A Precise Magnetic Walking Mechanism - IEEE Xplore

1412

IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014

A Precise Magnetic Walking Mechanism Jung Soo Choi and Yoon Su Baek

Abstract—Inspired by magnetically actuated precision motion devices based on magnetic levitation and those based on elastic suspension, we devised a novel magnetic actuation mechanism, namely, a magnetic walking mechanism. This biped precision mechanism is a legged precision mechanism that consists of two elastically connected rigid legs. We designed the mechanism to be driven by one horizontal actuator and two vertical actuators, each comprising a permanent magnet and an air-core coil, and to produce both fine motion and walking motion. We discuss the underlying principles of force generation, leg actuation, fine motion, and walking motion. To demonstrate that both fine motion and walking motion can be created with this mechanism, we built and experimentally tested a two-legged walker. We successfully created a legged precision device capable of long-range fine motion and long-step walking motion based on a magnetic walking mechanism. Index Terms—Bipedal walking, magnetic actuation, magnetic walking, micro/nanorobots, precision positioning.

I. INTRODUCTION EGGED precision positioning systems have been researched for applications such as micro- or nanoscale manipulation, assembly, automation, and microscopy [1]–[7]. A legged precision machine requires actuators to produce displacements at the micro- and nanoscales to realize movement of the machine. A general way to produce such small displacements is to use piezoelectric actuators built with piezoelectric materials. These piezoelectric actuators have been successfully used to create legged precision machines for microscale or nanoscale operations. An obvious way to achieve walking motion is to build a legged precision mechanism that uses piezoelectric actuators as legs, with no requirement for mechanical bearings such as ball or roller bearings. For instance, Binnig and Gerber [1] proposed a piezoelectric traveling support using an eight-legged lift-shift-place-lift-shift-place locomotion method. Uozumi et al. [2] constructed a piezoelectric walker based on a six-legged shrink-shear-expand-shrink-shear-expand walking mechanism. Breguet and Renaud [3] built a piezoelectric robot using a threelegged stick–slip locomotion mechanism. This type of mechanism has also been used in MINIMAN robots [4], [5]. Martel

L

Manuscript received November 8, 2013; revised March 29, 2014; accepted September 22, 2014. Date of publication October 14, 2014; date of current version December 3, 2014. This paper was recommended for publication by Associate Editor J. Dai and Editor B. J. Nelson upon evaluation of the reviewers’ comments. This work was supported by grants from the National Research Foundation of Korea (NRF) funded by the Korea government (MSIP) under Grant NRF-2008-0059827 and Grant NRF-2009-0080591. J. S. Choi is with the Department of Mechanical Engineering, Graduate School, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]). Y. S. Baek is with the School of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2014.2360492

et al. [6], [7] built a piezoelectric robot, called NanoWalker, whose design is based on a three-legged push-slip locomotion mechanism. All of these legged precision mechanisms are based on the piezoelectric actuation principle, and they are well suited for use in the manufacture of legged precision machines that walk on individually actuated deformable legs, i.e., piezoelectric legs, but such legs have short range of motion due to the limited displacements of piezoelectric actuators. To overcome this main disadvantage, a new legged precision mechanism is required. Another way to produce displacements at the micro- and nanoscales is to use magnetic actuators that comprise permanent magnets and air-core coils. These magnetic actuators have been successfully used to create precision positioning systems that require micro- or nanopositioning capability. There are two main types of magnetically driven precision positioners: positioners based on magnetic levitation (maglev) [8]–[11] and positioners based on spring or flexure suspension [12]–[15]. These devices provide frictionless fine motion with the help of noncontact actuation and frictionless suspension; therefore, they do not suffer from mechanical friction and the resulting wear. In the case of the former, mechanical friction between a moving part and a stationary part is eliminated by magnetically levitating the moving part off the stationary part. In the case of the latter, mechanical friction between a moving part and a stationary part is eliminated by supporting the moving part by means of springs or flexures. Examples of magnetically driven precision positioners based on magnetic levitation are as follows. Hollis et al. [8] designed a maglev wrist driven by six actuation units, each of which consisted of four magnets, one coil, and two steel plates. Kim and Trumper [9] developed a maglev stage driven by four permanent magnet linear motors that consisted of Halbach magnet arrays and coil sets. Jung and Baek [10] presented a maglev positioner driven by a surface actuator that comprised multiple magnets and multiple coils. Kim and Verma [11] described a maglev positioner driven by three actuation units, each of which consisted of one magnet and two coils. Examples of magnetically driven precision positioners based on spring or flexure suspension are as follows. Smith et al. [12] designed a flexure-based actuation mechanism consisting of two magnets and two coils. Wang and He [13] developed a spring-suspended positioner driven by an actuation mechanism that comprised three groups of coils and 12 magnetic shoes built with multiple magnets and multiple steel plates. Culpepper and Anderson [14] presented a flexure-based manipulator driven by three actuation units built with magnets and coils. Teo et al. [15] described a flexure-based linear actuator built with four magnets and one coil. The aforementioned precision positioners consist of two parts: a magnet assembly and a coil assembly. According to

1552-3098 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

CHOI AND BAEK: PRECISE MAGNETIC WALKING MECHANISM

Fig. 1.

Conceptual design of our proposed magnetic walking mechanism.

which of the two assemblies is movable, actuation mechanisms used for these precision positioners can be categorized into two groups: moving-magnet type [9]–[11], [14] or moving-coil type [8], [13], [15]. In a moving-magnet-type actuation mechanism, the magnet assembly is movable, and the coil assembly is stationary. In a moving-coil-type actuation mechanism, the coil assembly is movable, while the magnet assembly is stationary. Both types are suitable for use in the manufacture of micropositioning or nanopositioning systems, but it is difficult to see how these mechanisms’ range of horizontal motion could be increased without redesigning them or increasing their size. To overcome this difficulty, a new actuation mechanism is required. Inspired by magnetically actuated and levitated precision positioners and by magnetically actuated and elastically suspended precision positioners, we devised a magnetic actuation mechanism that is capable not only of frictionless fine motion over a limited range but of walking motion as well. Here, we describe our novel magnetic actuation mechanism. The conceptual design of our mechanism is provided in Fig. 1. Its principle of walking motion is illustrated in Fig. 2. Table I briefly compares our mechanism with existing magnetic actuation mechanisms [11], [15]. The magnetic actuation mechanism that we propose here consists of two elastically connected parts: a magnet assembly and a coil assembly. It is a combination of a moving-magnet-type actuation mechanism and a moving-coil-type actuation mechanism. Namely, we designed this mechanism to act as a movingmagnet-type mechanism to move the magnet assembly and to act as a moving-coil-type mechanism to move the coil assembly. Our proposed mechanism has two modes of operation: fine motion and walking motion. When the mechanism is in fine motion mode, one of this mechanism’s two assemblies can be moved with the other planted on a base. When the mechanism is in walking motion mode, it can walk using its two assemblies along the x-axis. This walking motion enables it to incrementally extend a range of travel in the x-axis direction without the requirement for extra actuation elements or increases in size, making it both compact and simple. We designed this magnetic actuation mechanism for use in the manufacture of a legged precision walker, which we, hereafter, refer to as a magnetic walking mechanism. We designed it not to suffer from friction that would limit its precision of motion; this frictionless design helps to produce precise motion, and therefore, we refer to our mechanism as a precise magnetic walking mechanism. Table II compares our mechanism with existing precision walking mechanisms [2], [3], [6], [7].

1413

The novel features of our mechanism in comparison with existing legged precision mechanisms are as follows. 1) Magnetic actuation: Movement is based on the principle of magnetic actuation. Motion is driven by one horizontal magnet-coil force actuator and two vertical magnet-coil force actuators. Each actuator consists of a permanent magnet and an air-core coil. 2) Rigid leg: It has two rigid frames, each of which is used as a rigid leg. 3) Biped: It has two elastically connected rigid legs—an upper leg and a lower leg. These two rigid legs are connected by springs to support the weight of the upper leg and to provide mechanical stiffness; therefore, the biped structure is stable during the displacement of each leg, which helps to achieve a stable walking motion. The upper leg is suspended by springs (like the moving part of an existing positioner based on elastic suspension); therefore, there is no mechanical friction between the upper leg and the base on which the mechanism walks. The lower leg, which is elastically connected to the upper leg, can be magnetically lifted off the base (like the moving part of an existing positioner based on magnetic levitation); therefore, mechanical friction between the lower leg and the base can be eliminated. 4) Bipedal walking: It locomotes via the alternation of two elastically connected rigid legs. That is, it walks by attracting its two legs to each other and repelling them from each other so as to eliminate the effect of mechanical friction between each leg and the base. It achieves walking motion by repeating this attract-propel-repel-propel cycle. More specifically, the mechanism takes one step in the following sequence. First, the upper leg is attracted down to the lower leg, and then the lower leg is attracted up to the attracted upper leg. Second, the attracted lower leg is propelled forward. Third, the propelled lower leg is repelled down from the attracted upper leg, and then, the attracted upper leg is repelled up from the repelled lower leg. Finally, the repelled upper leg is propelled forward. 5) Actuator sharing: Its rigid legs share actuators. Horizontal and vertical actuators used to drive the upper leg can be used to drive the lower leg, when the upper leg is lowered and the lower leg is lifted by the magnetic attraction forces provided by the vertical actuators; therefore, there is no need to add extra actuators to drive the lower leg. We built a magnetic walker based on the mechanism described previously and tested it experimentally to demonstrate the feasibility of the mechanism. To demonstrate the proof of concept, the walker built was a macroscopic device operated in open-loop (manual) control; this allowed us to build the device simply and to see it without using a microscope. In this study, magnetic actuation was chosen over piezoelectric actuation because piezoelectric actuators are limited in their maximum displacement; piezoelectric actuators have the advantage that they can be driven at high frequencies, but this advantage is not needed in a manually controlled positioning or manipulation system. The main contribution of this paper is the proposal of a magnetic actuation mechanism capable of frictionless fine motion

1414


Fig. 2. Magnetic walking mechanism. (a) Principle of walking motion (for clarity, the coils and magnets are not shown. A detailed description of the principle of walking motion is given in Section VI). (b) Currents applied to the vertical and horizontal coils in each transition. (c) Directions of the applied currents. The trailing superscript u denotes that the upper leg is in transition from one state to the next. The trailing superscript l denotes that the lower leg is in transition from one state to the next. The initial distance between the supports of the upper leg and the base when no currents are applied to the actuators is represented as gs 0 .

TABLE I COMPARISON OF OUR MECHANISM WITH EXISTING MAGNETIC ACTUATION MECHANISMS

Actuation configuration

Kim and Verma’s mechanism [11]

Teo et al.’s mechanism [15]

Our proposed mechanism

MOVING-MAGNET-TYPE ACTUATION

MOVING-COIL-TYPE ACTUATION

Combination of moving-magnet-type actuation and moving-coil-type actuation No redesign or increase in size is required to increase the range of horizontal motion Complex multiple-rigid-body dynamics

Advantages

Simple single-rigid-body dynamics

Disadvantages

Increasing the range of horizontal motion requires redesign or an increase in size.

and walking motion, namely, the magnetic walking mechanism and the experimental demonstration of this mechanism’s feasibility. The remainder of this paper is organized as follows. In Section II, we describe the basic structure and force-generation principles of the proposed mechanism. Vertical and horizontal actuators of the mechanism are described in Section III, while the principle of shared leg actuation is described in Section IV. In Section V, we describe the fine motion of each leg and, in Section VI the walking motion of the mechanism. A prototype built based on the mechanism is presented in Section VII, and experimental results are provided in Section VIII. Section IX concludes this paper. II. BASIC STRUCTURE AND FORCE-GENERATION PRINCIPLE The magnetic walking mechanism consists of two elastically connected moving parts (see Fig. 1), each of which is a rigid leg. In other words, this mechanism has two elastically connected rigid legs. One leg is the upper body. The upper leg is a magnet

assembly that has an upper platform to which permanent magnets and mechanical supports are attached. The other leg is the lower body. The lower leg is a coil assembly that has a lower platform on which air-core coils are mounted. These legs are connected and guided by springs. The upper leg is suspended by springs, and the lower leg is supported by a base. To generate forces in the x- and z-axes and a torque about the y-axis, a minimum of three single-axis actuators must be incorporated. Because the mechanical and control complexity of the mechanism, as well as its size, increase with the number of actuators used, we incorporated this minimum number of actuators in our mechanism to make it as simple and small as possible. Three magnet–coil pairs are used to actuate the two legs. Two magnet–coil pairs on the left and right generate vertical forces on the upper leg and reaction forces on the lower leg. Both legs are raised and lowered by applying currents to the coils on the left and right of the lower leg. These two magnet– coil pairs are used as vertical actuators to raise and lower the two legs. The vertical actuators are based on magnetic attraction and repulsion. The other magnet–coil pair in the middle generates


1415

TABLE II COMPARISON OF OUR MECHANISM WITH EXISTING PRECISION WALKING MECHANISMS Uozumi et al.’s mechanism [2]

Breguet and Renaud’s mechanism [3]

Martel et al.’s mechanism [6], [7]

Actuation type Leg type Structure

PIEZOELECTRIC ACTUATION DEFORMABLE LEGS A RIGID FRAME TO WHICH DEFORMABLE LEGS ARE ATTACHED WITHOUT THE REQUIREMENT FOR MECHANICAL BEARINGS SUCH AS BALL BEARINGS THAT SUFFER FROM MECHANICAL FRICTION.

Leg actuation means

INDIVIDUAL LEG ACTUATION: EACH LEG HAS ITS OWN MEANS OF INDIVIDUAL ACTUATION.

Locomotion technique

Differences

HEXAPOD SHRINK-SHEAREXPAND-SHRINK-SHEAR-EXPAND LOCOMOTION • Well-suited for use in the manufacture of a walker that walks by shrinking each of two sets of three legs and expanding them before shrinking the other set so as to eliminate the effect of mechanical friction between the legs and a walking surface.

TRIPOD STICK-SLIP

TRIPOD PUSH-SLIP

Magnetic actuation Rigid legs Two rigid frames that are elastically connected to each other without the requirement for mechanical bearings such as ball bearings that suffer from mechanical friction. The rigid frames act as legs. Shared leg actuation: legs share an actuation means. BIPED ATTRACT-PROPEL-REPEL-PROPEL

LOCOMOTION

LOCOMOTION

LOCOMOTION

• Well-suited for use in the manufacture of a walker that is driven at a rate of thousands of steps per second so as to walk with fast steps.

• They are well suited for use in the manufacture of a walker that walks by means of deformable legs, which are actuated individually, with short steps (of tens of micrometers or less) along the x- and y-axes. • Not suitable for use in the manufacture of a walker that walks • Not suitable for use in the by lifting its legs properly off a walking surface so as to eliminate manufacture of a walker that is the effect of mechanical friction between the legs and the walking driven at a rate of thousands of surface. steps per second so as to walk with fast steps.

Advantages Disadvantages

Our proposed mechanism

• Not designed for use in the manufacture of a walker that walks by means of rigid legs, which share actuators, with long steps. Simple single-rigid-frame structure. Legs have short ranges of motion.

a horizontal force on the upper leg and a reaction force on the lower leg. Both legs are propelled by applying a current to the coil in the middle of the lower leg. This magnet–coil pair is used as a horizontal actuator to propel the two legs. The horizontal actuator is based on magnetic propulsion. A detailed description of each actuator is provided in the following section.

III. VERTICAL AND HORIZONTAL ACTUATORS As described briefly in Section II, the magnetic walking mechanism uses two vertical actuators, which generate z-axis magnetic forces to move the upper and lower bodies in the z-axis direction and one horizontal actuator, which generates an x-axis magnetic force to move the legs in the x-axis direction. As depicted in Fig. 1, the three magnets of the actuators are attached to the upper leg, and the three coils of the actuators are mounted on the lower leg. Each vertical actuator comprises one magnet magnetized in the positive z-axis direction and one coil whose air-core axis is parallel to the z-axis direction. This magnet–coil pair is a vertical actuator. Magnets on the left and right of the upper leg interact magnetically with the current-carrying coils on the left and right of the lower leg, respectively. Each interaction generates magnetic force in the z-axis direction.

• Well-suited for use in the manufacture of a walker that walks by attracting two legs to each other and repelling them from each other so as to eliminate the effect of mechanical friction between the legs and a walking surface.• Well-suited for use in the manufacture of a walker that walks by means of rigid legs, which share actuators, with long steps (hundreds of micrometers) along the x-axis.

• Not designed for use in the manufacture of a walker that is driven at a rate of thousands of steps per second so as to walk with fast steps. • Not designed for use in the manufacture of a walker that walks by means of deformable legs, which are actuated individually, with short steps along the x- and y-axes.

Legs have long ranges of motion. Complex multiple-rigid-frame structure.

In contrast, the horizontal actuator is composed of one magnet that is magnetized in the positive z-axis direction and one coil whose air-core axis is parallel to the x-axis direction. This magnet–coil pair is a horizontal actuator. The magnet in the middle of the upper leg interacts with the current-carrying coil in the middle of the lower leg. The magnetic interaction between them generates magnetic force in the x-axis direction. Theoretically, the magnetic force generated by interaction between magnetic elements can be expressed by the modified Lorentz’s law, which is expressed in vector form [16], [17] as F = (m · ∇)B

(1)

where m is the magnetic dipole moment, and B is the magnetic flux density. If a cylindrical permanent magnet has a magnetic ˆ where mx = 0, j + mz k, dipole moment m (m = mxî + my ˆ my = 0, and mz = constant), the rectangular components of the force can be expressed as ∂Bx ∂z ∂By F y = mz ∂z ∂Bz F z = mz . ∂z

F x = mz

(2) (3) (4)

1416


We approximated the value of the magnetic dipole moment of the permanent magnet using the following expression developed by Wangsness [16] for the magnetic flux density along the axis of a cylindrical solenoid: mz =

2Bm a μ0 Vm

⎤ 2 2 2 + (h − z 2 rm + zm rm m mp) p ⎦ ·⎣ 2 2 2 2 (hm − zm p ) rm + zm p + zm p rm + (hm − zm p ) ⎡

(5) where hm is the height of the magnet, rm is the radius, Vm is the volume, zm p is the distance from the center to a point on the axis, Bm a is the measured magnetic flux density on the center of the magnet’s surface, and μ0 is the permeability of free space (4π × 10−7 N·A−2 ). To calculate the forces of vertical and horizontal actuators, the magnetic field produced by the air-core coil has to be determined; we describe how we determined the field below.

Fig. 3. Vertical actuator. (a) Vertical coil. (b) Distance from the center of the vertical coil to the center of the permanent magnet. (c) Cross section of the vertical coil. (d) jth square wire loop in the vertical coil.

A. Magnetic Field of an Air-Core Coil The magnetic flux density B of a current-carrying wire at point P can be expressed by the Biot–Savart law, as shown as follows in vector form [16]: μ0 i ds × R (6) B(p) = 4π R3

where i is the current that the wire carries, ds is an element of length along the wire, and R is the position vector from the current element ids to the point P. Using the Biot–Savart law, we calculated the magnetic flux density of the air-core coil in each actuator. The z-axis air-core coil in the vertical actuator is shown in Fig. 3. The symbols and ⊗ denote the direct current applied to the coil points out of and into the paper, respectively. We approximated the value of the total magnetic flux density B V of the z-axis air-core coil at a point P1 by integrating the contribution of each square loop carrying current iV as follows: D V /2 L V/2 μ0 NV iV BV = bV j dz10 daj (7) 4π( 12 DV − 12 dV )LV d V/2 −L V/2 where NV is the number of windings. Here, bVj is contributed by the jth square loop shown in Fig. 3(d), whose dimensions are 2aj × 2aj and can be expressed as bVj = bV 1 + bV 2 + bV 3 + bV 4 where

bV 1 =

−a j

bV 2 =

aj

−a j

bV 3 =

aj

aj

−a j

(8)

ˆ j − Y1 dx1 k Zdx1 ˆ 2 2 [(x1p − x1 ) + Y1 + Z 2 ]3/2

(8a)

ˆ −Zdy1î + X2 dy1 k 2 2 [X2 + (y1p − y1 ) + Z 2 ]3/2

(8b)

ˆ j + Y2 dx1 k −Zdx1 ˆ 2 2 [(x1p − x1 ) + Y2 + Z 2 ]3/2

(8c)

bV 4 =

aj

−a j

[X12

ˆ Zdy1î − X1 dy1 k . 2 + (y1p − y1 ) + Z 2 ]3/2

(8d)

Here, Z, Y1 , Y2 , X1 , and X2 denote (z1p −z10 ), (y1p −aj ), (y1p + ˆ are unit j, and k aj ), (x1p −aj ), and (x1p + aj ), respectively; î, ˆ vectors parallel to the x-, y-, and z-axes, respectively, where the xz plane is parallel to the x1 − z1 plane; the center of mass of the air-core solenoid in the vertical actuator is located at the origin of the vertical actuator O1 ; and the center of mass of the permanent magnet is located at point P1 , whose coordinates are (0, 0, z1p ) in reference to the origin of the vertical actuator O1 when the magnet center is placed on the z1 axis, as shown in Fig. 3(b). The rectangular components of the magnetic flux density of the z-axis air-core coil in (7) can be expressed as

(BV)x =

(BV )y =

(BV )z =

μ0 NV iV 4π( 12 DV − 12 dV )LV μ0 NV iV 4π( 12 DV − 12 dV μ0 NV iV 1 4π( 2 DV − 12 dV

D V /2

d V /2

L V /2

/2

(bv j )x dz10 daj (9)

L V /2

−L V /2

d V /2

)LV

DV

−L V /2

d V /2

)LV

D V /2

(bv j )y dz10 daj (10)

L V /2

−L V /2

(bv j )z dz10 daj . (11)

The x-axis air-core coil in the horizontal actuator is shown in Fig. 4. We approximated the value of the total magnetic flux density B H of the x-axis air-core coil at point P2 by integrating


1417

We expressed the rectangular components of the magnetic flux density of the x-axis air-core coil in (12) as follows: D H/2 L H/2 μ0 NH iH (bH k )x dx20 dak (BH )x = 4π( 12 DH − 12 dH )LH d H/2 −L H /2

(BH )y =

(BH )z =

μ0 NH iH 4π( 12 DH − 12 dH )LH μ0 NH iH 1 4π( 2 DH − 12 dH )LH

D H/2

d H/2

(14) L H/2

−L H/2

(bH k )y dx20 dak (15)

D H/2 L H/2

d H/2

−L H/2

(bH k )z dx20 dak . (16)

B. Magnetic Force Generated by Each Magnet–Coil Pair Fig. 4. Horizontal actuator. (a) Vertical coil. (b) Distance from the center of the horizontal coil to the center of the permanent magnet. (c) Cross section of the horizontal coil. (d) kth square wire loop in the horizontal coil.

fz = mz

the contribution of each square loop carrying current iH : BH

μ0 NH iH = 1 4π( 2 DH − 12 dH )LH

D H /2

d H /2

L H /2

−L H /2

bH k dx20 dak

(12) where NH is the number of windings. Here, bH k , contributed by the kth square loop shown in Fig. 4(d), whose dimensions are 2ak × 2ak , can be expressed as bH k = bH 1 + bH 2 + bH 3 + bH 4

The magnetic dipole moment of the permanent magnet and the magnetic flux density of the z-axis air-core coil are given by (5) and (11), respectively. Inserting these expressions into the right-hand side of (4) gives the following expression for the z-axis force generated by the vertical actuator:

fx = mz bH1 =

−a k

bH2 =

ak

−a k

bH4 =

ak

−a k

bH3 =

ak

ˆ f2 dy2 î − h dy2 k 2 2 [h + (y2p − y2 ) + f22 ]3/2

(13a)

j −g1 dz2 î + h dz2 ˆ 2 [h + g12 + (z2p − z2 )2 ]3/2

(13b)

ˆ −f1 dy2 î + h dy2 k [h2 + (y2p − y2 )2 + f12 ]3/2

(13c)

ak

−a k

[h2

j g2 dz2 î − h dz2 ˆ . 2 + g2 + (z2p − z2 )2 ]3/2

(13d)

Here, f1 , f2 , g1 , g2 , and h denote (z2p −ak ), (z2p + ak ), (y2p −ak ), (y2p + ak ), and (x2p −x20 ), respectively; the xz plane is parallel to the x2 z2 plane; the center of mass of the air-core solenoid in the horizontal actuator is located at the origin of the horizontal actuator O2 ; and the center of mass of the permanent magnet is located at point P2 with coordinates (0, 0, z2p ) in reference to the origin of the horizontal actuator O2 when the magnet center is placed on the z2 axis, as shown in Fig. 4(b).

(17)

The magnetic dipole moment of the permanent magnet and the magnetic flux density of the x-axis air-core coil are given by (5) and (14), respectively. The permanent magnet in the horizontal actuator has the same magnetization direction as that of the permanent magnet in the vertical actuator. Inserting these expressions into the right-hand side of (2) gives the following expression for the x-axis force generated by the horizontal actuator:

(13)

where

∂(BV )z . ∂z1p

∂(BH )x . ∂z2p

(18)

C. Magnetic Force Quantization and Experimental Characterization We calculated the magnetic forces using (17) and (18) and conducted experiments on magnet-coil forces. Forces were measured using a six-axis force/torque sensor (Nano17, ATI) mounted on an xyz stage (BSS76-60C, Suruga Seiki). For each magnet–coil pair, the magnet was attached to the sensor, and the coil was fixed to a base plate. As shown in Table III, the magnet and coil used in the vertical actuator were the same as those used in the horizontal actuator, except that the coil of the vertical actuator had a z-axis air core, and the coil of the horizontal actuator had an x-axis air core. The vertical actuator force and the horizontal actuator force were theoretically and experimentally obtained under two individual conditions. The first condition was that the distance from the center of the coil to the center of the magnet was variable within a fixed range of travel in the z-axis direction when the current applied to the coil was fixed. The second condition was that the applied current was variable within a fixed range of current when the distance from the coil center to the magnet center was

1418


TABLE III MAGNETIC ELEMENTS USED IN THE VERTICAL AND HORIZONTAL ACTUATORS Dimensions & Material

Element Vertical coil

Horizontal coil

Permanent magnet

∗

20 mm × 20 mm × 20 mm (L × W × H)∗ Air-core size: 12 mm × 12 mm × 20 mm Number of windings: 280 Copper wire 20 mm × 20 mm × 20 mm Air-core size: 20 mm × 12 mm × 12 mm Number of windings: 280 Copper wire 10 mm in diameter and 10 mm in height Neodymium-iron-boron (NdFeB) Measured magnetic flux density on the center of the magnet surface (B ma ): 0.475 T (4.75 kG)

Fig. 7. Actuation forces acting on the upper leg: (a) Vertical forces. (b) Horizontal force. For clarity, the magnets, supports, and springs are not shown.

L: Length, W: Width, H: Height.

Fig. 8. Actuation forces acting on the lower leg: (a) Vertical forces. (b) Horizontal force. For clarity, the coils and springs are not shown.

as the current decreased, which means that the magnetic force was proportional to the current. Magnetic force was a function of both applied current and distance. These results indicated that the calculated forces of the vertical actuator were in good agreement with the measured forces and that the derived forces of the horizontal actuator were also in good agreement with the measured forces. From the experimental data depicted in Figs. 5 and 6, the following models of magnetic force were derived: Fig. 5.

Calculated and experimental force values of the vertical actuator.

fz (iV , z1p ) = (C1 − C2 z1p )iV

(19)

fx (iH , z2p ) = (C3 − C4 z2p )iH .

(20)

In (19) and (20), z1p and z2p are, respectively, the distance between the center of the magnet of the vertical actuator and the center of the coil of the vertical actuator and the distance between the center of the magnet of the horizontal actuator and the center of the coil of the horizontal actuator; iV and iH are, respectively, the current applied to the coil of the vertical actuator and the current applied to the coil of the horizontal actuator; and C1 , C2 , C3 , and C4 are constants, which are calculated from the experimental data as C1 = 0.85 N·A−1 , C2 = 25.5 N·m−1 ·A−1 , C3 = 0.344 N·A−1 , and C4 = 14.96 N·m−1 ·A−1 . Fig. 6.

Calculated and experimental force values of the horizontal actuator.

fixed. The magnet center and the coil center were placed on the z-axis. Comparisons between theoretical and empirical values are shown in Figs. 5 and 6. The magnetic force of each actuator increased as the distance decreased, whereas the magnetic force decreased as the distance increased, indicating that magnetic force was dependent on distance. Furthermore, the magnetic force increased as the current that was applied to the coil of each actuator increased, whereas the magnetic force decreased

IV. SHARED LEG ACTUATION The horizontal actuator exerts one horizontal force on the rigid legs, while the vertical actuators exert two vertical forces on the rigid legs. The actuation forces acting on the upper and lower legs are depicted in Figs. 7 and 8, respectively. The mechanism uses the action forces, i.e., moving-magnet-type actuation forces created by the vertical and horizontal actuators to move the upper leg; the mechanism uses reaction forces, i.e., moving-coil-type actuation forces created by the vertical and horizontal actuators to move the lower leg. The total magnetic forces on the upper or lower leg in the xand z-axes and the torque about the y-axis are


FHq = fHq

(21)

FVq

(22)

=

Tyq = where FHq

1419

fVq L

+ fVq R −rm L × fVq L

+ rm R × fVq R

(23)

and FVq

are the total magnetic forces in the horizontal (x-axis) and vertical (z-axis) directions, respectively, on leg q, where q = u denotes the upper leg and q = l denotes the lower leg; Tyq is the torque about the y axis on leg q; fHq is the horizontal magnetic force on leg q created by the middle actuator; fVq L and fVq R are the vertical magnetic forces on leg q created by the left and right actuators, respectively; and rm L and rm R are the distances along the x-axis from the center of mass of the upper leg to the centers of mass of the left and of the right magnets, respectively, which are, respectively, equal to the distances along the x-axis from the center of mass of the lower leg to the centers of mass of the coils on the left and on the right. The total magnetic forces in the x- and z-axes on the upper leg have opposite directions to those on the lower leg. The directions of horizontal and vertical magnetic forces on both the upper and lower legs are determined by the directions of currents applied to the horizontal and vertical actuators, respectively. The torque about the y-axis on the upper leg is in the opposite direction to that on the lower leg. These opposite relationships result from both the action and reaction forces arising from magnetic interaction between the magnets and the coils. When a torque about the y-axis due to vertical actuator forces and a magnetic torque about the y-axis due to the interaction between the magnet in the middle of the upper leg and the coil in the middle of the lower leg, which is not required for functioning of the mechanism, have equal magnitudes and opposite directions, these torques cancel each other out; therefore, the resulting rotation about the y-axis (pitch motion) can be set to be zero. We designed the mechanism to have two actuation configurations—moving-magnet type and moving-coil type— so that the legs could share actuators. The configuration can be changed from moving-magnet type to moving-coil type by attracting the legs to each other. Furthermore, the configuration can be changed from the moving-coil-type configuration to the moving-magnet-type configuration by repelling the legs from each other. When the mechanism is in the moving-magnet-type actuation configuration, the upper leg is movable, and the lower leg is stationary. When the mechanism is in the moving-coiltype actuation configuration, the upper leg is stationary, and the lower leg is movable. The actuators used to drive the upper leg are also used to drive the lower leg this way; therefore, there is no need to add extra actuators to drive the lower leg. V. FINE MOTION Fine motion of the two rigid legs is achieved using the shared leg actuation mechanism described in Section IV. When horizontal and vertical forces provided by the actuators are applied to the legs, one of the two legs creates two translations (x and z), whereas the other maintains contact with a base. Fig. 9 illustrates the magnetic and spring forces acting on the q q and FsV are the total spring upper and lower legs. Here, FsH forces in the horizontal (x-axis) and vertical (z-axis) directions,

Fig. 9. Actuation forces and spring forces (a) acting on the upper leg that is suspended by the springs, when counterclockwise currents are applied to the vertical coils, and a counterclockwise current is applied to the horizontal coil, and (b) acting on the lower leg that is elastically connected to the upper leg and is lifted off the base when counterclockwise currents are applied to the vertical coils, and a counterclockwise current is applied to the horizontal coil. For clarity, the magnets, coils, springs, and supports are not shown.

Fig. 10. Dynamic model of the upper leg. When the mechanism is in the moving-magnet-type configuration (a), it generates horizontal motion of the upper leg (b) and vertical motion of the upper leg (c).

Fig. 11. Dynamic model of the lower leg. (a) When no currents are applied to the coils, the mechanism has a moving-magnet-type configuration. (b) When counterclockwise currents are applied to the vertical coils, the vertical actuators exert attraction forces on the upper leg, which causes the upper leg to be attracted down to the lower leg, i.e., to be lowered and placed on the base. (c) As the currents applied to the vertical coils increase, the attraction forces increase, which causes the lower leg to be attracted up to the attracted upper leg, i.e., to be lifted off the base. (d) When the upper leg is placed on the base and the lower leg is lifted off the base, i.e., when the mechanism has a moving-coil-type configuration, the mechanism generates horizontal and vertical motions of the lower leg.

respectively, on leg q. To obtain a tractable mathematical model for the mechanism, we assumed that we could neglect disturbances such as vibrations and acoustic noises, damping effects due to magnetic interactions between the magnetic elements, and air friction between the legs. The equations of motion for the upper or lower leg that sum the forces on the upper or lower leg in the x- and z-directions (see Figs. 10 and 11), derived from Newton’s law for translational motion [18], are q ¨q + FsH = FHq mq x

mq z¨q +

q FsV

=

FVq

(24) (25)

1420


Fig. 12. State machine that tracks the walking behavior of the mechanism. The trailing superscript u denotes that the upper leg is in transition from one state to the next or in a state between one transition and the next. The trailing superscript l denotes that the lower leg is in transition from one state to the next or in a state between one transition and the next.

where xq and zq are the displacements of leg q in the x- and z-directions, respectively, and mq is the mass of leg q. When the mechanism has the moving-magnet-type actuation configuration [see Fig. 10(a)], the two translations of the upper leg and its other DOF motions are stabilized by the springs. The upper leg, which is suspended by springs and to which no mechanical bearings such as ball bearings are connected, moves through magnetic actuation; therefore, there is no mechanical friction between the upper leg and the base. This elimination of mechanical friction helps the upper leg to move precisely. The upper leg moves in the x- and z-directions when the lower leg is stationary on the base. When the mechanism has the movingcoil-type actuation configuration [see Fig. 11(c)], the weight of the upper leg is canceled out by the total z-axis magnetic force on the lower leg, and the two translations of the lower leg and its other DOF motions are stabilized by the springs. The lower leg is lifted off the base by forces of the vertical actuators so that mechanical friction between the lower leg and the base can be avoided. This avoidance of mechanical friction helps the lower leg to move precisely. The lower leg moves in the x- and z-directions when the upper leg is stationary on the base. VI. WALKING MOTION As depicted in Fig. 2 and briefly described in Section I, walking motion of the mechanism is achieved by repeating an attract-propel-repel-propel cycle. During one cycle of walking, the mechanism takes one step. The walking sequence of our mechanism is represented in Fig. 12 based on the concept of a state diagram as described in [19] and [20]. The action of walking is modeled as six states and six transitions between states. The transition from one state to the next occurs due to a change in input to the actuators, i.e., currents applied to the coils of the actuators. Examples of how the currents excite the coils of the actuators for each transition are provided in Fig. 2(b) and (c). Our walking model includes

two differential equations for each of the transitions and a state diagram to sequence the walking cycle. To obtain a tractable mathematical walking model for the mechanism, we made the same simplifying assumption as when we derived the equations of motion for each leg in Section V. The walking sequence starts in State 0 with the lower leg on the base and the upper leg suspended by springs. State 0 represents the stationary mechanism with no applied current. No actuation forces exist in State 0. The initial distance between the supports of the upper leg and the base is determined in State 0, which is represented as gs0 [see Fig. 2(a)]. There is no motion in State 0, and thus, the relative positions of the upper and lower legs with respect to the initial positions are (0, 0, 0) and (0, 0, 0), respectively. As the centers of mass of the two legs are on the z-axis in the initial conditions, the initial positions of the centers of mass of the upper and lower legs defined from the common origin of the fixed x, y, z reference frame are (0, 0, zu 0 ) and (0, 0, zl0 ), respectively. The modified positions of the upper and lower legs due to their motion in the fixed reference frame can be expressed as (xu , yu , zu 0 + zu ) and (xl , yl , zl0 + zl ), respectively. Therefore, the relative positions of the upper and lower legs with respect to their initial positions becomes (xu , yu , zu ) and (xl , yl , zl ), respectively. We define w δxq and w δzq as small perturbations of leg q in the x- and z-directions for Transition w, respectively, while w q FH and w FVq are the total magnetic forces in the horizontal (x-axis) and vertical (z-axis) directions for Transition w, respectively, on leg q. The leading superscript w denotes the Transition number, w = 1, 2, 3, . . . , 6. For instance, 1 δxu and 1 δzu represent perturbations of the upper leg in the x- and z-directions for Transition 1, respectively. For Transition 1 from State 0 to State 1 (Attractionu in Figs. 2 and 12), the equations of motion of the mechanism become mu 1 x ü + KH 1 xu = 0

(26)

mu zü + KV zu = − 1

1

1

FVu

(27)

where KH and KV are the spring constants in the horizontal and vertical directions, respectively. This transition occurs by application of counterclockwise currents to the vertical coils in State 0, as shown in Fig. 2(b). Applying the currents to the vertical coils causes the vertical actuators to exert attraction forces on the upper leg, which break the vertical balance of forces maintained in State 0. These forces move the upper leg downward by an amount 1 δzu (= gs0 ). The lower leg is stationary. When 1 zu = −1 δzu , i.e., zu = −1 δzu , the upper leg is in an equilibrium state of Landingu (State 1). In State 1, the relative positions of the upper and lower legs with respect to the initial positions are (0, 0, −1 δzu ) and (0, 0, 0), respectively. For Transition 2 from States 1 to 2 (Attractionl ), the equations of motion of the mechanism become ml 2 x ¨l + KH 2 xl = 0

(28)

ml 2 z¨l + KV 2 zl = 2 FVl − KV 1 δzu − ml g

(29)

where g is the acceleration due to gravity. This transition occurs by increasing the currents applied to the vertical coils in State


Fig. 13. coils.

1421

Dimensions of components of the prototype designed based on the magnetic walking mechanism. Table III shows dimensions of the magnets and the

1. Increasing the currents applied to the vertical coils causes the vertical actuators to exert attraction forces on the lower leg, which break the vertical balance of forces on the lower leg maintained in State 1. These forces move the lower leg upward by an amount 2 δzl . The upper leg is stationary. When 2 zl = 2 δzl , i.e., zl = 2 δzl , the lower leg is in an equilibrium state of Flightl (State 2). In State 2, the relative positions of the upper and lower legs with respect to the initial positions are (0, 0, −1 δzu ) and (0, 0, 2δzl ), respectively. For Transition 3 from States 2 to 3 (Propulsionl ), the equations of motion of the mechanism become ¨l + KH 3xl = 3 FHl ml 3 x

(30)

ml 3 z¨l + KV 3zl = 0.

(31)

This transition occurs by applying a clockwise current to the horizontal coil in State 2. Applying the current to the horizontal coil causes the horizontal actuator to exert a thrust force on the lower leg, which breaks the horizontal balance of forces on the lower leg maintained in State 2. This force propels the lower leg forward by the amount 3 δxl . The upper leg is stationary. When 3 xl =3 δxl , i.e., xl =3 δxl , the lower leg is in the equilibrium state of Thrustl (State 3). In State 3, the relative positions of the upper and lower legs with respect to the initial positions are (0, 0, −1 δzu ) and (3 δxl , 0,2 δzl ), respectively. For Transition 4 from States 3 to 4 (Repulsionl ), the equations of motion of the mechanism become ¨l + KH 4 xl = 4 FHl − KH 3δxl ml 4 x 4

4

ml z¨l + KV zl =

4

FVl

(32)

− KV δzl − KV δzu − ml g. (33) 2

1

This transition occurs by decreasing the currents applied to the vertical coils in State 3. Decreasing the currents applied to the vertical coils reduces the total z-axis magnetic force on the lower leg, which breaks the vertical balance of forces on the lower leg maintained in State 3. This total z-axis magnetic force reduction causes the gravitational force of the lower leg and

the total z-axis spring force produced by the potential energy stored in the springs of State 3 to repel the lower leg by the amount 4 δzl (=2 δzl ) in the negative z-direction. The upper leg is stationary. When 4 zl = −4 δzl , i.e., zl =2 δzl − 4 δzl = 0, the lower leg is in the equilibrium state of Landingl (State 4). As the lower leg moves down, the vertical distance between the horizontal coil and the magnet increases. With the current fixed, increasing this vertical distance decreases the total x-axis magnetic force of the horizontal actuator, as shown in Fig. 6, which breaks the horizontal balance of forces on the lower leg maintained in State 3. This total x-axis magnetic force decrease causes the total x-axis spring force produced by the potential energy stored in the springs of State 3 to move the lower leg backward by the amount 4 δxl . Thus, when the lower leg is in State 4, 4 xl = −4 δxl , i.e., xl =3 δxl −4 δxl . The step length of the lower leg is (3 δxl −4 δxl ). This step length can be increased by increasing the current applied to the horizontal coil because increasing the current applied to the horizontal coil increases the total x-axis magnetic force of the horizontal actuator, as shown in Fig. 6. In State 4, the relative positions of the upper and lower legs with respect to the initial positions are (0, 0, −1 δzu ) and (3 δxl − 4 δxl , 0, 0), respectively. For Transition 5 from States 4 to 5 (Repulsionu ), the equations of motion of the mechanism become ü + KH 5 xu = −5 FHu + KH (3 δxl − 4 δxl ) (34) mu 5 x mu 5 zü + KV 5 zu = −5 FVu + KV 1 δzu .

(35)

This transition occurs by decreasing the currents applied to the vertical coils in State 4. Decreasing the currents applied to the vertical coils reduces the total z-axis magnetic force on the upper leg, which breaks the vertical balance of forces on the upper leg maintained in State 4. This total z-axis magnetic force reduction causes the total z-axis spring force produced by the potential energy stored in the springs of State 4 to repel the upper leg by the amount 5 δzu (=1 δzu ) in the positive z-direction. The lower leg is stationary. When 5 zu = 5 δzu , i.e.,

1422


zu = −1 δzu + 5 δzu = 0, the upper leg is in the equilibrium state of Flightu (State 5). As the upper leg moves up, the vertical distance between the horizontal coil and the magnet increases. As stated above, with the current fixed, increasing this vertical distance decreases the total x-axis magnetic force of the horizontal actuator, which breaks the horizontal balance of forces on the upper leg that is maintained in State 4. This decrease in the total x-axis magnetic force causes the total x-axis spring force produced by the potential energy stored in the springs of State 4 to move the upper leg forward by the amount 5 δxu . Thus, when the upper leg is in State 5, 5 xu = 5δxu , i.e., xu = 5δxu . In State 5, the relative positions of the upper and lower legs with respect to the initial positions are (5 δxu , 0, 0) and (3 δxl − 4δxl , 0, 0), respectively. For Transition 6 from States 5 to 6 (Propulsionu ), the equations of motion of the mechanism become

Fig. 14. built.

(a) Rendered model of the design. (b) Picture of the prototype walker

Fig. 15.

Experimentally determined total z-axis spring force curve.

Fig. 16.

Experimentally determined total x-axis spring force curve.

ü + KH 6 xu = −6 FHu + KH (3 δxl − 4 δxl − 5 δxu ) mu 6 x (36) mu 6 zü + KV 6 zu = 0.

(37)

This transition occurs by decreasing the current applied to the horizontal coil in State 5. Decreasing the current applied to the horizontal coil reduces the total x-axis magnetic force on the upper leg, which breaks the horizontal balance of forces on the upper leg maintained in State 5. This total x-axis magnetic force reduction causes the total x-axis spring force produced by the potential energy stored in the springs of State 5 to propel the upper leg by the amount 6 δxu in the positive x-direction. The lower leg is stationary. When 6 xu = 6δxu , i.e., xu = 5δxu +6 δxu , the upper leg is in the equilibrium state of Thrustu (State 6). The step length of the upper leg is (5 δxu + 6δxu ). In State 6, the relative positions of the upper and lower legs with respect to the initial positions are (5 δxu + 6δxu , 0, 0) and (3 δxl − 4δxl , 0, 0), respectively. State 6 indicates the return of the mechanism to State 0 after having moved one step forward, and the procedure is completed. This corresponds to one cycle of walking motion. As this cycle is repeated, the mechanism walks forward. Applying a counterclockwise current to the horizontal coil allows it to walk backward. VII. PROTOTYPE To demonstrate that the proposed mechanism is possible, we built a magnetically driven bipedal walker based on the mechanism. Dimensions of the components of the prototype are shown in Fig. 13. A rendered model of the design and a picture of the prototype are provided in Fig. 14. The walker had two rigid legs. The upper leg comprised an upper plate of aluminum to which three neodymium–iron–boron permanent magnets and four stainless-steel supports were attached. The mass of this upper leg was 0.054 kg. The lower leg comprised a lower plate of aluminum to which three air-core solenoids of copper wire were attached. Mass of the lower leg was 0.116 kg. The lower leg contained four holes for the four supports through which to move.

The two rigid legs were connected and guided by four leaf springs made of polyvinyl chloride film bent into the shape of a semicircle. The upper leg was suspended by C-shaped springs, and the lower leg was supported and guided by an aluminum base plate. The elastic connection between the two legs helped to stabilize their motions in open-loop control. The stiffer the springs, the greater the stability. Experimental total z-axis spring force results are presented in Fig. 15. The linear fitted line had a slope of 208 N·m−1 . Total x-axis spring force experimental results are shown in Fig. 16. The linear fitted line had a slope of 61 N·m−1 . VIII. EXPERIMENTAL SETUP AND TEST RESULTS A. Experimental Setup A photograph of the experimental setup used for this study is shown in Fig. 17. It consisted of the biped shown in Fig. 14(b), a laser displacement sensor (LK-H025 and LK-G5000, Keyence),


1423

Fig. 19. Experimentally determined relationship between the current applied to the horizontal coil and the microdisplacement of the upper leg. Fig. 17.

Experimental setup.

Fig. 18.

Control block diagram of the biped. Fig. 20.

three nonprogrammable dc power supplies (HY3005D, Mastech), a personal computer (Lenovo), and an optical table (Newport). The sensor, which operates based on the principle of optical triangulation, was used to measure the horizontal position of the upper leg of the biped: An aluminum block was placed on and fixed to the optical table with bolts, and a sensor head (LK-H025) was mounted on the block with bolts and nuts. Information about the horizontal position of the upper leg collected by the sensor was stored on the computer. The power supplies applied currents to the coils of the biped. Magnitudes of the currents applied to the coils were adjusted by turning rotary knobs of the power supplies by hand, and the directions of currents applied to the coils were changed by switching the two leads of the power supplies. The biped was placed on an optical table for external vibration isolation. A control block diagram of the biped is shown in Fig. 18. The biped, or manually operated legged precision motion machine, was operated in open-loop control. B. Fine Motion Experiments We performed two experiments to determine the relationship between displacement of the upper leg and the current applied to the horizontal coil to evaluate the fine motion capability of the biped in terms of range and repeatability. We conducted the first experiment to show that it was possible for the biped to achieve micromotion. Micromotion of the upper leg was achieved (see Figs. 19 and 20). Fig. 19 shows the experimentally determined relationship between current and microdisplacement. Measurement of the microdisplacement of the upper leg as a function of the current applied to the coil pro-

Response of the upper leg to a 0.43-A step input.

duced a line. For displacements less than 600 μm and currents less than 0.5 A, the current applied to the coil was proportional to the microdisplacement of the upper leg. The upper leg had a long range of 500 μm (see Fig. 20). This indicated that the biped could achieve long-range motion without walking. The speed of fine motion was about 10 mm·s−1 . The biped’s response to the step input revealed that its open-loop performance was characterized by a large overshoot in step response. The response of the upper leg to the 0.43-A step input is similar to the response of an underdamped second-order system to a step input. The damping ratio of an underdamped second-order system can be determined from the response of the system to a step input [21], [22]; based on the experimental plot, we calculated that the damping ratio was 0.87. We conducted the second experiment to show that it was possible for the biped to achieve nanomotion. Nanomotion of the upper leg was achieved (see Figs. 21 and 22). Fig. 21 shows the experimentally determined relationship between current and nanodisplacement. Measurement of the nanodisplacement of the upper leg as a function of the current applied to the coil produced a line. For displacements less than 1 μm and currents less than 2.5 mA, the current applied to the coil was proportional to the nanodisplacement of the upper leg. The upper leg was capable of producing a 378-nm displacement, as shown in Fig. 22. In this experiment, a ramp current, instead of a step current, was applied to the coil, which resulted in smoother motion. This nanomotion experiment was repeated 50 times to test the repeatability of the biped’s fine motion. Repeatability test results are shown in Fig. 23. Although no quantitative estimate of the influence of

1424


Fig. 21. Experimentally determined relationship between the current applied to the horizontal coil and the nanodisplacement of the upper leg.

Fig. 22.

Response of the upper leg to a 0.6-mA ramp input.

Fig. 24. Experimentally determined relationship between the current applied to the horizontal coil and the step length of the upper leg of the walking biped.

Fig. 25. Step length of the upper leg of the biped after one step. When a 2.2-A ramp current was applied to the horizontal coil, the step length of the upper leg was 374 μm.

C. Walking Motion Experiments

Fig. 23. Repeatability tests for nanopositioning of the upper leg position for repeated trials of a 0.6-mA ramp current. The mean (m) equals 365 nm. The standard deviation (σ) equals 74.75 nm.

environmental disturbances such as vibration and acoustic noise was made, the experimental data shown in Fig. 23 indicated that the effect of environmental disturbances on the biped was small, as the biped was able to achieve nanopositioning. As can be observed from Fig. 23, the repeatability was within ±250 nm. The resolution of the biped’s fine motion was better than 400 nm (see Fig. 23): The resolution of the fine motion depended on how finely we turned the rotary knobs of the power supplies by hand. To summarize, our prototype is sensitive to disturbances, but requires no complex control mechanism and is cost-effective because it is open-loop controlled and manually operated. The above experimental results on fine motion demonstrated that our mechanism is suitable for creating a biped capable of long-range fine motion.

We next performed walking motion experiments to evaluate the walking capability of the biped in terms of step length and repeatability of step length. The first experiment was conducted to show that it was possible for the biped to walk with long steps. Long-step walking was achieved, as shown in Figs. 24–26. Fig. 24 shows the experimentally determined relationship between the current applied to the horizontal coil and the step length of the upper leg of the walking biped. Measurement of the step length of the upper leg as a function of the current applied to the coil produced a line. For step lengths less than 400 μm and currents less than 2.2 A, the current applied to the coil was proportional to the step length of the upper leg of the walking biped. It is clear from Fig. 24 that a high current of 1 A or more was required to achieve long-step walking. The main reason for this is that the leads of the power supplies, which were connected to the coils attached to the lower leg of the biped, behaved like springs. The influence of these leads could not be avoided because the prototype was tethered, but no embedded batteries or processors were required. Fig. 25 shows that the step length of the upper leg of the walking biped was 374 μm. This indicates that the biped could achieve long-step walking. The displacement of 374 μm was achieved when a ramp current of 2.2 A was applied to the horizontal coil. It would be ideal if the upper leg moved only downward when currents were applied to the vertical coils. In practice, the upper leg tended to not only move downward but


Fig. 26. Step lengths that the upper leg of the biped which took two steps had. When a 1.8-A ramp current was applied to the horizontal coil, the first step length of the upper leg was 327 μm, and the second step length of the upper leg was 310 μm.

backward as well when currents were applied to the vertical coils. The backward movement of the upper leg that we observed was likely due to the fact that the upper leg tended to move diagonally when currents were applied to the vertical coils, and there was misalignment between the magnet and the coil of the vertical actuator, resulting in generation of an inclined force. Fig. 25 shows that backward movement of the upper leg occurred; nonetheless, the biped was capable of taking a long step forward when a clockwise current was applied to the horizontal coil, as depicted in Fig. 2(b), after backward movement of the upper leg. A walking speed of about 12 μm·s−1 was achieved when a ramp current of 2.2 A was applied to the horizontal coil. The speed of walking depended on how fast we turned the rotary knobs of the power supplies by hand. Step lengths of the upper leg when the biped took two steps are shown in Fig. 26. In this trial, a 1.8-A ramp current was applied to the horizontal coil. The first step length was 327 μm, and the second step length was 310 μm. These results indicated that the biped was capable of continuous walking with long steps of similar lengths. We conducted a second experiment to assess the repeatability of long-step walking of the biped in terms of step length. In this experiment, when the biped took one step, the achieved step length was measured. More specifically, when a 1.8-A ramp current, which was randomly selected, was applied to the horizontal coil, the length of one step of the upper leg of the biped was measured. This experiment was repeated 50 times, showing that the repeatability of the step length was within ±50 μm (see Fig. 27). The repeatability of the step length depended on how finely we turned the rotary knobs of the power supplies by hand. Overall, our experimental results demonstrated that our mechanism is suitable for creating a biped capable of both long-range fine motion and long-step walking motion. IX. CONCLUSION We have proposed a novel magnetic actuation mechanism, i.e., a magnetic walking mechanism. We briefly compared our legged precision mechanism with existing mechanisms. We discussed the underlying principles of force generation, shared leg actuation, fine motion, and walking motion and described the dynamics of each rigid leg of the mechanism and the walking

1425

Fig. 27. Repeatability tests for long-step walking of the biped. A 1.8-A ramp current was applied to the horizontal coil for each trial. The mean (m) equals 310 μm. The standard deviation (σ) equals 22.62 μm.

dynamics of the mechanism. We built a bipedal walker based on the mechanism and performed a series of experiments to characterize its motion. Our results demonstrated that it is possible to use our mechanism to build a nanobiped, namely, a two-legged machine with nanomotion and walking capabilities. Our work indicates the feasibility of creating legged machines with fewer legs than predecessors that are capable of nanoscale motion. Our mechanism allows realization of a legged robot with precision positioning capability whose legs share actuators and who walks by attraction and repulsion of two elastically connected rigid legs. Our mechanism represents one possible solution to the problem of how to build a legged precision machine capable of long-range nanopositioning and long-step walking. Our ongoing work is focused on introducing feedback. This effort includes modification of the experimental setup used in this study, such as by replacing the nonprogrammable power supplies used for manual control herein with programmable power amplifiers suitable for automatic control and by adding more sensors. An obvious way to improve an open-loop control system that is highly sensitive to disturbances and to parameter variations is to introduce feedback, although this does have the disadvantage of increasing cost and system complexity [21], [22]. Nevertheless, we anticipate that introducing feedback will improve the control system of the nanobiped. Furthermore, we anticipate that this modified experimental setup will increase the speed of walking and improve the repeatability of the step length and will enable us to perform walking motion experiments to evaluate the walking capability of the biped in terms of range, without the burden of turning the rotary knobs of the nonprogrammable power supplies finely or quickly by hand. This will allow us to experimentally verify the dynamic model of the biped and to experimentally characterize the effect of a magnetic torque due to the interaction of a permanent magnet and a current-carrying coil on the motion of the biped or the effect of certain parameters on the performance of the biped. In an attempt to advance the development of spatial nanobipeds, in future work, we intend to extend the proposed mechanism to enable the biped to move in 3-D space. Its future applications may include mobile positioners for controlling the position of a submillimeter-sized object with micrometer or nanometer resolution. In addition, there would be several applications for specialized nanobiped tools with arbitrarily shaped

1426


needles attached to their moving parts: mobile manipulators for handling nanosized and microsized objects such as graphene flakes and biological cells; mobile separators for separating objects that attract each other and whose sizes are up to hundreds of micrometers; and mobile graspers for grasping objects with sizes of up to hundreds of micrometers. REFERENCES [1] G. Binnig and C. Gerber, “Piezo travelling support,” IBM Tech. Disclosure Bull., vol. 23, no. 7B, pp. 3369–3370, 1980. [2] K. Uozumi, K. Nakamoto, and K. Fujioka, “Novel three-dimensional positioner and scanner for the STM using shear deformation of piezoceramic plates,” Jpn. J. Appl. Phys., vol. 27, no. 1, pp. L123–L126, 1988. [3] J.-M. Breguet and P. Renaud, “A 4-degrees-of-freedom microrobot with nanometer resolution,” Robotica, vol. 14, no. 2, pp. 199–203, 1996. [4] S. Fahlbusch, S. Fatikow, J. Seyfried, and A. Buerkle, “Flexible microrobotic system MINIMAN: Design, actuation principle and control,” in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatron., Atlanta, GA, USA, Sep. 19–23, 1999, pp. 156–161. [5] F. Schmoeckel and H. Wörn, “Remotely controllable mobile microrobots acting as nano positioners and intelligent tweezers in scanning electron microscopes (SEMs),” in Proc. IEEE Int. Conf. Robot. Autom., Seoul, Korea, May 21–26, 2001, pp. 3909–3913. [6] S. Martel, P. Madden, L. Sosnowski, I. Hunter, and S. Lafontaine, “NanoWalker: A fully autonomous highly integrated miniature robot for nano-scale measurements,” Proc. SPIE, vol. 3825, pp. 111–122, 1999. [7] S. Martel, “Fundamental principles and issues of high-speed piezoactuated three-legged motion for miniature robots designed for nanometer-scale operations,” Int. J. Robot. Res., vol. 24, no. 7, pp. 575–588, 2005. [8] R. L. Hollis, S. E. Salcudean, and A. P. Allan, “A six-degree-of-freedom magnetically levitated variable compliance fine-motion wrist: Design, modeling, and control,” IEEE Trans. Robot. Autom., vol. 7, no. 3, pp. 320–332, Jun. 1991. [9] W.-J. Kim and D. L. Trumper, “High-precision magnetic levitation stage for photolithography,” Precis. Eng., vol. 22, no. 2, pp. 66–77, 1998. [10] K. S. Jung and Y. S. Baek, “Study on a novel contact-free planar system using direct drive dc coils and permanent magnets,” IEEE/ASME Trans. Mechatronics, vol. 7, no. 2, pp. 35–43, Mar. 2002. [11] W.-J. Kim and S. Verma, “Multiaxis maglev positioner with nanometer resolution over extended travel range,” J. Dyn. Syst. Meas. Control-Trans. ASME, vol. 129, no. 6, pp. 777–785, 2007. [12] S. T. Smith, D. G. Chetwynd, and S. Harb, “A simple two-axis ultraprecision actuator,” Rev. Sci. Instrum., vol. 65, no. 4, pp. 910–917, 1994. [13] W. Wang and T. He, “A high precision micropositioner with five degrees of freedom based on an electromagnetic driving principle,” Rev. Sci. Instrum., vol. 67, no. 1, pp. 312–317, 1996. [14] M. L. Culpepper and G. Anderson, “Design of a low-cost nanomanipulator which utilizes a monolithic, spatial compliant mechanism,” Precis. Eng., vol. 28, no. 4, pp. 469–482, 2004.

[15] T. J. Teo, I.-M. Chen, G. Yang, and W. Lin, “A flexure-based electromagnetic linear actuator,” Nanotechnol., vol. 19, no. 31, pp. 315501-1– 315501-10, 2008. [16] R. K. Wangsness, Electromagnetic Fields. New York, NY, USA: Wiley, 1979. [17] D. J. Griffiths, Introduction to Electrodynamics. Englewood Cliffs, NJ, USA: Prentice-Hall, 1999. [18] R. C. Hibbeler, Engineering Mechanics: Principles of Statics and Dynamics. Englewood Cliffs, NJ, USA: Prentice-Hall, 2006. [19] M. H. Raibert, Legged Robots that Balance. Cambridge, MA, USA: MIT Press, 1986. [20] P. J. McKerrow, Introduction to Robotics. Sydney, Australia: AddisonWesley, 1991. [21] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Englewood Cliffs, NJ, USA: Prentice-Hall, 2002. [22] R. C. Dorf and R. H. Bishop, Modern Control Systems. Englewood Cliffs, NJ, USA: Prentice-Hall, 2011.

Jung Soo Choi received the B.S. degree in mechanical engineering from Yeungnam University, Gyeongsan, Korea, in 2000 and the M.S. degree in mechanical engineering from Yonsei University, Seoul, Korea, in 2002. He is currently working toward the Ph.D. degree in mechanical engineering with Yonsei University, Seoul. His research interests include friction-free positioners and conveyors based on the principle of magnetic levitation and nanoprecision biped robots.

Yoon Su Baek received the B.S. and M.S. degrees in mechanical engineering from Yonsei University, Seoul, Korea, in 1979 and 1981, respectively, and the M.S. and Ph.D. degrees in mechanical engineering from Oregon State University, Corvallis, OR, USA, in 1986 and 1990. From 1990 to 1993, he was the Head Research Engineer with the Mechatronics Research Center, Samsung Heavy Industry Ltd. Since 1994, he has been a Faculty Member with the School of Mechanical Engineering, Yonsei University. His research interests include multidegree-of-freedom actuators and robotics, especially relating to fine motion control. He is also interested in data storage devices such as optic disk drives, laser application devices such as bar code readers, and motorized exoskeletons for the elderly or paralyzed.