Chapter 6: Introduction of Uniform Model Structures for Partial, Temporal, and
A.4. Simulation Results for Section “6.2.3. Application Example for the Use of
Mechanics”
In the forthcoming simulation examples the same robot arm structure was used as that of Fig. A.4.1. [its Euler-Lagrange equations of motion are given in (A.4.1) and (A.4.2)].
q
Figure A.4.1. The particular paradigm considered
( )
qq k( ) ( )
q q gq QThe endpoint of the arm was connected to a dashpot producing elastic spring forces (with stiffness of 600 [N/m]) and viscous damping (100 [Ns/m]) as external perturbations. This manipulator arm consists of a vertical rod of 5 kg moving up and down (q1 in m), rotating around itself as a vertical axis (q2 in rad), and a second rod joined to it by a wrist tilting around a horizontal axis (q3 in rad). This latter joint also was translated by q1 and rotated by q2. The second rod had negligible mass but carried a point-like small body of variable mass. It also had constant length (R0=3 [m]). The three axes were controlled by drives exerting force for q1 and torque for q2 and q3 prescribed by the control strategy. In each case considered the end-point of the robot arm was desired to be moved with circular frequency Ω [rad/s] along a circle of 0.5 m radius lying in a vertical plane at a distance of 2 m from the vertical axis. In each case the "initial rough estimation" of the dynamic model consisted of a non-singular, constant inertia matrix and a constant gravitational term. No quadratic velocity coupling was taken into account. Making all the further corrections was the task of the Symplectizing Algorithm.
The “canonical coordinates” without system identification were [qT,(Mdq/dt)T]T with the estimated model inertia M. Simulations were run for pure application of the Symplectizing Algorithm and with complementary tuning only one of the Symplectic matrices in (6.2.5), namely matrix B by matrix P.
-14.8 -7.3 0.1 7.6 -30
-20 -10 0 10 20 30
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.1 -6.2 0.7 7.6
-30 -20 -10 0 10 20 30
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-3 -2 -1 0 1
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
-14.8 -7.3 0.1 7.6
-30 -20 -10 0 10 20 30
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.1 -6.2 0.7 7.6
-30 -20 -10 0 10 20 30
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-3 -2 -1 0 1
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
Figure A.4.2. Simulation results for “slow” motion Ω=5 [rad/s] without (LHS) and with (RHS) external perturbation without complementary tuning (the first two rows) and with complementary tuning of step length 10-8 [dimensionless] (3rd and 4th rows)
[q1:black, q2:blue, q3:green lines]
(For the sake of simplicity P*≡I was investigated only with independent tuned variables ϕ in (6.2.21) and ψ in (6.2.24).) In the simulations in the first half of the time considered no any Symplectic identification was applied, only the rough initial model was in use. In the second half of the time of the investigations the [qT,(Mdq/dt)T]T “canonical coordinates” were transformed by Symplectic matrices obtained by the Symplectizing Algorithm and the additional tuning if it was applied.
For trajectory tracking the purely kinematically formulated
(
R N) (
R N)
N
D q bq q cq q
q& =&& − & −& − −
& (A.4.3)
error relaxation was prescribed with b=30 [1/s], and c=0.8×(b2)/4 [1/s2] that guarantees oscillation-free desired tracking (the superscripts “R”, “N” and “D”
corresponds to the realized, the nominal, and the desired quantities).
In the calculations Ω=5 [rad/s], Ω=10 [rad/s], Ω=20 [rad/s] and Ω=25 [rad/s]
nominal motions were considered referred to as “slow”, “normal”, “fast”, and “very fast” nominal motions. The controller’s cycle time was supposed to be 1 [ms].
The phase trajectories and the tracking errors with and without external perturbation (i.e. the dashpot) and with and without complementary tuning for “slow”
motion are given in Fig. A.4.2. It is clear that in each case turning on the symplectic identification considerably improves the tracking accuracy and the phase trajectory, too. Due to the essentially exponential nature of the generators of the fine tuning in (6.2.21) and (6.2.24) small step length in fine tuning (10-8 [dimensionless]) was found to be reasonable. At slow motion no essential improvement by fine tuning was achieved.
0.000 2.67 5.33 8.00
10 20 30 40
Phenomenology Test: Tail -- Time [10^0 vs Time s]
0.00 2.67 5.33 8.00
0 10 20 30 40
Phenomenology Test: Tail -- Time [10^0 vs Time s]
0.00 2.67 5.33 8.00
-20 -10 0 10 20
The Greneralized Forces Q [10^2] Q1[N], Q2,Q3[Nm] vs Time s
0.00 2.67 5.33 8.00
-30 -20 -10 0 10 20 30 40
The Greneralized Forces Q [10^2] Q1[N], Q2,Q3[Nm] vs Time s
Figure A.4.3. Simulation results for “slow” motion Ω=5 [rad/s] without (LHS) and with (RHS) external perturbation using only the Symplectizing Algorithm: the norm of the “truncated generalized force components” (1st row) and the generalized forces
(2nd row) [q1:black, q2:blue, q3:green lines]
Since the main “tool of system identification” is the Symplectizing Algorithm in these simulations obtaining precisely and exceptionally “phenomenologically correct” block diagonal transformations was not guaranteed. On this reason the illegally nonzero components of the transformed generalized forces have been simply truncated from the resulting force after executing the multiplication by the symplectic matrix. In the sequel this “truncated” part (more precisely its norm according to Frobenius) is referred to as “tail” and is described in the charts called
“Phenomenology Test”.
-14.2 -6.9 0.3 7.6
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.0 -6.1 0.8 7.6
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-4
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-3.0
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
-14.2 -6.9 0.3 7.6
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.0 -6.1 0.8 7.6
Phase Spaces [10^-1 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-4
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-3.0
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
Figure A.4.4. Simulation results for “normal” motion Ω=10 [rad/s], without (LHS) and with (RHS) external perturbation without complementary tuning (the first two rows) and with complementary tuning of step length 10-6 [dimensionless] (3rd and 4th
rows) [q1:black, q2:blue, q3:green lines]
In Fig. A.4.3. the norm of the truncated components and the generalized forces are described for “slow” motion and the use of the “pure Symplectizing Algorithm”. It can be seen that while the generalized forces are in the ≈500 or ≈1000 [N] or [Nm], the norm of the truncated components is small, about ≈5 ≈8 [N] or [Nm]
only (with the exception of certain “extreme points” in which they achieve ≈40 [N]
or [Nm] that is also small in comparison with the full force components). (On this
reason was allowed in the fine tuning the use of non-block-diagonal generators, too.) The generally increased force component at the right hand side of the figures in Q2
reveals the significance of the external perturbations. It is worth noting that the fine tuning did not essentially influence these charts in Fig. A.4.3., therefore, for saving room, these charts are not described here.
The appropriate counterpart of Fig. A.4.2. for “normal” speed of Ω=10 [rad/s] is given in Fig. A.4.4. It can well be seen that in this case the fine tuning definitely improved the tracking accuracy. Figure A.4.5. reveals some details on the variation of the generalized forces and the tuned parameters versus time. It has to be noted that for 10 consecutive steps only parameter ϕ, and following that, for the next 10 steps only parameter ψ was tuned by the Simplex Algorithm.
0.00 2.67 5.33 8.00
-40 -30 -20 -10 0 10 20 30
The Greneralized Forces Q [10^2] Q1[N], Q2,Q3[Nm] vs Time s
0.00 2.67 5.33 8.00
-60 -40 -20 0 20 40 60 80
The Greneralized Forces Q [10^2] Q1[N], Q2,Q3[Nm] vs Time s
0.00 2.67 5.33 8.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Variable phi [10^-3] vs Time s
0.00 2.67 5.33 8.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Variable phi [10^-3] vs Time s
0.00 2.67 5.33 8.00
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Variable psi [10^-4] vs Time s
0.00 2.67 5.33 8.00
0 1 2 3 4 5 6 7
Variable psi [10^-4] vs Time s
Figure A.4.5. Simulation results for “normal” motion Ω=10 [rad/s] without (LHS) and with (RHS) external perturbation (1st row) and the variation of the tuned
parameters (2nd row and 3rd row ) [q1:black, q2:blue, q3:green lines]
Similar observations can be done in the case of “fast” motion of Ω=20 [rad/s]
when the fine tuning rather “smoothes” the phase trajectories and has less influence on the tracking errors (Fig. A.4.6.).
In the case of the “very fast” motion (Fig. A.4.7.) it can well be observed that the fine tuning keeps the tracking errors “at bay”, i.e. makes their variation less chaotic than without fine tuning.
-12.7 -5.8 1.1 8.0
-15 -10 -5 0 5 10 15
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.2 -6.1 1.0 8.0
-15 -10 -5 0 5 10 15
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-5 -4 -3 -2 -1 0 1 2
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-4 -2 0 2
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
-12.8 -5.9 1.0 8.0
-15 -10 -5 0 5 10 15
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13 -6 1 8
-15 -10 -5 0 5 10 15
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-5 -4 -3 -2 -1 0 1 2
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-4 -2 0 2
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
Figure A.4.6. Simulation results for “fast” motion Ω=20 [rad/s], without (LHS) and with (RHS) external perturbation without complementary tuning (the first two rows)
and with complementary tuning of step length 5×10-6 [dimensionless] (3rd and 4th rows) [q1:black, q2:blue, q3:green lines]
-16.5 -8.3 -0.0 8.2 -20
-15 -10 -5 0 5 10 15
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.9 -6.1 1.7 9.5
-15 -10 -5 0 5 10 15 20 25
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-5 -4 -3 -2 -1 0 1 2 3
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-4 -2 0 2 4
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
-14.9 -7.1 0.6 8.3
-20 -10 0 10 20
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
-13.7 -6.3 1.0 8.3
-15 -10 -5 0 5 10 15 20
Phase Spaces [10^0 vs 10^-1] q1[m/s vs m], q2,q3[rad/s vs rad]
0.00 2.67 5.33 8.00
-5 -4 -3 -2 -1 0 1 2 3
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
0.00 2.67 5.33 8.00
-4 -2 0 2
Tracking Error [10^-1 q1[m], q2,q3[rad] vs Time s]
Figure A.4.7. Simulation results for “very fast” motion Ω=25 [rad/s], without (LHS) and with (RHS) external perturbation without complementary tuning (the first two rows) and with complementary tuning of step length 10-5 [dimensionless] (3rd and 4th
rows) [q1:black, q2:blue, q3:green lines]
A.5. Simulation Results for Section “7.3. Simulation Example for Potential