[SOLVED] Psl432 theoretical physiology assignment 1

Your job is to simulate a simplified, two-joint arm and design a policy
with Hurwitz desired dynamics and internal feedback that makes the
arm reach to and track a moving target.
In continuous time, the arm’s dynamics (apart from joint-bounding,
described below) are described by the differential equation,
1 2 2 3 2 2 2 1 2
2 2
3 2 2 3 1
( ) ( , )
2 cos( ) cos( ) ( )
sin( ) ,
cos( ) 0
M q q q q q
ψ ψ q ψ ψ q q q q
q ψ q q
ψ ψ q ψ q
   
        
          
where q is the configuration vector containing the shoulder and elbow
angles in that order, q
and
q
are velocity and acceleration, τ is the
vector of torques at the two joints, and ψ1, ψ2, and ψ3 are constants: ψ1
= 1.88, ψ2 = 0.6, and ψ3 = 0.48.
The torque vector τ itself depends on the neural command or action according to the equation,
    a ,
where ζ = 0.75.
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In your code, use the name torque for the τ vector, and zeta for ζ, and
integrate the dynamics using Euler’s method, with ∆t = 0.01 s.
You should also bound the joint angles to keep them inside their motion ranges. That is, define q_min = [-2; 0], q_max = [1.5; 2.5],
and after Euler integration write:
q = max(q_min, min(q_max, q));
q_vel = q_vel – q_vel.*( (q == q_max).*(q_vel > 0) + …
(q == q_min).*(q_vel < 0) );
Program the arm’s reaching target (the desired joint-angle vector q*)
so that it jumps, every 1 s, to a new, random location inside the arm’s
joint ranges and then glides at a random, constant velocity between
jumps, with speeds ranging between 0 and about 2.5 to 3.5 rad/s, always staying inside the joint ranges, as shown by the dotted lines in
the plot on the next page.
Choose Hurwitz desired dynamics that get both joints to their target
values within about 0.75 s or quicker and then keep them on the gliding target until the next jump. Design a policy that implements those
desired dynamics and keeps both elements of a smaller than about
25,000 in absolute value (you needn’t prove mathematically that a
never crosses those bounds — just look at your simulations to check
that it usually stays in about that range).
Your agent must rely on internal feedback. The only things it can
know directly are its own action a, the position and velocity of the target, and the forms of the dynamics equations including the joint
bounds and where the various constants and variables (ψ1, ψ2, ψ3, q,
q
(1)
, a and so on) plug into those equations. Give your agent exactly
correct estimates of ψ1, ψ2, ψ3, and ζ, and have it use internal simulations to estimate q and q
(1) and any other necessary variables. Give the
agent correct initial estimates of those variables, and reset those estimates to their correct values every time the target jumps. You can use
the sample code Saccadic_internal.m as a partial guide, but your agent
should not assume that actual velocity equals desired velocity.
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The agent doesn’t know anything else about the target except its current position and velocity. For instance, it doesn’t know that the target
will jump at 1-s intervals or maintain a constant velocity between
jumps.
In the handout notes, we derived policies using scalar operations, but
here you will need vectors and matrices, because q has more than one
element. For vector and matrix operations anywhere in your code, use
Matlab’s built-in functions, e.g. if you need to multiply A and B or invert or transpose A, then write A*B or inv(A) or A’ — don’t write out
the complicated, element-wise formulae for these operations.
Display your results in a figure with 2 panels. In the upper panel, plot
shoulder and elbow angles (as red and blue solid lines) and their targets (as red and blue dotted lines) versus time, with y-axis limits set
using ylim(1.05*[min(q_min), max(q_max)]). In the lower panel,
plot both elements of a versus time. Simulate 10 s in all, or in other
words 10 reaches to 10 targets. If all goes well, your upper-panel plot
should look something like this:
Please name your variables as in these pages, the notes, and the sample
code: q, q_star, q_vel, q_acc, psi, zeta, M, GAMMA, a, and so on.
Submit an m-file called Yourgivennameinitial_Yoursurname_A1, as
for example D_Tweed_A1.m.