Improve cart-pole example initial guess

Author: ivanightingale

class05/class05.jl

@@ -1,5 +1,5 @@
 ### A Pluto.jl notebook ###
-# v0.20.18
+# v0.20.24
 
 using Markdown
 using InteractiveUtils
@@ -493,16 +493,16 @@ begin
 	# initial guess: uniform motion, no control
 	@variable(cp_model, -d_max .<= q_1[i=1:N] .<= d_max, start=i/N * d)
 	@variable(cp_model, q_2[i=1:N], start=i/N * pi)
-	@variable(cp_model, d_q_1[i=1:N], start=0)
-	@variable(cp_model, d_q_2[i=1:N], start=0)
+	@variable(cp_model, d_q_1[1:N], start=d/T)
+	@variable(cp_model, d_q_2[1:N], start=pi/T)
 	@variable(cp_model, -u_max .<= u[1:N] .<= u_max, start=0)
 
 	# collocation points
-	@variable(cp_model, -d_max .<= q_1_c[1:N-1] .<= d_max)
-	@variable(cp_model, q_2_c[1:N-1])
-	@variable(cp_model, d_q_1_c[1:N-1])
-	@variable(cp_model, d_q_2_c[1:N-1])
-	@variable(cp_model, -u_max .<= u_c[1:N-1] .<= u_max)
+	@variable(cp_model, -d_max .<= q_1_c[i=1:N-1] .<= d_max, start=(2*i + 1)/(2 * N) * d)
+	@variable(cp_model, q_2_c[i=1:N-1], start=(2*i + 1)/(2 * N) * pi)
+	@variable(cp_model, d_q_1_c[1:N-1], start=d/T)
+	@variable(cp_model, d_q_2_c[1:N-1], start=pi/T)
+	@variable(cp_model, -u_max .<= u_c[1:N-1] .<= u_max, start=0)
 
 	@expression(cp_model, x[i=1:N], [q_1[i], q_2[i], d_q_1[i], d_q_2[i]])
 	@expression(cp_model, x_c[i=1:N-1], [q_1_c[i], q_2_c[i], d_q_1_c[i], d_q_2_c[i]])