Research on Speed and Acceleration of Hand Movements as Command Signals for Anthropomorphic Manipulators as a Master-Slave System

: Due to threats to human safety, remotely controlled manipulators are more and more often used to carry out rescue tasks in hazardous zones. To ensure high efﬁciency and productivity of their work, intuitive control systems are necessary, e.g., master-slave and drive systems that maximize the speed of working movements by copying the movements of the operator’s hands and are adapted to human perception and capabilities. Proper design of manipulator drive and control systems, therefore, requires knowledge of the acceleration and velocity of hand movements as signals controlling manipulators. This paper presents the results of tests of speed and acceleration in the implementation of the hand when making precise movements and moving objects over short distances (0.4–0.5 m) and during relatively long-distance reaching movements (0.73–0.93 m). Research has shown that, at short distances, the hand movements do not reach the maximum speed, while at longer distances, there is a period of constant maximal speed. In addition, studies have shown that the maximum speed of manipulation movements (longitudinal, lateral, and vertical) does not depend on the direction of movement. Moreover, precise movements were performed at a much slower velocity than reaching movements.


Introduction
Military, nuclear, biological, chemical, and environmental threats (e.g., noise, vibration, temperature) have resulted in a growing interest in manipulators for intervention activities in recent years, performing tasks in the conditions of teleoperation. IED (Improved Explosive Device) threats and the disaster in Fukushima demonstrated the need to have robots equipped with such manipulators, capable of quick interventions in unfavorable conditions. Due to the threats, robots should quickly remove obstacles to the side using manipulators (in this case, high precision of movement is not required) and carry out reconnaissance and precise interventions consisting in picking up delicate objects, opening doors, turning devices or installations on and off, etc.
Achieving high efficiency requires, in addition to a very good environmental imaging system to ensure a high level of situational and actional awareness, an intuitive control system that reduces operator training time and increases the speed of work movements.
The joystick control systems for manipulators commonly used in such robots are not very intuitive, they require extensive training to develop appropriate habits and may limit the effectiveness of robots under stressful conditions. For these reasons, the number of degrees of freedom of manipulators is limited in intervention robots-for example, the manipulator of the Talon MK IV robot has only 4 DOF. Master-slave systems controlled by hand movements (Figure 1) are definitely a better solution because they use the natural and previously acquired skills, habits, and reflexes of the operator. This allows one to Research related to the broadly understood dynamics of manipulator operation and their control is currently the subject of interest for many scientists [1][2][3][4][5]. Particular attention has been given to anthropomorphic manipulators [6][7][8][9][10] and redundant manipulators [11][12][13] because these constructions have a very high potential for use in areas requiring the implementation of unique movements changing over time and subject to human decisions and control. The conducted research concerns their application in space missions [14][15][16][17], underwater [4,18], surgery [10,19,20], telerobotic in medicine [21] and rehabilitation [22][23][24][25][26][27][28][29][30][31][32]. Many works also concern research on the cooperation of such manipulators and factors that have a decisive impact on their effectiveness [1,[33][34][35][36][37][38].
The most common intuitive control is to track the location of the characteristic points of the shoulder, elbow, and wrist [52,53], or the hand itself [10,54,79]. In the first case, all parts of the manipulator repeat hand movements, while in the second, only the effector is given the movement parameters, and the remaining parts of the manipulator adjust their position on the basis of established algorithms. This method is also used to control redundant manipulators [8,9,80]. Depending on the size of the controlled manipulator and its work area and the range of used hand movement, the measured displacements may be multiplied or reduced in order to ensure the intuitiveness of the control room and high work efficiency [81,82]. The control signals can also be modified in order to increase the accuracy of the manipulator's movement or to limit the dynamic loads [11,12].
To ensure the high efficiency of the manipulators, it is necessary to provide them with the highest possible speed of movement. The perception and the ability to carry out human movements should be the limitations. Therefore, designing an appropriate manipulator control system and ensuring high work efficiency requires knowledge of the control signals that can be generated by a human hand. Therefore, it is necessary to know the maximum speed of movement of hands in different directions and occurring accelerations. This will allow for the proper shaping of the manipulator drive and control.
The studies of the dynamics of the human hand usually concern the kinematic structure, trajectory of movement, and problems in tracking the position and mathematical description of the dynamics of its operation by equations [83][84][85]. There is a lack of information on the actual hand speeds and accelerations occurring during manipulation, which should inform the control systems used in manipulators in a master-slave tracking control system. It should be noted that these may vary depending on the activities carried out. The reaching movements are usually faster, while the precise movements require lower speeds.
In order to better understand the dynamic processes taking place during the control of manipulators in a master-slave system, research was carried out to determine the speed and acceleration of the human hand during reaching movements and precise movements. It is expected that, thanks to the appropriate design of the drive and control system, the manipulator's foundry and salvage will be able to copy reaching movements of the human hand and slower movements requiring high precision, providing the operator with a sense of full and conscious control of the manipulator [85].
Most of the conducted research concerns the possibility of increasing the precision of manipulators' movements in conditions of limited data transmission and large time delays. In these studies, waveforms representing relatively slow movements with a limited range are most often used as control signals. Conducting effective rescue operations in areas of destruction and catastrophes requires the ability to implement not only precise movements but also fast, long-range movements in the entire manipulator's field of operation. Similar requirements apply to, e.g., foundry manipulators, segregation manipulators, or mining manipulators for crushing boulders that are too large. The development of new 5G data transmission technologies indicates the possibility of significantly reducing problems with data transmission [86][87][88][89][90][91]. Therefore, there is a need to know the control signals that can be generated by a human while controlling manipulators in hazardous areas. Their knowledge will allow for proper design of drive systems (achieving the expected speeds of movements), control of rescue manipulators (ensuring the expected precision and stability of movement), and defining the existing dynamic loads of the manipulator structure.
Analyzing the real inputs from human hand movements to the control system, the following null hypotheses were also adopted:

1.
Speeds of human hand movements, as inputs introduced into the manipulator control system, strongly depend on the direction of movement (longitudinal, lateral, and vertical)-there are significant differences between speed and direction of movement, which significantly affects the design of the drive system and manipulator control system, and different control procedures are necessary depending on the direction of movement.

2.
The maximum speeds of human hand movements during the execution of delivery and precision tasks, which are to be copied in real time by intervention manipulators, do not differ significantly.

Experimental Setting and User Task
The primary purpose of the research was to learn about the mean maximum velocity and mean maximum acceleration of human hand movement during the implementation of manipulation tasks and the influence of the direction of movement on the achieved speeds and accelerations.
The study was divided into two stages. The first stage concerned the analysis of the reaching movements, which did not require high precision of the final position. They were made in three directions-longitudinal, lateral, and vertical-and were rectilinear movements. In the second stage, the hand movement during a task requiring precision and obtaining the required accuracy when moving the object between designated areas was examined.
All subjects performed reaching and precise movements with their dominant hand. Each subject performed trials with four reaching movements for each. Therefore, the number of recorded, reaching movements was 360 (30 subjects × 4 movements × 3 directions), and for precise movements, it was 150 (30 subjects × 5 movements × 1 direction). This test was measured after two practice trials. The operator's task during the study of reaching movements was to move the handle ((2) in Figure 2) from the adopted initial position to the fixed end position, defined by flexible bumpers mounted on the guides. The operator's task was to make a one-way movement and return to the position close to the initial position, with a normal hand movement speed. For each type of movement, the position of the guides (3) and bumpers (5) (Figure 2) was changed. Due to the ergonomics of hand movements [92,93], it was assumed that the distance between the bumpers for longitudinal movement (x i ) is 430 mm, for transverse (y i ) it is 730 mm, and for vertical (z i ) it is 930 mm (Figure 2). The basic measuring element of the test stand was a linear encoder with a measuring range of 0-1.25 m and an accuracy of 0.625 mm [94].
1. Speeds of human hand movements, as inputs introduced into the manipulator control system, strongly depend on the direction of movement (longitudinal, lateral, and vertical)-there are significant differences between speed and direction of movement, which significantly affects the design of the drive system and manipulator control system, and different control procedures are necessary depending on the direction of movement. 2. The maximum speeds of human hand movements during the execution of delivery and precision tasks, which are to be copied in real time by intervention manipulators, do not differ significantly.

Experimental Setting and User Task
The primary purpose of the research was to learn about the mean maximum velocity and mean maximum acceleration of human hand movement during the implementation of manipulation tasks and the influence of the direction of movement on the achieved speeds and accelerations.
The study was divided into two stages. The first stage concerned the analysis of the reaching movements, which did not require high precision of the final position. They were made in three directions-longitudinal, lateral, and vertical-and were rectilinear movements. In the second stage, the hand movement during a task requiring precision and obtaining the required accuracy when moving the object between designated areas was examined.
All subjects performed reaching and precise movements with their dominant hand. Each subject performed trials with four reaching movements for each. Therefore, the number of recorded, reaching movements was 360 (30 subjects × 4 movements × 3 directions), and for precise movements, it was 150 (30 subjects × 5 movements × 1 direction). This test was measured after two practice trials. The operator's task during the study of reaching movements was to move the handle ((2) in Figure 2) from the adopted initial position to the fixed end position, defined by flexible bumpers mounted on the guides. The operator's task was to make a one-way movement and return to the position close to the initial position, with a normal hand movement speed. For each type of movement, the position of the guides (3) and bumpers (5) (Figure 2) was changed. Due to the ergonomics of hand movements [92,93], it was assumed that the distance between the bumpers for longitudinal movement (xi) is 430 mm, for transverse (yi) it is 730 mm, and for vertical (zi) it is 930 mm ( Figure 2). The basic measuring element of the test stand was a linear encoder with a measuring range of 0-1.25 m and an accuracy of 0.625 mm [94]. During the study of precise movements, the operator's task was to transfer the cylinder from circle A to circle B ( Figure 3). The distance between the centers of the circles was 500 mm. The circle in which the object should be placed was equal to its diameter. For displacement measurement, the position of optical markers (3) was recorded by stereovision cameras ((4) in Figure 4). The displacement in the two axes was recorded so that it was possible to assess the correctness and accuracy of putting down the object.  During the study of precise movements, the operator's task was to transfer the cylinder from circle A to circle B ( Figure 3). The distance between the centers of the circles was 500 mm. The circle in which the object should be placed was equal to its diameter. For displacement measurement, the position of optical markers (3) was recorded by stereovision cameras ((4) in Figure 4). The displacement in the two axes was recorded so that it was possible to assess the correctness and accuracy of putting down the object. During the study of precise movements, the operator's task was to transfer the cylinder from circle A to circle B ( Figure 3). The distance between the centers of the circles was 500 mm. The circle in which the object should be placed was equal to its diameter. For displacement measurement, the position of optical markers (3) was recorded by stereovision cameras ((4) in Figure 4). The displacement in the two axes was recorded so that it was possible to assess the correctness and accuracy of putting down the object.   Stand for examining precise hand movements: 1-transferred object, 2-initial area in which the transferred object is located, 3-position marker, 4-stereovision camera, 5-device for data acquisition and processing, 6-area in which the transferred object should be located.
The basic measuring elements of the test stand based on the MyoVideo (Noraxon U.S.A. Inc., Scottsdale, AZ, USA) system were two stereovision cameras with a horizontal field of view of 57.5°, a vertical field of view of 43.1°, and an accuracy of less than 0.1 mm [95]. To assess the accuracy of the task, it was assumed that the error of putting the object down should not exceed 2 mm on both the x and y axes. The value of the setdown error corresponded to 5% of the diameter of the transferred object.

Participants
Two-step testing of speed and acceleration of movement of a human hand was carried out in a group of 30 people (male) aged 21 to 37 years and with a height of 165-194 cm. The most numerous groups of respondents (22 people) were students aged 21-22. Five of these volunteers were left-handed.

Statistical Analyses
The values of velocity and acceleration of the human hand for both types of tests were determined based on the backward differential quotient. The relationship that us allows to determine the speed takes the form [96,97]: where ni, the displacement depending on the type of motion carried, was xn for the precise transport task, xi for longitudinal movement, yi for lateral movement, and zi for vertical movement; is the time corresponding to the movement ni. The acceleration was calculated via the following equation [96,97]: (2) In order to determine the level of variability of the values of velocity and accelerations of the human hand depending on the operator who carries out the movement, the standard deviation σ was determined and the coefficient of variation Cv. Standard deviation was determined using the following equation [98][99][100]: Stand for examining precise hand movements: 1-transferred object, 2-initial area in which the transferred object is located, 3-position marker, 4-stereovision camera, 5-device for data acquisition and processing, 6-area in which the transferred object should be located.
The basic measuring elements of the test stand based on the MyoVideo (Noraxon U.S.A. Inc., Scottsdale, AZ, USA) system were two stereovision cameras with a horizontal field of view of 57.5 • , a vertical field of view of 43.1 • , and an accuracy of less than 0.1 mm [95]. To assess the accuracy of the task, it was assumed that the error of putting the object down should not exceed 2 mm on both the x and y axes. The value of the setdown error corresponded to 5% of the diameter of the transferred object.

Participants
Two-step testing of speed and acceleration of movement of a human hand was carried out in a group of 30 people (male) aged 21 to 37 years and with a height of 165-194 cm. The most numerous groups of respondents (22 people) were students aged 21-22. Five of these volunteers were left-handed.

Statistical Analyses
The values of velocity and acceleration of the human hand for both types of tests were determined based on the backward differential quotient. The relationship that us allows to determine the speed takes the form [96,97]: where n i , the displacement depending on the type of motion carried, was x n for the precise transport task, x i for longitudinal movement, y i for lateral movement, and z i for vertical movement; t i is the time corresponding to the movement n i . The acceleration was calculated via the following equation [96,97]: ..
In order to determine the level of variability of the values of velocity and accelerations of the human hand depending on the operator who carries out the movement, the standard Appl. Sci. 2022, 12, 3863 7 of 20 deviation σ was determined and the coefficient of variation C v . Standard deviation was determined using the following equation [98][99][100]: where µ i is the successive values of a given random variable in the sample, µ is the arithmetic mean of the sample, n is the number of elements in the sample. The coefficient of variation C v was determined from the following equation [98][99][100]: As part of the statistical analysis, we checked whether the obtained test results have a similar distribution to the norm and determined the test power. In order to test the normality of the distribution, the Shapiro-Wilk test was performed. The parameters given in the analysis of the normality of the distribution included the maximum velocities and RMS velocities as well as the maximum accelerations, and the RMS acceleration in the subsequent phases of the hand movement for all the carried tests. In performance research, there are backward and outward movements. In precision movement, only the movement to the target can be distinguished. Each of the movements can be divided into phases related to acceleration, steady motion, and deceleration, which are characterized by different values of speed and acceleration.

Results
The obtained results of the Shapiro-Wilk test [101] indicate that the distribution of the obtained test results is close to normal. The results of the ANOVA [102] power analysis (for α = 0.05; RMSSE (Root Mean Square Standardized Effect) is 0.566) indicate that the number of trials is sufficient to show whether there are significant differences in the speeds of reaching and precision movements and to ensure the test power at the level of 80%. The number of trials is also sufficient (test power > 80%) to determine the influence of the reaching movement's direction on the obtained speed (ANOVA for α = 0.05; RMSSE = 0.978).

Results of Research on Reaching Movements
Exemplary waveforms of changes in speed and acceleration for one of the operators obtained as a result of the study of the hand reaching movements depending on the direction of movement performed are shown in Figures 5-7 where is the successive values of a given random variable in the sample, ̅ is the arithmetic mean of the sample, n is the number of elements in the sample. The coefficient of variation was determined from the following equation [98][99][100]: As part of the statistical analysis, we checked whether the obtained test results have a similar distribution to the norm and determined the test power. In order to test the normality of the distribution, the Shapiro-Wilk test was performed. The parameters given in the analysis of the normality of the distribution included the maximum velocities and RMS velocities as well as the maximum accelerations, and the RMS acceleration in the subsequent phases of the hand movement for all the carried tests. In performance research, there are backward and outward movements. In precision movement, only the movement to the target can be distinguished. Each of the movements can be divided into phases related to acceleration, steady motion, and deceleration, which are characterized by different values of speed and acceleration.

Results
The obtained results of the Shapiro-Wilk test [101] indicate that the distribution of the obtained test results is close to normal. The results of the ANOVA [102] power analysis (for α = 0.05; RMSSE (Root Mean Square Standardized Effect) is 0.566) indicate that the number of trials is sufficient to show whether there are significant differences in the speeds of reaching and precision movements and to ensure the test power at the level of 80%. The number of trials is also sufficient (test power > 80%) to determine the influence of the reaching movement's direction on the obtained speed (ANOVA for α = 0.05; RMSSE = 0.978).

Results of Research on Reaching Movements
Exemplary waveforms of changes in speed and acceleration for one of the operators obtained as a result of the study of the hand reaching movements depending on the direction of movement performed are shown in Figures 5-7.   Analyzing the presented waveforms ( Figures 5-7), two stages of movement can be distinguished. The first is related to the movement to achieve the goal (point B in Figure 2); the other is for getting back to the starting position (point A). At each stage of movement, one can observe: an acceleration phase-pAkl, a steady motion phase-pSkl, and a deceleration phase-pDkl, where l refers to the type of movement (longitudinal-x, lateral-y, and vertical-z), and k designates the movement (1-motion to achieve the target, 2-return movement). The steady motion phase is considered a period when the speed does not change by more than 5%. It should be noted that, with the shortening of the length of motion, the steady state phase decreases. For small displacements, it may disappear.
The values of the duration of the first (movement to the target) and second stage (return movement) of the hand movement during the tests for all operators are summarized in Figure 8. A box plot to show the distribution of the dataset was used. In a box plot, numerical data are divided into quartiles, and a box is drawn between the first and third quartiles, with an additional line drawn along the second quartile to mark the median. The minima and maxima outside the first and third quartiles are depicted with lines, which are often called whiskers. The mean value is marked by an x. They show that the average time for the return movement is usually longer-differences amounted to 9% for lateral movement, 10% for longitudinal movement, and 20% for vertical movement. The ANOVA F [101]    Analyzing the presented waveforms ( Figures 5-7), two stages of movement can be distinguished. The first is related to the movement to achieve the goal (point B in Figure 2); the other is for getting back to the starting position (point A). At each stage of movement, one can observe: an acceleration phase-pAkl, a steady motion phase-pSkl, and a deceleration phase-pDkl, where l refers to the type of movement (longitudinal-x, lateral-y, and vertical-z), and k designates the movement (1-motion to achieve the target, 2-return movement). The steady motion phase is considered a period when the speed does not change by more than 5%. It should be noted that, with the shortening of the length of motion, the steady state phase decreases. For small displacements, it may disappear.
The values of the duration of the first (movement to the target) and second stage (return movement) of the hand movement during the tests for all operators are summarized in Figure 8. A box plot to show the distribution of the dataset was used. In a box plot, numerical data are divided into quartiles, and a box is drawn between the first and third quartiles, with an additional line drawn along the second quartile to mark the median. The minima and maxima outside the first and third quartiles are depicted with lines, which are often called whiskers. The mean value is marked by an x. They show that the average time for the return movement is usually longer-differences amounted to 9% for lateral movement, 10% for longitudinal movement, and 20% for vertical movement. The ANOVA F [101]   Analyzing the presented waveforms ( Figures 5-7), two stages of movement can be distinguished. The first is related to the movement to achieve the goal (point B in Figure 2); the other is for getting back to the starting position (point A). At each stage of movement, one can observe: an acceleration phase-p Akl , a steady motion phase-p Skl , and a deceleration phase-p Dkl , where l refers to the type of movement (longitudinal-x, lateral-y, and vertical-z), and k designates the movement (1-motion to achieve the target, 2-return movement). The steady motion phase is considered a period when the speed does not change by more than 5%. It should be noted that, with the shortening of the length of motion, the steady state phase decreases. For small displacements, it may disappear.
The values of the duration of the first (movement to the target) and second stage (return movement) of the hand movement during the tests for all operators are summarized in Figure 8. A box plot to show the distribution of the dataset was used. In a box plot, numerical data are divided into quartiles, and a box is drawn between the first and third quartiles, with an additional line drawn along the second quartile to mark the median. The minima and maxima outside the first and third quartiles are depicted with lines, which are often called whiskers. The mean value is marked by an x. They show that the average time for the return movement is usually longer-differences amounted to 9% for lateral movement, 10% for longitudinal movement, and 20% for vertical movement. The ANOVA F [101] analysis showed that there are no significant differences between the time of movement to the target and the time of return for longitudinal movement The summary of the recorded values of the maximal speed of movement and th effective RMS (Root Mean Square) speed obtained during the study of reaching hand movements in the acceleration phase and the deceleration phase is shown in Figures 9-12.    The summary of the recorded values of the maximal speed of movement and the effective RMS (Root Mean Square) speed obtained during the study of reaching hand movements in the acceleration phase and the deceleration phase is shown in Figures 9-12.    The summary of the recorded values of the maximal speed of movement and the effective RMS (Root Mean Square) speed obtained during the study of reaching hand movements in the acceleration phase and the deceleration phase is shown in Figures 9-12.                Analyzing the graphs (Figures 9-12), it can be noticed that the average maximal values of the hand movement speed towards the target (point B) were at the level of 1.01-1.13 m/s, while the average maximal speeds of the return movement were slightly lower at 0.93-1.05 m/s. The highest average maximal speed in the acceleration phase recorded for the vertical motion was 1.13 m/s. The highest average values of maximal acceleration of the human hand were obtained during the longitudinal movement in the first stage of movement in the acceleration phase and were close to 6 m/s 2 .   Analyzing the graphs (Figures 9-12), it can be noticed that the average maxi ues of the hand movement speed towards the target (point B) were at the level 1.13 m/s, while the average maximal speeds of the return movement were slight   Analyzing the graphs (Figures 9-12), it can be noticed that the average maxi ues of the hand movement speed towards the target (point B) were at the level 1.13 m/s, while the average maximal speeds of the return movement were slight at 0.93-1.05 m/s. The highest average maximal speed in the acceleration phase r Analyzing the graphs (Figures 9-12), it can be noticed that the average maximal values of the hand movement speed towards the target (point B) were at the level of 1.01-1.13 m/s, while the average maximal speeds of the return movement were slightly lower at 0.93-1.05 m/s. The highest average maximal speed in the acceleration phase recorded for the vertical motion was 1.13 m/s. The highest average values of maximal acceleration of the human hand were obtained during the longitudinal movement in the first stage of movement in the acceleration phase and were close to 6 m/s 2 .
ANOVA analysis showed that the direction of the performed movement (vertical, lateral, and longitudinal) influences the maximal speed of movement: F (2; 357) = 5.35 for p = 0.006. Based on the results of the research on reaching movements, the average values of maximal speeds and maximal accelerations, as well as average values of RMS speed and RMS accelerations for the acceleration phase-p A , steady motion phase-p S, and deceleration phase-p D were determined. The calculated parameters are summarized in Table 1. The median of maximal speeds in the acceleration phase is 1.0 m/s; in the steady phase it is 1.04 m/s, and in the deceleration phase it is 1.01 m/s. The median speed of the effective RMS speed of reaching movements compared to the median of maximal speeds is 62% for the acceleration phase and 56% for the deceleration phase. The maximal speeds for the three quartiles are higher and amount to 1.45 m/s in the acceleration phase, 1.33 m/s in the steady phase, and 1.30 m/s in the deceleration phase.
The median of the maximal accelerations of all reaching movements in the acceleration phase is 4.83 m/s 2 , and in the deceleration phase, it is 3.68 m/s 2 . The median RMS acceleration of reaching movements in relation to the median of maximal accelerations is 56% for the acceleration phase and 51% for the deceleration phase. The maximal accelerations for the three quartiles are higher: 6.4 m/s 2 in the acceleration phase and 5.7 m/s 2 in the deceleration phase.
The average maximal speed value for all phases and directions of movements is similar and is about 1.0 m/s for two quartiles and approx. 1.3 m/s for three quartiles. The average maximal acceleration for all phases and directions of movements is approx. 4.8 m/s 2 for two quartiles and approx. 5.5 m/s 2 for three quartiles.
It is worth noting that the average maximal acceleration in the deceleration phase is 24% lower than the acceleration in the acceleration phase. The coefficient of variation for maximal speed and RMS speed ranges from 11% to 25%. This indicates a high homogeneity of the studied community. The coefficient of variation for maximum acceleration and RMS acceleration ranges from 18% to 40%. This indicates a low or average variability of accelerations in the studied group.

Results of Research on Rectilinear Precise Movements
An example of the course of displacement, velocity, and acceleration obtained from the study of precise hand movements for one of the operators is shown in Figure 17. As in the case of reaching movements, three phases of movement can be distinguished: p Aacceleration phase, p S -steady motion phase, and p D -deceleration phase. The duration of the deceleration phase is about 25% longer than that of the acceleration phase. The summary of the values of maximal speeds and maximal accelerations as well as RMS speeds and RMS accelerations obtained during the tests of precise hand movements are presented in Figures 18 and 19. ppl. Sci. 2022, 12, x FOR PEER REVIEW of the deceleration phase is about 25% longer than that of the acceleration ph mary of the values of maximal speeds and maximal accelerations as well a and RMS accelerations obtained during the tests of precise hand movements in Figures 18 and 19.   ppl. Sci. 2022, 12, x FOR PEER REVIEW of the deceleration phase is about 25% longer than that of the acceleration mary of the values of maximal speeds and maximal accelerations as we and RMS accelerations obtained during the tests of precise hand moveme in Figures 18 and 19.   Analyzing the graphs (Figures 17-19), it can be seen that the median value of maximal accelerations in the acceleration phase (2.3 m/s 2 ) is less than half the median of maximal acceleration values in the acceleration phase obtained in the study of reaching movements. In contrast, the median value of the maximal acceleration in the deceleration phase (1.3 m/s 2 ) is close to 3-fold lower than the median value of the maximal acceleration in the deceleration phase of reaching movements. The median value of maximal speed in the acceleration phase (0.58 m/s) is half the median value of maximal speed obtained in the acceleration phase of reaching movements. The maximal speeds in the acceleration phase for the three quartiles reach 0.65 m/s, and the accelerations 2.5 m/s 2 . Thus, they are 12% and 9% higher, respectively, than the median value.  Analyzing the graphs (Figures 17-19), it can be seen that the median value of maximal accelerations in the acceleration phase (2.3 m/s 2 ) is less than half the median of maximal acceleration values in the acceleration phase obtained in the study of reaching Based on the results of the study of precise movements, the average values of maximal speeds and maximal accelerations, as well as average values of RMS speed and RMS accelerations for each movement phase (p A , p S , and p D ), were determined. The calculated parameters are summarized in Table 2. The average maximal speed value for all movement phases is similar and is about 0.6 m/s-approx. 40% lower than the average maximal speed of reaching movements. The average RMS speed in the acceleration and deceleration phase is approx. 45% lower than the maximal speed values in these phases of motion.
The average maximal acceleration in the acceleration phase is approx. 2.3 m/s 2 and is about 50% of the value of the average maximal acceleration in the acceleration phase in the study of reaching movements. The coefficient of variation for maximal speed and effective RMS speed ranges from 4% to 15%. This indicates a high homogeneity in the studied population.
The compilation of the dropout errors in relation to the x and y axes obtained during the tests by all operators is shown in Figure 20. Analyzing the obtained test results (Figure 20), it can be noticed that the average error of putting the object off does not exceed 0.5 mm. The first quartile of the withdrawal error concerning the x axis is −1.8 mm, and the third quartile is 0.5 mm. The yaw-off error to the y-axis for the first quartile is 1.2 mm, and for the third is 1.3 mm. Therefore, no less than 50% of attempts to complete the task were carried out with the assumed accuracy. the tests by all operators is shown in Figure 20. Analyzing the obtained test results (Figure 20), it can be noticed that the average error of putting the object off does no 0.5 mm. The first quartile of the withdrawal error concerning the x axis is −1.8 m the third quartile is 0.5 mm. The yaw-off error to the y-axis for the first quartile is and for the third is 1.3 mm. Therefore, no less than 50% of attempts to complete were carried out with the assumed accuracy.

Discussion
The obtained test results are characterized by a normal distribution close to the Gaussian curve (p > α). The power of the test (>80%) was sufficient to show the differences between the tested parameters of human upper limb movements. Most of the parameters determined in the studies of human hand movements are characterized by a low coefficient of variation (C v < 25%). Therefore, they can be applied to most people, and give very important information about the dynamics of the human hand as an element controlling the movement of the anthropomorphic manipulator effectors and about the possibilities of controlling anthropomorphic manipulators by means of hand movements, controlling the position of the effector in a master-slave system.
The values of averaged maximal speeds and averaged maximal accelerations obtained as a result of the conducted tests indicate that there are significant differences between the reaching and precise movements of the human limb. The direction of the movement (longitudinal, lateral, and vertical) has no significant influence on the maximal speed achieved. The median value of the maximal speed of reaching movements is 1 m/s, while the maximal speed for the three quartiles is approximately 1.3 m/s.
The speed values obtained as a result of precise hand movement tests are nearly 50% lower than the speeds of reaching movements. A significant reduction in the speed value, in relation to the reaching movements, indicates that a task that requires strictly defined precision forces the operator to work at lower speeds, which significantly extends the time needed to complete the task. The results are consistent with the test results presented in [103]; however, the presented research covers a much larger range of motion and various directions.
The conducted research showed that there are no significant differences between the velocity of motion to achieve the target and return movement-longitudinal movement- Figure 7, lateral movement- Figure 6, and vertical movement- Figure 5.
The median value of maximal acceleration during reaching movements (all phases) was 4 m/s 2 and was about 50% higher than in the case of precise movements. It should be noted that the acceleration values are not fixed in the acceleration and deceleration phases. The maximal acceleration values usually occur in the initial phase and decrease quickly, although there are incidental hard deceleration cases. It is confirmed by a large difference between the values of averaged maximal accelerations and the effective values of RMS accelerations. Direct copying of such movements can result in significant dynamic loads on the manipulator. In the case of mechanical hand tracking systems, it is possible to reduce acceleration by increasing the resistance to hand movement by introducing friction or using servos with haptic feedback. In the case of optical tracking systems, it may be necessary to optimize the control signals in the master-slave control system.

Conclusions
Thanks to the use of an innovative method of measuring hand movements with the use of a system of stereovision cameras, high accuracy of hand displacement measurements was achieved. These allowed us to determine the maximum values of velocity and acceleration of longitudinal, lateral, and vertical movements of the human hand. Research has shown that regardless of the direction of movement (longitudinal, lateral, vertical), the maximum velocity values are close and amount to approximately 1 m/s. Moreover, they showed that during a movement longer than 0.4 m, there is a steady phase in which the speed of movement is constant.
The conducted research has shown that there are no relations between the direction of hand movement (longitudinal, lateral transverse, vertical) and the obtained speed of movement. The test results also show a significant difference in the achieved speeds of reaching and precise movements. Therefore, the null hypotheses made should be rejected.
The obtained research results allow for a better design of manipulator control systems operating in the master-slave system. The measured speed values can be used as inputs in the master-slave manipulator control system. Moreover, they allowed us to determine the accelerations that can be generated by the operator during steering with hand movements.
To control a manipulator with a much larger working area in relation to human reach, with speeds multiplied in relation to the speeds obtained from the tests. It is likely that operating at such high speeds may not be possible due to limited operator perception. Therefore, it is necessary to conduct further research on the possibilities of controlling manipulators with extended range, which depend mainly on the perception of the operator and the mobility of the limbs.