Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007
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Neuroengineering model of human limb control - Gainscheduled feedback control approach Kazutaka Takahashi and Steve G. Massaquoi Abstract— Recent neuroengineering modeling suggests that human control of limb movements may take advantage of simple linear control modules that are gainscheduled according to limb state, intended limb state, or both. The scheduling variable(s) appear to consist of limb position and possibly velocity. Thus it is possible that limb control can be modeled effectively using Linear Parameter Varying (LPV) techniques for designing gainscheduled linear control systems. We demonstrate the efficacy of an LPV controller based on a model of cerebrocerebellar neuroengineering models in simulating human arm control of moderate speed, direction-changing movements in the horizontal plane. The technique shows how position dependent dynamics can be stably and realistically controlled using a simple convex interpolation of linear controller modules. The finding could provide a systematic method of representing human motor control using an established engineering technique. It may also support the controversial view that the motor control system does not require internal models of body dynamics to achieve satisfactory performance.
I. I NTRODUCTION The cerebrocerebellar system is central to motor control [1], [2], [3] and has been characterized in terms of its anatomical connections among the areas in the system [1], [3] and in terms of the physiology of each area or network of cerebral cortical areas [4], [5], [6], [7]. The cerebellum is intimately connected to almost all major motor and sensory areas. Furthermore, cerebellar pathology usually results in uncoordinated movements [8] or errors of directions, force [9], amplitudes and delayed movement initiations [10]. Therefore, an importance of the cerebellum in motor control should be emphasized in the analysis and modeling of cerebrocerebellar system. A number of studies have examined simple spike (SS) firing in cerebellar Purkinje cells (PCs) [11], [12], [13]. Those analyzing the firing patterns in anterior intermediate and lateral cerebellum (motor cerebellum) have generally found correlations between PC SS frequency and position and/or velocity. However, a functional model that accounts for the temporal details of these signals during arm movement control is still lacking. In regard to modeling, it is important to note that PCs have been found to fire in relation to both passive and active motion of the body part, though perhaps not as vigorously in the former condition as in the latter [14], [15]. This suggests that PCs may be involved in both monitoring and controlling K. Takahashi is with Department of Organismal Biology and Anatomy, University of Chicago, 1027 E. 57th St. Rm. 202, Chicago, IL 60637, USA
[email protected] S. G. Massaquoi is with the Department of EECS, LIDS, CSAIL, MIT-Harvard Division of Health Sciences and Technology, Massachusetts Institute of Technology, 32 Vassar St. Rm. 214, Cambridge, MA 02139, USA
[email protected]
1-4244-1498-9/07/$25.00 ©2007 IEEE.
body parts. Because the cerebellum is a site of considerable convergence of both peripheral sensory information via the spinocerebellar tracts [16] and brainstem counterparts, and copies of motor outflow via pontine nuclei [17] it is natural to consider that PC activity may be a function of both sensory information and motor outflow. This would be consistent with the observation that interpositus (IP) and dentate firing activity modulates during point-to-point movement control [18], [19], passive body movement [15] and postural maintenance [20]. Thhis observation is not of trivial consequence because important motor control models emphasizing feedforward inverse dynamics control based on desired, rather than sensed, movement trajectories (e.g., [21], [12], [22]) do not predict that sensory signals in the principal cerebellar movement control signals. Other formulations [23], [24], [25] suggest sensory signals could be prominently represented in at least some PCs. Importantly [12] showed that PC SS firing activity can be fit by a linear combination of signals needed for dynamic control of the eyes, the data do not specifically exclude the possibly strong dependence of the PC signal on sensory information. Thus, while the precise mechanism of cerebellar motor control has not been established, it is very likely to have a strong dependence on feedback signals. However, if, as has been conjectured more recently [23], [26], [27], [28] the control does not involve feedforward models of inverse dynamics, it remains to be demonstrated how a heavily feedback-dependent control system alone could afford sufficient control despite signal transmission delays. A. RIPID and RICSS models Recently, the Recurrent Integrator PID (RIPID) cerebrocerebellar control model posits that certain recurrent signals from cerebellum stabilize long-loop proprioceptive responses by processing error-like signals so that they may participate strongly in both postural maintenance [29], [28] and pointto-point movement control. Combining the studies of intracortical connectivity [30] with recent data on cerebrocerebellar anatomy [3], and established concept of distributed signal representation as population vectors [4], a slightly more detailed but still simple picture of the cerebrocerebellar architecture is proposed as in Fig. 2. The Recurrent Integrator Cerebellar Simple Spike (RICSS) model (Figs 2, 3) is demonstrated to account well for SS activity in PCs of behaving primates [26] while remaining functionally consistent with the RIPID model. Fig 3. depicts the cerebellar component. An array of PCs as would lie within a microzone [31] projects to a single group of deep cerebellar nuclei to form a functional corticonuclear microcomplex [32]. PCs are
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Fig. 1. One version of RIPID model. Colored circles designate functional anatomic areas related to the RICSS Model (Fig 2). Inner feedback loop from area 1 (green) containing block I2/s (”recurrent” integrator) provides phase lead that stabilizes the long outer feedback loop.
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2 units based on behavioral intent (”motor set” by [2]). Thus, it is conceived that different subpopulations of PCs are engaged according to the activity of the group of selPFs. If PCs implement the control gains Gb, Gk, I1 and I2 in Fig 1, and selPF activity reflects the state of the plant and of behavioral intent, then the circuitry in Fig 3 can be viewed as a general mechanism for context-dependent scheduling of control gains according to the scheduling signals ρ(t).
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Fig. 3. Cerebellar cortical architecture proposed to underlie PC SS activity.
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Fig. 2. Cerebral cortical component of RICSS model from the perspective of a single cerebellar PC. Recurrent integrator loop is not shown. 8 cerebral cortical columns in Sensorimotor Cortical Area 3a (SMC-1, area 1 in Fig.1) implementing a neural population-based representation, e.g. [4], of the tracking error-like vector (red arrow) and subsequent distribution after cerebellar processing.
assumed to receive two types of parallel fiber (PF) input. Descending signals edir· cb (t) from area 3a or 4 travel by mossy fibers designated here as signal mossy fibers (sigMF) to reach signal PFs (sigPF) that are considered here to be those whose ascending axons synapse multiply on proximal PC dendrites to afford a strong excitatory connection [33], [34]. A given set of sigPF inputs is presumed to synapse on many PCs within the microzone and on the associated deep nuclei. On the other hand, the distal dendrites of each PC are influenced more subtly by passing PFs [33], [34]. These are termed here selector PFs (selPFs). The principal hypothesized action of selPFs is to inhibit laterally adjacent PCs via basket cells (solid red in Fig. 3). Thus, each PC is potentially suppressible by ’beams’ of active PFs to either side. Conversely, PC activity along an active beam of selPFs is comparatively preserved. This mechanism is consistent with the experimental observations of active centers and inhibitory surrounds [35]. The quantitative formulation is very similar to that in [29] but updated to be more consistent with recent work [33], [34]. SelPFs are assumed to be supplied by spinocerebellar tracts among other pathways and thus to carry body state information [36] and other context variables. Based on the structure above, PC SS firing depends on both descending sigPF inputs and modulation by coincident selPF activity. It is assumed that in general the descending cortical inputs may come from different SMC-
As it appears that scheduling control gains may be a mechanism by which the nominally linear RIPID/RICSS circuitry could affect general control throughout extended workspaces, an analytical input-output description is desirable. Linear parameter varying (LPV) formulations have a similar structure to that suggested by the RICSS model and possesses theoretical tractability. Therefore, LPV models were tested on data from human subjects performing double step tasks that required a rapid change in movement direction. The LPV synthesis procedure discussed here applies to affine parameter-dependent plants described below: x˙ P (·, ρ) = y z
= A(ρ)x + B 1 (ρ)w + B 2 u = C 1 (ρ)x + D11 (ρ)w + D12 u = C 2 x + D 21 w + D 22 u
(1)
where
ρ(t) = [ρ1 (t), . . . , ρn (t)],
ρi ≤ ρi (t) ≤ ρi
(2)
is a time-varying vector of scheduling variables each component of which is bounded by its minimum and maximum values ρi and ρi respectively, n is the number of the scheduling variables, A(·), B1 (·), C1 (·), and D11 (·), are affine functions of ρ(t). Note that ρ(t) may contain part of the state vector itself , assuming that the corresponding states are accessible to measurement. If the scheduling parameter vector ρ(t) takes values in a box of Rn with corners {Πi }, i = 1, . . . , N = 2n , the plant P (·, ρ) ranges in a matrix polytope with vertices V (Πi ). Namely, given any convex decomposition
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
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of ρ over the corners of the parameter box, then the parameter dependent system is given by P (·, ρ) =
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Fig. 4. A realization of RIPID/RICSS structure as a gain-scheduling control system. For the current study, R(y(t)) = ρ(θe ) as shown in Eqn’s 20, 21.
Thus, we would like to find a set of parameter dependent controllers taking the following form: G(·, ρ) =
(
ξ˙ = AG (ρ)ξ + B G (ρ)y , u = C G (ρ)ξ + D G (ρ)y
(5)
with the following vertex property: PN Given the convex decomposition ρ(t) = i=1 αi Πi of the current parameter value ρ(t), the values of AG (ρ), B G (ρ), C G (ρ), and D G (ρ) are expressed as a linear combination of AG (Πi ), B G (Πi ), C G (Πi ), and DG (Πi ), i = 1, · · · , N at the corners of the parameter box by
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N 21 0 ˜ 21 = N , 0 I A(Πi ) B(Πi ) C 1 (Πi ) D11 (Πi )
∞
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II. LPV FORMULATION FOR THE TWO - LINK ARM PLANT Formulation here considers the nonlinearity only due to inertia and viscosityi.e., no nonlinearity in muscle dynamics:
.
the vertex property (Eq.(6)) the closed-loop system is stable for all admissible parameter trajectories ρ(t) a guaranteed L2 -gain bound γ > 0 for the closed loop system from the generalized disturbance signal w to the error signal z, i.e., Z T Z T z T z dt ≤ γ 2 wT w dt, ∀T ≥ 0 (7) 0
N 12 0 , 0 I Ai B 1i = C 1i D 11i
for i = 1, . . . , N and N 12 and N 21 are bases of the null spaces of (B T2 , DT12 ) and (C 2 , D 21 ), respectively. Recall that N = 2n where n is the number of the scheduling parameters. Thus, there will be 2n + 1 linear matrix inequalities (LMI) such as ones in Eqn’s 8 ∼ 10. The assumption that B 2 , C 2 , D12 , and D 21 to be independent of ρ(t) will be satisfied by placing a lowpass filter, an excitation-contraction (EC) coupling filter to characterize a gradual force development [27], so that the overall plant is realized as shown in Eq (1). To enforce the performance and robustness requirements, we can use the loop shaping criterion summarizing RMS gain constraint
T
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(6) The objective is to design a gain-scheduled controller G(·, ρ) that satisfies: •
τ (t) τ (t)
¨ + C(θ(t), θ(t)) ˙ ˙ = H(θ(t))θ(t) θ(t) = Ku(t)
(12) (13)
u(t)
= EC (r(t) − θ(t − taf f ))
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˙ are the inertia and viscosity where H(θ) and C(θ, θ) matrices respectively, and K is a constant muscle stiffness matrix, and the torque τ is a linear function of the input signal u which is the output of the EC filter. Eq. (14), should be interpreted as a mapping of the difference between the reference signal r(t) and the delayed afferent joint angles θ(t − taf f ), where taf f is the afferent delay amount, and the output of the EC filter, u through the LTI system EC. The inertia matrix can be seen as an affine function of the elbow angle θe :
Theorem [37]: Suboptimal scaled H∞ problem is solvable H(θ) = H 0 + cos θe H 1 , (15) if and only if there exist pairs of symmetric matrices L ∈ where Rn×n and M ∈ Rn×n such that h1 + h2 + m2 l12 h2 Ai L + LATi LC T1i B 1i H0 = , (16) T h2 h2 C 1i L −γI D 11i N˜12 < 0, (8) N˜12 2 1 B T1 DT11i −γI . H 1 = m2 l 1 l 2 T 1 0 T Ai M + M Ai M B 1i C 1i T B T1i M −γI D T11i N˜21 < 0, (9) N˜21 Following the similar manner, the viscosity matrix ˙ can be affinely decomposed as follows: C 1i D11i −γI C(θ, θ) L I ≥ 0, (10) ˙ = C 0 + sin θe θ˙s C 1 + sin θe θ˙e C 2 , (17) C(θ, θ) I M
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(18)
Thus, the arm dynamics can be expressed as: ¨ = −C(θ, θ) ˙ θ˙ + Ku, H(θ)θ ˙ = Ap (ρ)Θ + B p (ρ)u, Θ = θ˙ ⇒ Θ θ = C p Θ.
θ
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The input to the EC filter, uEC is related to the output, u as in the following transfer function κ2 1 0 U EC (s), U (s) = (s + κ)2 0 1
where U (s) and U EC (s) are the Laplace transforms of u and uEC respectively. Its relation in the state space form is: ζ˙ = AEC ζ + B EC uEC u
= C EC ζ.
Thus the augmented parameterized plant from the input of the EC filter to the joint angles is AP (ρ) B P (ρ)C EC 0 ˙ ζa = uEC ζa + 0 AEC B EC | | {z } {z } Ba
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where only Aa (ρ) is parameter dependent. In the analysis below, only the configuration dependency, i.e., the effect of the elbow angle, θe through the inertial matrix H(θ) is examined. This is because of the following reasons. First, including angular velocity terms requires exponential addition of LMI’s to be solved and implementation of corresponding controllers. As there are two parameters in the viscosity matrix in Eqn 17, i.e., sin θe θ˙s and sin θe θ˙e , total increase of the number of the scheduling parameters would be 2 × 2 where additional 2 comes from the inversion of the inertial matrix. Secondly, the viscosity matrix contains terms with joint velocities, sin θe θ˙s and sin θe θ˙e . Thus, the size of workspace alone is not sufficient to estimate the bounds of the scheduling parameters. Therefore, in the case under the consideration, i.e., θ˙e = θ˙s = 0, there is effectively only one parameter, θe characterizing the scheduling variables. Thus, Ap (ρ) is now a constant matrix, i.e., Ap (ρ) = [ 0I 00 ] . Yet, H(θ)−1 in B p (ρ) contains two parameters both of which are functions of θe : h2 −h2 −1 H(θ) = ρ1 (θe ) −h2 h1 + h2 + m2 l12 0 −m2 l1 l2 (19) +ρ2 (θe ) −m2 l1 l2 2m2 l1 l2 1 ρ1 (θe ) = , (20) 2 h1 h2 + h2 m2 l1 − (m2 l1 l2 cos θe )2 cos θe ρ2 (θe ) = . (21) h1 h2 + h2 m2 l12 − (m2 l1 l2 cos θe )2
FrC01.5 Thus, R(·) in Fig. 4 for this particular formulation is T R(y(t)) = [ρ1 (θe (t)); ρ2 (θe (t))] . Note that this formulation does not account for the rate variation of ρ1 and ρ2 directly in addition to excluding the joint velocities. Thus, the arm dynamics here is yet a subset of the arm dynamics only containing the inertial effect. In order to use a polytope based LPV synthesis, we need to set, or find, the lower and upper bounds for all the scheduling parameters, ρi to define the vertices. To find the bounds on ρi , the minimum and maximum values of ρi are taken over the possible range of the elbow angles computed from the size of the workspace through the inverse kinematics such that ρi = min ρi , ρi = max ρi , ρi ≤ ρi (t) ≤ ρi , ∀i , ∀t. Low frequency tracking performance can be enhanced with a choice of the weighting W 1 being a lowpass filter. The following parameter independent frequency weighting is used: 25 I. W 1 (s) = (s + 5)2 The weighting filter for the complementary sensitivity function, W 2 (s) is set to be unity. To compare the performance of a resultant LPV controller, one H∞ controller based on a linearized plant around the center of the workspace was designed with the same weightings. III. H UMAN DOUBLE STEP MOVEMENT EXPERIMENT Each subject sat on a chair with his/her body stabilized by a back support as well as a four-point seat belt that restricted body motion substantially to two degrees of freedom of the arm in a horizontal plane. Each subject held a handle of the InMotion2TM two-link manipulundum to control the location of a yellow cursor shown on a monitor.
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Human double-step reaching protocol.
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Hand speed
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Fig. 6. An example of kinematic fit of the data with response of the LPV system. Data from a subject performing S-C-R task. Experimental data (blue), LPV fit (red), and Reference command for the LPV system (green dashed). Left: Hand path. Right: Hand speed. Hand speed
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After 40 ∼ 120 practice trials there were four blocks of 140 test trials. Within each block the starting point, either North (N) or South (S), and the final target, either on the Right (R) or the Left (L) side, all relative to the center (C), remained fixed. Both starting targets are located 15 cm vertically away from the C respectively. The task was to move sequentially from a start location toward the center then toward a final target according to visual cues. An example of an action sequence in each trial is shown in Fig. 5 and is explained here. Each trial was initiated by a subject moving the small yellow cursor to a larger yellow circle at one of the two start locations, N or S (Fig. 5-I,-II). The subject was instructed to hold the cursor in the green circle for 1 ∼ 1.5 seconds until the circle disappeared (Fig. 5-III), and the first central target turned red to cue a brisk movement to C (Fig.5IV). The second target which was blue appeared randomly 100 to 400 ms later and one at one of sevan possible locations (dashed circles) (Fig.5-V) such that the number of trials to each possible target was equal to 20. Kinematic data, hand position and velocity, from each trial was recorded. Then, with obtained or estimated parameters for the arm plant for each subject, an LPV controller is designed. A series of two minimum jerk (MJ) profiles in series was fed as a reference command to the closed loop LPV system. Then, in order to fit the experimental hand position and velocity with the output from the closed loop LPV system, genetic algorithm command in Matlab, ga was used to find a set of coefficients characterizing two MJ profiles in series [38].
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Data were recorded from four right handed subjects in this study. Each data set consisted of 560 trials total, 140 trials for each of the combinations of the starting point (N,S) and final targets (L,R). Due to various kinematic variability, only five trials from each subject for the S-C-R task and another set of five trials for the S-C-L task, total of 10 trials were used to explore the quality of the fit of the LPV controller. Out of 40 trials, ga algorithm converged for 39 trial data. Fig.6 shows an example of a data fit well by the LPV model. A minor curvature of the hand path in the first segment is not extremely accurately captured and there is a slight offset at the end point of the movement, but overall the hand path is fit by the LPV model well. The extrema of the hand speed slightly deviate from the data, but the LPV response faithfully captures the hand speed of the data whose second segment command starts roughly at the middle of the first segment. Fig. 7 illustrates an example of a fit of the movement of the target sequence S-C-L. Both the hand path and speed are reproduced in S-C-L movements as well as in S-C-R movements. The contribution of LPV gainscheduling to dynamic control is illustrated in Fig. 8 which shows an performance comparison between H∞ and LPV responses. During the first segment of the movement, the responses from both systems do not differ much either in the hand path or speed,
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Fig. 7. An example of kinematic fit of the data with response of the LPV system. Data from a subject performing S-C-L task. Line types follow Fig.6.
but in the second segment it appears that H∞ response yields slightly late initiation and slight directional offset to the left in the second segment. Furthermore, the end point error of H∞ controller is much larger than that of the LPV. Therefore, even though the task was not very demanding dynamically, the LPV controller usually resulted in a better fit to the experimental data. V. D ISCUSSION AND C ONCLUSIONS The simplicity and linearity of the RIPID formulation suggests that some method of modifying its function to address nonlinearities and temporal variation in plant dynamics would be required for it to account for human limb control. The RICSS model suggests a particular multiplicative relation between the error-like signal and state information at the PCs. This mechanism was abstracted to a well established LPV gainscheduling control scheme. To test the feasibility of such a formulation, the double step task was performed
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Fig. 8. Difference between the single H∞ controller response and the LPV response. Data from a subject performing S-C-L task. Experimental data (blue), LPV fit (red), and H∞ fit (yellow). Left: Hand path. Right: Hand speed.
to human subjects. An LPV model that was gainscheduled according to limb state accounted fairly well for some of the human data. This provides a preliminary indication that simple scheduling of the RIPID gains according to limb state may be adequate to manage limb dynamics. Moreover, it is intriguing to note the structural similarity between an affine representation of the current schedule variable in an LPV formulation and a hypothesized selPF which is an affine function of afferent limb states and their derivatives. Thus, it is possible that limb control can be modeled effectively using LPV techniques. The finding potentially provides a systematic method of representing human control using an established and flexible engineering technique. It also supports the controversial view that the human motor control system does not require internal models of body dynamics to achieve satisfactory control of body motion. In order to critically test if an LPV or gainscheduling system is better than a single H∞ , or even a necessity of gainscheduling to explain behavioral data, more data with faster speeds would be required. In addition, the schedule variables in Eqn’s (20,21) are nonlinear functions of a limb state, θe (t) which is a natural choice as an argument for the scheduling variables here because it is the variable that survives linearization around equilibrium points. However, the RICSS model suggests that selPF’s are thresholded affine functions of limb states and their derivatives. Thus, it is critical to approximate the limb dynamics as an affine function of θe for an LPV model. In addition, although it is not necessary at speeds tested here, scheduling based on velocity may be necessary. The data from [39] implies that some form of velocity gainschedule may be present as PCs appear to be sensitive to different movement speeds. How to incorporate both position and velocity as scheduling variables in an optimal control synthesis will await further study. Furthermore, to avoid ambiguity of the effects between the responses of dynamical systems and those of optimization
FrC01.5 procedures to minimize the difference between the simulation responses and the experimental data, a better method of combining dynamical controller design and data fitting need to be developed without exhaustively searching finely gridded spaces of command and controller parameters. That the kinematic data obtained here can be often fit adequately by a feedback-based control structure does not in itself exclude the possibility of other feedforward inverse dynamics control that has been proposed for cerebellar function by many [40], [25]. However, taken together with the observation by the physiologists that most PC units responded to passive manipulation argues strongly for the presence of feedback signals in PC firing activity, as used by the RICSS model, and against purely feedforward cerebellar control models [21], [40]. The possibly multiplicative relationship between kinematics and cerebellar signals used in RICSS had not been emphasized before [41], although purely linear formulations such as in [42], [12], do not appear to consider linearity as a fundamental requirement. Moreover, other proposals [43], [44] are sufficiently general to be consistent with PC data used here. However, these models have not yet been explicitly reconciled with cerebrocerebellar circuitry and cerebellar signals recorded during arm movement. The debate on whether cerebellum acts like a feedforward inverse dynamics controller [25], [43] or feedback controller remains still active. Two recent physiological studies attempted to answer the question by applying force fields. One recent study [45] showed that the SS firing of majority of PCs was not significantly modulated by the force nor was their spatial tuning affected as long as kinematic performance remained the same. Thus, these results do not support the hypothesis that PCs represent the output of an inverse dynamics model of the arm. Instead these neurons provide a kinematic and possibly kinematic error representation of arm movements. The other study [46] showed that, by applying two different force fields, SS activities in many PCs differed distinctly depending on the type of force field. Furthermore, difference in averaged PC SS activities increased approximately 40 ms before the difference appeared in EMG activities. The authors conclude that PC SS in the intermediate part of cerebellar lobules V-VI encode movement dynamics. As we are not sure about the tracking error in these studies, it is still difficult to clearly reconcile these findings with the RIPID/RICSS formulation. More detailed information regarding the internal signals will be needed. In any case, the current study suggests that scheduled linear control based on kinematic tracking error may be adequate to achieve satisfactory control of body motion. Internal models of body dynamics may not be required. ACKNOWLEDGMENT The authors would like to thank Prof. Tim Ebner at Univ. of Minnesota and Dr. Alex Roitman at UCSF for the primate physiology and kinematic data, and Prof. Munther Dahleh at MIT for discussion on LPV and its application to limb control.
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