Vehicle operation is highly variable causing parameter and states data to change unpredictably. A way to deal with this data is Markov Analysis.
A Markov Process is defined a having a finite number of states and the probability of transitioning from the current state to another state is between 0 and 1. A Markov transition matrix is used to count these transitions, table 1.
In this example the current state is B and the item transitions to the next state D. The count of 30 will be incremented to 31. For each current state, the transition counts in each cell can be converted into a percentage, reflecting the probability of transition to the next state. These percentages are shown in figure 2.
Note that given any current state, the row-wise sum of probability of arriving in the next states add to 1.
Transmission Data Example
Consider the transmission usage. There is a current gear state at all times, but a transition to another state can occur at any time. Because the CAN data is recorded at a rate of 1 per second, the most likely next state is the current state. However, occasioally, there is a gear state shift and a cell is incremented off the upper left to lower right diagonal. The probablity of a gear shift increases with time. The transmission shift is a response to vehicle usage or environmental factors. For the one-second telematics data, an 8-speed transmission markov data table could look like.
In table 3, the numbers indicate the forward gear. The current state is the position the transmission is leaving and the next state the next gear engaged. In the cells, the count is stored. The cells are filled with non-negative integers.
The probability of a transition from any transmission state to any other transmission state is calculated by dividing the cell value by the total number of transitions. The probabilities would be stored in a similar table. The sum of these probabilities must equal 1. For a fleet of vehicles, their respective tables could be analyzed to provide an average probability of transitioning from any gear state to another gear state.
Gear transitions are stressful events that may result in failures. This information could be used to design a transmission reliability test. A shift schedule can be developed from the probability matrix. Alternatively, the shift schedule could be a randomly selected variable controlled by the probability table. Either approach could be used to develop vehicle focused transmission tests.
A primary failure mode of electronics is thermal cycling. There are cumulative damage failure models that can be used to analyze the damage created by different cycles. To use these damage models, the thermal cycles within vehicles need to be considered. The important factor for these models is the minimum and the maximum cycles for each temperature swing. The Markov model can capture this data.
For a thermal model, define the start and next states as temperature bins, perhaps with a constant width. For example, define bins with 20°C interval widths centered on -30°C, -10°C, 10°C …110°C, 120°C. While a very large table, the counts will capture a vehicles temperature history.
One must be understand that this approach captures half cycles, rising temperatures and falling temperatures, but is not defining cycles. A comparison of the Markov transition counting approach to rain flow analysis showed a 2:1 relationship. The Markov counts were twice the rain flow counts. The temperature data analyze with rain flow software to get a cycle count table. That may be projected to 10 years of operation, figure 1.
Figure 1 was published in “Accelerated Life Test – Simulation Integration”, by James McLeish, in the SAE Automotive Excellence, Spring, 2017. These thermal cycle counts provides a basis for modeling damage to the electronics.
Accelerator pedal assemblies can wear out from usage. The assembly includes a spring that returns the pedal from a depressed state. Because the assembly is under stress, metal can fatigue and bushings can wear. The design engineer needs to subject the assembly to reliability verification tests.
The pedal positions are measured as a voltage on the CAN bus. When the driver depresses the accelerator pedal, it produces an up-stroke from some initial low voltage to a final higher voltage at maximum stroke. When the driver releases the accelerator pedal, a down-stroke from a high initial voltage to a lower voltage results.
To apply the Markov model, first define voltage intervals. The time series is analyzed to determine the initial and final voltage states, and the appropriate counter is incremented. Then the data was converted to a percentage and plotted, figure 2.
This figure shows a driving pattern where the pedal is at 0 voltage at either the start or end of pedal stroke. This may imply the driver was heavy footed and the pattern should be compared with other drivers. Averaging over a fleet produces a representative pattern for design and verification testing.
Markov methods may be used to analyze some telematics data for components and subsystems. The analysis provides design and test engineers with information that can be used for design, simulation, and for verification testing.