I-RIM Conference 2019 October 18 - 20, Rome, Italy ISBN: 9788894580501 Workspace Analysis Of Fully Routed Two-Link Tendon-Driven Manipulators Vishal Ramadoss*, Keerthi Sagar*, Rezia Molﬁno, Dimiter Zlatanov and Matteo Zoppi PMAR Robotics Group University of Genoa Via allOpera Pia, 15/a - 16145 Genoa, Italy {keerthi, molﬁno, zlatanov, zoppi}@dimec.unige.it, austinvishal@gmail.com Abstract—This paper presents workspace analysis of tendon- driven planar serial kinematic chains which allows single-point and multi-point routing with bundling per rigid link. Target- ing a musculoskeletal two-link system with different tendon arrangements as a case study, the wrench-closure and wrench- feasible workspace are evaluated for different conﬁgurations. It is observed that, the single-point internally routed chain has more feasible workspace than the externally routed robot with 3 tendons. Index Terms—tendon-driven, routing, bundling, wrench- closure I. INTRODUCTION Musculoskeletal systems are inherently redundant which poses challenges in kinematic modeling. A generalized mod- eling of multilink unilateral manipulators was studied in [1]. Tendon driven system [2] eliminates the need of rigid links and mechanical joints making the system light weight. The ability of a system to maintain the equilibrium with a set of positive cable tension (force-closure) is a vital tool in workspace analysis and minimization of actuators [3]–[10]. In this work, we evaluate the workspace of two-link tendon driven robot with 3 to 6 tendons. II. FORMULATION a1 p1 m1 m2 l5 a2 b2 b1 p2 l3 l6 l4 l1 l2 P (a) e1 e4 e3 a1 b1 a2 C3 C1 C2 l22 l21 l31 l11 l12 P l1q e11 e12 l1r (b) Fig. 1: a) Human arm model b) with multipoint internal routing and bundling The human arm model with both biarticular and monoartic- ular tendons is shown in in Fig. 1(a). The kinematic model of two-link tendon driven robot with different routing topologies * The ﬁrst two authors contributed equally to this work. was discussed in [11]. For 2DOF tendon driven robot, having n + 1 cables routed to its links as shown in Fig. 1(b), boundaries of the workspace are calculated analytically using Cramer rule. A is of order (2 × 3). A can be expressed using ith column, Ai as follows: τ = At (1) t = A∗τ + η(A)z (2) η(A) denotes the null space of the structure matrix and z is an arbitrary vector. A2×3 = A1 A2 A3 (3) A1t1 + A2t2 + A3t3 = τ2×1 (4) With the linear system obtained From Eq. 4, the constraint that the set of poses have to satisfy to be part of WCW is given by, ∀τ > 0 : At = τ (5) From Eq. 2 A1N1 + A2N2 + · · · + AmNm = 0 (6) A1N1 + A2N2 + A3N3 = 0 (7) A1 A2 2×2  N1 N2 2×1 = − A3N3 2×1 (8) The solution of Eq. 8 can be computed using Cramer’s rule. N1 = −A3N3 A2  A1 A2  (9) N2 = −A1 A3N3  A1 A2  (10) η = ⎡ ⎣ N1 N2 N3 ⎤ ⎦= ⎡ ⎣ A3 A2  A1 A3  − A1 A2  ⎤ ⎦ (11) To ﬁnd WCW, Eq. (11) can be used to ﬁnd regions where N1, N2 and N3 are of same sign. If N − i denotes negative values of Ni negative and N + i denotes positive values of Ni, then WCW can be expressed as the union of the regions which are formed from the intersection of N i i and N + i . The equation can be formed as below: WCW = (N − 1 ∩N − 2 ∩N − 3 ) ∪(N + 1 ∩N + 2 ∩N + 3 ) (12) The Wrench feasible workspace is the subset of WCW which is evaluated within nominal tension limits. The DOI: 10.5281/zenodo.4811592