#### 2.1. Regularity of the Occurrence of the Maximum Value of the Induced Current, the Zero-Crossing of the Induced Current, and the Maximum Value of the Change Rate of the Induced Current

When the hand moves linearly near the spherical electrode, an induced current is generated on it. Suppose that the hand charge quantity is Q, the distance between the trajectory of the hand movement and electrode is z_{0}, and the velocity of the hand is v for a very short time when the hand moves near the electrode.

The left-hand coordinate system is established with the ball center as the origin. For the convenience of calculation, let the trajectory of the hand movement be in the

XOZ plane and parallel to the

x-axis, as shown in

Figure 2. Take the time when the hand is on the

z-axis as time zero, and the angle between the

x-axis and the line passing through the hand and the origin is

α.

The induced current signal of the spherical electrode can be expressed as:

where

R_{0} is the radius of the spherical electrode, and

r is the distance from the trajectory of the hand to the center of the spherical electrode.

In addition, the

α can be calculated as:

Take in the parameters (set

v = 2 m/s,

z_{0} = 0.3 m,

Q = −10

^{−9} C,

R_{0} = 0.004 m) to simulate the current

i, and the current waveform is shown in

Figure 3.

The waveform of the induced current

i in the figure has two extreme points and one zero-crossing point. In order to calculate the position relationship between the hand and the electrode when the extreme points occur, differentiate the induced current

i to

t to obtain the change rate of the induced current as follows:

The simulated waveform of the change rate of the induced current

${i}^{\prime}$ is shown in

Figure 4.

When the maximum value of the induced current

i occurs, the change rate of the induced current is equal to zero, that is:

That is, when

α = 54.7°, the maximum value of the induced current

i appears. The waveform of the change rate of the induced current has three extreme points. Similarly, let the second derivative of the induced current

i to

t be equal to zero.

When $\mathrm{cos}\alpha =\sqrt{\frac{3}{5}}$, that is, α = 39.2°, the first maximum value of the change rate of the induced current ${i}^{\prime}$ appears. When $t=0$, the hand is directly above the electrode, α_{3} = 90°, the minimum value of the change rate of the induced current ${i}^{\prime}$ appears, and the zero-crossing of the induced current i appears.

In summary, when α = 39.2°, the first maximum value of the change rate of the induced current ${i}^{\prime}$ appears; when α = 54.7°, the maximum value of the induced current i appears; and when α = 90°, the zero-crossing of the induced current i appears. For the convenience of the introduction later, let α_{1}= 39.2°, α_{2} = 54.7°, and α_{3} = 90°.

#### 2.2. Algorithm for Measuring the Distance between the Trajectory of a Hand Movement and the Straight Line of Two Electrodes

Assume that the trajectory of a hand movement is parallel to the straight line of two electrodes; the distance between them can be calculated based on the values mentioned above. Take the midpoint of the connection of electrodes

P_{1} and

P_{2} as the origin, and the line of two electrodes is the

x-axis. Set the trajectory of the hand movement in the

XOZ plane and establish the coordinate system as shown in

Figure 5. Set the electrode spacing as

L.

According to the order of the maximum values of the induced current, the maximum values of the change rate of the induced current, and the zero-crossings of the induced current on the two electrodes, we can classify the range of the hand trajectory in the upper half of the

XOZ plane, as shown in

Figure 6.

In

Figure 6,

P_{1} and

P_{2} are spherical electrodes placed on the

x-axis. Make lines

l_{11} and

l_{21} have an angle of α

_{1}= 39.2° with the

x-axis through

P_{1} and

P_{2}. When the hand moves along the

x-axis, the trajectory of the hand intersects with

l_{11} and

l_{21} successively, while the maximum values of the change rate of the induced current on

P_{1} and

P_{2} occur successively. Let their occurrence time be

T_{11} and

T_{21}. Make lines

l_{12} and

l_{22} have an angle of α

_{2} = 54.7° with the

x-axis through

P_{1} and

P_{2}. When the hand moves along the

x-axis, the trajectory of the hand intersects with

l_{12} and

l_{22} successively, while the maximum values of the induced current on

P_{1} and

P_{2} occur successively. Let their occurrence time be

T_{12} and

T_{22}. Make lines

l_{13} and

l_{23} have an angle of α

_{3} = 90° with the

x-axis through

P_{1} and

P_{2}. When the hand moves along the

x-axis, the trajectory of the hand intersects with

l_{13} and

l_{23} successively, while the zero-crossings of the induced current on

P_{1} and

P_{2} occur successively. Let their occurrence time be

T_{13} and

T_{23}.

F_{1},

F_{2}, and

F_{3} are the intersections of

l_{13} and

l_{21},

l_{13} and

l_{22}, and

l_{12} and

l_{21}, respectively.

l_{1},

l_{2}, and

l_{3} are straight lines passing through

F_{1},

F_{2}, and

F_{3} and parallel to the

x-axis. Their distances from the

x-axis are

D_{1},

D_{2}, and

D_{3}, respectively.

In the right-angled triangle

F_{1}P_{1}P_{2} ∠

F_{1}P_{2}P_{1} = α

_{1},

P_{2}P_{1} =

L, so:

In the right-angled triangle

F_{2}P_{1}P_{2}, ∠

F_{2}P_{2}P_{1} =

${\alpha}_{2}$,

P_{2}P_{1} =

L, so:

In the triangle

F_{3}P_{1}P_{2}, make the line segment

F_{3}G perpendicular to the

x-axis, ∠

F_{3}P_{2}P_{1} = α

_{1}, ∠

F_{3}P_{2}P_{1} = α

_{2},

P_{2}P_{1} =

L, so:

It can be calculated that:

l_{1}, l_{2}, and l_{3} divide the upper part of the x-axis into four areas of different heights. For convenience, they are named “Area 1”, “Area 2”, “Area 3”, and “Area 4”.

When the trajectory of a hand movement is located in different areas above the electrode, the occurrence time of the maximum values of the induced current, the zero-crossings of the induced current, and the maximum values of the change rate of the induced current on the two electrodes are different. Take the trajectory of the hand movement in Areas 3 and 2 as examples. When the hand moves along the x-axis, the trajectory intersects with l_{11}, l_{12}, l_{21}, l_{22}, l_{13}, and l_{23} successively if it is in Area 3; in this case, T_{11} < T_{12} < T_{21} < T_{22} < T_{13} < T_{23}. When the hand moves along the x-axis, the trajectory intersects with l_{11}, l_{12}, l_{21}, l_{13}, l_{22,} and l_{23} successively if it is in Area 2. The difference is, in this case, T_{11} < T_{12} < T_{21} < T_{13} < T_{22} < T_{23}. By sorting the six moments, the position of the trajectory of a hand movement parallel to the x-axis is divided into four areas. Based on this, the distance d between the trajectory of the hand movement and the x-axis can be calculated by the following method.

Suppose that the hand moves along the

x-axis; no matter which area the trajectory is located in, it passes through

l_{13} first, and then

l_{23}, after the time difference

Δt_{0}. In the very short time that the hand passes the electrode, the hand movement speed

v can be regarded as a fixed value:

Taking the height

D_{2} as a reference, let the time difference between the hands passing

l_{13} and

l_{22} be

Δt_{2} (when

d <

D_{2},

Δt_{2} is positive; when

d =

D_{2},

Δt_{2} is zero; when

d >

D_{2},

Δt_{2} is negative), then

d can be calculated as:

Substituting Equation (12) into Equation (13):

The distance d between the trajectory of a hand movement and the x-axis can be calculated through Equation (14). In practical applications, in order to accurately calculate d, first determine the area in which the hand is located, and calculate d with one or more of D_{1}, D_{2}, and D_{3} as a reference.

If taking the height

D_{1} as a reference, let the time difference between the hands passing

l_{13} and

l_{21} be

Δt_{1} (when

d <

D_{1},

Δt_{1} is positive; when

d =

D_{1},

Δt_{1} is zero; when

d >

D_{1},

Δt_{1} is negative), then

d can be calculated as:

If taking the height

D_{3} as a reference, let the time difference between the hands passing

l_{12} and

l_{21} be

Δt_{3} (when

d <

D_{3},

Δt_{1} is positive; when

d =

D_{3},

Δt_{3} is zero; when

d >

D_{3},

Δt_{3} is negative), then

d can be calculated as:

In the actual application scenario, we can first determine the area where the hand is, then select the appropriate formula to calculate the value of d. If it is determined that the trajectory of the hand movement is in Area 1, calculate d by Equation (14); if it is determined that the trajectory of the hand movement is in Area 2, calculate d by Equations (14) and (15), respectively, and take their average value as the final result; if it is determined that the trajectory of the hand movement is in Area 3, calculate d by Equations (15) and (16), respectively, and take their average value as the final result; if it is determined that the trajectory of the hand movement is in Area 4, calculate d by Equation (16).

#### 2.3. Measurement Algorithm for the Height of Hand Movements Based on Human Body Electrostatics

The height of a hand movement can be calculated by making up a planar array of multiple electrodes, that is, the vertical distance between the trajectory of the hand movement and the planar array of electrodes. Using an array of four spherical electrodes, the coordinate system as shown in

Figure 7 is established by taking the midpoint of the square with four electrodes as the vertex as the origin. The electrodes

P_{1},

P_{2},

P_{3}, and

P_{4} are all in the

XOY plane, and the distance of each electrode from the

x and

y axes is

l. The height of the hand movement is the distance

h between the trajectory of the hand movement and the

XOY plane.

For convenience, the eight directions on the plane are numbered in the order as shown in

Figure 7. In most cases, the distance

d measured by two electrodes is not equal to the height of the hand movement

h. Take direction ① as an example.

AB is the trajectory of the hand movement parallel to the

x-axis, and

AD is perpendicular to point

D on

P_{1}P_{3}. The length of segment

AD is the height of the hand movement

h. The distance measured by electrodes

P_{1} and

P_{2} between the trajectory and the straight line

P_{1}P_{2} is

d_{1} and the distance measured by electrodes

P_{3} and

P_{4} between the trajectory and the straight line

P_{3}P_{4} is

d_{2}.

In the triangle

AP_{1}P_{3}, the lengths of the three sides are

d_{1},

d_{2}, and 2

l, respectively. According to Heron’s formula, the area of the triangle

AP_{1}P_{3} is:

where

p is the half perimeter of the triangle

AP_{1}P_{3}, that is,

p = (

d_{1} +

d_{2} + 2

l)/2.

The value of the height of the hand movement can be calculated by:

When the direction of hand movement is ②, ④, ⑥, or ⑧, there is only one set of electrodes in these directions, so there will be an error between the electrode measurement result

d and the actual height

h. This can be improved by adding electrodes, such as electrodes

P_{5} and

P_{6} shown in

Figure 8, which are located on the

x-axis, and

P_{4}P_{6} =

P_{3}P_{5} =

P_{2}P_{6} =

P_{1}P_{5} =

$\sqrt{2}l$.

For example, when it is determined that the direction of the hand movement is direction ②, calculate the distance

d_{1} from the hand trajectory to line

P_{2}P_{3} through Equation (14) by using the signals on electrodes

P_{2} and

P_{3}. The distance between electrodes

P_{2} and

P_{3} is

$L=2\sqrt{2}l$. Then, calculate the distance

d_{2} from the hand trajectory to line

P_{4}P_{6} through Equation (15) by using the signals on electrodes

P_{4} and

P_{6}. The distance between electrodes

P_{4} and

P_{6} is

$L=\sqrt{2}l$, and the height of the hand movement

h is:

The flowchart for obtaining the height of the hand movement is shown in

Figure 9.

First, determine the direction of the hand movement and select the corresponding electrode. Then, by analyzing the order of the maximum values of the induced current, the maximum values of the change rate of the induced current, and the zero-crossings of the induced current on the two electrodes, we can know which area the hand is located in order to calculate d_{1} and d_{2}.