damped spring with a variable rest length which is under the direct control of ... of the spring represents the virtual trajectory (x0) and the position of the mass (x) .... system which does not exhibit an attractor trajectory, consider a hockey puck ... ice towards the goal â if a perturbation is applied to it (such as contact with a ...
Hodgson AJ and Hogan N (1999) A Model-Independent Definition of Attractor Behaviour Applicable to Interactive Tasks. IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Review, in press, February 2000
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A Model-Independent Definition of Attractor Behaviour Applicable to Interactive Tasks Antony J. Hodgson, Member, IEEE, and Neville Hogan
ABSTRACT Both in designing teleoperators or haptic interfaces and in fundamental biological motor control studies, it is important to characterize the motor commands and mechanical impedance responses of the operator (or subject). Although such a characterization is fundamentally impossible for isolated movements when these two aspects of motor behaviour have similar time scales (as is the case with humans), it is nonetheless possible, if we are dealing with repeated movements, to measure a trajectory which is analogous to the current source in Norton-equivalent electrical circuits.
We define the attractor trajectory to be this equivalent source and show that it
rigorously embodies the notion of the attractor point of a time-evolving system. We demonstrate that most previous attempts to test a controversial motor control hypothesis known as the equilibrium point or virtual trajectory hypothesis are based on inadequate models of the neuromuscular system and we propose here a model-independent means of testing the hypothesis based on a comparison of measurable attractor trajectories at different levels of the motor system. We present and demonstrate means of making such measurements experimentally and of assigning error bounds to the estimated trajectories.
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INTRODUCTION Experimenters in the fields of teleoperation, haptic interfaces and biological neuromotor control share a common problem: how to characterize the behaviour of a human subject performing an interactive task. In teleoperator design, there is a fundamental trade-off between stability and performance, but stability is determined in large part by the impedances of the systems (i.e., the human operator and the environment) with which the teleoperator interacts [1,2]. The designer of a teleoperator often models the human as a static impedance, but this is rarely an accurate model; more typically, the human uses the teleoperator to perform dynamic tasks and so is providing it with time-varying inputs. If designers are to ensure that the teleoperator remains stable in typical interactions, they must be able to account for the possible influences of the operator’s dynamic inputs, and to do this it is helpful to have a model for the operator’s behaviours. In motor control studies, such a model is at the heart of the experiment – the goal is to relate measurements of the motor behaviour of a subject (i.e., trajectories and interaction forces) to patterns of neural activity in order to test hypotheses related to the structure of the neural computational processes. If it is important, then, to characterize a subject’s motor performance, what aspects of their behaviour are relevant to these questions? Impedance1 estimation is an important element in highperformance teleoperators and Hannaford [2] notes that this is a very difficult problem because of numerical conditioning and noise.
1
He proposes using “world knowledge” to improve his
By impedance, we mean the relationship between a displacement imposed on a system and the force the system
produces in response. For a linear spring with the constitutive relation F = kx, the impedance is given by the ratio of the change in force, δF, to the displacement, δx, which caused it. This impedance is simply the stiffness, k.
3 estimates, but in the absence of such knowledge, one of the critical difficulties in identifying a system’s impedance is separating its impedance response from time-varying changes in its internal commands, especially when the characteristic time scales of these command changes are comparable to those of the impedance responses, as is generally the case with humans [3]; in principle, multiple-trial system identification methods are required to separate these two elements. In contrast, machine controllers are typically designed with high position gains, so that the impedance responses have very short characteristic times and single-trial system identification methods which assume quasistatic inputs are applicable. In motor control studies, a major focus over the past 25 years has been how the brain represents a motor task. One controversial hypothesis, the so-called equilibrium point or virtual trajectory hypothesis [4-8], claims that the central2 representation of a movement is primarily a sequence of position commands (possibly coupled with a representation of stiffness); this hypothesis further argues that the peripheral neuromuscular structure maps these commands into equilibrium points of attractor fields defined by the mechanical response of activated muscles. Two important advantages of such a control scheme would be a radical simplification of the motor computations (due to the absence of a need to explicitly represent arm or muscle dynamics) and an ability to exploit the intrinsic viscoelastic properties of the muscles to ensure stable interactions with the environment. However, this scheme relies on the attractor field being sufficiently strong to dominate other forces affecting the movement (such as the dynamic inertial forces). Several recent studies have brought this claim into question by showing that the linearized stiffness about an actual arm trajectory is often relatively low [9-14]. However, although such studies are
2
Central in the sense of being related to the central nervous system (i.e., the brain).
4 important in examining the virtual trajectory hypothesis, the fundamental motor control question described earlier concerns the trajectory followed by the attractor point rather than the impedance itself; the studies mentioned above require impedance measurements only as a means to an end – used in conjunction with a model of the muscles, they lead to an estimate of the location of the attractor point. Since the attractor point itself is the crucial element in the hypothesis, we would like to measure it directly. Interestingly, recent simulations by Gribble and Ostry have shown that low stiffnesses during execution can be reconciled with the virtual trajectory hypothesis [15]. The purpose of this paper, therefore, is to formalize our notion of what the attractor of a physical system is and to present an experimental technique by which the attractor can be measured. We will show that existing estimates of the location of the attractor are based on models of the neuromuscular system which are, of necessity, oversimplified; the conclusions of studies which use such models are therefore vulnerable to criticism that the model used is inadequate. We will put forward a definition of attractor behaviour which, instead of relying on an imperfect model, will depend only on variables such as force and position which can be measured at the point where a subject interacts with the environment. We will explain under what circumstances the attractor can be defined and measured, and will present an experimental methodology for making such measurements. Finally, we will use this measurement technique to find an attractor trajectory in a task which could be used to test the virtual trajectory hypothesis. The primary significance to motor control studies of the attractor trajectory concept developed in this paper is that it provides us with a fundamentally different approach to testing this controversial hypothesis.
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DEFINITION OF ATTRACTOR BEHAVIOUR Definition of Virtual Trajectory Since it is important both in teleoperator design and in motor control studies to be able to identify the efforts and impedances exhibited by a human subject or operator, we must carefully define these terms. In the context of the motor control theories discussed earlier, the mechanical response of the subject’s hand during a planar arm movement can be envisioned as a force field F = F ( t , x ( t ), x ( t ), x( t ),..., p(t )) determined by the histories of the central commands (p(t)) and the hand’s trajectory (x(t)). In the quasistatic case, if the central commands were held constant for a time and we slowly moved the subject’s hand to different positions, we would map out a position-dependent force field. As long as we were sure that the central commands remained constant, we could identify the stiffness with this force field and the attractor point with the equilibrium at the bottom of the force bowl (assuming that such a minimum exists). When the central commands are changing and the arm moves, however, the analysis becomes more complicated. Since the brain can only directly control the activation of certain efferent 3 nerves rather than the force field itself, the resulting movement is affected by the dynamics of local reflex loops, muscle activation and inertial and environmental loadings. If, as the virtual trajectory hypothesis asserts, the efferent signals encode an equilibrium position for the limb, then there will
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Efferent (descending) nerve signals flow from the central nervous system towards the periphery, while afferent
(ascending) nerve signals flow towards the brain.
6 in general be a discrepancy between this equilbrium position and the actual (or current) limb position. Feldman has proposed that a good model for the dynamics of muscle is a nonlinear damped spring with a variable rest length which is under the direct control of descending neural signals [4,8,16]. According to Feldman’s model, we can picture arm movements as arising from the movement of an imaginary (or virtual) massless arm linked to the real arm by nonlinear springs. Any discrepancy between the positions of the imaginary and real arms would elicit muscle activity both through the intrinsic length-tension properties of the muscle and through the effect of local feedback circuits; this activity would act to draw the real arm towards the imaginary arm and the actual trajectory which results would depend on the interaction between the muscle activity, the arm’s inertia, and the dynamics of the environment. The position of the imaginary arm is called the virtual position [17], and the time-evolution of the virtual position is known as the virtual trajectory.
Model-Dependence of Virtual Trajectory Proponents of Feldman-style models have argued that the system’s attractor is the virtual position and that the impedance is the force field about that virtual position. In mathematical terms, the motion of the arm would be given by H ( q)q + C( q, q )q = Fm (t , q − q0 , q − q0 ) + Fe (⋅)
Eq. 1.
where q is the vector of generalized coordinates describing the arm’s position, H is a positiondependent inertia matrix, C is a vector of dynamic effects (e.g., coriolis and centripetal effects), Fm is the force due to the neuromuscular system, and Fe is the force from interactions with the environment. The trajectory q0 represents the virtual trajectory which is viewed as being under
7 the control of the central nervous system.
In order to find the virtual trajectory, we must
therefore formulate a model for Fm, estimate the impedance parameters, and invert the model to locate the virtual trajectory. Figure 1
Unfortunately, however, the “virtual trajectory” we find through this inversion process is critically
near here
dependent on the model we have of the system and need not correspond to any signals which exist in the actual system. Consider, for example, a pair of one-dimensional mechanical systems which can both be represented by the form of Equation 1 (see Figure 1). In these models, the free end of the spring represents the virtual trajectory (x0) and the position of the mass (x) represents the position of the arm. The first model is the one which has traditionally been used in interpreting experimental results related to tests of the virtual trajectory hypothesis [12,14,18-20], while McIntyre & Bizzi [21] have advanced the second (which changes the damping term from groundreferenced to virtual trajectory-referenced) in an attempt to improve the predictive power of the model. It is straightforward to show that these two versions of the model are functionally equivalent, at least in the case where the stiffnesses and dampings are linear. If we were to write out the equations of motion for these two systems, we would have: Mx + bg x + k1 ( x − x 01 ) = Fe Mx + br ( x − x 02 ) + k 2 ( x − x 02 ) = Fe Expanding the terms in parentheses and gathering all the common terms and terms dependent on the actual trajectory on the left, we have: Mx + bg x + k1 x − Fe = k1 x 01 Mx + br x + k 2 x − Fe = k2 x 02 + br x 02
8 The trajectories at the interaction ports4 with the environment will be the same for these two systems if bg = br, if k1 = k2, and if the righthand sides of these equations are equivalent; i.e., if k1 x 01 = k 2 x 02 + br x 02 . In other words, for arbitrary manipulations (Fe) at the interaction port, these two systems will respond with precisely the same port trajectories if their differing virtual trajectories are related by the differential equation given above;
no experiment limited to
perturbing and measuring at the interaction port will allow us to distinguish between the two systems. Since we cannot uniquely define the system’s virtual trajectory, it cannot be considered a system property. If we wish to characterize a system’s attractor behaviour as a measurable property, we must base our definition on interaction port behaviour alone, as described in the next section. In particular, we will formally show that the hand trajectory which causes the muscles to exert no net effort during the limb’s motion5 is the property that the virtual trajectory concept was designed to represent.
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An interaction port is an interface between two physical systems across which energy flows during their
interaction. In the context of this paper and the motor control studies it references, we would consider the interface between the subject’s hand and the perturbing device (generally a robotic manipulandum) to be the interaction port; the energy exchanged is the product of force and velocity. Since we consider motion in a plane, which has two dimensions, we call this a 2-port system. 5
We obtain this trajectory experimentally by applying external forces to the arm which balance the inertial loads
during the movement, leaving the muscles with no net work to perform.
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The Attractor Trajectory The concept of the virtual trajectory was originally developed to capture the notion that a system with attractor properties is tending towards some point at each instant. That point was variously called the equilibrium or virtual position and was claimed to be a controllable parameter [8,22]. However, since the mapping from experimental measurements to virtual trajectories is not unique, but depends on our model of the peripheral neuromuscular system, we cannot convincingly claim that any particular virtual trajectory we compute correctly represents the meaning of real neural signals. Any such argument as to the meaning of neural representations must be rooted in an unambiguous definition of attractor behaviour, which in turn must be derived without reference to a system’s internal structure from measurements made at its interaction ports. We therefore consider how we might define “attractor behaviour” in terms of interaction port behaviour. We begin with the intuition that a stable n-port system at rest and isolated (i.e., with no efforts imposed at its interaction ports) will exhibit attractor behaviour about its vector of (static) port positions.6 If the interaction port position subsequently begins to move, it is then logical to identify the attractor trajectory with the port trajectory as it evolves in the absence of interaction forces (it is important to note that the attractor trajectory so defined is not the virtual trajectory because the attractor trajectory’s definition is model-independent).
More rigorously, and to
include the requirement of stability in our definition, we propose the following as the formal definition of an attractor trajectory:
10 Definition 1. Let S be a time-varying state-determined n-port system whose interaction port behaviour can be characterized by the vector pair {x(t),F(t)}, which represents the generalized positions and forces at each interaction port. Definition 2. A system S is said to be in a state of internal equilibrium if all internal effort and flow sources are constant, all internal states are unchanging with time and all interaction port forces and velocities are zero. Definition 3. Let S be in internal equilibrium at t = ti.
G is the set of all interaction port
trajectories, x(t), t ≥ ti, for which F(t) is identically zero for t ≥ ti. Definition 4. A trajectory xG(t) in G is said to be uniformly stable if for any to>ti and for any r > 0 independent of to, there exists a positive scalar R(r) such that ||x(to)-xG(to)|| < R and F(t) = 0, ∀ t ≥ to => ||x(t)-xG(t)|| < r, ∀ t ≥ to. Definition 5. A trajectory xG(t) in G is said to be uniformly asymptotically stable if it is uniformly stable, and if there exists a positive scalar R independent of to such that ||x(to)-xG(to)|| < R and F(t) = 0, ∀ t ≥ to => ||x(t)-xG(t)|| → 0 as t → ∞ and, for any R1, R2 satisfying 0 < R2 < R1 < R, there exists T(R1,R2) > 0 such that ||x(to)-xG(to)|| < R1 and F(t) = 0, ∀ t ≥ to => ||x(t)-xG(t)|| < R2, ∀ t ≥ to + T(R1,R2).
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It is this vector which has erroneously been called the virtual position (“erroneously” because it presumes a
model whose input equals its output when at rest; one could propose an infinite number of physical systems, indistinguishable in terms of port behaviour, which do not satisfy this requirement).
11 Definition 6. The set A is the set of all uniformly asymptotically stable trajectories in G. If A is the null set, no attractor trajectory is said to exist; if A is not null, any trajectory in A is said to be an attractor trajectory. In effect, the attractor trajectory is defined to be any uniformly asymptotically stable trajectory of the interaction ports of the system which evolves from an equilibrium point in the absence of interaction forces. The attractor trajectory of a nonlinear system is therefore analogous to the current source in the Norton equivalent representation of a linear system. As an example of a system which does not exhibit an attractor trajectory, consider a hockey puck travelling across the ice towards the goal – if a perturbation is applied to it (such as contact with a defender’s stick), its trajectory will be irrevocably changed; without additional intervention, the puck will not return to its original course. Interestingly, it is impossible to determine during a single movement trial whether or not a given system exhibits an attractor trajectory because the definition relies on a comparison of an unperturbed and a perturbed trajectory measured simultaneously (see Definitions 4 & 5 above). Since this is impossible, no interaction port measurement, taken on its own, can provide any information relevant to the question of stability. We can use a multiple-trial experimental protocol to decide this question if we are willing to make the ergodic assumption, which states that observations made on a single system during different time intervals can be compared in the same manner in which we would compare observations of several identical systems made during a single time interval. This assumption is reasonable if we are willing to concede that we can bring the system back to a state which it was in previously and have all time-varying parameters of the
12 system follow the same time-course during each trial.
All ensemble-based impedance
identification techniques make the ergodic assumption [9,23].
EXPERIMENTAL IDENTIFICATION OF ATTRACTOR TRAJECTORIES Problem Formulation In the previous section, we described what an attractor trajectory is. For an arbitrary system, if we are justified in making the ergodic assumption and if we have direct access to the interaction port of interest, then we can both test for the existence of an attractor trajectory and measure it directly.
For the kinds of motor control studies described earlier, however, there are two
interaction ports of interest to us: (1) at the hand itself (we will refer to this as the postinertial port) and (2) at the output of the muscles (the preinertial port). Since we cannot directly measure the forces generated by the muscles (especially in humans), this latter port is hidden behind the inertial response of the limb. In the following section, we describe how we cancel the inertial dynamics of the arm in order to measure the preinertial attractor trajectories in an experiment in which subjects use their arm to interact with a planar manipulandum; we also discuss the various assumptions and parameter computations which are required. In brief, we apply forces to the arm during movement which cancel the forces normally applied by the muscles; in doing so, the interaction forces at the muscle port are eliminated and the resulting trajectory can be interpreted as an attractor trajectory. This preinertial attractor trajectory is an unambigous and measurable representation of the motor plan and, while conceptually different
13 from the virtual trajectory defined by Feldman [8], Hogan [24] and McIntyre [21], can be used in a similar manner to assess motor control hypotheses. To develop our technique for measuring the preinertial attractor trajectory, we begin by considering a model of our subjects which allows us to represent interactions between them and their environment through a mechanical interaction port at which we can measure force and velocity (as well as position and acceleration). In this model, all of the internal details are hidden – subjects are capable of initiating and executing motor actions, but all we are able to observe is Figure 2 near here
the final result of those internal processes as they are made manifest at the interaction port. We do know, however, that if the subject is sufficiently constrained (i.e., if their wrist is immobilized, shoulder position fixed, and arm confined to a plane), then the position of their arm can be inferred directly from a measurement of their hand position; in other words, if the number of degrees of freedom we allow the subject is equivalent to the number of coordinates needed to express the position of the interaction port, then the subject can be characterized as kinematically non-redundant.
This assumption is expressed in the bond graph in Figure 2 by the direct
connection of the inertia of the subject’s arm to the interaction port (1-junction); as discussed in the previous section, the mechanical behaviour of the remaining components of the arm (neuromuscular system, joint dynamics, etc.) is represented by a Norton-equivalent subsystem (see dotted box) consisting of a flow source term, vo(t), connected through an impedance, Zo(t), to the interaction port.
By assumption, this subsystem has attractor properties, so we can in
principle determine its attractor trajectory; this will be the path the interaction port will follow if Fz(t) = 0, ∀t. That is, if we find an interaction port trajectory for which Fz(t) = 0, then vi(t) = vo(t), the preinertial attractor trajectory we wish to measure.
14 To compute the impedance force, Fz. we can use the simple equation Fi = Fe + Fz if we know the net force (Fi) applied to the inertial linkage; to estimate Fi, we multiply the estimated inertia of the subject by the accelerations we measure. Note that since Fe is largely under the experimenter’s control, we can influence the achieved trajectories, vi(t), by applying different force histories or supplying different environments for the subject to interact with. Our goal is to adjust the interaction forces until the achieved trajectory is equal to the attractor trajectory. It is also important to note that the criterion for assessing whether or not a given trajectory is the attractor trajectory is applied after an experimental run – i.e., it is a post-test assessment: if, after applying a history of interaction forces, Fe(t), we found that Fz(t) = 0, ∀t, we would conclude that vo(t) = vi(t); i.e., that the path we had just measured was the attractor trajectory.
To be
absolutely sure that we have found a subject’s attractor trajectory, we would have to cause their hand to actually track it.
However, since the commands the subjects issue vary with each
execution of the task, we are not likely to have applied the correct Fe to make Fz = 0 for any particular run; the best we can hope for is a series of runs in which Fz ≈ 0 (relative to the estimated inertial forces, Fi). We must therefore consider how to derive an estimate for the virtual trajectory from a collection of executed trajectories, vi, and estimated impedance forces, Fz.
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Attractor Trajectory Estimation Procedure If we do indeed have a series of runs in which Fz ≈ 0, then we could quite reasonably take as our estimate of the attractor trajectory the mean of the collection of the executed trajectories. However, since the attractor trajectory is not known prior to the experiment, it is not likely that our first series of runs will yield Fz ≈ 0, so we must adopt an iterative procedure in which the data we obtain from a given series of runs allows us to refine our estimate of the attractor trajectory and adjust the forces applied to the subjects so as to take them closer to their attractor trajectories in subsequent experimental runs. This necessarily involves extrapolating from (or, if we have been fortunate enough to have bracketed the attractor trajectory, interpolating between) the measured trajectories. In principle, we could perform this extrapolation by casting the problem as a function mapping problem and solving it using, for example, a neural network. From a function mapping point of view, we are trying to find the trajectory, vo, which maps to Fz(t) = 0. One could envision training a neural network on the collection of (vi, Fz) and then taking as our estimate of the attractor trajectory the output of the network when Fz = 0 is supplied as the input. If the average Fz were approximately zero throughout the movement, we would expect the output of the network to be a good estimate of the attractor trajectory. However, in those regions of the training set where none of the Fz are near zero, the neural network is projecting rather than interpolating, and the network's estimate is likely to be incorrect because of the strong nonlinearities in the limb impedance. Nonetheless, if the correction is in the right direction and the step is not too large, the set of impedance forces after the next round of experiments will be closer to zero and we can repeat the procedure until we achieve convergence.
16 Figure 3
Since the convergence tests are applied directly to the experimental measurements, it is immaterial
near here
how we choose to generate the perturbations which produce the measurements, so long as the corrections ultimately result in convergence. We have found it helpful to use a linearized model of the impedance based on postural measurements to generate the corrections; this generally results in acceptably rapid convergence of the attractor trajectory estimates. The experimental procedure is outlined in the flowchart shown in Figure 3.
Using Linear Model for Convergence As discussed above, we can assess proximity to the attractor trajectory independently of the means by which we generate the intermediate estimates of the attractor trajectory in the process of convergence. We can therefore use any model of the subject’s arm we like to generate a new attractor trajectory estimate without jeopardizing our claim that the final estimate is independent of any modelling assumptions. Since there is a reasonable body of data on the linearized stiffness and damping properties of the arm at rest in configurations similar to those used in our experiments [7,25,26], we use a linearized model of the arm’s impedance to generate the new estimates of the attractor trajectories. Figure 4
Figure 4 is a schematic of a two-link arm which shows the generalized coordinates used in the
near here
equations of motion. Note that this schematic allows for the possibility that the center of mass of the forearm does not lie on a direct line between the elbow and the endpoint; although this may be strictly true for arbitrary kinematic linkages, for the human arm, the center of mass is very near the limb’s axis, so we will make this approximation (i.e., α = 0). The equations of motion for this arm model (inertial effects only) can be written as
17 H ( q )q + C( q, q ) = τ . If we model the net effect of the neuromuscular and other non-inertial arm dynamics as shown above in Figure 4, and if we linearize the impedance, Zo, and assume that it is time-invariant, then the net torque applied to the inertial linkage is given by
τ = τ e + τ z = τ e + Zo (q o − q, qo − q ) = τ e + B( qo − q ) + K ( qo − q ) , where the subscript on the position and velocity terms refers to the attractor trajectory, the absence of subscript implies that they are the actual endpoint positions and velocities, and B and K are 2x2 matrix approximations of the subject’s impedance expressed in angular coordinates. Given knowledge of the inertial parameters in the matrices H and C, and having measured τe, we can compute τz for any given trial. Assuming that the subject starts at rest and has been at rest for long enough that the attractor trajectory and the actual hand position are identical, then both
δq(0) = 0 and δq ( 0) = 0 . We can therefore write a differential equation for the deviation between the attractor and the actual trajectories as Bδq + Kδq = τ z = τ − τ e = H (q )q + C( q, q ) − τ e . Since all the terms on the right hand side of this equation are known functions of time, it is straightforward to integrate it, and the result is an estimate of the attractor trajectory based on a linear projection from the trajectory executed during a single movement. The accuracy of the new estimate will clearly depend on several factors, most notably the accuracy and applicability of the linearized impedance matrices, although if the impedance torques are zero throughout the movement, then the deviation between the attractor and actual trajectories will also be zero and the estimate will be independent of both the values of B and K and of the assumption of linearity.
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Error Estimates Sources of Error Figure 5
On a single trial, there are three major sources of error in the estimate of the attractor trajectory;
near here
these can be understood most readily in reference to Figure 5, which shows the force-length relationship of a nonlinear spring. The attractor point for this simple system is the length at which the force becomes zero, whereas in the two-dimensional case the attractor point is the combination of positions and velocities at which the two-dimensional forces become zero noninstantaneously.7 The first source of error is the degree to which the linear model is a poor approximation to the actual impedance behaviour. Although we know that in general the behaviour of muscles is highly nonlinear, a number of studies have nevertheless had reasonable success in accounting for the response to small or moderate perturbations using a linear model; often, the variance in the force records accounted for by the linear model exceeds 70% [9]. Since we will be concerned about error bounds on our attractor trajectory estimate only at the final stage when we have achieved convergence, we will be projecting over relatively short distances where the linearized approximation has been shown to be reasonable, so that the errors associated with the nonlinearities will be negligible.
7
At any instant it is possible for the impedance force to become zero without actually being on the attractor
trajectory. Consider the linear case in which Fz = B(v-vo) + K(x-xo). It is clear that for any (xo,vo,x), if Fz = 0, there will be a value of v for which this equation is satisfied. Only along the attractor trajectory, however, will this equation be satisfied at all times.
19 A second major source of error is in computing the inertial torques; these torque estimates are affected in part by errors in estimating accelerations and velocities, but more significantly by errors in the inertial parameter estimates themselves. In the context of Figure 5, these errors manifest themselves as errors in the force measurements, which has the effect of shifting all the measured points up or down. We will shortly describe how we estimate and ascribe error bounds to the inertial parameters. The final major source of error is inaccuracy in the linearized impedance estimates. An overestimate of stiffness, for example, will clearly lead to an underestimate of the deviation between the attractor and achieved trajectories. Again, in the context of Figure 5, errors in the impedance parameters correspond to errors in the estimate of the slope of the impedance relation in the vicinity of the data points. If we can bound the impedance parameters, however, we will be able to compute bounds on the location of the function’s zero (and, by analogy, on the location of the attractor trajectory) – see, for example, the region of the X axis between the two lines projecting from the top right data cluster in Figure 5. We now outline this procedure. Assigning Error Bounds Consider again the equation we are integrating: Bδq + Kδq = τ z . Assume for the moment that the problem is one-dimensional (i.e., B, K, τz and the position and velocity elements are all scalars). For a given combination of B, K, and τz, we can write an expression for δq(t) using the convolution operator t
δq( t ) = ∫ G ( t − T )τ z (T )dT , 0
20 where G(t) is the impulse response function of the system whose Laplace transform is G( s) =
1 . Bs + K
If we know that B and K are bounded above and below their nominal values, then G(t) will have a maximum and minimum value for each time, t, which we can compute knowing these bounds on the impedance parameters. If we define a new impulse response operator, G ( t ), which can take on any value in the range associated with the given bounds on B and K, and if we define a new impedance torque function, τ z ( t ), which can take on any value in the range given by the bounds on τz(t), then we can ascribe a minimum and maximum value for δq(t) by performing the convolution shown above, choosing at each instant T in the integration interval [0,t] the combination of values of G ( t ) and τ z ( t ) which minimize or maximize that instant’s contribution to δq(t). This will yield a conservative bound on δq(t). The above analysis is easily generalized to two dimensions. In 2-D, τz and δq are 2-vectors and B, K, and G(s) are 2x2 matrices. To compute the bounds on δq, we simply compute the maximum or minimum dot product of each row vector in G(s) with the impedance torque vector.8 We
8
To avoid assigning excessively large (conservative) error bounds, we do make the assumption that each element
of G(s) is adjusted in the same direction. For example, if we decide to set K(1,1) to its maximum value, then we also set K(1,2) to its maximum; we do not allow them to vary independently. This assumption is based on the observation of Mussa-Ivaldi that the stiffness ellipses are essentially invariant in shape, even under voluntary modulation of overall stiffness levels. Recently, Gomi and Osu showed that different tasks can elicit changes in the shape and orientation of impedance ellipses; since our tasks do not have strong directional differences, we expect this effect to be modest [45].
21 compute the bounds in joint coordinates because the impedance matrices are relatively constant in that frame when compared to a cartesian frame [7], and then convert the bounds to cartesian space by computing the coordinates of the corners of the rectangle defined in joint space by the various combinations of δqmin and δqmax. Combining Multiple Trials The procedure outlined above generates a bounded estimate of the attractor trajectory for a single movement, but it does not describe how we might combine multiple experimental trials. Since subjects do not perform tasks the same way each time (i.e., they may start earlier or later, take a longer or shorter time to make the movement, start and stop in slightly different positions, or take a somewhat different path to the target), we must combine a series of bounded attractor trajectory estimates into a bounded estimate of the mean attractor trajectory. This process is complicated by the fact that we are concerned primarily with the average shape of the trajectories (a spatial measure), rather than their time-averaged location. In order to avoid washing out spatial features, therefore, we must normalize trajectories in both position and time so as to preserve as much of their shapes as possible.9 Since each trajectory begins and ends at a well-defined location, we
9
Note that if we do not normalize in time, then the averaging process distorts the shape of a trajectory. Consider
averaging two ramps of exactly the same duration, but starting at slightly different times. The average will start rising with half the slope of the ramps when the first ramp rises, will switch to rising at the same slope as both ramps once the second ramp begins rising, and will finish with a third segment, again of half the ramp slope, which ends when the second ramp ends. Non-normalized averaging therefore creates a time-stretched, threeregion “average” trajectory out of two perfectly well-defined single-region ramps. Similarily, failing to normalize in position can wash out smaller features of the movement.
22 linearly remap each of them so that they begin at (0,0) and end at (1,0). To normalize them in time, we compute the optimal match between each trajectory and some reference (analytical) function whose transition time is easily characterized (such as a ramp or minimum-jerk transition, for example); each trajectory is then normalized in time by dividing through by the transition time of the optimally-matched reference function. Once the trajectories have been normalized, we compute a weighted-average attractor trajectory by combining all the estimates at each sampling instant in proportion to the inverse of the area enclosed by the error bounds – the smaller the error bound, the greater the weight given to that estimate. This averaging technique is similar to the generalized Procruste’s method [27] for averaging spatial hand paths, except that our technique also accounts for temporal scaling and variable measurement error associated with each trajectory included in the estimate. To compute an overall error bound on a mean attractor trajectory estimate, we account for both the variability inherent in repeated executions of a trajectory as well as the uncertainty due to inaccurate estimates of the inertial and impedance parameters. If we can characterize the error bound associated with parametric uncertainty at each instant of trial i by a characteristic length, Li(t), then the overall error bound can be given by the following expression:
σ (t ) σ (t ) = + N
∑ Li ( t )wi ( t ) i
∑ wi (t )
,
i
where σ(t) is the estimated variability at each instant, N is the number of trials used to compute the average attractor trajectory, and the weighting factor for each trial, wi ( t ) = 1 / L2i ( t ) , is in inverse proportion to the area of the parametric uncertainty error bounds. This weighting scheme makes the parametric uncertainty component of the averaged error bounds no smaller than the
23 smallest bound and no larger than the largest, which is the desired effect, since no increase in the number of trials can reduce this uncertainty. Basis for Selecting Linearized Impedance Parameters In order to implement the scheme outlined above to update the attractor trajectory estimates and to assign error bounds, we have to specify the linearized impedance parameters. There are a few direct studies of the impedance of an arm moving in a plane [10,14,23], and of single-joint motion [3,9,28]. Although these studies of single-joint motion show a fairly wide range of stiffnesses, they have generally been performed in an experimental setting in which the subject is able to relax almost completely. In contrast, the studies described here require the subject to support the weight of their own arm and to maintain an active grip on the manipulandum, as would be the case if the subject were controlling a teleoperator or using certain tools. Under these conditions, the range of achievable stiffnesses is significantly diminished. For example, Hodgson [29] found that in a task which required the subject to resist a force applied perpendicular to an outstretched arm, the total stiffness exhibited by the subject varied by a factor of only two or three from minimum to maximum.
Gomi’s study of stiffnesses during movement [10] reveals that the
minimum stiffness is of the same order as the postural stiffness, while Bennett [9] claims a reduction during motion to a minimum of approximately 50% of the postural levels. Mussa-Ivaldi et al. [7] measured the static stiffness matrices of a number of subjects in the same configuration we used in this study; they found striking regularities in the shape and orientation of the stiffness ellipses across the workspace. Dolan, in a series of postural measurements [25,30], confirmed these results regarding shape and orientation, and found that the majority of his subjects exhibited
24 stiffness magnitudes which differed by less than a factor of two from one another, so for the most part, we can expect our subjects to exhibit relatively similar postural stiffnesses. For the purposes of updating the attractor trajectory estimates, therefore, and to avoid overshooting the attractor trajectory, we selected a nominal stiffness derived from a postural experiment we performed on one of our subjects: . − 251 . Nm 453 . K= . 43.7 rad − 251 This value is somewhat greater than that of Mussa-Ivaldi’s Subject A and is at the high end of the group of subjects in Dolan’s study. For the damping matrix, we selected a value equivalent to this stiffness matrix multiplied by a factor of 0.07 s, which is a typical ratio determined by Dolan in his postural measurement tests.10 For the purposes of assessing the error bounds on the attractor trajectory estimates, we selected minimum and maximum impedance matrices equal to the nominal matrices plus and minus 50%; this range spans the normal range of variability between subjects.
Estimating Inertial Parameters The technique of measuring preinertial attractor trajectories described above relies on estimates of the inertial parameters of the human subject.
Dolan has performed perhaps the most
comprehensive set of inertial measurements to date [30]; in his approach, he fitted a linear impedance model to the response to a sequence of small (
