TRANSACTIONS ON MOBILE COMPUTING,VOL.17,NO.10,OCTOBER 2018 05 05 -0.45 -0.55 0.55 0.6 0.55 -06 -0.6 -0.65 ( 0.65 0. Real Initial 0.7 Real Initial Real Initial Position 2 Position -0.7 Position -0.75 -0.75 -0.75 08 0 .50.550.60.650.70.750.8 085050.6065、0707508 .450.50.550.60.650.70.750.8 X(m) X(m) X(m) (a)Horizontal (b)Vertical (c)Real experiment Figure 16.Initial position simulation and experimental results To better understand this method,we illustrate the process threshold th and thy,respectively.We set both th and thy as of initial position refinement via simulation.Suppose that a user 0.05 m in the following experiments.Next,we find two closest moves his hand along X-axis for 30 cm.We use a maximum candidate positions in the two directions and derive the new likelihood scheme to determine the initial position based on the estimation by linearly combining these two positions.The iteration measured distance change.Generally,the gesture trajectory (y) stops if the distance between these two candidate positions is related to candidate initial position (x',y)can be determined by smaller than an empirical threshold thz,which is set as 0.1 m. solving the following two equations: V2+y2+Vx-L1)2+y 4.4 Successive 2D Tracking =V2+y2+V(x-L1)2+2+△d山 As the lengths of the two initial paths from the transmitter to two receivers in Figure 12(di and d2)are estimated in Section 4.3. (12) the instantaneous length of the paths d and d can be calculated Vx2+y2+V2+(-L2) by updated distance change measurements (i.e..Ad and Ad2). =Vx2+y2+V2+(y-L2+△d2 Then,by solving Eq.(11)with the conditions of 0 and y<0, where Ad and Ad2 are the path length changes corresponding to the realtime position of hand can be located,and the successive two receivers at (L1,0)and (0,L2),while the source is at (0,0). position will be derived from this iterative method. Each path contains two segments,e.g.,V2+y2(from source to candidate initial position)and v(z'-L1)2+y2(from the 4.5 Trajectory Correction candidate initial position to RXI at (0,L1)).Eq.3 shows how The estimated trajectory of the hand still has large error due to the path length changes when the hand moves from candidate persistent systematic noise.To filter out such noise and further initial position (x',y)to (x,y).Therefore,the point (,y)on improve the accuracy of the trajectory,we propose to use the KF the gesture trajectory is on the intersections of two elliptic curves based on a 2D Continous Wiener Process Acceleration(CWPA) determined by the position of receivers and the measured path model [31],which basically handles the case where the object's length change.Figure 16(a)shows the simulating result for hori- acceleration is perturbed by Gaussian noise,to correct the traject- zontal movement.The color represents deviation of the trajectory ory.The state vector of 2D CWPA model is along Y-axis for various initial positions (ie.,arg max) Sk=[rk张王k弧k] (13) Similar to the horizontal movement,Figure 16(b)shows deviation of the trajectory along X-axis for the vertical movement (i.e., where (xk,y),(ik,and (k,are the movement dis- arg max).From Figure 16(a)and Figure 16(b).we find tances,velocity and acceleration of the hand at time k,respect- that the red point is the real initial position in our simulation. ively.The KF model assumes the true state at time k is evolved However,in reality,the real initial position may not be the red from the state at (k-1)according to point,since the distance measurement may have some errors.As Sk=Ask-1+qk (14) shown in Figure 16(c).the real initial position is the second darkest red point of all the candidate positions.Thus,we combine X-axis where only the movement acceleration has the process noise,q and Y-axis directions to locate the initial point more accurately. N(0,Q).Q is the covariance matrix of the process noise.Based Algorithm 2 shows the description of locating the initial point. on the physical laws of motion,the transition matrix is We first select the candidate area around coarse position estimation [10t0号t201 (e,ye)that is a square with side-length of L.Since the majority 010t0号2 of the estimation error is less than 0.5 m as shown in Figure 0010t 0 A 13,we set L as 0.5 m.We use an iterative candidate region (15) 00010 t selection algorithm that converges until the final initial position 00001 0 is determined.We calculate the tracking trajectory corresponding 00000 1 to all positions in the candidate region to find a limited number of candidates for the two directions separately.These candidates The measurement vector at time k in our system is should have movement deviations smaller than two empirical k=k VA]T (16)TRANSACTIONS ON MOBILE COMPUTING, VOL. 17, NO. 10, OCTOBER 2018 8 0.5 0.55 0.6 0.65 0.7 0.75 0.8 X(m) -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 Y(m) 1 2 3 4 5 6 7 104 Real Initial Position (a) Horizontal 0.5 0.55 0.6 0.65 0.7 0.75 0.8 X(m) -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 Y(m) 1 2 3 4 5 6 7 104 Real Initial Position (b) Vertical 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 X(m) -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 Y(m) 1 2 3 4 5 6 7 104 Real Initial Position (c) Real experiment Figure 16. Initial position simulation and experimental results To better understand this method, we illustrate the process of initial position refinement via simulation. Suppose that a user moves his hand along X-axis for 30 cm. We use a maximum likelihood scheme to determine the initial position based on the measured distance change. Generally, the gesture trajectory (x, y) related to candidate initial position (x 0 , y0 ) can be determined by solving the following two equations:    p x 02 + y 02 + p (x 0 − L1) 2 + y 02 = p x 2 + y 2 + p (x − L1) 2 + y 2 + ∆d1 p x 02 + y 02 + p x 02 + (y 0 − L2) 2 = p x 2 + y 2 + p x 2 + (y − L2) 2 + ∆d2 (12) where ∆d1 and ∆d2 are the path length changes corresponding to two receivers at (L1, 0) and (0, L2), while the source is at (0, 0). Each path contains two segments, e.g., p x 02 + y 02 (from source to candidate initial position) and p (x 0 − L1) 2 + y 02 (from the candidate initial position to RX1 at (0, L1)). Eq.3 shows how the path length changes when the hand moves from candidate initial position (x 0 , y0 ) to (x, y). Therefore, the point (x, y) on the gesture trajectory is on the intersections of two elliptic curves determined by the position of receivers and the measured path length change. Figure 16(a) shows the simulating result for hori￾zontal movement. The color represents deviation of the trajectory along Y-axis for various initial positions (i.e., arg maxy0 1 |y0−y| ). Similar to the horizontal movement, Figure 16(b) shows deviation of the trajectory along X-axis for the vertical movement (i.e., arg maxx0 1 |x0−x| ). From Figure 16(a) and Figure 16(b), we find that the red point is the real initial position in our simulation. However, in reality, the real initial position may not be the red point, since the distance measurement may have some errors. As shown in Figure 16(c), the real initial position is the second darkest red point of all the candidate positions. Thus, we combine X-axis and Y-axis directions to locate the initial point more accurately. Algorithm 2 shows the description of locating the initial point. We first select the candidate area around coarse position estimation (xe, ye) that is a square with side-length of L. Since the majority of the estimation error is less than 0.5 m as shown in Figure 13, we set L as 0.5 m. We use an iterative candidate region selection algorithm that converges until the final initial position is determined. We calculate the tracking trajectory corresponding to all positions in the candidate region to find a limited number of candidates for the two directions separately. These candidates should have movement deviations smaller than two empirical threshold thx and thy, respectively. We set both thx and thy as 0.05 m in the following experiments. Next, we find two closest candidate positions in the two directions and derive the new estimation by linearly combining these two positions. The iteration stops if the distance between these two candidate positions is smaller than an empirical threshold thz, which is set as 0.1 m. 4.4 Successive 2D Tracking As the lengths of the two initial paths from the transmitter to two receivers in Figure 12 (d1 and d2) are estimated in Section 4.3, the instantaneous length of the paths d 0 1 and d 0 2 can be calculated by updated distance change measurements (i.e., ∆d1 and ∆d2). Then, by solving Eq.(11) with the conditions of x > 0 and y < 0, the realtime position of hand can be located, and the successive position will be derived from this iterative method. 4.5 Trajectory Correction The estimated trajectory of the hand still has large error due to persistent systematic noise. To filter out such noise and further improve the accuracy of the trajectory, we propose to use the KF based on a 2D Continous Wiener Process Acceleration (CWPA) model [31], which basically handles the case where the object’s acceleration is perturbed by Gaussian noise, to correct the traject￾ory. The state vector of 2D CWPA model is sk = [xk yk x˙k y˙k x¨k y¨k] T (13) where (xk, yk), ( ˙xk, y˙k) and (¨xk, y¨k) are the movement dis￾tances, velocity and acceleration of the hand at time k, respect￾ively. The KF model assumes the true state at time k is evolved from the state at (k − 1) according to sk = Ask−1 + qk (14) where only the movement acceleration has the process noise, qk ∼ N (0, Q), Q is the covariance matrix of the process noise. Based on the physical laws of motion, the transition matrix is A =         1 0 t 0 1 2 t 2 0 0 1 0 t 0 1 2 t 2 0 0 1 0 t 0 0 0 0 1 0 t 0 0 0 0 1 0 0 0 0 0 0 1         . (15) The measurement vector at time k in our system is zk = [xk yk] T (16)