colgate 1993 robustimpedanceshapingtelemanipulation

374 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION. VOL. 9, NO. 4, AUGUST 1993

Robust Impedance Shaping Telemanipulation J. Edward Colgate

Abstract-When a human operator performs a task via a bilateral manipulator, the “feel” of the task is embodied in the mechanical impedance of the manipulator. ’Ikaditiody, a bilateral manipulator is designed for transparency; i.e., so that the impedance reflected through the manipulator closely approx- imates that of the task. “Impedance shaping bilateral control,” introduced here, differs in that it treats the bilateral manipulator as a means of constructively altering the impedance of a task. This concept is partidarly valuable if the characteristic dimensions (e.g., force, length, time) of the task impedance are very different from those of the human limb. It is shown that a general form of impedance shaping control consists of a conventional power- scaling bilateral controller augmented with a real-time interactive task simulation (i.e., a virtual environment). An approach to impedance shaping based on kinematic similarity between tasks of different scale is introduced and illustrated with an example. It is shown that an important consideration in impedance shaping controller design is robustness; i.e., guaranteeing the stability of the operatodmanipulatodtask system. A general condition for the robustness of a bilateral manipulator is derived. This condition is based on the structured singular value (p). An example of robust impedance shaping bilateral control is presented and discussed.

I. INTRODUCTION

HUMAN performing a manual task will typically asso- A ciate a “feel” with the task. For instance, the feel of carving wood is very different when performed with a sharp knife than with a dull one. Feel relates to the manner in which the environment responds to the motions and forces generated by the human. As such, feel can be described by a causal dynamic operator’ that relates a response (output) variable to a stimulus (input) variable. For instance, if the output is taken to be force and the input velocity, the appropriate operator is an impedance [24]. Different selections of input and output variables may lead to admittance, hybrid, scattering, or other descriptions of task feel. In this paper, unless otherwise specified, the term “impedance” will refer generally to an operator that embodies feel.

When a bilateral manipulator is inserted between a human and environment, it will alter the feel of the task. In most traditional applications of bilateral manipulation, a design goal is to achieve transparency; i.e., minimize the extent to which the bilateral manipulator alters the feel of the task. But it is also possible to make intentional use of a bilateral manipulator as a means of altering feel. This concept is particularly valuable if the characteristic dimensions (e.g., force, length, time) of

Manuscript received November 13, 1991; revised March 4, 1993. This work was supported in part by the Engineering Foundation, Grant RI-A-90-4 and in part by the National Science Foundation, Grant MSS-9022513.

The author is with the Department of Mechanical Engineering, Northwest- em University, 2145 Sheridan Road, Evanston, IL 60208.

IEEE Log Number 9212605. A causal dynamic operator is one in which the present value of the output

may depend on present and past values of the input.

the task impedance are very different from those of the human limb.

In this paper, the use of a bilateral manipulator to “shape”* the perceived impedance of the environment is discussed. An approach to impedance shaping is presented and illustrated with an example. The example, however, also illustrates a potential difficulty with impedance shaping. A bilateral manipulator designed to shape the impedance of a particular environment in a particular frequency range may, for another environment or another frequency range, exhibit non-robust behavior. Results from robust control theory, including the structured singular value (p), are used to develop a powerful stability criterion. Finally, an example of a robust impedance shaping controller is given.

A. Power and Impedance Scaling in Bilateral Manipulation

This work is motivated largely by its application to the class of bilateral manipulators in which master and slave operate on very different length, force, and power scales. This class includes strength-increasing “man-amplifiers” [3 11 or “extenders” [27] as well as dexterity-increasing “macro-micro bilateral manipulators” (MMBMs) [9]. [20], [25]. Unlike more conventional master-slave manipulators, the major impetus for such devices is not to enable remote manipulation, but rather is to extend the capabilities of the human limb. As such, these devices usually exhibit some measure of power and impedance scaling.

A basic understanding of power and impedance scaling may be gained from Fig. 1. In this figure, an “ideal” bilateral manipulator is shown connecting a 1-port operator and a 1- port environment. The power transfer from the operator to the manipulator is the product of effort and flow, $ 1 ~ 1 , while the power transfer from the environment to the manipulator is 4 2 ~ 2 [32]. These are related by the scale factors that define the bilateral control law:

(1)

The ratio, k E / k + , therefore, is known as the “power scaling factor.” The “impedance scaling factor,” k,k+, is derived from the relation of the impedance felt by the operator to the impedance of the environment:

k, k+

41E1 = - - h e 2

Zapp(41) = k,k+Ze ( 4 2 ) (2)

where Zap, stands for “apparent impedance.” It is evident that, with proper selection of k, and k+, the power and impedance scale factors may be independently set.

2The term “shape” originates from the effect that the bilateral manipulator has on the frequency response plots of linear impedance-in general, these plots will be not only shifted, but reshaped as well.

1042-296X/93$03.00 0 1993 IEEE

COLGATE: ROBUST IMPEDANCE SHAPING TELEMANIPULATION 315

Pon I ,-I Port2

Operator Bilateral Manipulator Environment

Fig. 1 . Block diagram of a one degree-of-freedom teleoperator system with an “ideal” (dynamics-less) power scaling bilateral manipulator. q5 is a flow variable and E is an effort; Yo is the admittance of the operator, and 2, is the (linear) impedance of the environment. k+ and k, are dimensionless, static scaling factors. The power flow convention used throughout this paper can be seen in this figure: the efforts and flows associated with all physical subsystems (i.e., operator, manipulator, etc.) are defined such that their inner products are the power flows into the subsystems. Subsystems are connected with common-flow energic junctions (the bond graph I-junction 1321 ) to assure correct signs.

Before proceeding, it is perhaps worth noting that ordinary passive hand tools may be classified as manipulators capable of scaling impedance, but not power. For instance, the feel of tightening a nut is certainly different with and without a wrench. Also in this class are conventional micromanipulators which employ transmissions (e.g., fine-pitch lead screws) to scale down human motion^.^ The utility of such relatively simple devices is evident as they are routinely used for micropuncture of arteries less than 100 pm in diameter [21].

The idea of using bilateral manipulation to provide power scaling appears to date to the early 1960s and the “man amplifiers” built at General Electric [31]. Man amplifiers were intended to give the ordinary man extraordinary strength, presumably supplanting the need for an array of powered tools, such as the hydraulic lift. The same period gave rise to cybernetic prostheses, which were to give amputees benefits beyond those of body-powered prostheses, by providing greater strength and more natural means of control [34]. Neither man amplifiers nor cybernetic prostheses met with much initial success, judged in terms of user acceptance. Common to both seemed to be the problem of mental fatigue: they simply required too much concentration on the part of the operator.

In the 1980s, the design of more “natural” human-machine interfaces gave new life to both concepts. Kazerooni introduced the “extender,” which is similar to the man amplifier; however, by being intimately connected to the powered limb of the extender, the operator communicates with it via borh power and information [27]. In effect, the extender is a power-assist, not unlike a power steering system. In a careful set of experiments, Doubler and Childress demonstrated that a very similar concept, called “extended physiological proprioception” (EFT), provided significant benefits for the control of upper-extremity prostheses [15]. It is important to recognize that neither extenders nor EPP seek to provide a limb replacement; rather, both seek to provide a very natural limb extension, in much the same sense that a tennis racket is an extension to the arm of an accomplished player. Other approaches to achieving “natural” control of power-assistive

3The apparent impedance in this case is dominated by manipulator’s intrinsic impedance (inetia, friction) and is not at all indicative of environment impedance.

devices were also studied in the 1980s; notably, the development of improved myoelectric controllers for prosthetic limbs

Power attenuating bilateral manipulators (MMBMs) appear to be of more recent vintage. In 1987, Fukuda, er al. [20] described a one degree of freedom system in which the master and slave were parallel-jaw grippers. Stable gripping of both hard and soft objects was demonstrated. In 1989, Hunter, et al. [25] described a six degree of freedom system that was developed for the mechanical testing of individual muscle cells. In the author’s laboratory, a four degree of freedom MMBM is now under development as a tool for the study of dexterity enhancement. The master manipulandum is described in [29], and the slave is described in [22].

H I , [261.

11. IMPEDANCE S W I N G

The historical review above suggests that the development of power scaling bilateral manipulators may be considered an ongoing effort to improve scaled manipulation via enhanced sensory feedback. For instance, an MMBM is a natural step beyond a fine-pitch lead screw: while both devices enable high resolution positioning, only the MMBM provides useful kinesthetic feedback. It has been rigorously demonstrated that the information contained in scaled kinesthetic feedback is useful (i.e., improves performance), at least in the case of EPP control [ 151.

Impedance shaping is offered here as a potential evolu- tionary step beyond power scaling manipulation. The idea underlying impedance shaping bilateral control is that a priori information, in the form of a task model, may be used to enhance performance further. In other words, the impedance shaping bilateral controller seeks to improve performance by adding to the sensor-based feedbacks (vision and force) a model-based feedback (force). This idea has one clear precedent in the work of Herzog [23], who demonstrated improved manual control when the impedance of a joystick was designed to approximate that of the plant being controlled. This, however, was not an example of bilateral control, because sensor-based force feedback was absent. Another precedent is model-based control (including loopshaping techniques [ 1 SI), although the analogy to impedance shaping cannot be taken very far. For instance, the ultimate design goals in impedance shaping and loopshaping are rather different. In one respect, however, there is notable similarity: an essential tradeoff in both methods is between aggressiveness and robustness.

Impedance shaping is an approach to bilateral manipulation, but is not a design methodology. It lacks, for instance, a general approach to performance specification (this depends on the task at hand and what measure of performance is to be optimized). Nonetheless, a design methodology which incorporates impedance shaping is entirely feasible. As an illustration of this, a specific methodology based upon kinematic similarity is developed below and in Section IV.

A. Similarity-A Basis for Impedance Shaping Design

Similarity rules are commonly used in model testing [35]. For instance, a physical model and full-scale prototype are

. _. .._ .

376 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 9, NO. 4, AUGUST 1993

said to be geometrically similar if all length scale ratios (e.g. height over width) are the same in both. Kinematic similarity requires that length scale ratios are the same in model and prototype and time scale ratios are also the same in model and prototype. Dynamic similarity requires that this be true of length, time, and force scale ratios. For linear systems represented by transfer functions, dynamic similarity implies that the poles and zeros of the model are the same as those of the prototype.

Geometric similarity is enforced in most implementations of telemanipulation. In other words, the length ratios experienced at the master are in fixed proportion to those of the task. This is consistent with the use of visual feedback, which produces an image that is geometrically similar to the task. Most implementations of bilateral manipulation attempt to enforce dynamic similarity as well. For instance, the "ideal" power-scaling manipulator illustrated in Fig. 1 generates an apparent impedance which is simply that of the environment times a constant. If these impedances are linear, it is clear that the poles and zeros are unaffected. Of course, dynamic similarity is impossible to achieve in practice due to the effect of manipulator dynamics on the apparent impedance (this point will be clarified in the examples that follow), but it is commonly taken as a design goal, termed "transparency" [5 ] .

An obvious limitation of dynamic similarity, however, is that the poles and zeros of the apparent impedance cannot be altered relative to those of the environment. Stated differently, the forces contributed by various physical effects (e.g., inertia, stiffness, friction) cannot be independently scaled. Yet, in the context of micromanipulation, it may sometimes be desirable to scale inertial forces, for instance, to a greater extent than, say, surface tension forces. This may be done in an effort to make the microdynamical system feel like a macrodynamical system. One rational approach to this type of scaling problem is to enforce (via impedance shaping bilateral control) kinematic similarity but not dynamic similarity between the apparent impedance and the actual impedance.

The example below illustrates the distinction between conventional (dynamically similar) bilateral manipulation and impedance shaping (kinematically similar) bilateral manipulation. It also provides the beginnings of a design methodology for impedance shaping control.

B. Design Example

To begin, assume that the MMBM may be represented, as in Fig. 1, by the two scale factors k, and k+. Suppose, further, that the environment consists simply of a small block (mass me) immersed in a viscous liquid, and that the Reynolds number is low enough that drag forces scale linearly with the block's length scale and velocity [35] (viscous drag coefficient be) . The MMBM is to be used for manipulating the block.

If a visual display system is used to magnify the image of this microenvironment by a certain factor a, then a good choice of the velocity scale factor is k+ = l / a . This choice would ensure that the perception of a 1 : 1 relationship between the position of the slave's endpoint and the position of the master's handle is maintained, as in EPP (of course, in actual implementation, position control would be required).

~~~~1 E1 E2

k E +

Environment "Ideal" Impedance Shaping Bilateral Controller

Fig. 2. Block diagram of an "ideal" impedance shaping bilateral controller attached to an environment.

The force relation, however, is more complicated. Because mass scales as length cubed, one sensible choice of the force scale might be k, = a4, which would result in an impedance scaling of k+kE = a3. This would make the perceived mass equal to the expected mass of the object seen in the display, assuming equivalent densities. The viscous forces on the mass, however, vary according to length, so that the appropriate scaling of the viscous component of the impedance would be k+k, = a. The question of which of these scalings to select has no clear answer. Any fixed scaling has the disadvantage that the apparent impedance will be dynamically similar to the environment impedance, and therefore will feel more sluggish than a kinematically similar macrosystem. In other words, a fixed scaling cannot alter the inherent time constant ( T ~ = me/be) of the environment.

Now suppose ?at the MMBM incorporates an impedance shaping term, AZe(s), as in Fig. 2. The following procedure will

a

a

a

a

a

a

enforce kinematic similarity but not dynamic similarity. Select a geometric scale factor, a. The time scale factor is unity (because manipulation must occur in real-time, any time scaling is prohibited). Construct a model of the environment. In this case:

Ze(s) = h e s + it? (3)

where he and ie are estimates of me and be. Determine the kinematically similar, scaled impedance. In this case:

(4)

Select k+ = l / a . Select the minimum k, necessary to match one term of k,k+Ze(s) to the corresponding term of Z,"'(s). In this case, k, = a2 leads to:

( 5 )

The damping term matches that of Z,"'(s). Select the impedance shaping term AZe(s) according to:

(6)

Z p ( s ) = a3yjLes + abe

k,k+Z,(~) = ( Y ~ , s + ab,

AZe(s) = k~'k~'Z~'(s) - Ze(s)

In this case:

AZe(s) = (a2 - l)yjLes

zapp(s) = t k + ( A Z e ( s ) + z e ( s ) )

(7)

The result of this design procedure is an apparent impedance having the following form:

= a ( @ - l)yjLes + Z,(s)). (8)


Slave Controller Master Bilateral Controller Slave

1 Environment

Impedance Shaping Bilateral Controller

0: = 25 n = 100 m, = 0.5 kg bm = 1 .0 N-dm mS = 0.005 kg bs = 0.001 N-dm Bs = 282 N-dm Ks = 8ooO N/m

Fig. 3. A one degree-of-freedom impedance shaping MMBM featuring master and slave dynamics.

As the estimate Ze (5) approaches the actual impedance Z, (s), the apparent impedance will approach the kinematically similar impedance. It is important to recognize, however, that a precise model of the actual impedance is not essential to the success of the method. The apparent impedance in (8) consists of two parts: the actual environment impedance scaled by a fixed factor of a, and the impedance of a simulated inertia ((a2 - l ) h e s ) , also scaled by a. The latter may be recognized as a simple form of “virtual environment”; i.e., an interactive, real-time simulation [2]. Note that this construction of Zapp(s) will produce the desired result of amplifying inertia to a greater extent than damping (thereby increasing the apparent time constant of the environment), even if the-estimate he is considerably in error, and even if the estimate be is completely in error.

C. Discussion

The example above demonstrates that impedance shaping depends on the availability of an environment model. The force generated by the environment is, in general, the outcome of multiple physical effects. If each of these effects is subject to a distinct scaling law, then they must be separated by some means in order to implement impedance shaping. It is a reasonable conjecture that a model is the only such means, unless internal states of the environment can be measured directly.

The need for a model highlights a potential weakness of the impedance shaping concept: a standard reason for choosing teleoperation rather than autonomous operation is that the environment is unstructured. Nonetheless, every environment will exhibit some structure, and what information is available should be used. For instance, in microelectronics assembly, masses and coefficients of friction may well be known a priori, and in microsurgery, tissue properties may be known. In general, impedance shaping may be made more or less aggressive, according to the knowledge of the environment. Furthermore, so long as the bilateral controller meets a non- restrictive robustness criterion presented in Section 111, its behavior will be benign, even if the estimated and actual environment dynamics differ radically (as may occur when the slave suddenly contacts a rigid surface).

Even if unknown a priori, the environment dynamics may be estimated on-line. It is not uncommon for a person performing a delicate task to begin by gently probing the environment. A micromanipulator might do something similar, in order to build an environment model. Another possibility is that the bilateral controller include a parameter adaptation scheme that runs continuously. One difficulty with adaptive techniques is that they are inherently nonlinear, making performance and stability analyses difficult. In this paper only linear time- invariant bilateral controllers are considered-the design and analysis of nonlinear, time-varying controllers are subjects of future research.

111. ROBUSTNESS In the example below, a somewhat more realistic MMBM

model involving master and slave dynamics is considered. This example illustrates the difficulty of maintaining robustness.

A. Example-Poor Robustness

A block diagram of a somewhat realistic impedance shaping implementation is shown in Fig. 3. The master is modeled as a controlled force source acting on an inertia m, with viscous damping b,. The slave is modeled as a controlled force source acting on an inertia m, with viscous damping b,, connected to its load via a transmission of gear ratio n. A PI velocity controller (effectively a PD position controller) is closed around the slave. The impedance shaping controller is similar to that in Fig. 2; however, the velocity feedback term is taken from the actual velocity of the slave which will, in general, differ from the commanded velocity.

Fig. 4 compares the apparent admittance (inverse of impedance) of an environment (the one considered above with me = 5 grams, be = 1 N-s/m) felt through the bilateral manipulator to the actual admittance of the environment, the admittance that would be felt through the ideal impedance shaping controller of Fig. 2, and the admittance that would be felt if the impedance shaping were turned off. Note that the impedance shaping controller has the effects of reducing the admittance (amplifying the impedance) and reducing the comer frequency (this corresponds to amplifying inertia more than damping); the non-shaping controller does only the

..*"I.


-10 - -20

-30 -

-

1 4- d

-50-

-# - 2.

- 'O i -80 . :1__ - . . .

frequency (dm-)

(b)

Fig. 4. Environment admittance (upper dashed line); apparent admittance via bilateral manipulator with no shaping (dot dash); apparent admittance via ideal impedance shaping bilateral manipulator pictured in Fig. 2 (lower dashed); apparent admittance via bilateral manipulator pictured in Fig. 3 (solid). Notice that all apparent admittances are far less than the actual admittance, however, the comer frequency (at which inertial effects become important) is moved significantly only if impedance shaping is used. Notice that a realistic impedance shaping design (solid) is close to ideal (dashed) for a finite frequency range; at high enough frequency, master inertia dominates the admittance of the realistic design.

former! The impedance shaping controller appears to work well: its admittance is comparable to that of the ideal controller over a substantial frequency range, though at high enough frequency it is dominated by the inertia of the master (mm). Of course, the high-frequency dominance of inertia is an effect which must occur in any real hand tool or bilateral manipulator.

It is important, however, to assess the robustness of the impedance shaping controller before passing final judgment.

4The nonshaping controller does in fact change the shape of the admitance frequency response plots, as seen in Fig. 4. This change is due to the inherent dynamics of the master and slave manipulators which affect the apparent impedance, but do not contribute to intentional shaping.

A precise definition of robustness is provided below, but it is easily illustrated that this design is not robust. Although the impedance shaping controller is designed for a specific microenvironment, the characteristics of this environment are subject to rapid and dramatic change. Perhaps the best illustration of this is making or breaking contact with a rigid surface. To extend this example, suppose that contact is made with a surface of stiffness I C , = 5 x lo4 N/m. The isolated MMBM with impedance shaping control will remain stable when contacting this surface; however, when the inertia of the operator's arm is included, the entire system may lose stability. As a specific example, suppose that the arm introduces an inertia of m, = 2 kg. The system will exhibit unstable oscillations at a frequency of about 125 rad/= (20 Hz) which the operator will find disturbing, but in all likelihood, absolutely uncontrollable.

B. Approach to Robustness

The ideas presented in this section draw heavily on recent (since 1982) advances in the field of robust control. The basic approach is to identify the class of manipulator/environment behaviors (admittances) with which a human can be expected to interact successfully, and to identify the class of all possible environment behaviors. Then conditions may be placed on the manipulator behavior to ensure that, for environment behaviors within the latter class, manipulator/environment behaviors will always fall within the former class.

It is a matter of common observation that human operators have no difficulty maintaining stability when interacting with all manner of passive tools and environments. Indeed, even greater skill can be expected: some tools, such as rotary sanders and floor waxers, exhibit what are obviously not passive impedances, but can, with some training, be operated quite effectively. Nonetheless, the broadest class of tooYenvironment behaviors which can, at this point, be analytically characterized as stable under human control, is the set of passive behaviors. A reasonable choice for the class of environment impedances is also that of passive impedances. This has been motivated elsewhere [13]. Therefore, the following intuitive condition for stability/robustness will be used: a bilateral manipulator is said to be robust iJ; when coupled to any passive environment, it presents to the operator an impedance (admittance) which is passive. Thus, while the apparent impedance felt through the bilateral manipulator may exhibit a radically different magnitude, or even shape in the frequency domain, than the environment impedance, it must at least be passive. The mathematical statement of this condition, and the derivation of a simple test for it, follow the preliminaries below.

C. Mathematical Preliminaries

Impedances, Admittances, and Hybrid Matrices -The terms impedance and admittance may be applied to nonlinear systems as well as linear systems. Impedances and admittances are causal dynamic operators that map vectors of inputs (4 and E , respectively) to vectors of outputs ( E and 4, respectively):

E = Z(+) + = Y(&) (9)

. .,..-.


In the case of linear, time-invariant (LTI) systems, these operators reduce to transfer function matrices:

4 s ) = Z ( S ) 4 ( S ) 4 ( s ) = Y ( S ) E ( S ) . (10)

Associated with each pair ( E ; , 4i), where E; and 4i are the ith components of the vectors E and 4, respectively, is the notion of a “port.” A physical system connects to its environment at ports; the product ~ i d i is the power flowing into the system at port i.

Not all multiport physical systems can be represented by impedances or admittances. An example is the two-port trans- former, which can be represented by the “hybrid matrix:”

but not by any impedance or admittance matrix [33]. It has been demonstrated that, for any LTI multiport, a hybrid matrix will exist [3]. The general form of a hybrid matrix is:

where qTqi is the total power input.

is one for which, given zero energy storage at t = 0 [14]: Passivity and Positive Real Matrices-A passive multiport

I ’Q: (T)d+T 2 0 VQdt ) , t L 0. (13)

This definition applies equally to nonlinear and linear systems. Its interpretation is simply that finite energy may never be extracted from a passive system. The hybrid matrix of a passive LTI multiport must be positive real:

H ( s ) stable ; H * ( s ) + H ( s ) 2 0 for Re{s} > 0. (14)

A test for positive realness is that the multivariable Nyquist plots of H ( s ) must lie in the closed right half plane [8].

Scatrering Matrices-The interaction between physical systems is fundamentally bilateral; therefore, an effort ( E ) may never exist independent of a flow ($), and vice versa. Efforts and flows, however, are by no means the only pair of variables that may be used to describe bilateral interaction. Any pair related to E and 4 through a nonsingular linear transformation is equally legitimate. Another useful set of variables is (G, 20 ), the “inwave” and “outwave” or “scattering variables” [33]. Scattering variables relate to efforts and flows by

t

1 1 1 [E] = 3 [ 1 -1][;]

LTI multiports may always be represented by scattering matrices:

(16)

Passivity and Bounded Real Matrices: In terms of scattering variables, a passive multiport is one for which, given zero energy storage at t = 0 :

t w (s) = S ( s ) G ( s ) .

(GT7i? - f T 5 ) & 2 0 V $ ( t ) , t 2 0. (17)

Fig. 5. (a) A generalized description of an operator-MMBM-environment system in terms of scattering variables and scattering operators. (b) An equivalent system description (except that the operator has been replaced with a pseudo-operator).

This condition applies equally to linear and nonlinear systems. The scattering matrix of a LTI multiport must be’bounded real:

S(s) stable; I - S*(s)S(s) 2 0 for Re{s}. (18)

A test for bounded realness is that the maximum singular value of S ( j w ) be less than or equal to one at all frequencies, or equivalently, that the infinity norm of S ( j w ) be less than or equal to one.

D. Robustness Criterion The robustness criterion to be developed will apply to

bilateral manipulators that may be described by hybrid or scattering matrices; the assumption will also be made (temporarily) that the environment is also linear. Suppose that the bilateral manipulator and environment are described by the following scattering matrices (see also Fig. 5(a)):

bilateral manipulator: f ‘ ( s ) =

environment: Se (s) (19)

where P(s ) is partitioned according to master ports (subscript 1) and slave ports (subscript 2). The intuitive robustness criterion given at the beginning of this section may be stated in terms of these matrices as shown as follows.

Robustness Criterion-Version I The P ( s ) contains no poles in the closed right half plane,

and:

llp11+ P12Se(I - p22Se)-1pz111m I 1 v llSelloo I 1. (20) 0

The dependence upon the Laplace variable, s, is understood. The expression on the left is simply the infinity norm of the


scattering matrix “felt” at the operator ports; it is a linear fractional transformation of Se [17]. This statement of the robustness criterion, however, is of little computational value, because the norm must be computed for all n x n Se of infinity norm less than or equal to one. A condition in terms of P ( s ) alone would be preferable.

The key to finding such a condition lies in recognizing that the conditions sought are equivalent to the conditions for the stability of the telemanipulator/environment when coupled to a pseudo-operator which is passive, but otherwise arbitrary. This can be understood by replacing the operator block in Fig. 5(a) with an arbitrary passive n-port, then asking the question, under what conditions on P(s ) is stability guaranteed? The conditions are the same as those sought here, because a necessary and sufficient condition to guarantee the stability5 of a LTI system coupled to an arbitrary passive n-port, is that the system itself appear to be passive [8]. Thus, by

passive, block diagonal 2n-port (Sstrct (s), above) iff

(23)

0

A third version of the robustness criterion follows immedi- ately.

Robustness Criterion-Version 3

plane, and: The matrix P ( s ) contains no poles in the closed right half

(24)

0

proving coupled stability-the stability of the telemanipulator coupled to an environment and pseudo-operator which are both passive but otherwise arbitrary-the apparent passivity of the telemanipulator/environment is implicitly proved. This reasoning leads to the revised block diagram in Fig. 5(b), and to a second statement of the robustness condition:

The proof of the coupled stability theorem, therefore the robustness condition, is straightforward with the assistance of the multivariable Nyquist Theorem [28], because the definition of p is essentially tautological. It may appear, in fact, that the introduction of the structured singular value has been counterproductive, as all “perturbations” A E X, must be considered. Certain properties of p ( P ) , however, make it possible to arrive at a condition that may be computed in terms of P(s ) alone.


the following criteria on structure and infinity norm: The P ( s ) is stable when coupled to any 2n-port that meets

The useful properties are the following:

0 ii) p(DPD-’) = p ( P ) where D has the form:

(26)

where 8 again refers to the maximum singular value. The first

second from the fact that D-’AD = ” where A E xm’ Together’ these properties

Notice that IISstrctIlm 5 1 (the 2n-port is passive) if and only D = [ d l I n x n 0 , j I n X n ] > d l > d z 0 >’ if IISpseudol(, 5 1 and IISellm 5 1 (the pseudo-operator and environment are both passive).

a condition on p ( s ) done than Version

uncertainty problem, which may be addressed by Doyle’s structured singular value [16] or Boyd’s structured Lyapunov

Version is no 1; however, the problem c m now be recognized as a structured Property follOws from the definition Of p(‘>? the

functions [6]. The structured singular value of P,p(P), is defined as p ( M ) 5 a(DMD-1) .

0 if no A E X, solves det(I + PA) = 0 PL(p) = { ( A;;m J - ~ ( A ) I det(l + PA) = 0) )-I

(22) where X,(s) is defined as the class of 2n-port block diagonal matrices (same structure as Sstrct, above) with no restriction on infinity norm, and F(A) is the maximum singular value of A. A useful coupled stability condition can be stated in terms of p:

While this result is only a modest advance, it can be shown that, if the right hand side is minimized over all allowable D, the equality holds. This result was first proved by Doyle, who also showed that the optimization problem is convex, ensuring the tractable computation of p ( P ) 1161. Thus, a useful form Of the robustness condition is finally achieved as fOllOWS..


The P ( s ) contains no poles in the closed right half plane, Coupled Stability Criterion for Block Diagonal 2n-Port Environments and

The 2n-port system with scattering matrix G(s) will be guaranteed to remain stable when coupled to an arbitrary,

in the closed left half plane. This includes what is sometimes known as

passive” according to criterion given in [12].

sup( w p > o inf .( [ ;;il 21)) 5 1 (28)

0 51n this paper, stability means that all the poles of the LTI system lie

marginal stability, and does not guarantee asymptotic stability. A guarantee of asymptotic stability requires that the pseudo-operator environment be “strictly

where p = dl/dz.


Remark 1: For p = 1, the condition above would be equivalent to the passivity of P. In macro-micro manipulation, P will never be passive due to the amplification inherent in the scale factors IC, and IC+(IC,/k+ > 1). Thus, the freedom provided by the parameter ,l3 is essential to the utility of the above condition.

In [6], Boyd and Yang demonstrate via Lyapunov and absolute stability theory, that if D exists such that DPD-’ is bounded real, then P will remain stable when coupled to any nonlinear, passive, block-diagonal 2n-port; i.e., a causal dynamic operator satisfying the passivity condition of (1 3), and the structure condition of (21). This condition is more conservative than Doyle’s only in that D is not frequency dependent. The corresponding robustness condition is shown as follows.

Robustness Criterion-Version 5 (Nonlinear Environments)

plane, and p > 0 exists such that: The matrix P ( s ) contains no poles in the closed right half

[ $PZ1 21 U is bounded real.

Boyd and Yang also show that an equivalent condition exists in terms of hybrid matrices. It is perfectly analogous, and can be obtained by replacing P with H , and “bounded real” with “positive real‘’ in the statement above. A similar result, which does not use input-output descriptions at all, but instead relies on a network representation of an operator-telemanipulator- environment system, is described by Anderson in [4].

Remark 2: In addition to nonlinear environments, the condition is useful in dealing with nonlinear manipulator dynamics. It is often possible to treat a bilateral manipulator as a pair of 2n-port nonlinear admittances (master and slave mech- anisms) sandwiching a 2n-port LTI controller. This breakdown allows the nonlinear mechanism dynamics to be absorbed into the operator and environment descriptions.

In the next section, these results will be used to analyze impedance shaping controllers.

Iv. EXAMPLE OF ROBUST IMPEDANCE SHAPING BILATERAL CONTROL

The nonrobustness of the MMBM considered in Section III- A can be verified by computing the structured singular value of its scattering matrix. The result is displayed in Fig. 6. An attempt was made to improve the robustness properties of this MMBM by including various forms of compensation in both the feedforward (velocity) and feedback (force) paths. Fairly extensive exploration failed to produce any design that would satisfy the robustness criterion over all frequencies, although violations could be pushed to quite high frequency. This problem appears to stem from two factors: noncollocation [ 191 and nonbackdriveability. Noncollocation describes a situation in which significant dynamics separates sensor and actuator signals; nonbackdriveability describes the property of low output admittance in a manipulator. In terms of these factors, the difficulty may be understood as follows: the admittance that the bilateral manipulator presents to the environment

frequency (radsec)

5

Fig. 6. Plot of the structured singular value ( p ) associated with the bilateral manipulator pictured in Fig. 3. p must be less than or equal to one at all frequencies to ensure robustness. This design is not robust.

( + ~ ( s ) / E ~ ( s ) in Fig. 3) should, for connection to all admissible pseudo-operators, be positive real. This admittance consists of the sum of two parts: the intrinsic admittance of the servo-controlled slave, and the admittance of the feedback loop via the bilateral controller, maskdoperator, and slave. The former is typically positive real while the latter, due to noncollocation, cannot be. However, in part due to the large gear ratio, the admittance of the slave is very small (i.e., it is nonbackdriveable), thus the total admittance is dominated by that part which is not positive real. This difficulty is exactly analogous to the compliance control of a nonbackdriveable robot, which is discussed in [7].

Of the two factors, the more difficult to contend with is generally nonbackdriveability. In [lo], a similar example which includes noncollocation, but no gear ratio, is considered. It is shown that a low pass filter in the force feedback path is sufficient to provide robustness. Even so, the design process employed is ad hoc and not readily generalized to more complex situations.

Further strides can be made by eliminating both nonbackdriveability and noncollocation. An example of such a design is illustrated in Fig. 7. The placement and role of the impedance shaping term are now quite intuitive: as in the “ideal” case (see Fig. 2), the shaping term is a real-time environment simulation that augments the physical environment. This term is readily implemented as part of the collocated feedback of the slave; i.e., the slave control law becomes

(29) 1 K 8

E,o,trol(S) = &PI & A Z ( ~ ) = - B8 - AZ,(s) 42(5) [ s

where 4 2 is the velocity of the slave, and E control is the control force applied to the slave which is the sum of PI control (&PI) and impedance shaping ( E A Z ) terms.

The force feedback signal is based on the PI control effort rather than the force at the slave/environment interface, making it effectively collocated with the velocity of the master, 41. Note that, due to the low inertia and backdriveability of the slave, the PI control effort will, under most circumstances,

. ., ... *


Ze

Environment Slave Controller Slave Plant Master Bilateral Controller Slave Shaping

Impedance Shaping Bilateral Controller

a = 25 m, = 0.5 kg bm = 1 .O N-s/m ms = 0.005 kg br = 0.001 N-dm Bs = 400 N-s/m Ks = 4OOO N/m

Fig. 7. An architecture for robust impedance shaping bilateral control.

approximate the interface force (&PI x €2 - E A Z ) . Collocated force feedback has little effect on performance: the performance of this design is shown in Fig. 8, which compares quite well to Fig. 4.

The real benefit of this approach is robustness, as illustrated in Fig. 9. It is interesting to note that the optimal scaling p is found to be p = a-1.5 independent of frequency, so that the design is robust for even nonlinear environments. It may appear from Fig. 9, however, that the design is only marginally robust, as the structured singular value is approximately unity across the frequency spectrum. Somewhat surprisingly, this is the desired result. As an illustration of this point, consider a bilateral manipulator that behaves as a rigid body (many hand tools behave approximately as rigid bodies). Because such a manipulator would be lossless, its structured singular value would be one at all frequencies. Greater robustness could be achieved by adding internal dissipation, but this is not necessarily desirable.

A final point regarding the architecture f f Fig. 7 is that robustness will be guaranteed whenever AZ, is positive real (it can then be lumped together with 2, without altering the assumption of passivity). Thus, the shaping term should be se- lected to supplement the passive dynamics of the environment, but not subtract from those dynamics.

V. CONCLUSION

The principal contributions of this paper have been the introduction of impedance shaping bilateral control and the development of a powerful robustness criterion for bilateral manipulation. A design approach based on kinematic similarity has emerged through the examples. This approach may be summarized as follows.

Select a scale factor a and design an impedance shaping term (AZ,) according to the procedure given in Section 11-B. The shaping term should be positive real. Using a backdriveable slave manipulator, implement bilateral control as indicated in (29) and Fig. 7 (collocated position and velocity feedback for servo control of slave; collocated feedback for impedance shaping; and constant scale factors).

While this approach may serve as the basis for a formal methodology, issues stemming from master and slave dynam-

frrquency ( d s e c )

(b)

Fig. 8. Environment admittance (upper dashed line); apparent admittance via ideal impedance shaping bilateral manipulator pictured in Fig. 2 (lower dashed); apparent admittance via bilateral manipulator pictured in Fig. 7 (solid). Notice similarity to Fig. 4; the revised control architecture has not degraded performance.

ics, multiple degrees of freedom, and discrete time controller implementation need to be thoroughly addressed. Moreover,


X I O ’

Ti-0-2 10-1 100 101 100-2 103 104 1 0 5

frequency (ndlsec)

Fig. 9. Structured singular value plot for bilateral manipulator pictured in Fig. 7. This design is robust, because p is slightly less than 1 at all frequencies.

the notion of kinematic similarity is only one, and by no means an optimum, approach to impedance shaping.

Ultimately the utility of the impedance shaping concept, and of any other approaches to enhancing operator dexterity, must be borne out experimentally. An MMBM comprising a four degree-of-freedom master manipulandum, a six degree- of-freedom slave manipulator, and a transputer-based real time controller, is being developed for this purpose [30]. Because the impedance shaping term is essentially a “virtual environment”, this system is also able to execute real-time virtual environment simulations, outputting forces of interaction via the master manipulandum.

The robustness criterion presented here is broadly appli- cable to problems in telemanipulation; however, it remains conservative in the following respects: it assumes that the environment is unstructured, and it requires that the apparent impedance felt through the bilateral manipulator be passive. As pointed out in the discussion of impedance shaping, some understanding of environment structure generally does exist, and should be exploited. The appropriate restriction on the apparent impedance merits experimental investigation, and it may be found that the restriction to passivity can be relaxed. Unfortunately, removing that which is conservative in the problem statement does not mean a more powerful robustness criterion will be found. The mathematics may simply not be tractable. To address this issue, a generalized description of physical (bilateral) interaction, based on neither hybrid nor scattering descriptions, is being developed. The idea is to select that description which optimally encodes the available information (e.g., passivity, environment dynamics, etc.) into a simple measure such as the infinity norm. This would enable the robustness theory presented here to be extended to a significantly larger class of environment, operator, and manipulator behaviors. Preliminary results have been presented in [ 1 I].

ACKNOWLEDGMENT The author also thanks Ambarish Goswami for his careful

reading of this manuscript.

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J. Edward Colgate received the S.B. degree in physics and the S.M. and Ph.D. degrees in Mechan- ical Engineering from tie Massachusetts Institute of Technology, Cambridge, MA.

Since 1988, he has been an Assistant Professor in the Department of Mechanical Engineering at Northwestem University, Evanston, IL. His areas of interest include haptic interface to teleoperators and virtual environments, robotic tools for microsurgery, electrostatic actuators, and modeling and control of dynamic systems.

colgate 1993 robustimpedanceshapingtelemanipulation

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