Largely because of the MEMS technology, inertial sensors (accelerometers and gyroscopes) are becoming smaller and cheaper [1], which makes it possible to use inertial sensors in many applications. Inertial sensors are used in motion trackers [2], personal navigation systems [3] and remote control systems [4].

In some applications such as body motion trackers (for example, the product ‘Moven’ by XSENS), inertial sensors are used to track body movement. As the number of inertial sensors increases, the size of sensor data increases accordingly. The sensor data is transmitted to the microprocessor board through wired or wireless communication channels. In a wireless communication channel, the transmission speed is relatively low compared with a wired communication channel. The size of the sensor data needs to be reduced if it exceeds the capacity of the communication channel. For example, the transmission rate of raw sensor data for one MTx (commercial inertial and magnetic sensor) could be as high as 72 Kbps: 9 sensor outputs (3 accelerometers, 3 gyroscopes, and 3 magnetic sensors) × 500 Hz (maximum sampling rate) × 16 bits (16 bit A/D conversion for each sensor). If four MTx are used, the size of the sensor data exceeds the capacity of Zigbee (maximum rate is 250 Kbps [5]). More applications are expected as networked control and monitoring systems are becoming more popular [6].

One way to reduce the size of the sensor data is to compress the sensor data before transmission and decompress the received data in the microprocessor board. Data compression has been extensively studied in many areas [7]. In applications such as body motion trackers, real-time compression is preferable, in order to avoid delayed sensor data transmission and consequently the delay in motion estimation. One of the most popular real-time compression methods is ADPCM (Adaptive Differential Pulse Coded Modulation) [8], which is optimized for voice data. In [9], a simplified ADPCM method is used for inertial sensor data compression, where the maximum error (the difference between the original data and the compressed-and-then-decompressed data) is only relatively bounded (e.g., 1% of the sensor data).

The performance indices of data compression are the bit rate and the quality of compressed data. In voice data compression, the quality of compressed data is evaluated by listening to the compressed-and-then-decompressed voice data. This rather subjective evaluation makes sense since the final destination of compressed data is the human ear. On the other hand, the final destination of compressed inertial sensor data is usually a filter (such as a Kalman filter), where orientation is estimated. Thus the quality of compression should be judged by how the compression affects the estimation error.

In this paper, a modified ADPCM method is proposed, where the absolute maximum error bound is explicitly given. Also, we investigate how this error affects the estimation error. A part of this paper was presented in [10].

The overall process of compression and estimation is given in Figure 1, where k is a discrete time index. The objective is to estimate some states x(k) (attitude, heading, position, etc.) using inertial sensor data y(k) at a limited data transmission rate. The inertial sensor data y(k) is compressed into d̃(k) and transmitted to the microprocessor board. The compressed data is decompressed into ŷ(k) and the state x(k) is estimated using a filter.

Since the objective is to find a good estimator of x(k), the quality of compression is considered good if the estimation error x(k) − x̂(k) is small. The quality of the compression algorithm is evaluated using the following estimation error covariance: (1) P error = E { ( x − x ^ ( y ^ ) ) ′ ( x − x ^ ( y ^ ) ) }where x̂(ŷ) is an estimator when ŷ is used as an output.

Note that P_error depends on the filter algorithm used to compute x̂(ŷ) in addition to the compression algorithm.

The ideal compression algorithm minimizes both P_error and the bit rate. However, usually if P_error is small, the bit rate tends to be large. In Section 4, we propose a compression algorithm where the maximum compression error is bounded. The maximum compression error bound plays a role of design parameter to adjust P_error and the bit rate.

The ADPCM block scheme is given in Figure 2. We assume that y(k) is the output of n_y bit uniform quantizer, where y(k) satisfies

Let the quantization size δ of y(k) be defined by (3) δ = y max 2 n y − 1If there is more than one sensor, we need one encoder for each sensor.

The sensor data y(k) is compared with the predictor output ỹ(k). The difference d(k) is coded into d̃(k) and this d̃(k) is transmitted to the estimator board. In the standard ADPCM, d̃(k) is a quantization index i(k). In this paper, d̃(k) consists of one bit mode information m(k) and a quantization index i(k): (4) d ˜ ( k ) = [ m ( k ) i ( k ) ]In the decoder, the decompressed data is ŷ(k) = ỹ(k)+ d̂(k). The predictor output ỹ(k) can be computed from d̂(k) and thus does not need to be transmitted.

The adaptive predictor uses the same pole-zero configuration as that in CCITT G.726 ADPCM, which is an ADPCM speech compressor/decompressor protocol proposed in 1990 [11] : (5) y ˜ ( k ) = ∑ i = 1 2 a i ( k − i ) y ^ ( k − i ) + ∑ i = 1 6 b i ( k − 1 ) d ^ ( k − i )From the assumption (2), if ỹ(k) > y_max from (5), we set ỹ(k) = y_max. Similarly, if ỹ(k) < −y_max, we set ỹ(k) = −y_max.

The adaptive algorithm in the G.726 protocol is used to adjust a_i and b_i and the detail is given in [11]; the tone and transition detector part was omitted since the part is only for voice data.

The compression error e_c(k) is the difference between the original signal y(k) and the decompressed signal ŷ(k): (6) e c ( k ) = y ( k ) − y ^ ( k ) = ( y ˜ ( k ) + d ( k ) ) − ( y ˜ ( k ) + d ^ ( k ) ) = d ( k ) − d ^ ( k )

Standard ADPCM algorithms [8] will be modified so that the maximum error is bounded as follows: (7) | e c ( k ) |     ≤     e max

Now d̃(k) coding is explained. The mode bit m(k) in d̃(k) is used to ensure (7). If the compression error e_c(k) of a standard ADPCM method satisfies (7), then the mode is 0 (i.e., m(k) = 0). As will be seen in Section 3.1, this is true if |d(k)| < 2^y_s(k)δ, where y_s(k) is an adaptive scaling factor. On the other hand, if the compression error e_c(k) of a standard ADPCM method does not satisfy (7), then the mode is 1 (i.e., m(k) = 1) and a special uniform quantizer is used as in Section 3.2. Thus the mode m(k) is given by (8) m ( k ) = { 0 , if | d ( k ) | 2 y s ( k ) δ ≤ 1 1 , otherwisewhere y_s(k) is an adaptive scaling factor.

The quantization index i(k) is defined differently when m(k) = 0 and when m(k) = 1.

In Section 3, a compression method is proposed, where the maximum compression error is e_max. Also if m(k) and i(k) are known, bounds of the compression error are given by (13) and (15). In this section, we use this information in a Kalman filter.

We assume that y(k) is generated by a linear system (16) x ( k + 1 ) = A x ( k ) + w ( k ) y ( k ) = C x ( k ) + u ( k )where x ∈ Rⁿ is the state, y ∈ R^p is the output, and w(k) and v(k) are uncorrelated, zero-mean, white Gaussian noises that satisfy E { w ( k ) w ′ ( k ) } = Q , E { v ( k ) v ′ ( k ) } = R

In the standard Kalman filter, x(k) is estimated using y(k). If y(k) is compressed, ŷ(k) = y(k) −e_c(k) is used instead. From (13) and (15), ē_c(k) can be computed, which is used to reduce the estimation error. If we assume e_c,i(k) (i-th element of e_c(k)) has a uniform distribution, E { e c , i 2 ( k ) } = 1 3 e ¯ c , i 2 ( k ). By treating the compression error e_c(k) as measurement noise in y(k), the following model can be used for an estimator. (17) x ( k + 1 ) = A x ( k ) + w ( k ) y ^ ( k ) = C x ( k ) + v ¯ ( k )where (18) R ¯ ( k ) = E { v ¯ ( k ) v ¯ ′ ( k ) } = R + 1 3 [ e ¯ c , 1 2   ( k ) ⋱ e ¯ c , p 2   ( k ) ]

Since the compression error is compensated in the estimation algorithm, we can expect a smaller estimation error, which is verified through the simulations in Section 5. When a Kalman filter is used for (17), the estimation error covariance P(k) = E{(x(k) − x̂(k))(x(k) − x̂(k))′} can be computed from a Riccati equation [12]. (19) P ( k + 1 ) = A P ( k )   A ′ + Q − A P ( k ) C ′ ( C P ( k ) C ′ + R ¯ ( k ) ) − 1 C P ( k )   A ′Since R̄(k) depends on ŷ(k), we cannot compute P(k) before simulation. To evaluate the estimation error covariance without simulation, we use an upper bound of R̄(k): R ¯ ( k ) ≤ R + 1 3 [ e max , 1 2 ⋱ e max , p 2 ] = R ¯ maxUsing this R̄_max in place of R̄(k), we can find a steady-state solution to (19). (20) P ¯ = A P ¯ A ′ + Q − A P ¯ C ′ ( C P ¯ C ′ + R ¯ max ) − 1 C P ¯ A ′P̄ can be considered as an upper bound of P(k) in (19). From P̄, we can see how e_max,i affects the estimation error.

A similar idea is used in [13], where a networked estimation problem is considered.

We compared three data sets using the proposed compression algorithm. Original data is 1600 bits/s for each sensor : 16 bit A/D converted data (i.e., n_y = 16) with the sampling rate being 100 Hz. We used n_b = 5: that is, the number of bits for the quantization index when m(k) = 0 is 5. e_max for each sensor is chosen so that e_max = 300δ, where δ is different for each type of sensors. ȳ_s = 12 is found to satisfy (12).

Bit rates for the three compressed data sets are given in Table 2. All three data sets are obtained using XSENS MTx (3 accelerometer, 3 gyroscopes, and 3 magnetic sensors). Holding MTx with a hand, we moved MTx slowly (data set 1) and fast (data set 2). Data set 3 is obtained from a personal navigation system, where MTx is attached on the shoe of a pedestrian [3].

The bit rates of data set 3 is the largest because the change of data is the most abrupt. In particular, when the shoe contacts the floor, there is a large change in the accelerometer and gyroscope outputs, and consequently the compression algorithm becomes m(k) = 1 more often.

To see how the compression error affects the estimation error, a simple one dimensional attitude estimation problem is considered. An attitude (θ) is estimated using two outputs: y_i is an inclinometer output and y_g is a gyroscope output, where (21) y i = θ + v i y g = θ ˙ + v gwhere v_i and v_g are measurement noises and q i = E { v i 2 } = 0.13 × 10 − 1 and q g = E { v g 2 } = 0.78 × 10 − 5.

An indirect Kalman filter ([12]) is used to estimate θ using y_i and y_g. In the indirect Kalman filter, the gyroscope output is integrated to compute θ_i, i.e., θ ˙ i = y gDefining θ_δ as θ δ = θ i − θwe obtain (22) θ ˙ δ = v gIn the indirect Kalman filter, θ_δ is first estimated and θ̂ is obtained indirectly from θ̂ = θ_i − θ̂_δ.

By discretizing (22) with the sampling period T (T = 0.01sec), we have (23) θ δ ( k + 1 ) = θ δ ( k ) + T v g θ i − y a = θ δ ( k ) − v i

Now the proposed compression algorithm is used to compress y_i and y_g. The maximum compression errors e_max,i (e_max for y_i) and e_max,g (e_max for y_g) affect both the bit rate and the estimation error. If e_max is small, the chance that the mode becomes 1 increases. Thus the bit rate becomes large. How the bit rate changes with the changing e_max is given in Figure 4. The data y_i and y_g are generated using Matlab.

From Figure 4, we can see that the bit rate of the inclinometer outputs increases rapidly as e_max,i is decreased. On the other hand, the bit rate of the gyroscope outputs does not change much as e_max,g is decreased. The bit rate depends on how often the mode becomes 1: note that n_i (the number of bits needed for the quantized data when the mode is 1) is generally larger than n_d (the number of bits needed when the mode is 0). If the original signal is sufficiently smooth, d(k) is small since the predicted value ỹ(k) is very close to y(k). Thus even if we decrease e_max, d(k) still satisfies |d(k)| ≤ 2^y_s(k)δ condition in (8).

The y_i and y_g signals and the compressed signals are given in Figures 5 and 6, where e_max,i = 0.2876 and e_max,g = 0.0122. We can see that the gyroscope output is relatively smooth compared with the inclinometer output. Thus the bit rate of the gyroscope output is relatively insensitive to the changes in e_max,g.

The effects of changes in e_max on the estimation error are given in Figure 7, where the estimation error is predicted using (20). Actual estimation error from simulation is given in Figure 8, where ∑ ( θ ( k ) − θ ^ ( k ) ) 2 is computed.

The relationship between bit rates and estimation error is presented in Figure 9, where data are from Figures 4, 7 and 8. In the left graph, the points 1–6 have similar bit rates but different P_error. Thus in the following simulation, we chose the point 1, which corresponds to e_max,i = 0.2876 and e_max,g = 0.0122. We compared three different filters: (a) a standard Kalman filter using y_i and y_g; (b) a Kalman filter using ŷ_i and ŷ_g with compression error compensation (proposed in Section 4); (c) a Kalman filter using ŷ_i and ŷ_g without compression error compensation. We randomly generated 12 data sets and the results are given in Table 3.

It is not surprising that the P_error of the standard Kalman filter is the smallest because the original data y_i and y_g are used for measurements. We can also see that the P_error of the proposed filter is smaller than that of the filter (c). On average, the estimation error of the proposed filter is smaller by 18.81% compared with that of the filter (c). Note that the proposed filter (b) and the filter (c) use the same decompressed data ŷ_i and ŷ_g for measurements. However, in the proposed filter, the compression error information (13) and (15) are explicitly used in (18), whereas they are ignored in the filter (c). In summary, the estimation error reduction was possible because of two facts: (1) the proposed compression method provides the compression error bound (13) and (15), and (2) the proposed filter algorithm explicitly uses the error compression bound.

In this paper, we have proposed a compression method for inertial and magnetic sensor data. The proposed compression method guarantees that the compression error is bounded by a prescribed e_max value. Smaller e_max value usually increases the bit rate and reduces the estimation error of the filter when the decompressed data is used. Thus e_max plays the role of a trade-off parameter between the bit rate and the estimation error. Also we have seen that by using a bound on compression error, the estimation error can be reduced.