Velocity scaling of cue-induced smooth pursuit acceleration obeys constraints of natural motion

Abstract

Information about the future trajectory of a visual target is contained not only in the history of target motion but also in static visual cues, e.g., the street provides information about the car’s future trajectory. For most natural moving targets, this information imposes strong constraints on the relation between velocity and acceleration which can be exploited by predictive smooth pursuit mechanisms. We questioned how cue-induced predictive changes in pursuit direction depend on target speed and how cue- and target-induced pursuit interact. Subjects pursued a target entering a ±90° curve and moving on either a homogeneous background or on a low contrast static band indicating the future trajectory. The cue induced a predictive change of pursuit direction, which occurred before curve onset of the target. The predictive velocity component orthogonal to the initial pursuit direction started later and became faster with increasing target velocity. The predictive eye acceleration increased quadratically with target velocity and was independent of the initial target direction. After curve onset, cue- and target-induced pursuit velocity components were not linearly superimposed. The quadratic increase of eye acceleration with target velocity is consistent with the natural velocity scaling implied by the two-thirds power law, which is a characteristic of biological controlled movements. Comparison with linear pursuit models reveals that the ratio between eye acceleration and actual or expected retinal slip cannot be considered a constant gain factor. To obey a natural velocity scaling, this acceleration gain must linearly increase with target or pursuit velocity. We suggest that gain control mechanisms, which affect target-induced changes of pursuit velocity, act similarly on predictive changes of pursuit induced by static visual cues.

Appendix

In this appendix we analyze pursuit systems that are able to generalize their response to a target moving along a fixed path at a variety of target velocities. Proper generalization is here defined by the property that the position error is, for any given target trajectory, invariant to temporal scaling of target motion. It will be shown that this ability to generalize across different target speeds implies a quadratic relation between target speed and the acceleration generated by the pursuit system. For simplicity we first (in Appendix 1) restrict the consideration to constant tangential velocities, which is a quite natural situation when the movement of a target is diverted by passive constraints such as borders of a mechanically defined path. In Appendix 2, we generalize the conclusion to a certain type of scaling of tangential velocities that change along the target path. It is important to note that this type of scaling is compatible with the type of velocity scaling that is, according to the two-thirds power law, applied in natural biologically controlled motion. Finally, in Appendix 3, we show that a simple 2D velocity servo controller that has the ability to generalize across different target velocities as defined above must increase its feed-forward gain linearly with the tangential target velocity.

Appendix 1: Constant tangential velocity

Let $\underline{{\dot{p}}} (t,\,v_{0})$ be the 2D velocity of the target moving with constant tangential velocity v ₀ along a fixed path $\underline{q} (l),$ parameterized according to its path length l. Moving with the velocity v ₀, the target trajectory $\underline{p} (t,\,v_{0})$ can then be computed by $\underline{p} (t,\,v_{0}) = \underline{q} (v_{0} \cdot t).$

Using the scaling factor $a: = \frac{v}{{v_{0}}},$ we can now compute the 2D target position and velocity for any other tangential target velocity v as follows:

$$ \underline{p} (t,\,v) = \underline{q} (v \cdot t) = \underline{q} (v_{0} \cdot a \cdot t) = \underline{p} (a \cdot t,\,v_{0}). $$

(4a)

Differentiating two times with respect to time (t) yields

$$ \underline{\dot{p}} (t,\,v) = a \cdot \underline{\dot{p}} {\left({a \cdot t,\;v_{0}} \right)}\,\hbox{and}\; \underline{\ddot{p}} (t,\,v) = a^{2} \cdot \underline{\ddot{p}} {\left({a \cdot t,\;v_{0}} \right)}. $$

(4b)

This means that changes of the tangential velocity result only in temporal scaling of the 2D-position, but in both temporal and metric scaling of the 2D-velocity. Invariance of the position error to changes of the tangential target velocity implies that the position output $\underline{x} (t,v)$ of a pursuit system is scaled in the same way as $\underline{p} (t,\,v).$ Thus, for an invariant position error, we obtain the following relationship for the position output:

$$ \underline{x} (t,\,v) = \underline{x} (a \cdot t,\,v_{0}). $$

(5a)

By differentiating this equation two times with respect to time, we obtain

$$ \underline{{\dot{x}}} (t,\,v) = a \cdot \underline{{\dot{x}}} {\left({a \cdot t,\;v_{0}} \right)}\,\hbox{and}\,\underline{{\ddot{x}}} (t,\,v) = a^{2} \cdot \underline{{\ddot{x}}} {\left({a \cdot t,\;v_{0}} \right)}. $$

(5b)

Thus, invariance of the position error to changes of the tangential target velocity implies that the output velocity $\underline{{\dot{x}}}$ is scaled linearly, and the output acceleration $\underline{{\ddot{x}}}$ is scaled quadratically with the tangential target velocity v ₀. This holds for any system generating a response x following the input p, not only for the velocity servo shown in Fig. 8, and corresponds exactly to the velocity dependence of the cue-induced, predictive pursuit change (see Fig. 7).

Appendix 2: Time-dependent tangential velocity

If the tangential target velocity is not a constant, but changes along the target path as in our experiment, it is obvious that the derivation shown above will not hold for any change of the time course of the tangential velocity, because any arbitrary change cannot uniquely be described by a single scaling factor a. However, this section shows that the conclusion of (a) still holds for arbitrary time courses of the tangential velocity and arbitrary target paths, when they are subject to a certain type of velocity scaling. The following condition for the scaling of the tangential velocity is sufficient: The position of the target measured along its path D(t,v) and the tangential velocity V(t,v) must fulfill a differential equation of the type

$$ V(t,\,v) = \dot{D} = h(D) \cdot v; \,D(t=0, v)=0 $$

(6)

with any function h(D) that meets the two following restrictions

• 1. h(D = 0) = 1, and

• 2. Eq. 6 must have a unique solution.

The constant parameter v is the tangential velocity at the beginning of the path, it was used in our experiment to specify the target velocity. It is easy to see that the trajectories generated on the basis of the two-thirds power law applied in our experiment (Eq. 2) meet this differential equation, because in Eq. 2, the factor K is proportional to the velocity at the beginning of the curve, and the factor ${\left(\frac{r}{{1 + \alpha \cdot r}} \right)}^{{1 - \beta}}$ can for any path be expressed as a function of D. When solving equation 6 by integration

$$ D(t,\,v) = {\int\limits_{u = 0}^t {h(D) \cdot v\,du}} = {\int\limits_{u = 0}^t {h(D) \cdot v_{0} \cdot a\,\;du}}\;\left(\hbox{with}\; a: = \frac{v}{{v_{0}}}\right), $$

substitution of a· u by y leads to

$$ D(t,\,v) = {\int\limits_{y = 0}^{a \cdot t} {h(D) \cdot v_{0} \,dy}}, $$

showing that

$$ D(t,\,v) = D(a \cdot t,\,v_{0}).\,\hbox{and} $$

$$ V(t,v) = a \cdot V(a \cdot t,\,v_{0})\,\hbox{with}\,V(t=0,v)=v. $$

With these properties, we obtain for the 2D-position of the target:

$$ \underline{p} (t,\,v) = \underline{q} (D(t,\,v)) = \underline{q} (D(a \cdot t,\,v_{0})) = \underline{p} (a \cdot t,\,v_{0}). $$

These equations show that the 2D target position complies with the same type of velocity scaling as already found for changes of constant tangential velocity in Eq. 4a. Because this scaling type is a characteristic of the two-thirds power law describing natural movements, we call this type “natural” velocity scaling. In analogy to the last paragraph, it follows from Eq. 4a that a position error that is invariant to natural velocity scaling according to Eq. 6 implies a quadratic relation between target speed and the output acceleration of the system.

3. An implication of scaling invariance in simple velocity servos and other linear systems

Let $\underline{{\dot{x}}} (t,\,v_{0})$ be the velocity output of a simple velocity servo (Fig. 8) in response to the velocity input $\underline{{\dot{p}}} (t,\,v_{0}).$ This velocity output satisfies the following differential equation:

$$ \underline{{\ddot{x}}} (t,\,v_{0}) = g(v_{0}) \cdot {\left({\underline{{\dot{p}}} (t,\,v_{0}) - \underline{{\dot{x}}} (t,\,v_{0})} \right)} $$

(7)

We assume that the gain depends on the tangential velocity v ₀ with an unknown function g(v ₀). The goal of the following derivation is to identify this gain controller under the constraint that the position error of the servo is invariant to the natural velocity scaling described in (a) and (b). Thus, we assume that the servo output satisfies Eqs. (5a) and (5b).

We set up the differential equation of the velocity servo for any arbitrary tangential velocity v,

$$ \underline{{\ddot{x}}} (t,\,v) = g(v) \cdot {\left({\underline{{\dot{p}}} (t,\,v) - \underline{{\dot{x}}} (t,\,v)} \right)}, $$

and rewrite it using Eqs. 4a, 4b and 5a, 5b:

$$ a^{2} \cdot \underline{{\ddot{x}}} (a \cdot t,\,v_{0}) = g(v) \cdot a \cdot {\left({\underline{{\dot{p}}} (a \cdot t,\,v_{0}) - \underline{{\dot{x}}} (a \cdot t,\,v_{0})} \right)}\,(\hbox{again with}\; a: = \frac{v}{{v_{0}}}). $$

After dividing by a ² we obtain

$$ \underline{{\ddot{x}}} (a \cdot t,\,v_{0}) = \frac{1}{a}g(v) \cdot {\left({\underline{{\dot{p}}} (a \cdot t,\,v_{0}) - \underline{{\dot{x}}} (a \cdot t,\,v_{0})} \right)}. $$

(8)

Since the differential equation of the original servo Eq. 7 holds for any time, we can replace t by a· t in Eq. 7. This yields

$$ \underline{{\ddot{x}}} (a \cdot t,\,v_{0}) = g(v_{0}) \cdot {\left({\underline{{\dot{p}}} (a \cdot t,\,v_{0}) - \underline{{\dot{x}}} (a \cdot t,\,v_{0})} \right)}. $$

(9)

Finally, by inserting Eq. 9 in Eq. 8, we see that the modified servo leading to the same position error as the original one is identified by

$g(v) = a \cdot g(v_{0}) = \frac{v}{{v_{0}}} \cdot g(v_{0}),$ i.e., the servo gain must be adjusted proportionally to the tangential target velocity in order to make its position error invariant to natural velocity scaling.

Using the Laplace theory of linear differential equations, we can easily verify that the accuracy of any linear system characterized by the transfer function F(s) (where s denotes the complex frequency) is invariant to natural velocity scaling, only if F(s) depends on the target velocity in a very specific way. Given that the input (p(t,v)) and output signals (x(t,v)) of the system are scaled according to Eqs. 4a and 5a, the corresponding Laplace-transformed functions are scaled as follows:

$$ \underline{P} (s,v) = \frac{1}{a} \cdot \underline{P} {\left({\frac{s}{a},v_{0}} \right)}\,\hbox{and}\;\underline{X} (s,v) = \frac{1}{a} \cdot \underline{X} {\left(\frac{s}{a},v_{0} \right)}. $$

(10)

Since in the Laplace domain the input–output relation is given by a multiplication,

$$ \underline{X} (s,v) = {{\mathbf{F}}}(s,v) \cdot \underline{P} (s,v),\,\hbox{and}\;\underline{X} {\left({s,v_{0}} \right)} = {{\mathbf{F}}}(s,v_{0}) \cdot \underline{P} (s,v_{0}), $$

(11)

inserting Eq. 10 in Eq. 11 reveals the required scaling property of the transfer function F:

$$ {{\mathbf{F}}}(s,v) = {\mathbf{F}}\left(\frac{s}{a},v_{0} \right).$$

(12)

Thus, the invariance of the accuracy of the system to natural velocity scaling requires frequency scaling of the Laplace transfer function. Since such frequency-scaling involves a dynamic change of the coefficients of the linear differential equations defining the transfer function, invariance to natural velocity scaling cannot be achieved by any time-invariant linear system. Applying Eq. 12 to the Laplace transfer function of the velocity servo shown in Fig. 8 reveals that this example is a special case of the more general formulation of Eq. 12. If the integrator gain is assumed to be unity at target velocity v ₀, the transfer function of the servo becomes

$$ {{\mathbf{F}}}(s,v_{0}) = \frac{{\frac{1}{s}}}{\frac{1}{s}+ 1}. $$

To achieve invariance of the accuracy of the servo to natural velocity scaling of the input, the complex frequency s must be divided by the scaling factor a (Eq. 12)

$$ {{\mathbf{F}}}(s,v) = {{\mathbf{F}}}{\left({\frac{s}{a},v_{0}} \right)} = \frac{{\frac{a}{s}}}{\frac{a}{s} + 1}. $$

Since the transfer function of the integrator equals $\frac{1}{s},$ dividing s by a is identical with multiplication of the integrator gain with a.