Formulating the Optimization Problem for Stopping BLOODHOUND

In my last blog post, Dr John Davis from BLOODHOUND provided 5 additional pieces of the puzzle to help us figure out how to stop BLOODHOUND using a combination of brakes and parachutes. 


In this post, we analyse the additional pieces of information in turn and introduce labels for the main quantities and key points in time. 

1)    We want to stop at exactly 10 miles so we are ready to turn around to go back

This is the main objective or goal of the problem. To precisely formulate the problem, let's express this objective in symbols. We also need to convert to metric units (e.g. metres) rather than imperial (miles), and to decide where to measure from. The most obvious choice is to measure from the start of the 10 mile track, although as this problem is focussed on the stopping phase an alternative would be to measure distance from the end of the measured mile. For now, we'll let s be the distance (in metres, m) from the start line, and v be the velocity (in metres per second m/s) at time t (in seconds). We can use an online calculator to work out that 10 miles is 16093 metres (or 16.093 km). We can then write this objective in symbols as
                                    
                                              v = 0 m/s, when s = 16093 m

2) We can not deploy the parachutes above a certain speed, definitely not supersonic

This is a constraint on the timing of when the parachutes can be deployed. We'll label these points in time so we can refer to them more easily. Let t1 be the time in seconds when the first parachute is deployed and t2 be the time in seconds when the second parachute is deployed. The velocity at t1 is written v(t1) and must be less than supersonic (i.e. less than the speed of sound). The speed of sound  depends on the environmental conditions, but is approximately 343 m/s. This constraint can be written in symbols as


                                                v( t1 ) < 'a certain speed' < 343 

 3)    We can stop at 10 miles by two methods 

      a)    Airbrake at the end of the measured mile and wheel brakes at about 150 mph. This is the preferred option. 

      b)    Parachutes: first just sub-sonic, second almost immediately afterwards, followed by wheel brakes maybe a bit faster than 150 mph

This piece of additional information suggests that we should split the problem into 2 cases, with and without the air-brake. 


Let t0 be the time when the air-brake is deployed, and t3 be the time when the wheel brakes are applied. We have chosen the time labels to reflect the expectation that t0 <= t1 <= t2 <= t3. Let s0 be the distance (in metres) of the end of the measured mile from the start line.  


In case (a), BLOODHOUND is stopped using the air-brake at the end of the measured mile, followed by the wheel brakes when the velocity is approximately 150 mph (or 67.06 m/s ). In symbols these constraints can be written as


                  s(t0) = s0  and v(t3) is approximately 67.06 m/s  


In case (b), the constraints can be written as 
                 v(t1) = 343 - (a bit), v(t2) = 343 - (a bit more), v(t3) = 67.06 + (another bit)


where 'a bit','a bit more' and 'another bit' are small positive velocities that need to be decided.  

4)    If the airbrake only partly deploys we will need one parachute, but not necessarily at high speed. This will be very dynamic

This sounds like an interesting case. Let's call it case (c). In this case we need to find the velocity when the first parachute is deployed, that is v(t1). 

5)    The air-brake has one good mode of deployment and four possible failure modes, three with a reduced function, one with no function

This point introduces more logic for us to handle. Based on the language John uses it sounds like the good mode of deployment will put us into case (a), the 'no function' mode puts us into case (b), and the 3 reduced function failure modes all put us into case (c). Let's call the 3 reduced function cases (c1), (c2) and (c3).


Analysing these additional pieces of information has highlighted the importance of understanding the velocity, v(t), and the distance s(t). In the next blog, we'll start building up a model for the velocity of BLOODHOUND. 

The plot thickens on how to stop BLOODHOUND

After the tantalizing first description of the problem of how to deploy the parachutes effectively to stop the BLOODHOUND Super Sonic Car, I asked Dr John Davis for more details. Specifically, I asked what is the objective of the choice of chute opening times and what constraints apply. 

John replied with the following information: 

1)    We want to stop at exactly 10 miles so we are ready to turn around to go back

2)    We can not deploy the parachutes above a certain speed, definitely not supersonic

3)    We can stop at 10 miles by two methods 

      a)    Airbrake at the end of the measured mile and wheel brakes at about 150 mph. This is the preferred option. 

      b)    Parachutes: first just sub-sonic, second almost immediately afterwards, followed by wheel brakes maybe a bit faster than 150 mph

4)    If the airbrake only partly deploys we will need one parachute, but not necessarily at high speed. This will be very dynamic

5)    The air-brake has one good mode of deployment and four possible failure modes, three with a reduced function, one with no function

Like many real-world problem (or Dr Who's tardis!), the problem gets bigger once you get beyond the front door. How would you interpret the additional information? What additional questions would you ask? Comments below please.

In my next blog post, I'll write about how I interpreted the above information and the questions I asked next.

Gathering more information on the problem of How to Stop BLOODHOUND?

Last week I introduced a vague initial problem statement for how to stop BLOODHOUND, and asked for comments on what questions I should ask John Davis to start gathering more information.

On Twitter, @printerelf, an aerospace engineer, suggested a couple of good questions. He asked

Q1. How much KE (Kinetic Energy) will the car have when the chutes are deployed?
Q2. How big are the parachutes?

Let's take these questions in turn. Firstly, the definition of kinetic energy is 0.5*mass*velocity², therefore, in order to know the kinetic energy of BLOODHOUND, we need to know its mass and velocity. We can look up that the mass of the car plus the driver Andy Green (once the fuel has burnt off) will be approximately 5000 kg, with more details available from http://www.bloodhoundssc.com/car/facts_and_figures.cfm. However, the velocity of the car when the chutes are deployed is unknown, in fact, the velocities can be thought of as design parameters that we can adjust to find the best way of stopping BLOODHOUND. Hence, after a bit of thought, we see that Q1 is just an alternative way of phrasing the part of the initial problem (i.e. where to place the markers around the Speedo to give Andy guidance of when to deploy the parachutes). 

Re-phrasing vaguely worded problems can often help to understand the problem better, or to suggest possible routes to a solution. In this case, the thought of looking at the problem from an energy perspective strikes me as a promising one. It prompts me to recall that many physics-related problems can be solved through applying the laws of conservation of energy, and indeed in a recent phone call with John, he'd mentioned that an energy-based analysis may be relevant. We should keep this inkling in mind as we explore the problem further.  

Q2 is also pertinent. The size of the parachutes will relate to the amount of drag (the force acting to slow down BLOODHOUND): the bigger the area of the parachute, the larger the drag. However, the parachute size isn't known as the supplier for the parachutes hasn't been fixed yet. This is another typical feature of real-world problems - the numbers used in the calculations change as the project progresses. For now, John tells me we should use a 'D/q value' (also known as a drag coefficient) of 0.8 for both parachutes. I remind myself (using a spreadsheet Ron Ayers once gave me) that it is common in aerodynamics to model drag as the product of the drag coefficient 'D/q' and q, the dynamic pressure, i.e. DragChute = 0.8*q. Here, DragChute is the drag force from the chute in Newtons (N), and q is the dynamic pressure in m². 

  

Both @printerelf's questions revealed more information about the problem in hand. What other questions might you want to ask at this stage? Suggestions welcome in the comments below. 

The first questions I actually asked John reflected my background as a mathematician and were aimed at trying to cast the problem as the solution to an optimization problem. I asked: 

What is the objective of the choice of chute opening times? E.g. Is it to stop at exactly 10 miles, or for them to open at defined speeds? What constraints apply? 

Tune in next week for John's reply...  

Initial Problem Statement - How to Stop BLOODHOUND?

Here is the initial problem statement in the first email I received from John Davis on this topic.

On the slow down from the high speed runs it is important that the staging of the parachutes and wheel brakes ensure that the car is brought to a halt by the end of the 10 mile track.

Andy is keen to have markers around the Speedo which gives him guidance of when to deploy the parachutes, Chute 1 then Chute 2. These pointers need to be dynamic and automatically adjust should the previous one be late or less efficient than expected.

The basic mathematics is based around a simple Drag/Mass equals Acceleration model with the Airbrakes and Chutes having an assumed Drag Coefficient.

What I need is some simple equations to put into the Simulink Code to work out the required speed at which the next deployment is required.

If you feel you have some time to help I would like to talk through the information that is required to complete this model.

This is a typical introduction to a real-world problem. Unlike word problems you may remember from maths class in school, the problem is ill-defined, no numbers are included, and much more detail is needed before the 'simple equations' can be identified.

However, what the email does convey is:

• It is a problem worth solving
• The customer for the solution is defined: Andy Green the pilot/driver
• A qualitative description of what is needed: simple equations and dynamic speed markers
• Pointers to the mathematics/physics that will be needed: aerodynamics and Newton's 2nd law of motion 
• The software for implementing the equations will be Simulink  
• John is starting the problem-solving process by figuring out who can help him solve it  

How would you respond to the email? What questions would you ask? Suggestions in the comments below....

First Blog - An Introduction

Well, here goes with my first personal blog post. The blogging impetus came from a conversation I just had with Dr John Davis, Senior Controls System Engineer for the Bloodhound Super Sonic Car. Bloodhound aims to go 1000 mph and in the process inspire the next generation of scientists, engineers and mathematicians.

John asked for my help to solve the problem of when and if to use the parachutes in order to decelerate Bloodhound from > 1000 mph at the end of the measured mile to stop exactly at the 10 mile marker. Rather than just solve the problem and then document a neatly packaged answer, we are hoping to open up our thinking process in order to give students and Bloodhound enthusiasts an insight into how 'real-world' engineering problem-solving works.  I'll be posting regular updates on our progress. Perhaps we may end up crowd-sourcing an even better solution through the comments!

By way of introduction, I am the Application Engineering Manager at MathWorks in the UK. The MathWorks tools are being used for several elements of the design of Bloodhound, including the control system and aerodynamics. I plan to write this blog during the remainder of my maternity leave and any views expressed are my own. You can follow me @tanyammorton and the Bloodhound project @bloodhound_ssc on Twitter. The hashtag for the problem is #stopbloodhound.