Effects of model inaccuracies on reaching movements with intermittent control

Background and objectives Human motor control (HMC) has been hypothesized to involve state estimation, prediction and feedback control to overcome noise, delays and disturbances. However, the nature of communication between these processes, and, in particular, whether it is continuous or intermittent, is still an open issue. Depending on the nature of communication, the resulting control is referred to as continuous control (CC) or intermittent control (IC). While standard HMC theories are based on CC, IC has been argued to be more viable since it reduces computational and communication burden and agrees better with some experimental results. However, to be a feasible model for HMC, IC has to cope well with inaccurately modeled plants, which are common in daily life, as when lifting lighter than expected loads. While IC may involve event-driven triggering, it is generally assumed that refractory mechanisms in HMC set a lower limit on the interval between triggers. Hence, we focus on periodic IC, which addresses this lower limit and also facilitates analysis. Theoretical methods and results Theoretical stability criteria are derived for CC and IC of inaccurately modeled linear time-invariant systems with and without delays. Considering a simple muscle-actuated hand model with inaccurately modeled load, both CC and IC remain stable over most of the investigated range, and may become unstable only when the actual load is much smaller than expected, usually smaller than the minimum set by the actual mass of the forearm and hand. Neither CC nor IC is consistently superior to the other in terms of the range of loads over which the system remains stable. Numerical methods and results Numerical simulations of time-delayed reaching movements are presented and analyzed to evaluate the effects of model inaccuracies when the control and observer gains are time-dependent, as is assumed to occur in HMC. Both IC and CC agree qualitatively with previously published experimental results with inaccurately modeled plants. Thus, our study suggests that IC copes well with inaccurately modeled plants and is indeed a viable model for HMC.

where x ∈ R n is the state of the plant, u(t) ∈ R m is the control signal, w(t) ∈ R n is the 6 process noise, and x IM ∈ R n is the state of the internal model.

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The compound effect of process and measurement delay is accounted for by 8 introducing measurement delay τ : where y(t) ∈ R q is the measurement and v(t) ∈ R q is the measurement noise.

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Observer combines the internal model (Eq. (2)) and delayed measurement (Eq. (3)) 11 to generate the estimated statex according to: Predictor predicts the current state, x p (t), given the estimated state,x(t − τ ), and 13 the control signal u(σ) for σ ∈ [t − τ, t), based on the internal model (Eq. (2)): LTI systems, i.e., LTI plants with time-invariant observer and controller gains, L 15 and K, can be described by the overall state

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Combining Eq.s (1), (3) and (4) yields: where A o and B o are defined in the main text, and w ov ( is the overall process noise.

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Intermitent control performs predictions at discrete times t m , which can be evenly 20 spaced (periodic IC, t m = mh, where h is the sampling period) or event driven (not 21 considered in this work). At t m , the predictor receivesx(t m − τ ) from the observer and 22 October 13, 2019 1/3 generates x p (t m ) according to Eq. (5). The latter provides the initial condition for the 23 hold state, x h (t) that determines the control signal: Between samples, x h (t) evolves continuously according the feedback matrix 25 (A F (t) = A − BK(t)), defining the SMH: To facilitate the stability analysis of continuous time-delayed systems, they are 28 converted to equivalent discrete-time systems using the standard zero-order hold [1].

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Given the delay τ , the discretization time ∆ is selected so k τ = τ /∆ is an integer 30 number. Thus, the equivalent discrete-time system of the plant, (1) and (3), and 31 internal model, (2), is described by the following difference equations: The covariance matrices of the discrete process and 36 measurement noise are W d = W ∆ and V d = V /∆, respectively [1].

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Stability is analyzed for time-invariant systems with constant observer and controller 38 gain matrices, L d and K d , respectively. These gain matrices can be computed using 39 standard optimal estimation and control tools under the assumption of accurate model 40 for an infinite horizon cost function. In this case, the optimal Kalman gain matrix is Due to the measurement delay, the current measurement y(k) depends on the 47 delayed state x(k − k τ ). Hence the observer updates the estimated statex(k − k τ ) 48 according to: The predicted state is the solution of (11), given the estimated statex(k − k τ ): The control signal is proportional to the predicted state: Equations (13) and (14) imply that x p (k) depends on x p (k − k τ ), ...., x p (k − 1). This 52 dependence is captured by defining the extended state 53 x exv (k) = [x(k − k τ ) x(k − k τ ) x p (k − 1) . . . x p (k − k τ ) ] . Thus, the dynamics of the 54 overall discrete system described by (9) -(14) can be expressed as 55