store preffered velocity w_alpha separately from the actual v_alpha

now we track w_alpha across steps to preserve speed over time memory (equation 10 and 11)

In the pedestrian 2.pass (integrate) we now integrate dw/dt = F instead of dv/dt = F

Co-authored-by: Copilot <copilot@github.com>

main.cpp

@@ -78,10 +78,12 @@ int main(int argc, const char *argv[]) {
     for (int i = 0; i < n_p; i++) {
       Pedestrian &p = pedestrians[i];
       // # 4 flops = 2 ADDs, 2 MULTs
-      Vector2d w_alpha = p.curr_vel + p.f_i * dt;
+      // integrate preferred velocity:  dw/dt = F           (Eq. 10)
+      p.w = p.w + p.f_i * dt;
+      // Vector2d w_alpha = p.curr_vel + p.f_i * dt; // commented since this was using p.curr_vel instead of p.w (different from paper)
 
       // # 8 flops = 2 MULTs, 2 DIVs, 1 SQRT, 2 MULTs, 1 ADD
-      p.curr_vel = w_alpha * g(p.v_max, w_alpha.norm());
+      p.curr_vel = p.w * g(p.v_max, p.w.norm());
 
       // # 4 flops = 2 ADDs, 2 MULTs
       p.curr_pos = p.curr_pos + p.curr_vel * dt;

pedestrian.hpp

@@ -7,7 +7,8 @@ enum Color { BLACK, WHITE };
 
 struct Pedestrian {
   Color color;
-  Vector2d curr_vel;
+  Vector2d curr_vel; // v_alpha  (actual, ≤ v_max)
+  Vector2d w;        // w_alpha  (preferred, integrates force)
   Vector2d curr_pos;
   Vector2d e;
   double v_max;
@@ -17,5 +18,7 @@ struct Pedestrian {
 
   Pedestrian() : color(WHITE), curr_vel(0.0, 0.0), curr_pos(0.0, 0.0), e(0.0, 0.0), v_max(0.0), desired_v0(0.0), destination(0.0, 0.0), f_i(0.0, 0.0) {}
 
-  Pedestrian(Color clr, Vector2d c_v, Vector2d c_p, Vector2d e, double v_max, double desired_v0, Vector2d d, Vector2d f) : color(clr), curr_vel(c_v), curr_pos(c_p), e(e), v_max(v_max), desired_v0(desired_v0), destination(d), f_i(f) {}
+  Pedestrian(Color clr, Vector2d c_v, Vector2d c_p, Vector2d e, double v_max, double desired_v0, Vector2d d, Vector2d f)
+      : color(clr), curr_vel(c_v), w(c_v), // initialize w = v at t=0
+        curr_pos(c_p), e(e), v_max(v_max), desired_v0(desired_v0), destination(d), f_i(f) {}
 };