Research

Research Interests

FASN's research interests relate to the intelligent design and implementation of multi-dimensional mechanical behavior (kinematics and admittance/impedance) for application in physically interacting systems. This research is fundamentally important to achieve improved dexterity in contact manipulation task (e.g. sanding, polishing, assembly, etc). Depending on the nature of the task, the appropriate mechanical behavior may be constant or time-varying. Implementation of the appropriate linear and nonlinear multi-dimensional mechanical behavior is achieved through 1) passive mechanism design, 2) programmable (selectable or semi-active) passive mechanical component design, and/or 3) active controller design. The figure below illustrates the research topics and activities related to implementing the appropriate mechanical behavior to successfully perform contact interaction tasks.

Research Topics

• Admittance selection for robust assembly
• Contact interaction analysis
• Admittance characterization
• Mechanism design for compliance realization
• Variable stiffness actuator design
• Path planning for redundant manipulator motion in a combined position/compliance space

Our strategy for executing contact interaction tasks is to use a gross motion plan (a sequence of commanded nominal end-effector positions) to execute the task and to use a fine motion plan in which position discrepancy between the nominal position and the contact-surface is corrected by complying with the contact constraint. Proper admittance design reduces the contact forces, preventing the robot or workpiece from becoming damaged, and ensures that the gross motion plan results in reliable completion of the task. In general, proper admittance design is not trivial; it involves contact interaction analysis (of the motion resulting from contact forces) and a mathematical admittance characterization.

The selected mechanical behavior may be implemented actively using sensor feedback and an admittance (or impedance) control law to control the actuators to emulate the desired mechanical behavior. In active admittance control, the forces at the end-effector are measured and are used to determine joint motions that approximate the desired mechanical behavior. In active impedance control, the motion (displacement, velocity, acceleration) of the end-effector is measured and used to determine joint torques that approximate the desired mechanical behavior. One disadvantage of active implementation is that the control law may yield unstable interaction. Unstable interaction is usually avoided by carefully selecting the desired mechanical behavior such that the robot behaves as a passive system. Another disadvantage of active implementation is that the control loop may not be fast enough to yield the correct dynamic response, especially during collisions.

Passive implementation (with physical components) does not have these limitations. However, it is much more difficult to realize an arbitrary desired admittance with a passive mechanism design. A pasive mechanism may be integrated into the end-effector design and attached to a traditional robot arm. The fine motion plan is realized in the end-effector and the gross motion plan is realized by the robot arm. The robot arm itself may be redundant (having more joints than required to complete the task), or non-redundant. The gross motion plan must be converted to a joint motion plan; the conversion is non-unique for redundant robots. The disadvantage of the compliant end-effector approach is that the end-effector is only suitable for tasks needing that particular admittance. Some tasks may require a time-varying admittance and cannot be executed using only a compliant end-effector.

Programmable and time-varying passive admittance realization can be implemented with variable impedance actuated robots. In this case, the robot arm itself is the redundant mechanism realizing the desired end-effector admittance. Unlike a passive end-effector mechanism, the robot “mechanism” geometry depends on the end-effector position. Kinematic redundancy is used to increase the control on the mechanism geometry to realize a larger space of admittances in the dexterous workspace of the robot. The redundant robot arm is responsible for both the gross motion and fine motion (admittance). The joint admittance plan must be determined simultaneously with the joint motion plan while addressing redundancy resolution.

The authors address the design of manipulator admittance for reliable force-guided assembly. Their goal is to design the admittance of the manipulator so that, at all possible bounded part misalignments, the contact forces always lead to error-reducing motions. If this objective can be accomplished for a given pair of mating parts, the parts are called force-assemblable. As a testbed application of manipulator admittance design for force-guided assembly, the authors investigate the insertion of a workpiece into a fixture consisting of multiple rigid fixture to be one for which there exists an admittance matrix that ensures the unique positioning of a workpiece despite initial positional error. It is shown that, in the absence of friction, all deterministic fixtures are linearly force-assemblable. How to design an admittance matrix that guarantees that the workpiece will be guided into the deterministic fixture by the fixel contact forces alone is shown.

Previously, force-assembly has been defined as an assembly process for which a single admittance control law (i.e., a single nominal velocity in conjunction with a single mapping of forces to motions) can guarantee the proper assembly of a given pair of mating parts. As a testbed application of force-assembly, the insertion of a workpiece into a fixture consisting of multiple rigid fixture elements (fixels) is addressed. Previous work in this area has shown that, when workpiece/fixture contact is frictionless and positional error is infinitesimal, there always exists an admittance control law that will ensure the proper insertion of a workpiece into a deterministic fixture. When workpiece/fixture contact is frictionless, the workpiece/fixture contact force contains the relative positional information required to identify error-reducing motions. Friction between the workpiece and fixture, however, provides a disturbance to the geometrical information contained in the contact force. This paper addresses: 1) the identification of the conditions that must be satisfied for force-assembly with friction, and 2) the formulation and results of an optimization of the admittance control law to obtain the maximum value of friction that will satisfy the force-assembly conditions for a given workpiece/fixture combination. Results indicate that force-assembly fails when the contact forces are no longer characteristic. Forces are characteristic if the possible contact forces that may occur for one type of misalignment are unique to that type of misalignment.

Force-assembly has been defined as an assembly process for which the use of a single, properly designed, admittance control law will guarantee the proper assembly of a given pair of mating parts. In previous work in workpart-into-fixture insertion, the conditions on a manipulators accommodation control law that ensure proper insertion despite infinitesimal positional error and finite (but bounded) friction have been identified. Through the use of an optimization routine, a control law that satisfies these force-assembly conditions at or below a friction maximum value can be obtained. This single control law, however, is not unique-there exists many other control laws that will satisfy the conditions of force-assembly at the same value of friction. This paper addresses the identification and construction of a linear space of accommodation control law parameters that ensure force-assembly with friction. First, linear sufficient conditions that ensure force-assembly with friction are identified. These linear sufficient conditions are then modified to separate the N^2+N dimensional space of accommodation control law parameters into N+1 different N-dimensional subspaces. A means of efficiently generating basis nominal velocity vectors and basis accommodation matrices is presented. A nominal velocity selected using any positive linear combination of the nominal velocity basis vectors and an accommodation matrix selected using any positive linear combination of the accommodation basis matrices will guarantee force-assembly (for any value of friction less than that used in generating the basis matrices). A planar example of the construction of each accommodation control law subspace is presented and illustrated in the geometry of the fixturing task.

Admittance control approaches show significant promise in providing reliable force-guided assembly. An important issue in the development of these approaches is the specification of an appropriate admittance control law. This paper identifies procedures for selecting the appropriate admittance to achieve reliable planar force-guided assembly for single-point contact cases. A set of conditions that are imposed on the admittance matrix is presented. These conditions ensure that the motion that results from contact reduces part misalignment. We show that for bounded misalignment, if the conditions are satisfied for a finite number of contact configurations, the system ensures that force guidance is achieved for all intermediate configurations.

An important issue in the development of force guidance assembly strategies is the specification of an appropriate admittance control law. This paper identifies conditions to be satisfied when selecting the appropriate admittance to achieve force-guided assembly of polygonal parts for multipoint contact with friction. These conditions restrict the admittance behavior for each of the various one-point and two-point contact cases and ensure that the motion that results from contact reduces part misalignment for each case. We show that, for bounded friction and part misalignments, if the identified conditions are satisfied for a finite number of contact configurations and friction coefficients, the conditions ensure that force guidance is achieved for all configurations and values of friction within the specified bounds.

By judiciously selecting the admittance of a manipulator, the forces of contact that occur during assembly can be used to guide the parts to proper positioning. This paper identifies conditions for selecting the appropriate spatial admittance to achieve reliable force-guided assembly of polyhedral parts for cases in which a single feature (vertex, edge, or face) of one part contacts a single feature of the other, i.e., all single principal contact cases. These conditions ensure that the motion that results from frictionless contact always instantaneously reduces part misalignment. We show that, for bounded misalignments, if an admittance satisfies the misalignment-reducing conditions at a finite number of contact configurations, then the admittance will also satisfy the conditions at all intermediate configurations.

The admittance of a manipulator can be used to improve robotic assembly. If properly selected, the admittancewill regulate a contact force and use it to guide the parts to proper positioning. In previous work, procedures for selecting the appropriate admittance for single principal contact (PC) cases were identified. This paper extends this research for some of the two PC cases—those for which each contact occurs at a single point. The conditions obtained ensure that the motion that results fromfrictionless contact always instantaneously reduces part misalignment. We show that, for bounded misalignments, if an admittance satisfies the misalignmentreducing conditions at a finite number of contact configurations, then the admittancewill also satisfy the conditions at all intermediate configurations.

Robots are not typically used for assembly tasks in which positioning requirements exceed robot capabilities. To address this limitation, a significant amount of work has been directed toward identifying desirable mechanical behavior of a robot for force-guided assembly. Most of this work has been directed toward the 'standard' peg-in-hole assembly problem. Little has been done to identify the specific behavior necessary for reliable assembly for different types of polygonal parts, and little has been done relating assembly characteristics to classes of part geometries. This paper presents the best passive admittance and associated maximum coefficient of friction for planar force-assembly of a variety of different polygonal parts, specifically pegs with rectangular, trapezoidal, triangular, and pentagonal cross sections. The results show that force-guided assembly can be reliably achieved at higher values of friction when parts are shorter and wider. For all geometries considered, force-guided assembly is ensured for any value of friction less than 0.8 when the optimal admittance is used; and, for some geometries, for any value of friction less than 15.

Current robots lack the precise relative positioning necessary to complete automatic assembly tasks. Several solutions have been proposed. Some approaches use complex vision and force sensing systems to generate corrective motion if misalignment is present in the assembly task. Other solutions rely on generating elastic behavior, known as compliance, between the end eector and the held movable part. This compliant mechanism helps guide the movable part of the assembly into its proper position. The project focuses on designing a process by which passive compliant systems can achieve successful assembly for a range of misalignment and generate error-reducing motion that is considered of high quality. This is accomplished by using a velocity metric as the goal of a constrained optimization. The metric uses the average discrepancy of all the particle motion from an established "best motion". This motion minimizes the discrepancy in the velocity of all particles motion from their ideal motion towards their proper position. This procedure identifies the best worst case scenario for a representative set of configurations. The results obtained for optimization over polygonal geometries of 3, 4, and 5 vertices, demonstrate the effectiveness of the procedure in designing passive compliant behavior resulting in high quality error-reducing motion. Results also show that high quality motion is not only achieved for a set of finite configurations but also for all intermediate ones.

This two-part paper presents a method for both improving the positioning capability and increasing the effective stiffness (bracing) of a robotic manipulator through multidirectional compliance and constraint. Improved relative positioning and increased stiffness are obtained through the use of: (1) an end-effector mounted jig capable of establishing a workpart-based reference frame through multipoint contact with the workpart, and (2) a manipulator compliance designed to provide a specified form of directional coupling in its mapping of forces to deflections. The directional coupling in the compliance is shown to be important in establishing multipoint contact (during insertion) and in maintaining contact (while edge tracking) despite finite positional errors. Improved manipulator positioning is demonstrated in the context of workpart edge deburring.

In this paper, we study the quasi-static motion of an elastically suspended, unilaterally constrained rigid body. The motion of the rigid body is determined, in part, by the position controlled motion of its support base and by the behavior of the elastic suspension that couples the part to the support. The motion is also determined, in part, by contact with a frictional surface that both couples the rigid body to the unilateral constraint and generates a friction force. The unknown friction force, however, is determined in part by the unknown direction of the rigid-body motion. We derive an analytically solvable set of equations that simultaneously determines both the friction force and the resulting rigid-body motion. We also address the issues of whether a solution to these equations exists and whether the obtained solution is unique. We show that, for any passive compliant system in which the nominal motion imposes contact, a solution to the set of motion equations always exists. We also show that, for any passive system with an upper bounded friction coefficient, the solution is unique. Two sufficient conditions that guarantee the uniqueness of the solution are presented.

The reliable automation of assembly is important for improved product manufacture. One approach to achieving reliable automated assembly is known as force assembly. A robot's motion response to a contact force is determined by its mechanical admittance. An admittance is the relationship between the applied force and the resulting motion. The force assembly approach uses the manipulator admittance to compensate for part misalignment. The design strategy is that by imposing a set of sufficient conditions on the admittance at some sample configurations (a configuration describes the relative position between two objects), misalignment reduction is achieved for all the possible intermediate configurations. The number of configurations for a general assembly task is infinite; whereas, the number of contact states, which characterize the contact status between two object, is finite. Therefore, the approach to identifying the sample configurations contains the following two issues: 1) all contact states for an assembly task are first obtained; 2) a finite number of "extrenal" configurations (the configurations that have extreme values in misalignment) are extracted within each contact state. These finite number of extrenals are selected as the set of representative configurations. The approach to identifying contact states and the associated extrenal configurations is from the less constrained contact states (the contact states that constrain less degrees of freedom for the parts) to the more constrained contact states (the contact states that constrain more degrees of freedom for the parts). The approach has the following three components: Obtains the unique contact state description by considering topological and geometrical conditions. This component eliminates the redundant description for the same contact features. Identifies whether a contact state description is valid for an assembly task. The procedure first uses simple topological and geometrical conditions to evaluate the existence of a contact state description. Then an optimization procedure is applied to evaluate the existence of the contact state by finding whether there is valid configuration within the contact state. Determine the extrenal configurations for the contact states by involving constrained optimizations. In the approach, a strategy based on the contact state generation procedure, external configurations and the topological relationships among contact states is used to improve the robustness of the algorithm.

In this paper, the motion of an elastically suspended rigid body unilaterally constrained by contact at multiple locations is studied. In this problem, each individual contact may or may not constrain the body’s motion. Whether a contact provides an active constraint or not is determined by: 1) the commanded motion of the body’s compliant support, 2) the direction of the constraining surface normal, 3) the number and direction of other potentially constraining surfaces, and 4) the elastic properties of the support. Here, the investigated problem is restricted to quasistatic motion and frictionless contact. We show that the set of possible constraints that are active for a given set of contacts, elastic behavior, and commanded motion is unique. We also develop a procedure to determine the set of active constraints. This procedure is then used as part of a larger procedure to determine the compliant motion of the rigid body.

Two motions of motion quality have been developed for planar motion. They are the "best motion measure" and the "velocity metric". The "best motion measure" identifies the best motion for a given displacement. The "velocity metric" quantifies the discrepancy between two planar motions for the same rigid body. The best motion measure compares the motion of each particle on the body to an "ideal", but usually unobtainable, motion. This ideal motion moves each particle from its current position to its desired position on a straight-line path. Although the ideal motion is not a valid rigid body motion, this does not preclude its use as a reference standard in evaluating valid rigid body motions. It is shown that the best motion measure can be reduced to the product of two components: the average distance from a point in the plane to the body, and a term based on configuration parameters. The optimal instantaneous planar motion for general rigid bodies in translation and rotation is characterized. This optimal motion is defined as the geodesic motion of a frame located at the geometric center of the body. In other words, the geometric center of the body will move in a straight-line path to its desired position, while the body rotates about the axis perpendicular to the plane. The optimal angular velocity is a function of the discrepancy between the current configuration and the desired configuration.

We identify the space of spatial compliant behavior that can be achieved through the use of simple springs connected in parallel to a single rigid body. Here, the expression "simple spring" refers to the set of compliant relations associated with passive translational springs and rotational springs. The restriction on the stiffness matrices is derived using the screw theory by investigating the compliant behavior of individual simple springs. We show that the restriction results from the fact that simple springs can only provide either a pure force or a pure torque to the suspended body. We show that the 20-dimensional subspace of "realizable" spatial stiffness matrices achieved with parallel simple springs is defined by a linear necessary and sufficient condition on the positive semidefinite stiffness matrix. A procedure to synthesize an arbitrary full-rank stiffness matrix within this realizable subspace is provided. This procedure requires no more than seven simple springs.

We address the spatial elastic behavior that can be achieved through the use of a serial chain of revolute and prismatic elastic joints. We show that, regardless of the number of joints or the configuration of the links, there exists a subspace within the 21-dimensional compliance matrix space that cannot be reached by a simple serial elastic mechanism. This restriction is shown to be dual to the restriction on the stiffness matrices associated with simple parallel mechanisms. Although analogous to each other, the two restrictions correspond to different elastic behaviors. A procedure to synthesize any realizable compliance matrix with a simple serial mechanism is provided. The dualities and differences between the parallel and serial cases are discussed.

A manipulator system is modeled as a kinematically unconstrained rigid body suspended by elastic devices. The structure of spatial stiffness is investigated by evaluating the stiffness matrix "primitives"-the rank-1 matrices that compose a spatial stiffness matrix. Although the decomposition of a rank-2 or higher stiffness matrix into the sum of rank-1 matrices is not unique, one property of the set of matrices is conserved. This property, defined as the stiffness-coupling index, identifies how the translational and rotational components of the stiffness are related. Here, we investigate the stiffness-coupling index of the rank-1 matrices that compose a spatial stiffness matrix. We develop a matrix decomposition that yields a set of rank-1 stiffness matrices that identifies the bounds on the stiffness-coupling index for any decomposition. This decomposition, referred to as the eigenscrew decomposition, is shown to be invariant in coordinate transformation. With this decomposition, we provide some physical insight into the behavior associated with a general spatial stiffness matrix.

This article presents a new method for the synthesis of an arbitrary spatial elastic behavior with an elastic mechanism. The mechanisms considered are parallel and serial mechanisms with concurrent axes. We show that any full‐rank spatial stiffness matrix can be realized using a parallel mechanism with all spring axes intersecting at a unique point. It is shown that this intersection point must be the center of stiffness. We also show that any full‐rank spatial compliance matrix can be realized using a serial mechanism with all joint axes intersecting at a unique point. This point is shown to be the center of compliance. Synthesis procedures for mechanisms with these properties are provided. The realizations are shown to be minimal in the sense that both the number of screw components and the total number of components are minimum.

Previously, we have shown that, to realize an arbitrary spatial stiffness matrix, spring components that couple the translational and rotational behavior along/about an axis are required. We showed that, three such coupled components and three uncoupled components are sufficient to realize any full-rank spatial stiffness matrix and that, for some spatial stiffness matrices, three coupled components are necessary. In this paper, we show how to identify the minimum number of components that provide the translationalrotational coupling required to realize an arbitrarily specified spatial stiffness matrix. We establish a classification of spatial stiffness matrices based on this number which we refer to as the ‘‘degree of translational–rotational coupling’’ (DTRC). We show that the DTRC of a stiffness matrix is uniquely determined by the spatial stiffness mapping and is obtained by evaluating the eigenstiffnesses of the spatial stiffness matrix. The topological properties of each class are identified. In addition, the relationships between the DTRC and other properties identified in previous investigations of spatial stiffness behavior are discussed.

Spatial elastic behavior is characterized by a 6x6 positive definite matrix, the spatial stiffness matrix, or its inverse, the spatial compliance matrix. Previously, the structure of a spatial stiffness matrix and its realization using a parallel elastic system have been addressed. This paper extends those results to the analysis and realization of a spatial compliance matrix using a serial mechanism and identifies the duality in spatial stiffness and compliance associated with parallel and serial elastic mechanisms. We show that, a spatial compliance matrix can be decomposed into a set of rank-1 compliance matrices, each of which can be realized with an elastic joint in a serial mechanism. To realize a general spatial compliance, the serial mechanism must contain joints that couple the translational and rotational motion along/about an axis. The structure of a spatial compliance matrix can be uniquely interpreted by a 6-joint serial elastic mechanism whose geometry is obtained from the eigenscrew decomposition of the compliance matrix. The results obtained from the analysis of spatial compliant behavior and its realization in a serial mechanism are compared with those obtained for spatial stiffness behavior and its realization in a parallel mechanism.

A significant amount of research has been directed toward developing a more intuitive appreciation of spatial elastic behavior. Results of these analyses have often been described in terms of the elastic behavior (stiffness or compliance) centers. This paper investigates the properties of centers of stiffness and compliance and provides a fresh view of elastic center locations, specifically, the locus of centers associated with a given mechanism’s topology and geometry. We show that the location of the center of stiffness (compliance) for a set of elastic components connected in parallel (in serial) can be described in terms similar to the location of the center of mass for a set of mass particles. This provides a physical interpretation of the centers associated with a compliant behavior, and a useful guide in the design of mechanisms that realize desirable compliant behaviors.

This paper presents the design of a passive robotic wrist that is capable of establishing and maintaining an accurate position relative to a workpart edge through compliance and constraint (force guidance). In previous work, we have shown that, through proper selection of a manipulator's impedance, a manipulator's end-effector can be guided to its desired relative position despite errors in its commanded position. The selected proper impedance is attained here through the design of a passive micromanipulator that is mounted on the end-effector of a conventional manipulator. The micromanipulator consists of three linkages connected by revolute joints and torsional springs. The outermost linkage contacts the workpart at multiple locations providing multidirectional unilateral kinematic constraint. This kinematic constraint in conjunction with the compliance provided by the torsional springs causes the linkage to be re-positioned so that any existing misalignment (that inevitably occurs) is eliminated and a unique planar position/orientation with respect to the workpart edge is attained. Here, we present the procedure used in the parametric design of this mechanism. The desired compliant properties identified in task space (using Cartesian variables (x, y, and θ) for force and motion) are extended here to joint space (using joint variables (θ1, θ2), and θ3) for torque and motion). The appropriate micromanipulator link lengths, initial linkage angles, and the appropriate torsional spring constants are selected using an optimization procedure. Computer simulation of the constrained manipulator/workpart interaction demonstrates that the desired force guidance behavior is attained.

In this paper, the synthesis of an arbitrary spatial stiffness matrix is addressed. We have previously shown that an arbitrary stiffness matrix cannot be achieved with conventional translational springs and rotational springs (simple springs) connected in parallel regardless of the number of springs used or the geometry of their connection. To achieve an arbitrary spatial stiffness matrix with springs connected in parallel, elastic devices that couple translational and rotational components are required. Devices having these characteristics are defined here as screw springs. The designs of two such devices are illustrated. We show that there exist some stiffness matrices that require 3 screw springs for their realization and that no more than 3 screw springs are required for the realization of full-rank spatial stiffness matrices. In addition, we present two procedures for the synthesis of an arbitrary spatial stiffness matrix. With one procedure, any rank-m positive semidefinite matrix is realized with m springs of which all may be screw springs. With the other procedure, any positive definite matrix is realized with 6 springs of which no more than 3 are screw springs.

In this paper, the synthesis of diagonally dominant stiffness matrices is addressed. It is known that the space of dominant matrices is a polyhedral cone in the n(n+1)/2 dimensional space and every matrix in the cone is a non-negative combination of the “edges” of the polyhedral cone, the basis matrices. An elastic parallel mechanism is designed based on the spring realization of the basis matrices so that every dominant matrix can be realized with a mechanism of constant geometry having variable spring constants.

In this paper, we address the synthesis and realization of a subset of spatial elastic behaviors, those with compliant axes, in compact parallel mechanisms. Using the procedures developed here, a geometric description of the layout of a compact mechanism of springs connected in parallel is obtained. In each of these mechanisms, each spring axis is restricted to intersect a single point in space. The variation in the orientation of the intersecting spring axes is also restricted. The degree of restriction in axis orientation is determined by the number of “compliant axes” associated with the specified elastic behavior. Also, as part of this work, a physical appreciation of the different classes of stiffnesses that have compliant axes are identified and interpreted in different types of concurrent parallel mechanisms.

This paper presents a new method for the realization of a planar compliant behavior with an elastic mechanism. The mechanisms considered are parallel with symmetric geometry. We show that any planar stiffness matrix can be realized using a parallel mechanism with four line springs connected symmetrically. Among the four springs, two are identical parallel springs equidistant from the stiffness center, and the other two identical springs intersect at the stiffness center. A synthesis procedure based on geometry is presented and mechanism compactness is discussed.

This paper presents methods for the realization of 2×2 translational compliance matrices using serial mechanisms having only revolute joints, each with selectable compliance. The link lengths of the mechanism and the location of the compliant frame relative to the mechanism base are arbitrary but specified. The realizability of a given compliant behavior is investigated, and necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. These realization conditions are interpreted in terms of geometric relationships among the joints. We show that, for an appropriately sized 3R serial mechanism, any single 2×2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. Requirements on mechanism geometry to realize every particle planar elastic behavior at a given location just by changing the mechanism configuration are also identified.

This paper presents methods for the realization of 2×2 translational compliance matrices using serial mechanisms having three joints, each either revolute or prismatic and each with selectable compliance. The geometry of the mechanism and the location of the compliance frame relative to the mechanism base are each arbitrary but specified. Necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. We show that, for an appropriately constructed serial mechanism having at least one revolute joint, any single 2×2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. For each type of three-joint combination, requirements on the redundant mechanism geometry are identified for the realization of every point planar elastic behavior at a given location, just by changing the mechanism configuration and the joint compliances.

This paper addresses the passive realization of any selected planar elastic behavior with a parallel or a serial manipulator. Sets of necessary and sufficient conditions for a mechanism to passively realize an elastic behavior are presented. These conditions completely decouple the requirements on component elastic properties from the requirements on mechanism kinematics. The restrictions on the set of elastic behaviors that can be realized with a mechanism are described in terms of acceptable locations of realizable elastic behavior centers. Parallel–serial mechanism pairs that realize identical elastic behaviors (dual elastic mechanisms) are described. New construction-based synthesis procedures for planar elastic behaviors are developed. Using these procedures, one can select the geometry of each elastic component from a restricted space of kinematically allowable candidates. With each selection, the space is further restricted until the desired elastic behavior is achieved.

This paper addresses the passive realization of any selected planar elastic behavior with redundant elastic manipulators. The class of manipulators considered are either serial mechanisms having four compliant joints or parallel mechanisms having four springs. Sets of necessary and sufficient conditions for mechanisms in this class to passively realize an elastic behavior are presented. The conditions are interpreted in terms of mechanism geometry. Similar conditions for nonredundant cases are highly restrictive. Redundancy yields a significantly larger space of realizable elastic behaviors. Construction-based synthesis procedures for planar elastic behaviors are also developed. In each, the selection of the mechanism geometry and the selection of joint/spring stiffnesses are completely decoupled. The procedures require that the geometry of each elastic component be selected from a restricted space of acceptable candidates.

In this paper, a geometric approach to the passive realization of any planar compliance with a redundant compliant mechanism is presented. The mechanisms considered are either simple serial mechanisms consisting of five elastic joints or simple parallel mechanisms consisting of five springs. For each type of mechanism, realization conditions to achieve a given compliance are derived. The physical significance of each condition is identified and graphically interpreted. Geometry based synthesis procedures to achieve any given compliance are developed for both types of mechanisms. Since each realization condition imposes restrictions solely on the mechanism geometry, the procedures allow one to choose the geometric properties of each component (from a set of admissible options) independently from the selection of the elastic properties of each component.

In this paper, the synthesis of any planar compliance with a 6-component compliant mechanism is addressed. The mechanisms studied are either serial mechanisms with six elastic joints or parallel mechanisms with six springs. For each type of mechanism, conditions on the mechanism configurations that must be satisfied to realize a given compliance are developed. The geometric significance of each condition is identified and graphically represented. Geometric construction based synthesis procedures for both types of mechanism are developed. These procedures allow one to select each elastic component from a restricted space based on its geometry.

This paper addresses the realization of spatial elastic behavior with a parallel or a serial manipulator. Necessary and sufficient conditions for a manipulator (either parallel or serial) to realize a specific elastic behavior are presented and interpreted in terms of the manipulator geometry. These conditions completely decouple the requirements on component elastic properties from the requirements on mechanism kinematics. New construction-based synthesis procedures for spatial elastic behaviors are developed. With these synthesis procedures, one can select each elastic component of a parallel (or serial) mechanism based on the geometry of a restricted space of allowable candidates. With each elastic component selected, the space of allowable candidates is further restricted. For each stage of the selection process, the geometry of the remaining allowable space is described.

In this article, the synthesis of any specified planar compliance with a serial elastic mechanism having previously determined link lengths is addressed. For a general n-joint serial mechanism, easily assessed necessary conditions on joint locations for the realization of a given compliance are identified. Geometric construction-based synthesis procedures for five-joint and six-joint serial mechanisms having kinematically redundant fixed link lengths are developed. By using these procedures, a given serial manipulator can achieve a large set of different compliant behaviors by using variable stiffness actuation and by adjusting the mechanism configuration.

In this paper, the realization of any specified planar compliance with two 3R serial elastic mechanisms is addressed. Using the concept of dual elastic mechanisms, it is shown that the realization of a compliant behavior with two serial mechanisms connected in parallel is equivalent to its realization with a 6-spring fully parallel mechanism. Since the spring axes of a 6-spring parallel mechanism indicate the geometry of a dual 3R serial mechanism, a new synthesis procedure for the realization of a stiffness matrix with a 6-spring parallel mechanism is first developed. Then, this result is extended to a geometric construction-based synthesis procedure for two 3-joint serial mechanisms.

In this paper, a geometric construction based means of realizing any specified planar compliance for an object held by a compliant hand is developed. It is shown that the elastic behavior of an object held by a multi-serial parallel mechanism (a multi-finger compliant hand) is more simply and equivalently modeled by a fully-parallel dual elastic mechanism. Synthesis procedures are developed for the realization of an arbitrary compliance with compliant hands using geometric constraints on the fully-parallel elastic dual. Kinematic topologies addressed are those associated with hands having 2 or 3 fingers for which each finger has 2 joints.

Redundant manipulators have an infinitely large set of joint paths that yield a desired end-effector path in the task space. A unique joint path can be obtained by minimizing a global cost function. Prior optimal control methods minimize a global cost function to find a local minimum within a homotopy class. Many possible locally optimal joint paths are in different homotopy classes. This paper presents an algorithm that effectively searches the solution space and finds many locally optimal paths in all relevant homotopy classes. The path with the lowest cost is very likely the globally optimal path. The algorithm is demonstrated in a case study for which the globally optimal path would be impossible to find using traditional methods.

Passive compliance control is an approach for controlling the contact forces between a robotic manipulator and a stiff environment. This paper considers passive compliance control using redundant serial manipulators with real-time adjustable joint stiffness. Such manipulators can control the elastic behavior of the end-effector by adjusting the manipulator configuration and by adjusting the intrinsic joint stiffness. The end-effector's time-varying elastic behavior is a beneficial quality for constrained manipulation tasks such as opening doors, turning cranks, and assembling parts. The challenge in passive compliance control is finding suitable joint commands for producing the desired time-varying end-effector position and compliance (task manipulation plan). This problem is addressed by extending the redundant inverse kinematics (RIK) problem to include compliance. This paper presents an effective method for simultaneously attaining the desired end-effector position and end-effector elastic behavior by tracking a desired variation in both the position and the compliance. The set of suitable joint commands is not unique; the method resolves the redundancy by minimizing the actuator velocity norm. The method also compensates for joint deflection due to known external loads, e.g., gravity.

Variable stiffness actuators (VSAs) provide realtime adjustable joint compliance. Redundant serial manipulators with VSAs are capable of passive compliance control, in which the elastic behavior of the end-effector is controlled for robust interaction with a stiff environment. The challenge of passive compliance control is finding an appropriate joint manipulation path (sequence of joint positions and compliances) that yields a desired task manipulation path (sequence of end-effector positions and compliances). This paper addresses the problem of finding a joint path, if one exists, that follows the task manipulation path in the best way. More specifically, it finds the globally optimal joint path based on an integral cost criterion. The space of admissible joint paths can be very complex, with multiple locally optimal joint paths in multiple homotopy classes. The number of homotopy classes depends on both the manipulator structure and the task manipulation path. This paper provides means of finding the best joint path for manipulation with one degree of redundancy in the combined position and compliance space. Concepts are illustrated in a case study for particle-planar passive compliance control with a 3R-VSA manipulator.

The high stiffness of conventional robots is beneficial in attaining highly accurate positioning in free space. High stiffness, however, limits a robot's ability to perform constrained manipulation. Because of the high stiffness, geometric conflict between the robot and task constraints during constrained manipulation can lead to excessive forces and task failure. Variable stiffness actuators can be used to adjust the stiffness of robot joints to allow high stiffness in unconstrained directions and low stiffness in constrained directions. Two important design criteria for variable stiffness actuation are a large range of stiffness and a compact size. A new design, the Arched Flexure VSA, uses a cantilevered beam flexure of variable cross-section and a controllable load location. It allows the joint to have continuously variable stiffness within a finite stiffness range, have zero stiffness for a small range of joint motion, and allow rapid adjustment of stiffness. Using finite element analysis, flexure geometry was optimized to achieve high stiffness in a compact size. A proof-of-concept prototype demonstrated continuously variable stiffness with a ratio of high stiffness to low stiffness of 55.

Antagonistically actuated variable stiffness actuators (VSAs) take inspiration from biological muscle structures to control both the stiffness and positioning of a joint. This paper presents the design of an elastic mechanism that utilizes a cable running through a set of three pulleys to displace a linear spring, yielding quadratic spring behavior in each actuator. A joint antagonistically actuated by two such mechanisms yields a linear relationship between force and deflection from a selectable equilibrium position. A quasi-static model is used to optimize the mechanism. Testing of the fabricated prototype yielded a good match to the desired elastic behavior.

This thesis presents the design, implementation, and testing of a passive robotic wrist that is capable of establishing and maintaining an accurate position relative to a workpart edge through force guidance. The surfaces adjacent to the workpart edge are used to define the edge's true geometry. The wrist consists of three links in series connected by revolute joints and three linear compression springs offset from the revolute joints to create a torsion-like compliance. The third link contacts the workpart surfaces a three locations providing multiple unilateral constraints. These constraints along with the compliance created by the springs cause the third link to be repositioned so that any positional error will be eliminated and unique position/orientation will be obtained. The wrist will mechanically compensate for bounded robot trajectory error as well as bounded positional variation in the workpart edge location. The test results have shown that the desired force guidance behavior is achieved despite large positional error.