Modeling and Analysis of Manned Robotic Systems

SUPERVISORY CONTROL

Copyright © IFAC Man-Machine Systems. The Hague. The Netherlands. 1992

MODELING AND ANALYSIS OF MANNED ROBOTIC SYSTEMS P.H. Wewerinke Department of Applied Mathematics. Systems and Control Group and Mechatronics Research Centre Twente, University of Twente, P.O. Box 217,7500 AE Enschede , The Netherlands

Abstract

2.

In this paper a modelling approach is presented to describe complex manned robotic systems. The robotic system is modelled as a (highly) nonlinear, possibly time-varying dynamic system including any time delays in terms of optimal estimation-, control- and decision theory. The role of the human operator(s) is modelled varying from supervisor of the automated (part of the) system to controller in

In general the task to be performed with a manned robotic system can be described in terms of the various components involved. This is indicated in the block diagram of Fig. 2.1.

terms of the various

functions

The first important aspect of the task are the goals to be achieved under given boundary conditions. Realistic operations may involve a complex goal hierarchy (interrelated, or even conflicting goals, subgoals, procedures, etc.). These goals dictate the tasks to be performed. The complexity of the task hierarchy will correspond with the complexity of the goal hierarchy. The defined task will be affected by the operational

involved to perform

goal-oriented tasks. In the paper it is indicated how the model differs from previous optimal control models. Furthermore, the use of the model is illustrated in discussing typical manned robotic tasks and applications.

environment.

Next, the functions can be derived to perform the defined task. The motives to fulfill these functions originate from the goals to be achieved. The human operator (HO) will perform these functions utilizing the available resourses, being separate items or elements of the system. These functions are the result of a function allocation procedure to the man and the machine, taking into account the human capabilities and limitations and the possibilities of the system.

It may be expected that the model is capable of answering questions related to reliability and efficiency, design alternatives, function allocation,

1.

MANNED ROBOTIC SYSTEMS

automation, etc.

INTRODUCTION

Robotic systems are more and more applied in many areas. Examples of industrial operations are part assembly, material transfer, repair of parts and inspections. In addition, robotic systems play an important role in many teleoperations, e. g. in space applications (space stations, serviceable sattelites, material processing platforms) and

Finally, the result of HO functions are actions, taken on the basis of drives (derived from his motives, etc.). Simple stimulus response behavior takes place at this level only. The actions affect the system resulting in a certain system behavior. Based on performance criteria and measures of this

operations in a risky or unaccessible environment.

behavior, total system behavior can be evaluated with respect to the goals to be achieved.

Autonomous systems can meet the safety, reliability and especially economic requirements for specific tasks, but many operations involve the interacting contribution of both human operator(s) and robotic system(s). This concerns especially

More specifically, a manned robotic system can

be analyzed or designed based on the assumption that a goal-oriented operation can be defined, e.g. controlling the robot from A to B, with given constraints ( due to robot dynamics, control limits, environment, etc.) in a given (disturbance)

complex, non-standard operations in an unstructured

environment. One example is shared compliant control, especially in applications with telemanipulators where time delays are considerable. The role of the human operator(s) may vary from direct controller to supervisor of the automated (part of the) system. This depends on the goals to be achieved and the related functions to be fulfilled.

environment.

Furthermore,

the

robotic

system

is

described as a (highly) nonlinear, possibly time-varying dynamic system including any transport or communication time delays. The role of the HO may vary from supervisor of the automated (part of the) system to controller, or combinations. This involves HO functions such as perception of sytem outputs provided by (e.g.) displays and for the visual scene, information processing to assess the task- and system variables of interest, decision making involved in monitoring the autonomous (sub)system and involved in

In the next section manned robotic systems are discussed in more specific terms. One approach to design and analyze a manned robotic system is based on mathematical models of this complex man-machine system. This is contained in section 3. The paper is concluded with some remarks about how the model can be utilized.

intermittent control, and control.

In this context,

control is used in a broad sense including planning and compensatory actions.

151

Goal

)

ment

hierarchy

hierarchy

6

Environ-

Task

1

1

1 )

1

[3

Resourses

Functions

I

1

{~::J

B

HUMAN OPERATOR

Fig.2.l

Manned robotic system components

As indicated before, the role of the HO may vary from supervisor of the automated (part of the) robotic system to controller. In this paper the manual control task will be assumed. The human functions involved include state estimation and decision making about (ab)normal system behav ior (supervising). So the supervisor task is part of the manual control task. This will be discussed in the following.

There are various ways to analyze a manned robotic system. One approach utilizes a variety of system performance and HO measures obtained in an experimental situation. The drawback of an experimental approach is that it involves per se comlex man-in-the-loop simulation. This implies that experiments mainly serve to verify system performance rather than to predict the performance of new systems. This limits their utility, especially early in the design stage. An alternative method to describe manned robotic system behavior is the use of mathematical models . The main advantage of mathematical models is that they provide a precise (quanti ta ti ve) formulation of task aspects such as goals and system dymanics and basic concepts of HO functioning. Therefore, models potentially have a predictive capability, rendering a basis for selecting design alternatives. Furthermore, models provide a powerful tool to analyze operational problems of existing robotic systems systematically.

The following elements are included in the model as indicated in the block diagram of the manned robotic system model shown in Fig. 3.1. The HO perceives (e.g. via a TV-camera/monitor system) the outputs of the system state (characteristic system variables such as position, velocity, attitude, etc.), which is perturbed by external disturbances. The HO utilizes these (inaccurate) observations of the state to estimate the state recursively. Based on this estimate and the task specifications a control input is selected and executed, resul t ing in a closed loop manual control. In addition, the future desired state is observed (e. g . from ·pictorial information) and estimated. Based on this estimate and the system dynamics a pre-programmed control is selected. The resulting open loop control is combined wi th the feedback control to compensate for deviations from the planned trajectory due to random disturbances.

In the next section a model structure is presented

describing the foregoing manned robotic system.

3.

MODEL STRUCTIJRE

3.1 General In this section a model of the manned robotic systems discussed in section 2 is presented. For details the reader is referred to Ref.l. Basically, the task considered is to control the robotic system from an initial state Xo at time k - 0 to a final state Xn at time k - N following a given desired trajectory Xd(k) . Strictly speaking, the time aspect involved in this formulation is absent in most of the control tasks of interest. For example, the task is to arrive at a certain point in the state space,

3.2 System dynamics A robotic system may be represented nonlinear, time-varying dynamic system by

irrespective of

as

a

X(k) - f(X(k-l) ,V(k-l),W(k-l) ,k)

(la)

Y(k)

(lb)

g(X(k) ,V(k) ,k)

where X(k) is the n-dimensional state vector at time k, f is the n- dimensional vector function, V is the r-dimensional control vector, W is the l-dimensional disturbance vector, Y is the m-dimensional system output vector and g is the

time, but following a given sequence (e.g. a spatial trajectory). This is known in control theory as a space constrained control problem. Theoretically this is a difficult problem , certainly for robotic (i.e . nonlinear) systems. Thus, it is attractive to prescribe to desired trajectory for each time step and solve for the corresponding optimal control at each time step. This seems for many control tasks a meaningful approach.

m-dimensional vector function.

The standard procedure is followed to describe the nonlinear system behavior X in terms of a state reference Xo and a "small perturbation x around this reference; thus X - Xo + K, V - Vo + U, etc. This lineari- zation scheme yields a time varying reference model ll

152

disturbance,

The so lution of thi s optimal control probl em is discussed in the fo llowing subsection dealing with hwnan control behavior.

W

1 task specification

3.4 Human observer and controller The model of t he HO func tional aspects, which are diagram of Fi g. 3. 1 .

comprises various shown in the block

Perception The p e rcep tual model describes how th e system outputs y are r e l ated to the perceived variables yp ' It i s assumed that t he HO perceives these system outputs with a certain inaccuracy and with a certa in time delay

a llo cat ion of attention

yp(k) - y(k- i) + v(k-i)

Here v is an independent, Gaussian, purely random observation noise sequence, representing the var i ous sourses of hwnan randomness (unpredictable in other than a statistical sense) . Each e l ement v j , therefore, is specified by its covar iance V j • This covariance is functionally related to the signal level, hwnan attention allocation and threshold phenomena. The l wnped time delay i can be associated with the HO's internal time de l ays related to perceptual, central processing , and neuromotor pathways. For systems with r e l atively large time constan ts, these delays can be neglected. Also system- related delays, for instance the communi cation or transport delays of remotely controlled systems may be modelled as a lumped equiva l ent "perceptual" time delay involved in eq(5) .

desired state

1\ Xd

contro l se l ection (planning) and execution control Fig. 3.1

(5)

model of the Hwnan Operator

Block diagram of the Manned Robotic System model

Information processing

Xo(k) - f(Xo(k-l) ,Uo(k-l) ,Wo (k- l ),k)

(2a)

Yo (k)

(2b)

Based on the perceived data yp up to time k (corresponding to data y up to k-i) and the known (learned, thus asswning that the HO is well-trained) dynamics of the system, the best (minimum variance) estimate ~ of the system state x can be made corresponding to time k - i. The resulting Kalman-Bucy filter equations are given by

and a time - varying linear system description

x(k) -

~x(k-l)

+

~u(k-l)

+ rw(k-l)

(3a) (3b)

y(k) - Hx(k)

with ~ - ~(k,k-l) being the Jacobian matrix of f with respect to X, etc.; w is assumed to be a zero mean, Gaussian, purely random seque n ce, with covariance matrix R, representing disturbances and other system uncertainty.

with

This linearization scheme holds for relatively small x, u and w. This fact dictates the update rate of the reference and, therefore, of the various Jacobian matrices.

and

The task considered is to contro l the system state x o f eq (3a) over some fixed interval of t ime [0 ,N}, so as to follow the desired state x d , by realizing a contro l sequence lu (k) , k - 0,1 , ... ,N-l) that minim izes t he p erformance index

(x(i) -xd (i)) 'Qx(i) ( x (i) -xd(i))

i -1

+ u ' (i- l )Qu(i -l )u(i -l )

~(k- i/k-i-l) - ~~(k-i -l ) + ~u(k- i -l )

(7)

K(k-i) - P(k- i)H'V·'(k- i)

(8)

n(k) - yp(k) - H~(k- i/(k-i-l).

( 9)

Th e best estimate of x at time k, ~(k), is obtained on the basis of ~(k-i) and the known system dynamics by means of an opti mal linear prediction process. The resulting predi ction equation becomes

N

I

(6)

Hence ~(k- i/(k- i- l ) indicates the est imate of x at time k - i based on the data y up to time k - i - 1, K represents the optimal trade-off be tween system uncertainty (in terms of the estimation error covariance P) and reliability of the data (in terms of the observation noise covariance V), and the innovation sequence n represents the new information.

3 . 3 Control task

IN(u) - El

~(k- i) - ~(k-i/k-i- l) + K(k-i)n(k)

(4)

~(k/k-i) - ~i~(k_i) +

i-l

I

2~o

where Qx and Qu are weighting mat ric es .

153

~i.' . 2~u(k_i+ £) (10)

which corresponds prediction.

to

eq(7)

for

the

one

u(k) - S(k)~(k) + Sm(k)z(k+l)

step

(14)

with Perception and estimation of the future desired state xd is described in a similar way. It is assumed that visual cues Yo can be observed that are related to the difference between the present state and the future desired state. Estimation of the future desired state is depending on the

S(k) -

assumed a priori knowledge that the HO may have about x d . Three cases may be considered: no prior knowledge of x d • only statistical knowledge of xd and imperfect knowledge of x d . The latter case results in the same filter equations as above. For the other cases the reader is referred to Ref. 1.

Sm(k)W(k+l)~

(lSa)

Sm(k) ~ -(w'W(k+l)w + Qu)-lW'

(lSb)

W(k)

(lSc)

~

Qx(k) +

~'W(k+l)[~

+ wS(k)]

and

z(k) - [~' + S'(k)w' ]z(k+l) - Qx(k)~d(k) (16)

with The HO's allocation of attention among all the system outputs y and Yo is described by a visual scanning model. The basic assumption is that the HO divides his attention among the visual cues such that the performance index of eq (4) is minimized (Ref. 1).

k - N-l • .... 0.

Thus. the control is composed of two parts: a feedback control operating on the state estimate and a feedforward (open loop) control operating on the estimate of the desired state Xd and computed recursively backwards in time.

Sequential decision making After the finite time interval for which the control task defined in subsection 3.3 has been performed. the decision has to be made about what to do next. This amounts to the binary decision as to whether the system behaves according to the small perturbation model of eq(3). corresponding to a given state reference. or a systematic discrepancy between both necessitates a correcting action of the HO and an update of the system model. In the first case the HO continues to control this system steady-state. In the second case. the HO initiates another maneuver to track the systematic deviation (xd ) over some fixed interval of time. after which the HO updates his system model. It might be necessary to update the sytem model more often. The extreme is to update the system model each time step. The reference is adjusted based on the estimated state (deviation from the old reference). This is known as the extended Kalman filter.

3.5 Constraints So far. the estimation and control problem of the nonlinear system is solved by linearizing the nonlinear

T

around

an

estimated

reference

Both types of constraints can be handled in a similar way. Conceptually. the procedure is (as discussed in Ref. 2) to "adjoin" the constraints to the performance index I N • given by eq(4). by a set of (socalled Lagrange) multipliers, which are chosen in such a way. that an optimal control is obtained (corresponding with an minimal I N ) given the constraints. Such optimization problems with constraints are conceptually straightforward. Numerical solutions. however, can require considerable computational cost. This can be aggrevated if the control problem is stochastic by nature. In that case the state has to be estimated requiring the solution of a nonlinear estimation problem (e.g. based on an extended Kalman filter).

°1

~

system

yielding a linear estimation and control problem. Control comprises permanent feedback control and intermittent open loop control based on sequential decision making. Several constraints may play a role in a given task. Control is, in principle. constrained because of limited control authority. hardware limits. etc. Generally these constraints have to be included a priori in formulating and solving the optimal control problem. Secondly. it may be necessary to consider (hard) constraints in the state space. Examples are the requirement to realize precisely a final destination of the system state (apart from stochastic effects). the limited space available to go from A to B (partly due to fixed obstacles). and limited space because of moving obstacles.

The comparison of system behavior as observed by the HO in terms of Yo and the expected behavior on the basis of the present system model is made by the HO in terms of the innovation sequence no' A systematic deviation of the zero mean sequence x due to a change in the desired state (with respect to the present state reference) results in a non-zero mean innovation sequence (Ref. 1). This can be tested by comparing (the log of) a generalized likelihood ratio with a threshold T according to

L(k)

(17)

(11)

with

(12) and (13)

no

For example. the solution of the optimal control task inclusive collision avoidance of moving obstacles requires: - estimation of own system state and the state of relevant obstacles; - definition of the constraints that have to be met in terms of the estimated states (e.g. the estimated distance to obstacles); computation of the optimal control while meeting the constraints. A more simple (engineering) approach to solve this and similar problems is given in Ref. 3.

where is a moving average of no No is the covariance of no and PM and P F are the decision error probabilities ("miss" and "false alarm'l). I

Human control behavior The control task discussed in subsection 3.3 is defined in terms of I N (u) of eq(4). Optimal human control corresponds to the minimal I N • The resulting optimal control sequence (u(k) . k - O.l •...• N-ll is derived in Ref. l. The result is given by

154

3.6 Comparison with control models

previous

optimal

of interest, e . g. the effect of time delays in teleoperations, the visual environment, the use of graphic-enhanced video images and additional cues or complete graphic substitution, system dynamics and HO related aspects. For the linearized model structure the model outputs are obtained in terms of means and variances of the system variables providing all statistical information of the process (assuming Gaussian disturbances). In case nonlinear system dynamics or non-Gaussian distributed disturbances are assumed, Monte - Carlo simulations can be performed resulting in time sequences of the system variables.

An important class of man-machine system models may be referred to as control theoretic models. These models are based on the modeling principles described previously. An important result of this approach is the so-called optimal control model (Ref. 4). The basic optimal control model structure has been extended in several directions (Ref. 1). Several of these extensions are combined in a model (PROCRU) for analyzing flight crew procedures in the approach to landing (Ref. 5). Instead of trying to review and evaluate the control theoretic modeling literature, it is indicated which aspects of the model of this paper are unique. First of all the model considers the general task to control a system from an initial state to a final state, following a given desired trajectory. Such a tracking task implies not only regulating control (to mlnlmlze deviations from a given reference) but also an open loop control to track the desired state. In the present model this desired state has to be estimated based on outside world cues possibly in combination with an internal model of the desired state.

Real time simulations can be carried out using the model to assist the HO in performing the task. For example, the model can provide an improved (graphic-enhanced) visual environment or additional visual information. In addition, the model dicision and control outputs can be used for a decision support system. In this case of real time operations, the model structure must be evaluated (and possibly simplified) to check the numerical requirements for real-time use.

Secondly the model deals with nonlinear, time-varying system dynamics by assuming a state reference and "small" perturbations around this reference. In case the reference is unknown it can be estimated by means of an extended Kalman filter. Both state and control constraints will generally be involved in robotic tasks. These are accounted for in the model by adjoining the constraints to the performance index resulting in an optimal control while meeting the constraints.

In this paper a model structure is presented of manned robotic systems in terms of optimal estimation-, control- and decision theory. The resulting model can be utilized to analyze, design and evaluate these systems by establishing the effect of task variables of interest on model outputs. In case the model is used in a time simulation mode the results are in terms of time histories (sequences) of interesting system- and HO- related variables. For the linearized model version, statistical measures can be obtained (i.e. ensemble mean values, covariances and probabilities) of all variables of interest. In the latter case the model provides a very cost-effective tool to assess the performance and reliability of manned robotic systems. It may be expected that the model is capable of answering questions related to design alternatives, function allocation, automation, system reliability and efficiency , etc.

4.

CONCLUDI NG REMARKS

I

Furthermore, the model assumes finite time intervals for which the control (sub)task is specified. After each interval the decision has to be made about what to do next. This sequential binary decision process is modelled in terms of a generalized likelihood ratio test. The result is intermittent open loop control (apart from the standard continuous feedback control). Finally, a model is included for visual scanning .

It provides an optimal strategy to allocate the human attention among all system outputs y (related to the state x) and Yo (related to the desired state x d ).

5.

REFERENCES

1. Wewerinke, P.H. Models of the human observer and controller of a dynamic system. Ph.D . thesis, University of Twente, the Netherlands, 1989.

3.7 Model applications The model presented in this paper can be applied to a number of manned robotic systems. Industrial applications tend to autonomous operations, such as part assembly, material transfer, spot welding and inspection, but for many operations the role of the human operator is (still) required (e.g. shared compliant control). An important area concerns teleoperations. Applications in space concern (e.g.) space stations, serviceable sattelites and material processing platforms. Also important are operations in a risky or unaccessible environment (deep water, dangerous radiation, etc . ) .

2. Bryson, A.E. and HO Y.C. control. Halsted Press, 1975.

Applied

optimal

3. Hoogland, M. Modeling vessel traffic (in Dutch). Thesis, University of Twente, the Netherlands,

1991. 4. Kleinman, D.L., Baron, S. and Levison, W.H. An optimal control model of human response . Part I: Theory and validation, Automatica, Vol. 6, 1970. 5. Baron, S. A control theoretic approach to modelling human supervisory control of dynamic systems. In: Advances in Man-Machine Systems Research (ed. Rouse), Vol. I, JAI Press,

For all these applications the model can be used to simulate the task (generally to control the system from state A to state B), the system dynamics and the environment. In addi tion HO functioning is included to adequately perform the task . For analytic purposes, tasks can be simulated fast time, i.e . including the HO model. This allows a straightforward investigation of all task variables

1989.

155