Question

Astronauts in training are required to practice a docking maneuver under manual control. As part of this maneuver, they are required to bring an orbiting spacecraft to rest relative to another orbiting craft. The hand controls provide for variable acceleration and deceleration, and there is a device on board that measures the rate of closing between the two vehicles. The following strategy has been proposed for bringing the craft to rest. First, look at the closing velocity. If it is zero, we are done. Otherwise, remember the closing velocity and look at the acceleration control. Move the acceleration control so that it is opposite to the closing velocity, and proportional in magnitude. After this time, look at the closing velocity again and repeat the procedure. Under what circumstances would this strategy be effective?

In the example from the textbook, there were no assumptions made with regards to any of the reaction times, but to make this a bit more concrete, we are going to assume that the astronaut’s reaction time is five seconds, that he or she waits 10 seconds before making the next observation of closing velocity, and that the constant of proportionality between the closing velocity and manual acceleration is 0.02.

Reconsider the docking problem of Example 4.3, and now assume that c=5 sec, w=10 sec, and k=0.02.

a. Assuming an initial closing velocity of 50 m/sec, calculate the sequence of velocity observations v₀, v₁, v₂. . ., predicted by the model. Is the docking procedure successful?

b. An easier way to compute the solution in part (a) is to use the iteration function G(x)=x+F(x), with the property that x(n+1)=G(x(n)). Compute the iteration function for this problem, and use it to repeat the calculation in part (a).

c. Calculate the solution x(1), x(2), x(3), . . ., starting x(0)=(1,0). Repeat, starting at x(0)=(0,1). What happens as n→∞? What does this imply about the stability of the equilibrium (0,0)? [Hint: Every possible initial condition x(0)=(a,b) can be written as a linear combination of the vectors (1,0) and (0,1) and G(x) is a linear function of x].

d. Are there any states x for which G(x)=λx for some real λ? If so, what happens to the system if we start with this initial condition?

Answer 1