Analysis of Actuator Redundancy Resolution Methods for Bi-articularly Actuated Robot Arms

Transcription

1 The 1th IEEE International Workshop on Advanced Motion Control March 5-7, 01, Sarajevo, Bosnia and Herzegovina Analsis of Actuator Redundanc Resolution Methods for Bi-articularl Actuated Robot Arms Valerio Salvucci Department of Advanced Energ Engineering The Universit of Toko Kashiwanoha, Kashiwa, Chiba, Japan Sehoon Oh and Yoichi Hori Department of Electrical Engineering Universit of Toko, -7-1 Hongo Bunkoku Toko, Japan Abstract Bi-articular actuators actuators that span two joints are rising interest in robot application because the increase stabilit, optimize force production, and reduce the non-linearit of the end effector force as a function of force direction. In this paper, we propose an approach to resolve actuator redundanc for bi-articularl actuated robot arms in which the three actuators produce maimum joint actuator torques that differs among each other. A closed form solution based on the infinit norm is derived. The proposed infinit norm based approach is compared with the conventional orm and -norm based methods. Under the same actuator limitations, the maimum end effector force produced with the proposed method is significantl greater than the one produced b the conventional methods. The proposed closed form solution is suitable for redundant sstems with three inputs and two outputs, bringing the advantage of an higher maimum output without the need for iterative algorithms. I. INTRODUCTION Robot arms presenting animal musculo-skeletal characteristics such as bi-articular actuators actuators that span two joints have been proposed for more than two decades [1]. In recent ears there has been increasing attention on such animal inspired robot arms, both in hardware and control design aspects.regarding the hardware design, bi-articularl actuated robots have been realized b means of pneumatic actuators [], [], and motors with transmissions sstems based on pulles [4], [5], planetar gears, [6], [7], wires [8], [9], and passive springs [10], [11]. All these robots are driven b more actuators than joints, resulting therefore in actuator redundanc. In order to resolve the actuator redundanc problem due to the presence of biarticular actuators, man approaches have been proposed. In [1] several animal inspired approaches such as fatigue minimization, muscle force minimization, total muscle metabolic energ consumption, total muscle stress minimization are compared among each others. Among these approaches, the muscle force minimization [1] is implemented on robot applications using the 1 norm. Approaches based on pseudo-inverse matrices are used in the control design for kinematicall redundant robot arm [14], [15]. Pseudo-inverse matrices are also used for actuator redundanc resolution [5], [16]. Moore-Penrose is the simplest pseudo-inverse matri, and correspond to the minimization of the euclidean norm ( norm) [17]. Iterative algorithms based on norm optimization criteria have been used to resolve redundanc in kinematicall redundant robot arms [18], [19]. In this paper the norm norm is used to resolve the actuator redundanc for bi-articularl actuated robot arms. Differentl from our previous work [0], the here proposed norm based approach is etended also to the case in which the three actuators produce different maimum joint actuator torque among each other. A closed form solution based on a piecewise linear function for the infinit norm approach is proposed. The norm approach allows to minimize the necessar maimum torque at each joint for a given force at end effector. Therefore it is an approach to optimize actuators design for robot arm equipped with b bi-articular actuators. In addition, the proposed norm is analticall compared with the traditional 1 norm and norm approaches both in terms of joint actuator input torques and maimum output force at the end effector. In Section II main features and statics of robot arms equipped with bi-articular actuators are described. Then, in Section III, three approaches for torque distribution resolution minimization of muscle force (1 norm), norm and norm are introduced. In Section IV the characteristics of 1 norm, norm and norm approaches are analzed in terms of joint actuator input torques and maimum output force at the end effector. Finall, in Section V, the advantages of the proposed optimization criteria are summarized. II. CHARACTERISTICS AND MODELING OF ROBOT ARM WITH BI-ARTICULAR ACTUATORS In conventional robot arms each joint is driven b one actuator. On the contrar, animal limbs present a comple musculoskeletal structure based on two tpes of muscles: 1) Monoarticular: the contraction of one of these muscles produces a torque on one joint. ) Bi-articular: the contraction of one of these muscles produces the same torque on two consecutive joints at the same time. Gastrocnemius is an eample of biarticular muscle in the human leg. A simplified model of the comple animal musculoskeletal sstem is shown in Fig. 1. This model is based on 6 contractile /1/$ IEEE

2 Fig. 1. Two-link arm with four mono- and two bi-articular muslces: model and resulting forces at the end effector Fig.. Statics of two-link arm with four mono- and two bi-articular actuators actuators etensors (e1, e and e) and fleors (f1, f and f) coupled in three antagonistic pairs: e1 f1 and e f: couples of mono-articular actuators that produce torques about joint 1 and, respectivel. e f: couple of bi-articular actuators that produce torque about joint 1 and contemporaneousl. Robot arms driven b bi-articular actuators have numerous advantages: dramatical increase in range of end effector impedance which can be achieved without feedback [1], realization of path tracking and disturbance rejection using just feedforward control [1], improvement of balance control for jumping robots that do not use force sensors []. Moreover, multi-joints actuators such as tri-articular actuators, increase the efficienc in the output force for robot arm [4]. Another advantage of robot arm equipped with bi-articular actuators is the abilit to produce a more homogeneousl distributed maimum output force at the end effector [0], []. III. ACTUATOR REDUNDANCY PROBLEM AND RESOLUTION METHODS The resulting statics of the bi-articularl actuated arm of Fig. 1 are shown in Fig., where: T 1 = τ 1 + τ (1) T = τ + τ () The total torques about joint 1 and are T 1 and T, respectivel. The torques produced b mono-articular actuators about joints 1 and are τ 1 and τ, respectivel. The are calculated from the actuator input forces e i and f i for i = (1,) as: τ 1 = ( f 1 e 1 )r () τ = ( f e )r (4) where r is the distance between the joint ais and the point where the muscle force is applied, consider to be the same for all the muscles and all joint angles. The bi-articular torque produced about both joints is τ : τ = ( f e )r (5) F is a general force at the end effector with magnitude F and direction θ f. A two-link robot arm with the statics shown in Fig. presents actuator redundanc. Given τ 1, τ, and τ, it is possible to determine T, and therefore F b using the Jacobian: [ ] [ ] T1 = J T F (6) F where J = T [ l1 sin(θ 1 ) l sin(θ 1 + θ ) l sin(θ 1 + θ ) l 1 cos(θ 1 ) + l cos(θ 1 + θ ) l cos(θ 1 + θ ) and F and F are the orthogonal projection of F on the -ais and -ais, respectivel. On the other hand, given F, and therefore T, it is generall not possible to determine uniquel τ 1, τ, and τ (see (1) and ()) due to the actuator redundanc. The problem represented b (1) and () is referred in the following as the redundanc actuator problem. A. 1 norm approach The actuator redundanc is resolved using the 1 norm b solving the following problem: min ( τ m 1 + τ τ m ) + τ m ] (7) s.t. T 1 = τ 1 + τ T = τ + τ (8) where τ m i,i = (1,,) is the maimum joint actuator torque that the actuator i can produce. The problem is solved using an iterative algorithm. Software tools as MATLAB can solve such problems.

3 B. -norm based approach The actuator redundanc is resolved using the norm b solving the following problem: (τ min 1 ) ( m + (τ ) ) (τ m + (τ ) ) ( m) (9) s.t. T 1 = τ 1 + τ T = τ + τ The solution of the problem epressed b (9) is: τ 1 = (T 1 T )(τm 1 ) (τ m ) + T 1 (τ m 1 ) (τ m ) (τ m 1 ) (τ m ) + (τ m 1 ) (τ m ) + (τ m ) (τ m ) (10) τ = T (τ m 1 ) (τ m ) + (T T 1 )(τ m ) (τ m ) (τ m 1 ) (τ m ) + (τ m 1 ) (τ m ) + (τ m ) (τ m ) (11) T τ = 1 (τ m) ( m) + T ( m) ( m) ( m) (τ m) + ( m) ( m) + (τ m) ( m (1) ) Proof of (10), (11), and (1) is reported in Appendi A. If m = τm = τm the solution becomes [0]: C. Infinit norm based approach τ 1 = T 1 1 T (1) τ = 1 T 1 + T (14) τ = 1 T T (15) The actuator redundanc is resolved using the norm b solving the following problem: min ma m, τ τ m, τ m s.t. T 1 = τ 1 + τ (16) T = τ + τ The fact that three torque values are scaled b the respective maimum torque guarantees that the solution, when eists, does not violate an of the three constraints: Let us define: τ m 1 τ 1 τ m 1 (17) τ m τ τ m (18) τ m τ τ m (19) c 1 = τm τm 1 τ m + τm c = τm + τm τ m + τm 1 c = τm τm τ m + τm 1 (0) (1) () The three parameters c 1, c, and c are defined for an maimum joint actuator torque, and are constant. A closed form solution of the problem (16) is determined on the basis of the values of T 1 and T as follows: (T 1 T ) m τ m if case 1 +τm 1 τ 1 = T 1 T m τ m if case +τm () T m 1 m if case +τm τ (T T 1 ) m τ m if case 1 +τm 1 τ τ = T m τ m if case +τm (4) T T m 1 m if case +τm T 1 τ m +T τm 1 τ m if case 1 +τm 1 τ = T m τ m if case +τm (5) if case where T 1 τ m τ m 1 +τm case 1 = (T 1 c 1 T and T c T 1 ) or (T 1 > c 1 T and T < c T 1 ) case = (T 1 c 1 T and T c T 1 ) or (T 1 < c 1 T and T < c T 1 ) case = (T c T 1 and T c T 1 ) or (T > c T 1 and T < c T 1 ) Proof of (), (4), and (5) is reported in Appendi B. It is trivial to verif that the three linear piecewise functions (), (4), and (5) are continuous in all the domain D = (T 1,T ). In summar, the values of τ 1, τ, and τ that produce a given F at the end effector, are determined as follows: 1) Calculate the desired joint torques T = J T F. ) According to calculated T 1 and T, the three desired joint actuator torques are directl determined using the three piecewise linear function (), (4), and (5). When all the actuators produce the same maimum joint actuator torque, that is m = τm = τm, c 1 = c = 0 and c = 1, and the solution becomes as in the following [0]: T 1 T if T 1 T 0 τ 1 = T 1 T if T 1 T > 0 and T 1 T (6) T 1 if T 1 T > 0 and T 1 > T T T 1 if T 1 T 0 T τ = if T 1 T > 0 and T 1 T (7) T T 1 if T 1 T > 0 and T 1 > T T 1 +T if T 1 T 0 T τ = if T 1 T > 0 and T 1 T (8) T 1 if T 1 T > 0 and T 1 > T IV. RESULTS In the following a two-links robot arm with l 1 = 15 m and l = 1 m is taken into account. The maimum joint actuator torques are m = 16 N, τm = 14 N, and τm = 18 N.

4 Torque [Nm] Torque [Nm] F [N] , -n, F [N] (a) Desired output force ate the end effector ( F = 1 N) -n θ f [ ] (b) τ 1 m + τ τ m + τ m for 1 norm, norm and norm approaches n θ f [ ] (τ (c) 1 ) ( m + (τ ) ) (τ m + (τ ) ) ( m for 1 norm, norm and norm approaches ) Torque [Nm] n θ f [ ] (d) ma m, τ τ m, τ m for 1 norm, norm and norm approaches Fig.. Comparison of joint actuators torque inputs among 1 norm, norm and norm approaches F [N] n F [N] Fig. 4. Comparison of maimum output force at end effector θ = (0,60,90,10,150 ) A. Joint actuator torques comparison Fig. shows a comparison of actuator joint torques input for three actuator redundanc resolution methods, 1 norm, norm and norm. The desired output is an output force F at the end effector with magnitude 1 N in all the directions for θ = 90. The compared values are τ 1 + τ τ m + τ (τ m (Fig. (b)), 1 ) ( m + (τ ) ) (τ m + (τ ) ) ( m (Fig. (c)), and ) ma m, τ τ m, τ m (Fig. (d)). The advantage of 1 norm approach is that requires the lowest sum of actuator joint torques (Fig. (b)). The advantage of norm is that allows to minimize the maimum actuator joint torques (Fig. (d)). norm is in between the other two approaches. The same results can be obtained for θ 90. B. Maimum output force Fig. 4 shows the maimum output force at end effector for θ = (0,60,90,10,150 ). Given the same maimum actuator joint torques norm can produce the greatest force at end effector. This is the great advantage of the norm approach. 1 norm produces the lower output force, while norm is in between the other two approaches. V. CONCLUSIONS In this paper, we propose an approach to resolve actuator redundanc for bi-articularl actuated robot arms in which the three actuators produce maimum joint actuator torques that differs among each other. Moreover, the proposed norm based approach is compared with the muscle force minimization approach (1 norm) and with the Moore-Penrose pseudoinverse matri ( norm) approaches. Main results are: The proposed norm approach allows the maimization of output force at the end effector. The 1 norm approach minimizes the total muscle force input, but the maimum force space is the smallest. If a circular output force the end effector is desired, this approach requires the highest joint actuator torque. The norm approach is in between 1 norm and norm in terms of both maimum force space and total torque input. τ m 1

5 ACKNOWLEDGMENT This work is supported b Inamori Foundation APPENDIX A PROOF OF CLOSED FORM SOLUTION FOR THE -NORM APPROACH The problem epressed b (9) is written for a simpler notation as: () min + () + (z) (m) () () (9) s.t. T 1 = + z T = + z where T 1 and T are the desired joint torques (known),,, and z are the desired joint actuator torques τ 1, τ, and τ (unknown), respectivel; m = m, = τm, and = m. Taking into account R, the solution (,, z) which () satisf + () + (z) has to meet the following three (m) () () requirements: 1) To be on the line defined b T 1 = + z (0) T = + z (1) ) To be on the ellipsoid surface defined b: m + + z = k () where k is a constant. ) The plane passing through the line defined b (0) and (1) has to be tangent to the ellipsoid defined b (). Hence: 1 m z z = 0 () Combining (0), (1), (), () straightforward follows the solution of the problem (9): = (T 1 T )m + T 1 m m + m + (4) = T m + (T T 1 ) m + m + (5) z = T 1 + T m m + m + (6) Equations (4), (5), and (6) correspond to (10), (11), and (1), respectivel. APPENDIX B PROOF OF CLOSED FORM SOLUTION FOR THE INFINITY-NORM APPROACH The problem epressed b (16) is written for a simpler notation as: min ma m,, z s.t. T 1 = + z (7) T = + z where T 1 and T are the desired joint torques (known),,, and z are the desired joint actuator torques τ 1, τ, and τ (unknown), respectivel; m = m, = τm, and = τm. A closed form solution of (7) is determined in the following. The searched solution has to satisf at least one of the three equations m =, = z, m = z. In fact, when one of three variable s absolute value decreases at least one of the other two increases. Therefore for an solution of the sstem with m z it is possible to decrease the higher value among the three so to be equal to at least one of the other two. Therefore the searched solution is one among the following si: 1) m = + z = T 1 + z = T m = ) = z ) m = z + z = T 1 + z = T = z + z = T 1 + z = T m = z 4) m = + z = T 1 + z = T m = 5) = z 6) m = z Let us define: + z = T 1 + z = T = z + z = T 1 + z = T m = z m = (T 1 T ) = (T T 1 ) z = T (T T 1 ) = T 1 T = T z = T + + m = T 1 = T T 1 z = T 1 +m +m +m + +m +m +m m = (T T 1 ) = (T T 1 ) z = T (T T 1 ) = T 1 T = T z = T m m m = T 1 = T T 1 z = T 1 m c 1 = + + c = + m c = + m m m m m (8) (9) (40) (41) (4) (4) (44) (45) (46) These values depend onl on the hardware characteristics of the arm, therefore are constant. Among the si possible solutions the searched one is directl selected on the basis

6 of T 1 and T as follows (the variable subscript represents the respective equation number): If (T 1 c 1 T 1 and T c T 1 ) or (T 1 > c 1 T and T < c T 1 ): = (8) m z (8) (47) (8) (9) (48) (8) (8) (40) (49) (8) (41) (50) (8) (4), if (51) (8) (4), if < (5) (8) (4), if m (5) (8) (4), if m < (54) Therefore solution is (8). In this case, τ 1 in (), τ in (4), and τ in (5), are equal to in (8), in (8), and z in (8), respectivel. If (T 1 c 1 T and T c T 1 ) or (T 1 < c 1 T and T < c T 1 ): (9) = z (9) (9) (55) m z (9) z (8) (56) (9) (40) (57) (9) (41), if m (58) z (9) z (41), if < m (59) (9) (4) (60) (9) (4), if m (61) z (9) z (4), if m < (6) Therefore solution is (9). In this case, τ 1 in (), τ in (4), and τ in (5), are equal to in (9), in (9), and z in (9), respectivel. 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