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A multistable tensegrity structure with a gripper application
Mechanism and Machine Theory 114 (2017) 204–217
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
A multistable tensegrity structure with a gripper application Susanne Sumi∗, Valter Böhm, Klaus Zimmermann Ilmenau University of Technology, Technical Mechanics Group, Germany
a r t i c l e
i n f o
Article history: Received 15 December 2016 Revised 7 April 2017 Accepted 8 April 2017
Keywords: Compliant tensegrity structure Multistable tensegrity structure Multiple states of self-equilibrium Gripper application Compliant gripper
a b s t r a c t Multistable tensegrity structures are a new interesting class of compliant prestressed structures. Due to their beneficial properties, these structures are attractive for robotic applications. In this paper a gripper is introduced, which is based on a mechanical compliant, multistable tensegrity structure. The underlying tensegrity structure of the considered gripper is investigated in detail. The influence of the member parameters on the existence of multiple states of self-equilibrium and the mechanical compliance is discussed with the help of static geometric nonlinear analyses, based on the Finite Element Method. The dynamical behaviour of the structure, during the change between the equilibrium configurations, is considered. Therefor the dynamical equations of motion are derived. Then gripper arms are added to the tensegrity structure to obtain a gripper. Different actuation principles for the gripper are discussed. Additionally, a prototype of the gripper has been built and is presented, as well as, selected experimental results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The use of mechanically prestressed compliant structures in robotic applications is a recently discussed topic [1–4]. One specific class of these structures are compliant free standing tensegrity structures. They consist of a set of disconnected compressed members connected with a continuous net of compliant tensioned members. Robots based on these structures are deployable, lightweight, have a very high strength to weight ratio and shock absorbing capabilities. An overview of recent developments and development directions can be found in [5–7]. Known robotic systems use conventional tensegrity structures with only one state of self-equilibrium (equilibrium configuration) [8–11]. A new specific type of tensegrity structures are structures with multiple states of self-equilibrium, and different mechanical properties within these equilibrium configurations. Such structures are called multistable tensegrity structures. In literature only few structures of this kind are reported [12–14]. The consideration of these structures with members of pronounced elasticity is an attractive new research topic. Their potential use in robotic applications is promising in several aspects: • Shape change can be realised by transformation between the equilibrium configurations. • The structures have distinguished mechanical properties in the different equilibrium configurations. In this article the potential use of multistable tensegrity structures in gripper applications is discussed. As an example a two-finger-gripper based on a planar tensegrity structure with two equilibrium configurations is considered. The two equilibrium configurations correspond to the opened and closed states of the gripper. For that system, the gripping force results merely from the prestress of the structure. ∗
Corresponding author. E-mail addresses: [email protected] (S. Sumi), [email protected] (V. Böhm), [email protected] (K. Zimmermann).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.04.005 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Topology and three possible stable equilibrium configurations of the considered planar tensegrity structure (tensegrity structure I) for the selected mechanical parameters of the members.
In [15,16] preliminary studies have been done and this article is an extension of these works: the tensegrity structure and the gripper are considered in more detail. Moreover, dynamical analyses of the tensegrity structure and experimental results of the gripper are considered and further possible topologies for gripper applications are presented. This article is composed as follows: After the introduction, in chapter 2 the topology of the structure and its equilibrium configurations in relation to the member parameters (stiffness and free-length of the members) are described. Then fundamental studies on the mechanical compliance are done. After that the dynamical behaviour of the tensegrity structure is investigated. Moreover the design of the gripper arms and the actuation of the gripper are discussed with the help of theoretical analyses. This is followed by experimental results in chapter 3. Then, in chapter 4, a miniaturisation of the gripper and further topologies that may be used for gripper applications are discussed. Finally some conclusions and further research directions are given. 2. Theory In this paper the planar tensegrity structure (tensegrity structure I), which is depicted in Fig. 1 is considered. The topology of the structure results from a systematic study of a planar basic topology with 6 nodes and 13 members, as described in [17]. With 6 nodes it is possible to have 3 compressive members, which are isolated from each other. Then the nodes of the compressive members are indirectly connected with the nodes of the other compressive members through tensile members. Furthermore, with this basic topology, it is possible to derive tensegrity structures which have only two, from each other isolated (only connected with tensile members), groups of compressive members. Each group of compressive members may contain one or more compressive members. It is possible to choose another basic topology with more members. But in a first phase of the investigations this simple basic topology is chosen. The structure consists of three members of large axial stiffness (members 1–3, compressive members) and six members of low axial stiffness (members 4–9, tensile members). To determine the equilibrium configurations of a tensegrity structure with given member parameters (stiffness and freelength) the so-called form finding procedure has to be applied in a first step. The form finding procedure and further theoretical analyses are performed with the static Finite Element Method including geometric nonlinearity. For that a program code under Matlab® R2009b was developed. In this code each member is assumed as a massless 2D linear spring element with constant axial stiffness (n nodes; member parameters: ki – axial stiffness, L0i – free-length, i = 1,..,m; for the element stiffness matrix see [18]). The characteristic equation of the structure reads
F ( u ) = K ( u ) u.
(1)
Here F(u ) ∈ is the vector of nodal forces, u ∈ is the vector of nodal displacements and K(u ) ∈ is the tangent stiffness matrix. This nonlinear equation (with the geometric boundary conditions: ux1 = uy1 = uy2 = 0 mm) is solved with an incremental-iterative procedure. The algorithm starts in an initial configuration, which is defined by the positions of the nodes. The algorithm is able to find more than one equilibrium configuration of a tensegrity structure by varying the initial configuration. For further information about this form finding procedure see [17]. The static stability of the equilibrium configurations is determined with the help of the eigenvalues of K(u), [19]. For proper selected member parameters the structure possesses three stable equilibrium configurations A, B, and C (see Fig. 1), in the planar case. These parameters are obtained by varying them individually systematically and the form finding procedure is applied repeatedly (the initial configurations are varied as well). In all three equilibrium configurations the R2n
R2n
R2nx2n
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Fig. 2. Different geometrical shape of the equilibrium configurations A and B (black), compared to Fig. 1 (here depicted in grey). Changed parameters to Fig. 1: L04 = L07 = 51.37 mm, L05 = L06 = 69.57 mm.
Fig. 3. Existence and stability of both equilibrium configurations, depending on the member parameters. Black: equilibrium configuration does not exist or no convergent result; white/grey: stable/unstable equilibrium configuration; shaded: at least one member has not the requested stress.
members of large axial stiffness are compressed and the members of low axial stiffness are tensioned. Corresponding to the tensegrity principle, the two compressed member groups (member 1 as the first, and members 2 and 3 as the second group) are only indirectly connected through tensile members. Since the compressive members 2 and 3 are directly connected, the structure is a class 2 [20] tensegrity structure. In the following only the equilibrium configurations A and B are considered, because only these two are needed for the specific application. 2.1. Existence of equilibrium configurations In this chapter the influence of the member parameters of the tensegrity structure I with respect to the existence of the equilibrium configurations A and B and to its geometrical shape is investigated. Equilibrium configuration C is not considered, because it is not needed for the gripper application. If the parameters of the members (stiffness and free-length) are varied, the geometrical shape of the equilibrium configurations may change, see Fig. 2. However, these equilibrium configurations are qualitatively the same in both parameter sets. Additionally, the equilibrium configurations in Fig. 2 are stable. If the parameters are varied on a larger scale, or more parameters are changed, the corresponding equilibrium configuration may get unstable or even vanish. So a suitable choice of the parameters is necessary to ensure that the considered tensegrity structure has more than one stable equilibrium configuration. The relation between the existence and the stability of equilibrium configurations is shown by two examples. Within the first example the stiffnesses of the tensile members 4 and 7, as well as the stiffnesses of the tensile members 5 and 6 are varied: k4 = k7 = 0.2,…,1.0 N/mm, k5 = k6 = 0.2,…,1.0 N/mm with a step size of 0.002 N/mm. To obtain results, which exceed the assumed symmetric case (k8 = k9 ), the stiffnesses of the tensile members 8 and 9 are set to 0.25 N/mm and 0.3 N/mm. All other parameters are chosen as listed in Fig. 1. The chosen initial configurations are the node positions of equilibrium configuration A and B, which are depicted in Fig. 1. For every step the form finding procedure (see chapter 2 and [17]) is done with both initial configurations. After applying the form finding procedure, the resulting equilibrium configuration has to be identified. Both equilibrium configurations, A and B, may be identified with the help of the coordinates of the nodes. There are ranges defined instead of precise values for the nodes, since slightly different geometrical shapes are permitted (see the difference in Figs. 1 and 2. Both are versions of equilibrium configuration A and B). For A and B the inequalities |x5 – x3 | < ε , |y3 + y5 | < ε have to be satisfied, where ε denotes a computational tolerance. Additionally, node 3 lies off the x-axis: |y3 | > δ . In this example δ = 1 mm is chosen. The difference of both A and B is defined by the relative position of node 3 and 4: for equilibrium configuration A x3 – x4 > δ and in contrast, x4 – x3 > δ for B. The results (see Fig. 3) show: The parameter ranges, where A and B are stable (white) or unstable (grey) equilibrium configurations, are different. In addition there are regions in which some tensile members are compressed and some compressive members are tensioned (shaded regions in Fig. 3). The parameter range, for which both A and B are stable equilibrium configurations with the requested stress of the members, is small. In the second example the free-lengths of the tensile members 4 and 7, as well as the free-length of the tensile member 8 are varied: L04 = L07 = 20,…,80 mm, L08 = 20,…,80 mm with a step size of 0.2 mm. All other parameters are set as listed in
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Fig. 4. Existence and stability of both equilibrium configurations, depending on the member parameters. Black: equilibrium configuration does not exist or no convergent result; white/grey: stable/unstable equilibrium configuration; shaded: at least one member has not the requested stress.
Fig. 5. Flexibility figures with an external load of 1.5 N on node 5 (first row) and node 4 (second row), with two different prestresses (middle and right column).
Fig. 1. The initial configurations are the same as in the previous example. Again the form finding procedure is applied for both initial configurations in every step. And the results are interpreted as in the first example. The results (see Fig. 4) are qualitatively similar to the results of the first example: There are parameter regions, where only one of the requested equilibrium configurations exists. As well it is possible to have both equilibrium configurations A and B. When there are both equilibrium configurations, they can be stable or unstable. So the possible region of the freelengths, so that both equilibrium configurations are stable, is small. It can be observed that both, the stiffnesses and the free-lengths of the members, have an influence on the existence and stability of the equilibrium configurations. In conclusion: The parameters of the members have to be chosen carefully to get the two stable equilibrium configurations A and B. Both, the stiffnesses and the free-lengths of the members influence the existence and stability of the equilibrium configurations. 2.2. Mechanical compliance For the planned application some statements about the mechanical compliance in both equilibrium configurations A and B are needed. Factors which influence the compliance of the tensegrity structure are presented in the following example based on equilibrium configuration A. For this purpose a force (|F| = 1.5 N) is applied to node 4 or 5 and its angle (γ = 0,…,2π ) is varied. The deformed state is characterised by the related positions of the nodes and presented with flexibility figures (Fig. 5), see [21,22]. There are two different prestress states considered (stiffnesses listed in Fig. 1 and the same stiffnesses reduced by the factor 0.65). With a symmetric change of the prestress the shape of both stable equilibrium configurations A and B stays the same (see Fig. 5, first row). The mechanical compliance of tensegrity structure I depends on the member properties and the quantity, location and direction of external loads. If the stiffnesses of the members are larger, the flexibility figures are smaller. This means, that the mechanical compliance is smaller. Furthermore the figure shows, that the position of the external load affects the deformation of the tensegrity structure (compare first and second row). With a suitable change (e.g. scaling all stiffnesses with the same factor) of the prestress the mechanical compliance can be modified without changing the equilibrium configurations A and B. 2.3. Dynamic behaviour of the tensegrity structure Tensegrity structure I has two equilibrium configurations (A and B) which are needed for the planned gripper application. For the application it has to be possible to change between these two equilibrium configurations. Different actuation principles are discussed in chapter 2.5. For the design of the actuation, independent from the chosen kind of actuation, the properties of the change between the equilibrium configurations are important. To examine the
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Fig. 6. Notations for the equations of motion.
behaviour of the tensegrity structure during actuation the dynamical model of tensegrity structure I is determined in the first part of this chapter. After that two examples are presented which examine the change between the equilibrium configurations. The equations of motion can be determined with the Lagrange’s equations of the second kind for non-conservative forces:
d ∂L − dt ∂ q˙ a
∂L ∂D =− + Qa , a = 1, . . . , n ∂ qa ∂ q˙ a
(2)
where • • • • • • •
qa are the generalised coordinates and q˙ a are the generalised velocities, n is the degree of freedom of the tensegrity structure, L = T − U is the Lagrangian function, T is the kinetic energy of the tensegrity structure, U is the potential energy of the tensegrity structure, Qa are the generalised forces of the tensegrity structure (e.g. actuation forces) and D is the damping of the tensegrity structure.
If nodes 1 and 2 of the tensegrity structure have fixed bearings, the structure has a degree of freedom of 4, so the generalised coordinates are
(q1 , q2 , q3 , q4 ) := (x4 , y4 , ϕ2 , ϕ3 ).
(3)
The used notations and values are depicted in Fig. 6. Additionally, the stiffnesses and free-lengths are chosen as listed in Fig. 1. The tensile members are assumed to be massless. Parallel to each tensioned member a linear damper is modelled. The compressive members are assumed to be stiff and to have a constant mass (m2 and m3 respectively). The centre of gravity of each stiff member is located in its middle. The potential energy of the tensile members can be calculated with
U=
9 1 kf (Lf − L0f )2 , 2
(4)
f=4
where Lf ∈ R are the current lengths of the tensile members. The damping of the tensegrity structure may be determined with
D=
9 1 di r˙ Tpi r˙ pi , 2
(5)
i=4
where r˙ pi ∈ R2 are the relative velocities of the dampers and di are the damping coefficients of the dampers. Furthermore, Qqa ∈ R are the generalised forces. Let Fk ∈ R2 be external loads (for example actuation forces) applied to the tensegrity structure and let rk ∈ R2 be the position vector to the corresponding external load Fk . The generalised forces Qqa are calculated with
Qqa =
k
FTk
∂ rk , with qa ∈ {x4 , y4 , ϕ2 , ϕ3 }. ∂ qa
(6)
Starting with (2) and exploiting (3), (4), (5) and (6) the equation of motion is given by:
⎛ ⎞ x¨ 4
⎜ y¨ 4 ⎟ = b, M ∈ R4x4 , b ∈ R4 ϕ¨ 2 ⎠ ϕ¨ 3
M⎝
(7)
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with
and
⎛
m2 + m3
0
− 12 m2 L2 cos (ϕ2 )
⎜ 0 ⎜ M := ⎜ ⎝− 12 m2 L2 sin (ϕ2 )
1 m2 L2 2
cos (ϕ2 )
1 m2 L22 3
− 12 m3 L3 sin (ϕ3 )
1 m3 L3 2
cos (ϕ3 )
0
⎛1
m2 + m3
1 m2 L2 2
cos (ϕ2 )
209
⎞
− 12 m3 L3 sin (ϕ3 ) 1 m3 L3 2
cos (ϕ3 ) ⎟ ⎟
0
⎟ ⎠
(8)
1 m3 L23 3
⎞
1 ∂U ∂D m3 L3 cos (ϕ3 )ϕ˙ 32 − − + Qx4 2 ∂ x4 ∂ x˙ 4 ⎜2 ⎟ ⎜1 ⎟ ⎜ m2 L2 sin (ϕ2 )ϕ˙ 2 + 1 m3 L3 sin (ϕ3 )ϕ˙ 2 − ∂ U − ∂ D + Qy ⎟ 4 2 3 ⎜2 ⎟ 2 ∂ y4 ∂ y˙ 4 ⎟. b := ⎜ ⎜ ∂U ⎟ ∂ D ⎜− ⎟ − + Qϕ2 ⎜ ∂ϕ2 ∂ ϕ˙ 2 ⎟ ⎝ ⎠ ∂U ∂D − − + Qϕ3 ∂ϕ3 ∂ ϕ˙ 3 m2 L2 cos (ϕ2 )ϕ˙ 22 +
(9)
Observe that (7) is an implicit, coupled, nonlinear differential equation system. With the help of the dynamical model several actuation principles to change between the equilibrium configurations can be investigated. Different actuation principles are discussed in chapter 2.5. In this chapter one kind of actuation is considered as an example: a time-dependent actuation force at node 4 along the x-axis. There are two basic possibilities to change between the equilibrium configurations: 1. a forced change: node 4 is guided into the other equilibrium configuration 2. an unforced change: the inertia of the structure is exploited. Since there are two stable equilibrium configurations, there has to be an unstable equilibrium configuration on the boundary between both stable equilibrium configurations A and B. It can be determined with the potential energy of the tensile members. The tensegrity structure is in an equilibrium configuration, if the potential energy of the spring elements has a local minimum. The position of the unstable equilibrium configuration (critical position) is:
(x4 , y4 , ϕ2 , ϕ3 ) = (xcrit , ycrit , −ϕcrit , ϕcrit ) = (68.584 mm, 0 mm, − 1.5463, 1.5463 ).
(10)
This is approximately in the middle between equilibrium configuration A and B. If node 4 is slightly over the critical position (xcrit , ycrit ) or if the angles ϕ 2 and ϕ 3 are slightly over the critical position ϕ crit a snap-through happens and no further force is needed to reach the other equilibrium configuration. In the unforced change, the actuation force is zero before the tensegrity structure reaches the critical position. The change into the other equilibrium happens because of the inertia of the structure. In the forced change the actuation force is not zero in the critical position, so it is a guided change between the equilibrium configurations. First Example In the first of two examples the second kind of actuation is investigated: It is examined how big the actuation force has to be, that a change between equilibrium configuration A and B happens. The actuation force is assumed to be continuous. It is symmetric and starts and ends with 0 N. More precisely the actuation force is F1 := (F(t ), 0)T with maximal force Fmax ∈ R and actuation time tmax ∈ R:
F(t ) :=
Fmax 2π π sin t − 2 tmax 2
+
Fmax 2
(11)
and r1 := (x4 , y4 )T . The initial position is equilibrium configuration A: (x4 , y4 , ϕ2 , ϕ3 ) = (41.02 mm, 0.00 mm, − 0.7652, 0.7652) and the initial velocities are set to 0. In the considered case the actuation force has to be zero before the snap-through takes place. Otherwise the tensegrity structure would be pushed directly into the other equilibrium configuration (This would be the forced change, which is not considered in this example). In this example the actuation force defined in Eq. (11) is applied to node 4 with tmax = 0.05 s and a varied Fmax ∈ {8.5, 9.5, 10.5, 10.9} N. The obtained results are depicted in Fig. 7. The second diagram in the first row shows, that the actuation force is zero if angle ϕ 3 reaches ϕ crit . Only angle ϕ 3 has to be considered, because due to symmetry it applies that ϕ2 = −ϕ3 . In the third diagram a projection of the phase diagram is depicted. It can be seen that a maximal force of 9.5 N is too small to switch from equilibrium configuration A to equilibrium configuration B (see first picture in the second row, too). But with a maximum force of 10.5 N (or more) there is the snap-through into equilibrium configuration B. The movement of the nodes during the actuation and snap-through is depicted in the second row in Fig. 7.
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Fig. 7. Behaviour of the tensegrity structure with different actuation forces. From left to right and top to down: actuation force in dependence of the time; angle ϕ 3 in dependence of the time; projection of the phase diagram (velocity of node 4 in dependence of position of node 4 along the x-axis); trace of the nodes with a maximal actuation force of 8.5 N; trace of the nodes with a maximal actuation force of 10.9 N.
Fig. 8. Behaviour of the tensegrity structure with different actuation forces; grey: no change between equilibrium configurations; black: unforced change into equilibrium configuration B; white: actuation force is not zero if angle ϕ 3 reaches ϕ crit .
Second Example In the first example is has been shown that it is possible to change between the equilibrium configurations without pushing it directly into the other equilibrium configuration. But, if the force is too small, there is no change into the other equilibrium configuration. This relation between the amount and duration of force and the change between the equilibrium configurations is investigated in the second example. Again the actuation force defined in Eq. (11) is used again. There is a variation of Fmax = 0 . . . 60 N (stepsize of 0.1 N) and a variation of tmax = 0.01 . . . 0.1 s (stepsize 0.001 s) considered. For every combination the dynamical analysis is done and the results are depicted in Fig. 8: The grey area shows that if the time of the actuation force is too short or the maximal actuation force is too small, there is no snap-through. In this cases the equilibrium configuration after the actuation is again equilibrium configuration A. If time and maximal force are large enough (for concrete values see Fig. 8, black regions), there is a change between the equilibrium configurations. This region (black) is small compared to the other two regions. But if the time or the force are too large, the actuation force is not zero in the unstable equilibrium configuration between equilibrium configuration A and B (white in Fig. 8). As described above, this case is not considered in this article. It can be concluded that it is possible to actuate the tensegrity structure with only one actuation force along one axis to change between equilibrium configuration A and B. It is also possible to stop the actuation force before the snap-through takes place. Because of the same type of actuation the change from B to A shows the same behaviour. 2.4. Design of the gripper arms Recall that the aim is to use tensegrity structure I as a tensegrity gripper. In this section gripper arms are added to the structure. Then the equilibrium configurations A and B represent the closed and opened state of the two-finger-gripper, respectively. In a possible variant stiff gripper arms are fixed connected to the compressive members 2 and 3, see Fig. 9, first row. Observe that the space requirement in the opened state is large. This problem can be solved by a more complex realisation of the gripper arms: an additional joint and two additional tensile members are used in both arms. So the space requirement in the opened state and during the opening / closing of the gripper is reduced, see Fig. 9, second row. In the following this kind of gripper arms is considered more detailed.
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Fig. 9. Two different variants of gripper arms and the change between equilibrium configuration A (closed gripper) and B (opened gripper).
Fig. 10. Detailed design of the gripper arms and the used truss frameworks for the form finding procedure.
Due to the symmetry in the x-axis of the tensegrity structure, it suffices to describe only the upper part of the tensegrity structure. The notations of the new nodes and members can be seen in Fig. 10. The gripper shall grip an object in front of node 2, at node 11. To reduce the space requirement of the gripper arms the arm (member 15) in the closed gripper shall be parallel to member 6 and in the opened gripper parallel to member 1. For this reason a change of the angle between member 11 and 19 is necessary, so a pin joint at node 7 is used. The other added joints (connection between member 3 and 11 and member 19, 13 and 15) are stiff. So there have to be added truss frameworks to the structure to use the form finding procedure with stiff connections between members. The used truss frameworks can be seen in Fig. 10, right. The stiffnesses of the added tensioned members 17 and 21 have been determined so that the gripper arm is parallel to member 6 or 1 (as described above). To increase the gripping force, the rotation space of the pin joint in node 7 can be reduced so that the angle between member 11 and 19 cannot get smaller than needed in the closed space. When using there smaller pin joints, the stiffnesses of members 17 and 21 can be lower than the stiffnesses of the tensioned members in the underlying tensegrity structure. The gripping force depends only on the underlying tensegrity structure. Since the added tensile members have a low stiffness, the changes of the geometrical shape and the mechanical properties of both equilibrium configurations are negligible. 2.5. Actuation of the gripper In this section different actuation principles are described. After that one actuation principle is chosen, which is considered more detailed. The actuation of the gripper to open and to close the gripper (transition between equilibrium configuration A and B) may be realised by three different principles: • Change of the position of selected nodes or members • Change of the relative orientation between members • Change of the prestress of selected members (by changing the member properties: stiffness and free-length) To illustrate these principles, there are three examples presented in Fig. 11. In the left variant node 4 is moved along the x-axis. The second actuation changes the angle between the compressive members 2 and 3. In the third variant (right)
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Fig. 11. Three different kinds of actuation.
Fig. 12. Relation between the force on node 4 (F4x ) and the position of node 4 along the x-axis (x4 ). Black: members’ parameters from Fig. 1; bright / dark grey: all stiffnesses increased / decreased by the factor 1.5 / 0.65.
the actuation is done through the change of the prestress of member 8 (to close the gripper) and member 9 (to open the gripper). The first variant (left) has some advantages: Only one actuator is needed and with a suitable design of the actuator it can be spatially fixed. For those reasons the left variant of actuation is considered in the following. Altering the position of node 4 by a force along the x-axis, there is a snap-through between the equilibrium configurations (both from A to B and from B to A). The snap-through is similar to the snap-through that has been investigated in Section 2.3. The relation between the force at node 4 (F4x ) and the position of node 4 along the x-axis (x4 ) is depicted in Fig. 12. It is presented for three different states of prestress. To open the gripper a larger force is needed (Fopen ) than to close the gripper (Fclose ). The positions (xopen , xclose ), where the change between the equilibrium configurations takes place, are the same for all different states of prestress, but the amount of force is smaller (low prestress) or larger (high prestress) depending on the prestress. Additionally, this shows that the mechanical compliance of the gripper depends on the prestress of the system.
3. Experiments To verify the theoretical results, a gripper has been built with the parameters from Fig. 1. Additionally, the parameters of the added springs (see member 17 and 21 in Fig. 10), have been set to: L017 = 29.4 mm, k17 = 0.033 N/mm, L021 = 13.2 mm, k21 = 0.05 N/mm. Since the tensegrity structure is a planar structure and some members are on top of each other (e.g. member 1 and member 8), some modifications had to be applied to the construction of the prototype. Further, due to the spatial stability, members 2 to 9 have been doubled in the gripper prototype. In Fig. 13 the isometric view of the gripper can be seen. And in Fig. 14 the gripper in the opened and closed state is depicted. In the closed state the gripper holds an object with a size of 70 mm x 30 mm x 30 mm. The connecting pieces are 3D printed and the members with a high stiffness are made of aluminium. Since the opened and closed state of the gripper are stable equilibrium configurations, the actuation force is merely needed to open or to close the gripper. But while the object is gripped there is no actuation force needed. The actuation is realised with a Linear DC-Servomotor (LM 2070-080-01, Faulhaber Inc.), with a maximum peak force of 27.6 N, a continuous force of 9.2 N, a stroke length of 80 mm and a maximum speed of 2.3 m/s. In Fig. 15 the process of opening the gripper is shown with a picture sequence obtained with a high-speed camera. There is a video provided, showing the transition between the opened and the closed state in real time and in slow motion. The movement of the nodes in the high-speed recording have been extracted and compared with the theoretical movement of the nodes. For the theoretical movement the static calculations from the beginning of chapter 2 have been used. The comparison between the calculation and measurement is depicted in Fig. 16. The black nodes are calculated and the
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Fig. 13. Isometric view of the gripper.
Fig. 14. Opened and closed gripper, gripping an object.
Fig. 15. Process of opening the gripper (from left to right).
red nodes are the measured positions. The shape of the trajectory is nearly the same in all 3 nodes. Node 3 and 5 have an offset of approximately 3 mm. Additionally in Table 1 the positions of the nodes in the prototype (measured) and the theoretical positions of the nodes (calculated) are compared. The differences are small. The positions of node 4 are the best. The positions of node 3 are better than the positions of node 5, since the prototype is not completely symmetric. The differences in the trajectory and the positions of the nodes can be explained by inaccuracies in the manufacturing process and by slight differences in the parameters of the used springs.
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Fig. 16. Movement of the nodes 3, 4 and 5 calculated (black) and measured (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 17. Mechanical compliance.
There are differences of the positions of node 4 compared to Fig. 7, since in Fig. 7 the gripper arms are not included in the dynamical model. The actuator is coupled with node 4, as described in Section 2.5. Additionally the coupling is done with a connecting piece, containing an elongated hole. Node 4 lies inside the elongated hole. So the snap-though is not influenced by the actuator. Another advantage of the connecting piece is that the mechanical compliance of the gripper is maintained. The gripper is mechanical compliant in all directions, see Fig. 17. So it can grip objects that are not exactly in the correct position and small obstacles do not impede the movement of the gripper. The gripper has a maximum gripping force of approximately 2 N and can grip objects with a width up to 60 mm. These properties can be changed by changing the length of the gripper arms (member 15 in Fig. 10).
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Table 1 Calculated and measured positions of the nodes and the calculated distance between both variants. Closed gripper
Opened gripper
Calculated
Measured
Distance
Calculated
Measured
Distance
Node 4 x Node 4 y
39.8 mm 0 mm
39.6 mm 0 mm
0.1 mm
102.5 mm 0 mm
103.6 mm 0.2 mm
1.1 mm
Node 3 x Node 3 y
77.4 mm −32.8 mm
78.2 mm −35.6 mm
2.9 mm
60.9 mm −27.7 mm
62.9 mm −32.5 mm
5.2 mm
Node 5 x Node 5 y
77.4 mm 32.8 mm
78.7 mm 35.7 mm
3.2 mm
60.9 mm 27.7 mm
62.8 mm 34 mm
6.6 mm
Fig. 18. Miniaturised prototype of the tensegrity structure presented in this article with two stable equilibrium configurations (bounding box approximately 70 mm x 50 mm x 30 mm).
4. Outlook For further investigations with the focus on the miniaturisation of the gripper another prototype has been built, see Fig. 18. It has both stable equilibrium configurations A and B. Member 1 has a length of approximately 70 mm and members 2 and 3 of 30 mm. In this prototype the tensile members are merged to a continuous monolithic net. The tensile members are made of silicone elastomer. With this design method the realisation of further miniaturised grippers is conceivable. The introduced gripper is based on a class 2 tensegrity structure. Further preliminary investigations show, that it is possible to build gripper based on planar class 1 tensegrity structures. Two possible class 1 tensegrity structures and our first prototypes are depicted in Figs. 19 and 20. In both variants the gripper arms are formed by two compressive members. The gripper arms are only indirectly connected (through tensile members) with each other and with the frame (compressive member in y = 0).
Fig. 19. Two topologies of planar class 1 tensegrity grippers (upper and lower row), left the closed state, right the opened state.
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Fig. 20. Two prototypes (side view) of class 1 tensegrity grippers (upper and lower row), left the closed state, right the opened state.
5. Conclusions In this paper a new gripper mechanism based on a compliant multistable tensegrity structure has been introduced. Therefor a mechanical compliant, multistable tensegrity structure has been investigated in detail. It has been demonstrated that the parameters of the members have to be chosen carefully to get two stable equilibrium configurations. Further it has been shown that the mechanical compliance depends on the member properties and the quantity, location and direction of external loads. With a suitable change of the prestress the mechanical compliance can be modified without changing the shape of the equilibrium configurations. Then gripper arms have been added to the tensegrity structure to obtain a gripper. It has been demonstrated that it is possible to build compliant grippers based on tensegrity structures with members of pronounced elasticity. To use the introduced gripper in a selected application, the needed specific properties can be achieved by customizing only few systemparameters. One possible actuation principle has been investigated. The dynamical analysis of the tensegrity structure has shown that it is possible to actuate the tensegrity structure with only one actuation force along one axis to change between equilibrium configuration A and B. Also it is possible to stop the actuation force before the snap-through takes place. The investigations showed as well, that a larger force is needed to open than to close the gripper. With a prototype and first experiments it has been shown, that such a gripper can be built. Future work will focus on additional experiments with the gripper presented in this paper (e.g. gripping force in dependence of the size of the objects and the length of the gripper arms). In further theoretical investigations a more general consideration of the used topology will be done. In these investigations we will consider the existence and stability of the equilibrium configurations in relation to the member parameters. Also the class 1 tensegrity grippers from Figs. 19 and 20 will be considered in more detail. Acknowledgements This work is supported by the Deutsche Forschungsgemeinschaft (DFG project BO4114/2-1). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2017.04.005. References [1] Q. Boehler, et al., Definition and computation of tensegrity mechanism workspace, J. Mech. Robot. 7 (4) (2015) 4 Paper No: JMR-14-1168. [2] V. Böhm, et al., Vibration-driven mobile robots based on single actuated tensegrity structures, in: Proc. of the IEEE Int. Conf. on Robotics and Automation, 2013, pp. 5455–5460.
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[3] K. Caluwaerts, et al., Design and control of compliant tensegrity robots through simulation and hardware validation, J. R. Soc. Interf. 11 (2014) 13 Paper No: 20140520. [4] J.A. Rieffel, et al., Morphological communication: exploiting coupled dynamics in a complex mechanical structure to achieve locomotion, J. R. Soc. Interf. 7 (2010) 613–621. [5] V. Böhm, et al., An approach to the dynamics and control of a planar tensegrity structure with application in locomotion systems, Int. J. Dyn. Control 3 (1) (2015) 41–49. [6] J.M. Mirats Tur, et al., Tensegrity frameworks: Dynamic analysis review and open problems, Int. J. Mech. Mach. Theory 44 (1) (2009) 1–18. [7] R.E. Skelton, et al., Tensegrity Systems, Springer, 2009. [8] S. Lessard, et al., A bio-inspired tensegrity manipulator with multi-DOF, structurally compliant joints, in: Proc. of the Int. Conf. on Intelligent Robots and Systems 2016, Daejeon, Korea, 2016. [9] J.M. Friesen, et al., The second generation prototype of a duct climbing tensegrity robot, DuCTTv2, in: IEEE Int. Conf. on Robotics and Automation, 2016, pp. 2123–2128. [10] D. Hustig-Schultz, et al., Morphological design for controlled tensegrity quadruped locomotion, IEEE Int. Conf. on Intelligent Robots and Systems (2016) 4714–4719. [11] L. Chen, et al., Soft spherical tensegrity robot design using rod-centered actuation and control, ASME Int. Design Engineering Technical Conferences and Computers and Information in Engineering Conf. (2016). [12] M. Defossez, Shape memory effect in tensegrity structures, Mech. Res. Commun. 30 (4) (2003) 311–316. [13] A. Micheletti, Bistable regimes in an elastic tensegrity system, Proc. R. Soc. A 469 (2154) (2013). [14] X. Xu, et al., Multistable tensegrity structures, J. Struct. Eng. 137 (1) (2011) 117–123. [15] S. Sumi, et al., Compliant gripper based on a multistable tensegrity structure, in: New Trends in Mechanism and Machine Science, Springer, 2017, pp. 143–151. [16] S. Sumi, et al., A novel gripper based on a compliant multistable tensegrity mechanism, in: Microactuators and Micromechanisms, Springer, 2017, pp. 115–126. [17] V. Böhm, et al., Compliant multistable tensegrity structures with simple topologies, in: New Trends in Mechanism and Machine Science, Springer, 2017, pp. 153–162. [18] S. Guest, The stiffness of prestressed frameworks: a unifying approach, Int. J. Solids Struct. 43 (3–4) (2006) 842–854. [19] J.Y. Zhang, et al., Tensegrity Structures - Form, Stability, and Symmetry, Springer, 2015. [20] R.E. Skelton, et al., An introduction to the mechanics of tensegrity structures, The Mechanical Systems Design Handbook: Modeling, Measurement, and Control, CRC Press, 2001. [21] S. Dalilsafaei, et al., Stiffness visualization for tensegrity structures, in: Proc. of the 23rd Nordic Sem. on Comp. Mech., Stockholm, 2010, pp. 130–133. [22] D. Ströbel, et al., Flexibilitätsellipsoide zur Beurteilung von Tragwerken, Der Bauingenieur 78 (2003) 509–516.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
A multistable tensegrity structure with a gripper application Susanne Sumi∗, Valter Böhm, Klaus Zimmermann Ilmenau University of Technology, Technical Mechanics Group, Germany
a r t i c l e
i n f o
Article history: Received 15 December 2016 Revised 7 April 2017 Accepted 8 April 2017
Keywords: Compliant tensegrity structure Multistable tensegrity structure Multiple states of self-equilibrium Gripper application Compliant gripper
a b s t r a c t Multistable tensegrity structures are a new interesting class of compliant prestressed structures. Due to their beneficial properties, these structures are attractive for robotic applications. In this paper a gripper is introduced, which is based on a mechanical compliant, multistable tensegrity structure. The underlying tensegrity structure of the considered gripper is investigated in detail. The influence of the member parameters on the existence of multiple states of self-equilibrium and the mechanical compliance is discussed with the help of static geometric nonlinear analyses, based on the Finite Element Method. The dynamical behaviour of the structure, during the change between the equilibrium configurations, is considered. Therefor the dynamical equations of motion are derived. Then gripper arms are added to the tensegrity structure to obtain a gripper. Different actuation principles for the gripper are discussed. Additionally, a prototype of the gripper has been built and is presented, as well as, selected experimental results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The use of mechanically prestressed compliant structures in robotic applications is a recently discussed topic [1–4]. One specific class of these structures are compliant free standing tensegrity structures. They consist of a set of disconnected compressed members connected with a continuous net of compliant tensioned members. Robots based on these structures are deployable, lightweight, have a very high strength to weight ratio and shock absorbing capabilities. An overview of recent developments and development directions can be found in [5–7]. Known robotic systems use conventional tensegrity structures with only one state of self-equilibrium (equilibrium configuration) [8–11]. A new specific type of tensegrity structures are structures with multiple states of self-equilibrium, and different mechanical properties within these equilibrium configurations. Such structures are called multistable tensegrity structures. In literature only few structures of this kind are reported [12–14]. The consideration of these structures with members of pronounced elasticity is an attractive new research topic. Their potential use in robotic applications is promising in several aspects: • Shape change can be realised by transformation between the equilibrium configurations. • The structures have distinguished mechanical properties in the different equilibrium configurations. In this article the potential use of multistable tensegrity structures in gripper applications is discussed. As an example a two-finger-gripper based on a planar tensegrity structure with two equilibrium configurations is considered. The two equilibrium configurations correspond to the opened and closed states of the gripper. For that system, the gripping force results merely from the prestress of the structure. ∗
Corresponding author. E-mail addresses: [email protected] (S. Sumi), [email protected] (V. Böhm), [email protected] (K. Zimmermann).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.04.005 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Topology and three possible stable equilibrium configurations of the considered planar tensegrity structure (tensegrity structure I) for the selected mechanical parameters of the members.
In [15,16] preliminary studies have been done and this article is an extension of these works: the tensegrity structure and the gripper are considered in more detail. Moreover, dynamical analyses of the tensegrity structure and experimental results of the gripper are considered and further possible topologies for gripper applications are presented. This article is composed as follows: After the introduction, in chapter 2 the topology of the structure and its equilibrium configurations in relation to the member parameters (stiffness and free-length of the members) are described. Then fundamental studies on the mechanical compliance are done. After that the dynamical behaviour of the tensegrity structure is investigated. Moreover the design of the gripper arms and the actuation of the gripper are discussed with the help of theoretical analyses. This is followed by experimental results in chapter 3. Then, in chapter 4, a miniaturisation of the gripper and further topologies that may be used for gripper applications are discussed. Finally some conclusions and further research directions are given. 2. Theory In this paper the planar tensegrity structure (tensegrity structure I), which is depicted in Fig. 1 is considered. The topology of the structure results from a systematic study of a planar basic topology with 6 nodes and 13 members, as described in [17]. With 6 nodes it is possible to have 3 compressive members, which are isolated from each other. Then the nodes of the compressive members are indirectly connected with the nodes of the other compressive members through tensile members. Furthermore, with this basic topology, it is possible to derive tensegrity structures which have only two, from each other isolated (only connected with tensile members), groups of compressive members. Each group of compressive members may contain one or more compressive members. It is possible to choose another basic topology with more members. But in a first phase of the investigations this simple basic topology is chosen. The structure consists of three members of large axial stiffness (members 1–3, compressive members) and six members of low axial stiffness (members 4–9, tensile members). To determine the equilibrium configurations of a tensegrity structure with given member parameters (stiffness and freelength) the so-called form finding procedure has to be applied in a first step. The form finding procedure and further theoretical analyses are performed with the static Finite Element Method including geometric nonlinearity. For that a program code under Matlab® R2009b was developed. In this code each member is assumed as a massless 2D linear spring element with constant axial stiffness (n nodes; member parameters: ki – axial stiffness, L0i – free-length, i = 1,..,m; for the element stiffness matrix see [18]). The characteristic equation of the structure reads
F ( u ) = K ( u ) u.
(1)
Here F(u ) ∈ is the vector of nodal forces, u ∈ is the vector of nodal displacements and K(u ) ∈ is the tangent stiffness matrix. This nonlinear equation (with the geometric boundary conditions: ux1 = uy1 = uy2 = 0 mm) is solved with an incremental-iterative procedure. The algorithm starts in an initial configuration, which is defined by the positions of the nodes. The algorithm is able to find more than one equilibrium configuration of a tensegrity structure by varying the initial configuration. For further information about this form finding procedure see [17]. The static stability of the equilibrium configurations is determined with the help of the eigenvalues of K(u), [19]. For proper selected member parameters the structure possesses three stable equilibrium configurations A, B, and C (see Fig. 1), in the planar case. These parameters are obtained by varying them individually systematically and the form finding procedure is applied repeatedly (the initial configurations are varied as well). In all three equilibrium configurations the R2n
R2n
R2nx2n
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Fig. 2. Different geometrical shape of the equilibrium configurations A and B (black), compared to Fig. 1 (here depicted in grey). Changed parameters to Fig. 1: L04 = L07 = 51.37 mm, L05 = L06 = 69.57 mm.
Fig. 3. Existence and stability of both equilibrium configurations, depending on the member parameters. Black: equilibrium configuration does not exist or no convergent result; white/grey: stable/unstable equilibrium configuration; shaded: at least one member has not the requested stress.
members of large axial stiffness are compressed and the members of low axial stiffness are tensioned. Corresponding to the tensegrity principle, the two compressed member groups (member 1 as the first, and members 2 and 3 as the second group) are only indirectly connected through tensile members. Since the compressive members 2 and 3 are directly connected, the structure is a class 2 [20] tensegrity structure. In the following only the equilibrium configurations A and B are considered, because only these two are needed for the specific application. 2.1. Existence of equilibrium configurations In this chapter the influence of the member parameters of the tensegrity structure I with respect to the existence of the equilibrium configurations A and B and to its geometrical shape is investigated. Equilibrium configuration C is not considered, because it is not needed for the gripper application. If the parameters of the members (stiffness and free-length) are varied, the geometrical shape of the equilibrium configurations may change, see Fig. 2. However, these equilibrium configurations are qualitatively the same in both parameter sets. Additionally, the equilibrium configurations in Fig. 2 are stable. If the parameters are varied on a larger scale, or more parameters are changed, the corresponding equilibrium configuration may get unstable or even vanish. So a suitable choice of the parameters is necessary to ensure that the considered tensegrity structure has more than one stable equilibrium configuration. The relation between the existence and the stability of equilibrium configurations is shown by two examples. Within the first example the stiffnesses of the tensile members 4 and 7, as well as the stiffnesses of the tensile members 5 and 6 are varied: k4 = k7 = 0.2,…,1.0 N/mm, k5 = k6 = 0.2,…,1.0 N/mm with a step size of 0.002 N/mm. To obtain results, which exceed the assumed symmetric case (k8 = k9 ), the stiffnesses of the tensile members 8 and 9 are set to 0.25 N/mm and 0.3 N/mm. All other parameters are chosen as listed in Fig. 1. The chosen initial configurations are the node positions of equilibrium configuration A and B, which are depicted in Fig. 1. For every step the form finding procedure (see chapter 2 and [17]) is done with both initial configurations. After applying the form finding procedure, the resulting equilibrium configuration has to be identified. Both equilibrium configurations, A and B, may be identified with the help of the coordinates of the nodes. There are ranges defined instead of precise values for the nodes, since slightly different geometrical shapes are permitted (see the difference in Figs. 1 and 2. Both are versions of equilibrium configuration A and B). For A and B the inequalities |x5 – x3 | < ε , |y3 + y5 | < ε have to be satisfied, where ε denotes a computational tolerance. Additionally, node 3 lies off the x-axis: |y3 | > δ . In this example δ = 1 mm is chosen. The difference of both A and B is defined by the relative position of node 3 and 4: for equilibrium configuration A x3 – x4 > δ and in contrast, x4 – x3 > δ for B. The results (see Fig. 3) show: The parameter ranges, where A and B are stable (white) or unstable (grey) equilibrium configurations, are different. In addition there are regions in which some tensile members are compressed and some compressive members are tensioned (shaded regions in Fig. 3). The parameter range, for which both A and B are stable equilibrium configurations with the requested stress of the members, is small. In the second example the free-lengths of the tensile members 4 and 7, as well as the free-length of the tensile member 8 are varied: L04 = L07 = 20,…,80 mm, L08 = 20,…,80 mm with a step size of 0.2 mm. All other parameters are set as listed in
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Fig. 4. Existence and stability of both equilibrium configurations, depending on the member parameters. Black: equilibrium configuration does not exist or no convergent result; white/grey: stable/unstable equilibrium configuration; shaded: at least one member has not the requested stress.
Fig. 5. Flexibility figures with an external load of 1.5 N on node 5 (first row) and node 4 (second row), with two different prestresses (middle and right column).
Fig. 1. The initial configurations are the same as in the previous example. Again the form finding procedure is applied for both initial configurations in every step. And the results are interpreted as in the first example. The results (see Fig. 4) are qualitatively similar to the results of the first example: There are parameter regions, where only one of the requested equilibrium configurations exists. As well it is possible to have both equilibrium configurations A and B. When there are both equilibrium configurations, they can be stable or unstable. So the possible region of the freelengths, so that both equilibrium configurations are stable, is small. It can be observed that both, the stiffnesses and the free-lengths of the members, have an influence on the existence and stability of the equilibrium configurations. In conclusion: The parameters of the members have to be chosen carefully to get the two stable equilibrium configurations A and B. Both, the stiffnesses and the free-lengths of the members influence the existence and stability of the equilibrium configurations. 2.2. Mechanical compliance For the planned application some statements about the mechanical compliance in both equilibrium configurations A and B are needed. Factors which influence the compliance of the tensegrity structure are presented in the following example based on equilibrium configuration A. For this purpose a force (|F| = 1.5 N) is applied to node 4 or 5 and its angle (γ = 0,…,2π ) is varied. The deformed state is characterised by the related positions of the nodes and presented with flexibility figures (Fig. 5), see [21,22]. There are two different prestress states considered (stiffnesses listed in Fig. 1 and the same stiffnesses reduced by the factor 0.65). With a symmetric change of the prestress the shape of both stable equilibrium configurations A and B stays the same (see Fig. 5, first row). The mechanical compliance of tensegrity structure I depends on the member properties and the quantity, location and direction of external loads. If the stiffnesses of the members are larger, the flexibility figures are smaller. This means, that the mechanical compliance is smaller. Furthermore the figure shows, that the position of the external load affects the deformation of the tensegrity structure (compare first and second row). With a suitable change (e.g. scaling all stiffnesses with the same factor) of the prestress the mechanical compliance can be modified without changing the equilibrium configurations A and B. 2.3. Dynamic behaviour of the tensegrity structure Tensegrity structure I has two equilibrium configurations (A and B) which are needed for the planned gripper application. For the application it has to be possible to change between these two equilibrium configurations. Different actuation principles are discussed in chapter 2.5. For the design of the actuation, independent from the chosen kind of actuation, the properties of the change between the equilibrium configurations are important. To examine the
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Fig. 6. Notations for the equations of motion.
behaviour of the tensegrity structure during actuation the dynamical model of tensegrity structure I is determined in the first part of this chapter. After that two examples are presented which examine the change between the equilibrium configurations. The equations of motion can be determined with the Lagrange’s equations of the second kind for non-conservative forces:
d ∂L − dt ∂ q˙ a
∂L ∂D =− + Qa , a = 1, . . . , n ∂ qa ∂ q˙ a
(2)
where • • • • • • •
qa are the generalised coordinates and q˙ a are the generalised velocities, n is the degree of freedom of the tensegrity structure, L = T − U is the Lagrangian function, T is the kinetic energy of the tensegrity structure, U is the potential energy of the tensegrity structure, Qa are the generalised forces of the tensegrity structure (e.g. actuation forces) and D is the damping of the tensegrity structure.
If nodes 1 and 2 of the tensegrity structure have fixed bearings, the structure has a degree of freedom of 4, so the generalised coordinates are
(q1 , q2 , q3 , q4 ) := (x4 , y4 , ϕ2 , ϕ3 ).
(3)
The used notations and values are depicted in Fig. 6. Additionally, the stiffnesses and free-lengths are chosen as listed in Fig. 1. The tensile members are assumed to be massless. Parallel to each tensioned member a linear damper is modelled. The compressive members are assumed to be stiff and to have a constant mass (m2 and m3 respectively). The centre of gravity of each stiff member is located in its middle. The potential energy of the tensile members can be calculated with
U=
9 1 kf (Lf − L0f )2 , 2
(4)
f=4
where Lf ∈ R are the current lengths of the tensile members. The damping of the tensegrity structure may be determined with
D=
9 1 di r˙ Tpi r˙ pi , 2
(5)
i=4
where r˙ pi ∈ R2 are the relative velocities of the dampers and di are the damping coefficients of the dampers. Furthermore, Qqa ∈ R are the generalised forces. Let Fk ∈ R2 be external loads (for example actuation forces) applied to the tensegrity structure and let rk ∈ R2 be the position vector to the corresponding external load Fk . The generalised forces Qqa are calculated with
Qqa =
k
FTk
∂ rk , with qa ∈ {x4 , y4 , ϕ2 , ϕ3 }. ∂ qa
(6)
Starting with (2) and exploiting (3), (4), (5) and (6) the equation of motion is given by:
⎛ ⎞ x¨ 4
⎜ y¨ 4 ⎟ = b, M ∈ R4x4 , b ∈ R4 ϕ¨ 2 ⎠ ϕ¨ 3
M⎝
(7)
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with
and
⎛
m2 + m3
0
− 12 m2 L2 cos (ϕ2 )
⎜ 0 ⎜ M := ⎜ ⎝− 12 m2 L2 sin (ϕ2 )
1 m2 L2 2
cos (ϕ2 )
1 m2 L22 3
− 12 m3 L3 sin (ϕ3 )
1 m3 L3 2
cos (ϕ3 )
0
⎛1
m2 + m3
1 m2 L2 2
cos (ϕ2 )
209
⎞
− 12 m3 L3 sin (ϕ3 ) 1 m3 L3 2
cos (ϕ3 ) ⎟ ⎟
0
⎟ ⎠
(8)
1 m3 L23 3
⎞
1 ∂U ∂D m3 L3 cos (ϕ3 )ϕ˙ 32 − − + Qx4 2 ∂ x4 ∂ x˙ 4 ⎜2 ⎟ ⎜1 ⎟ ⎜ m2 L2 sin (ϕ2 )ϕ˙ 2 + 1 m3 L3 sin (ϕ3 )ϕ˙ 2 − ∂ U − ∂ D + Qy ⎟ 4 2 3 ⎜2 ⎟ 2 ∂ y4 ∂ y˙ 4 ⎟. b := ⎜ ⎜ ∂U ⎟ ∂ D ⎜− ⎟ − + Qϕ2 ⎜ ∂ϕ2 ∂ ϕ˙ 2 ⎟ ⎝ ⎠ ∂U ∂D − − + Qϕ3 ∂ϕ3 ∂ ϕ˙ 3 m2 L2 cos (ϕ2 )ϕ˙ 22 +
(9)
Observe that (7) is an implicit, coupled, nonlinear differential equation system. With the help of the dynamical model several actuation principles to change between the equilibrium configurations can be investigated. Different actuation principles are discussed in chapter 2.5. In this chapter one kind of actuation is considered as an example: a time-dependent actuation force at node 4 along the x-axis. There are two basic possibilities to change between the equilibrium configurations: 1. a forced change: node 4 is guided into the other equilibrium configuration 2. an unforced change: the inertia of the structure is exploited. Since there are two stable equilibrium configurations, there has to be an unstable equilibrium configuration on the boundary between both stable equilibrium configurations A and B. It can be determined with the potential energy of the tensile members. The tensegrity structure is in an equilibrium configuration, if the potential energy of the spring elements has a local minimum. The position of the unstable equilibrium configuration (critical position) is:
(x4 , y4 , ϕ2 , ϕ3 ) = (xcrit , ycrit , −ϕcrit , ϕcrit ) = (68.584 mm, 0 mm, − 1.5463, 1.5463 ).
(10)
This is approximately in the middle between equilibrium configuration A and B. If node 4 is slightly over the critical position (xcrit , ycrit ) or if the angles ϕ 2 and ϕ 3 are slightly over the critical position ϕ crit a snap-through happens and no further force is needed to reach the other equilibrium configuration. In the unforced change, the actuation force is zero before the tensegrity structure reaches the critical position. The change into the other equilibrium happens because of the inertia of the structure. In the forced change the actuation force is not zero in the critical position, so it is a guided change between the equilibrium configurations. First Example In the first of two examples the second kind of actuation is investigated: It is examined how big the actuation force has to be, that a change between equilibrium configuration A and B happens. The actuation force is assumed to be continuous. It is symmetric and starts and ends with 0 N. More precisely the actuation force is F1 := (F(t ), 0)T with maximal force Fmax ∈ R and actuation time tmax ∈ R:
F(t ) :=
Fmax 2π π sin t − 2 tmax 2
+
Fmax 2
(11)
and r1 := (x4 , y4 )T . The initial position is equilibrium configuration A: (x4 , y4 , ϕ2 , ϕ3 ) = (41.02 mm, 0.00 mm, − 0.7652, 0.7652) and the initial velocities are set to 0. In the considered case the actuation force has to be zero before the snap-through takes place. Otherwise the tensegrity structure would be pushed directly into the other equilibrium configuration (This would be the forced change, which is not considered in this example). In this example the actuation force defined in Eq. (11) is applied to node 4 with tmax = 0.05 s and a varied Fmax ∈ {8.5, 9.5, 10.5, 10.9} N. The obtained results are depicted in Fig. 7. The second diagram in the first row shows, that the actuation force is zero if angle ϕ 3 reaches ϕ crit . Only angle ϕ 3 has to be considered, because due to symmetry it applies that ϕ2 = −ϕ3 . In the third diagram a projection of the phase diagram is depicted. It can be seen that a maximal force of 9.5 N is too small to switch from equilibrium configuration A to equilibrium configuration B (see first picture in the second row, too). But with a maximum force of 10.5 N (or more) there is the snap-through into equilibrium configuration B. The movement of the nodes during the actuation and snap-through is depicted in the second row in Fig. 7.
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Fig. 7. Behaviour of the tensegrity structure with different actuation forces. From left to right and top to down: actuation force in dependence of the time; angle ϕ 3 in dependence of the time; projection of the phase diagram (velocity of node 4 in dependence of position of node 4 along the x-axis); trace of the nodes with a maximal actuation force of 8.5 N; trace of the nodes with a maximal actuation force of 10.9 N.
Fig. 8. Behaviour of the tensegrity structure with different actuation forces; grey: no change between equilibrium configurations; black: unforced change into equilibrium configuration B; white: actuation force is not zero if angle ϕ 3 reaches ϕ crit .
Second Example In the first example is has been shown that it is possible to change between the equilibrium configurations without pushing it directly into the other equilibrium configuration. But, if the force is too small, there is no change into the other equilibrium configuration. This relation between the amount and duration of force and the change between the equilibrium configurations is investigated in the second example. Again the actuation force defined in Eq. (11) is used again. There is a variation of Fmax = 0 . . . 60 N (stepsize of 0.1 N) and a variation of tmax = 0.01 . . . 0.1 s (stepsize 0.001 s) considered. For every combination the dynamical analysis is done and the results are depicted in Fig. 8: The grey area shows that if the time of the actuation force is too short or the maximal actuation force is too small, there is no snap-through. In this cases the equilibrium configuration after the actuation is again equilibrium configuration A. If time and maximal force are large enough (for concrete values see Fig. 8, black regions), there is a change between the equilibrium configurations. This region (black) is small compared to the other two regions. But if the time or the force are too large, the actuation force is not zero in the unstable equilibrium configuration between equilibrium configuration A and B (white in Fig. 8). As described above, this case is not considered in this article. It can be concluded that it is possible to actuate the tensegrity structure with only one actuation force along one axis to change between equilibrium configuration A and B. It is also possible to stop the actuation force before the snap-through takes place. Because of the same type of actuation the change from B to A shows the same behaviour. 2.4. Design of the gripper arms Recall that the aim is to use tensegrity structure I as a tensegrity gripper. In this section gripper arms are added to the structure. Then the equilibrium configurations A and B represent the closed and opened state of the two-finger-gripper, respectively. In a possible variant stiff gripper arms are fixed connected to the compressive members 2 and 3, see Fig. 9, first row. Observe that the space requirement in the opened state is large. This problem can be solved by a more complex realisation of the gripper arms: an additional joint and two additional tensile members are used in both arms. So the space requirement in the opened state and during the opening / closing of the gripper is reduced, see Fig. 9, second row. In the following this kind of gripper arms is considered more detailed.
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Fig. 9. Two different variants of gripper arms and the change between equilibrium configuration A (closed gripper) and B (opened gripper).
Fig. 10. Detailed design of the gripper arms and the used truss frameworks for the form finding procedure.
Due to the symmetry in the x-axis of the tensegrity structure, it suffices to describe only the upper part of the tensegrity structure. The notations of the new nodes and members can be seen in Fig. 10. The gripper shall grip an object in front of node 2, at node 11. To reduce the space requirement of the gripper arms the arm (member 15) in the closed gripper shall be parallel to member 6 and in the opened gripper parallel to member 1. For this reason a change of the angle between member 11 and 19 is necessary, so a pin joint at node 7 is used. The other added joints (connection between member 3 and 11 and member 19, 13 and 15) are stiff. So there have to be added truss frameworks to the structure to use the form finding procedure with stiff connections between members. The used truss frameworks can be seen in Fig. 10, right. The stiffnesses of the added tensioned members 17 and 21 have been determined so that the gripper arm is parallel to member 6 or 1 (as described above). To increase the gripping force, the rotation space of the pin joint in node 7 can be reduced so that the angle between member 11 and 19 cannot get smaller than needed in the closed space. When using there smaller pin joints, the stiffnesses of members 17 and 21 can be lower than the stiffnesses of the tensioned members in the underlying tensegrity structure. The gripping force depends only on the underlying tensegrity structure. Since the added tensile members have a low stiffness, the changes of the geometrical shape and the mechanical properties of both equilibrium configurations are negligible. 2.5. Actuation of the gripper In this section different actuation principles are described. After that one actuation principle is chosen, which is considered more detailed. The actuation of the gripper to open and to close the gripper (transition between equilibrium configuration A and B) may be realised by three different principles: • Change of the position of selected nodes or members • Change of the relative orientation between members • Change of the prestress of selected members (by changing the member properties: stiffness and free-length) To illustrate these principles, there are three examples presented in Fig. 11. In the left variant node 4 is moved along the x-axis. The second actuation changes the angle between the compressive members 2 and 3. In the third variant (right)
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Fig. 11. Three different kinds of actuation.
Fig. 12. Relation between the force on node 4 (F4x ) and the position of node 4 along the x-axis (x4 ). Black: members’ parameters from Fig. 1; bright / dark grey: all stiffnesses increased / decreased by the factor 1.5 / 0.65.
the actuation is done through the change of the prestress of member 8 (to close the gripper) and member 9 (to open the gripper). The first variant (left) has some advantages: Only one actuator is needed and with a suitable design of the actuator it can be spatially fixed. For those reasons the left variant of actuation is considered in the following. Altering the position of node 4 by a force along the x-axis, there is a snap-through between the equilibrium configurations (both from A to B and from B to A). The snap-through is similar to the snap-through that has been investigated in Section 2.3. The relation between the force at node 4 (F4x ) and the position of node 4 along the x-axis (x4 ) is depicted in Fig. 12. It is presented for three different states of prestress. To open the gripper a larger force is needed (Fopen ) than to close the gripper (Fclose ). The positions (xopen , xclose ), where the change between the equilibrium configurations takes place, are the same for all different states of prestress, but the amount of force is smaller (low prestress) or larger (high prestress) depending on the prestress. Additionally, this shows that the mechanical compliance of the gripper depends on the prestress of the system.
3. Experiments To verify the theoretical results, a gripper has been built with the parameters from Fig. 1. Additionally, the parameters of the added springs (see member 17 and 21 in Fig. 10), have been set to: L017 = 29.4 mm, k17 = 0.033 N/mm, L021 = 13.2 mm, k21 = 0.05 N/mm. Since the tensegrity structure is a planar structure and some members are on top of each other (e.g. member 1 and member 8), some modifications had to be applied to the construction of the prototype. Further, due to the spatial stability, members 2 to 9 have been doubled in the gripper prototype. In Fig. 13 the isometric view of the gripper can be seen. And in Fig. 14 the gripper in the opened and closed state is depicted. In the closed state the gripper holds an object with a size of 70 mm x 30 mm x 30 mm. The connecting pieces are 3D printed and the members with a high stiffness are made of aluminium. Since the opened and closed state of the gripper are stable equilibrium configurations, the actuation force is merely needed to open or to close the gripper. But while the object is gripped there is no actuation force needed. The actuation is realised with a Linear DC-Servomotor (LM 2070-080-01, Faulhaber Inc.), with a maximum peak force of 27.6 N, a continuous force of 9.2 N, a stroke length of 80 mm and a maximum speed of 2.3 m/s. In Fig. 15 the process of opening the gripper is shown with a picture sequence obtained with a high-speed camera. There is a video provided, showing the transition between the opened and the closed state in real time and in slow motion. The movement of the nodes in the high-speed recording have been extracted and compared with the theoretical movement of the nodes. For the theoretical movement the static calculations from the beginning of chapter 2 have been used. The comparison between the calculation and measurement is depicted in Fig. 16. The black nodes are calculated and the
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Fig. 13. Isometric view of the gripper.
Fig. 14. Opened and closed gripper, gripping an object.
Fig. 15. Process of opening the gripper (from left to right).
red nodes are the measured positions. The shape of the trajectory is nearly the same in all 3 nodes. Node 3 and 5 have an offset of approximately 3 mm. Additionally in Table 1 the positions of the nodes in the prototype (measured) and the theoretical positions of the nodes (calculated) are compared. The differences are small. The positions of node 4 are the best. The positions of node 3 are better than the positions of node 5, since the prototype is not completely symmetric. The differences in the trajectory and the positions of the nodes can be explained by inaccuracies in the manufacturing process and by slight differences in the parameters of the used springs.
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Fig. 16. Movement of the nodes 3, 4 and 5 calculated (black) and measured (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 17. Mechanical compliance.
There are differences of the positions of node 4 compared to Fig. 7, since in Fig. 7 the gripper arms are not included in the dynamical model. The actuator is coupled with node 4, as described in Section 2.5. Additionally the coupling is done with a connecting piece, containing an elongated hole. Node 4 lies inside the elongated hole. So the snap-though is not influenced by the actuator. Another advantage of the connecting piece is that the mechanical compliance of the gripper is maintained. The gripper is mechanical compliant in all directions, see Fig. 17. So it can grip objects that are not exactly in the correct position and small obstacles do not impede the movement of the gripper. The gripper has a maximum gripping force of approximately 2 N and can grip objects with a width up to 60 mm. These properties can be changed by changing the length of the gripper arms (member 15 in Fig. 10).
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Table 1 Calculated and measured positions of the nodes and the calculated distance between both variants. Closed gripper
Opened gripper
Calculated
Measured
Distance
Calculated
Measured
Distance
Node 4 x Node 4 y
39.8 mm 0 mm
39.6 mm 0 mm
0.1 mm
102.5 mm 0 mm
103.6 mm 0.2 mm
1.1 mm
Node 3 x Node 3 y
77.4 mm −32.8 mm
78.2 mm −35.6 mm
2.9 mm
60.9 mm −27.7 mm
62.9 mm −32.5 mm
5.2 mm
Node 5 x Node 5 y
77.4 mm 32.8 mm
78.7 mm 35.7 mm
3.2 mm
60.9 mm 27.7 mm
62.8 mm 34 mm
6.6 mm
Fig. 18. Miniaturised prototype of the tensegrity structure presented in this article with two stable equilibrium configurations (bounding box approximately 70 mm x 50 mm x 30 mm).
4. Outlook For further investigations with the focus on the miniaturisation of the gripper another prototype has been built, see Fig. 18. It has both stable equilibrium configurations A and B. Member 1 has a length of approximately 70 mm and members 2 and 3 of 30 mm. In this prototype the tensile members are merged to a continuous monolithic net. The tensile members are made of silicone elastomer. With this design method the realisation of further miniaturised grippers is conceivable. The introduced gripper is based on a class 2 tensegrity structure. Further preliminary investigations show, that it is possible to build gripper based on planar class 1 tensegrity structures. Two possible class 1 tensegrity structures and our first prototypes are depicted in Figs. 19 and 20. In both variants the gripper arms are formed by two compressive members. The gripper arms are only indirectly connected (through tensile members) with each other and with the frame (compressive member in y = 0).
Fig. 19. Two topologies of planar class 1 tensegrity grippers (upper and lower row), left the closed state, right the opened state.
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Fig. 20. Two prototypes (side view) of class 1 tensegrity grippers (upper and lower row), left the closed state, right the opened state.
5. Conclusions In this paper a new gripper mechanism based on a compliant multistable tensegrity structure has been introduced. Therefor a mechanical compliant, multistable tensegrity structure has been investigated in detail. It has been demonstrated that the parameters of the members have to be chosen carefully to get two stable equilibrium configurations. Further it has been shown that the mechanical compliance depends on the member properties and the quantity, location and direction of external loads. With a suitable change of the prestress the mechanical compliance can be modified without changing the shape of the equilibrium configurations. Then gripper arms have been added to the tensegrity structure to obtain a gripper. It has been demonstrated that it is possible to build compliant grippers based on tensegrity structures with members of pronounced elasticity. To use the introduced gripper in a selected application, the needed specific properties can be achieved by customizing only few systemparameters. One possible actuation principle has been investigated. The dynamical analysis of the tensegrity structure has shown that it is possible to actuate the tensegrity structure with only one actuation force along one axis to change between equilibrium configuration A and B. Also it is possible to stop the actuation force before the snap-through takes place. The investigations showed as well, that a larger force is needed to open than to close the gripper. With a prototype and first experiments it has been shown, that such a gripper can be built. Future work will focus on additional experiments with the gripper presented in this paper (e.g. gripping force in dependence of the size of the objects and the length of the gripper arms). In further theoretical investigations a more general consideration of the used topology will be done. In these investigations we will consider the existence and stability of the equilibrium configurations in relation to the member parameters. Also the class 1 tensegrity grippers from Figs. 19 and 20 will be considered in more detail. Acknowledgements This work is supported by the Deutsche Forschungsgemeinschaft (DFG project BO4114/2-1). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2017.04.005. References [1] Q. Boehler, et al., Definition and computation of tensegrity mechanism workspace, J. Mech. Robot. 7 (4) (2015) 4 Paper No: JMR-14-1168. [2] V. Böhm, et al., Vibration-driven mobile robots based on single actuated tensegrity structures, in: Proc. of the IEEE Int. Conf. on Robotics and Automation, 2013, pp. 5455–5460.
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