Generation of Quadratic Potential Force Fields From Flow Fields for ...
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Generation of Quadratic Potential Force Fields From Flow Fields for Distributed Manipulation Konstantinos Varsos, Student Member, IEEE, Hyungpil Moon, Member, IEEE, and Jonathan Luntz, Member, IEEE
Abstract—Distributed manipulation systems induce motions on objects through the application of many external forces. Many of these systems are abstracted as planar programmable force fields. Quadratic potential fields form a class of such fields that lend themselves to analytical study and exhibit useful stability properties. This paper introduces a new methodology to build quadratic potential fields with simple devices using the naturally existing phenomena of airflow, which is an improvement to the traditional use of the complicated programmable actuator arrays. It also provides a basis for the exploitation, in distributed manipulation, of natural phenomena like airflow, which require rigorous analysis and display stability difficulties. A demonstration and verification of the theoretical results for the special case of the elliptic field with airflows is also presented. Index Terms—Airflow, distributed manipulation, nonprehensile manipulation, parts feeding, programmable force fields, quadratic potential fields.
I. INTRODUCTION
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ISTRIBUTED manipulation systems induce motions on objects through the application of many external forces. The general task is to impart some desired motion to an object, either in space or restricted to a plane, by using either a large number of independently controlled actuators acting at many points on the object, or by exploiting the dynamics of some continuous medium which is in contact with the manipulated object. Many forms of distributed manipulation systems exist, including those based on vibrating plates [1], [2], actively controlled air jets [3], passive airflow fields [4], and planar micro[5], [6], mini- [7], and macro-mechanical actuator arrays [3], [8], [9]. By nature, such systems generally involve redundant actuation, and hence provide tremendous manipulation power. In addition, by distributing the application of force, these devices allow the manipulation of thin, flexible, and delicate objects. The concept of a programmable force field abstracts many distributed manipulators, including actuator arrays, which explicitly generate arbitrary forces at every point in the planar field. Manuscript received December 23, 2004; revised May 7, 2005. This paper was recommended for publication by Associate Editor K. Lynch and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported in part by the National Science Foundation under CAREER award RHA-0093312. This paper was presented in part at the International Conference on Robotics and Automation, Barcelona, Spain, April 2005. K. Varsos and J. Luntz are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2121 (e-mail: [email protected]; [email protected]). H. Moon was with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2121. He is now with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213 USA. Digital Object Identifier 10.1109/TRO.2005.858858
An object experiences net force and moment, which are the integral of the pointwise force field over the area of the object. The study of programmable force fields reduces to the determination of the net force and moment as a function of the shape of the field and of the object, and the object’s position and orientation relative to the field. Typical work in this area revolves around the particular problem of bringing an object to a unique position and orientation in a sensorless manner. Based on minimalistic manipulation concepts [10]–[12], Böhringer et al. [5] began examining provably useful force fields for orienting objects. Improving upon this, Kavraki [13] analyzed an elliptic potential field which orients most parts to one of two orientations (within symmetry). Finally, these researchers combined a unit radial field and constant (gravity) field to bring parts to a single orientation [14]. This line of research trades off feedback and control complexity for actuation complexity, using redundant actuation to produce natural equilibria without sensing and control. In previous work [14], [15], generating elliptic fields (and other powerful distributed manipulation fields) required pointwise actuation (i.e., a dense array of individually addressable actuators) which leads to complicated devices. This paper introduces a new methodology to build a readily analyzable class of quadratic potential force fields (a generalization of Kavraki’s elliptic potential field) with simple devices, using naturally existing phenomena such as airflows. The benefit of the proposed method is that it simplifies the actuation system without losing the benefits of quadratic potential force fields. This methodology could also be employed in several other phenomena, such as electro-magnetic, dielectrophoretic, thermophoretic, etc., to produce such fields. In this paper, we focus on generating quadratic potential force fields using planar airflows. We first, in Section II, categorize and compare a spectrum of methodologies that can be used to construct force fields, balancing expressive power against complexity of implementation to put this work in context. In Section III, we briefly define and classify quadratic potential force fields and their action on objects, and give a description of the use of airflow fields in distributed manipulation. In Section IV, we present a methodology of building quadratic fields from airflow fields to reduce implementation complexity while retaining the manipulation power of quadratic fields. Finally, in Section V, we demonstrate these results by presenting experiments from a prototype device quantitatively verified against theoretical predictions.
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Fig. 1. Spectrum of field-generating approaches. Brute-force artificial means tend to produce a wider array of fields than natural means, but are generally more complex to implement.
Fig. 2. Artificial approaches to producing force fields. (a) Setting the force at each grid. (b) Setting the potential at each grid (the force is the gradient of the potential).
II. GENERATION OF FORCE FIELDS The generation of force fields can be accomplished by a variety of approaches falling under two categories: via artificial means, where some quantitative parameter (force or potential) is set explicitly at each point in the field, and via natural means, where the natural behavior (i.e., a differential equation) of a medium is set over the field, and the resulting force field is shaped either by setting the differential equation at each point or just by setting boundary conditions. The choice of field-generation approach typically represents a tradeoff between the complexity of implementation and the expressive power of the field (meaning the variety of fields which can be generated by the approach). Fig. 1 shows this spectrum of approaches, which we discuss below. A. Artificial Means—Constructing the Field Constructing a field artificially requires the brute-force application of a desired behavior at each point in space. We distinguish two different methodologies: directly setting the force, and directly setting the potential. 1) Setting the Force: By explicitly setting the traction force to be induced on the object at each point in space, we directly construct the shape of the force field [see Fig. 2(a)]. The net force and torque on the object is the discrete summation of the pointwise forces over the area of the object. The main benefit of this method is the freedom of producing forces of any magnitude and direction at each point in space. This allows the production of any type of desired force fields. It is counterbalanced, though, by the complexity of the design and construction of such devices. That arises from the fact that we need an independent programmable actuator at each application point. Such actuator array devices are the most common form of distributed manipulation seen in the literature [3], [5]–[9], and
many analyzed fields require this level of programmability [5], [13]–[15]. Another issue that arises in these cases is discretization. Since we can generally only set the field in a discrete manner in space, the actual field that is being produced is just an approximation of the desired one [15]. Finer array grids produce better approximations of the desired force field, but require larger numbers of actuators. 2) Setting the Potential: Rather than setting the force at each point, we can directly set the potential at each point in space, and therefore shape the potential field. The force at each point will be the negative gradient of the field [see Fig. 2(b)]. A major benefit of this method is that in many cases, it is simpler to set the potential rather than the force at each point (for example, by applying voltages via an array of electrodes to produce an electrostatic field). There is no need for direct actuation, which makes these devices simpler than the actuator arrays. This sort of method can produce a wide variety of force fields, but is limited, of course, to the class of potential (i.e., conservative) fields. However, many of the fields analyzed in the literature are, in fact, potentials [5], [13]–[15]. Discretization issues still arise, since we must enforce the potential at each point, although densely packing potential generators is generally easier than densely packing force generators. This class of methods is currently underexplored. The authors of this paper are currently examining approaches based on electrode arrays to manipulate microscale objects using dielectrophoretic forces. B. Natural Means In contrast with the previous methodologies, we exploit the dynamics of some natural phenomena (airflow, electric fields, vibrating plates, etc.) to physically produce forces in space. The goal is to shape the action of a medium to produce a desired force field. In most cases, these phenomena follow some physical law which is mathematically expressed by a differential equation (e.g., Poisson’s or Laplace’s equation). Natural approaches fall into two categories: directly setting terms in a differential equation at each point, and setting the boundary conditions. 1) Setting Differential Equation Terms at Each Point: By adjusting the dynamics of the physical medium at each point, a potentially wide range of fields can be produced. For example, by constructing the material through which air flows with varying density, or by varying the permittivity of an electric-field-producing medium, the resulting force field can be shaped. These adjustments generally take the form of the adjustment of one or more terms in the differential equation describing the dynamics.
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While this sort of approach is limited to the production of fields only within the solution space of the differential equations, since terms can be set at each point, the expressive power is still relatively high. It requires a medium which can be constructed with varying properties, but this is much less complex than the active arrays of field generators required to produce fields artificially. In addition, many useful fields could be produced by making piecewise constant variations in the dynamics. This is the approach taken in this paper: a uniform density of airflow is set over specially shaped regions of the plane. This class of methods has not previously been explored in the literature. 2) Setting the Boundary Conditions: Besides directly setting the force, setting the boundary conditions for the differential equation describing the dynamics of a natural medium is the only other commonly examined approach to producing force fields. For example, small numbers of airflow suction points define point boundary conditions for planar airflows described by Laplace’s equation [16], and clamped points along the edges of a vibrating plate set the boundary conditions for the differential equation corresponding to either out-of-plane [17] or in-plane [18] vibrations. The most trivial case is the generation of a uniform field via gravity; simply tilting the plane produces a uniform force on the object. These approaches are the simplest means to generate force fields. All that is required is a natural medium with, typically, some physical condition enforced at only the edges, or at a small number of points in the field. Correspondingly, the expressive power of such field generators is relatively low; the aforementioned airflow and vibratory-plate fields are relatively simple, and require switched sequences of fields to manipulate objects, where more general fields do not.
Fig. 3. Coordinate transformation to the field, specifying the shape (K and K ), position ~x , and orientation of the (in this example) elliptical field.
Fig. 4. Elliptic (upper left), hyperbolic (upper right), critical (lower left), and constant (lower right) potential fields.
where is a point in global coordinates. The force is the negative gradient of (1), which acting on a point with an eigenvalue decomposition becomes (2)
C. Hybrid Designs and Superposition A simple way to produce a force field or to actively change its action is by superimposing more than one field together. These fields can be produced by any combination of the previously mentioned methods. In the case where the fields can be simultaneously active without distorting each other, we can directly combine them in the same plane, such as an actuator array and a gravitational field, as was done in [14]. If the fields are time-independent, a way to combine is by time averaging. Each of multiple fields are activated over short time periods to produce a possibly quite different net field averaged over time.
III. ANALYSIS OF QUADRATIC FIELDS AND AIRFLOW FIELDS
where
is a rotation matrix (with angle determining the orientation and are the eigenvalues and represent the of the field), strengths of the fields in the two directions, and is the force produced by the field at the origin, and it encodes the location of the field defined uniquely as the point where . Thus, any quadratic field can be represented by exactly five parameters (see Fig. 3) (3)
A. Quadratic Potential Force Fields In this subsection, we briefly define and classify quadratic potential force fields and discuss their action on objects. This prior work can be found in [19]. The general form of a quadratic potential is (1)
We can classify the types of quadratic potential fields based on the shape of their equipotential lines, and their stability properand [equivties determined by the stiffness coefficients and the alently, the orientation strength positioning strength ], as shown in Fig. 4. and have the same sign 1) Elliptic Field: When , the equipotential lines of the field , then the forces act form ellipses. If
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toward the center of the field, forming a stable center of , then the forces point attraction. If outwards from the unstable center of the field. Note that if , the semimajor axis of the ellipse will be oriented along the direction (and vice-versa). In the special , the equipotential lines will have case where a circular shape (circular field). Any rotation of the circular field about its origin produces the same circular field, thus, a circular field can be simply represented by a stiffness matrix , where the identity matrix corresponds to a unit circular field. and have different signs 2) Hyperbolic Field: When , the equilibrium of the field is a saddle. The force field pulls in toward the equilibrium along one principal axis and pushes out along the other. We call the special a balanced hyperbolic field with case where stiffness matrix , where signifies the unit hyperbolic field. or , 3) Critical Field: When the field only applies a force in the nonzero-strength direction. Equilibrium in this case lies everywhere along a line, aligned with the sole eigenvector of the field. A critical field closely resembles the action of a squeezing field. , 4) Constant Field: When the only contribution to force is , which is constant over the . field. A special case of this is the null field, where B. Effect on Object The net force and torque on the object can be computed by integration of the pointwise forces over the area of the object (this was done in [20]). For the force, we have (4) where is a unit hyperbolic field and is a unit circular field. The net moment is obtained by integrating the scalar cross where is the position relative to product ). The moment about the object the centroid (i.e., centroid with its principal axis rotated at an angle is (5) Note that appears as a scaling factor on the net moment, hence, the term orientation strength. The final forms of (4) and (5) are in a decomposed form, which provides a significant benefit to the analysis and design of quadratic fields for a manipulation tasks. Since a circular field does not apply any net moment to the object, we can use a circular field to directly set the positioning strength , and use a hyperbolic field to set the orientation strength of the field . For a stable quadratic field1 aligned with the global frame , an object will come to translational equilibrium with its centroid coincident with the origin. The object will have two stable orientations and two unstable orientations, and will come to rest at one of two stable equilibria with the principal axis aligned with the field axis. In addition, any quadratic field satisfies the following useful properties.
K
1Without
). If
S
loss of generality, we assume the convention that = ), orientation is not possible.
= 0(
K
K
S > 0(K >
1) The net applied force is not a function of the orientation of the object, only the position of the centroid relative to the field-fixed frame. 2) The net applied moment is not a function of the position of the centroid of the object, only the orientation of the object relative to the field-fixed frame. This decoupling of force and moment allows for a separate analysis and design of the position and orientation components of a task, coupled only by the fact that both force and moment deand , which we will pend on the field strengths select once for a given task. C. Airflow Fields In two-dimensional flow, a “point” flow sink (we later introduce a flow-sink “region”) sucks in fluid and makes the fluid flow toward it. The drag force between the airflow and small flat objects is considerable enough to manipulate them on a low-friction surface. A distributed manipulation system was introduced in [4], which uses a few point airflow sinks. Flat objects are manipulated using airflow sinks which are placed over the object surface. An air bearing applied under the object provides the low-friction surface. The flow rate of the air bearing is small enough not to affect the manipulating flow. Reynolds number analysis indicates that the manipulating flow is within the laminar flow limit and is fully developed. Assuming that edge effects are negligible, that the pointwise force exerted by the airflow is proportional to its velocity at each point for low-speed flow, and that the flow is irrotational, potential flow theory can be employed for modeling the force field generated by a sink [16] and
(6)
where is the flow-sink strength. The equipotential lines of the field due to a point flow-sink form concentric circles, and the fluid flows directly toward the sink with increasing velocity. While this potential flow model is based on many generous assumptions, it still properly predicts both equilibrium poses for flat objects of arbitrary shape and the quasi-static motion of objects in airflow fields [16], [21], [22]. The net force exerted on the object due to the airflow can be obtained by taking the integral of the pointwise drag force over the object surface [16]. The net force on an object in a potential field can be obtained from taking the derivative of its lifted potential, which is the integral of the pointwise potential over the object surface. This commutative property of the partial differentials and the integrals is also proposed by previous researchers, such as in [23] for polygon objects and in [24] for general objects. Commutativity is derived from smoothness of the lifted potential and integrability of the net force for potential functions which are continuous almost everywhere
(7) where the integral is over the object surface is a pointwise potential, is called the “lifted” potential, is the pointwise represents the boundary of , and force, is the net force,
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is the normal vector to . Notice that the third equality implies commutativity between the partial differentials and the integral [24], which is valid even for integrable singularities (proved by taking an detour around the singularity [25]). The last term is a line integral form of the force equation based on virtual work analysis in [16]. The point flow-sink can position small flat objects, but cannot orient them, due to the radial symmetry of the field. For this purpose, we can generate more complex flow fields by linearly superimposing the logarithmic potential functions. The potential of a point sink field is a solution to the Laplace equation everywhere except at the sink, which equivalently means that the divergence of the force field is zero everywhere, except at the sink, where it is integrable (8)
Fig. 5. Duality of an object in a point sink field and a point object in a set of uniformly distributed sinks. The set of sinks is called a “region” sink, whose shape is identical to the object .
O
unit flow-sink density (i.e., unit suction flow rate per unit area), the divergence is equal to negative one, and the force field satisfies Poisson’s equation (9)
Due to this property, an object on a field generated by one or more sinks cannot be in stable equilibrium unless it is covering at least one sink. However, equilibria generally are not unique. Using point sinks to form a distributed manipulation system gives us very simple system configurations, but produces hard analysis problems, such as finding all the equilibria and their stability. In [16], it is shown that, for example, an -shaped object in a three-flow sink field has six stable equilibria, and the number of equilibria changes, depending on the configuration of sinks and the distance between sinks relative to the size of the object. Although that paper provides an efficient algorithm to find all the equilibria, doing so is still computationally intensive compared with calculating axes of inertia of an object, which is done in the case of quadratic fields. In [21], a manipulation algorithm for convex objects is proposed using two airflow sinks, but requires that objects have a unique position equilibrium on a single sink, which is not guaranteed in general cases. To make use of the algorithm, it is necessary to go through a rigorous analysis to select objects which have a unique equilibrium. While the fields generated by flow sinks have a seemingly different character from the readily analyzable quadratic fields, we propose in the next section a methodology for constructing quadratic fields from continuous regions of flow sinks, rather than from individual sinks, taking advantage of the integrable nature of a flow sink’s singularity. The result is a field which has the desirable equilibria properties of a quadratic field (single position and two orientation equilibria), while keeping a simple device configuration (not requiring explicit actuation at each point in the field).
IV. BUILDING QUADRATIC FIELDS FROM AIRFLOW FIELDS Stable quadratic potential force fields have a constant negative divergence. On the other hand, a flow sink has an infinite but integrable negative divergence at a point (and zero elsewhere). By uniformly distributing a continuum of flow sinks, we can create the constant uniform negative divergence required to build a quadratic field, where the divergence singularities integrate to a finite value. More specifically, inside a region with
Constructing a particular quadratic field from a flow-sink region requires the following. 1) Match the desired divergence of the field by setting the appropriate density of sinks over the region. 2) Shape the region to impose a linear form on the generated net force and torque, matching that of the desired quadratic field (since divergence alone does not fully describe the nature of the generated field.) In effect, we are selecting regions where we impose one of two differential equations: Laplace’s or Poisson’s equation, such that the resulting flow produces the desired force field. To design a sink region which generates a desired quadratic field, we approach the dual problem, rather than the original problem, i.e., we select a shape for an object in a point sink field whose lifted potential is quadratic, rather than finding a region of uniformly distributed sinks which generates a quadratic field. acting on an object Lemma 1: (Duality): The net force by a point sink at is equivalent to the force acting by a set of uniformly distributed sinks that cover on the point ). the shape of the mirrored object (i.e., See Fig. 5. Proof: The net force is
where the subscript denotes the object and the argument denotes the location of the sink. and a uniformly distributed sink Consider a point object at . The net force is the integral of the pointwise force region over (10) to and negating Mirroring the region of integration from each point in the integral preserves the result. Therefore (11)
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Fig. 6. Circular and critical fields (described by equipotential lines and gradient vectors) produced by regions of uniformly distributed sinks (grey).
O
Fig. 7. Linearity of the elliptic shape. The white ellipse is the entire ellipse “cut by” point ~ r , and the dashed ellipse is ellipse O scaled by and is equal to the ellipse cut by ~ r . The gray part does not contribute to the force r on the ellipse O . from a sink at ~
O We now apply this to design a set of useful quadratic fields. For the rest of the analysis, we assume with no loss of generality that the flow rate through any sink region is of unit flow strength. 1) Circular Field: To produce a circular field (see Fig. 6), we set the sink region to be a circular disc , with radius and without loss of generality, its center located at the center of our frame. Due to the dual property of flow fields, the force on an arbitrary point at inside the sink region is the same as the force acting on a circular object by a point sink located at . Using the mean value theorem of harmonic functions, which states that the average value of the potential over a circular surface is equal to the value at the center of the disc, each component of the net force on the object is found to be [21]
Instead, we prove the linearity of the force field inside the elliptic sink region using the follow propositions. be a force field and be the Proposition 1: Let same force field shifted by . Let a compact object and be the object scaled by , i.e., . Let be the net force on from the force field . If the field , then , i.e., satisfies the net force scales linearly when the object and field position are scaled. Proof: The net force on an object is found by direct integration of the pointwise forces over the object’s area
(12) (15) As the strengths are equal, the generated force field is a circular field. The positioning strength of field is con. This implies that the strength is not a function of stant, the size of the circular region, and thus, we can only change the strength of the field by adjusting the flow rate per unit area of the sink region. 2) Critical Field: To produce a critical field, we solve the to be a strip dual problem where we choose the object between two parallel lines that extend to infinity (see Fig. 6). Without loss of generality, we assume that the parallel lines are and , where is the distance from the given by axis. The force from a sink located at can be found by direct integration of (7)
Scaling the object and the field position object in Fig. 7)
by
(dotted elliptic
(16) Note that integrating by over is equivalent to inteby over , and thus, we can write grating
and since
, we have
(13)
(17)
(14)
Proposition 2: The net force on an elliptic shell defined by two concentric ellipses generated by a single sink located at an arbitrary point inside the shell is zero. Proof: The proof has been done in [26], based on a geometric argument used in electrostatic fields for a uniformly charged ellipsoid that imposes a force of the form . The same method applies here in the 2-D case, but it is omitted. Proposition 3: The net force induced by a point sink is located at an arbitrary point inside an elliptic object proportional to the net force induced by an identical sink located on the same object , i.e., the at a point net force on the elliptic object is linear with the distance from the sink to the center of the object.
In this case, , and , which is a critical field. Thus, based on the duality property, we can produce a critical field inside a sink region confined by two infinite parallel lines. To implement this field, we use as a flow region, a finite strip with frictionless walls on the ends, which produces the same field with that of an infinite strip. 3) Elliptic Field: To produce an elliptic field, we choose the , where is the length of the region to be an ellipse major axis and is the length of the minor axis. To demonstrate that this indeed produces an elliptic field requires computing the pointwise forces inside the ellipse directly. This requires computing an elliptic integral which does not have a closed form.
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Proof: Define as a point in the interior of an elliptic ob(as in Fig. 7). Define as the interior elliptic part of with same aspect ratio as , with its boundary crossing through (white colored elliptic object in Fig. 7). We call the the ellipse “cut by” . Consider a point such ellipse still in the interior. that , 1) Since we have that we have from Proposition 1 that ject
(18) 2) Since for a point sink, we scale the object by , we have , and thus, (18) becomes that . 3) Finally, since Proposition 2 applies in our case, we can include the outer portion of the object without changing the net force on , and thus, . The expression for the net force becomes (19) Thus, it is evident that the force inside source varies linearly, and can be expressed in two components in the form and . We can directly compute the slopes and by integrating the derivative of the pointwise forces for the special case where the sink is located at the center of the elliptic object. This special case has a closed form (20) (21) Finally, using the duality lemma, we can state the following corollary. Corollary 1: An elliptic field is produced inside an elliptic region with density sink (see Fig. 6). The elliptic field is (22) This elliptic field exhibits some interesting properties. 1) The positioning stiffness is always constant, , for unit strength airflow. 2) The force is independent of the size of the ellipse, as the field can be written in terms of its aspect ratio . Note that . 3) The elliptic field equipotential lines have a different aspect . ratio from that of the boundary , the ellipse degenerates to the circular case 4) When and . with 5) When or , the ellipse degenerates to a crit(stable), and ical field with (unstable). The fact that we have a bounded positioning and orientation strength implies a bounded net force and torque that can be induced on the object limited by the obtainable flow density. 4) Balanced Hyperbolic Field: A balanced hyperbolic field is a divergenceless field (since its positioning strength is
Fig. 8. Left: An elliptic field is generated inside an elliptic shaped region sink . Right: The hyperbolic field is generated in the circular region. The elliptic field acts as a sink, while the circular field acts as a source. The two fields cancel each other.
O
O
O
), and thus, cannot be constructed in the same manner as the previous quadratic fields. Instead, it is a solution to Laplace’s equation, and thus, we can construct it by selecting a boundary which does not contain any sinks. An alternative design is based on the superposition theory of quadratic fields [19]. We can construct a hyperbolic field by superimposing different combinations of circular, critical, or elliptic fields. For example, we can use an elliptic field in which a circular source region (with posfield (circular region sink) has been itive divergence) superimposed concentric with an elliptic field (elliptic region sink). Inside the circular boundary, the net positioning strength , while . Thus, the will be zero region does not act as a sink, and we have a naturally produced hyperbolic field (Fig. 8). Though hyperbolic fields directly affect the torque on the object and provide a significant analytical tool when designing using decomposition, due to their unstable nature, their realization is not practical. 5) Constant Field: The only practical use of a constant field is to shift the equilibrium position. It is a divergenceless field, and, as in the case of the hyperbolic field, it cannot be built in the interior of a sink region. A constant field can be produced, though, simply between a line source and a parallel line sink. Since there are easier methods of producing constant fields with other physical phenomena like gravity, etc., we will not provide any further analysis in this section. V. EXPERIMENTAL RESULTS We have done a series of experiments which demonstrate both that we can bring an object to a single equilibrium point with one of the two possible orientations, and that the net force and torque are consistent with the theoretical results, i.e., net force is linear to the position with proper proportions of their slopes, and net torque is sinusoidal with the orientation. A. Hardware Testbed We have constructed a hardware testbed capable of creating a quadratic field, shown in Figs. 9 and 10. To achieve a uniform flow rate through the elliptic opening (sink region) we put “honeycomb” filters in the flow chamber, so that the flow breaks evenly after passing through them. The quadratic field is generated within the boundary of the sink region, and its strength depends on the airflow rate through the sink region. As the damping of the system is rather low, we generate intermittent “hopping” of an object by applying a pulsed flow in the air bearing to emulate high damping. The manipulating flow is continuously on top of the object during the entire experimental process. The object is released at an arbitrary point within the
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Fig. 9. Experimental setup. The device is composed of three parts: the air-bearing table which lifts the object, the sink shaped according to the desired field, and the flow chamber which distributes the flow evenly across the sink. A honeycomb is also placed on the sink to even the flow.
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Fig. 11. Equilibrium pose photos of several objects under the action of a critical field on the experimental device. Left top and bottom shows the two stable orientation equilibria that were achieved for an object. The boundary of the sink region and its axes are superimposed on the photos, together with the major and minor axes of the objects in the equilibrium pose. The objects are well aligned, as predicted for a critical field. There is a consistent error to the position of the objects, which might be attributed to a slight tilt of the air table.
Four different-shaped objects were brought into position and orientation equilibrium. We present the results in Fig. 11. The major and the minor axis of the object are clearly marked. In equilibrium, the major axis is closely aligned to the midline of the critical field. The position equilibria lies along the horizontal midline of the rectangle. In Fig. 11, we observe that there is an error at the position of the object, as its centroid is not on the midline. Since this error is consistent to all the different objects that we tested, we conclude that the error is due to a constant field (possibly, a gravitational field due to a slight tilt in the table) that shifts the equilibrium of the critical field. C. Elliptic Field Fig. 10. Pictures of actual device. Top Left: The air table and the object lying on it. Lower left: The rectangular-shaped critical field sink with the honeycomb attached, laid above the air table. Right: The flow chamber has been added on top of the sink to distribute the flow.
effective field boundary, and its centroid moves toward the equilibrium at each hop. We present results for the cases of a critical and an elliptic field. B. Critical Field To produce a critical field, we used an elongated rectangular opening as the sink with its vertical (shortest) sides blocked to the flow. The object was kept well inside the shortest (horizontal boundary) width of the rectangle to avoid any boundary disruptions of the flow. We assume that the effects due to the vertical boundary are negligible, and that the elongated rectangle with blocked ends is a valid approximation of an opening bounded by two infinite parallel lines.
To generate an elliptic field, we set the shape of the sink region to be an ellipse. The elliptic field is produced below this sink region, and can bring the object to a unique position and one of the two possible stable orientation equilibria. To validate the theoretical results of the net force and torque, we examine both the final equilibrium position and orientation of the object and its dynamic behavior. The net force and torque due to an elliptic field are given by (4) and (5). To experimentally validate these equations, we consider the case where for each step, we activate the air bearing for a small interval in time (pulse), which causes a “hop” of the object. At each hop, the object moves a certain distance, due to the drag force of the manipulating flow. The sink region is active during the entire duration of the experiment, and can effectively move the object a noticeable amount during each hop. Each hop is ended by turning off the air bearing, and the object settles at a new location. The force due to the sink which is still active is not capable of overcoming the friction force between the object and the aluminum plate, and thus, cannot further move the object. Assuming a small time interval and a
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Fig. 12. Demonstration of the linearity of the force components F and F of the net force in an elliptic field, versus the x and y position of the object’s centroid, respectively.
Fig. 13. Experimental verification of an object’s torque curve in an elliptic field. The net torque follows a sinusoidal form, as predicted by the theoretical results.
constant acceleration for each hop, we have , and the amount that the centroid of the object is expected to . Thus, the force must be directly move is proportional to the distance that the object traveled at each hop (23) where is the area of the object and is the aspect ration of the sink region. Thus, force can be measured by measuring dis, and a plot of over should be a straight placement line for a quadratic field. In this experiment, we selected the as. Fig. 12 shows the linearity pect ratio of the ellipse to be of the and components of force. We estimated the aspect ratio of this field by dividing the estimated slopes of the fitted , which has an % lines, and it was found to be error. Similarly, the net torque is proportional to the amount of the , and thus, a plot of over the orienangular movement would have a sinusoidal form tation of the object (24) Fig. 13 shows the results for the torque measurements. We observe that the acquired data have a sinusoidal form, as was expected from the theoretical results. A sinusoidal curve was least-square fitted to the data. The period of the sinusoid was well observed to be , as to the . There is a shift in the phase by theoretically expected and a dead-zone of around the equilibrium, which may be attributed to static friction at the beginning and end of each hop. Finally, in Fig. 14, we present the position and orientation equilibria of three different objects. The major axes of the objects are closely aligned with those of the elliptic field. In the upper left and lower left figures, we present the two different orientation equilibria for the same object. The position equilibria
Fig. 14. Equilibrium pose of several objects. Left top and bottom shows the two stable orientation equilibria achieved for an object. The elliptic boundary of the sink and its axes are shown, together with the major and minor axes of the objects in equilibrium.
show some error, which may be due to the presence of a gravitational (constant) field (i.e., a tilt in the air-bearing plate) or to the dead zone that we observe in Figs. 12 and 13. D. Limitations of the Experimental Device The experimental device was modeled using a generous set of assumptions. Since the experimental results are well-predicted by the theory, we conclude that the model is sufficiently accurate, although certain properties of the experimental device limit the accuracy. First, while the large low-pressure plenum (flow chamber) and honeycomb tend to distribute the flow, there is no
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guarantee that the flow is exactly uniform over the sink region. In addition, the honeycomb may discretize the flow, although this effect is likely to be small, since the cell density is high relative to the size of the object. Second, several local effects may disrupt the flow, including the edges of the object itself which has significant thickness, the air-bearing flow which is small but not negligible, and 3-D effects, particularly due to vertical flow near the edges of the sink region. Third, producing a viscous surface via hopping causes inconsistencies in friction. For example, during the time that the air bearing takes to develop, large manipulation forces can overcome static friction, while smaller ones cannot.
VI. CONCLUSION This paper is concerned with the development of methods for constructing planar force fields for distributed manipulation applications. Most distributed manipulation research focuses on either extreme of a range of possible methodologies for producing force fields, either by using explicit generation of forces at each point, or by setting boundary conditions for a dynamic manipulation medium through which forces are generated. In this paper, we focus on the relatively unexplored midrange, where we explicitly set terms in the differential equation describing the medium at each point. In this way, we gain back some of the expressive power of pointwise-explicit fields, while maintaining the ease of implementation of manipulation through natural medium dynamics. We review the set of quadratic fields which previously could only be generated through explicit force actuation at each point. In addition to being readily analyzable, these fields naturally produce predictable equilibria, thus simplifying implementation since feedback and control may not be necessary. The proposed novel approach is to generate quadratic potential force fields using simple devices based on passive planar airflow. In this way, we are taking advantage of the ease of implementation of this natural medium, while alleviating the typical problems of multiple equilibria and intensive computational planning inherent in airflow approaches. We demonstrate analytically how to produce particular quadratic fields from passive flow sink regions, and provide experimental demonstration and quantitative verification of the theoretical results. There are numerous applications of the methodology that has been developed in this paper. These applications mainly concern the simplified construction of micro or mesoscale distributed manipulation devices, which have so far been done by complicated structures. Additional benefits are that the action on the object is realized in a noncontact manner, and the system exhibits smooth dynamics, unlike typical approaches in nonprehensile manipulation. The same approach could potentially apply to other similar phenomena, such as electromagnetics, thermophoretics, etc. to produce useful forms of conservative force fields for sensorless distributed manipulation.
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REFERENCES [1] K. Böhringer, K. Goldberg, V. Bhatt, and B. Donald, “Algorithms for sensorless manipulation using a vibrating surface,” Algorithmica, vol. 26, pp. 389–429, Mar. 2000. [2] P. Frei, M. Wiesendanger, R. Büchi, and L. Ruf, “Simultaneous planar transport of multiple objects on individual trajectories using friction forces,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 49–64. [3] M. Yim, J. Reich, and A. Berlin, “Two approaches to distributed manipulation,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 237–261. [4] J. Luntz and H. Moon, “Distributed manipulation with passive air flow,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2001, pp. 195–201. [5] K. Böhringer, B. Donald, R. Mihailovich, and N. MacDonald, “Sensorless manipulation using massively parallel microfabricated actuator arrays,” in Proc. IEEE Int. Conf. Robot. Autom., 1994, pp. 826–833. [6] J. Suh, R. Darling, K. Böhringer, B. Donald, H. Baltes, and G. Kovacs, “CMOS integrated organic ciliary actuator array for general-purpose micromanipulation tasks,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 191–215. [7] S. Tadokoro, S. Fuji, T. Takamori, and K. Oguro, “Distributed actuation devices using soft gel actuators,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 217–235. [8] J. Luntz, W. Messner, and H. Choset, “Parcel manipulation and dynamics with a distributed actuator array: The virtual vehicle,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 1541–1546. [9] T. Murphey and J. Burdick, “Feedback control for distributed manipulation systems that involve mechanical contacts,” Int. J. Robot. Res., vol. 23, no. 7, pp. 763–781, Jul. 2004. [10] S. Akella, W. Huang, K. Lynch, and M. Mason, “Sensorless parts feeding with a one joint robot,” in Algorithms for Robotic Motion and Manipulation, J.-P. Laumond and M. Overmars, Eds. Wellesley, MA: A. K. Peters, 1997, pp. 229–237. [11] M. Erdmann, “An exploration of nonprehensile two-palm manipulation,” Int. J. Robot. Res., vol. 17, no. 5, 1998. [12] K. Goldberg, “Orienting polygonal parts without sensors,” Algorithmica: Special Issue Comput. Robot., vol. 10, pp. 201–225, 1993. [13] L. Kavraki, “Part orientation with programmable vector fields: Two stable equilibria for most parts,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 2446–2451. [14] K. Böhringer, B. Donald, L. Kavraki, and F. Lamiraux, “Part orientation with one or two stable equilibria using programmable force fields,” IEEE Trans. Robot. Autom., vol. 16, no. 2, pp. 157–170, Apr. 2000. [15] J. Luntz, W. Messner, and H. Choset, “Distributed manipulation using discrete actuator arrays,” Int. J. Robot. Res., vol. 20, no. 7, pp. 553–583, Jul. 2001. [16] H. Moon and J. Luntz, “Prediction of equilibria of lifted logarithmic radial potential fields,” Int. J. Robot. Res., vol. 23, no. 7-8, pp. 747–762, Jul.–Aug. 2004. [17] K. Böhringer, V. Bhatt, and K. Goldberg, “Sensorless manipulation using transverse vibrations of a plate,” in Proc. IEEE Int. Conf. Robot. Autom., 1995, pp. 1989–1996. [18] D. Reznik, E. Moshkoich, and J. Canny, “Building a universal planar manipulator,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA, 2000, pp. 147–171. [19] K. Varsos and J. Luntz, “Analysis, decomposition, and superposition of quadratic potential force fields for distributed manipulation,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2000, pp. 3168–3173. [20] K. Varsos and J. Luntz, “Distributed manipulation along trajectories using open-loop force fields,” in Proc. IEEE Int. Conf. Robot. Autom., 2002, pp. 1203–1209. [21] H. Moon and J. Luntz, “Synthesis bounds for distributed manipulation using logarithmic-radial potential fields,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2003, pp. 2908–2913. [22] H. Moon and J. Luntz, “Toward sensorless manipulation using airflow,” in Proc. IEEE Int. Conf. Robot. Autom., 2004, pp. 1574–1579. [23] K. Böhringer, B. Donald, and N. C. MacDonald, “Programmable vector fields for distributed manipulation, with applications to MEMS actuator arrays and vibratory parts feeders,” Int. J. Robot. Res., vol. 18, no. 2, pp. 168–200, Feb. 1999. [24] F. Lamiraux and L. Kavraki, “Positioning of symmetric and nonsymmetric parts using radial and constant fields: Computation of all equilibrium configurations,” Int. J. Robot. Res., vol. 20, no. 8, pp. 635–659, Aug. 2001. [25] H. Moon and J. Luntz, “Distributed manipulation by superposition of logarithmic-radial potential field,” in Proc. IEEE Int. Conf. Robot. Autom., 2002, pp. 1197–1202. [26] W. Thomson, Papers on Electricity and Magnetism. London, U.K.: Macmillan, 1884, pp. 178–179.
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Konstantinos Varsos (S’00) received the diploma from the Department of Electrical Engineering and Computer Technology, University of Patra, Patra, Greece, in 2000, and the Masters degrees in 2001 from the Department of Industrial and Operations Engineering and the Department of Mechanical Engineering, University of Michigan, Ann Arbor, where he is currently working toward the Ph.D. degree in mechanical engineering. His research interests include distributed manipulation, medical robotics, and industrial automation.
Hyungpil Moon (S’01–M’06) received the B.S. and M.S. degrees from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1996 and 1998, respectively, and the Ph.D. degree in mechanical engineering from the University of Michigan, Ann Arbor, in 2005. He is now a Visiting Research Scientist with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA. His research interests include distributed manipulation, coordination and control of distributed systems, SLAM, and nonlinear control theory with applications in mechanical systems.
Jonathan Luntz (M’00) obtained the B.S. degree in mechanical engineering from the State University of New York at Buffalo in 1992, and the M.E. and Ph.D. degrees in mechanical engineering from Carnegie Mellon University, Pittsburgh, PA, in 1994 and 1999, respectively. He joined the University of Michigan, Ann Arbor, as an Assistant Professor of Mechanical Engineering in 2000, and in 2004 joined the research faculty as an Assistant Research Scientist. He currently works in the areas of distributed manipulation, cooperative mobility, smart material actuation (particularly with shape memory alloy), and in reconfigurable logic control of combined hardware and software resources in manufacturing systems.
IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006
Generation of Quadratic Potential Force Fields From Flow Fields for Distributed Manipulation Konstantinos Varsos, Student Member, IEEE, Hyungpil Moon, Member, IEEE, and Jonathan Luntz, Member, IEEE
Abstract—Distributed manipulation systems induce motions on objects through the application of many external forces. Many of these systems are abstracted as planar programmable force fields. Quadratic potential fields form a class of such fields that lend themselves to analytical study and exhibit useful stability properties. This paper introduces a new methodology to build quadratic potential fields with simple devices using the naturally existing phenomena of airflow, which is an improvement to the traditional use of the complicated programmable actuator arrays. It also provides a basis for the exploitation, in distributed manipulation, of natural phenomena like airflow, which require rigorous analysis and display stability difficulties. A demonstration and verification of the theoretical results for the special case of the elliptic field with airflows is also presented. Index Terms—Airflow, distributed manipulation, nonprehensile manipulation, parts feeding, programmable force fields, quadratic potential fields.
I. INTRODUCTION
D
ISTRIBUTED manipulation systems induce motions on objects through the application of many external forces. The general task is to impart some desired motion to an object, either in space or restricted to a plane, by using either a large number of independently controlled actuators acting at many points on the object, or by exploiting the dynamics of some continuous medium which is in contact with the manipulated object. Many forms of distributed manipulation systems exist, including those based on vibrating plates [1], [2], actively controlled air jets [3], passive airflow fields [4], and planar micro[5], [6], mini- [7], and macro-mechanical actuator arrays [3], [8], [9]. By nature, such systems generally involve redundant actuation, and hence provide tremendous manipulation power. In addition, by distributing the application of force, these devices allow the manipulation of thin, flexible, and delicate objects. The concept of a programmable force field abstracts many distributed manipulators, including actuator arrays, which explicitly generate arbitrary forces at every point in the planar field. Manuscript received December 23, 2004; revised May 7, 2005. This paper was recommended for publication by Associate Editor K. Lynch and Editor I. Walker upon evaluation of the reviewers’ comments. This work was supported in part by the National Science Foundation under CAREER award RHA-0093312. This paper was presented in part at the International Conference on Robotics and Automation, Barcelona, Spain, April 2005. K. Varsos and J. Luntz are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2121 (e-mail: [email protected]; [email protected]). H. Moon was with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2121. He is now with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213 USA. Digital Object Identifier 10.1109/TRO.2005.858858
An object experiences net force and moment, which are the integral of the pointwise force field over the area of the object. The study of programmable force fields reduces to the determination of the net force and moment as a function of the shape of the field and of the object, and the object’s position and orientation relative to the field. Typical work in this area revolves around the particular problem of bringing an object to a unique position and orientation in a sensorless manner. Based on minimalistic manipulation concepts [10]–[12], Böhringer et al. [5] began examining provably useful force fields for orienting objects. Improving upon this, Kavraki [13] analyzed an elliptic potential field which orients most parts to one of two orientations (within symmetry). Finally, these researchers combined a unit radial field and constant (gravity) field to bring parts to a single orientation [14]. This line of research trades off feedback and control complexity for actuation complexity, using redundant actuation to produce natural equilibria without sensing and control. In previous work [14], [15], generating elliptic fields (and other powerful distributed manipulation fields) required pointwise actuation (i.e., a dense array of individually addressable actuators) which leads to complicated devices. This paper introduces a new methodology to build a readily analyzable class of quadratic potential force fields (a generalization of Kavraki’s elliptic potential field) with simple devices, using naturally existing phenomena such as airflows. The benefit of the proposed method is that it simplifies the actuation system without losing the benefits of quadratic potential force fields. This methodology could also be employed in several other phenomena, such as electro-magnetic, dielectrophoretic, thermophoretic, etc., to produce such fields. In this paper, we focus on generating quadratic potential force fields using planar airflows. We first, in Section II, categorize and compare a spectrum of methodologies that can be used to construct force fields, balancing expressive power against complexity of implementation to put this work in context. In Section III, we briefly define and classify quadratic potential force fields and their action on objects, and give a description of the use of airflow fields in distributed manipulation. In Section IV, we present a methodology of building quadratic fields from airflow fields to reduce implementation complexity while retaining the manipulation power of quadratic fields. Finally, in Section V, we demonstrate these results by presenting experiments from a prototype device quantitatively verified against theoretical predictions.
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Fig. 1. Spectrum of field-generating approaches. Brute-force artificial means tend to produce a wider array of fields than natural means, but are generally more complex to implement.
Fig. 2. Artificial approaches to producing force fields. (a) Setting the force at each grid. (b) Setting the potential at each grid (the force is the gradient of the potential).
II. GENERATION OF FORCE FIELDS The generation of force fields can be accomplished by a variety of approaches falling under two categories: via artificial means, where some quantitative parameter (force or potential) is set explicitly at each point in the field, and via natural means, where the natural behavior (i.e., a differential equation) of a medium is set over the field, and the resulting force field is shaped either by setting the differential equation at each point or just by setting boundary conditions. The choice of field-generation approach typically represents a tradeoff between the complexity of implementation and the expressive power of the field (meaning the variety of fields which can be generated by the approach). Fig. 1 shows this spectrum of approaches, which we discuss below. A. Artificial Means—Constructing the Field Constructing a field artificially requires the brute-force application of a desired behavior at each point in space. We distinguish two different methodologies: directly setting the force, and directly setting the potential. 1) Setting the Force: By explicitly setting the traction force to be induced on the object at each point in space, we directly construct the shape of the force field [see Fig. 2(a)]. The net force and torque on the object is the discrete summation of the pointwise forces over the area of the object. The main benefit of this method is the freedom of producing forces of any magnitude and direction at each point in space. This allows the production of any type of desired force fields. It is counterbalanced, though, by the complexity of the design and construction of such devices. That arises from the fact that we need an independent programmable actuator at each application point. Such actuator array devices are the most common form of distributed manipulation seen in the literature [3], [5]–[9], and
many analyzed fields require this level of programmability [5], [13]–[15]. Another issue that arises in these cases is discretization. Since we can generally only set the field in a discrete manner in space, the actual field that is being produced is just an approximation of the desired one [15]. Finer array grids produce better approximations of the desired force field, but require larger numbers of actuators. 2) Setting the Potential: Rather than setting the force at each point, we can directly set the potential at each point in space, and therefore shape the potential field. The force at each point will be the negative gradient of the field [see Fig. 2(b)]. A major benefit of this method is that in many cases, it is simpler to set the potential rather than the force at each point (for example, by applying voltages via an array of electrodes to produce an electrostatic field). There is no need for direct actuation, which makes these devices simpler than the actuator arrays. This sort of method can produce a wide variety of force fields, but is limited, of course, to the class of potential (i.e., conservative) fields. However, many of the fields analyzed in the literature are, in fact, potentials [5], [13]–[15]. Discretization issues still arise, since we must enforce the potential at each point, although densely packing potential generators is generally easier than densely packing force generators. This class of methods is currently underexplored. The authors of this paper are currently examining approaches based on electrode arrays to manipulate microscale objects using dielectrophoretic forces. B. Natural Means In contrast with the previous methodologies, we exploit the dynamics of some natural phenomena (airflow, electric fields, vibrating plates, etc.) to physically produce forces in space. The goal is to shape the action of a medium to produce a desired force field. In most cases, these phenomena follow some physical law which is mathematically expressed by a differential equation (e.g., Poisson’s or Laplace’s equation). Natural approaches fall into two categories: directly setting terms in a differential equation at each point, and setting the boundary conditions. 1) Setting Differential Equation Terms at Each Point: By adjusting the dynamics of the physical medium at each point, a potentially wide range of fields can be produced. For example, by constructing the material through which air flows with varying density, or by varying the permittivity of an electric-field-producing medium, the resulting force field can be shaped. These adjustments generally take the form of the adjustment of one or more terms in the differential equation describing the dynamics.
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While this sort of approach is limited to the production of fields only within the solution space of the differential equations, since terms can be set at each point, the expressive power is still relatively high. It requires a medium which can be constructed with varying properties, but this is much less complex than the active arrays of field generators required to produce fields artificially. In addition, many useful fields could be produced by making piecewise constant variations in the dynamics. This is the approach taken in this paper: a uniform density of airflow is set over specially shaped regions of the plane. This class of methods has not previously been explored in the literature. 2) Setting the Boundary Conditions: Besides directly setting the force, setting the boundary conditions for the differential equation describing the dynamics of a natural medium is the only other commonly examined approach to producing force fields. For example, small numbers of airflow suction points define point boundary conditions for planar airflows described by Laplace’s equation [16], and clamped points along the edges of a vibrating plate set the boundary conditions for the differential equation corresponding to either out-of-plane [17] or in-plane [18] vibrations. The most trivial case is the generation of a uniform field via gravity; simply tilting the plane produces a uniform force on the object. These approaches are the simplest means to generate force fields. All that is required is a natural medium with, typically, some physical condition enforced at only the edges, or at a small number of points in the field. Correspondingly, the expressive power of such field generators is relatively low; the aforementioned airflow and vibratory-plate fields are relatively simple, and require switched sequences of fields to manipulate objects, where more general fields do not.
Fig. 3. Coordinate transformation to the field, specifying the shape (K and K ), position ~x , and orientation of the (in this example) elliptical field.
Fig. 4. Elliptic (upper left), hyperbolic (upper right), critical (lower left), and constant (lower right) potential fields.
where is a point in global coordinates. The force is the negative gradient of (1), which acting on a point with an eigenvalue decomposition becomes (2)
C. Hybrid Designs and Superposition A simple way to produce a force field or to actively change its action is by superimposing more than one field together. These fields can be produced by any combination of the previously mentioned methods. In the case where the fields can be simultaneously active without distorting each other, we can directly combine them in the same plane, such as an actuator array and a gravitational field, as was done in [14]. If the fields are time-independent, a way to combine is by time averaging. Each of multiple fields are activated over short time periods to produce a possibly quite different net field averaged over time.
III. ANALYSIS OF QUADRATIC FIELDS AND AIRFLOW FIELDS
where
is a rotation matrix (with angle determining the orientation and are the eigenvalues and represent the of the field), strengths of the fields in the two directions, and is the force produced by the field at the origin, and it encodes the location of the field defined uniquely as the point where . Thus, any quadratic field can be represented by exactly five parameters (see Fig. 3) (3)
A. Quadratic Potential Force Fields In this subsection, we briefly define and classify quadratic potential force fields and discuss their action on objects. This prior work can be found in [19]. The general form of a quadratic potential is (1)
We can classify the types of quadratic potential fields based on the shape of their equipotential lines, and their stability properand [equivties determined by the stiffness coefficients and the alently, the orientation strength positioning strength ], as shown in Fig. 4. and have the same sign 1) Elliptic Field: When , the equipotential lines of the field , then the forces act form ellipses. If
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toward the center of the field, forming a stable center of , then the forces point attraction. If outwards from the unstable center of the field. Note that if , the semimajor axis of the ellipse will be oriented along the direction (and vice-versa). In the special , the equipotential lines will have case where a circular shape (circular field). Any rotation of the circular field about its origin produces the same circular field, thus, a circular field can be simply represented by a stiffness matrix , where the identity matrix corresponds to a unit circular field. and have different signs 2) Hyperbolic Field: When , the equilibrium of the field is a saddle. The force field pulls in toward the equilibrium along one principal axis and pushes out along the other. We call the special a balanced hyperbolic field with case where stiffness matrix , where signifies the unit hyperbolic field. or , 3) Critical Field: When the field only applies a force in the nonzero-strength direction. Equilibrium in this case lies everywhere along a line, aligned with the sole eigenvector of the field. A critical field closely resembles the action of a squeezing field. , 4) Constant Field: When the only contribution to force is , which is constant over the . field. A special case of this is the null field, where B. Effect on Object The net force and torque on the object can be computed by integration of the pointwise forces over the area of the object (this was done in [20]). For the force, we have (4) where is a unit hyperbolic field and is a unit circular field. The net moment is obtained by integrating the scalar cross where is the position relative to product ). The moment about the object the centroid (i.e., centroid with its principal axis rotated at an angle is (5) Note that appears as a scaling factor on the net moment, hence, the term orientation strength. The final forms of (4) and (5) are in a decomposed form, which provides a significant benefit to the analysis and design of quadratic fields for a manipulation tasks. Since a circular field does not apply any net moment to the object, we can use a circular field to directly set the positioning strength , and use a hyperbolic field to set the orientation strength of the field . For a stable quadratic field1 aligned with the global frame , an object will come to translational equilibrium with its centroid coincident with the origin. The object will have two stable orientations and two unstable orientations, and will come to rest at one of two stable equilibria with the principal axis aligned with the field axis. In addition, any quadratic field satisfies the following useful properties.
K
1Without
). If
S
loss of generality, we assume the convention that = ), orientation is not possible.
= 0(
K
K
S > 0(K >
1) The net applied force is not a function of the orientation of the object, only the position of the centroid relative to the field-fixed frame. 2) The net applied moment is not a function of the position of the centroid of the object, only the orientation of the object relative to the field-fixed frame. This decoupling of force and moment allows for a separate analysis and design of the position and orientation components of a task, coupled only by the fact that both force and moment deand , which we will pend on the field strengths select once for a given task. C. Airflow Fields In two-dimensional flow, a “point” flow sink (we later introduce a flow-sink “region”) sucks in fluid and makes the fluid flow toward it. The drag force between the airflow and small flat objects is considerable enough to manipulate them on a low-friction surface. A distributed manipulation system was introduced in [4], which uses a few point airflow sinks. Flat objects are manipulated using airflow sinks which are placed over the object surface. An air bearing applied under the object provides the low-friction surface. The flow rate of the air bearing is small enough not to affect the manipulating flow. Reynolds number analysis indicates that the manipulating flow is within the laminar flow limit and is fully developed. Assuming that edge effects are negligible, that the pointwise force exerted by the airflow is proportional to its velocity at each point for low-speed flow, and that the flow is irrotational, potential flow theory can be employed for modeling the force field generated by a sink [16] and
(6)
where is the flow-sink strength. The equipotential lines of the field due to a point flow-sink form concentric circles, and the fluid flows directly toward the sink with increasing velocity. While this potential flow model is based on many generous assumptions, it still properly predicts both equilibrium poses for flat objects of arbitrary shape and the quasi-static motion of objects in airflow fields [16], [21], [22]. The net force exerted on the object due to the airflow can be obtained by taking the integral of the pointwise drag force over the object surface [16]. The net force on an object in a potential field can be obtained from taking the derivative of its lifted potential, which is the integral of the pointwise potential over the object surface. This commutative property of the partial differentials and the integrals is also proposed by previous researchers, such as in [23] for polygon objects and in [24] for general objects. Commutativity is derived from smoothness of the lifted potential and integrability of the net force for potential functions which are continuous almost everywhere
(7) where the integral is over the object surface is a pointwise potential, is called the “lifted” potential, is the pointwise represents the boundary of , and force, is the net force,
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is the normal vector to . Notice that the third equality implies commutativity between the partial differentials and the integral [24], which is valid even for integrable singularities (proved by taking an detour around the singularity [25]). The last term is a line integral form of the force equation based on virtual work analysis in [16]. The point flow-sink can position small flat objects, but cannot orient them, due to the radial symmetry of the field. For this purpose, we can generate more complex flow fields by linearly superimposing the logarithmic potential functions. The potential of a point sink field is a solution to the Laplace equation everywhere except at the sink, which equivalently means that the divergence of the force field is zero everywhere, except at the sink, where it is integrable (8)
Fig. 5. Duality of an object in a point sink field and a point object in a set of uniformly distributed sinks. The set of sinks is called a “region” sink, whose shape is identical to the object .
O
unit flow-sink density (i.e., unit suction flow rate per unit area), the divergence is equal to negative one, and the force field satisfies Poisson’s equation (9)
Due to this property, an object on a field generated by one or more sinks cannot be in stable equilibrium unless it is covering at least one sink. However, equilibria generally are not unique. Using point sinks to form a distributed manipulation system gives us very simple system configurations, but produces hard analysis problems, such as finding all the equilibria and their stability. In [16], it is shown that, for example, an -shaped object in a three-flow sink field has six stable equilibria, and the number of equilibria changes, depending on the configuration of sinks and the distance between sinks relative to the size of the object. Although that paper provides an efficient algorithm to find all the equilibria, doing so is still computationally intensive compared with calculating axes of inertia of an object, which is done in the case of quadratic fields. In [21], a manipulation algorithm for convex objects is proposed using two airflow sinks, but requires that objects have a unique position equilibrium on a single sink, which is not guaranteed in general cases. To make use of the algorithm, it is necessary to go through a rigorous analysis to select objects which have a unique equilibrium. While the fields generated by flow sinks have a seemingly different character from the readily analyzable quadratic fields, we propose in the next section a methodology for constructing quadratic fields from continuous regions of flow sinks, rather than from individual sinks, taking advantage of the integrable nature of a flow sink’s singularity. The result is a field which has the desirable equilibria properties of a quadratic field (single position and two orientation equilibria), while keeping a simple device configuration (not requiring explicit actuation at each point in the field).
IV. BUILDING QUADRATIC FIELDS FROM AIRFLOW FIELDS Stable quadratic potential force fields have a constant negative divergence. On the other hand, a flow sink has an infinite but integrable negative divergence at a point (and zero elsewhere). By uniformly distributing a continuum of flow sinks, we can create the constant uniform negative divergence required to build a quadratic field, where the divergence singularities integrate to a finite value. More specifically, inside a region with
Constructing a particular quadratic field from a flow-sink region requires the following. 1) Match the desired divergence of the field by setting the appropriate density of sinks over the region. 2) Shape the region to impose a linear form on the generated net force and torque, matching that of the desired quadratic field (since divergence alone does not fully describe the nature of the generated field.) In effect, we are selecting regions where we impose one of two differential equations: Laplace’s or Poisson’s equation, such that the resulting flow produces the desired force field. To design a sink region which generates a desired quadratic field, we approach the dual problem, rather than the original problem, i.e., we select a shape for an object in a point sink field whose lifted potential is quadratic, rather than finding a region of uniformly distributed sinks which generates a quadratic field. acting on an object Lemma 1: (Duality): The net force by a point sink at is equivalent to the force acting by a set of uniformly distributed sinks that cover on the point ). the shape of the mirrored object (i.e., See Fig. 5. Proof: The net force is
where the subscript denotes the object and the argument denotes the location of the sink. and a uniformly distributed sink Consider a point object at . The net force is the integral of the pointwise force region over (10) to and negating Mirroring the region of integration from each point in the integral preserves the result. Therefore (11)
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Fig. 6. Circular and critical fields (described by equipotential lines and gradient vectors) produced by regions of uniformly distributed sinks (grey).
O
Fig. 7. Linearity of the elliptic shape. The white ellipse is the entire ellipse “cut by” point ~ r , and the dashed ellipse is ellipse O scaled by and is equal to the ellipse cut by ~ r . The gray part does not contribute to the force r on the ellipse O . from a sink at ~
O We now apply this to design a set of useful quadratic fields. For the rest of the analysis, we assume with no loss of generality that the flow rate through any sink region is of unit flow strength. 1) Circular Field: To produce a circular field (see Fig. 6), we set the sink region to be a circular disc , with radius and without loss of generality, its center located at the center of our frame. Due to the dual property of flow fields, the force on an arbitrary point at inside the sink region is the same as the force acting on a circular object by a point sink located at . Using the mean value theorem of harmonic functions, which states that the average value of the potential over a circular surface is equal to the value at the center of the disc, each component of the net force on the object is found to be [21]
Instead, we prove the linearity of the force field inside the elliptic sink region using the follow propositions. be a force field and be the Proposition 1: Let same force field shifted by . Let a compact object and be the object scaled by , i.e., . Let be the net force on from the force field . If the field , then , i.e., satisfies the net force scales linearly when the object and field position are scaled. Proof: The net force on an object is found by direct integration of the pointwise forces over the object’s area
(12) (15) As the strengths are equal, the generated force field is a circular field. The positioning strength of field is con. This implies that the strength is not a function of stant, the size of the circular region, and thus, we can only change the strength of the field by adjusting the flow rate per unit area of the sink region. 2) Critical Field: To produce a critical field, we solve the to be a strip dual problem where we choose the object between two parallel lines that extend to infinity (see Fig. 6). Without loss of generality, we assume that the parallel lines are and , where is the distance from the given by axis. The force from a sink located at can be found by direct integration of (7)
Scaling the object and the field position object in Fig. 7)
by
(dotted elliptic
(16) Note that integrating by over is equivalent to inteby over , and thus, we can write grating
and since
, we have
(13)
(17)
(14)
Proposition 2: The net force on an elliptic shell defined by two concentric ellipses generated by a single sink located at an arbitrary point inside the shell is zero. Proof: The proof has been done in [26], based on a geometric argument used in electrostatic fields for a uniformly charged ellipsoid that imposes a force of the form . The same method applies here in the 2-D case, but it is omitted. Proposition 3: The net force induced by a point sink is located at an arbitrary point inside an elliptic object proportional to the net force induced by an identical sink located on the same object , i.e., the at a point net force on the elliptic object is linear with the distance from the sink to the center of the object.
In this case, , and , which is a critical field. Thus, based on the duality property, we can produce a critical field inside a sink region confined by two infinite parallel lines. To implement this field, we use as a flow region, a finite strip with frictionless walls on the ends, which produces the same field with that of an infinite strip. 3) Elliptic Field: To produce an elliptic field, we choose the , where is the length of the region to be an ellipse major axis and is the length of the minor axis. To demonstrate that this indeed produces an elliptic field requires computing the pointwise forces inside the ellipse directly. This requires computing an elliptic integral which does not have a closed form.
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Proof: Define as a point in the interior of an elliptic ob(as in Fig. 7). Define as the interior elliptic part of with same aspect ratio as , with its boundary crossing through (white colored elliptic object in Fig. 7). We call the the ellipse “cut by” . Consider a point such ellipse still in the interior. that , 1) Since we have that we have from Proposition 1 that ject
(18) 2) Since for a point sink, we scale the object by , we have , and thus, (18) becomes that . 3) Finally, since Proposition 2 applies in our case, we can include the outer portion of the object without changing the net force on , and thus, . The expression for the net force becomes (19) Thus, it is evident that the force inside source varies linearly, and can be expressed in two components in the form and . We can directly compute the slopes and by integrating the derivative of the pointwise forces for the special case where the sink is located at the center of the elliptic object. This special case has a closed form (20) (21) Finally, using the duality lemma, we can state the following corollary. Corollary 1: An elliptic field is produced inside an elliptic region with density sink (see Fig. 6). The elliptic field is (22) This elliptic field exhibits some interesting properties. 1) The positioning stiffness is always constant, , for unit strength airflow. 2) The force is independent of the size of the ellipse, as the field can be written in terms of its aspect ratio . Note that . 3) The elliptic field equipotential lines have a different aspect . ratio from that of the boundary , the ellipse degenerates to the circular case 4) When and . with 5) When or , the ellipse degenerates to a crit(stable), and ical field with (unstable). The fact that we have a bounded positioning and orientation strength implies a bounded net force and torque that can be induced on the object limited by the obtainable flow density. 4) Balanced Hyperbolic Field: A balanced hyperbolic field is a divergenceless field (since its positioning strength is
Fig. 8. Left: An elliptic field is generated inside an elliptic shaped region sink . Right: The hyperbolic field is generated in the circular region. The elliptic field acts as a sink, while the circular field acts as a source. The two fields cancel each other.
O
O
O
), and thus, cannot be constructed in the same manner as the previous quadratic fields. Instead, it is a solution to Laplace’s equation, and thus, we can construct it by selecting a boundary which does not contain any sinks. An alternative design is based on the superposition theory of quadratic fields [19]. We can construct a hyperbolic field by superimposing different combinations of circular, critical, or elliptic fields. For example, we can use an elliptic field in which a circular source region (with posfield (circular region sink) has been itive divergence) superimposed concentric with an elliptic field (elliptic region sink). Inside the circular boundary, the net positioning strength , while . Thus, the will be zero region does not act as a sink, and we have a naturally produced hyperbolic field (Fig. 8). Though hyperbolic fields directly affect the torque on the object and provide a significant analytical tool when designing using decomposition, due to their unstable nature, their realization is not practical. 5) Constant Field: The only practical use of a constant field is to shift the equilibrium position. It is a divergenceless field, and, as in the case of the hyperbolic field, it cannot be built in the interior of a sink region. A constant field can be produced, though, simply between a line source and a parallel line sink. Since there are easier methods of producing constant fields with other physical phenomena like gravity, etc., we will not provide any further analysis in this section. V. EXPERIMENTAL RESULTS We have done a series of experiments which demonstrate both that we can bring an object to a single equilibrium point with one of the two possible orientations, and that the net force and torque are consistent with the theoretical results, i.e., net force is linear to the position with proper proportions of their slopes, and net torque is sinusoidal with the orientation. A. Hardware Testbed We have constructed a hardware testbed capable of creating a quadratic field, shown in Figs. 9 and 10. To achieve a uniform flow rate through the elliptic opening (sink region) we put “honeycomb” filters in the flow chamber, so that the flow breaks evenly after passing through them. The quadratic field is generated within the boundary of the sink region, and its strength depends on the airflow rate through the sink region. As the damping of the system is rather low, we generate intermittent “hopping” of an object by applying a pulsed flow in the air bearing to emulate high damping. The manipulating flow is continuously on top of the object during the entire experimental process. The object is released at an arbitrary point within the
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Fig. 9. Experimental setup. The device is composed of three parts: the air-bearing table which lifts the object, the sink shaped according to the desired field, and the flow chamber which distributes the flow evenly across the sink. A honeycomb is also placed on the sink to even the flow.
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Fig. 11. Equilibrium pose photos of several objects under the action of a critical field on the experimental device. Left top and bottom shows the two stable orientation equilibria that were achieved for an object. The boundary of the sink region and its axes are superimposed on the photos, together with the major and minor axes of the objects in the equilibrium pose. The objects are well aligned, as predicted for a critical field. There is a consistent error to the position of the objects, which might be attributed to a slight tilt of the air table.
Four different-shaped objects were brought into position and orientation equilibrium. We present the results in Fig. 11. The major and the minor axis of the object are clearly marked. In equilibrium, the major axis is closely aligned to the midline of the critical field. The position equilibria lies along the horizontal midline of the rectangle. In Fig. 11, we observe that there is an error at the position of the object, as its centroid is not on the midline. Since this error is consistent to all the different objects that we tested, we conclude that the error is due to a constant field (possibly, a gravitational field due to a slight tilt in the table) that shifts the equilibrium of the critical field. C. Elliptic Field Fig. 10. Pictures of actual device. Top Left: The air table and the object lying on it. Lower left: The rectangular-shaped critical field sink with the honeycomb attached, laid above the air table. Right: The flow chamber has been added on top of the sink to distribute the flow.
effective field boundary, and its centroid moves toward the equilibrium at each hop. We present results for the cases of a critical and an elliptic field. B. Critical Field To produce a critical field, we used an elongated rectangular opening as the sink with its vertical (shortest) sides blocked to the flow. The object was kept well inside the shortest (horizontal boundary) width of the rectangle to avoid any boundary disruptions of the flow. We assume that the effects due to the vertical boundary are negligible, and that the elongated rectangle with blocked ends is a valid approximation of an opening bounded by two infinite parallel lines.
To generate an elliptic field, we set the shape of the sink region to be an ellipse. The elliptic field is produced below this sink region, and can bring the object to a unique position and one of the two possible stable orientation equilibria. To validate the theoretical results of the net force and torque, we examine both the final equilibrium position and orientation of the object and its dynamic behavior. The net force and torque due to an elliptic field are given by (4) and (5). To experimentally validate these equations, we consider the case where for each step, we activate the air bearing for a small interval in time (pulse), which causes a “hop” of the object. At each hop, the object moves a certain distance, due to the drag force of the manipulating flow. The sink region is active during the entire duration of the experiment, and can effectively move the object a noticeable amount during each hop. Each hop is ended by turning off the air bearing, and the object settles at a new location. The force due to the sink which is still active is not capable of overcoming the friction force between the object and the aluminum plate, and thus, cannot further move the object. Assuming a small time interval and a
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Fig. 12. Demonstration of the linearity of the force components F and F of the net force in an elliptic field, versus the x and y position of the object’s centroid, respectively.
Fig. 13. Experimental verification of an object’s torque curve in an elliptic field. The net torque follows a sinusoidal form, as predicted by the theoretical results.
constant acceleration for each hop, we have , and the amount that the centroid of the object is expected to . Thus, the force must be directly move is proportional to the distance that the object traveled at each hop (23) where is the area of the object and is the aspect ration of the sink region. Thus, force can be measured by measuring dis, and a plot of over should be a straight placement line for a quadratic field. In this experiment, we selected the as. Fig. 12 shows the linearity pect ratio of the ellipse to be of the and components of force. We estimated the aspect ratio of this field by dividing the estimated slopes of the fitted , which has an % lines, and it was found to be error. Similarly, the net torque is proportional to the amount of the , and thus, a plot of over the orienangular movement would have a sinusoidal form tation of the object (24) Fig. 13 shows the results for the torque measurements. We observe that the acquired data have a sinusoidal form, as was expected from the theoretical results. A sinusoidal curve was least-square fitted to the data. The period of the sinusoid was well observed to be , as to the . There is a shift in the phase by theoretically expected and a dead-zone of around the equilibrium, which may be attributed to static friction at the beginning and end of each hop. Finally, in Fig. 14, we present the position and orientation equilibria of three different objects. The major axes of the objects are closely aligned with those of the elliptic field. In the upper left and lower left figures, we present the two different orientation equilibria for the same object. The position equilibria
Fig. 14. Equilibrium pose of several objects. Left top and bottom shows the two stable orientation equilibria achieved for an object. The elliptic boundary of the sink and its axes are shown, together with the major and minor axes of the objects in equilibrium.
show some error, which may be due to the presence of a gravitational (constant) field (i.e., a tilt in the air-bearing plate) or to the dead zone that we observe in Figs. 12 and 13. D. Limitations of the Experimental Device The experimental device was modeled using a generous set of assumptions. Since the experimental results are well-predicted by the theory, we conclude that the model is sufficiently accurate, although certain properties of the experimental device limit the accuracy. First, while the large low-pressure plenum (flow chamber) and honeycomb tend to distribute the flow, there is no
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guarantee that the flow is exactly uniform over the sink region. In addition, the honeycomb may discretize the flow, although this effect is likely to be small, since the cell density is high relative to the size of the object. Second, several local effects may disrupt the flow, including the edges of the object itself which has significant thickness, the air-bearing flow which is small but not negligible, and 3-D effects, particularly due to vertical flow near the edges of the sink region. Third, producing a viscous surface via hopping causes inconsistencies in friction. For example, during the time that the air bearing takes to develop, large manipulation forces can overcome static friction, while smaller ones cannot.
VI. CONCLUSION This paper is concerned with the development of methods for constructing planar force fields for distributed manipulation applications. Most distributed manipulation research focuses on either extreme of a range of possible methodologies for producing force fields, either by using explicit generation of forces at each point, or by setting boundary conditions for a dynamic manipulation medium through which forces are generated. In this paper, we focus on the relatively unexplored midrange, where we explicitly set terms in the differential equation describing the medium at each point. In this way, we gain back some of the expressive power of pointwise-explicit fields, while maintaining the ease of implementation of manipulation through natural medium dynamics. We review the set of quadratic fields which previously could only be generated through explicit force actuation at each point. In addition to being readily analyzable, these fields naturally produce predictable equilibria, thus simplifying implementation since feedback and control may not be necessary. The proposed novel approach is to generate quadratic potential force fields using simple devices based on passive planar airflow. In this way, we are taking advantage of the ease of implementation of this natural medium, while alleviating the typical problems of multiple equilibria and intensive computational planning inherent in airflow approaches. We demonstrate analytically how to produce particular quadratic fields from passive flow sink regions, and provide experimental demonstration and quantitative verification of the theoretical results. There are numerous applications of the methodology that has been developed in this paper. These applications mainly concern the simplified construction of micro or mesoscale distributed manipulation devices, which have so far been done by complicated structures. Additional benefits are that the action on the object is realized in a noncontact manner, and the system exhibits smooth dynamics, unlike typical approaches in nonprehensile manipulation. The same approach could potentially apply to other similar phenomena, such as electromagnetics, thermophoretics, etc. to produce useful forms of conservative force fields for sensorless distributed manipulation.
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REFERENCES [1] K. Böhringer, K. Goldberg, V. Bhatt, and B. Donald, “Algorithms for sensorless manipulation using a vibrating surface,” Algorithmica, vol. 26, pp. 389–429, Mar. 2000. [2] P. Frei, M. Wiesendanger, R. Büchi, and L. Ruf, “Simultaneous planar transport of multiple objects on individual trajectories using friction forces,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 49–64. [3] M. Yim, J. Reich, and A. Berlin, “Two approaches to distributed manipulation,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 237–261. [4] J. Luntz and H. Moon, “Distributed manipulation with passive air flow,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2001, pp. 195–201. [5] K. Böhringer, B. Donald, R. Mihailovich, and N. MacDonald, “Sensorless manipulation using massively parallel microfabricated actuator arrays,” in Proc. IEEE Int. Conf. Robot. Autom., 1994, pp. 826–833. [6] J. Suh, R. Darling, K. Böhringer, B. Donald, H. Baltes, and G. Kovacs, “CMOS integrated organic ciliary actuator array for general-purpose micromanipulation tasks,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 191–215. [7] S. Tadokoro, S. Fuji, T. Takamori, and K. Oguro, “Distributed actuation devices using soft gel actuators,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 217–235. [8] J. Luntz, W. Messner, and H. Choset, “Parcel manipulation and dynamics with a distributed actuator array: The virtual vehicle,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 1541–1546. [9] T. Murphey and J. Burdick, “Feedback control for distributed manipulation systems that involve mechanical contacts,” Int. J. Robot. Res., vol. 23, no. 7, pp. 763–781, Jul. 2004. [10] S. Akella, W. Huang, K. Lynch, and M. Mason, “Sensorless parts feeding with a one joint robot,” in Algorithms for Robotic Motion and Manipulation, J.-P. Laumond and M. Overmars, Eds. Wellesley, MA: A. K. Peters, 1997, pp. 229–237. [11] M. Erdmann, “An exploration of nonprehensile two-palm manipulation,” Int. J. Robot. Res., vol. 17, no. 5, 1998. [12] K. Goldberg, “Orienting polygonal parts without sensors,” Algorithmica: Special Issue Comput. Robot., vol. 10, pp. 201–225, 1993. [13] L. Kavraki, “Part orientation with programmable vector fields: Two stable equilibria for most parts,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 2446–2451. [14] K. Böhringer, B. Donald, L. Kavraki, and F. Lamiraux, “Part orientation with one or two stable equilibria using programmable force fields,” IEEE Trans. Robot. Autom., vol. 16, no. 2, pp. 157–170, Apr. 2000. [15] J. Luntz, W. Messner, and H. Choset, “Distributed manipulation using discrete actuator arrays,” Int. J. Robot. Res., vol. 20, no. 7, pp. 553–583, Jul. 2001. [16] H. Moon and J. Luntz, “Prediction of equilibria of lifted logarithmic radial potential fields,” Int. J. Robot. Res., vol. 23, no. 7-8, pp. 747–762, Jul.–Aug. 2004. [17] K. Böhringer, V. Bhatt, and K. Goldberg, “Sensorless manipulation using transverse vibrations of a plate,” in Proc. IEEE Int. Conf. Robot. Autom., 1995, pp. 1989–1996. [18] D. Reznik, E. Moshkoich, and J. Canny, “Building a universal planar manipulator,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA, 2000, pp. 147–171. [19] K. Varsos and J. Luntz, “Analysis, decomposition, and superposition of quadratic potential force fields for distributed manipulation,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2000, pp. 3168–3173. [20] K. Varsos and J. Luntz, “Distributed manipulation along trajectories using open-loop force fields,” in Proc. IEEE Int. Conf. Robot. Autom., 2002, pp. 1203–1209. [21] H. Moon and J. Luntz, “Synthesis bounds for distributed manipulation using logarithmic-radial potential fields,” in Proc. IEEE Int. Conf. Intell. Robot. Syst., 2003, pp. 2908–2913. [22] H. Moon and J. Luntz, “Toward sensorless manipulation using airflow,” in Proc. IEEE Int. Conf. Robot. Autom., 2004, pp. 1574–1579. [23] K. Böhringer, B. Donald, and N. C. MacDonald, “Programmable vector fields for distributed manipulation, with applications to MEMS actuator arrays and vibratory parts feeders,” Int. J. Robot. Res., vol. 18, no. 2, pp. 168–200, Feb. 1999. [24] F. Lamiraux and L. Kavraki, “Positioning of symmetric and nonsymmetric parts using radial and constant fields: Computation of all equilibrium configurations,” Int. J. Robot. Res., vol. 20, no. 8, pp. 635–659, Aug. 2001. [25] H. Moon and J. Luntz, “Distributed manipulation by superposition of logarithmic-radial potential field,” in Proc. IEEE Int. Conf. Robot. Autom., 2002, pp. 1197–1202. [26] W. Thomson, Papers on Electricity and Magnetism. London, U.K.: Macmillan, 1884, pp. 178–179.
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Konstantinos Varsos (S’00) received the diploma from the Department of Electrical Engineering and Computer Technology, University of Patra, Patra, Greece, in 2000, and the Masters degrees in 2001 from the Department of Industrial and Operations Engineering and the Department of Mechanical Engineering, University of Michigan, Ann Arbor, where he is currently working toward the Ph.D. degree in mechanical engineering. His research interests include distributed manipulation, medical robotics, and industrial automation.
Hyungpil Moon (S’01–M’06) received the B.S. and M.S. degrees from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1996 and 1998, respectively, and the Ph.D. degree in mechanical engineering from the University of Michigan, Ann Arbor, in 2005. He is now a Visiting Research Scientist with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA. His research interests include distributed manipulation, coordination and control of distributed systems, SLAM, and nonlinear control theory with applications in mechanical systems.
Jonathan Luntz (M’00) obtained the B.S. degree in mechanical engineering from the State University of New York at Buffalo in 1992, and the M.E. and Ph.D. degrees in mechanical engineering from Carnegie Mellon University, Pittsburgh, PA, in 1994 and 1999, respectively. He joined the University of Michigan, Ann Arbor, as an Assistant Professor of Mechanical Engineering in 2000, and in 2004 joined the research faculty as an Assistant Research Scientist. He currently works in the areas of distributed manipulation, cooperative mobility, smart material actuation (particularly with shape memory alloy), and in reconfigurable logic control of combined hardware and software resources in manufacturing systems.
