Guy on Simulink

Simulink & Model-Based Design

Hyperloop: Not so fast! 30

Posted by Guy Rouleau,

This week Matt Brauer is back to describe work he did to analyze the trajectory of the Hyperloop proposal. This is a key input to the Hyperloop Simulink models we are building.

Matt's conclusion is surprising:
From the perspective of geographic constraints and rider comfort, the 760 mph peak speed is not an issue. It’s the 300 mph section through the suburbs of San Francisco that requires closer consideration.

Matt Brauer, guest blogger and Hyperloop enthousiast

Where is the Hyperloop going?

As mentioned in a previous blog post, we’ve begun implementing Simulink models of the Hyperloop concept. To exercise those models, a core piece of information is the trajectory. We need to know where the Hyperloop is going and at what speed. I analyzed the proposal to derive this information, and what I found was interesting. From the perspective of geographic constraints and rider comfort, the 760 mph peak speed is not an issue. It’s the 300 mph section through the suburbs of San Francisco that requires closer consideration.

the Hyperloop
Potential Hyperloop route, including image from and Google Earth

The basic idea put forth in the proposal is to follow existing highways as much as possible. In order to limit lateral accelerations experienced by the passengers to 0.5g, there needs to be “minor deviations when the highway makes a sharp turn”. I remember from Freshmen Physics class that centripetal acceleration in a curve is proportional to the square of velocity. This means that increasing your velocity from 76 to 760 mph is actually a 100x multiplication of g forces.

Is it really possible to average 600 mph across the state of California on available land without making passengers nauseous? Or would a reasonable trajectory require wide loops, impeding on private citizens’ backyards? I put the technical computing power of MATLAB to work on answering these questions.

Technical Computing of a Route

I won’t go into all the details because this is a Simulink blog, but here is an overview of the steps I followed:

  • First, I used Google Earth to get a set of longitude and latitude points along the California highways I-5 and I-580. After getting directions, saved a KML file with the data.
  • I used the read_kml submission from the MATLAB Central File Exchange to import the data in MATLAB.
  • It is possible to use functions like wmsfind, wmsupdate and wmsread from the Mapping Toolbox to add topography data to the route and surrounding area. However, I only used this information for plotting. For this first study, the derived route is only 2-dimensional.
  • This gave me a set of discrete points along the highways, to which I associated a desired velocity based on the Hyperloop document.
  • From these target points, I created a smoothed trajectory using the fit function from the Curve Fitting Toolbox.
  • Using this smoothed trajectory, I wrote a simple script to calculate and plot the lateral acceleration along this trajectory.
  • For the portions of the trajectory that were exceeding 0.5g's, I adjusted some target points to drive lateral accelerations below 0.5g’s while staying as close as possible to the original route.

In the end, I got the results below:

Velocity and acceleration patterns during the trajectory

The final route
Velocity and Acceleration along derived Hyperloop route (created using the Mapping Toolbox)

You can see that the travel time is very close to the advertised 35 minutes. The accelerations were kept within reason, although it looks like a pretty dynamic ride.

How much does the Hyperloop need to deviate from the existing highways in order to achieve these results?

The red line shows the targeted highways. The yellow portions show sections of the derived Hyperloop route that deviate from the mean highway route by more than 50m. The yellow, "off-highway", portions account for 113 miles of the 347 mile trip.

To review in depth, I wrote the trajectory back to KML using kmlwriteline from the Mapping Toolbox. Now it's possible to view the derived route in Google Earth.

The final route seen in Google Earth
No major issues seen along I-5 (Image created using Google Earth)

Again, the red line is the mean highway route. Here, the yellow line shows the complete derived Hyperloop route. Upon reviewing the details, the route seems pretty reasonable up until I-580 outside of the bay area.

When I-580 turns west and starts going through the San Francisco suburbs, it becomes difficult to stay on the road. Here’s a snapshot of a particularly rough curve:

A rough part of the trajectory
Re-routing along I-580 to avoid excessive g-forces (Image created using Google Earth)

Below is the route data for the area surrounding the above curve.

MATLAB Central submission to explore the computed route

Despite the issues highlighted above, the final conclusion is quite positive. It seems that the first 295 miles of the route can be accomplished in about 27 minutes without excessive g’s on the passengers or encroaching on private property. “Landing” in the Bay Area may take a bit longer than originally advertised, but that seems like splitting hairs at this point. Further investigation is certainly warranted. So, let’s get those Simulink models running!

Now it's your turn

What do you think? Is the Hyperloop going in the right direction?

Given Elon Musk's statement that the Hyperloop should be an open design concept, would you be interested use Matt's trajectory and begin filling some of the boxes in our Hyperloop model architectures?

Let us know what you think by leaving a comment here.

30 CommentsOldest to Newest

Sean Robert Meaney replied on : 1 of 30

Firstly its the whole corporations must control this and charge fifty dollars for a one way trip that concerns me. That means it’s out of bounds for the poorest workers who might have been able to live in adjacent suburb-city and hyperloop in to their work city. Suddenly it is costing you a hundred dollars a day to come and go – that isn’t conducive to future where you can go where the work is on minimal wage.

Consider a lottery selling two billion tickets at 500 dollars each. That’s 250 billion in taxes, 50 billion in lottery management, 300 billion in prizes (1×100 billion, 10x 10 billion, 100x 1 billion) and 400 billion in the constriction of a hyperloop network. Assuming 250 miles of hyperloop cost 100 billion to build – it provides us with sufficient funds to build a hyperloop that links Fresno, San Diego, and Phoenix to Vegas as ‘Hyperloop Suburbs’ and the remaining hundred billion is sufficient to manage that as a free service (assuming 5% interest on the hundred billion). That means that Vegas can expand as a ‘resort’ with its residential populace relocating to the City-suburbs that would be thirty minutes away by hyperloop. As a free service it means workers can come and go from vegas at no cost to them. That is an incredible concept.
It also means that twenty such lotteries can be conducted over the next twenty years financing 15,000 miles of hyperloop network across the USA As a free public service.

That would be a golden age for the USA.

Paul Metcalf replied on : 2 of 30

Very cool post. How about going the vertical analysis next? I suppose that will be easier to mitigate by varying the height of the track using bridges etc… I would also think about introducing a cost-benefit model at some stage of the process.
I’m wondering, if you were the only car on the road, and there were no speed limits, and you drove in a safe way, how quickly could you make the trip with the existing roads/infrastructure?
Perhaps try to calculate the cost to reduce drive times, such as lane widening, ride sharing etc, project that data and a basic cost-benefit could be made…?

@Sean Robert Meaney – Thanks for the comment. You dove deep into one of the more challenging aspects of Hyperloop proposal, the business model and funding. It’s a bit beyond the scope of this blog, so we are going to keep our focus on the engineering and design of the Hyperloop.

@Paul Metcalf – I think Matt is interested in adding a few more axes to his model to incorporate a full 6-degree-of-freedom trajectory (x,y,z,roll,pitch,yaw). The roll component should lessen some of the lateral accelerations.

Modeling a cost-function for the Hyperloop could prove very challenging but also very useful. If we had a good cost model the route could be optimized to minimize cost and travel time.

Sean Meaney replied on : 4 of 30

OK. Lets look at the effects the economics of it has on the LA to SF hyperloop. At an annual two billion cost that is fifty people per pod at fifty dollars a person every minute. That is a very different hyperloop vehicle.

As I recall, taking the original proposal at face value, the passenger throughput is simply abysmal (and that ignores the enormous downtime associated with anything going even mildly wrong with the system during operation). It seems to me that the key to improving public transport is simply making the speed of it relatively unimportant (if you’re comfortable and can work (including video conferencing) or entertain yourself while you travel, does it really matter whether you’re traveling enormously fast?) It seems to me that hyperloop is the wrong solution to the wrong problem.

Thank you for this post!

What you described as the g factor is square of velocity change. This is a well known physics of friction factors for pipe flow. Pipes carrying fluid need to be designed in such a way that fluid friction is kept to a reasonable level. Hyperloop is nothing but a pipe carrying load (at 700mph its a pipe carrying fluid).

Another thought about choosing I-5 or I-580. Why not choose the Coastal terrain (not Highway 1). The idea here is a straight path will avoid all of these obstacles. I know there are lots of cliffs etc around the coastal area. There could be other ways to get around it.

James Westgate replied on : 7 of 30

It would seem to me that as you reach built up areas that going underground might be the best option? How this works in a seismically active area like SF I don’t know, but the precedent for the creation of new underground transportation using tunnelling machines in well established in areas like London (cross rail) and Seattle. Whilst increasing cost, this could also solve problems around siting of convenient terminal buildings and facilities.

Why not leave the fiddly bits at each end ’till later? Terminate the route as soon as it hits the suburbs, at least initially. I imagine it would make the system vastly cheaper to build, and the expensive and troublesome parts of the journey could be added later if the economics warrant it, allowing more expensive options (like underground tunnelling) to be considered.

How about starting the solution for cargo transportation before really using for the people? However design the loop to carry both people and cargo in the first place. This removes lot of questions.

Jesse McGraw replied on : 10 of 30

(I’m not an engineer so forgive comments which have probably already been covered a million times or are patently dumb)

If the car can roll within the tube then the curve can be taken into the 3rd dimension with the occupants none the wiser, no? What does the path look like if g is kept to 1 but that can be in any direction (eg passengers are actually upside down in the tube but don’t notice because there are no windows and their weight is normal)

e replied on : 11 of 30

Even at $100/day, with 20 work days a week–you would only be at $2,000/month, which is less than a lot of people’s rent in SF.

pwb replied on : 12 of 30

According to the hyperlood document, it is not scheduled to make the turn that you have pictured. Instead, it continues straight along Hwy 238 (later making a relatively sharp turn north onto 880).

pwb replied on : 13 of 30

Where are people getting $50 per ride? The document suggests $20 which is pretty amazing for SF to LA. LA to Vegas might be more like $10 or $15 (can you imagine?).

Baxter replied on : 14 of 30

In calculating g-load the author has actually discounted one valuable tool available to the eventual builders of the hyperloop; the passengers will experience 1 g of acceleration on the vertical axis due to gravity in addition to up to .5 g of gravity centripedally.
Obviously, banking the track allows us to exert less lateral acceleration on the passengers, but at the expense of more (perceived) vertical acceleration, and this vertical acceleration (pitching motion) is the primary cause of seasickness.
However, since the hyperloop will be built on pylons, and it is possible to vary the height of those pylons, clever construction will allow the designer to bank the track inward in conjunction with a parabolic vertical ‘dip’ in the track which will temporarily relieve the force of gravity. This will then allow the designer to build track which incorporates up to 1.5 lateral gs while maintaining passenger percecption of 1 g vertical acceleration and only .5g lateral acceleration.

Travelling at 700 mph, the horizontal velocity in feet per second 1027. This means that to relieve the perception of gravity for 1 second would require dipping the track by an average of 30% grade for 1027 feet.
This would allow 1.5 g turns for around 1/5th of a mile with the perception of just .5gs of lateral acceleration

I’d love to hear your thoughts regarding this idea.

Alan replied on : 16 of 30

I say who’s eve confident enough in their understanding of the challenges and is willing to spend their own money on this, feel free.

When government makes these decisions, it seems to either do more of what it understands. Or subsidize technologies which aren’t economically feasible yet.

In the private sector, bold but feasible choices are made all the time. And there are failures too. But these failures are paid for entirely by those taking the risks.

When will the Sean Meanie feller understand that we are discussing solely engineering here, and that his populist proposals are not welcome?

Bob replied on : 18 of 30

Why not just start the Hyperloop in Oakland instead of SF? Oakland has more room to grow, so in the long term it will be a more important city. And SF is already connected to Oakland by BART.

Steve replied on : 19 of 30

What would a LA to Vegas hyperloop look like?
That would be a bit shorter and probably a popular route/

Joe Sohn replied on : 20 of 30

I counter you, Sean. Most Subway systems in the world before the 1920’s started outforproft . who is to say that faster traspot using new technology worst have the Same fate?

Ziggy replied on : 21 of 30

Maybe I’m missing something but why can’t it just slow down a bit around those few curves to keep g-forces within a reasonable range, similar to what trains do? It will increase the trip by a few minutes, but seems to be the simplest, most cost-effective solution.

Randy replied on : 22 of 30

The issue of passenger experienced accelerations is more complex than a simple radial acceleration. Capsules are supported by air bearings conforming to the cylindrical interior of the tube and will “bank” in a turn such that the vector sum of radial and vertical accelerations will be perpendicular to the capsule floor. It follows that in a 0.5 g radial acceleration turn a passenger will experience an apparent increased vertical acceleration of only 11.8% because the radial acceleration vector is orthogonal to the gravity vector.

Accelerations due to “vertical turns” are more problematic because the acceleration vector and gravity vector are aligned.

Matt Brauer replied on : 23 of 30

@Ziggy – as conceived in the proposal, active braking is accomplished with the linear actuators, which are only placed along certain sections to reduce cost. Additional infrastructure costs to provide more opportunity to actively brake and accelerate does seem like a reasonable trade-off that should be considered.

As several comments point out (thanks Randy, Baxter, Jessie McGraw), there are additional degrees of freedom available to mitigate perceived accelerations by the passengers.

I am skeptical of how much can be achieved with vertical grades (as proposed by Baxter) to essentially suspend gravity. If I understand the example, it would require about 300 feet of descent. As earthquakes are already a concern, pillars of that height could be problematic for structural rigidity. But,it is an interesting approach that I would expect to be utilized in a final design.

Randy’s point about the effects of the vehicle banking are also spot on. I chose not to include this in the analysis and instead used the same methodology and limits described in the proposal itself. I expect that this analysis method is a simplification in line with the simplified limit of 0.5 g. This limit seems high to me for general passenger comfort. It seems that we should have a solid understanding of reasonable passenger comfort limits for this type of application before performing the multi-dimensional analysis. I’d be interested in reference limits available from other transportation industries (air, rail, bus).

@pwb – thanks for pointing out this oversight. The proposed route actually turns at 880 to head North. It looks like that turn (and the subsequent turn West) is just as problematic as the one shown here. So, the conclusions are not affected.

Randy replied on : 24 of 30

I believe commercial airline operators attempt to limit bank angles to +/-30 deg. This implies lateral accelerations of 0.58 g more or less. Hyperloop’s 0.5 g limit looks to be in the same ballpark.

A perhaps more critical consideration for passenger comfort than the simple radial acceleration limit may be roll rate. As a capsule enters a turn it will roll to keep the acceleration vector nornal to the cabin floor – what in the airplane business is thought of as a “coordinated turn”. The rate at which tube curvature varies along the path, in concert with Capsule speed, implies a “roll rate”. High performance aircraft may achieve roll rates of 720 deg/sec. Airliners on the other hand treat passengers to perhaps 10 deg/s roll rates. Limiting radial accelerations to 0.5g implies a 26.6 deg bank angle. Limiting roll rate to 10 deg/s at a capsule speed of 1220 KPH implies a “spiral entry distance” of 900 m.

Designing a route (path) that will limit both accelerations and roll rates will have significant geometric consequences.

Peter replied on : 25 of 30

Unless you are in some kind of hyper cocoon of crash protection, I am not real sure how safe 600 to 700mph would be for a human body. At least in a jetliner you may have minutes to react to failure and there is a vast amount of free air space around the speeding vessel, but where you are sliding next to solid object so fast (ala toboggan run style) I am thinking something like bug on windshield effect. Not a lot of room for error – mechanical failures would have exponentially tragic effects, including as someone mentioned earthquakes, shifting ground… How about super human cannons with 1 mile barrels, shoot you out and land in a bid net at the next station (any birds you catch on the way consider as your in-flight meal, lol…)

RichMac replied on : 27 of 30

One important point about the route is that the Hyperloop Alpha proposal has a very limited length of expensive linear motors, hence the 1g acceleration and coasting between. The 0.5g lateral acceleration is also due to limitations to the bank angle due to engagement with the linear motors, 1g lateral is quite acceptable with correct banking.

I propose the use of wheels for traction, but there is a limit to the power of the onboard motors. Alpha uses a massive 30,000 kW for 1g at full speed, I have reduced this to only 0.25g up to a max of 3,500 kW. Amazingly it only adds 2 minutes to the whole trip, but this is recovered by being able to accelerate between the corners anywhere along the route.

Matt Brauer replied on : 28 of 30

@Aaron, the Alpha proposal terminates across the bay bridge, well past Hayward. If you have information that shows Hayward as the current plan, please share it. Maybe the terminal would be built at SpaceX headquarters?

@RichMax, this design “variant” looks really interesting. Our model architectures should be able to accommodate this concept using the Variant Subsystems. Your concept also makes it clear that the “optimal” route depends on the control authority of the propulsion system. The route I derived was based on the assumption that there was limited opportunities for acceleration and deceleration. Your concept would enable a route with tighter curves.

z940912 replied on : 29 of 30

Great job, Guy!

Taking on challenges like this and Mars colonization is what the US is supposed to be all about. Reducing acceleration G’s is just a matter of Musk’s buddies (like the Senate Majority leader, Governor, half the media et al.) helping extend existing easements and some additional eminent domain for a few miles of pylon that stay above existing developments.

If we ever start telling our children that people like Elon Musk just had a lucky streak with reinventing the yellow pages (zip2), financial transactions (payal), solar power for the masses (solar city), electric cars (tesla), and rockets (spacex) we should just sign things over to the Chinese and join everyone else in bitching about how the world’s problems suck.

Nathan replied on : 30 of 30

One of the problems with having a dip to alleviate the g forces of a corner is that the opposite direction will now have an rapid rise, increasing the g forces. This might not be a problem as pods going in one direction may be going substantially faster than the opposite direction, but it still is going to be a clunky solution.

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