Stiffness analysis of a metamorphic parallel mechanism with three configurations

Mechanism and Machine Theory 142 (2019) 103595

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Research paper

Stiffness analysis of a metamorphic parallel mechanism with three configurations Hai-bo Tian∗, Hong-wei Ma, Jing Xia, Kun Ma, Zi-zhuo Li College of Mechanical Engineering, Xi’an University of Science and Technology, Xi’an 710054, China

a r t i c l e

i n f o

Article history: Received 13 February 2019 Revised 4 June 2019 Accepted 11 August 2019

Keywords: Metamorphic parallel mechanism Rotatable-axis revolute joint Configuration transformation Workspace Stiffness

a b s t r a c t Compared with series mechanisms, a parallel mechanism has the merits of high stiffness, fast response, and centralised layout of electric cylinders, which is more suitable for the mechanical leg of a rescue robot. However, it is not flexible enough to improve the environment adaptability of the robot. Adding the abilities of reconfiguration and mobility change to general parallel mechanisms, a novel metamorphic parallel mechanism for robotic legs is proposed based on an innovative rotatable-axis revolute joint. The inverse kinematic model of the new mechanism is established and the constraint conditions and ranges of the key parameters are explored, as well as obtaining the cloud picture of its workspace. The stiffness model is formulated with the consideration of the deformation of the main components caused by the actuation and constraints, utilising the screw theory. The stiffness matrices of the home positions of configurations I and III are then acquired and the stiffness distributions in the sub-workspace around the desired trajectory are evaluated, which are expected to lay a good theoretical foundation for the application of this novel mechanism. © 2019 Elsevier Ltd. All rights reserved.

Introduction There are two major categories of mechanical legs of mobile robots, from the point of view of their mechanism types: serial mechanical legs and parallel mechanical legs. Compared with series mechanical legs, a parallel mechanical leg has the advantages of high stiffness, fast response speed, high bearing capacity, and high modularisation, as well as centralised layout of electric cylinders. These enable the realisation of dustproof, waterproof, and explosion-proof designs, making the parallel mechanism a more reasonable choice for the development of a rescue robot’s legs [1]. In the 1990s, a six-degree-offreedom (6-DOF) parallel mechanism was first used in the Para-walker robot by Shigeo Hirose [2]. WL-16, a biped robot with parallel legs, was developed by Sugahara et al. [3]. Cheng [4], Pan [5], Guo [6], Rushworth [7], and Russo et al. [8]. studied the performance of a certain type of parallel mechanical legs. Nevertheless, these mechanical legs are not suitable choices for the mechanical leg of a rescue robot. A rescue robot often encounters various complex conditions in field environment, therefore a mechanical leg of it must have the capability of both walking on the flat ground and assisting the robot to overcome high obstacles. However the aforementioned mechanical legs are not flexible enough to meet both the needs at the same time. Pisla et al. [9]. proposed a 5-DOF hybrid robot for laparoscopic surgery, and hybrid mechanisms were introduced into bionic robots by Tian et al. [10]. and Gao et al. [11]. The flexibility of these mechanisms is better than that of parallel mechanisms, but their stiffness and bearing capacity are weakened; moreover, the dustproof, waterproof, and ∗

Corresponding author. E-mail address: [email protected] (H.-b. Tian).

https://doi.org/10.1016/j.mechmachtheory.2019.103595 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

explosion-proof designs are more difficult to implement. In order to cope with the above problems, a metamorphic parallel mechanism with three configurations is brought forward in this paper. A metamorphic mechanism is a type of mechanism that can restructure and reconstruct itself to meet different environmental or functional needs [12,13]. A metamorphic parallel mechanism is the combination of a metamorphic mechanism and a parallel mechanism, which has better adaptability than a general parallel mechanism. A 4-x Px Rx Rx Ry RN Parallel mechanism with variable mobility was proposed by Li et al. [14]. Using a metamorphic kinematic joint, a multi-loop metamorphic mechanism which can perform a topological phase change was developed by Zhang et al. [15]. Ding et al. [16]. introduced a cellular parallel mechanism on the basis of the topology and kinematics analysis of a colour-changing ball. Gan et al. introduced several new metamorphic parallel mechanisms based on two invented joints: the reconfigurable revolute (rR) joint [17] and the reconfigurable Hook (rT) joint [18-20]. Chang et al. [21]. analysed the kinematics of a parallel spherical metamorphic mechanism applied to the design of spherical bionic joints. The stiffness of a mechanism refers to the end-effector displacement caused by the deformation of its parts under the action of external forces, which plays a decisive role in the motion accuracy of the mechanism under loads. In the past 30 years, the stiffness problem has been a hot topic in the research of parallel mechanisms. To date, there exist four main categories of stiffness analysis approaches for parallel mechanisms [22]: the finite element approach, structural matrix approach, strain energy approach, and virtual work principle approach. Compared with other approaches, the finite element approach is more accurate. However, the finite element model needs to be re-meshed when the parallel mechanism configuration changes, which renders the solving process elaborate and time-consuming [23]. The structural matrix approach is not suitable for stiffness analysis, owing to its tedious computation. The strain energy approach has been widely used in recent years. Rezaei [24] analysed the stiffness of a parallel manipulator using this approach, and Yan et al. [25]. analysed the total deformation of a general parallelogram-type parallel manipulator. From these works, it can be seen that this approach is of clear physical meaning, high accuracy, and low computational expense. However, the solving process of the wrench compliant module Jacobian matrix is troublesome. At present, the virtual work principle approach is used more commonly for the stiffness analysis of various parallel mechanisms. Gosselin [26] first investigated the stiffness of the Stewart mechanism; however, his model only included the compliance of the driving parts. Huang [27] studied the model of a tripod-based parallel mechanism with flexible joints and a leadscrew-nut mechanism. Dai et al. [28]. proposed the compliance matrices of a platform with three leaf-spring legs. Majou et al. [29]. analysed the influence of geometrical parameters on the stiffness of a mechanism with compliant limbs. Xu [30] studied the stiffness of a 3-PRC mechanism considering the compliances of the driving systems and limbs. Hu [31] investigated the kinetostatic model of a 2-RPU+UPR manipulator, which covers the stiffness of its actuators and constraints. Sun et al. [32]. analysed the stiffness of an over-constrained rotational parallel mechanism with flexible limbs. Klimchik et al. [33]. analysed the stiffness of an over-constrained parallel mechanism, which takes the compliances of the links and joints into account. Motivated by developing a new mechanical leg structure to satisfy the needs of rescue robots, a novel metamorphic parallel mechanism is proposed based on an innovative rotatable-axis revolute joint. The focus of this study is on the workspace and stiffness analysis of the proposed mechanism, and the remaining sections are organised as follows. After an introduction of the mechanism, its inverse kinematic model is analysed and the cloud picture of its workspace is demonstrated in Section 2. Next, in Section 3, the stiffness model is formulated with the consideration of the deformation of the main components caused by the actuation and constraints. In Section 4, the stiffness matrices of the home positions of configurations I and III are obtained and the stiffness distributions in the sub-workspace around the desired trajectory is evaluated. Finally, the conclusions of the above work are summarised in Section 5. 1. Description of a novel metamorphic parallel mechanism 1.1. Structure of the mechanism The solid model of the proposed mechanism is illustrated in Fig. 1. It consists of a mobile platform, a fixed platform, and three supporting limbs. Two of the limbs connect the mobile platform with the fixed platform by a prismatic joint (P) actuated by a linear actuator, an innovative rotatable-axis revolute joint (rR) which axis is rotatable, and a spherical joint (S) in turn. The other limb is composed of an active prismatic joint (P), a revolute joint (R), a lockable prismatic joint ((P)) and a spherical joint (S) successively. Thus, the type of the mechanism is 2-PrRS+PR(P)S. The linear actuators of the three limbs are installed on the fixed platform to reduce the motion inertia of the limbs. The linear actuator pushing directions of the two PrRS linkages are opposite and the actuator pushing direction of the PR(P)S linkage is perpendicular to them, all of which are parallel to the plane of the fixed platform. The centre points of the three spherical joints form an equilateral triangle. The position and attitude transformation of the mobile platform is achievable by co-controlling the expansion or contraction of the three actuators. The reconfigurability of this mechanism stems from the configuration change of the rR joint. The axis of a traditional revolute joint can no longer rotate after installed. The direction of the rotation axis of the rR joint in this metamorphic parallel mechanism can be altered yet. As shown in Fig. 2, the rR joint is mainly composed of an inner support body, a rotating shaft and an outer mount. One side of the inner support body is connected with the output end of the P pair, that is, the output shaft of the linear actuator, by a bearing. The other side of the inner support body is U-shaped and has a

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

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Fig. 1. 2-PrRS+PR(P)S metamorphic parallel mechanism.

Fig. 2. Solid model of the rotatable-axis revolute joint.

Fig. 3. Structure of the mounting body.

through hole that the rotating shaft passes through to coupled with the master link. The outer mount that two grooves is inside, is seated outside of on the inner support body. Every groove has two straight sections and one helix section (see Fig. 3). The rotating shaft can change its position by sliding freely along the grooves so as to alter the direction of the rotation axis of the rR joint. In order to prevent any interference with the master link, an open slot is provided in the outer mount. For ease of installation, the outer mount is divided into a mounting body, a left baffle and a right baffle. The left and right baffles are connected to the mounting body by screws. The rR joint can have various configurations as its axis can have various configurations when the rotating shaft sliding along the grooves, leading to some special and useful configuration phases of the mechanism to achieve variable mobility. The following study will reveal the configurations of the mechanism in accordance with the three phases of motion of the rR pair.

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Fig. 4. Schematic representation of 2-PrRS+PR(P)S metamorphic parallel mechanism.

1.2. Process of the configuration transformations As described in Fig. 4, the reference coordinate system O-XYZ is set up at the midpoint of the bottom edge of the fixed platform triangle, which is the intersection point of three actuator axes. The Y-axis points to the vertex of the fixed triangle base, the Z-axis points upward along the vertical direction, and the X-direction is generated by the right-hand screw rule. A moving coordinate system o1 -x1 y1 z1 is set up on the point o1 , which is the centre point of the mobile platform triangle. In the original position, the x1 , y1 , and z1 axes are coincident with the X, Y, and Z axes, respectively. For the sake of convenience, Si (i = 1, 2, 3) denotes the centre points of the ith S points and Bi are the centre points of ith R joints. According to the shape of the grooves, the motion phase of the rR joint can be divided into one axis-translation phase, one axis -rotation phase, and one axis-translation phase. Correspondingly, the stroke of the active prismatic joint (P joint) in the two PrRS limbs is divided into the initial, middle, and end segments. Therefore, 9 configurations of the mechanism could be obtained. However, only three configurations are available due to the structural constraints. When the strokes of the active (P) joints in the two PrRS limbs are at the initial segment, the (P) joint of the PR (P) S limb is locked, and the axes of the rR joints in the two PrRS limbs cannot be rotated. The position and posture of the mobile platform is achieved by co-controlling the extension length of the three actuator output shafts. In this state, the metamorphic parallel mechanism is in the first stage (configuration I), the type of which is 3-PRS. According to previous literatures[34], it has 3 degrees of freedom(DOF), that are the rotational DOF around the X-axis, the rotational DOF around the Y-axis and translational DOF along the Z-axis. When the strokes of the active (P) joints in the two PrRS limbs are at the middle segment, the axes of the rR joints in the two PrRS limbs is rotated with the expansion of the active P joint, which drives the corresponding upper link to rotate, and the locked (P) joint in the PR(P)S limb is unlocked under the tension from the mobile platform, causing the limb to elongate and rotate. In this state, the mechanism is in the second stage (configuration II), which is a 2-PHS+PRPS parallel mechanism, where H represents a helical joint. The degree of freedom of the configuration II can be calculated as follows using the spiral theory. The twist system of the PHS limb can be given as

⎧ $11 = (0 ⎪ ⎪ ⎨$12 = (0 $13 = (1 ⎪ ⎪ ⎩$14 = (0 $15 = (0

0 1 0 1 0

0; 0; 0; 0; 1;

1 0 0 −b1 0

0) a1 ) 0) c1 ) 0)

0 h b1 0 −c1

(1)

where h represents the pitch of the helical pair, a1 ,b1 ,c1 are common notations which values do not affect the later analysis. Then the reciprocal screw to (1) is

$11 = (0(c1 −a1 )h; −b1 (c1 −a1 ) − hc1

c1 (c1 −a1 ))

(2)

For the PRPS limb, the driving joint has no output, therefore the limb changes to a RPS limb, and twist system of it can be given as

⎧ $11 = (0 ⎪ ⎪$ = (0 ⎨ 12 $13 = (1 ⎪ ⎪ ⎩$14 = (0 $15 = (0

1 0 0 1 0

0; 0; 0; 0; 1;

0 ( c1 − a1 ) 0 −b1 0

0 0 b1 0 −c1

a1 ) b1 ) 0) c1 ) 0)

(3)

The reciprocal screw to (3) is

$11 = (0

1

0;

−b1

0

c1 )

(4)

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

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(2) and (4), respectively gives a constraint force acting along a line passing through the spherical joint centre, which constrain 3 DOF of the mobile platform. Therefore, the mobile platform has 3 DOF, including the rotational DOF around the X,Y-axis and translational DOF along the Z-axis. The DOF of the mechanism can be verified by G–K criterion in Eq. (5). Configuration II is underactuated as it has 3 DOF but only two driving joint.

M = d ( n − g − 1 )+

g 

fi +ν = 6 × (8 − 9 − 1 ) + 5 × 3 = 3

(5)

1

When the strokes of the active (P) joints in the two PrRS limbs are at the end segment, the axes of the rR joints in the two PrRS limbs cannot be rotated any longer with the restriction of their construction, and the (P) joint of the PR (P) S limb is locked again. The position and posture of the mobile platform in another space could be achieved by co-controlling the extension length of the three actuator output shafts, and the mechanism is in the third stage (configuration III), the type of which is 3-PRS as well. It has 3 degrees of freedom (DOF), that are the rotational DOF around the X-axis, the rotational DOF around the Z-axis and translational DOF along the Y-axis. It can be seen that the metamorphic parallel mechanism has three configurations. These configurations can be changed in sequence to achieve a large scale of movement of the mobile platform and to change the appearance of the working space of the end link, so as to meet the movement requirements under different conditions. The most apparent feature of the mechanism is that the driving mode remains unchanged under these configuration transformation processes, which brings great convenience to the design of the control system. 2. Inverse kinematic and workspace analysis 2.1. Inverse kinematic model Kinematics deals with the motion of the parallel mechanism as restricted by geometric constraints. The following inverse kinematic analysis consists of obtaining the input parameters of the driving system based on the expected position and posture of the mobile platform. The vector of the link Bi Si can be expressed as follows:

Bi Si = OSi − OBi = ROO1 · O1 Si − OBi

(6)

O1 S

0 R01

where is the matrix transformation from frame i is the vector of the centre point of the ith S joint in o1 -x1 y1 z1 , o1 -x1 y1 z1 to frame O-XYZ. The inverse kinematic model of configuration I is built first. The DOF number of configuration I is 3, and the Euler angles 0 can be denoted as α and β determine the pose of the mobile platform [35]. The orientation matrix R01



ROO1

= Rot(Y, β ) · Rot(X, α ) =

cos β 0 − sin β

sin α sin β cos α sin α cos β

cos α sin β − sin α cos α cos β



(7)

As mentioned above, for configuration I, the rR joint can be regarded as a revolute joint. Therefore, the unit vector μi of revolute joint Bi is perpendicular to the Bi Si vector. The following equation can be obtained:

μi · Bi Si = 0

( i = 1, 2, 3 )

(8)

Note that the length of the link Bi Si is constant, which leads to

|Bi Si | = li

( i = 1, 2, 3 )

(9)

Based on Eqs. (6)–(9), the inverse kinematic problem can be solved. To determine the selection of the ‘±’ sign in the equations, we substitute the initial values α = 0, β = 0 into the equation for a trial calculation. Thus, the inverse kinematic solution can be written as

⎧

2

√ ⎪ 2 ⎪ ⎪ D = l1 − 2a sin β − 63 a sin α cos β + zc − − 2a cos β − 1 ⎪ ⎪ ⎪ ⎨

2

√ 2 D2 = l1 − − 2a sin β − 63 a sin α cos β + zc + 2a cos β − ⎪ ⎪ ⎪

√ 2 √ ⎪ ⎪ ⎪ ⎩D3 = l 2 − 33 a cos β sin α + zc + 23 a cos α

√

3 a 2

√

3 a 2

sin α sin β sin α sin β





(10)

where l1 is the length of the links B1 S1 and B2 S2 , l is the length of the link B3 S3 in configuration I, a is the distance between any two centre points of the three spherical joints, zc is the Z-direction displacement of the centre point of the mobile platform relative to the reference coordinate system O-XYZ, Di (i = 1,2,3) are the distance between the end of the output shaft of the actuator and the origin O of the reference coordinate system. By subtracting from Di (i = 1,2,3) the distance between the end of the output shaft when the actuator output is zero and the origin O, the actuation parameters can be obtained.

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H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595 Table 1 Architectural parameters of the metamorphic parallel mechanism. Parameter

Value (mm)

Parameter

Value (mm)

Parameter

Value (mm)

a l1 l L dc

280 280 260 357 40

b Sd 0 Sd Sd 1

180 250 150 100

Sd 2 Sd 3

20 30 15°≤θ ≤165° 15°≤δ ≤165°

θ δ

Similarly, the inverse kinematic model of configuration III can be obtained

⎧

⎪ ⎪ ⎪ D1 = l12 − − 2a sin γ − ⎪ ⎪ ⎪ ⎨ D =

2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩D3 =

a 2

cos γ +

√

3 a cos 3

√ 3 a cos 2

√ 3 a cos 6

α  cos γ + yc

α  sin γ + l12 −

√

α  cos γ + yc +

L2 −

a 2

2

+

a 2

sin γ −

3 a sin 2

α

cos γ −

√

3 a cos 6

√

3 a cos 2

α  sin γ 2

α  cos γ + yc

(11)

2

where L is the length of the link B3 S3 in configuration III,α ’ is the angle of the mobile platform rotating around the X-axis, γ is the angle of the platform rotating around the Z-axis, yc is the Y-direction displacement of the centre point of the mobile platform relative to the reference coordinate system, and the other symbols have the same meanings as before. It should be noted that configuration II is the bond that realises the transformation between configurations I and III, the workspace of which is very small and cannot be used as the motion range of the mechanism; therefore, the kinematics and stiffness model of configuration II is not explored in this study. 2.2. Geometric constraints of the mechanism To obtain the mechanism’s reachable workspace, the reachable extent of its drives and joints should be determined. (1) Strokes of active P-joints. The active joints in all configurations of the mechanism are the three linear electric cylinders. From the point of view of the workspace, the maximum lengths of the linear actuators are the main constraints. (2) Rotational angle range of S joint. Every S joint in the structure has the same maximum and minimum rotational angles. Suppose the direction vector of the base of the ith S joint is represented by Sc i , the direction vector of the link Bi Si is represented by li in the reference coordinate system, and the rotation angle of the ith S joint is represented by θ i . Then, the constraint is represented as

θmin ≤ θi = arccos

li · Sci

|li ||Sci |

≤ θmax

(12)

(3) Rotational angle range of R joint. Similar to the S joints, every R joint has the same maximum and minimum rotational angles. Suppose the direction vector of the base of the ith electric cylinder is represented by bi , the direction vector of the link Bi Si is represented by li in the reference coordinate system, and the rotation angle of the ith R joint is represented by δi . Then, the constraint is represented as

δmin ≤ δi = arccos

li · bi

|li ||bi |

≤ δmax

(13)

(4) Interferences between limbs. Suppose the cross-sectional shape of each link is circular with a maximum diameter d and the shortest distance between the centre lines of every two links is expressed by di , then, the condition of avoiding interference of every two links is

di > d

(i =1, 2, 3 )

(14)

2.3. Workspace analysis The architectural parameters of the metamorphic parallel mechanism with Configuration Ⅰ and Ⅲ are shown in Table 1. where, b is the length of the end link, Sd 0 is the length of the linear actuator when its stroke is 0, Sd is the maximum stroke of all the linear actuator, Sdi (i = 1,2,3) is the value of the initial, middle, and end segments of the linear actuator stroke respectively, dc is the distance between the end of the linear actuator of PR(P)S limb and the centre line of the linear actuator of PrRS limb, and the other symbols have the same meanings as before. (Fig. 5.) If the endpoint of the end link is specified as the reference point of the metamorphic parallel mechanism, the workspace can be analysed by the Monte Carlo method. The random value N = 30 0,0 0 0 was taken to obtain the workspace shape, as shown in Fig. 6a. The red point area represents the workspace of configuration I, and the blue point area represents the

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Fig. 5. Three configurations of 2-PrRS+PR(P)S metamorphic parallel mechanism.

workspace of configuration Ⅲ. For comparison, the workspace of a parallel mechanism with the same parameters but no metamorphic characteristics was analysed by the same method, as shown in Fig. 6b. The random value was still taken as N = 30 0,0 0 0. As shown in Fig. 6a and Fig. 6b, there are significant differences between the workspace shape of a metamorphic parallel mechanism and that of a general parallel mechanism without any metamorphic characteristics. The workspace of the metamorphic parallel mechanism can be regarded as the integration of that of two parallel mechanisms. Although the workspace shape is irregular, it can meet two different demands at different times, as long as the design of its motion trajectory is reasonable. If the metamorphic parallel mechanism is applied to the design of mechanical legs, the advantages would be brought fully into play. Fig. 7 shows a wheel-tracked robot using the abovementioned mechanism as its mechanical leg, which can realise walking on the flat ground, steering or maintaining stability using the rotational DOF around the Y-axis and translational DOF along the Z-axis of configuration I, as shown in Fig. 7a. In addition, the robot can use the rotational DOF around the X-axis and translational DOF along the Z-axis of configuration Ⅲ to assist itself to overcome obstacles, as shown in Fig. 7b. It should be noted that the mechanical leg shown in Fig. 7 changes the end link into a wheel to improve the robot’s trafficability. 3. Stiffness model As mentioned above, the basic functions required of the mechanical leg are to be able to walk and steer on a flat ground, and to assist the robot to overcome obstacles. Therefore, the mobile platform needs to realize a wide range of rotation, and can still move after rotating to a certain configuration, for accomplishing some fine adjustment of the posture. In the following research, the stiffness model is formulated with the consideration of the deformation of the main components caused by the actuation and constraints, then the sub-workspaces are determined after the motion trajectories of the endpoint of the mechanism are given according to the requirements. Based on them, the stiffness distributions of the sub-workspaces is unfolded.

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Fig. 6. Workspace shape comparison.

Fig. 7. Motion modes of a wheel-tracked robot with metamorphic parallel leg.

3.1. Basic assumptions The stiffness model of the entire mechanism is built by a static analysis. First, some basic assumptions are made. (1) The effects of gravity are negligible. (2) The deformations of the components in the mechanism meet the requirements of the linear superposition theory. (3) The rigidities of the linear actuators, mobile platform, and fixed platform are much higher than those of the other components and therefore can be considered as infinite. (4) All the joints are considered ‘ideal’, which means they have no clearance, no friction, and no deformation. (5) The end link is considered as a rigid body, the stiffness of which is not considered below. 3.2. Derivation of the stiffness matrix As seen from the structural features of a limited-DOF parallel mechanism, each limb not only provides an active force but also transfers a constraint force to the mobile platform in order to remove some undesired DOFs. Consequently, both should be taken into consideration. According to the description above, the distribution of external forces acting on the mobile platform is closely related to the position and posture of the mechanical leg and the contact situation between the end of the leg and the ground at a certain instant. The external forces (or torques) applied to the centre point of the mobile platform are equivalent to three forces: Fx , Fy , and Fz , and three torques: Mx , My , and Mz , which can be denoted as

τ= F

M

T



= Fx

Fy

Fz

Mx

My

Mz

T

(15)

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

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The displacement and virtual displacement of the mobile platform can be denoted as



r= x

z

α

β

δy

δz

δα

y

δr = δx

γ

T

δβ

(16a)

δγ

T

(16b)

According to the literature [34], in configurations I and III, the reciprocal screw of every PRS limb acts on the centre of its S joint and is perpendicular to the plane of the limb. In addition, the active joint of each PRS limb also applies a driving screw along the link Bi Si to the mobile platform. Based on the virtual work principle, Eq. (11) can be generated as follows:

τ T δ r = τaT δ ra + τcT δ rc

(17)

where τ a represents the reaction forces/torques of the actuations, τ c represents the reaction forces/torques of the constraints, δ ra represents the virtual displacements of the actuations, and δ rc represents the virtual displacements of the constraints. δ ra and δ rc can be derived as

δ ra = Ja δ r

(18a)

δ rc = Jc δ r

(18b)

δ r is the virtual displacement of the mobile platform, Ja is the Jacobian matrix of the actuations, and Jc is the Jacobian matrix of the constraints. Then, by substituting Eqs. (12a) and (12b) into Eq. (11), one can obtain

τ T δ r = τaT Ja δ r + τcT Jc δ r

(19)

Because δ r is arbitrary, we can conclude that

τ = JaT τa + JcT τc

(20)

In order to analyse the small change in the generalised displacement of the mobile platform when there is a small change in the external forces, the time derivative of both sides of Eq. (14) is calculated

∂τ ∂ JaT ∂τ ∂ JT ∂τ = τa + JaT a + c τc + JcT c ∂t ∂t ∂t ∂t ∂t

(21)

According to Hooke’s theorem, one can obtain that

τ = K · δr

(22a)

τa = Ka · δ ra

(22b)

τc = Kc · δ rc

(22c)

where K represents the overall stiffness matrix of the entire mechanism; Ka and Kc represent the stiffness matrices of the actuations and constraints, respectively. Ka = diag[Ka1 Ka2 Ka3 ] and Kc = diag[Kc1 Kc2 Kc3 ]. Here, Kai and Kci (i = 1,2,3) are the stiffnesses of the actuations and constraints of the ith limb, respectively. The time derivative of both sides of the above equations can be calculated

∂τ ∂ K ∂ (δ r ) = δr + K ∂t ∂t ∂t ∂ τa ∂ Ka ∂ ( δ ra ) = δ r + Ka ∂t ∂t a ∂t ∂ τc ∂ Kc ∂ ( δ rc ) = δ r + Kc ∂t ∂t c ∂t

(23a) (23b) (23c)

Substituting Eqs. (17a), (17b), and (17c) into Eq. (15) yields

∂K ∂ (δ r ) ∂ JaT ∂K ∂ (δ ra ) ∂ JcT ∂K ∂ ( δ rc ) δr + K = τ + JT a δr + JT K + τ + JT c δr + JT K ∂t ∂t ∂t a a ∂t a a a ∂t ∂t c c ∂t c c c ∂t

(24)

The time derivative of both sides of Eqs. (12a) and (12b) can be calculated

∂ (δ ra ) ∂ Ja ∂ (δ r ) = δ r + Ja ∂t ∂t ∂t ∂ (δ rc ) ∂ Jc ∂ (δ r ) = δ r + Jc ∂t ∂t ∂t

(25a) (25b)

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H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

Substituting Eqs. (19a) and (19b) into Eq. (18) yields

∂K ∂ (δ r ) δr + K = ∂t ∂t



 ∂ Ja ∂ JcT ∂ Jc ∂ (δ r ) ∂ JaT T ∂ Ka T T ∂ Kc T K ·J +J J +J K + K ·J +J J +J K δ r + (JaT Ka Ja + JcT Kc Jc ) ∂t a a a ∂t a a a ∂t ∂t c c c ∂t c c c ∂t ∂t (26)

The conclusion can be obtained by comparing the similar terms on both sides of the above equation

K = JaT Ka Ja + JcT Kc Jc = J T K¯ J

(27)

where J denotes the overall Jacobian matrix and can be expressed as J = [JJa ]6×6 , and K¯ denotes the components of the c

stiffness matrix of all the limbs, such that K¯ = [

Ka 03×3

03×3 ] . Kc 6×6

The elements of the matrix should not be less than zero. When the eigenvalue of the matrix is zero, the matrix is singular, which means that the mobile platform loses its rigidity in a certain direction and is in an uncontrollable state. Therefore, the configuration that leads to a singular stiffness matrix should be avoided in applications. In order to obtain K, we should first solve for J and K¯ . 3.3. Development of the overall Jacobian matrix 3.3.1. Jacobian matrix of constraints As mentioned above, both configurations I and III are 3-PRS mechanisms. Because the S joint can be equivalently replaced by three orthogonal unit screws, the instantaneous twist number of every limb of the 3-PRS mechanism is 5. The twist $m of the mobile platform could be represented as the linear superposition of the twists,

$m = d˙i $ˆ 2i + θ˙ 2i $ˆ 2i + θ˙ 3i $ˆ 3i + θ˙ 4i $ˆ 4i + θ˙ 5i $ˆ 5i ,

i = 1, 2, 3

(28)

where $ˆ ji represents a unit screw affiliated with the jth joint of the ith limb relative to the instantaneous reference frame, and d˙i and θ˙ ji (j = 1–5) denote the intensity of the relevant joint screw. The five unit screws of every PRS limb can be expressed as

 















0 ˆ s2i s3i s4i s5i , $ 2i = , $ˆ 3i = , $ˆ 4i = , $ˆ 5i = s1i O Bi × s2i O Si × s3i O Si × s4i O Si × s5i

$ˆ 1i =

 i = 1, 2, 3

(29)

Then, the screw that is reciprocal to all the above screws of one PRS limb may be expressed as



$ˆ ri =

s2i O Si × s2i



i = 1, 2, 3

(30)

For the twist $m and the screw $ˆ ri , they satisfy the condition

$ˆ r1, j ◦ $m = 0

(31)

where “°” denotes the reciprocal product. Eq. (25) produces equations that can be written in matrix form as

JC $m = 0

(32)

Here, Jc is known as the Jacobian matrix of constraints and is represented as

⎡ ⎢

sT21

Jc = ⎣sT22 sT23

(OS1 × s21 )T

⎤

⎥ (OS2 × s22 )T ⎦ (OS3 × s23 )

(33)

T 3×6

3.3.2. Jacobian matrix of actuations If we lock the active joint P, there exists one new reciprocal screw in each PRS limb, which can be expressed as



$ˆ ai =

ki OSi × ki



i = 1, 2, 3

(34)

where ki denotes the unit vector of the link Bi Si . Similarly, an equation set is obtained by taking the reciprocal products of both sides of Eq. (22) with Eq. (28)

Ja $ = q˙ 0

(35)

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

where q˙ 0 = [ d˙1 d˙2 and is expressed as

⎡ ⎢ ⎣

Ja = ⎢

d˙3 ] denotes the vector of the actuated joint rates; Ja is known as the Jacobian matrix of actuations

(OS1 ×k1 )T

k1T k1T ·s21 k2T k2T ·s22 k3T k3T ·s23

11

k1T ·s21

(OS2 ×k2 )T k2T ·s22

(OS3 ×k3 )

T

k3T ·s23

⎤ ⎥ ⎥ ⎦

(36)

3×6

As the moblie platform of the metamorphic parallel mechanism has not only translational DOF but also rotational DOF, the dimensions of the Jacobian matrix of constraints and Jacobian matrix of actuations are not homogeneous. If the distance a between any two centre points of the three spherical joints is used as the characteristic length, the dimensions of the two Jacobian matrices are homogenised, and then the overall Jacobian matrix can be obtained



  Ja

J=

=

Jc

kiT kiT ·s2i

sT2i

(OSi ×ki )T

a·kiT ·s2i (OSi ×s2i )T a



i = 1, 2, 3

(37)

6×6

3.4. Stiffness of actuations and constraints The PRS limb of the 3-PRS parallel mechanism is driven by the leadscrew–nut system, which can translate the rotational motion of a servo motor to the linear motion of a link. The force and linear displacement of the ith nut can be expressed as

fi =

2Ti

μc d s pTi Ksmi

ti =

i = 1, 2, 3

(38)

i = 1, 2, 3

(39)

where Ti represents the torque of the ith servo motor, Ksmi represents the torsional stiffness of the ith servo motor, ds and p represent the pitch diameter and lead of the leadscrew–nut system, and μc represents the coefficient of friction between the screw and nut. Hence, the compliance of the ith leadscrew–nut system in the direction of the link Bi Si can be derived as

Cai = (kiT · s1i )Ci = (kiT · s1i )

μc d s p 2Ksmi

i = 1, 2, 3

(40)

where the meaning of ki is shown in Eq. (28), s1 i is the unit vector of the ith actuator direction, and Ci is the compliance of the ith leadscrew–nut system. Apart from the active force, the PRS limb bears a constraint force Fci along the direction perpendicular to the limb plane and exerted at the position of point Si . The deformations of the ith limb produced by Fci can be expressed as

lai =

li Fci Aai E

i = 1, 2, 3

(41a)

lci =

li3 Fci 3E IZ

i = 1, 2, 3

(41b)

where li denotes the length of each link, Aai denotes the cross-sectional area of each link, E is Young’s modulus of each link, and IZ is the polar moment of inertia of each link. Assuming Cli to be the longitudinal compliances of the ith link, we can obtain

Cli =

li Aai E

i = 1, 2, 3

(42)

Consequently, the stiffness of the actuations and constraints may be acquired −1 −1 Kai = (Cai + Cli + K −1 ) sp + Kr p

Kci =

3E IZ li3

i = 1, 2, 3

i = 1, 2, 3

(43a) (43b)

where Ksp denotes the stiffness coefficients of the S joint and Krp denotes the stiffness coefficients of the R joint, which can be obtained by a finite element method.

12

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595 Table 2 Physical parameters of the metamorphic parallel mechanism. Parameter

Value

E A Iz

2.03 × 10 N/m 4.005 × 10−5 m2 3.73 × 10−10 m4 11

2

Parameter

Value

Ksp Krp Ksmi

2.5 × 10 N·m 0.8 × 108 N·m 1.45×106 N·m/rad 8

Parameter

Value

μc

0.25 20mm 5mm

ds p

4. Stiffness analysis of the mechanism Once the stiffness matrices of the parallel mechanism are identified, the stiffness within the workspace can be predicted so as to assess whether the structure parameters can meet the demand of stiffness. To that end, the stiffness evaluation index should be selected first. To date, certain individual values have been used as stiffness evaluation indices. The most direct way is to use the diagonal form of the stiffness matrix [36]. Additionally, the eigenvalue [26] or the determinant of the stiffness matrix [37], the trace of the compliance matrix [38], KSI [24,39], which is the ratio of the minimum to maximum eigenvalues of the stiffness matrix, and the inverse of the condition number of the integrated stiffness matrix [40] have been utilised as evaluation indices. Additionally, Yan [25], Wang [41], and Cao [22] proposed stiffness indices to measure the stiffness based on the strain energy. Among these indices, the eigenvalue, determinant, trace, and condition number of the stiffness matrix cannot clearly describe the stiffness in a certain direction to provide a reliable basis for the designer. However, the diagonal form of the stiffness matrix is the most simple and intuitive index for evaluating the stiffness matrix of a mechanism when the working conditions are known, which is adopted to assess the stiffness performance in the following. 4.1. Numerical identification of the stiffness model The architectural parameters of the metamorphic parallel mechanism with ConfigurationⅠand Ⅲ are shown in Table 1. The physical parameters of the metamorphic parallel mechanism with ConfigurationⅠand Ⅲ are shown in Table 2. The values of the parameters are given after referred to [30] and [36]. Where A is the cross-sectional area of all the links, and the other symbols have the same meanings as before. According to Eqs. (21), (31), (37a), and (37b), the stiffness of the mechanism can be obtained. Assuming that in the home position of configuration I, the strokes of the linear actuators of the two PrRS limbs are zero and the mobile platform is parallel to the fixed platform, the stiffness matrix of the position can be calculated

⎡

0.3928 ⎢ 0 ⎢ 0 K10 = ⎢ ⎢ 0 ⎣ 0.7450 −0.0 0 01

0 0.2065 −0.7441 −0.8292 0 0

0 −0.7441 4.2617 2.9903 0 0

0 −0.8292 2.9903 3.3317 0 0

0.7450 0 0 0 1.4133 −0.0 0 01

⎤

−0.0 0 01 0 ⎥ ⎥ 0 ⎥ × 108 ⎥ 0 ⎦ −0.0 0 01 0.0 0 01

Similarly, assuming that in the home position of configuration III, the stroke of the actuator of the PR(P)S limb is maximal and the strokes of the linear actuators of the two PrRS limbs are minimal at the end segment, the stiffness matrix of the position can be obtained

⎡

3.9277 ⎢ 0 ⎢ 0 K30 = ⎢ ⎢ 0 ⎣ 0.0 0 01 −2.7591

0 2.6548 −0.2910 −0.4780 0 0

0 −0.2910 0.0532 0.0849 0 0

0 −0.4780 0.0849 0.1384 0 0

0.0 0 01 0 0 0 0.0 0 05 0

⎤

−2.7591 0 ⎥ ⎥ 0 ⎥ × 107 ⎥ 0 ⎦ 0 1.9383

0 , K 0 , K 0 , K 0 , K 0 }, N/rad for {K 0 , K 0 , K 0 , K 0 , K 0 }, Nm/m for The units of the elements in K10 and K30 are N/m for {K11 22 23 32 33 15 16 24 34 35

0 , K 0 , K 0 , K 0 , K 0 }, {K42 43 51 53 61

0 , K 0 , K 0 , K 0 , K 0 }. and Nm/rad for {K44 55 64 65 66

4.2. Stiffness distributions in the workspace As revealed in Fig. 6a, the workspace of the mechanism is large enough. However, the mechanism need not work throughout the workspace. According to the requirements of the rescue robot, two straight segments are preset in the workspace of configuration I and Ⅲ as the motion trajectory of the endpoint of the mechanism. One straight trajectory ensures that the robot can travel and steer on a flat ground, and the other ensures that the robot can adjust the posture as it overcomes the obstacle. Considering that the mechanical leg should realize some fine adjustment of the posture, the areas

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

13

Fig. 8. Stiffness distributions of the mechanism in the sub-workspace of configuration I.

Fig. 9. Stiffness distributions of the mechanism in the sub-workspace of configuration III.

around the two straight trajectories are selected as the sub-workspace to be studied. The following analyses are developed for the two sub-workspaces. The sub-workspace of configuration I is assigned as

⎧ ⎨α ∈ [−0.018π , 0.018π ] β ∈ [−0.09π , 0.09π ] ⎩ zc = 195 + b − b cos β

(44)

14

H.-b. Tian, H.-w. Ma and J. Xia et al. / Mechanism and Machine Theory 142 (2019) 103595

Following the process above, the stiffness distributions of the mechanism in the sub-workspace can be easily obtained. In Fig. 8, kx , ky , and kz represent the linear stiffness along the x-, y-, and z-axes, and ku , kv , and kw represent the torsion stiffness around the x-, y-, and z-axes. As illustrated in Fig. 8, kz is significantly larger than kx and ky , and kw is much lower than ku and kv . All variations of the stiffness are symmetric about the plane defined by β = 0. Additionally, kx and ky have similar variation tendencies and show little change; kz , ku , and kv have similar variation tendencies. The stiffness distributions of the mechanism in the sub-workspace can be acquired if the sub-workspace of configuration III is assigned as

⎧α  ∈ −0.0 058π + π /6,0.0 058π + π /6 [ ] ⎪ ⎨ γ ∈ [−0.0175π ,0.0175π ]  √  ⎪ ⎩y = 3 a cos α  − l 2 − (S + S + S + dc/ + S3/ − a/ ) − b sin α  + b sin α  cos γ c 0 1 2 1 6 2 2 2

(45)

The meanings of kx , ky , kz , ku , kv , and kw in Fig. 9 are the same as those in Fig. 8. As shown in Fig. 9, the torsion stiffness around the y-axis kv is significantly lower than ku and kw . All variations of the stiffness are symmetric about the plane defined by γ = 0. kx and ky have antipodal variation tendencies, whereas kz , ku , and kv have similar variation tendencies. In addition, the variation tendency of ky is similar to that of kw . 5. Conclusions In order to satisfy the need for rescue robots, a novel metamorphic parallel mechanism for mechanical legs was proposed based on rotatable-axis revolute joints. The workspace and the stiffness of the proposed mechanism were addressed. The inverse kinematic model was established and the constraint conditions and ranges of the key parameters were explored; the cloud picture of its workspace was also obtained. The stiffness model was formulated with the consideration of the deformation of the main components caused by the actuation and constraints. The stiffness matrices of the home position of configurations I and III were obtained and the stiffness distributions in the sub-workspace around the desired trajectory was evaluated. The results show that the torsion stiffness around the z-axis (kw ) of configuration I and the torsion stiffness around the y-axis (kv ) of configuration III are much lower than those in the other two directions, which is notable for future applications of the metamorphic parallel mechanism. Acknowledgments The work was supported by the National Natural Science Foundation of China (NSFC) (Grant no. 51705412), the Natural Science Foundation of Shaanxi (Grant no. 2015JM5235 and 2018JQ5116), and the Key Laboratory of Embedded System and Service Computing, China Ministry of Education (ESSCKF 2015-04). References [1] H. Saafi, M.A. Laribi, S. 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