I am using GEKKO in Python to estimate trajectory of a bouncing ball. For that i need to estimate 2 variables: e_1(coefficient_of_restitution) and q_1(horizontal_velocity loss at each bounce).
I have written the following code for it but the parameters dose not seems to be updated, although the solver is executed successfully. The initial value of parameters are same as the final optimized value of parameters e_1= 0.8 and q_1= 1.

CODE:

import numpy as np
from gekko import GEKKO

# Actual trajectories
# actually y
act_x = np.array([0.019053200000000013, 0.08770320000000008, 0.1850550000000004, 0.2825550000000008, 0.38005500000000114, 0.4775550000000015, 0.5750549999999806, 0.6725549999999532, 0.7700549999999258, 0.8675549999998984, 0.965054999999871, 1.0625549999998436, 1.1600549999998162, 1.2575549999997888, 1.3550549999997614, 1.452554999999734, 1.5500549999997066, 1.6475549999996792, 1.7450549999996519, 1.8425549999996245, 1.940054999999597, 2.0375549999996125, 2.1234107142854164, 2.193053571428365, 2.2626964285713136, 2.332339285714262, 2.401982142857211, 2.4716250000001594, 2.541267857143108, 2.6109107142860566, 2.680553571429005, 2.750196428571954, 2.8198392857149024, 2.889482142857851, 2.9591250000007996, 3.028767857143748, 3.098410714286697, 3.1680535714296454, 3.237696428572594, 3.3073392857155426, 3.376982142858491, 3.44662500000144, 3.5162678571443884, 3.585910714287337, 3.6555535714302856, 3.725196428573234, 3.794839285716183, 3.8644821428591314, 3.93412500000208, 4.003767857145029, 4.073410714287977, 4.143053571430926, 4.212696428573874, 4.282339285716823, 4.351982142859772, 4.42162500000272, 4.491267857145669, 4.560910714288617, 4.630553571431566, 4.700196428574515, 4.769839285717463, 4.839482142860412, 4.90912500000336, 4.978767857146309, 5.048410714289258, 5.118053571432206, 5.187696428575155, 5.257339285718103, 5.326982142861052, 5.396625000004001, 5.466267857146949, 5.535910714289898, 5.605553571432846, 5.675196428575795, 5.744839285718744, 5.814482142861692, 5.884125000004641, 5.953767857147589, 6.023410714290538, 6.093053571433487, 6.162696428576435, 6.232339285719384, 6.301982142862332, 6.371625000005281, 6.44126785714823, 6.510910714291178, 6.580553571434127, 6.650196428577075, 6.719839285720024, 6.789482142862973, 6.859125000005921, 6.92876785714887, 6.998410714291818, 7.068053571434767, 7.137696428577716, 7.207339285720664, 7.276982142863613, 7.346625000006561, 7.41626785714951, 7.485910714292459, 7.555553571435407, 7.625196428578356, 7.694839285721304, 7.764482142864253])
# actually z
act_y = np.array([1.0379079699999998, 1.1754534700000003, 1.3602534700000006, 1.5209239699999955, 1.6570944699999859, 1.7687649699999706, 1.8559354699999533, 1.9186059699999365, 1.95677646999992, 1.970446969999904, 1.9596174699998887, 1.9242879699998736, 1.8644584699998585, 1.7801289699998444, 1.6712994699998283, 1.537969969999807, 1.3801404699997812, 1.19781096999975, 0.9909814699997134, 0.7596519699996719, 0.5038224699996257, 0.22349296999957416, 0.13930352847394978, 0.3499050645343172, 0.5360066005946793, 0.6976081366550367, 0.8347096727153887, 0.9473112087757359, 1.0354127448360804, 1.0990142808964258, 1.1381158169567716, 1.1527173530171173, 1.1428188890774638, 1.1084204251378103, 1.0495219611981568, 0.9661234972585043, 0.8582250333188504, 0.7258265693791917, 0.5689281054395274, 0.3875296414998584, 0.18163117756018424, 0.11887053865291446, 0.2798887215524012, 0.41640690445188283, 0.5284250873513592, 0.6159432702508338, 0.6789614531503088, 0.7174796360497837, 0.7314978189492595, 0.7210160018487353, 0.6860341847482112, 0.6265523676476875, 0.5425705505471646, 0.4340887334466397, 0.3011069163461103, 0.14362509924557573, 0.10758844178113687, 0.2208353837910192, 0.309582325800899, 0.37382926781077935, 0.41357620982065996, 0.42882315183054115, 0.4195700938404224, 0.385817035850304, 0.3275639778601862, 0.24481091987006875, 0.13755786187995, 0.07905142871570557, 0.1646809455657065, 0.22581046241570793, 0.2624399792657099, 0.2745694961157122, 0.26219901296571463, 0.2253285298157173, 0.16395804666572056, 0.07808756351572432, 0.09996825173824968, 0.1511630228129375, 0.17785779388762588, 0.1800525649623144, 0.15774733603700322, 0.11094210711169239, 0.05563702598852237, 0.09995862508974714, 0.11978022419097237, 0.1151018232921977, 0.08592342239342333, 0.06022248147389577, 0.08484848858531084, 0.0849744956967261, 0.06060050280814157, 0.06605277311641561, 0.06792022730639775, 0.05243051383233173, 0.060240252169206004, 0.0532114047328211, 0.05285089806699147, 0.052793140813351076, 0.05130362411928389, 0.05057861286791727, 0.049998536419320685, 0.049997973905616416, 0.049997920093917625, 0.049997920094010155])

def get_initial_trajectory(e_1, q_1):
    # Constants
    g = 9.81  # gravity (m/s^2)

    x0, y0 = 0, 1  # initial position (m)
    vx0 = 1.95     # Velocity in x direction
    vy0 = 4.4      # Velocity in y direction
    t_step = 0.01  # time step for simulation (s)
    t_max = 5  # max time for simulation (s)

    x_vals = [x0]
    y_vals = [y0]

    t = 0
    x, y = x0, y0
    vx, vy = vx0, vy0

    while t < t_max:
        t += t_step
        x += vx * t_step
        y += vy * t_step - 0.5 * g * t_step ** 2
        vy -= g * t_step

        if y <= 0:  # Ball hits the ground
            y = 0
            vy = -e_1 * vy
            vx = vx * q_1

        x_vals.append(x)
        y_vals.append(y)

        if len(x_vals) > 1 and x_vals[-1] == x_vals[-2] and y_vals[-1] == y_vals[-2]:
            break

    return np.array(x_vals, dtype='float64'), np.array(y_vals, dtype='float64')

def cost_function(e_1, q_1):
    x_vals, y_vals = get_initial_trajectory(e_1, q_1)

    y_temp = np.interp(act_x, x_vals, y_vals)

    return np.sum((act_y - y_temp)**2)

# Optimization with Gekko
m = GEKKO(remote=False)

# Define variables
e_1 = m.Var(value=0.8, name='e_1')
q_1 = m.Var(value=1, name='q_1')

m.Minimize(cost_function(e_1.value, q_1.value))

# Set the objective
m.options.SOLVER=3

# Solve the optimization problem
m.solve(disp=True)

# Print results
print(f"Estimated e_1: {e_1.value[0]}")
print(f"Estimated q_1: {q_1.value[0]}")

OUTPUT:

----------------------------------------------------------------
 APMonitor, Version 1.0.3
 APMonitor Optimization Suite
 ----------------------------------------------------------------
 
 
 --------- APM Model Size ------------
 Each time step contains
   Objects      :            0
   Constants    :            0
   Variables    :            2
   Intermediates:            0
   Connections  :            0
   Equations    :            1
   Residuals    :            1
 
 ________________________________________________
 WARNING: objective equation           1 has no variables
 ss.Eqn(1)
 0 = 12.347769475360362
 ________________________________________________
 Number of state variables:              2
 Number of total equations: -            0
 Number of slack variables: -            0
 ---------------------------------------
 Degrees of freedom       :              2
 
 solver            3  not supported
 using default solver: APOPT
 ----------------------------------------------
 Steady State Optimization with APOPT Solver
 ----------------------------------------------
    1  1.23478E+01  0.00000E+00
 Successful solution
 
 ---------------------------------------------------
 Solver         :  IPOPT (v3.12)
 Solution time  :   2.689999999711290E-002 sec
 Objective      :    12.3477694753604     
 Successful solution
 ---------------------------------------------------
 

Estimated e_1: 0.8
Estimated q_1: 1.0

Process finished with exit code 0

Let me know if any other information is required.