Mechanics of compliant serial manipulator composed of dual-triangle segments Wanda Zhao, Anatol Pashkevich, Damien Chablat, Alexandr Klimchik
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Wanda Zhao, Anatol Pashkevich, Damien Chablat, Alexandr Klimchik. Mechanics of compliant serial manipulator composed of dual-triangle segments. International Journal of Mechanical Engineering and Robotics Research, Dr. Bao Yang, 2021, 10 (4), pp.169-176. 10.18178/ijmerr.10.4.169-176. hal-03195185
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Wanda Zhao, Anatol Pashkevich and Damien Chablat Laboratoire des Sciences du Numérique de Nantes (LS2N), UMR CNRS 6004, Nantes, France Email: [email protected], [email protected], [email protected]
Alexandr Klimchik Innopolis University, Tatarstan, Russia Email: [email protected]
Abstract—This paper focuses on the mechanics of a applications [8]. However, the pure soft continuum robot compliant serial manipulator composed of the new type of received little attention, as its small output force and dual-triangle elastic segments. Both the analytical and design difficulty. Thus, combining rigid and elastic or numerical methods were used to find the stable and unstable soft components becomes a popular practice in designing equilibrium configurations of the manipulator, and to a robot manipulator. The typical earlier hyper-redundant predict the corresponding manipulator shapes. The stiffness analysis was carried on for both loaded and unloaded modes, robot designs and implementations can be date back to the stiffness matrices were computed using the Virtual Joint the 1970s [9], which includes a series of plates Method (VJM). The results demonstrate that either interconnected by universal joints and elastic control buckling or quasi-buckling phenomenon may occur under components for pivotable action to one another. the loading if the manipulator initial configuration is [10][11][12] straight or non-straight one. Relevant simulation results are Nowadays, a very promising trend in designing presented at last, which confirm the theoretical study. compliant robots is using a series of similar segments based on various tensegrity mechanisms, which are Index Terms—component, compliant manipulator, stiffness composed in an equilibrium of compressive elements and analysis, equilibrium, robot buckling, redundancy tensile elements (cables or springs) [13][14]. Some kinds of tensegrity mechanisms have been already studied carefully. Such as [15], the authors dealt with the I. INTRODUCTION mechanism composed of two springs and two length- Currently, compliant serial manipulators are used more changeable bars. They analyzed the mechanism stiffness and more in many applications (such as inspection in using the energy method, demonstrated that the constraint environment, medical fields etc.), because of mechanism stiffness always decreasing under external their sophisticated motions and low weights. loading with the actuators locked, which may lead to Conventional compliant manipulators are usually “buckling”. And in [16][17], the cable-driven X-shape composed of rigid links and compliant actuators, like tensegrity structures were considered, the authors hinges, axles, or bearings. However, there is a lot of investigated the influence of cable lengths on the research in this area dealing with some new mechanical mechanism equilibrium configurations, which may be structures [1][2][3][4], which achieve compliant motions both stable and unstable. The relevant analysis of the through tensegrity mechanisms. And one of them will be equilibrium configurations stability and singularity can be studied here. seen in [18]. In general, the robotic manipulators are usually A new type of compliant tensegrity mechanism was classified into three types [5], conventional discrete, proposed in our previous papers [19][20]. It is composed serpentine, and continuum robots. The first one is made of two rigid triangle parts, which are connected by a of traditional rigid components. The second one uses passive joint in the center and two elastic edges on each discrete joints but combine very short rigid links with side with controllable preload. The stiffness analysis of a large density joints, which produce smooth curves and basic dual-triangle was carried on, and the stable make the robot similar to a snake or elephant trunk [6]. condition of the equilibrium was obtained. The results While the continuum robots do not contain any rigid links also showed that there may be a buckling phenomenon. or joints, they are very smooth and soft, bending Usually, while designing a robot, researchers always try continuously when working [7]. Many researchers have to avoid buckling, but such behavior can make done studies on serpentine and continuum robots in improvements in some fields [21]. So this phenomenon recent years, designed flexible mechanisms for many must be taken into account. In this paper, we study a compliant serial manipulator composed of the dual- Manuscript received Sep, 2020; revised Sep 25, 2020; accepted Oct triangle segments mentioned above, concentrate on the 9, 2020. equilibrium configurations and their transformations Mext =0. As follows from the relevant analysis, the under the loading, which may be either continuous or function M(q) can be either monotonic or non-monotonic sporadic that leading to buckling phenomenon. Both one, so the single-segment mechanism may have multiple loaded and unloaded stiffness model of this manipulator stable and unstable equilibriums, which are studied in were analyzed. The simulation of the manipulator detail [19][20]. As Fig. 2 shows, the torque-angle curves behavior after buckling was obtained, which provides a M(q) that can be either monotonic or two-model one, the good base of the design and relevant control algorisms of considered stability condition can be simplified and such manipulator reduced to the derivative sign verification at the zero point, i.e. II. MECHANICS OF DUAL-TRIANGLE MECHANISM M q 0 (2) Let us consider first a single segment of the compliant q0 serial manipulator. It consists of two rigid triangles which is easy to verify in practice. It represents the connected by a passive joint whose rotation is constrained mechanism equivalent rotational stiffness for unloaded by two linear springs as shown in Fig. 1. It is assumed configuration with q=0. that the mechanism geometry is described by the triangle Let us also consider in detail the symmetrical case, for parameters (a , b ) and (a , b ), and the mechanism shape 0 0 1 1 2 2 which a1=a2=a, b1=b2=b, k1=k2, L = L . Then as follows is defined by the central angle, which is adjusted through i from the mechanism geometry, to distinguish the two control inputs influencing on the springs L and L . 1 2 monotonic and non-monotonic cases presented in Fig. 2, Let us denote the spring lengths in the non-stress state as we can omit some indices and present the torque-angle 0 0 ， L1 and L2 and the spring stiffness coefficients k1 and k2. relationship as well as the stiffness expression in more compact forms: M q 2 ck c cos sin q L0 cos( 2)sin( q 2) 12 12 (3) 0 M()q ck2cos c12 cos q L cos (2)12 ) c os( q 2 it is also necessary to compute M’(q) for unloaded equilibrium configuration q=0, that let us obtain the condition of the torque-angle curve monotonicity: L02 b 1( a b ) 2 for the further analysis. Figure 1. Geometry of a single dual-triangle mechanism.
To find the mechanism configuration angle q III. MECHANICS OF SERIAL MANIPULATOR corresponding to the given control inputs L0 and L0 , let 1 2 A. Manipulator Geometry and Kinematics us derive first the static equilibrium equation. From Hook’s law, the forces generated by the springs Let us consider a manipulator composed of three similar segments connected in series as shown in Fig. 3, are F kL( L0 ) (i =1, 2), where L and L are the i ii i 1 2 where the left hand-side is fixed and the initial spring lengths |AD|, |BC|. These values can be computed configuration is a “straight” one (q1=q2=q3=0). This 2 2 configuration is achieved by applying equal control using the formulas Lii( ) cc1 2 2 cc 1 2 cos( i ) (i=1, inputs to all the mechanism segments. For this 2 2 2). Here ci a i b i (i=1, 2), and the angles 1 , 2 are manipulator, it is necessary to investigate the influence of expressed via the mechanism parameters as q , the external force Fe=(Fx, Fy), which causes the end- 1 12 effector displacements to a new equilibrium location 2 12 q , and 12 atan(a 1 / b 1 )+atan( a 2 / b 2 ) . The T T (,)x y (6 b x ,) y , which corresponds to the torques M1=F1·h1, M1=F2·h2 in the passive joint O can be computed from the geometry, so we can get nonzero configuration variables (q1, q2, q3). It is also assumed here the external torque Mext applied to the end- 0 effector is equal to zero. It can be easily proved from the Mq1( ) k 1 (1 LL 1 1 ( 1 )) cc 1 2 sin( 1 ) 0 geometry analysis that the configuration angles satisfy the Mq2( ) k 2 (1 LL 2 2 ( 2 )) cc 1 2 sin( 2 ) following direct kinematic equations where the difference in signs is caused by the different x b2 bC 2 bC bC direction of the torques generated by the forces F1, F2. 1 12 123 (4) y2 bS 2 bS bS Further, taking into account the external torque Mext 1 12 123 applied to the moving platform, the static equilibrium where Ccos q q q , Ssin q q q , equation for the considered mechanism can be written as 123 1 2 3 123 1 2 3
M1(q)+ M2(q)+Mext =0. C12cos q 1 q 2 , S12sin q 1 q 2 , C1 cos q 1 , Let us now evaluate the stability of the mechanism S1 sin q 1 . These two equations include three unknown under consideration. In general, this property highly variables, and it allows us to compute two of them if the depends on the equilibrium configuration defined by the third one was known. For instance, if the angle q is angle q, which satisfies the equilibrium equation M(q)+ 1
0 0 Figure 2. The torque-angle curves and static equilibriums for L1 L 2 ( q0 0 ).
energy method. The latter is illustrated by combined plots assumed to be known, the rest two angles q2 , q3 can be computed from the classical inverse kinematics of the of the energy-torque curves computed for the initial two-link manipulator as follows “straight” configuration presented in Fig. 5, which shows that the max/min of the energy E(q1) correspond to zeros
q3 atan SC 3 3 of the torque Me(q1)=0. Further, to find the external y 2 bS bS forces corresponding to this end-point location, it is qatan(1 )atan( 3 ) q (5) 2 1 necessary to use the force-torque equilibrium equation x b2 bC1 2 b bC 3 M JFT 0 (8) where C x b2 bC2 y 2 bS 2 5 b2 4 b 2 , q 3 1 1 T T where M=(Mq1, Mq2, Mq3) , F=(Fx, Fy, Me) . They denote 2 S31 C 3 . The latter expressions provide two the internal torques Mq1, Mq2 and Mq3 in all manipulator groups of possible solutions, which correspond to the segments and the force/torque at the end-point. In this positive /negative configuration angles q 0 and q 0 . equation, the internal torques can be computed using the 3 3 previously derived expression from section II, To find a stable manipulator configuration under the loading, let us apply the energy method. It is clear that the 2 2 0 Mqi2 k ( b a )sin q i bL sin(0.5 q i ) ; i 1,2,3 (9) end-effector displacement caused by the external loading leads to the deflections of mechanism springs, which and the Jacobian matrix Jq can be computed using the allows us to compute the manipulator energy as standard technique for the three-link manipulator
3 2 presented as follows 1 0 2 E kLLij ij (6) 2 i1 j 1 2bS1 2 bS 12 bS 123 2 bS 12 bS 123 bS 123 J q 2bC1 2 bC 12 bC 123 2 bC 12 bC 123 bC 123 (10) o 1 1 1 where Lij and Lij are the spring lengths in current and initial (unextended) states respectively. The above energy where S and C with corresponding indices have the same can be expressed via one of the three variables q1, q2 or q3. meaning as in (4). Assuming that the Jacobian is non- Assuming that variable q1 is chosen as an independent singular (i.e. the loaded manipulator is already out of the one, the desired stable configurations can be found by straight configuration), the external force/torque can be computing local minima of the energy function T expressed directly as F JMq , where the transport -T E( q1 ) min (7) inverse matrix J q can be computed analytically. Then we q1 can get the following expression
Examples of such energy curves E( q1 ) for several typical M cases are presented in Fig. 4. Fx C12 CC 1 12 C 1 q1 1 Fy SSS12 1 12 SM 1 q 2 (11) B. Manipulator Stiffness Behavior 2bS2 bS bS bS2 bS bS Me 3 23 3 2 23 M q3 An alternative way to compute the configuration angles q1, q2, q3 at the equilibrium state is based on the The latter allows us to rewrite the system of the torque equation Me(q1)=0, which is implicitly used in the equilibrium equation (4) in the following extended form
0 0 Figure 3. The torque-angle curves and static equilibriums for L1 L 2 ( q0 0 ).
o Figure 4. Energy curves E( q1 ) for different combinations of manipulator geometric parameters a/b, L /b:
“blue curves”─ positive configuration with q3>0; “green curves” ─ negative configuration with q3<0; ● ─ stable equilibrium; ● ─ unstable equilibrium
variable, it is possible to express q1, q3 in the way b2 bC1 2 bC 12 bC 123 x 0 q1 1 q 2 , q3 3 q 2 , where 2bS1 2 bS 12 bS 123 y 0 (12) SM3q 1 S 23 SM 3 q 2 2 SSM 2 23 q3 0 1 21 11 20; 3 21 1 4 (14) whose solution (q , q , q ) may correspond to either stable 1 2 3 The latter gives us four possible manipulator geometric or unstable equilibriums of the manipulator configuration. configurations corresponding to the static equilibrium, Then, using expressions F (q , q , q ) and F (q , q , q ) x 1 2 3 y 1 2 3 two with U-shape and two with Z-shape (see Table 1). obtained from (11), one can get the external loading (F , x The corresponding external forces F , F can be F ) corresponding to the end-effector position (x, y), x y y linearized for small configuration angles, which yields which finally allows us to generate the desired force- deflection curves. Examples of such curves for several k F2( babLqqqF2 2 ) 0 ( 2 ); 0 (15) case studies are presented in Fig. 6, where it is assumed x2bq 1 3 2 y that under the loading the manipulator moves along with 2 x-axis, i.e. x var , y 0 . As follows from this figure, Further, taking into account (13) the desired critical force in general cases (Fig. 6a), the force-deflection curves are can be expressed in the following way quasi-linear, but some of them may do not pass through k o 2 2 0 the zero point. The latter means that the corresponding Fxlim F x 2( b a ) bL (16) qi 0 b manipulator possesses very specific particularity known as the “buckling” property [19][20][21], for which the where ( 21 14) 10 0.9417 for U-shape, and configuration angles may suddenly change while the external force increasing gradually. Besides, in the case ( 21 14) 10 1.8583 for Z-shape. presented in Fig. 6b, there is the “jumping” phenomenon, It is worth mentioning that the obtained expression because of the unstable geometrical parameters of the allows us to derive the static stability condition for the manipulator segment (see section II and stable condition ), straight configuration. In fact, this configuration is stable and the manipulator suddenly changes its shape even for 0 if and only if Fx 0 , which is equivalent to extremely small loading. 0 To compute the critical force Fx of the buckling, let us assume that the configuration angles (q1, q2, q3) are small enough but not equal to zero. This allows us to derive a linearized stiffness model in the neighborhood of qi=0 (i=1, 2, 3). Under such assumptions, the first and second equations from (3.15) can be presented in the following form
b( q2 q 2 0.5 q 2 ) x 1 12 123 (13) y 2b ( q1 q 12 0.5 q 123 ) which allows us to present the condition δy=0 as q1+q12+ q123/2=0. Applying similar linearization to the third equation from (12), one can get the additional relation of 2 2 the configuration angles q1 q 3 q 2 q 2 q 3 q 3 0 , which ensures the equality Me=0. Further, combining these two Figure 5. Correspondence between the maxima/minima of the obtained relations and considering q2 as an independent energy curves E( q1 ) and zeros of the external torque Me ( q1 ) .
Figure 6. Force-deflection curves and stiffness coefficients for the “straight” initial configuration.
2 2 0 max 2b a bL . It defining the monotonicity of the caused by the geometric constraints qi q i . However, torque-angle curves for the manipulator segments. the energy curves for the case (b) cannot be treated in the 0 Finally, let us compare the U-shape and Z-shape same way, because the combination of a, b, Li provides equilibrium configurations for their static stability. It can non-monotonic torque-angle curves for the segments and be easily proved that for the small configuration angles qi, even separate parts of the manipulator are unstable here. the end-effector deflection δx can be expressed in the It should be stressed that in the cases (a), each segment of following way the mechanism is statically stable. It should be also noted 2 that there are some unfeasible sections (black lines) x q2 (17) inside of the curve, where at least one of the angles q2 or q is out of the allowable geometric limits. where ( 21 21) 20 1.2791 for U-shape, and 3 The above-presented case studies, corresponding to the ( 21 21) 20 0.8209 for Z-shape. The latter end-effector initial position (x , y )T (5.5 b , 0) T , can be means that for the similar deflections δx, the U-shape has also illustrated by the force-deflection curves presented in the smaller configuration angles qi than the one of Z- Fig. 8. As follows, there is no buckling phenomenon in shape, which ensures smaller energy in agreement with the case (a), the curve is quasi-linear and passes through (7). the zero point. Besides, the buckling detected in the case Let us consider now when the manipulator initial (b) cannot be observed in practice because of the non- configuration is non-straight, which corresponds to the stability of the separate manipulator segments. 0 angles ( qi 0, i 1,2,3 ). Similar to the above section, To evaluate the manipulator stiffness matrix for the the equilibrium is defined by three equations (12), which non-straight configuration, let us first find the joint are derived from the direct kinematics and the zero torques for all manipulator segments using the method from section II, external torque assumption Me=0. It can be proved that the energy curves have the “∞-shape” similar to the M2( kba2 2 )sin() qkLaq 0 cos( 2) bq sin( 2) straight configuration considered before. However, qi i i1 i i kLa0 cos( q 2) bq sin( 2) ; i 1,2,3 depending on the initial end-effector location (x, y), these i2 i i energy curves may be non-symmetrical and can be even (18) discontinuous and include cusp points. Typical examples and compute the derivatives providing equivalent of such curves corresponding to the end-point location stiffness coefficients in the joints K dM dq (x , y )T (5.5 b , 0) T are presented in Fig. 7, where the qi qi i discontinuity caused by the geometric constraint is visible. 2 2 0 Kqi2( kba )cos() qkLbq i i1 cos( i 2) aq sin( i 2)2 In particular, the energy curve of cases (a) consists of two 0 kLaqsin( 2) bq cos( 2 ) 2; i 1, 2, 3 separate U-shape parts that yield two symmetrical stable i2 i i i equilibriums and four unstable ones. Such separation is (19)
TABLE I. POSSIBLE MANIPULATOR SHAPES IN STATIC EQUILIBRIUM
q1 q2 q3 Geometric configuration Stability q 1 q3 Stable ‒ + + U-shape: q2 Case of “+√” q2 + ‒ ‒ q1 q3 Stable U-shape: q q 3 1 Unstable ‒ + ‒ Z-shape: q2 Case of “-√” q2 q1 Unstable + ‒ + Z-shape: q3
o Figure 7. Energy curves E( q1 ) for different (a, b, L ) for non-straight initial configuration and displacement x, y b 2,0
“blue curves” ─ feasible configuration with q3>0; “green curves” ─ feasible configuration with q3<0; “black curves”─ unfeasible configuration; “red point ●”─ stable equilibrium; “black point ●” ─ unstable equilibrium..
T This allows us to apply the VJM method and to express variation F ( Fx , F y ) that corresponds to the joint the unloaded stiffness matrix of the considered angle variations q (, q q , q )T . As follows from manipulator as 1 2 3 the equilibrium equation MJ= T F , the corresponding 0 1 T 1 variation of the joint torque can be expressed as KJKJF o qo o (20)
T where the subscript “o” denotes the variables dJ T M qFJF (23) corresponding to the unloaded initial configuration. dq Further, if we express the 2x3 submatrix of the (10) for this configuration as where the part dJT dq , which includes the Jacobian derivative, can be rewritten as J J J J 11 12 13 (21) o J J J dJJT 3 T 21 22 23 2 3 qF= F qi Kq g (24) dq i1 qi The desired compliance matrix of the unloaded mode can be expressed analytically in the following way where K g is the 3×3 matrix describing the influence of 2 2 2 loading F on the manipulator Jacobian J J11 J 12 J13 * 0 1 T K K K T T T CJKJ q1 q 2 q 3 2 2 2 (22) JJJ F o qo o J J J K F FF (25) * 21 22 23 g q1 q 2 q 3 3 3 Kq1 K q 2 K q 3 that can be also written in the extended form as where K qo diagK(, q1 K q 2 , K q 3 ) is the matrix of size 3×3. For the loaded mode, the manipulator stiffness matrix JF21x JF 11 y JF 22 x JF 12 y JF 23 x JF 13 y (26) can be computed using the extended VJM technique K gJF22 xyxyxy JF 12 JF 22 JF 12 JF 23 JF 13 JF JF JF JF JF JF proposed in [22]. Within this technique, let us assume 23x 13 y 23 x 13 y 23 x 13 y 3 3 T that there is a non-negligible deflection (x , y ) Further, after expressing the virtual joint torque variation caused by the external force F (F , F )T , and there is a x y as MKq q and its substitution to (23), the variable small deflection ( x , y )T caused by this force q can be presented as
Figure 8. Force-deflection curves and stiffness coefficients for “non-straight” initial configuration with different parameters (a, b, Lo ) and displacement x, y b 2,0 . Newton’s Method). There are three combinations of the geometric parameters a/b ϵ{0.75; 0.9; 1.1}, relevant results are presented in Figs. 9 and 10. As follows from these figures, in most cases the manipulator stiffness essentially changes if the external loading is applied. In particular, the manipulator resistance in the x-direction becomes lower and lower while the force Fx is increasing (see Fig. 9a). In contrast, the resistance in the y-direction with respect to the force Fy becomes higher and higher while this force is increasing (see Fig. 9b). These results are also confirmed by the Kxx and Kyy plots presented in Fig. 10, which show an enormous loss of x-direction resistance under the Fx loading (it can be treated as a “quasi-buckling”, see Fig. 10a for the stiffness coefficient Kxx). On the other side, while increasing the force Fy, the stiffness coefficient Kyy is very small at the beginning, then it is increasing until reaches the maximum value, and then it is decreasing (see Fig. 10b). In this figure, an evolution of the manipulator configuration under the loading are also presented, with relevant stiffness Figure 9. Force-deflection relations of three-segment mechanism coefficients Kxx and Kyy plots (corresponding to the case x, y 5.5 b , 0 a/b=0.75). They demonstrate the above mention results for non-straight initial configuration with o . from the geometrical and physical point of view, which 1 are corresponded to the stiffness coefficient and force qKK JFT (27) q g relation. There are four representative configurations presented here, which showing the shapes of all segments which allows us to find the end-effector deflection and their position with respect to the joint limits. As J q , and finally to obtain the desired loaded follows from them, the observed sudden change of the compliance and stiffness matrices stiffness (see Figs. 9 and 10) occurs if one of the
1 segments is close to its joint limits, where the equivalent T CJKKJF q g rotational stiffness coefficient is very low. Hence, in 1 1 (28) T KJKKJF q g practice, it is necessary to avoid applying too high loading, or the manipulator will approach its joint limits It is worth mentioning that all the Jacobian and the joint and lose stiffness. stiffness matrices Kq, Kg must be computed for the Therefore, as follows from the above study, the loaded equilibrium configuration, which is different from mechanical properties of a serial manipulator based on the initial unloaded one (It requires relevant solutions of dual-triangle segments have several particularities, which the non-linear equations considered above). are different from a classical serial structure composed of To illustrate the importance of the loaded stiffness rigid links and compliant components. These analysis, the obtained expressions were applied to several particularities must be obligatory taken into account in cases study, which focusing on the manipulator stiffness control algorism, for ensuring desired motions of such changing under the external loading. For all considered manipulator, which is in the focus of our future research. cases, it was assumed that the initial manipulator configuration is a non-straight one, with the endpoint IV. CONCLUSION location (x0, y0)=(5.5b,0). Under the loading the The paper focuses on the compliant serial manipulator configuration angles corresponding to the external force T composed of a new type of dual-triangle tensegrity F=(Fx, Fy) were computed from (11) numerically (using mechanism, which is composed of rigid triangles
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