. Robotics, IEEE Transactions on, 22(3), pp. 434–445. [14] Zhou, X., Tang, C. P., and Krovi, V., 2013. “Cooperating mobile cable robots: Screw theoretic analysis”. In Redundancy in Robot Manipulators and Multi-Robot Systems. Springer, pp. 109–123. [15] Merlet, J.-P., 2012. Parallel robots, Vol. 74. Springer Science & Business Media. [16] Anson, M., 2015. “Cable-driven parallel manipulators with base mobility: A planar case study”. MS thesis, MAE Dept., SUNY Buffalo. [17] Diao, X., Ma, O., and Lu, Q., 2008. “Singularity analysis of planar cable-driven parallel robots”. In Robotics, Automation and Mechatronics, 2008 IEEE Conference on, IEEE, pp. 272– 277. [18] Pott, A., 2014. “An improved force distribution algorithm for over-constrained cable-driven parallel robots”. In Computational Kinematics. Springer, pp. 139–146. [19] Verhoeven, R., and Hiller, M., 2002. “Tension distribution in tendon-based stewart platforms”. In Advances in Robot Kinematics. Springer, pp. 117–124. [20] Borgstrom, P. H., Jordan, B. L., Sukhatme, G. S., Batalin, M., Kaiser, W. J., et al., 2009. “Rapid computation of optimally safe tension distributions for parallel cable-driven robots”. Robotics, IEEE Transactions on, 25(6), pp. 1271–1281. [21] Pham, C. B., Yeo, S. H., Yang, G., and Chen, I.-M., 2009. “Workspace analysis of fully restrained cable-driven manipulators”. Robotics and Autonomous Systems, 57(9), pp. 901–

Anson

18


LIST OF TABLES

List of Figures 1 Virtually Prototyping Cable-Driven Parallel Manipulators with Base Mobility in Multi-Domain Modeling and Simulation Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 (a) Rectangular Base Configuration, (b) Circular Base Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 (a) Kinematic singularities in the fixed-base system, architecture singularity (b) Kinematic singularities in the fixed-base system, Type I singularity (c) Type I singularities in systems with rectangular base mobility (d) Type I singularities in systems with circular base mobility, (e) Type III kinematic singularity in the circular base configuration . . . . . . . . . . 4 (a), (b) and (c) Common Type II Kinematic Singularities in the Rectangular Base Configuration (d), (e) and (f) Common Type II Kinematic Singularities in the Circular Base Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example Wrench-Closure Workspace and Tension Factor Map for Fixed-Base System. . . . . . . . . . . . . . . . . . . . 6 (a) Example Internal Wrench-Closure Singularities in the Fixed Configuration (b) Example Internal Wrench-Closure Singularities in the Rectangular Base Configuration (c) Example Internal Wrench-Closure Singularities in the Circular Base Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Approaching the Boundary - (a),(b) Fixed-base, (c),(d) Rectangular configuration, (e),(f) Circular configuration . . . . . . 8 Orientation Workspace: (a) Fixed-base, (b) Rectangular configuration, (c) Circular configuration . . . . . . . . . . . . . . 9 Wrench-Closure Orientation Workspace at x=0, y=0 - (a) Fixed-base, (b) Rectangular configuration, (c) Circular configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Position Control Scheme for Trajectory Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 graphical user interface (GUI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Trajectory Tracking Results: (a) Rectangular configuration, (b) Circular configuration . . . . . . . . . . . . . . . . . . . . 13 Relative Angle Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Eigenvalue and Objective Function Results for Sample System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Effect of Platform Dimensions on Optimal Relative Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Optimal Relative Angle for Sample Trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Unit Force Applied in x-Direction of Platform Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 (a) Tension requirements to evenly balance Fx , (b) Tension requirements to evenly balance Fy , (c) Tension requirements to evenly balance Mz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Tables 1 Trajectory tracking simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 System parameters and cable properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anson

19

2 3

5 5 8

8 9 9 10 10 11 11 14 15 15 15 16 17

11 14

Orientation Workspace and Stiffness Optimization of

Orientation Workspace and Stiffness Optimization of

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