## Sideways velocity control 1D

To investigate the consequences of sideways velocity control a configuration is chosen which in fact is a one dimensional reduction of the** complete model. **The gain for rotation control KrB has been chosen zero and the gain K_{r} for distance control has been chosen zero too (KrB=0 and K_{r}=0). The female is supposed to move with constant velocity V_{F} along a straight line. The male body axis is chosen perpendicular to the female path. Therefore in this case the male is moving parallel to the path of the female, as given in the figure below.

In the figure the female (indicated by green dots at time k*T) is moving with constant velocity V_{F} to the right. The male direction is –π/2. When the control system is stable, after some time K_{t}*θe_{filt}(k)=V_{F} and therefore K_{t}* θ_{e}(k)=V_{F} for k large.

*The algorithm for this configuration with KrB=0 and K*_{r}=0 is reduced from the 2D to the 1D case. x_{M} , v_{M} and x_{L} are real numbers now.

Define startconditions for k=1

for k=2 to N_{end}

x_{M}(k)=x_{M}(k-1)+ v_{M}(k-1)*T

θ_{e}(k)=asin((x_{F}(k)-x_{M}(k))/|r|)

θe_{filt}(k)=( exp(-T/τ_{f}))* θe_{filt}(k-1)+(1- exp(-T/τ_{f}))* θ_{e}(k).

v_{M}(k)=(exp(-T/τ))*v_{M}(k-1)+(1- exp(-T/τ))* K_{t}*θe_{filt}(k)

end

The stability of this control algorithm will now be investigated.

To be able to apply the linear theory, the expression θ_{e}(k)=asin((x_{F}(k)-x_{M}(k))/|r|)

is linearized using the approximation *asin*(α)=α for small α. Using this approximation θ_{e}(k)=(x_{F}(k)-x_{M}(k))/|r|. The continuous block diagram is given below.

This third order system will eventually oscillate with increasing K_{t}. Harmonic oscillation with frequency ω rad/s and constant amplitude will occur when

atan(ω.τ_{f})+atan(ω.τ)+π/2= π or atan(ω.τ_{f})+atan(ω.τ)= π/2 (1) and

Solution of (1) is ω^{2 }= 1/(τ.τ_{f}).

Substitution in the second relation delivers K_{t}=|r|*(1/τ+1/τ_{f}).