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Mechanics of Compliant Serial Manipulator Composed of Dual-Triangle Segments Wanda Zhao, Anatol Pashkevich, Damien Chablat, Alexandr Klimchik

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Mechanics of Compliant Serial Manipulator Composed of Dual-Triangle Segments Wanda Zhao, Anatol Pashkevich, Damien Chablat, Alexandr Klimchik Mechanics of compliant serial manipulator composed of dual-triangle segments Wanda Zhao, Anatol Pashkevich, Damien Chablat, Alexandr Klimchik To cite this version: 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 HAL Id: hal-03195185 https://hal.archives-ouvertes.fr/hal-03195185 Submitted on 10 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Mechanics of compliant serial manipulator composed of dual-triangle segments 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.
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