What is Planning? Engineering Foundation Conference 1996

What do we plan or control

when we perform a voluntary movement?

Gerald L. Gottlieb (617) 353-8984

NeuroMuscular Research Center 353-9757

Boston University fax 353-5737

44 Cummington St. glg@bu.edu

Boston MA 02215

Introduction

The quantitative analysis of multiple degree of freedom movements is a relatively recent practice in motor control. In the early ‘80s, Morasso, Lacquaniti and Soechting published studies of arm reaching that identified certain distinctive kinematic characteristics (Lacquaniti, et al., 1982; Morasso, 1981; Soechting and Lacquaniti, 1981). Morasso noted [p. 224] that "the common features among the different reaching movements are the single-peaked shape of the hand tangential velocity and the [straight] shape of the hand trajectory." Soechting and Lacquaniti further noted that these properties were unaffected by changes in the load held in the hand or by the intended speed of movement. These properties of straightness and "bell-shaped" velocity profiles have become defining features of unconstrained human reaching movements, even though Hollerbach (1982) noted that movements in the sagittal plane tended to be more curved than those in the horizontal plane. A model which captures many of these kinematic features in a parsimonious way, the minimum jerk trajectory, was proposed by Hogan (1984). This is widely used, although it is important to appreciate that it is a description of the movement trajectory and cannot be the exclusive basis for planning the trajectory.

We argue here for an alternative to kinematic planning. Voluntary movement is accomplished by the execution of motor programs for planned forces and corresponding EMG patterns in the muscles. However, rather than directly solving inverse dynamic equations, the CNS uses relatively simple coordination rules among muscles and joints that greatly simplify the problem of finding muscle activation patterns to satisfactorily approximate our kinematic goals. The well known kinematic features of movements result from a trial and error tuning of force profiles based upon visual and kinesthetic feedback. In this chapter we will present illustrative data for this hypothesis. Some data have been presented at greater length in (Gottlieb, et al., 1996a; Gottlieb, et al., 1996b; Gottlieb, et al., 1996c).

Methods and Results

Our studies have been of undisturbed movements of the arm in a sagittal plane. In one experiment, twelve targets 20 cm from center were positioned at the hours of a clock. Subjects were instructed be both fast and accurate (Gottlieb, et al., 1996c). The positions of infrared light emitting diodes taped over joint centers at the shoulder, elbow, wrist and on the finger tip were recorded using an OPTOTRAK 3010 (Northern Digital) system. The sampling rate was 200 per second.

Ten movements were made to each target. Trajectories were aligned on 5% of their peak tangential velocity and averaged. The joint torques were computed by inverse dynamical equations, given in eq. 1 in simplified form. The angles of the upper arm q_s and forearm q_e are defined relative to vertical. The interior angle of the elbow joint is given by . The net muscle torques at the elbow and shoulder are given by te and tm. Subscripts l and u refer to lower and upper arm segments. The mass and length of the segments are defined by m and l. r is the location of the segment center of mass. g is the acceleration of gravity.

t_e = eq 1

t_m = eq 2

Figure 1 illustrates movements in four of the twelve directions. All are fairly straight with bell shaped velocity profiles. The dynamic joint torques are all biphasic pulses. They are usually relatively symmetrical, deviating most from symmetry when the pulses are small as they are for the elbow torque at 2 and 9 o’clock and shoulder torque at 5 o'clock. These characteristics of the dynamic torque patterns, independent of the load or speed of the hand (Gottlieb, et al., 1996b), were noted by Soechting and Hollerbach (Hollerbach, 1982; Soechting and Lacquaniti, 1981) at about the same time that the distinctive kinematic properties of the movements were first described. They have received far less comment but are no less important than kinematic straightness and smoothness.

Figure 1. The figure illustrates four movements out of a series of 12, performed in different directions. The top two panels show that the kinematics are typically straight (a stick figure of the arm at the initial and final positions of the 1’oclock movement are drawn on the left) with bell-shaped tangential velocity profiles. The middle panels show that the joint angles change uniformly with the distance moved and that very different directions of movement may occur with similar motions of one or the other joint. The bottom left panel shows that the amplitude normalized dynamic muscle torques are all very similar in their temporal features. Heavy lines are shoulder, thin lines are elbow. On the right, the plot of one joint torque vs the other results in narrow profiles with long axiis that rotate with the direction of movement.

It is clear from equation 1 that the net muscle torque at one joint cannot be determined solely from the motion at that joint. Conversely, the motion at a joint cannot be determined exclusively by the torque at that joint. For example, elbow torques at 2 and 9 o'clock are small and almost the same while the elbow excursions are near maximal but in opposite directions. The relationship between muscle torque and direction of motion is less complex however, than it might seem thus far.

It is not sufficient to describe the behavior of individual joints. We must also describe the relationship between them. We have proposed that the relationship between joint torques can be approximated by a simple linear equation.

t_s = K_d t_e eq. 2

We call this rule linear synergy. Figure 1 also shows the dynamic elbow and shoulder torques for the four directions of movement, all scaled so that their first peaks have magnitudes of +1. Torques at both joints are biphasic, highly synchronized pulses that have nearly simultaneous peaks and zero crossings. Also plotted are the dynamic shoulder torques, plotted versus dynamic elbow torques. These form tight ellipses or figure eights rather than straight lines because of imperfect synchrony. The correlation coefficients between the two joints are greater than 0.92 for all four directions. These figures imply that the direction of movement can be controlled by the relative sizes of the torques at the two joints while preserving their individual and nearly identical temporal patterns. Preserving the relative sizes but changing the magnitude and timing properties of the biphasic pulses alters movement speed or distance or accommodates changes in the inertial load at the hand.

Figure 2. Two individual movements to the same target are illustrated. In the second movement (dashed lines), the hand did not remain at the target but promptly returned to its original position. Both movements are relatively straight but the latter has a double-peaked velocity profile. In this latter movement, there are reversals in the angular trajectories of both joints and the muscle torques have a triphasic rather than biphasic pattern. Heavy lines are shoulder, thin lines are elbow. Linear synergy between joint torques is preserved however.

The trajectories of these four movements are extremely simple both in external (Cartesian) and internal (joint angle) frames of reference. It might be asked whether the simple torque relationship of equation 2 is only possible with equally simple kinematics. To address this we have looked at a variety of other movements (and see Figures 1 and 2 of (Almeida, et al., 1995) and Figure 2B of (Gottlieb, et al., 1996b)). Figures 2 and 3 show pairs of individual movements. In Figure 2, one is a reach out to a target similar to the 1 o’clock movement in Figure 1. The other is a reversal movement in which the subject reached out to the same point and immediately returned to the initial position, similar to the movements studied in (Sainburg, et al., 1995). In the latter we a have triphasic torque pattern at the elbow. Never-the-less, there is linear synergy between elbow and shoulder torques (and wrist torques which are not illustrated).

In Figure 3 we have shown two point-to-point movements with no reversal of the finger tip but have chosen the initial and final points such that there is a reversal of one joint. Again there is linear synergy between shoulder, elbow and wrist torques.

Discussion

The first point to stress is that linear synergy, like straight paths, bell-shaped velocity profiles and biphasic torque pulses, is simply an observation of natural behavior. No one of these four mechanical features is a "constraint" that the motor control system is obligated to exhibit. We can change them. However we usually do not unless there are external constraints on performance that make it necessary. Furthermore, we probably cannot change the four features individually and independently because they are related by true constraints, Newton’s laws of motion. Those laws, for movements involving more than a single joint, are too complex to allow us to easily intuit the consequences of an arbitrary pattern of joint torques or to perform inverse dynamics "in our heads." It is their obvious complexity that makes us look askance at the idea of the brain "solving" equation 1 to generate torque patterns from a planned trajectory. Thus we look at linear synergy as a "discovered" rule that the CNS uses to coordinate the torques across multiple joints. This rule, along with biphasic patterns constitutes a torque based movement "plan" that leads to fairly straight, smooth motion.

Figure 3. Two individual movements are illustrated. The relative straightness of the of the hand paths requires a reversal of one or the other joint midway through the movement. Never-the-less, torques at both joints remain biphasic and linear synergy persists. Heavy lines are shoulder, thin lines are elbow. The largest deviation from linear synergy is seen for the shoulder reversal, a movement in which the linear component of the shoulder torque would be quite small.

The fact that these simple torque patterns lead to the observed kinematics is not obvious and in fact, is surprising. It prompted three questions. The first was had we made a programming error? Linear synergy has been independently reproduced in three other laboratories making that unlikely. The second is how general a rule is linear synergy? Work presented and cited here only begins to answer this.

The third is what do the four observed features of natural movements imply about the planning and control of those movements? Morasso also noted [p. 224] that "As a consequence [of the kinematic features], one may hypothesize that the central commands which underlie the observed movements are more likely to specify the trajectory of the hand than the motion of the joints...." In light of the features of the torque described above and the fact that the appearance of straightness appears to outweigh physical straightness (Flanagan and Rao, 1995; Wolpert, et al., 1994), we do not consider this argument compelling. If movement planning is done in that way, then it remains to be demonstrated how such plans are executed.

Straight, smooth movements require the CNS to produce appropriate muscle torques at the joints. The direct solution of the inverse dynamical equations is a possible but unlikely way to accomplish this. The use of force optimization criteria has been suggested (Uno, et al., 1989) but the generality of this approach over the work space has not been established. The often discussed l-equilibrium models assume that central commands from higher neural centers can be expressed in terms of a variable l, which has units of length and is the threshold of a muscle’s tonic stretch reflex. The difference between l and concurrent feedback from muscle proprioceptors defines a state in space, an "equilibrium point" (EP) at which all muscle and external torques are balanced. Voluntary movement is a matter of shifting this EP from the initial to the final position and allowing lower level neuromuscular mechanisms, especially those of the spinal cord, to drive the muscles to their new EPs. Feldman’s l-model (lF) posits a central command that shifts monotonically between its initial and final values, is "independent of current external conditions" and expressed in terms of "positional frames of reference" to control of both single and multiple muscles and degrees of freedom (Feldman and Levin, 1995). Latash has described a variant version of this hypothesis (lL) (Latash and Gottlieb, 1991a; Latash and Gottlieb, 1991b) in which the virtual trajectory of the EP is more complex (but see also (Gomi and Kawato, 1996)).

The central commands of both EP models include a moving EP or virtual trajectory command (R(l)) and a coactivation command (C(l)) plus others that affect reflex gain and reciprocal inhibition. In the lF model, R moves monotonically and quickly (relative to the movement itself) from the initial to the final EP. The changes in dynamical torques needed to accommodate changes in distance, speed, load and direction are reflex driven and require appropriate changes in C (and the others) but we do not yet know many details of how this command is planned. Some of our other reservations concerning the lF-hypothesis can be found in ((Gottlieb, 1995) and http://nmrc.bu.edu/MCL/Lambda_Thread/lambda.html).

The lL version proposes that for fast movements R has an "N-shape" which begins and ends at the same EPs as the R of the lF model but has intervening extrema. In both versions, muscle activation and ultimately muscle force (F) is graded by the difference between R and the actual limb position (x) and scaled by C according to equation 3 (see for example (Latash, 1992) or section 3.4 of (Latash and Zatsiorsky, 1993) or (Feldman and Levin, 1995; St-Onge, et al., 1993) for a more elaborate development that includes the velocity of motion).

F=ƒ(C)(x — R) eq. 3

This equation is not incompatible with biphasic torque pulses or linear synergy. The extrema of R (in the lL version) create the dynamic biphasic torque components while the end points of R establish the static torques required to balance external forces such as gravity which appear in equation 1. Thus, controlling in terms of R is similar to controlling in terms of F, combining dynamic and static components, and requiring the parallel and compatible planning of C at both joints. One difference is that the execution of an R/C plan for equation 3 would appear to require knowledge of a variable (x) that is only available by sensory feedback in order to create linear synergy while explicit planning in terms of F does not.

We conclude that movement planning is done explicitly in terms of muscle torques (or more precisely, the muscle activation patterns). Actual muscle torques will differ somewhat from those plans because of the compliant properties of the peripheral neuromuscular system. Actual EMG patterns will differ somewhat because of reflexes but feedback from muscle proprioceptors makes only a modest contribution (Gottlieb, 1996) to the execution of well planned movements. Movement emerges from the compliant interaction between muscles and their load according to Newtonian mechanics. This planning is based on a learned, internal model of limb and load dynamics that requires only a few parameters to create a biphasic torque pattern that matches the task (Gottlieb, 1993). In addition to the torque plan, we also have a trajectory plan, that is an "expectation" of what the trajectory should be, and by which we evaluate the kinematic outcome. This may be described as a trajectory of an EP (which exists by virtue of neuromuscular compliance) that is similar to the actual movement. If the movement deviates from what we expected because we planned poorly or were interfered with, segmental reflexes offer some compensation but we will not go straight to and may even miss our targets if the dynamic trajectory was important as in throwing for example. If the learned model was wrong (Shadmehr, et al., 1993), it will be revised, given sufficient practice, until the trajectory is restored by a new set of joint torques (see for example figure 3 of Shadmehr and Thoroughman, this section in which the EMG patterns after practice have changed dramatically). If the movement appears to deviate from what we expected because of experimental perversity (Flanagan and Rao, 1995; Wolpert, et al., 1994) then we modify both our force and our kinematic plan but not our model until the movement appears satisfactory. Generating curved movements requires deviation from linear synergy but often little more than small changes in the timing of one or the other joint torque are sufficient. Purposeful deviation from linear synergy with a precise kinematic objective may be more difficult to accomplish and require extensive practice.

Future directions

There are a number of questions that arise from this work.

1) How generally is linear synergy used by the nervous system? Knowledge of which movements violate it might tell us something about what are grace and skill and how we acquire them.

2) How might we experimentally distinguish between force and kinematic planning? As we indicated above, given the compliant nature of the peripheral neuromuscular system, it is difficult to infer from the outside what the unmeasurable central command might be. The same movement can, in principle, be produced by either one.

3) Does sensory feedback play an important role in the performance of well planned movements? One problem with this question is deciding what "important" means.

4) What is the effective limb compliance during movement? These compliant properties mean that the movement plan (in whatever terms it is formulated) will under many circumstances differ from the actual trajectory.

Question 1 is of general interest, regardless of the mechanisms responsible for it. Questions 2-4 are addressing the issue of how or if we can decide between EP hypothesis based models and force based models. The two EP versions make very different predictions about the location of the EP during a fast movement but both make strong demands on sensory feedback during the movements. Both versions require an appropriate degree of limb stiffness (e.g. ƒ(C) in equation 3) that must be specified along with R for every task and which remains to be independently measured during fast movements. Our force model would qualitatively agree with the lL version about where the trajectory of the EP but does not assume a major role for feedback or require any particular value of limb compliance to produce the joint torques. At present, the reader may look at the data base that has been published and apply Occam’s razor.

References Cited

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