Design and analysis of reconfigurable parallel robots with enhanced stiffness

Mechanism and Machine Theory 77 (2014) 92–110

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Design and analysis of reconfigurable parallel robots with enhanced stiffness Amin Moosavian a,⁎, Fengfeng (Jeff) Xi a,b a b

Department of Aerospace Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada Robotic Research Center, Shanghai University, Shanghai 200072, China

a r t i c l e

i n f o

Article history: Received 16 July 2013 Received in revised form 29 January 2014 Accepted 7 February 2014 Available online 12 March 2014 Keywords: Reconfigurable robots Parallel kinematic manipulators Kinetostatics Static redundancy Under-actuation Virtual alternating constraints Variable topology

a b s t r a c t Static redundancy in a parallel manipulator can enhance the stiffness of the end-effector, improve its fault tolerance, minimize its singularity loci, and reduce the internal loads experienced by the joints. Traditionally, this form of redundancy would be accompanied by actuation redundancy. Introduced in this paper is a new approach to statically enhance a manipulator without actuation redundancy. This is achieved through the use of lockable passive joints that are utilized in an alternating fashion to reconfigure the system into various isostatic and hyperstatic topologies without any external assistance. Although applicable to both kinematically non-redundant and constrained manipulators, this approach is especially effective for those with lower instantaneous mobility. The inherent redundancy in these reconfigurable robots is exploited to obtain full finite mobility with as few as one actuator through under-actuation with the use of virtual alternating constraints. The architecture design, kinematic analysis, and kinetostatic analysis of the proposed robots are addressed herein, followed by a case-study to demonstrate the effectiveness of the proposed design and analysis. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Reconfigurable parallel robots have attracted considerable interest both in the areas of machine design and motion control over the past two decades. Traditionally, the objective for reconfigurability has been the alteration of mobility [1,2], or motion characteristics [3,4]. However, with the exception of a few [4,5], reconfigurability for enhanced stiffness or static redundancy has not been extensively studied. Reconfigurability in parallel kinematic manipulators (PKMs) can be classified into geometric, topological, or a combination of the two [6]. Geometric morphing deals with the variation in size or orientation of the branches that make up the manipulator, without altering its kinematic architecture. For examples of geometric morphing see [4,5]. Topological morphing deals with variations in the kinematic architecture of the manipulator by changing the types or the sequence of the joints that make up the system. For examples of topological morphing see [1,2,7]. Analogous to the geometric and topological classifications of reconfigurability, the variation in the static and stiffness characteristics of a PKM can also be classified into similar groups (see Fig. 1). In this context, the geometric approach will take advantage of the change in the size and orientation of the branches to alter the stiffness of the end-effector or the internal loads experienced by the members, whereas the topological approach will utilize alterations in the connectivity of branches and joint types to do the same. Generally, only the topological approach has the potential to generate a statically redundant system, which makes it more suitable for autonomous and aerospace applications. ⁎ Corresponding author. E-mail addresses: [email protected] (A. Moosavian), [email protected] (F.(J.) Xi).

http://dx.doi.org/10.1016/j.mechmachtheory.2014.02.005 0094-114X/© 2014 Elsevier Ltd. All rights reserved.

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

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Nomenclature b C d h F f f J K k M N n p QP q q R U X XD XI θ

base vector branch connectivity Cartesian motion parameter module position vector external load vector internal load vector number of relative degrees of freedom Jacobian matrix stiffness matrix limb stiffness mobility summation scalar number of load-bearing limbs platform vector vector of lengths for the locked passive members limb vector limb length rotation matrix strain energy module pose vector vector of dependent Cartesian parameters vector of independent Cartesian parameters rotation angle

Subscripts A related to the actuated member C related to the Cartesian space hyp hyperstatic iso isostatic J/j related to the joint or the joint space l link P related to the passive member x along/about x-axis y along/about y-axis z along/about z-axis

Traditionally, in lower mobility manipulators, the topology is designed to accommodate the motion requirements. The static and stiffness characteristics are then addressed by altering the geometric parameters such as joint locations or the sizing of the actuators. By separating the kinematic and the static requirements, one can design manipulators with lower mobility and rather attractive static traits. Such lower mobility systems can exhibit stiffness characteristics comparable to that of a PKM with full mobility and superior to conventional lower mobility manipulators with permanently constraining legs. In addition to applications where varying the topology can enhance the static and stiffness characteristics of a PKM in motion, there exist applications where a manipulator may be required to act as a structure for a significant period of time during which it will be experiencing external loads far greater than those required for actuation. One example would be a PKM belonging to an autonomously reconfigurable structure. Although the manipulator may be in a stationary pose, any shift in the position of its payload, which may happen to be another PKM, could induce large internal loads on the manipulator. Another example is a PKM which would be experiencing large dynamic loads as it would be autonomously transported to reconfigure a larger structure. In any of these cases having a manipulator that can exhibit enhanced static and stiffness characteristics is desirable. Traditionally, this enhancement comes at the cost of having more actuators than kinematically required. In this paper we introduce a family of reconfigurable parallel robots with enhanced static characteristics with only as many actuators as the required instantaneous mobility. These robots belong to the topologically reconfigurable category. Although the focus in this paper is on robots with lower instantaneous mobility, the presented design and analysis methodologies are also applicable to enhance those with full mobility such as the Gough–Stewart parallel manipulators [8,9]. The proposed reconfigurable robots are fault tolerant and can provide enhanced static and stiffness characteristics with minimal number of

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Geometric Approach

Topological Approach

Fig. 1. Geometric and topological approaches to change the static and stiffness characteristics of a PKM.

actuators. In addition, they offer a larger freedom for their structural design through the separation of static and kinematic requirements. An example of such a robot is proposed in Fig. 2. The robot illustrated by Fig. 2 has an instantaneous mobility of four with four actuated limbs and four passive limbs which can be locked and unlocked to vary the topology of the manipulator. The four actuated limbs are distinguished using dashed lines in Fig. 2. This reconfigurable robot can achieve full finite mobility through under-actuation and has multiple degrees of fault tolerance. Design and analysis of such robots will be discussed in detail in the upcoming sections. However, to classify these robots one needs to first review various types of redundancies applicable to PKMs.

1.1. Manipulator redundancy A variable topology manipulator (VTM) can effectively reconfigure its topology to accommodate particular situations that may arise from either a static requirement, i.e. increased external loads, or a motion requirement, i.e. enlarged workspace. In either case, such a manipulator will have to be inherently redundant. Pierrot [10] classifies redundancies in parallel mechanisms into three types: a) actuation redundancy, b) kinematic redundancy, and c) sensor redundancy. We add d) static redundancy as the fourth type to this classification. Static redundancy occurs when a mechanism or a structure becomes hyperstatic. A robot or a configuration with a hyperstatic topology is one that is statically and kinematically indeterminate, or in other words it is redundantly rigid, whereas an isostatic topology is one that is both statically and kinematically determinate, or in other words, it is minimally rigid. Kinematic redundancy occurs when the mobility of the manipulator is greater than the number of Cartesian motion parameters. This type of redundancy

Fig. 2. Proposed variable topology robot with lower instantaneous mobility, full finite mobility, and enhanced static and stiffness characteristics; limbs with shading are actuated, whereas limbs with no shading are passive and lockable.

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is typically used to enlarge the workspace of a manipulator; for examples see [11,12]. Actuation redundancy occurs when the number of actuators is greater than the mobility of the manipulator. This type of redundancy can increase the stiffness or minimize the singularity loci and internal loads of a parallel manipulator; for examples see [13–15]. Sensor Redundancy occurs when the number of sensors is greater than the number of actuated joints. This type of redundancy is typically used for solving forward kinematic problems. See [16] for an example of this type of redundancy. Fig. 3 illustrates four examples belonging to each type of redundancy distinguishing them in terms of the number of actuators, NA, and mobility, M. Mobility of a system can be determined using Eq. (1) [17]: j
 X fi M ¼ d Nl −N j −1 þ

ð1Þ

i¼1

where the motion parameter, d, takes a value of three in ℝ2, and six in ℝ3. Nl, Nj, fi, and M are the number of links, the number of joints, the degrees of relative motion permitted by joint i, and the degrees of freedom (DOF) of the manipulator, respectively. The kinematically redundant manipulator (Fig. 3a) has 13 joints, and 11 links; it has a mobility of M = 4. The redundantly actuated manipulator (Fig. 3b) has 12 joints, and 10 links; it has a mobility of M = 3. The sensor redundant manipulator (Fig. 3c) has 12 joints, and 10 links; it has a mobility of M = 3. In the case of the statically redundant manipulator (Fig. 3d) with the lockable passive limb in the unlocked position, there are 12 joints and 10 links; this gives a mobility of M = 3. However, when the passive limb is locked, the number of joints and links become 11 and 9 respectively; this yields a mobility of M = 2. The manipulator depicted by d is statically and kinematically determinate when the passive limb is unlocked. In this case it would be similar to c in terms of determinacy. However, when the passive limb becomes locked the system becomes both statically and kinematically indeterminate. In this case, it would be similar to b in terms of determinacy and could be considered statically redundant. The kinematically redundant manipulator (a) does not offer static redundancy, that is, if one of the actuators were converted into a passive member the manipulator would become degenerate. The sensor redundant manipulator (c) is isostatic at all times; hence, it also offers no static redundancy. Therefore, the only two types that can deliver static redundancy are types b and d. To further distinguish between the applications of static redundancy, we introduce two terms referred to as the stagnant stiffness and the transitional stiffness. The stagnant stiffness refers to the stiffness of a kinematic module when it is not in motion, whereas the transitional stiffness simply refers to the stiffness of a kinematic module while it is in motion. Although, the stagnant and transitional stiffnesses are both static terms, the distinction between the two arises from a topological perspective. When transitional stiffness is achieved, the system is isostatic; however, when stagnant stiffness is achieved, it may remain isostatic or

Passive Revolute Joint

Rigid Platform Active Prismatic Joint

(a) Kinematic Redundancy

(b) Actuation Redundancy

M>d NA = M

M=d NA > M

Lockable Prismatic Joint

Passive Prismatic Joint

(c) Sensor Redundancy M=d NA = M

(d) Static Redundancy M≤d NA ≥ M

Fig. 3. Four types of redundancies associated with parallel manipulators.

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become hyperstatic. The distinction between the determinacy of the system has significant impacts on the actuation scheme of the manipulator. A redundantly actuated manipulator can either have joints that are passive, but can turn into actuators upon command or have redundant actuators that may or may not be part of the kinematic system but can be included when required. Other than the obvious inefficiency of having more actuators than kinematically required, the control implementation of such manipulators entails various complexities. In the case where redundant actuators are active for internal load reduction, due to the indeterminate nature of the system, additional measures from the control point of view must be taken to ensure that actuators are not working against each other or that unnecessary loads are not imposed on the system due to over-constraints. All published research in the area of parallel manipulator redundancy have been focused on kinematic, actuation, and sensor redundancies. However, static redundancy without redundant actuation has not been studied before. The manipulators introduced in this paper belong to this type and can improve their stagnant stiffness over a conventional non-redundant manipulator in most of their workspace. Additionally, they can generally improve their transitional stiffness for certain poses and areas in their workspace and potentially reduce their singularity loci. Generating static redundancy without actuation redundancy can improve the static characteristics of the system while avoiding the inherent complexities and inefficiencies of actuation redundancy for particular applications such as the ones introduced earlier. In addition, through under-actuation a high degree of actuation efficiency can be maintained while attaining static redundancy. In what follows, we address the design and analyses of the proposed family of reconfigurable robots.

2. Design The idea of using passive joints that could be locked and unlocked at particular times or sequences has been previously introduced and explored [1,3,7,18,19]. Up to this point the intention for using lockable passive joints have been for actuation efficiency [7,19] or functional/workspace improvements [1,3,18]. In the current work we use lockable passive joints to enhance the stiffness and static characteristics of the manipulator. Due to their sequential locking feature we refer to them as passive alternating lockable members (PALMs). Also, for brevity, any robot that falls in this variable topology family of manipulators will be referred to as a parallel robot with enhanced stiffness (PRES). Fig. 4 illustrates a planar PRES with two actuated limbs and two passive lockable limbs. In Fig. 4 the passive limbs (limbs 2 and 4) can be locked and unlocked to vary the topology of the manipulator. In total, one can have three distinct topologies for this system, two of which are isostatic (Fig. 5ii and iii) and one is hyperstatic (Fig. 5i). The isostatic configurations are used for actuation and enhanced transitional stiffness, whereas the hyperstatic configuration is used for enhanced stagnant stiffness. We will demonstrate the actuation process for this planar PRES through an illustrative example (Fig. 5). The goal is to go from some initial pose, 0X, to some final pose, 2X, with only two actuators. The left subscript notation refers to the stage associated with a pose, i.e. initial (0), intermediate (1), or final (2) in this case. With the module in the final pose, 2X, we have a vector 2Q = [2q1, 2q2, 2q3, 2q4] T that describes the lengths of the actuated, unlocked, and locked passive limbs for the final pose. At this point, we ignore the components associated with the actuated limbs since they can be directly controlled. Therefore, to attain 2X one must initially realize the components associated with the passive limbs, i.e. 2q2 and 2q4. The notation 2q4 refers to the length of the fourth limb in the second (final) pose. Without loss of generality, we will first obtain 2q4 and then 2q2. In order to achieve this we utilize an intermediate pose, 1X. The intermediate pose, which is an indirect approach to obtain 2q4, will need to satisfy an actual constraint which is the length of the second limb that remains locked from the initial pose, i.e. 0q2, and a virtual constraint, which is the desired final length of the fourth limb, i.e. 2q4. Once the desired final length for the fourth limb is obtained, it will become and remain locked, while the second limb will become unlocked

Limb 3 Link Passive Revolute Joint Active Prismatic Joint Locked Passive Prismatic Joint

Limb 4

Unlocked Passive Prismatic Joint

Limb 2 Limb 1 Fig. 4. A planar PRES with two actuators and two passive lockable limbs.

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0 0

q3

1

q4 2

0 0

2

0

q1 1

i) Initial Pose (0 X)

2

q3

q4

q2

97

q3

q4

q2

q1

ii) Intermediate Pose (1 X)

2

2

q2

q1

iii) Final Pose ( 2 X)

Fig. 5. Actuation procedure for a planar PRES with two actuators and two lockable passive limbs.

and the same procedure will be repeated using the other isostatic topology to obtain the desired final length for the second limb, i.e. 2q2. Then, by obtaining the lengths associated with the actuated limbs we attain 2Q and subsequently 2X. The optimal sequence of actuation and the choice for the independent Cartesian parameter for the intermediate pose can be determined through an optimization process which may further enhance the transitional stiffness of the system or minimize the internal loads during actuation. This will be explored and demonstrated in more detail in the upcoming sections and the case-study. Similarly, a planar manipulator with the same topology as the one in Fig. 4 with only one actuated and three passive lockable limbs (see Fig. 6a) can achieve the same in three stages. The additional stage is required to realize the final desired length for the additional passive member. In the case with only one actuator, one will have two actual constraints which correspond to two locked passive members to attain isostaticity and one virtual constraint corresponding to the final length of the unlocked limb. In this case, because the number of constraints is equal to the number of Cartesian parameters, there will generally exist only one reachable pose within the workspace of the manipulator. Therefore, the only optimization problem that could be set up is for the choice of sequence in obtaining the desired final lengths for the passive limbs. A similar actuation procedure can be applied to a spatial PRES. The generalization will be presented in Section 3.2. Fig. 6 illustrates a few examples of planar and spatial PRES's. The solid lines depicted in Fig. 6 ‘e’ represent rigid links. As it can be seen all these manipulators are either kinematically non-redundant or constrained. The degree of static redundancy can vary depending on the application. However, it is always equal to or greater than one. Although PRES's with NA b d actuators are systems with lower mobility and are primarily intended for application that require lower mobility but high stiffness, their inherent redundancy can be exploited to obtain full mobility through under-actuation. In essence, the proposed under-actuation control scheme is the means to maintain the enhanced stiffness characteristics and mobility without the use of actuation redundancy. 2.1. Under-actuation1 There are two types of DOFs with reference to the end-effector's pose for any mechanical system: instantaneous or velocity DOF and finite or configuration DOF [20]. Finite DOFs are the minimum number of geometric parameters required to uniquely identify the pose of a rigid body in space, i.e. three for planar and six for spatial cases. Instantaneous DOFs are those of a rigid body whose velocities and accelerations could be controlled. For fully-actuated PKMs the independent finite and the instantaneous DOFs are the same. However, if the system is under-actuated, one could have more independent finite DOFs than instantaneous ones. The finite and instantaneous mobilities of the manipulator could then be defined by the achievable independent DOFs in either case. There are various approaches to implement the under-actuation, such as by locking the passive joints [7,19] or using guiding racks [21]. Traditionally, the intent for the design of such manipulators has been to implement a control scheme for forward kinematics of serial manipulators with passive joints [19], or simply to minimize the number of actuators in the case of parallel manipulators [7,21]. However, to the authors' best knowledge, no under-actuated parallel manipulator has been designed for the purpose of redundant rigidity or enhanced stiffness. The variable topology approach which enables the under-actuation in the proposed system is made possible through the use of lockable passive joints. These lockable passive joints assist in the under-actuation by imposing kinematic constraints in a systematic fashion, as illustrated in the example shown in Fig. 5. There are several possible ways to implement such constraints in order to obtain the desired end results. From a kinematic perspective, the conversion of passive joints into fully rigid or partially rigid joints can be implemented on any type of joint. A fully rigid joint refers to a joint that loses all of its DOFs. A prismatic (P) or a revolute (R) joint would become fully rigid if it was to be locked. A partially rigid joint refers to a joint that may still have some DOFs left after being locked. A spherical (S), a universal (U), or a cylindrical (C) joint could potentially become partially rigid. The change in topology could also occur by locking multiple joints belonging to one limb. From a practical perspective, locking and 1

Under-actuation in the context of present application only deals with holonomically constrained systems.

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(a)

(b)

(c)

(d)

(e)

Fig. 6. Some examples of planar and spatial PRES's.

unlocking a passive P or R joint, or the axial component of a C joint is simpler to implement. Additionally, having limbs that are similar in architecture helps with maintaining modularity in design. Although having similar limbs with lockable P or R joints are preferred, they are not necessary features for the proposed design. Without loss of generality, we use lockable prismatic joints throughout this paper with similar RPR and SPS limbs for the planar and the spatial cases, respectively. 2.2. Class synthesis Based on the mobility, one can generalize PRES's into three classes as tabulated in Table 1. The first class is kinematically non-redundant, i.e. M = d, whereas the second and the third classes are kinematically constrained, i.e. M b d. All three classes could offer enhanced stiffness for particular applications over a conventional PKM. From a motion control point of view, any PRES that falls under class I is fully-actuated with full finite and instantaneous mobilities; a PRES belonging to class II is under-actuated with full finite and lower instantaneous mobilities; and a PRES belonging to class III is partially-actuated with lower finite and instantaneous mobilities. In all three cases, the redundant rigidity in the system is obtained through the use of PALMs. The manipulator shown in Fig. 3d is an example of class I; the manipulator depicted in Fig. 4 is an example of class II; and the manipulator illustrated by Fig. 6e is an example of class III. Manipulators that fall under class I could be used to only improve the stagnant stiffness. However, class II manipulators could enhance both the stagnant and the transitional stiffnesses while maintaining a higher level of actuation efficiency through under-actuation. Similarly class III manipulators can enhance both the stagnant and the transitional stiffnesses; however, they will not be able to attain full finite mobility. Although the third class is classified as a system without full finite mobility, the under-actuation scheme could be used to achieve a lower finite mobility which is still higher than the instantaneous mobility. Although all three classes vary from the mobility and actuation aspects, the same kinematic constraint and kinetostatic formulations can be applied to all three. The second class can be considered the most general of the three, due to the added Table 1 Classes of PRES's and their corresponding static and kinematic characteristics. Class

Kinematic characteristic

Static characteristic

Stagnant stiffness

Transitional stiffness

Finite mobility

Instantaneous mobility (M)

I II III

Non-redundant Constrained Constrained

Redundant Redundant Redundant

Enhanced Enhanced Enhanced

Not enhanced Enhanced Enhanced

=d =d bd

=da bd bd

a

Corresponding to the isostatic state.

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complexity introduced by under-actuation. Therefore, from the analysis point of view the fully-actuated class (class I) and the partially-actuated class (class III) can be treated as special cases of the under-actuated one (class II). Consequently, the focus of the remainder of the paper will be on the design and analysis of the second class. 2.3. Architecture design The architecture criteria define a series of design equations that could be used to generate planar and spatial PRES's capable of under-actuation. By inspecting Eq. (1), it becomes clear that the only viable limb architectures for the PRES are those with d non-redundant DOFs. Some examples are RPR and RRR for the planar case, and SPS, SRS, UPS, UCS, and SCS for the spatial case. When unlocked passive limbs with such architectures are added to an already isostatic manipulator they will have no effect on the mobility potential of the system, i.e. three for a planar manipulator and six for a spatial one. Without loss of generality, for the purposes of demonstration and presentation of the proposed design and analysis, RPR and SPS limb architectures are employed throughout the rest of this work, for planar and spatial cases, respectively. Furthermore, only one DOF is actuated/locked per limb. In the planar case, three RPR limbs would be required to maintain isostaticity. Similarly, in the spatial case, six SPS limbs would be required to do the same. This is generally true for systems with injective joints, i.e. those with limbs that are not sharing any joints. Even then, isostaticity is not guaranteed. An example of a singular configuration for a parallel manipulator that possesses six non-injective SPS limbs is the 6–6 Gough-Stewart platform with similarly oriented base and platform [22]. Therefore, due to the lack of a combinatorial condition that would hold for all architectures and configurations, we must rely on computational methods for detecting such singularities. This is a much larger concern in the spatial case, as we may be able to generate hyperstatic topologies that offer a significant increase in the stagnant stiffness of the robot, but in actuation, i.e. for isostatic topologies, the system may suffer from an abundance of singularities. With that said, we assume that meeting the necessary conditions for isostaticity is sufficient for generating the architecture criteria for the PRES, as the singularity issue is indeed a topology/geometry design problem and not one of mobility or enumeration. Based on that assumption, the addition of any load-bearing limbs to an isostatic topology will generally create a hyperstatic configuration. For a typical spatial PKM with six actuators, the six Cartesian parameters that represent the pose of the platform in the base coordinate frame (Fig. 7), i.e. X = [hx, hy, hz, θx, θy, θz]T, are usually independent of each other. By locking j prismatic joints, we are effectively converting j Cartesian parameters into dependent ones. For an isostatic topology, if we store all the independent and dependent Cartesian parameters into two vectors XI and XD, their ranks will be equal to the number of actuators, and the number of locked passive members, respectively. Therefore, to maintain isostaticity the following will have to be satisfied for any actuating topology: rankðXI Þ þ rankðXD Þ ¼ d:

ð2Þ

Fig. 7. Kinematic model of a PRES.

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If we let the total number of passive limbs be NP = rank(XD), then we will only have one possible isostatic topology. In this case, if rank(XD) = r ≠ 0 we will have r coupled Cartesian parameters that cannot be independently controlled in a finite sense. To gain full finite mobility through under-actuation we need more than one topology. Specifically, we require a minimum of Niso distinct isostatic topologies to achieve full finite mobility. To realize this, we need to have at least (rank(XD) + 1) passive lockable limbs. Hence, we can write  Niso ¼

 rankðXD Þ þ 1 : rankðXD Þ

ð3Þ

If a manipulator can provide a minimum of Niso distinct isostatic topologies, then it can be under-actuated. For example, in the case of the planar manipulator in Fig. 4, rank(XI) = 2 and rank(XD) = 1; this gives Niso = 2. Since we can generate two distinct isostatic topologies with the given architecture, under-actuation is possible in this case. Alternatively, one can express this requirement in terms of the number of lockable passive limbs, NP, as a function of the number of actuated limbs, NA, for any PRES as follows: NP ≥ ðd−NA Þ þ 1∀1≤N A ≤d:

ð4Þ

Eq. (4) is expressed with the underlying assumption that NA = rank(XI), since PRES's are either kinematically non-redundant or constrained. The choice for NA is driven by the tolerance of the application to utilize finite DOFs vs. instantaneous ones. To distinguish between the architectures we adopt a three-digit reference. The first letter will reference the operational dimensions of the robot, i.e. P for planar manipulators and S for spatial ones. The second digit will specify the number of actuated limbs and the third digit will specify the number of lockable passive limbs. For example the PRES depicted in Fig. 4 is P-2-2. This implies that it is a planar manipulator with two actuated limbs and two lockable passive limbs. The spatial PRES in Fig. 6c can be described as S-4-3, since it is a spatial manipulator with four actuated limbs and three lockable passive limbs. Table 2 summarizes the minimal architecture requirements for the planar and the spatial cases. M and NP,min are the instantaneous mobility of the system and the minimum number of lockable passive limbs, respectively. Naturally, the minimum number of limbs to become hyperstatic is four and seven for the planar and the spatial cases, respectively. However, this is only the minimum requirement. With the minimal number of limbs, there will potentially be one degree of static redundancy if the topology is designed properly. However, more passive members could be added to increase the degree of redundancy. For instance, the robot proposed in Fig. 2 has two degrees of static redundancy. 3. Analysis The goal of the proposed design is to improve the stiffness and static characteristics of the manipulator through reconfiguration. In the case of stagnant stiffness this improvement can be obtained through the instant locking of all lockable passive joints. This in turn provides a hyperstatic state which improves the stagnant stiffness of the manipulator. However, in the case of transitional stiffness, the realization of the improved stiffness is not trivial, as additional measures need to be taken to obtain the stiffness improvement. There are several solutions to the under-actuation problem which seeks to attain the final lengths of the passive members in a stepwise alternating fashion. Some of these solutions provide better transitional stiffness than others. In the following sections we present a motion methodology along with the accompanying analyses to obtain improved transitional stiffness. 3.1. Enhanced stiffness A problem could be posed to seek the optimal sequence of locking and unlocking as well as the optimal intermediate poses that can improve the transitional stiffness of the manipulator. Depending on the application, different components of stiffness, or

Table 2 Minimal architecture criteria for PRES's in ℝ3 and ℝ2.

ℝ3

ℝ2

Type

M

NA

NP,min

S-6-1 S-5-2 S-4-3 S-3-4 S-2-5 S-1-6 P-3-1 P-2-2 P-1-3

6 5 4 3 2 1 3 2 1

6 5 4 3 2 1 3 2 1

1 2 3 4 5 6 1 2 3

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the lowest directional stiffness may be used. Alternatively, the strain energy of the system can be used as a suitable measure for the compliance of the manipulator. The strain energy of the PRES can be defined as U¼

1 T δX KC δX 2

ð5Þ

where δX and KC represent the Cartesian displacement vector and the stiffness matrix of the manipulator, respectively. The strain energy of the system is a function of the loading, topology, and pose. For a given loading condition, one can determine the topology and pose that give the lowest strain energy while satisfying the required constraints to realize under-actuation. This problem can be posed as 8 min > 8 Strain Energy > < < Actuation Constraints ðEqualityÞ s:t: Pose Constraints ðInequalityÞ > > : : Stroke Constraints ðInequalityÞ

:

ð6Þ

The constraints can be classified into three groups: actuation, pose, and stroke constraints. The actuation constraints are required for realizing the under-actuation scheme. The total number of actuation constraints for each given intermediate pose would be equal to (d − NA + 1). The pose constraints are imposed to limit certain motions of the manipulator to account for workspace limitations, while the stroke constraints are imposed to account for the physical limits on the active and passive joints. To solve the problem posed by Eq. 6, one must be able to solve the kinematic constraints required for under-actuation and determine the stiffness for a given pose and topology. Therefore, the kinematic and the kinetostatic analyses of the manipulator are required to solve Eq. (6). These will be presented in the following sub-sections, and implemented through the simulations presented in Section 4. 3.2. Kinematic analysis The conventions and coordinates illustrated in Fig. 7 will be used to present the kinematic model for the PRES. As illustrated, a fixed coordinate frame, O − x y z is attached to the base, and a moving coordinate frame P0 − x′ y′ z′ is attached to the platform. Using the defined coordinate frames, the loop closure equations can be set up as follows: ′

b j þ q j ¼ h þ Rp j

ð7Þ

where h = [hx hy hz] T is the vector representing the coordinates of a point on the platform in the fixed coordinate frame; b j is the vector representing the position of the joint belonging to limb j on the base in the fixed coordinate frame; q j is the limb vector in the fixed coordinate frame; p′j represents the position of the joint belonging to limb j on the platform in the moving coordinate frame; and R is the rotation matrix of the moving coordinate frame with respect to the fixed coordinate frame. Expressed in terms of pitch (θx about the x axis), roll (θy about the y axis), and yaw (θz about the z axis), R could be written as follows:
 R ¼ R ðθx ÞR θy Rðθz Þ

ð8Þ

where 2

1 Rðθx Þ ¼ 4 0 0

0 cosðθx Þ sinðθx Þ


 2 cos θy
 6 R θy ¼ 4 0
 − sin θy

3 0 − sinðθx Þ 5 cosðθx Þ 0 1 0


3 sin θy 7 0
5 cos θy

2

3 cosðθz Þ − sinðθz Þ 0 Rðθz Þ ¼ 4 sinðθz Þ cosðθz Þ 0 5: 0 0 1 For a fully-actuated PRES the inverse kinematic relationship could be directly used to control the manipulator. However, an under-actuated or a partially-actuated PRES is a lower mobility system from an instantaneous point of view, in which case one needs to be able express the dependent Cartesian parameters in terms of the independent ones. Essentially, solving the kinematic constraint equations becomes a priori to solving the inverse kinematic problem.

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3.2.1. Kinematic constraints It was shown in Section 2.3 that in order to independently control all six Cartesian variables, we need Niso distinct topologies. In other words, to obtain full finite mobility, we will need (rank(XD) + 1) actuation stages, where each stage corresponds to a change in the pose. For instance, for a planar manipulator with two actuators, we will need a minimum of two passive lockable limbs and have to go through two stages to get to the final pose. The transitions from 0X to 1X and 1X to 2X illustrated by Fig. 5 represent the two aforementioned stages. For a spatial manipulator with four actuated limbs (see Fig. 6c), we would need a minimum of three passive lockable limbs and three stages to independently control any six finite DOFs, and so on. By locking one passive SPS limb we have changed its branch connectivity, Cj , from six to five, and effectively reduced the independent Cartesian parameters of the module form six to five. This is true since for any holonomically constrained kinematic system, we must have NL X j¼1

Cj ¼

Nj X

fi

ð9Þ

i¼1

where NL and Nj represent the total number of limbs and joints in the module, respectively. Therefore, in the case with one locked passive member we need to express one out of six parameters in terms of the other five. The number of locked passive members in the system directly corresponds to the number of dependent Cartesian parameters, which belong to vector XD. For every distinct isostatic topology, we need to express the dependent parameters in terms of the independent ones and the length(s) of the locked limbs. Symbolically we can write this as XD ¼ g ðXI ; Q P Þ

ð10Þ

where Q p is a vector of scalars containing the lengths of the locked passive members and g is the constraint function. The branch kinematic constraint is essentially the parametric equation of a circle and a sphere for the planar and the spatial cases, respectively, imposed by the fixed length of the locked passive member. We can write the constraint equation for the jth locked passive limb in terms of its Euclidian norm as 
2

T   2  2    ′   ′  h−b j  þ 2 h−b j Rp j þ p j  −q j  ¼ 0

ð11Þ

where ||qj|| represents a fixed length corresponding to the jth locked passive member. It is assumed that the positions of all the joints that lie on the platform with respect to the moving frame, as well as the length(s) of the locked passive member(s), are known. Therefore, the only variables to solve would be (hx, hy, hz, θx, θy, θz). In total there will be rank(XD) constraint equations, which implies rank(XD) unknown variables. The choice of the elements of XD is somewhat arbitrary from the finite DOF point of view, since we have full mobility with the under-actuated system as long as joint limits are not reached. Solving the constraint equations is useful in determining the dependent components of the pose; however, a systematic approach is needed to implement the under-actuation. We will refer to this method as under-actuation with virtual alternating constraints (UVAC). 3.2.2. Under-actuation with Virtual Alternating Constraints (UVAC) For any given pose in the Cartesian space of the manipulator, there would generally be a unique vector representing the lengths for all the limbs in the reachable workspace of the manipulator. To obtain that particular pose one needs to realize those lengths. The lengths of the actuated limbs can be simply varied through actuation; however, the lengths of the passive limbs must be realized indirectly since they cannot be actuated. We can achieve those lengths by imposing a virtual constraint in addition to the physical constraints needed to generate isostaticity. The virtual constraint will treat an unlocked passive member as if it were locked when solving the constraint equations. Therefore, any solution that satisfies the actual constraints, i.e. constraints required to maintain isostaticity, must also satisfy the length constraint imposed by the virtual constraint. The virtual constraint would be the desired length of a particular passive member in the final pose. By achieving the lengths in a sequential manner, one can obtain the final lengths of all the passive limbs without directly actuating any of them as demonstrated through the example in Section 2. The lengths of the locked passive members in the final pose must satisfy the constraint function g, which expresses the dependent Cartesian parameters in terms of the independent ones for going from the last intermediate pose to the final one. We will need a minimum of (rank(XD) + 1) constraint equations for every intermediate stage. Out of these, rank(XD) constraint equations are real, as they are imposed by the locked passive member(s) to maintain isostaticity and one is virtual as it corresponds to the one (or possibly more) passive member(s) that is (are) not locked. We can illustrate the two constraints using the intermediate pose from the example shown in Section 2 (Fig. 5ii). In this case, any pose that satisfies these two constraints can be a solution (Fig. 8). In the case depicted by Fig. 8, the norm of the vector that is formed between P2 and P4 can be used as a task variable to solve the constraint equations. Since (rank(XD) + 1) b d there will be an infinite number of solutions to choose from. After obtaining the final length for q4, the roles will be alternated, i.e. q4 will be locked and the final length for q2 will be realized. The final pose

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

B3

B4

Virtual Constraint Imposed by 2 q4

103

q4

Actual Constraint Imposed by 0 q2

q3

P3

P2

P4

q2 q1 B1

P1 B2

Fig. 8. Virtual and actual kinematic constraints used in UVAC.

can be simultaneously realized while obtaining the final length for q2. For a manipulator with more passive members, this alternation will be repeated until all the passive limbs have achieved the desired final lengths. While the kinematic constraint of Eq. (11) is presented for a lockable prismatic joint, the proposed scheme could be easily modified and applied to architectures with lockable R joints; for example a RRR limb for a planar PRES, or a SRS limb for a spatial PRES where the middle R joint is locked to generate the kinematic constraint. Although the discussion of joint placement is beyond the scope of this paper, it is worthwhile to note that in the case where P2 and P4 would be coincident for a P-2-2 robot, there would generally be one solution in the reachable workspace of the manipulator. In the case of the P-1-3 robot with coincident P2 and P4 nodes, generally no solutions for under-actuation would exist within the reachable workspace. As it can be observed, separating the joints associated with the passive limbs on the platform will lead to a wider range of solutions. Therefore, in order to optimize the actuation procedure, care must be taken from a layout design point of view to ensure that the passive limbs are not in very close proximities of each other. The idea of using active joints to drive lockable passive joints into a particular position in a closed-loop system has been previously introduced and investigated [3]. Similarly, the use of inverse kinematics to determine locking values of lockable passive joints for a parallel manipulator has also been investigated [18]. The robot in [3] is kinematically non-redundant, whereas the robot in [18] operates in two modes, one of which is redundantly actuated, and the other one is kinematically redundant. The robots introduced in the current work that utilize the proposed virtual constraints are kinematically constrained. What sets the proposed approach apart from the previous works is the utilization of under-actuation through an alternation of locking and unlocking of passive joints to give full finite mobility to an otherwise kinematically constrained robot. This alternating scheme, which could utilize the same passive member more than once in one actuation cycle, i.e. going from an initial pose to a final pose, enables the under-actuation and gives the UVAC approach its novelty compared to the previous works. 3.3. Kinetostatic analysis To obtain the strain energy of the robot in various topologies one must evaluate the transitional and stagnant stiffnesses/ internal loads for a variety of poses. Using screw theory [23–25], we apply the following relationship to develop the kinetostatic model for all PRES's: •

•

q ¼ JX

ð12Þ •

•

where q is the vector of joint rates and X is the vector of Cartesian rates, i.e. a twist vector containing the linear and angular velocities of a point on the platform, with three and six dimensions for the planar and the spatial cases, respectively. Matrix J, referred to as the Jacobian, is the mapping from the Cartesian velocity vector to the joint velocity vector. From Eq. (12) it could be concluded that δq ¼ JδX

ð13Þ

where δq and δX are the joint and Cartesian infinitesimal displacements, respectively. By applying the principle of kinematic/ static duality, we can write T

F¼ J f

ð14Þ

where f is the vector of actuator/locked passive member forces or torques defined in the joint space, and F is the vector of forces and torques acting on a point on the platform defined in the Cartesian space, i.e. a wrench vector with three and six dimensions

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A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

for the planar and the spatial cases, respectively. The actuator/locked passive member forces or torques can be related to their displacements using Hooke's law: f ¼ K J δq

ð15Þ

where K J ¼ diag ½k1 ; …; kn  ð16Þ is the joint stiffness matrix of the manipulator; n is the number of load-bearing limbs, i.e. actuator/locked passive member, corresponding to a particular topology; k1 through kn represent the stiffness of each load-bearing limb. By substituting Eq. (13) into Eq. (15) we have f ¼ K J JδX:

ð17Þ

Then, by substituting Eq. (15) into Eq. (12) we have T

F ¼ J K J JδX:

ð18Þ

Therefore, by assuming that the platform portion of the manipulator is rigid, the equivalent stiffness matrix of the platform in the Cartesian space becomes T

KC ¼ J K J J:

ð19Þ

The aforementioned Jacobian can be expressed as 2
T ′ h þ Rp1 −b1 6 −1 J ¼ ðdiag½q1 ; …; qn Þnn 6 ⋮ 4
T ′ h þ Rpn −bn


T 3 ′ ′ Rp1  h þ Rp1 −b1 7 7

⋮ T 5 ′ ′ Rpn  h þ Rpn −bn


:

ð20Þ

nm

To obtain a generalized form which could be applied to all isostatic and hyperstatic topologies for all PRES's, we decompose Eqs. (13), (14), and (19) as follows: 

δqA δqP

F¼

h



 ¼ T

JA

KC mm ¼

 JA δX JP T

i

JP h

T

JA

fA fP T

JP

ð21Þ  ð22Þ i mn



KJA 0

0 KJP

 nn



JA JP

 ð23Þ nm

Fig. 9. Schematic representation of the proposed robot (hyperstatic topology); solid lines represent actuated limbs; dashed lines represent locked passive limbs; units are in [m].

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

Isostatic Topology A

Isostatic Topology B

Isostatic Topology C

Isostatic Topology D

105

Fig. 10. Four isostatic topologies generated from the hyperstatic ground topology of the proposed PRES; solid lines represent actuated limbs; square dashed lines represent locked passive limbs; long dashed lines represent unlocked passive limbs.

where m = d and n ≥ d; A and P refer to ‘actuated’ and ‘passive’ limbs, respectively. Regardless of the architecture, the Cartesian stiffness matrix could then be classified into two categories: a) isostatic, where n = m; and b) hyperstatic, where n N m. In the isostatic case, J and KJ are square matrices; hence, the internal loads, fiso, could be simply solved for a given external load vector, F, using −T

f iso m1 ¼ Jmm Fm1 :

ð24Þ

To obtain the internal loads for the hyperstatic case, fhyp, while avoiding the inversion of a non-square matrix, we could substitute −1

ð25Þ

δX ¼ KC F

Table 3 Schematics of limb types/configurations. Schematic

Description Actuated limb Locked passive limb Unlocked passive limb without virtual constraint Unlocked passive limb with virtual constraint

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A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

Table 4 16 actuation sequences used for under-actuation of the S-4-4 PRES. Path

Stage 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Use " " " " " " " " " " " " " " "

Stage 2 A A A A B B B B C C C C D D D D

to get " " " " " " " " " " " " " " "

3qP3

Use " " " " " " " " " " " " " " "

3qP3 3qP4 3qP4 3qP2 3qP2 3qP4 3qP4 3qP1 3qP1 3qP3 3qP3 3qP1 3qP1 3qP2 3qP2

Stage 3 B D D C A C C D A B B D B A A C

To get " " " " " " " " " " " " " " "

3qP4 3qP1 3qP2 3qP3 3qP4 3qP1 3qP3 3qP2 3qP3 3qP2 3qP4 3qP1 3qP2 3qP3 3qP4 3qP1

Use " " " " " " " " " " " " " " "

D B C D C A D C B A D B A B C A

To get " " " " " " " " " " " " " " "

3qP1 3qP2 3qP1 3qP1 3qP1 3qP3 3qP1 3qP1 3qP2 3qP3 3qP1 3qP2 3qP3 3qP2 3qP1 3qP3

& " " " " " " " " " " " " " " "

3qP2 3qP4 3qP3 3qP2 3qP3 3qP4 3qP2 3qP3 3qP4 3qP4 3qP2 3qP4 3qP4 3qP4 3qP3 3qP4

into Eq. (21) to get −1

ð26Þ

δq ¼ JKC F: Finally, by substituting Eq. (26) into f hyp ¼ K J δq

ð27Þ

we obtain −1

f hyp n1 ¼ K J nn Jnm KC

mm Fm1 :

ð28Þ

Using the presented active/passive decomposition, for the isostatic cases where one or more of the passive limbs may become unlocked, the corresponding rows or columns can be simply deleted and the matrices can be reduced to a square form. This approach is used to generate the results for the case-study presented in Section 4. The presented kinetostatic model is applicable to all classes of PRES's for both isostatic and hyperstatic topologies. 4. Case-study: S-4-4 PRES To demonstrate the effectiveness of the design and analysis presented earlier, a case-study for a spatial PRES with four actuated and four passive limbs is presented herein. This robot, as illustrated by Fig. 2, has eight SPS limbs, four of which are actuated and the other four are passive. This PRES, which is categorized as S-4-4, has a lower instantaneous mobility of four and a finite mobility of six. For the purpose of the case-study, the topology can be schematically illustrated as shown in Fig. 9. By using the hyperstatic ground topology (Fig. 9) one can generate six potential isostatic topologies. However, two of these will be nearly degenerate. Those are the topologies with passive limbs 1, 4 or 2, 3 unlocked simultaneously. Fig. 10 summarizes the schematics of the four remaining isostatic topologies. The active limbs remain load-bearing in all configurations/topologies. However, the passive members can take one of two physical forms: locked and unlocked. Additionally, when unlocked the passive limbs can be subjected to a virtual constraint. The schematic representations of all four types/configurations of the limbs are summarized in Table 3. These schematics are used in the presented simulations. Given the number of topologies, two intermediate poses are required to realize under-actuation. From a motion realization perspective, one can start with any given isostatic topology in the first stage, which will have four actuators and two locked passive members. However, as established earlier, the locking sequence and choice of the intermediate poses will affect the Table 5 Final poses associated with each case. Case

Pose (3X)

i ii iii iv v vi

[0.5, 0, 1, 0, 0, 0]T [0, 0.5, 1, 0, 0, 0]T [0, 0, 1.5, 0, 0, 0]T [0, 0, 1, 45, 0, 0]T [0, 0, 1, 0, 45, 0]T [0, 0, 1, 0, 0, 45]T

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

Pose: 0X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0X Topology: A

Pose: 3X Topology: D

Step 2

Step 6

107

Pose: 1X Topology: A

Step 3

Pose: 2X Topology: D

Step 5

Pose: 1X Topology: B

Step 4

Pose: 2X Topology: B

Fig. 11. Stiffest path for case i is determined to be path 1.

Pose: 0X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0X Topology: B

Pose: 3X Topology: C

Step 2

Step 6

Pose: 1X Topology: B

Step 3

Pose: 2X Topology: C

Step 5

Pose: 1X Topology: A

Step 4

Pose: 2X Topology: A

Fig. 12. Stiffest path for case ii is determined to be path 5.

Pose: 0X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0X Topology: B

Pose: 3X Topology: D

Step 2

Step 6

Pose: 1X Topology: B

Step 3

Pose: 2X Topology: D

Step 5

Fig. 13. Stiffest path for case iii is determined to be path 7.

Pose: 1X Topology: C

Pose: 2X Topology: C

Step 4

108

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

Pose: 0 X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0 X Topology: A

Pose: 3X Topology: B

Step 2

Step 6

Pose: 1X Topology: A

Step 3

Pose: 2X Topology: B

Step 5

Pose: 1X Topology: D

Step 4

Pose: 2X Topology: D

Fig. 14. Stiffest path for case iv is determined to be path 2.

Pose: 0 X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0 X Topology: C

Pose: 3X Topology: D

Step 2

Step 6

Pose: 1X Topology: C

Step 3

Pose: 2 X Topology: D

Step 5

Pose: 1X Topology: B

Step 4

Pose: 2X Topology: B

Fig. 15. Stiffest path for case v is determined to be path 11.

Pose: 0 X Topology: Hyperstatic

Pose: 3X Topology: Hyperstatic

Step 1

Step 7

Pose: 0 X Topology: D

Pose: 3X Topology: A

Step 2

Step 6

Pose: 1X Topology: D

Step 3

Pose: 2X Topology: A

Step 5

Fig. 16. Stiffest path for case vi is determined to be path 16.

Pose: 1X Topology: C

Pose: 2X Topology: C

Step 4

A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

2.0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0 0

1

2

0.0 0

3

1

2

3

0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0 2

3

2

3

0.0 0

3

2 Pose (X)

2.0

1

1

Pose (X)

Pose (X)

0

109

1

2

3

0

1

Pose (X)

Pose (X)

Pose (X)

Fig. 17. Strain energy ratios between hyperstatic and isostatic topologies through under-actuation for cases i through vi.

transitional stiffness. By taking into account all the possible combinations and sequences, one can have 16 possible paths to get from an initial pose (0X) to a final one (3X), where the left subscript notation refers to the stage of the actuation. Subscript 0 refers to the initial pose before the actuation starts; subscripts 1 and 2 refer to the two intermediate poses; subscript 3 refers to the final pose. All 16 paths are summarized in Table 4. For example, with reference to Table 4, 3qP3 refers to the length of the third passive member in the third and the final pose. Each path presents a complete motion summary of how to get from some initial pose to some final pose. For instance if one uses path 1, isostatic topology A must be initially used to get the final desired length for passive member 3, i.e. 3qP3. Then, while keeping 3qP3 locked for the remainder of the actuation, isostatic topology B must be used to get 3qP4. With 3qP3 and 3qP4 locked one can only use isostatic topology D to obtain 3qP1 and 3qP2. During the last stage, the actuators will also achieve their desired final lengths. With all the passive members and the actuators at their final lengths, the final pose is simply realized. In addition to the choice for the sequence, the Cartesian parameters that express the intermediate poses are also included in the optimization problem for stiffness enhancement. Using this approach, we find the most optimal motions to move the platform along/about every one of the six Cartesian DOFs under a torsional moment, i.e. a positive moment about the z-axis. The home pose, 0X, is represented by [0, 0, 1, 0, 0, 0]T. All the translational components are in m, and all the rotational components are in deg; the directions are defined based on the coordinate frame illustrated in Fig. 9. Table 5 summarizes the final pose associated with each of the six cases presented here. All limbs are assumed to have a stiffness equivalent to (keq/q), where keq represents a stiffness constant, and q represents the length of the limb. Figs. 11 through 16 illustrate the results for the optimal motion control for enhanced transitional stiffness. Analyses presented for the kinematics (Section 3.2) and the kinetostatics (Section 3.3) of the PRES are used to obtain the simulations. There are a total of seven steps in each simulation, where each step represents a change in either the topology or the pose. A step is not to be confused with the previously defied stage, which only represents a change in the pose.

2.0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0 0

1

2

0.0 0

3

1

2

3

0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0 1

2 Pose (X)

3

2

3

2

3

Pose (X)

2.0

0

1

Pose (X)

Pose (X)

0.0 0

1

2 Pose (X)

3

0

1 Pose (X)

Fig. 18. Ratios between the largest magnitude of internal load experienced by a limb when hyperstatic and isostatic through under-actuation for cases i through vi.

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A. Moosavian, F.(J.) Xi / Mechanism and Machine Theory 77 (2014) 92–110

To validate the results, the ratio between the strain energy of the transitionally stiff isostatic topology in each pose and that of the hyperstatic topology in the same pose have been plotted as depicted in Fig. 17. Each plot captures eight data points, that is, two per pose. Each data point represents a snapshot in the actuation process where the eight points correspond to the eight instances depicted in each of Figs. 11 through 16. As it can be observed from the plots (Fig. 17), the ratio (Uhyp/Uiso) is very close to one for each case throughout the actuation. This is an indication of how close the transitional stiffness is to the highest attainable stagnant stiffness for that given pose. An additional benefit of using the path with the lowest strain energy is the reduction in the internal loads, since alternatively the strain energy can be expressed as U¼

1 T F δX: 2

ð29Þ

A comparison of the internal loads for each of the optimal intermediate poses and that of the hyperstatic topology validates this. Fig. 18 represents max(|fhyp|)/max(|fiso|) for each of the six cases. The results from Fig. 18 show the effectiveness of the proposed strain energy-based approach for generating an optimal path for enhanced stiffness. 5. Conclusion A new type of parallel robots that can enhance their static and stiffness characteristics by varying their topology while maintaining actuation efficiency has been presented in this paper. These robots can attain static redundancy without actuation redundancy. In addition, a class of these VTMs can deliver full finite mobility through under-actuation using a proposed scheme called UVAC. Design and analysis of these systems have been addressed in this paper, including the architecture design, the kinematic analysis, and the kinetostatic analysis. To display the effectiveness of the proposed design and analysis, a case-study was conducted using a spatial PRES. It was shown that the variable topology design can enhance the stiffness of the robot without the use of actuation redundancy. The static redundancy in the design of these reconfigurable robots improves their fault tolerance over conventional non-redundant ones, making them desirable for autonomous and aerospace applications. Additionally, the proposed robots can be reconfigured to change their motion characteristics to increase their workspace over conventional lower mobility systems with permanently constraining limbs. As an extension of this work, the future efforts should focus on the topological design and the joint placement for the proposed reconfigurable robots. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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