61 lines
3.2 KiB
Markdown
61 lines
3.2 KiB
Markdown
# Iterative Linear Quadratic Regulator
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This program is about iLQR (Iteratice Linear Quadratic Regulator)
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# Problem Formulation
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Two wheeled robot is expressed by the following equation.
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It is nonlinear and nonholonomic system. Sometimes, it's extremely difficult to control the
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steering(or angular velocity) and velocity of the wheeled robot. Therefore, many methods control only steering, like purepersuit, Linear MPC.
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However, sometimes we should consider the velocity and steering simultaneously when the car or robots move fast.
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To solve the problem, we should apply the control methods which can treat the nonlinear system.
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<a href="https://www.codecogs.com/eqnedit.php?latex=\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;x&space;\\&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;\cos(\theta)&space;&&space;0&space;\\&space;\sin(\theta)&space;&&space;0&space;\\&space;0&space;&&space;1&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_v&space;\\&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{B}\boldsymbol{U}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{d}{dt}&space;\boldsymbol{X}=&space;\frac{d}{dt}&space;\begin{bmatrix}&space;x&space;\\&space;y&space;\\&space;\theta&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;\cos(\theta)&space;&&space;0&space;\\&space;\sin(\theta)&space;&&space;0&space;\\&space;0&space;&&space;1&space;\\&space;\end{bmatrix}&space;\begin{bmatrix}&space;u_v&space;\\&space;u_\omega&space;\\&space;\end{bmatrix}&space;=&space;\boldsymbol{B}\boldsymbol{U}" title="\frac{d}{dt} \boldsymbol{X}= \frac{d}{dt} \begin{bmatrix} x \\ y \\ \theta \end{bmatrix} = \begin{bmatrix} \cos(\theta) & 0 \\ \sin(\theta) & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} u_v \\ u_\omega \\ \end{bmatrix} = \boldsymbol{B}\boldsymbol{U}" /></a>
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Nonliner Model Predictive Control is one of the famous methods, so I applied the method in the folder of this repository.
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(if you are interested, please look it)
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NMPC is very effecitive method to solve nonlinear optimal control problem but it is a handcraft method.
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This program is about one more other methods to solve the nonlinear optimal control problem.
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The method is iterative LQR.
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Iterative LQR is one of the DDP(differential dynamic programming) method.
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Recently, this method is used in IRL(inverse reinforcement learning), such as GPS(guided policy search)
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If you want to know more about the iLQR, please look the references.
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The paper and website is great.
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# Usage
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```
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```
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# Expected Results
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- static goal
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- track the goal
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# Applied other model
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# Requirement
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- python3.5 or more
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- numpy
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- matplotlib
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# Reference
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- study wolf
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https://github.com/studywolf/control
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- Sergey Levine's lecture
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http://rail.eecs.berkeley.edu/deeprlcourse/
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- Tassa, Y., Erez, T., & Todorov, E. (2012). Synthesis and stabilization of complex behaviors through online trajectory optimization. IEEE International Conference on Intelligent Robots and Systems, 4906–4913. https://doi.org/10.1109/IROS.2012.6386025
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- Li, W., & Todorov, E. (n.d.). Iterative Linear Quadratic Regulator Design for Nonlinear Biological Movement Systems. Retrieved from https://homes.cs.washington.edu/~todorov/papers/LiICINCO04.pdf |