On the Duration of Human Movement: From Self-paced to Slow/Fast Reaches up to Fitts’s Law

Abstract

In this chapter, we present a mathematical theory of human movement vigor. At the core of the theory is the concept of the cost of time. According to it, natural movement cannot be too slow because the passage of time entails a cost which makes slow moves undesirable. Within this framework, an inverse methodology is available to reliably and robustly characterize how the brain penalizes time from experimental motion data. Yet, a general theory of human movement pace should not only account for the self-selected speed but should also include situations where slow or fast speed instructions are given by an experimenter or required by a task. In particular, the limit case of a “maximal speed” instruction is linked to Fitts’s law, i.e. the speed/accuracy trade-off. This chapter first summarizes the cost of time theory and the procedure used for its accurate identification. Then, the case of slow/fast movements is investigated but changing the duration of goal-directed movements can be done in various ways in this framework. Here we show that only one strategy seems plausible to account for both slow/fast and self-paced reaching movements. By relying upon a free-time optimal control formulation of the motor planning problem, this chapter provides a comprehensive treatment of the linear-quadratic case for single degree of freedom arm movements but the principles are easily extendable to multijoint and/or artificial systems.

Appendix: Technical Details

1.1 Asymptotic Study

We describe in this asymptotic studies the behavior of the solutions of the linear quadratic model introduced in Sect. 2.2 when some parameters of the problem go to zero or infinity.

1.1.1 Asymptotic Study for Small/Large Time and Fixed Cost

Let us study the behavior of the optimal solutions when the final time $\tau$ varies, the quadratic cost $l(\mathbf {x},u)=u^{2}+\mathbf {x}^{T}Q\mathbf {x}+2\mathbf {x}^{T}Su$ and the terminal conditions $\mathbf {x}^{0}=(\theta ^{0},0,\dots ,0)$, $\mathbf {x}^{f}=0$ being fixed. For every $\tau >0$ we denote by $\mathbf {x}_{\tau }(t)= \left( \theta _{\tau }(t),\dots , \theta _{\tau }^{(n-1)}(t)\right) ,$ $t\in [0,\tau ],$ the solution of the free-time OC problem in fixed time $\tau$ whose expression is given by Eq. (6). Consider first the case of large times, that is the case where $\tau \rightarrow \infty$. Remind that in this case $e^{\tau A_{+}}$ and $e^{-\tau A_{-}}$ tend to zero.

Lemma 4

When $\tau \rightarrow \infty$, there holds

$$ \mathbf {x}_{\tau }(t)=e^{tA_{+}}\mathbf {x}^{0}+O(\Vert e^{\tau A_{+}}\mathbf {x}^{0}\Vert ). $$

As a consequence, there exists constants $c,\alpha >0$, and $\varepsilon \in (0,1)$, such that

$$ \theta _{\tau }(t)=c\theta ^{0}e^{-\alpha t}+O(e^{-\alpha \tau }),\qquad \hbox {for any }\,t\in [\varepsilon \tau ,\tau ]. $$

We thus recover a somewhat intuitive result: the solution of a LQ problem in fixed time converges to the solution of the same LQ problem with infinite horizon when the time goes to infinity (see Remark 2).

Proof

We deduce directly from the conditions of Eq. (7) the values of $\mathbf {p}_{1}=\mathbf {p}_{1}(\tau )$ and $\mathbf {p}_{2}=\mathbf {p}_{2}(\tau )$ in function of $\mathbf {x}^{0}$. By putting these values into Eq. (6), we obtain

$$ \mathbf {x}_{\tau }(t)=e^{tA_{+}}\left( I-e^{-\tau A_{-}}e^{\tau A_{+}}\right) ^{-1}\mathbf {x^{0}}-e^{(t-\tau )A_{-}}e^{\tau A_{+}}\mathbf {x^{0}}, $$

which is of the form $e^{tA_{+}}\mathbf {x}^{0}+O(\Vert e^{\tau A_{+}}\mathbf {x}^{0}\Vert )$ since $A_{+}$ is stable and $A_{-}$ is anti-stable. Now, $e^{tA_{+}}\mathbf {x}^{0}$ is a function of t which can be written as a sum of decreasing exponential terms. Denoting by $e^{-\alpha t}$ the less decreasing term in this sum, it appears that all other exponential terms in $e^{tA_{+}}\mathbf {x}^{0}$ are negligible in front of $e^{-\alpha \tau }$ for $t/\tau$ not too small and we obtain the formula for $\theta _{\tau }$ (note that in general $\alpha =\min \left\{ -\mathfrak {R}(\lambda )\ :\ \lambda \ eigenvalue\ of\ A_{+}\right\}$).   $\square$

Consider now the case of small times, i.e. the case where $\tau \rightarrow 0$. In that case we can prove the following result.

Lemma 5

Let p(s) be the polynomial function of degree $2n-1$ defined by $\left( p(0), p'(0),\dots ,p^{(n-1)}(0)\right) =\mathbf {x}^{0}$ and $\left( p(1),p'(1),\dots ,p^{(n-1)}(1)\right) =\mathbf {x}^{f}$. Then

$$ \theta _{\tau }(t)=p\left( \frac{t}{\tau }\right) +O(\tau ). $$

As a consequence, $\theta _{\tau }(t)\approx p(\frac{t}{\tau })$ for small times $\tau$: a change of the final time induces approximately a temporal rescaling of the solutions.

Remark 6

Note that since the terminal conditions are equilibriums, the polynomial $p(\cdot )$ satisfies $\dot{p}(t)=\dot{p}(1-t)$, which implies that the velocity profiles of $\theta _{\tau }$ have an almost symmetric shape for small times $\tau$. Indeed, the polynomial function $\widetilde{p}(t)=\theta ^{0}-p(1-t)$ satisfies the same conditions at $t=0$ and $t=1$ as p(t), which implies by unicity of the solution that $\widetilde{p}(t)=p(t)$, and so the conclusion.

Proof

Let us start with a preliminary remark on the optimal solution $\theta _{\tau }$. On one hand, it follows from Eq. (6) that $\theta _{\tau }(t)$ is an analytic function (i.e. it is equal to its Taylor series) which depends linearly on the vectors $\mathbf {p}_{1}=\mathbf {p}_{1}(\tau )$ and $\mathbf {p}_{2}=\mathbf {p}_{2}(\tau )$. Hence, all derivatives of $\theta _{\tau }$ at 0 depend linearly on the pair $(\mathbf {p}_{1},\mathbf {p}_{2})$. On the other hand, due to the particular properties of the matrices $A_{-},A_{+}$ (see [16, Lemma 1]), there is a one-to-one correspondence between $(\mathbf {p}_{1},\mathbf {p}_{2})$ and the 2n first derivatives of $\theta _{\tau }$ at 0, i.e. by $\theta _{\tau }^{(k)}(0),\ 0\le k\le 2n-1$. As a consequence, all derivatives of $\theta _{\tau }$ at 0 depend linearly on the 2n first ones: for every integer k there exists a constant $C_{k}$ such that, for any $\tau ,$ $|\theta _{\tau }^{(k)}(0)|\le C_{k}\varTheta _{\tau },$ where

$$ \varTheta _{\tau }=\max \left\{ |\theta _{\tau }^{(k)}(0)|,\ 0\le k\le 2n-1\right\} . $$

Set $\phi _{\tau }(t)=\theta _{\tau }(t)-p(\frac{t}{\tau })$. We have to prove that $\phi _{\tau }(t)=O(\tau )$. The above remark and the fact that $\left( p(0),\dots ,p^{(n-1)}(0)\right) =\left( \theta _{\tau }(0),\dots ,\theta _{\tau }^{(n-1)}(0)\right) =(\theta ^{0},0,\dots ,0)$ imply that the Taylor expansion of $\phi _{\tau }$ has the form,

$$\begin{aligned} \phi _{\tau }(t)=\sum _{k=n}^{2n-1}\frac{t^{k}}{k!}\left( \theta _{\tau }^{(k)}(0)-\frac{p^{(k)}(0)}{\tau ^{k}}\right) +\varTheta _{\tau }O(t^{2n}), \end{aligned}$$

(11)

where all $O(\cdot )$ are uniform with respect to $\tau .$ By definition of $p(\cdot )$ we have also $\phi _{\tau }^{(j)}(\tau )=0$ for $j=0,\dots ,n-1$, and from Eq. (11) we get

$$ \sum _{k=n}^{2n-1}\frac{1}{(k-j)!}\left( \tau ^{k}\theta _{\tau }^{(k)}(0)-p^{(k)}(0)\right) =\varTheta _{\tau }O(\tau ^{2n}),\qquad j=0,\dots ,n-1. $$

It follows that, for $k=n,\dots ,2n-1$ there holds $\tau ^{k}\theta _{\tau }^{(k)}(0)-p^{(k)}(0)=\varTheta _{\tau }O(\tau ^{2n})$, and thus from the definition of $\varTheta _{\tau }$ we obtain that $\varTheta _{\tau }\tau ^{2n}=O(\tau )$. This and Eq. (11) give $\phi _{\tau }(t)=O(\tau )$, which proves the lemma.   $\square$

1.1.2 Asymptotic Study for Fixed Time

Let us try to understand now how the optimal solutions behave when some coefficients in the cost function are modified. We fix an initial state $\mathbf {x}^{0}=(\theta ^{0},0,\dots ,0)$, a final one $\mathbf {x}^{f}=0$, and an infinitesimal CoT g(t). We consider a family of costs $l_{r}(\mathbf {x},u)$ depending on a parameter r of the form

$$ l_{r}(\mathbf {x},u)=u^{2}+r\theta ^{2}+\mathbf {x}^{T}Q_{0}\mathbf {x}+2\mathbf {x}^{T}Su, $$

that is, with a matrix $Q(r)=Q_{0}+re_{1}e_{1}^{T}$ ($e_{1}=(1,0,\dots ,0)$ denotes the first vector of the canonical basis of $\mathbb {R}^{n}$). We want to study the behavior when r tends to $\infty$ of the optimal solutions of the following free-time OC problem: minimize the cost

$$ C_{r}(u,t_{u})=\int _{0}^{t_{u}}\left( g(t)+{\displaystyle l_{r}\big (\mathbf {x}_{u}(t),u(t)\big )}\right) \,dt, $$

among all inputs $u(\cdot )$ and all times $t_{u}$ such that $\mathbf {x}_{u}(0)=\mathbf {x}^{0}$ and $\mathbf {x}_{u}(t_{u})=\mathbf {x}^{f}$. As we have seen previously, the time $\tau =\tau (r)$ is determined by Eq. (9) and the optimal solutions are the one of the OC problem $\min \int _{0}^{\tau }l_{r}(\mathbf {x},u)$ in fixed time $\tau$.

Lemma 7

For every $r>0$ we denote by $\mathbf {x}_{r}(t)= \left( \theta _{r}(t),\dots ,\theta _{r}{}^{(n-1)}(t)\right) ,$ $t\in [0,\tau (r)],$ the solution of the free-time OC problem associated with $C_{r}$. Assume that the infinitesimal cost of time $g(\cdot )$ is a bounded function. Then there exists constants $c,\alpha >0$, and $\varepsilon \in (0,1)$, such that, when $r\rightarrow \infty ,$ we have $r^{1/2n}\tau (r)\rightarrow \infty$ and

$$ \theta _{r}(t)=c\theta ^{0}e^{-\alpha r^{1/2n}t}+O\left( e^{-\alpha r^{1/2n}\tau (r)}\right) ,\qquad \hbox {for any }\,t\in [\varepsilon \tau (r),\tau (r)]. $$

Note that the boundedness assumption on g is very natural and seems to be verified experimentally since we obtain functions g(t) that are decreasing for large t.

Proof

To simplify the study, we give only the proof in the case where the matrices $Q_{0}$ and S are zero, and the dynamics (Eq. (5)) is of the form $\theta ^{(n)}=u$. The proof of the complete result can be obtained by showing that this case actually gives the highest order terms with respect to r. With the preceding hypothesis, $\theta _{r}(t)$ is the solution of the OC problem in fixed time $\tau =\tau (r)$ associated with the infinitesimal cost $u^{2}+r\theta ^{2}$, or equivalently with $\frac{1}{r}u^{2}+\theta ^{2}$. Set $\widetilde{\theta _{r}}(t)=\theta _{r}(tr^{-1/2n})$. Then $\widetilde{\theta _{r}}(t)$ is the solution of the OC problem in fixed time $r^{1/2n}\tau$ associated with the infinitesimal cost $u^{2}+\theta ^{2}$. In the latter problem, nothing depends on r except the duration $r^{1/2n}\tau$. It results from the analysis of Sect. 2.2.2 that there exists a universal function of time $\varphi (\cdot )$ such that $\widetilde{u_{r}}(r^{1/2n}\tau )=\theta ^{0}\varphi (r^{1/2n}\tau )$. Since we have $u_{r}(t)=r^{1/2}\widetilde{u_{r}}(r^{1/2n}t)$, we obtain

$$ u_{r}(\tau )=r^{1/2}\theta ^{0}\varphi (r^{1/2n}\tau ). $$

Now remember (see Eq. (8)) that the time $\tau$ must satisfy $g(\tau )=\left( u_{r}(\tau )\right) ^{2}$, which gives $g(\tau )=r\,\left( \theta ^{0}\varphi (r^{1/2n}\tau )\right) ^{2}$. Assume by contradiction that the quantity $r^{1/2n}\tau (r)$ is bounded as $r\rightarrow \infty$. Then $\varphi (r^{1/2n}\tau )$ is bounded away from zero ($\varphi$ is positive and continuous on $(0,+\infty )$, and converges to $+\infty$ as $t\rightarrow 0$, see Fig. 2), and therefore $g(\tau (r))\rightarrow \infty$ as $r\rightarrow \infty$, which contradicts the boundedness of g. Thus we get $r^{1/2n}\tau (r)\rightarrow \infty$.

Since $\widetilde{\theta _{r}}(t)$ is the solution of an OC problem in fixed time with a very large time $r^{1/2n}\tau (r)$, it results from Lemma 4 that $\widetilde{\theta _{r}}(t)=c\theta ^{0}e^{-\alpha t}+O\left( e^{-\alpha r^{1/2n}\tau (r)}\right)$ for t larger than $\varepsilon r^{1/2n}\tau (r)$ for some $\varepsilon \in (0,1)$. The conclusion follows from $\theta _{r}(t)=\widetilde{\theta _{r}}(tr^{1/2n})$.    $\square$

1.2 Model for Arm Reaching Movements

Single degree-of-freedom (dof) limb. For a 1-dof arm moving in the horizontal plane, the basic model used throughout the study was already described in numerous other studies (e.g. [4, 20, 21, 26, 49]) and is as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} I\ddot{\theta } &{} =\tau -b\dot{\theta }\\ \dot{\tau } &{} =u \end{array}\right. } \end{aligned}$$

(12)

where is $\theta$ the shoulder joint angle, $\tau$ is the muscle torque, b is the friction coefficient ($b=0.87$ here), I is the moment of inertia of the arm with respect to the shoulder joint (value estimated based upon Winter’s table for each participant; [57]) and u is the single control variable.

For the trajectory cost we typically considered canonical quadratic costs of the form $l(\mathbf {x},u)=u^{2}+\mathbf {x}^{T}Q\mathbf {x}+2\mathbf {x}^{T}Su$, where $\mathbf {x}=(\theta ,\dot{\theta },\ddot{\theta })\in \mathbb {R}^{3}$ denotes the system state. The two most famous examples are the minimum torque change corresponding to $l(\mathbf {x},u)=u^{2}$ [55] and the minimum jerk corresponding to $l(\mathbf {x},u)=\dddot{\theta }^{2}$ [18]. Other costs, possibly composite, may account for such planar movements in fixed time but such an investigation is out of the scope of the present chapter (but see [4,5,6,7, 19] for studies related to the trajectory cost identification).