As heart rate can be easily measured by using wireless portable sensors, it has been extensively used to monitor exercise intensity and estimate exercise response of key cardiovascular variables, such as oxygen consumption [1Su S, Celler B, Savkin A, et al. Transient and steady state estimation of human oxygen uptake based on noninvasive portable sensor measurements Med Biol Eng Comput 2009; 47(10): 1111-7., 2Fairbarn M, Blackie S, McElvaney N, Wiggs B, Pare P, Pardy R. Prediction of heart rate and oxygen uptake during incremental and maximal exercise in healthy adults Chest 1994; 105: 1365-9.], energy expenditure [3Su S, Wang L, Celler B, Ambikairajah E, Savkin A. Estimation of walking energy expenditure by using support vector regression Proceedings of the 27th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS), Shanghai, China 2005; 3526-9.] and cardiac output [4Astrand P, Cuddy T, Saltin B, Stenberg J. Cardiac output during submaximal and maximal work J Appl Physiol 1964; 9: 268-74., 5Freedman M, Snider G, Brostoff P, Kimelblot S, Katz L. Effects of training on response of cardiac output to muscular exercise in athletes J Appl Physiol 1955; 8: 37-47.].

The heart rate is a non-stationary signal, and its variation includes indicators of current disease or warnings about impending cardiac diseases [6Acharya R, Kumar A, Bhat IP, et al. Classification of cardiac abnormalities using heart rate signals Med Biol Eng Comput 2004; 42(3): 288-93.]. Recently, heart rate has been applied for the assessment of cardiovascular fitness and monitoring of rehabilitation exercise with a focus on cardiovascular abnormality detection [6Acharya R, Kumar A, Bhat IP, et al. Classification of cardiac abnormalities using heart rate signals Med Biol Eng Comput 2004; 42(3): 288-93., 7Wang L, Su S, Celler B. Time constant of heart rate recovery after low level exercise as a useful measure of cardiovascular fitness Proceedings of the 28th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS), New York, USA, September 2006.]. To effectively detect cardiac abnormalities and monitor rehabilitation exercise, an efficient way is to set up models for normal heart rate responses. Then, abnormal response detection can be treated as a fault detection problem. The most commonly used method of fault detection is model based residual analysis [8Blanke M, Kinnaert M, Lunze J, Staroswiecki M. Diagnosis and Fault-Tolerant Control. Berlin: Springer 2003.]. Based on the established models, abnormal responses can be detected by calculating the residual of heart rate response, which can be acquired by comparing the signals, measured by using sensor and estimated by using the model. There are plenty of papers [9Seliger V, Wagner J. Evaluation of heart rate during exercise on a bicycle ergometer Physiol Boheraoslov 1969; 18: 41., 10Chen Y, Lee Y. Effect of combined dynamic and static workload on heart rate recovery cost Ergonomics 1998; 41(1): 29-38.] about the analysis of steady state characteristics of heart rate response. Some papers investigate its dynamic characteristics by using linear or nonlinear models [11Hajek M, Potucek J, Brodan V. Mathematical model of heart rate regulation during exercise Automatica 1980; 16: 191-5., 12Cheng T, Savkin A, Celler B, Su S, Wang L. Nonlinear modelling and control of human heart rate response during exercise with various work load intensities IEEE Trans Biomed Eng 2008; 55(11): 2499-508.]. However, few papers investigate the variation of dynamic characteristic (such as time constant) of onsite and offsite walking exercises. For moderate exercise, literatures often assume heart rate dynamics can be described by linear time invariant models. However, in our previous studies, it was observed time constant of heart response to exercise depends on exercise intensity. It is also reported [13Su S, Wang L, Celler B, Savkin A, Guo Y. Identification and control for heart rate regulation during treadmill exercise IEEE Trans Biomed Eng 2007; 54(7): 1238-46.-15Eston R, Rowlands A, Ingledew D. Validity of heart rate, pedometry, accelerometry for predicting the energy cost of children's activities J Appl Physiol 1998; 84: 362-71.] that exercise effects can be optimised by regulating heart rate following a predefined exercise protocol. Therefore, it is worthwhile to establish a more accurate dynamical model to enhance the controller design of heart rate regulation. For this purpose, this study investigated the variations of time constant and steady state gain under different exercise intensity. More attention was paid on the difference of transient response of onsite and offset exercises. We designed a treadmill walking exercise protocol to analyse step response of heart rate. During experiments, ECG, body movement, and oxygen saturation were recorded by using portable non-invasive sensors: Alive ECG monitor, Micro Inertial Measurement Unit (IMU), and Alive Pulse Oximeter. It was confirmed that time constant are not invariant, especially when walking speed is faster than 3 miles/hour. Furthermore, time constant for offsite exercise is normally bigger than that of onset exercises. Steady state gain variation under different exercise intensity has also been visibly observed.

This paper is organised as follows. Section 2 describes the experimental equipments and exercise protocol. Data analysis and modelling results are given in Section 3. Finally, Section 4 gives conclusions.

Original signals of IMU, ECG, and SpO2 are shown in Figs. (3, 5, 6), respectively. It is well known that even in the absence of external interference the heart rate can vary substantially over time under the influence of various internal or external factors [13Su S, Wang L, Celler B, Savkin A, Guo Y. Identification and control for heart rate regulation during treadmill exercise IEEE Trans Biomed Eng 2007; 54(7): 1238-46.]. As mentioned before, in order to reduce the variance, designed experimental protocol has been repeated three times. Experimental data of these repeated experiments has been synchronised and averaged.

Fig. (5) Original ECG signal.

Fig. (6) The recording of SpO2.

A typical measured heart rate response is shown in Fig. (7). Paper [13Su S, Wang L, Celler B, Savkin A, Guo Y. Identification and control for heart rate regulation during treadmill exercise IEEE Trans Biomed Eng 2007; 54(7): 1238-46.] found that heart rate response to exercise can be approximated as first order process from a control application point of view. Therefore, we established first order model for six averaged step response data by using Matlab System Identification Toolbox [16Ljung L. System Identification Toolbox V40 for Matlab 1em plus 05em minus 04em MA: The MathWorks, Inc, 1995 ]. The identified time constants and steady state gain are shown in Table 3. Curve fitting results of all six cases are shown in Fig. (8).

Fig. (7) A measured heart rate step response signal.

Fig. (8) Curve fiting results for onset and offset exercises for all six exercise intensities as listed in Table 3.

Table 3

Summary of the Identified Time Constants and Steady State Gains

Based on the identified steady state gain (k) and time constant (T) as listed in Table 3, we can clearly see that both steady state gain and time constant vary when walking speed Va and Vb change. Furthermore, time constant of offsite exercise are noticeably bigger than those of onsite exercises. However, it should be pointed out that the variant of time constant is not distinctly dependent on walking speed when walking speed is less than 3 miles/hour. Overall, experimental results indicate that heart rate dynamics at onsite and offset exercise exhibited highly nonlinearity when walking speed is higher than 3 miles/hour.

This study mainly focuses on the analysis of dynamic nonlinear behaviour of heart rate response to treadmill walking exercises. Both steady state gain and time constant under different walking speeds have been identified and analysed by using the data from a healthy middle aged male subject. It was observed that both steady state gain and time constant are not invariant under different walking speeds. Especially, time constant for recovery stage is noticeably longer than that of onsite exercise. In order to verify this conclusion, we are planing to recruit more subjects in the next step of this study. Nonlinear control algorithm will also be developed for the established exercise intensity dependent nonlinear model. We believe that this study has great potential to improve exercise efficiency during treadmill exercises.