Comparison of kinematic symmetry index calculations and the effects of straight and circular trotting




Reasons for performing study: When assessing lameness in horses, left to right ratios of kinematic parameters are often used to quantify movement symmetry. Different methods of symmetry related measures have been proposed and inertial sensor data was used to evaluate the application of 3 methods of symmetry calculation during straight and circular trotting.

Objectives: To compare 3 sensor based methods of symmetry index calculation to assess; tuber coxae vs. sacrum motion, the effects of circular trotting and effect of using whole trials in place of individual stride calculations.

Methods: Inertial sensors were attached to the sacrum, left and right tuber coxae (LTC/RTC) of 21 non-lame horses. Straight and circular trotting data were collected. Symmetry indices based on vertical movement were calculated for each stride using 3 previously published methods.

Results:Method 2 (Fourier analysis) had significantly higher ratio values than method 1 (displacement amplitude ratio; tuber coxae) and methods 1 (displacement amplitude ratio; sacrum) and 3 (difference between LTC/RTC displacement). The effect of circular trotting within methods was highly variable, but was not significantly different between methods. No significant differences were found between whole trial and individual stride calculations. Method 2 when compared to method 1 underestimates the asymmetry present in a non-uniform way due to the use of squared amplitudes. Methods 2 and 3 become less accurate during circular trotting due to changes in amplitude timing.

Conclusions: On the same data substantial differences in symmetry indices are found when using different methods and locations. Further differences exist in accuracy when used on circular data. Interpretation and direct comparison of different symmetry indices should be approached with caution.


Unilateral lameness is characterised by asymmetry during the left and right stance phases of a stride. Many methods have been developed to enable objective analysis of kinematic asymmetry (Buchner et al. 1996; Peham et al. 1996, 1999; Uhlir et al. 1997; Keegan et al. 2001; Kramer et al. 2004). Quantitative analysis of lameness usually involves the calculation of symmetry ratios. A symmetry ratio refers to the calculation of a ratio aimed at describing parameter asymmetry between the left and right phases of a stride. Kinematic studies have calculated symmetry ratios to enable asymmetry to be quantified numerically (Peham et al. 1996; Pourcelot et al. 1997; Keegan et al. 2001, 2004; Audigie et al. 2002; Church et al. 2009).

Trotting is a symmetrical gait (Hildebrand 1965) with each of the diagonal pairs being dynamically coupled. Most methods of calculating kinematic symmetry ratios produce a ratio of 1 (100% symmetry) if the parameter under investigation is symmetrical (Pourcelot et al. 1997; Audigie et al. 2002; Church et al. 2009). If the ratio is ‘non-directional’ then an increase in asymmetry corresponds to a reduction in ratio towards zero regardless of the side of the lameness. Three published methods used for analysis of kinematic symmetry utilise different parameters or use a slightly different approach while maintaining similar overall principles (i.e. quantifying left/right symmetry). To date, no comparison between these methods has been performed; however, many studies utilise only one of the methods to define changes in symmetry during investigations, therefore limiting comparisons between studies.

The aim of this study was to quantify differences between 3 published methods of sensor-based kinematic symmetry ratio calculations. This was approached with 3 objectives: 1) Compare 3 methods of ratio calculation to assess the use of tuber coxae against os sacrum displacement data for calculation of symmetry ratios; 2) determine the effects of circular trotting on symmetry ratios and 3) investigate the use of whole trials in place of individual stride calculations.


We hypothesise that significant differences exist between symmetry ratios calculated using each method, that ratios calculated using the os sacrum differ significantly from ratios calculated using tuber coxae. We also hypothesise that no significant differences exist between symmetry indices calculated using individual strides or whole trials and that changes in symmetry ratio with circular trotting differ between methods (with circular trotting causing a reduction in symmetry ratios calculated using each method).

Materials and methods


The project detailed in this manuscript underwent review by the Royal Veterinary College ethics committee and was authorised as conforming to appropriate standards.


Twenty-one riding horses (1.32–1.72 m) identified as being free from lameness by their owners and with no history of lameness were used in this investigation. Any horses obviously suffering from painful limbs (lameness) were excluded from the study.

Inertial sensor system (MTx)

Three inertial sensors1 were attached to the skin overlying the following bony landmarks: os sacrum, left (LTC) and right (RTC) tuber coxae. Sensors were connected in series to a wireless transmitter (XBus1) unit attached to a roller. The unit synchronised data from all sensors and transmitted it wirelessly via a Bluetooth connection to a nearby laptop (within 100 m).

Data collection

Horses were led over a soft surface (sand or rubber/sand combination) by an experienced handler in a straight line (approximately 15–20 m) at trot until 6 good trials had been recorded and were subsequently lunged with a radius of 8–10 m on the same surface, while 4 trot trials of approximately 15 s were collected for each rein. Synchronised data from all sensors were collected at 100 Hz per sensor channel and recorded on a nearby laptop.

Data analysis

Recorded MTx data were first calibrated using the Xsens1 software to get 3-dimensional accelerations in m/s2 and 3-dimensional rate of turn in degrees/s. Calibrated data were then rotated into the horse system for double integration to displacement (Pfau et al. 2005) using a custom written MATLAB2 script, prior to cutting into strides. Data were cut into strides using the RTC displacement minima to identify mid-stance of the left hindlimb.

Once the data had been cut into strides symmetry ratios were calculated for each stride using 3 published methods (Pourcelot et al. 1997; Audigie et al. 2002; Church et al. 2009) each using displacements from different pelvic locations.

Method 1:Church et al. (2009)

Approximate mid-stance timings were identified for left and right stance phases for each stride using the tuber coxae minima used for stride cutting. Maximum and minimum vertical positions were extracted for each stride and sensor corresponding to the stance phase of each limb. Propulsion range (PR) i.e. os sacrum displacement amplitude, is defined as the difference between the minimal height reached near mid-stance and the maximal height reached just after the end of stance (Audigie et al. 2002). The same classification was used for LTC and RTC as os sacrum but to enable easier distinction between PR values measured in each sensor, a different name was assigned to the PR values for different sensors, namely OS1, OS2, LC1, LC2, RC1 and RC2 (see Fig 1 for illustration).

Figure 1.

Location of displacement amplitudes in all pelvic sensors. Minima and maxima associated with left hindlimb stance are used to calculate amplitude 1 (OS1, LC1 and RC1) while minima and maxima associated with right hindlimb stance are used to calculate amplitude 2 (OS2, LC2 and RC2) for the os sacrum, LTC and RTC, respectively.

The extracted amplitudes (Fig 1) were used to calculate ratios, giving an objective indication of symmetry present within the stride. Ratio calculation was a modification of the method used by Church et al. (2009); the directional component was removed and the calculation of the ratio altered so that the 2 amplitudes from each sensor were identified as being the greatest or smallest and then used as input for the calculations as shown below:

  • os sacrum: min (OS1, OS2)/max (OS1, OS2)

  • LTC: min (LC1, LC2)/max (LC1, LC2)

  • RTC: min (RC1, RC2)/max (RC1, RC2)

Method 2:Audigie et al. (2002)

Energy ratios use Fourier analysis to identify displacement symmetry within a stride. Three Fourier coefficients were used to reconstruct the os sacrum or tuber coxae displacement pattern of each stride - the DCoffset (direct current offset which for this study was 0 as we were investigating displacements), the symmetrical (amp2) component and the asymmetrical (amp1) component which were used for calculation of the vertical energy ratio (ERz) (Equation 1).


In previous studies horses were identified as sound/symmetrical if they had an energy ratio exceeding 95% i.e. ≤5% asymmetry during straight line trotting (Peham et al. 2001; Audigie et al. 2002).

Method 3:Pourcelot et al. (1997)

Data from the LTC and RTC were used to determine the difference in vertical displacement between the 2 sensor locations within a stride. Due to the asymmetric movement found at the tuber coxae (May and Wyn-Jones 1987), data from the RTC were shifted by 50% of the stride relative to the LTC to ensure stride data from each sensor started at mid-stance of the limb they were attached to. The difference in vertical displacement between the left and right phase shifted signal was then determined for each frame within a stride (Equation 2). Following individual frame calculations the mean kinematic symmetry index (KSI) was calculated. The KSI is the mean difference between LTC and RTC displacement signals throughout a stride i.e. the mean of each individual frame difference (Equation 3).


In the above equation:

Rj and Lj represent right and left displacement values at frame number j.

MeanR and MeanL are the mean displacement values across the stride for the RTC and LTC, respectively.


This method was also carried out on whole trial data. The calculations were similar except instead of taking the mean difference between signals for each stride individually the mean difference across a whole trial (containing multiple strides) was calculated.

Statistical analysis

Extracted symmetry ratios from each horse were tested for normality using a Kolmogorov-Smirnov3 test to determine whether mean/median values for each horse should be used for further analysis. Kolmogorov-Smirnov tests were also performed on the combined horse ratios to determine if data followed normal distribution, determining whether ANOVA3 (if normally distributed) or Kruskal-Wallis3 (if not normally distributed) were to be used to compare methods. P values <0.05 were considered significant.


Data from each horse and each method were not all normally distributed so median values for each horse and each method were used for further analysis (Table 1 in supplementary data). When the medians from all horses were combined the resultant data was normally distributed allowing parametric analysis.

Table 1. Change in variation of the tuber coxae ratios with circular trotting
MethodLTC (SD)RTC (SD)
StraightLeft reinRight reinStraightLeft reinRight rein
  1. The variation (SD) of both the LTC and RTC ratios calculated using methods 1 and 2 when trotting in a circle.

The effect of the method used on calculated symmetry ratios for straight trotting

A general trend was found between methods within each horse showing an increase in symmetry from method 12 and then a decrease from method 23. ANOVA analysis on combined horse data confirmed the differences between methods 1 and 2 and 2 and 3, but not 1 and 3 were significant (P<0.05) during straight line trotting.

Os sacrum vs. tuber coxae derived data

ANOVA analysis was used to investigate the differences between ratios calculated using os sacrum data (or combined RTC and LTC data - method 3) and those calculated using individual tuber coxae data (methods 1 and 2). We identified significant differences between methods 1 and 2 when os sacrum data or individual tuber coxae data were used.

Individual strides vs. whole trial data

Method 3 can be used for individual stride or whole trial ratio calculations. In this study (and to the author's knowledge for all published analyses) individual strides were used to enable direct comparison of the data with ratios from other methods. The final part of this study was, therefore, to determine if the use of individual strides had a significant effect on symmetry indices when compared to the use of whole trial calculations. No significant differences were found in symmetry ratios calculated using method 3 for any condition (straight or circle) irrespective of whether individual strides or whole trials were used.

Circular trotting

Significant differences were found between ratios calculated using method 2 and methods 1 and 3 during straight, left and right rein trotting (P<0.001) but not between methods 1 and 3.

Comparisons within methods were done to determine changes in symmetry ratio calculated on a circle with respect to the ratio calculated during straight line trotting. The difference in ratio change between methods was then compared for both left and right rein trotting to determine if circular movement affected each method equally (Fig 2) despite absolute values being different. No significant differences were found in the change calculated using each method for os sacrum or tuber coxae ratios as a result of circular trotting.

Figure 2.

Ratio change within each method when trotting on the left and right reins. Mean change in symmetry ratio during circular trotting calculated as a percentage of the straight symmetry ratio using methods 1–3. Vertical bars represent±1 s.d. No significant differences were found between methods or reins.


The aim of this study was to quantify differences between 3 published methods of kinematic symmetry ratio calculations. Large standard deviations were found in ratios across all horses within methods when the median from each horse was used. This was to be expected as our study consisted of 21 horses with multiple strides from each horse. Although the horses used were believed to be sound by their owners, upon close inspection of the data it was clear that many of the horses were asymmetric beyond the predefined threshold for sound horses (ER>95% [Audigie et al. 2002]). Many horses were found to exceed the previously published 95% energy ratio threshold. However, the inclusion of horses with slightly asymmetric movement patterns was not considered a problem as individual biological variation (variation between horses) might highlight differences between methods. An accurate quantification of the lameness grade was not the goal of this study which focussed on differences between methods. A soft surface was used for this study despite lameness often being more visible on a hard surface, to reduce the risk of slipping and increase the number of horses available for data collection as most yards have a suitable school surface.

Straight line trotting

During straight line trotting ratios calculated using methods 1 and 2 on the os sacrum sensor and method 3 with the LTC and RTC combined had median stride ratios (for individual horses) which differed in a consistent way. Ratios calculated using method 2 were always larger than those calculated using methods 1 and 3 suggesting increased symmetry. Methods 1 and 2 were used to calculate symmetry ratios from individual tuber coxae data with significant differences found between methods.

Circular trotting

When trotting in a circle on the left and right rein ratios calculated using the os sacrum (methods 1 and 2) or combined tuber coxae data (method 3) were found to differ in a similar way to straight trotting and each other, i.e. methods 1 and 3 differed significantly (P<0.001) from method 2 but not each other. Again, individual LTC and RTC ratios (methods 1 and 2) altered significantly (P<0.001) on both reins.

The consistently higher values using method 2 can be explained by the use of squared Fourier coefficients (amp1, amp2) while a simple amplitude ratio is used for method 1. The use of sound horses (i.e. ratios close to 1) resulted in the non-linear underestimation of asymmetry shown in Figure 3. Using the equations detailed in the methods section with a range of example values from 0–1 for amp1 (the asymmetric coefficient amplitude) and a constant value of 1 for amp2 (the symmetric coefficient) the symmetry ratios were calculated:

Figure 3.

Correlation between ratios calculated using methods 1 and 2. A correlation plot showing the non-linear relationship in symmetry ratios calculated using methods 1 and 2. The correlation differs to a greater extent at ratio values close to one (perfect symmetry).


When plotted against each other this demonstrates why method 2 produces consistently higher ratios with much smaller variation than methods 1 and 3.

Effect of circular trotting on symmetry ratios

The change in ratio within each method between straight and circular trotting was determined as a percentage of straight ratios before comparing between methods. No significant differences were found between methods, an unexpected finding since the mean change in ratio showed large differences between methods 1 and 3 when compared to method 2. This absence of significant differences is likely due to the large variations between horses within individual conditions for methods 1 and 3.

As with the os sacrum, no significant differences were found in tuber coxae ratio change with circular trotting. Tuber coxae ratios from methods 1 and 2 had similar increases in variation during circular trotting when compared to straight. Increased variation in method 2 is likely to be linked to the reduced accuracy of the signal reconstruction which was found to have ≥3% decrease during circular trotting when compared to straight (i.e. 96% compared to the 99% reconstruction found in straight os sacrum data). When combined with the previously described under-representation this deters the use of energy ratios during circular trotting.

Whole trial vs. individual strides

Unlike methods 1 and 2, method 3 did not use amplitudes of the 2 stride phases to determine symmetry; instead it determines the difference between the LTC and RTC displacement patterns at each frame and then uses the mean of this difference to provide a symmetry ratio. This enables whole trial ratio calculations in place of individual strides. We hypothesised no significant effect on symmetry ratio when compared to individual stride analysis providing the trial was steady state and there were no times where the horse stumbled or spooked. Following comparison of the corresponding results we found no significant changes in method 3 symmetry ratio whether individual strides or whole trials were used. This supports the use of method 3 for determining symmetry using whole trials.

Between method comparisons

Analysis detailed within this study confirms that the 3 methods differ significantly in many aspects and should therefore not be used interchangeably or for direct comparisons. Which method is the most appropriate to use is still a difficult question and depends on many factors including whether the data is from straight or circular trotting and the origin of the data being analysed (i.e. os sacrum or tuber coxae).

Method 1 has a relatively large standard deviation between horses but these remain fairly consistent between straight and circular trotting suggesting its accuracy is unaffected by changes in timing. In contrast, method 2 has a much smaller standard deviation when compared to both methods 1 and 3. Reduced accuracy of the signal reconstruction (method 2) was recorded at the tuber coxae when compared to the os sacrum. We only used 3 coefficients to reconstruct the signal. For an approximately symmetrical signal (os sacrum) this is sufficient but it appears here, for the asymmetric tuber coxae displacement signal the decomposition method is not sufficient leading to reduced sensitivity.

Method 3 has a variation similar to that found in method 1 during straight trotting. During circular trotting the ratio decreases showing, as expected, decreased symmetry with an associated increase in standard deviation. Each stride contains 2 minima corresponding to left and right hindlimb stance. As the tuber coxae are located away from the midline laterally, the vertical displacement measured during each phase of the stride differs in amplitude (May and Wyn-Jones 1987) allowing clear identification/classification of left hind (prior to the largest displacement amplitude) and right hind (prior to the smallest amplitude) stance phases. If the magnitude of the displacement differs between the inside and outside tuber coxae on a circle then this method, comparing LTC and RTC could wrongly imply lameness or asymmetry during circular trotting due to the effects of body tilt (Clayton and Sha 2006). The effects of skin movement at the attachment site is likely to further alter the movement symmetry at the tuber coxae (method 3) when compared to os sacrum movement. A further cause of error using this method is the 50% phase shift of RTC data enabling direct comparison with LTC data. Any error in this phase shift causes the strides being compared to differ in the time point from which they originate.

The points discussed above suggest method 2 is appropriate for determination of symmetry during straight line trotting from the os sacrum sensor as used in previous studies (Audigie et al. 2002; Church et al. 2009). Due to the poor reconstruction of the signal (using our method based on 3 Fourier coefficients) on non-symmetric displacement data it is an inappropriate method for tuber coxae analysis or circular os sacrum data due to the asymmetric timing of the signal in some horses caused by body tilt (Clayton and Sha 2006). Methods 1 and 3 have a larger change in ratio as a result of circular trotting, suggesting they are more sensitive to minor changes. This increased sensitivity makes method 1 more appropriate for use in analysis of circular trotting or for identifying the effect of lameness on circular trotting ratios. The potential inaccuracy with method 3 due to the different displacement magnitudes of the LTC and RTC with body tilt combined with potential errors associated with the 50% shift of the RTC signal, make method 1 the most appropriate for use in further studies. However, no previously published baseline data are available comparing ratios with visual gait scores for method 1. With this in mind, method 2 is considered appropriate for determining which horses are sound, using os sacrum ratios with a threshold of 95% during straight trotting. For all other investigations method 1 is the most appropriate for both os sacrum and tuber coxae sensors during straight and circular trotting investigations.


Many significant differences exist between methods and within methods when used on data from different sensor locations and under different conditions (straight or circle). It is unwise to directly compare symmetry ratios calculated using different methods if at all possible this should be avoided. From the 3 methods studied here we found method 1 to be the most appropriate to use for straight/circle investigations. Methods 2 and 3 were found to be greatly affected by asymmetric stride displacements making use on tuber coxae or circular data susceptible to errors.


We would like to thank all the owners for kindly allowing us to use their horses for this study. A.M. Wilson is a holder of a Royal Society Wolfson Research Merit award.

Conflicts of interest

The authors declare no potential conflicts.

Manufacturers' addresses

1 MTx; Xsens, Enschede, the Netherlands.

2 The Mathsworks INC, Natick, Massachusetts, USA.

3 SPSS Inc, Chicago, Illinois, USA.