Active Suspension: Kinematics and Control Part 1

Having previously modelled a potential active suspension in a very simplistic, static way, and proved that it will work, the next challenge is to accurately model the system dynamically. This will lead to fine-tuning of the control algorithms and more detailed predictions of its responses to different stimuli.

Modelling the control of the system, and the way that it interacts with the rest of the car, is no simple task because there are a series of mechanisms between the chassis and the ground which all require their own model. The first step modelling the system dynamically is to build these models and see how the suspension will work in principle.

Modelling the geometry of the system in SolidWorks

Conventional Suspension Modelling

Any active suspension system implemented on the Formula Student car will be in addition to a conventional system, so modelling the conventional system needs to be done first before the effect of the active suspension is added. This involves modelling every component that links the chassis to the ground.

These models will contain information about the geometry of the system, and the forces and loads carried by the linkages. Some parts will be purely geometric, while there are other models which will require parameters to be specified, such as the spring rate or the gearing of a system. In essence, for the whole car to be accurately modelled, each component within it must be modelled along with the relations between these parts.

There are three different classes of model. Firstly, there are the geometric models which are used for positioning and calculating motion paths. Secondly, there are models which are related to the way that forces are transferred through the system and between linkages. Calculating the forces in each bar has many advantages including optimisation of the structure and prediction of failure modes, but it will also be used for calculating the accelerations (and therefore positions) of all of the components dynamically. The third type of model is a dynamic model, where the parameters are a function of time. It will describe how the system behaves dynamically, and requires the static positions and predicted loads to calculate how the suspension will respond.

Geometric Models

The first model to be built is the wheel model, which specifies things like the wheelbase, track width, wheel diameter, and unsprung mass. This defines the position of all four corners of the car, and the size and shape of the footprint. The unsprung mass will also be important when it comes to modelling the way that the suspension behaves dynamically.

The wheel is connected to the chassis via six linkages. Four of these are part of the A-arms, or wishbones, and then there is a pull/push rod and a steering arm or toe link. The wishbones define the motion path that the wheel takes as it travels vertically, defining things like the roll centre and the instant centre. The pull/push rod controls the vertical position through a rocker linked to a spring, and the steering arm is connected to the upright to control the wheel’s position around the steering axis. The wishbones, and steering arm are purely geometric models, but the pull rod is more complex and is a hybrid of the geometric and kinematic models.

Kinematic Models

A Matlab plot showing how tyre spring rate varies with tyre pressure. Axis values are removed to protect intellectual property.

Kinematic models deal with the way that forces are transferred through the suspension system. Any deformable components, and the links between them, are models fitting in this category. This includes the tyre, which will be modelled as a spring. While the equations governing the spring are trivial, calculating parameters such as the spring rate is not so easy; extensive manipulation of tyre data provided by the manufactures has revealed how the spring rate varies according to a range of different parameters, such as pressure, and the tyre model will also require these parameters to be specified in order to determine the spring rate.

The pull rod and spring are part of the same model, because their displacements and the forces are closely interlinked. In particular, the position and force in one is fully defined by the position and force in the other, so it makes sense to combine the models. The two are linked by a rocker, which controls the motion ratio and relative spring rates.

The motion ratio is the distance moved by the spring for each unit distance moved by the rocker. This is a function of both the radius at which both parts attach to the rocker, and the direction the rod points in; it is based on a triple vector product of the rocker rotation axis, the direction of the rod, and the vector radius of the pickup point. Because it depends on the direction of the rod, it will vary as the suspension travels, which complicates the dynamic modelling substantially. The spring force is multiplied by the motion ratio twice; first the displacement is reduced and then the force itself is scaled, so the force ratio is the motion ratio squared.

The code for calculating the loads in the suspension members.

There is also a kinematic model for calculating the wishbone and steering arm loads. This is based on the solution to a matrix equation which states the force and moment equilibrium cases for wheel. It assumes that the wheel displacement must be held steady, but allows for vertical and lateral forces to be applied at the wheels to see how this affects the member loads. Using this also allows us to see the steady state force in the pull rod, and hence choose an appropriately stiff spring for the static case where the car is resting on the ground.

Dynamic Models

The dynamic models are models where the inputs and outputs are functions of time. The dynamic wheel model calculates the forces on the wheel at any given time as a function of body accelerations and spring positions. These forces will typically be imbalanced, hence accelerations, velocities and positions can be derived from an integral process.

As the position of the wheel updates, the spring position will change and hence the forces on the wheel will be different. In the next round of calculations, this will also affect the accelerations, velocities, and position of the wheel, so that over time the wheel behaviour can be analysed.

The active suspension component adds another level to this. If it is to be controlled as suggested before, based on the wheel position, it too will be a dynamic model. The active suspension control model will be fed the position of the wheel, mimicking the sensor inputs available in the car, and will respond with a command to the actuator.

The actuator force is therefore a function of time, and when it is added to the model, the actuator force will work alongside the spring force to control the position of the wheel. The actuator force will be fed back in to the dynamic wheel model to update the forces accordingly.

A range of different control algorithms can be tested to find the most effective one, defined as the one which produces the least overall unintentional wheel travel. By modelling the system computationally, the controller parameters can be optimised without doing any physical testing. It is also possible that this control model could lead to an unstable feedback loop, but if it does it will be discovered in the code before the physical parts are assembled, allowing time for design changes.

PID Controllers

The actuator force will be driven by a control algorithm which uses the vehicle ride height, or wheel position, as its input. However, purely driving the algorithm based on position is not necessarily effective. More effective controllers can be built which can reduce the overall error in the position by also taking into account the speed at which the position is changing, and the accumulation of errors over time.

This type of controller is known as a PID controller, with the terms standing for ‘Proportional-Integral-Derivative’. Three components make up the output of the controller, based on the sum of a value proportional to the error, a value proportional to the integral of the error over time, and a value proportional to the rate of change of the error. It is also possible to extend this to the second derivative of the error, which in this case represents the acceleration of the wheel, linked to the deficit or excess force provided by the passive system.

The PID controller needs to be tuned, by applying different multiples to each of the terms. Each term has a slightly different effect, and changing the weighting changes the response of the controller. This can be used to iron out high frequency inputs, or to try and pre-empt any significant roll or pitch, and correct before it starts.

Building the Models

There are a range of tools that can be used for this form of modelling. Microsoft Excel is a very powerful tool and is used for validation of the models, along with Matlab. Matlab is well suited to building models such as these, and integrates well with the requirements of the modelling: to perform vector and matrix multiplications and to draw graphs of the output. Matlab also includes a PID tuning library so is a very attractive prospect.

Validating the wishbone load model using Microsoft Excel.

MSC Adams is the industry standard system for modelling suspension systems. This can be used to show how the suspension deflects in response to loads, calculate the loads in members, and specify springs and rockers. It can also optimise these components. However, the support for a user-configured active suspension system is limited.

As a result, a bespoke system may be the answer, written and developed for the SUFST team in C#. This is the type of challenge I enjoy taking on. It will allow the model to be built with only the required functions, which is beneficial when it comes to ease of use and development. In addition, building a bespoke system lends a much better understanding of the way the dynamics work, and can be integrated easily with Solidworks.

The bespoke suspension model can show the geometry of the car projected on to the three main planes. Here, the front and rear suspension layout can be seen based on the 2014 SUFST car

The models are in development at the moment and future posts will show exactly how the dynamic models are used to evaluate and adapt the design.

Surveys, Specifications, and Statistics

Much of the information that will be used to determine the outcome of the VLE review will come directly from students. Data will be collected through surveys and direct interviews with focus groups, and will then be analysed to determine the general student feeling towards the VLE. The same process will be undertaken with academic staff as they are likely to have different views.

Collecting data via a survey is not as simple as just coming up with questions. In order to get effective responses to the survey and to draw significant conclusions, the survey needs to be designed with the intended responses in mind. It is also difficult to avoid inducing bias with the phrasing of the questions, and there is a balance to be had between being thorough and covering lots of information, and being brief enough to get plenty of responses.

Principle of the Review

The specification and surveys are used to put together a full picture of the current VLE status and inform the final outcome.

It has been determined that the review will focus on two main objectives, to make the outcome as clear as possible. Firstly, the review will determine criteria for a successful VLE, marking out the properties that the VLE should have in order to be most effective for learning and for teaching. Secondly, a range of possible VLEs will be evaluated against these criteria, including the current VLE, Blackboard. The degree to which Blackboard meets the specification is the benchmark; if other VLEs can beat this benchmark then a change can be considered.

Using this two-part structure allows at least some form of quantitative comparison between the alternatives, which will allow the systems to be ranked mathematically. This is important for removing bias – although human input is used to determine the ranking, the ranking is not specified by humans. This subtle difference is the fine line between a subjective decision and an objective decision. If the decision can be made objectively, it is more likely to be widely accepted, as opposed to a subjective decision where the reviewer can influence the result. An objective evaluation is more likely to determine the system that is the best for the most number of people.

Creating a Specification

The specification, or list of criteria, is part one of the review. Determining the specification points, and therefore coming up with a complete list of points on which the VLE will be judged, is something that can be done manually and subjectively. However, it needs to be sure to cover all of the possible points, because the importance of these points can then be established using the questionnaires.

The specification will include points such as the cost, which is easily measured and very objective, but also the ease of use, which is much harder to measure and is much more subjective. The importance of this point will therefore have to be evaluated in the questionnaire.

The draft specification is the initial set of criteria, and is created by the review team. Once the draft specification has been completed, it can be used as the basis of the questionnaire, which will aim to determine how important the subjective points are, and how well existing systems match these points.

Writing the Questions

An example of one of the survey questions in iSurvey

Writing effective questions means writing questions which have meaningful answers. In fact, it makes more sense to play a Jeopardy style game, where a relevant and objective statistic is given as the answer, and the question is created from this answer. The value of the statistic will later be determined from the range of responses. Creating questions in this way guarantees that every question can be used to produce an objective parameter.

For example, one of the specification points is that the VLE should be accessible in some form on mobile devices. To establish whether this is really important to the student group, we wish to be able to state that “x% of students would use the VLE on mobile devices, if the functionality was available”. To fill in the gap requires a set of answers: yes or no. So this leads to the question being written as something with a yes/no answer: and it becomes something along the lines of “Would you like to use the VLE on a mobile device?” From this, we can establish how important it is that a VLE is available on a mobile device, and present an objective, numerical answer.

This is one type of question. There is a second type of question that must also be considered – and this involves evaluating a VLE against a certain measure. It may be desirable to assess how useful a particular feature of a VLE is; the corresponding statistic would be “y% of students felt that this feature benefitted their studies, while z% had not used the feature”. It is important to include the second clause here, because if a feature has not been used it is not possible to tell if it is useful or not.

The answers to the question must therefore include at least ‘useful, not useful, never used’, and it would provide clearer definition if a range of usefulness was provided (‘very useful, useful, no benefit, unhelpful, very unhelpful, never used’). The question becomes: “How helpful did you find this feature to understanding your course?” Note that this is not “How useful did you find this feature”, or “How much did this benefit your studies”, as they are subjective questions and do not rank the feature clearly on how effective it is at improving learning in particular.

Analysing the Results

Once the questions have been written in the above way, with the intended responses linked to parameters that we wish to know, calculating the value of the parameter is left to the realm of statistics.

The survey will return a statistic based on the responses of a sample. Because we are not interviewing every single student at the university (known as the ‘population’), our results are only relevant to a ‘sample’, which is a small group of the population. The parameter that we wish to establish is a value which relates to the population, but because we only have data from the sample all that is known is a ‘statistic’. The statistic therefore needs to be converted into a parameter, and this requires knowledge of the sample, and some neat mathematics.

The sample will have returned a statistic, such as ‘72% of the sample would use the VLE on mobile devices’. In very simple terms, we could use this to estimate that 72% of the whole population would also use the VLE on mobile devices. However, it is possible that the sample consisted mostly of History students, who for some reason do not use the VLE on mobile devices. It does not matter what this reason is, or what the biased group is, but it can be corrected for.

Sampling

Stratified sampling: one sampling unit is taken from each stratum to form the sample, which contains proportional numbers from each group.

This means using a technique called stratified sampling. If History students make up 10% of the population, then even if they made up 70% of the statistic, their responses are only given a 10% weight in the final parameter. Other responses are given a greater weight, so that every faculty group has their relevant weight across the overall population. This makes the scaling process more accurate.

There will also be uncertainty in the value of the parameter, due to random fluctuations in the responses that cannot be ironed out. It is correct therefore to establish an uncertainty in the value, which is normally defined as 1 divided by the square root of n, the number of samples. In the case of 100 samples, this uncertainty is 10%. This is valid for the yes/no questions, where the response is binary and the result is a simple fraction.

However, for more complex questions, such as how useful the system is, the responses will follow a distribution. The nature of the distribution means it will have a mean and a variance – an average point, and an amount by which the value varies according to chance. The sample mean and variance both have these uncertainties in when scaled up to the population size, and as such stating an absolute value for the population can only be done with caution – the variance could be large.

Using the Data

Once the population parameters have been found, they can be used to fine-tune the VLE specification. In addition, the results will be used to evaluate the current VLE – Blackboard – against this specification and therefore set a benchmark that other VLEs can be compared against.

The methods described are essential in forming comparisons between the different systems in a way that is fair and unbiased to all. Measuring each against a well-defined, objectively written specification ensures that the review is rigorous and complete, and allows future re-evaluation if the review is conducted again in the future.