IROS2019国际学术会议论文集2478

Mathematic Modelling and Optimal Design of a Magneto-Rheological Clutch for the Compliant Actuator in Physical Robot Interactions Guangzeng Chen1, Yunjiang Lou2, Senior Member, IEEE and Tongyi Shang3 AbstractMagneto-rheological actuators (MRAs) equipped with magneto-rheological clutches (MRCs) are newly used in robots to provide intrinsic compliance and sense the human con- tacts. However, using the favorite fi nite element analysis (FEA) method to analyze and design the MRC is very computationally expensive and greatly depends on the engineering experiences and software skills of the researchers. In this paper, a novel mathematic and multidisciplinary modeling (magnetic and thermal modeling) method of an MRC is proposed. Benefi tting from the mathematic model, the MRC can be mathematically depicted with its input current and structural parameters. So, deep insights into the MRC can be easily achieved just within one second and fast and multidisciplinary optimal and global design can be realized combining the genetic algorithm. The proposed method is validated with both the FEA simulations and experiments on the optimally designed prototype. I. INTRODUCTION The motor-reducer system with high reduction-ratio reduc- ers and electric motors in series are widely used in robotic actuators. However, the high reduction ratio reducers greatly limit the compliance and backdrivability of the actuators, making the robots not compliant or unsafe in physical interactions 1. Addressing this challenge, the magneto- rheological actuators (MRAs) is one of the promising hard- ware solutions. The structure of the MRA is simple, where only one motor-reducer system is needed in serial with a magneto-rheological clutch (MRC) before connecting with the robot link. Then by controlling the input current of the MRC, the phases of the smart material inside the MRC, the magneto-rheological fl uid (MRF), can be continuously and reversibly changed from a free-fl owing liquid state to a solid-like state within several milliseconds 2, thus the brake torque of the MRC can be fast controlled to adjust the robot compliance. In such way, the MRC introduces variable compliance and backdrivability to the MRA and the slipping detection and the self-torque-sensing of the MRC can help to recognize the physical interactions. It has been proved that MRAs has many advantages in safety 3, compliance 4, high torque density 5, high-bandwidth and high-fi delity force control 6 and impact-tolerant 7. The MRC, serving as the compliance adjusting mod- ule in MRA and robots, has to be deeply studied and well designed at fi rst. Normally, the fi nite element analysis *This work was supported in part by the NSFC-Shenzhen Robotics Basic Research Center Program U1713202 and in part by the Shezhen Science and Technology Program under Grant JCYJ20180508152226630. The Authors are with the State Key Laboratory of Robotics and System, School of Mechatronics Engineering and Automation, Harbin Institute of Technology (Shenzhen), 518055 Shenzhen, China (hit.chen.gz; louyj; 1780319893) (FEA) method 814 and the magnetic equivalent circuit (MEC) method 1517 are used for the MRC analysis and design. In between the FEA method is much more favorable and accurate since the FEA method is able to achieve the nonlinear magnetic fl ux density everywhere, and realize the multidisciplinary (normally the magnetic circuit and the thermal circuit) analysis and design. However, it comes across great diffi culties in analyzing and designing the MRCs that utilized in robotic applications. Compared to the conventional MRCs with only one or a few millimeter- sized discs (1-13mm) and MRF gaps (0.5-2.0mm) 811, the MRCs utilized in robotic applications are with dozens of submillimeter-sized (0.2-1mm) discs and micro-sized (20- 100m) MRF gaps 12 13, in order to achieve a compact size and high torque density. To achieve an accurate design, the numerous and very thin discs and MRF gaps have to be meshed and analyzed more precisely. As a result, the FEA analysis and design of the MRCs become very time consuming, not to mention the multidisciplinary and optimal design. 16 took 8 hours to design a 2Nm MRC with only two discs by using the simplifi ed 2D FEA method. 14 took 11 hours to optimally design an MRC, and 10 took about 100 hours to have a global and multidisciplinary design of an MRC in the same way. It reveals that the FEA method is engineering experiences-depended, where various experience can lead to various results and lacking experiences may cause fails in design. And normally, details of the software usage and design, such as structure modeling, meshing, element and boundary choosing, etc, are not given in the papers. Therefore, it is not easy to duplicate the FEA design and researchers have to take times to learn software usage and accumulate experiences to realize a good design. On the contrast, the MEC method is an effi cient mathematic method and requires less engineering experiences. However, it oversimplifi es the MRC in modeling and ignores the nonlinearity and mutation of the magnetic fi eld, hence, only the average magnetic fi eld can be approximately computed, resulting in inaccurate modeling and being less popular in practices. And in the previous works, the MEC method is solely used in analyzing and designing the MRCs without considering the thermal safety, making it less reliable in designing a safely working MRC. In this paper, a novel mathematic and multidisciplinary modeling method of an MRC is proposed. Benefi tting from the modeling method, the MRC can be explicitly depicted with its input current and its structural parameters. So deep insights into the MRC can be easily achieved just within one second and fast and multidisciplinary optimal design can IEEE Robotics and Automation Letters (RAL) paper presented at the 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 Copyright 2019 IEEE be realized. The paper is organized as follows. In section II, preliminaries of designing an MRC are introduced. In section III, the novel mathematic and multidisciplinary model of an MRC is proposed in details and validated by the FEA simulations. In section IV, an optimal and multidisciplinary design is presented and the prototype is manufactured and tested to validate the proposed method. Finally, a conclusion is drawn in section V. II. PRELIMINARIES Figure 1 shows the typical structure of a multi-discs MRC. It consists of a rotor made of pure iron, an aluminum stator, coils, a discs supporter made of aluminum, many stator discs and rotor discs that are made of silicon iron and immersed in MRF. When current is applied to the coils, magnetic fl uxes are generated, passing along the path through MRF, as shown in the fi gure. Then MRF changes rapidly from liquid state to semisolid state and hence the stators and the rotors immersed in the MRF are engaged under magnetic forces 8, 9. As a whole, the MRC shows current-controllable brake torque between the stator and rotor. Fig. 1.The typical inner coil multi-discs type MRC structure In designing an MRC, the most challenging and time- consuming process is the computation of the nonlinear magnetic distribution in the MRF area. For the convenience of analysis, the magnetic distributions of a multi-discs MRC, with very thin discs and MRF gaps, is fi rstly achieved via 3D FEA method with the commercial software ANSYSr Workbench, when applying three individual input currents, 0.5A, 1A and 2A. The results are shown in Fig. 2. In the right sub-fi gure of Fig.2, “Radius of the MRC” refers to the radial length of the MRF area inside the MRC, i.e., from ri= 28mm to ro= 70mm. The B(r) curves in the fi gure represent the magnetic fi eld distributions along the radius direction (see also the x-axis in the left sub-fi gure of Fig.2) in the three operating cases respectively. As can be seen in the left sub-fi gure, the excited magnetic fl ux fl ows outside the core of the rotor and then passes through the MRF perpendicularly, and the magnetic fi eld in the MRF area concentrically and nonlinearly distributes along the radius direction of the MRC 10 11 18 19. The B(r) curve Fig. 2. Nonlinear magnetic fi eld distribution achieved by 3D FEA. The MRC are with the outer radius ro= 70mm, the inner radius ri= 28mm, the MRF gaps number N = 8, and the thickness of the discs ed= 0.5mm, and the thickness of MRF gap eF= 0.05mm. data will be imported into MATLAB to accurately compute the corresponding fi eld induced brake torque with (1) or approximately with (2) 11. T = 2N ro ri (B(r)r2dr(1) 2 3N( B)(r3 o r 3 i) (2) in which B(r) is the nonlinear magnetic fi eld distribution, (B) is the nonlinear fi eld-induced yield stress of the MRF material and N is the number of MRF gaps. Equation (2) is the approximation of (1) by treating the nonlinear magnetic fi eld B as constant, normally the average magnetic fi eld B over the MRF area (shown as the dash lines in the fi gure). In many works, the magnetic fl ux density is assumed to be uniform and treated as a constant, and the approximation equation (2) is used instead of (1) for the convenience of calculating and reducing the calculation time 912. However, using the approximation will inevitably result to less accurate torque computation. In the three cases, each 3D FEA simulation takes about 5.8 hours with a PC equipped with an IntelrCore i7- 6700K4GHz CPU and 8GB RAM. It takes a very long time because of the very thin 7 discs and 8 MRF gaps need to be precisely meshed and analyzed. The simulation or design time will be much more challengeable in practice in the multidisciplinary and the optimal design of the MRC with dozens of thin discs and MRF gaps. Because more physics has to be considered in the multidisciplinary design, and the above process needs to be repeated hundreds of times in the optimization. In order to achieve a fast and multidisciplinary (magnetic and thermal modeling) optimal design of the MRC, a novel mathematic multidisciplinary modeling method of an MRC is proposed in the next section. III. SYSTEMMODELLING A. Magnetic Circuit Modelling In this paper, some signifi cant improvements are made on the MEC method for MRC modeling. With this novel improved MEC method, the nonlinearity and mutation of the magnetic fi eld can be explicitly formulated with the input current and the MRC structure. The proposed model of an MRC is shown as Fig.3 and components are colored the same as in Fig.1. The notations in Fig.3 are described as in Table.I. In the fi gure, the blue arrows are the magnetic fl ux generated by the coil currents. Specially in this paper, the magnetic fl ux will be modeled as numbers of changeable bunches. The reluctance of each component are nonlinearly modeled according to its shape and size as well as the behavior (the passing path and the numbers) of the magnetic fl ux inside, and will be computed in the next subsection. The details of the modeling idea are described below. Fig. 3.The proposed model of an MRC in this paper TABLE I NOTATIONS AND DESCRIPTIONS INFIG.3 NotationDescription rhothe radius of the hollow rcthe radius of the core of the rotor rithe inner radius of the MRF area rothe outer radius of the MRF area rsthe radius of the stator tythe thickness of the yoke of the rotor hcthe height of the core/coils eFthe thickness of the MRF gap Nthe number of the MRF gaps edthe thickness of the discs lathe annular thickness between rcand ri lmthe annular thickness between riand ro lsthe annular thickness between roand rs 1) The cross-section area of the core is uniform, so the magnetic fl ux density is treated uniformly and reluctance is c. 2) At the corner, the magnetic fl ux is forced to change 90 along inside the yoke with non-uniform cross section. In this paper, the path of the magnetic fl ux is modeled with the elliptical path and the reluctance of the corner is represented by cc. 3) As the magnetic fl ux pass through the lapart, the main magnetic fl ux keeps passing along the radius direction with increasing cross section, with a few magnetic fl ux leaking into the air. So the reluctance of the lapart is represented by aand the leakage reluctance is 1. 4) As the magnetic fl ux pass through the lmpart, the behavior of the magnetic fl ux is complex. In this paper, its assumed that as the magnetic fl ux passing above the MRF area, the magnetic fl ux fl ows into the MRF area while it keeps passing along the radius direction in the yoke with increasing cross-section area but with less amount. Therefore, two kinds of reluctance should be considered in this part. The radius direction reluctance of lmpart is represented by m/r, the perpendicular direction reluctance is represented by m, and the sum of them is m. 5) Similar to a permanent magnet exposing in the air, the magnetic fl ux density is weaker farther away from the magnet. The magnetic fl ux density nonlinearly distributes along the radius direction inside and it is weaker farther away from the MRC center. The nonlinear magnetic fl ux density will be solved later. In this part, the reluctance of the total MRF gaps is Fand reluctance of the total discs is d. 6) Finally, the leakage outside the shell is considered and its reluctance is 2. B. Calculation of Average Magnetic Flux Density According to the proposed model in Fig.3, the total magnetic fl ux are generated by the coil turns n and operating current I. According to the Amperes circuital law nI = Hl = where H is the magnetic intensity, l is the length of the magnetic path is the total magnetic fl ux and is the total reluctance, we can have nI = c(c+ 2cc+ 2a) + F(F+ d+ 2m) where cand F are the total magnetic fl ux pass through the core and the MRF area respectively, and c= lc c(Bc)Sc = lc c(Bc)(r2 c r2ho) cc= 2 0 0.5 c( c (2rccos)(2rc cos) d a= ri rc dx c( c 2xty) 2xty F= lF F(B)SF ,d= ld d(B)SF m= m+ m/r,m= 0.5ty c(B)SF m/r= 0.5 tySF ro ri r2 o x2 xc( B(r2 ox2) 2xty ) dx are the nonlinear reluctance that modeled according to the descriptions in section A. lF= NeFis the thickness of the total MRF gaps, ld= (N 1)edis the total thickness of the discs, lc= lF+ld, SF= (r2 or2i) is the area of the MRF area, = (r c rho)cos2+ tysin2and is the nonlinear permeability of the material shown as Fig.4. Fig. 4.Permeability and yield stress of materials Similar to the electric circuit, F(F+ d+ 2m) = 11, in which 1is the leakage. Solving the equation will result in 1 F = F+ d+ 2m 1 , (3) Assuming that 2 1, then c= F+ 21 (1 + 2)F Therefore, the average magnetic fl ux density in the MRF area can be computed by B = F SF = nI SF (4) where = (1 + 2)(c+ 2cc+ 2a) + (2m+ F+ d) And the magnetic fl ux density in the leakage area is B= S where S is the area of the leakage fl ux pass through. C. Deduction of the Magnetic Nonlinearity As the magnetic fl ux pass along the yoke above the MRF area, the magnetomotive force occurs. The magnetomotive force along the yoke between the radius of r and R is F = 2 R r Bm(x) c(Bm(x)dx (5) where Bm(x) = BSF x ri B(t)2tdt 2xty is the magnetic fl ux density at the radius of x of the yoke induced by the remaining magnetic fl ux, and in which B(t) is the continuous magnetic fl ux density distribution inside the MRF that needs to be deduced later. On the other hand, the magnetomotive force F can also be calculated by F = HF(R) HF(r)lF+ Hd(R) Hd(r)ld = (1 + )HF(R) HF(r)lF(6) where = Hd(R) Hd(r) HF(R) HF(r) HdBF BdHF ? ? ? ? B 1 400, 1 1000 = ld lF By solving (5) and (6), its easy to have HF(R) HF(r) = 2 (1 + )lF R r Bm(x) c(Bm(x)dx (7) Then by multiplying the permeability of the MRF at its average magnetic fl ux density at both sides of (7), the the magnetic fl ux density difference between radius of r and R can be achieved as below B(R) B(r) = 2F(B) (1 + )lF R r Bm(x) c(Bm(x)dx (8) Recalling that f(b) f(a) = b a f(x)dx, therefore B(x) = 2F(B)Bm(x) (1 + )lFc(Bm(x) = F(B) (1 + )lF BSF x ri B(t)2tdt xtyc(Bm(x) = Q P B(x)(9) where Q = W(x,B)(BSF+ B(ri)r2 i + (0)B(x) P = W(x,B)x2 W(x,B) = F(B) (1 + )lFtyc(Bm(x)x (0)B(x) = x ri B(x)t2dt By solving (9), the formulation of the nonlinear distribution of magnetic fl ux density inside MRF is B(x) = e x riPdx(C + x ri Qe x riPdxdx) (10) where C is a constant. Its easy to know that if C is known, then the nonlinear distribution of magnetic fl ux density inside MRF can be accurately determined by (10). To establish a more accurate magnetic model, the magnetic mutation is considered in the modeling as well. The magnetic mutation easily occurs near the inner radius of the MRF, where is the boundary of the MRF and the air. The leakage fl ux in the air will enforce the magnetic fl ux density inside the MRF near the boundary. In this paper, the magnetic mutation is assumed to begin at the inner radius riand dissipated quickly along the radius direction. Motivated by the exponential form (10), the magnetic mutation is supposed as below b(x) = Be ro(xri) r2 i (11) where B is the magnetic fl ux density in the leakage area and is a factor determined by the MRC structure. In this paper, w h and w, hare width and height of the leakage area in Fig.3. With considering the magnetic mutation, the magnetic fl ux density distribution is B(x) =e x riPdx(C + x ri Qe x riPdxdx) + b(x) (12) Noting that the total magnetic fl ux pass through the MRF area is F= BSF= ro ri 2xB(x)dx + ro ri 2xb(x)dx Then the constant C can be solved as below C = BSF ro ri 2xb(x)dx m2 m1 (13) where m1= ro ri 2xe t riPdtdx m2= ro ri 2te t riPdt t ri Qe z riPdzdtdx In order to quickly solve