*4.1. Simulation Validation of Solution Functionality*

The functionality of the proposed solution was validated by simulations using the MATLAB software tool (R2018a, The MathWorks, Inc., Natic, MA, USA, 2018). The designed algorithms were processed by a script and the results were represented in graphical form.

Duration times *T1,j* of master manipulator *M1* operations were simulated as generated random values of a vector in the range of 0–5000 ms. Each value *T1,j* was modified during its processing with a random element from the range (−2%, +2%), that represents a stochastic process part. Because the synchronization algorithm for all slave manipulators *MSi* is identical, the case with one slave manipulator *MSi*, where *i* = 2, was validated by simulation. Operation durations *Ti,j* of this slave manipulator were simulated on the basis of Equation (6).

$$T\_{i,j} = \mathcal{S}\_{i,j} \mathcal{V}\_{i,j} \tag{6}$$

In Equation (6) the variable *Si,j* - *S* = *{S2,1,* ... *, Sn,m}* represents the path length of operation *Oj* performed by manipulator *MSi*. An idealized kinematic model of a manipulator with omission of the non-linear character of robot arm movement in acceleration and deceleration was used for simulation validation purposes. Elements of the *S* set were generated as random values of a vector in the range of 0–1000 mm in the simulation validation process. The default endpoint movement speed *Vi,j* of the slave manipulator was set to 200 mm/s. In processing of each operation duration *Ti,j*, an additional modification with the stochastic part from the interval (−2%, +2%) of *Ti,j* was also used.

4.1.1. Experiment 1a—Master Speed Change, Basic Algorithm

Simulation Experiment 1a includes:

	- in the 3rd production cycle by +20% of current speed value
	- in the 4th production cycle by +30% of current speed value
	- in the 7th production cycle by −50% of current speed value

4.1.2. Experiment 2a—Slave Speed Change, Basic Algorithm

Simulation Experiment 2a includes:

	- in the 2nd production cycle by −30% of current speed value
	- in the 4th production cycle by +30% of current speed value

#### 4.1.3. Summary of Experiment 1a and Experiment 2a

The obtained simulation results of the experiment with master speed change are depicted in Figures 4 and 5. Results of both experiments, experiment with master speed change and the experiment with slave speed change, is aggregated in Figure 6, and the results of the experiment with slave speed change in Figures 7 and 8. Figure 9 also presents results of both experiments, experiment with master speed change and the experiment with slave speed change.

**Figure 4.** Operation duration times of master and slave manipulators without feedforward synchronization—Experiment 1a.

**Figure 5.** Percentage proportion deviation without feedforward synchronization—Experiment 1a.

**Figure 6.** Brief summary of Experiment 1a results.

**Figure 7.** Operation duration times of master and slave manipulators without feedforward synchronization—Experiment 2a.

**Figure 8.** Percentage proportion deviation without feedforward synchronization—Experiment 2a.

**Figure 9.** Brief summary of Experiment 2a results.

4.1.4. Experiment 1b—Master Speed Change, Advanced Algorithm

The case with a 10% limit for the error of the difference between operation duration performed by master and slave manipulators and a 15% significance limit for the similarity of two successive deviations *Di,j* and *Di,j* <sup>+</sup> *<sup>1</sup>* was validated via simulation.

In simulation Experiment 1b, equal parameters of the whole simulated system were used as in simulation Experiment 1a, with the addition of the feedforward synchronization feature to the synchronization algorithm.
